FORMALIZED MATHEMATICS Vol.2, No.3, May–August 1991 Université Catholique de Louvain

A Construction of Analytical Ordered Trapezium Spaces ^{1}

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. We define, in a given real linear space, the midpoint operation on vectors and, with the help of the notions of directed parallelism of vectors and orthogonality of vectors, we define the relation of directed trapezium. We consider structures being enrichments of affine structures by a one binary operation, together with a function which assigns to every such structure its ”affine” reduct. Theorems concerning midpoint operation and trapezium relation are proved, which enables us to introduce an abstract notion of (regular in fact) ordered trapezium space with midpoint, ordered trapezium space, and (unordered) trapezium space.

MML Identifier: ANALTRAP.

The articles [11], [2], [4], [3], [13], [9], [12], [6], [7], [10], [8], [1], and [5] provide the notation and terminology for this paper. For simplicity we follow the rules: V will denote a real linear space, u, u_{1}, u_{2}, v, v_{1}, v_{2}, w, y will denote vectors of V, a, b will denote real numbers, and x, z will be arbitrary. Let us consider V, u, v, u_{1}, v_{1}. The predicate u, v |||| u_{1}, v_{1} is defined as follows:

(Def.1) u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1} or u, v \begin{matrix} \\ \\ ∥ \end{matrix} v_{1}, u_{1}.

The following propositions are true:

(1) If w, y span the space, then OASpace V is an ordered affine space.

(2) For all elements p, q, p_{1}, q_{1} of the points of OASpace V such that p = u and q = v and p_{1} = u_{1} and q_{1} = v_{1} holds p, q \begin{matrix} \\ \\ ∥ \end{matrix} p_{1}, q_{1} if and only if

u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1}.

(3) If w, y span the space, then for all elements p, q, p_{1}, q_{1} of the points of Λ(OASpace V) such that p = u and q = v and p_{1} = u_{1} and q_{1} = v_{1} holds p, q ∥ p_{1}, q_{1} if and only if u, v |||| u_{1}, v_{1}.

1Supported by RPBP.III-24.C2

(4) If w, y span the space, then for all elements p, q, p_{1}, q_{1} of the points of AMSp(V, w, y) such that p = u and q = v and p_{1} = u_{1} and q_{1} = v_{1} holds p, q ∥ p_{1}, q_{1} if and only if u, v |||| u_{1}, v_{1}.

Let us consider V, u, v. The functor u#v yielding a vector of V is defined by: (Def.2) u#v + u#v = u + v.

One can prove the following propositions:

(5) u#u = u.

(6) u#v = v#u.

(7) There exists y such that u#y = w.

(8) u#u_{1}#(v#v_{1}) = u#v#(u_{1}#v_{1}).

(9) If u#y = u#w, then y = w.

(10) u, v \begin{matrix} \\ \\ ∥ \end{matrix} y#u, y#v.

(11) u, v |||| w#u, w#v.

(12) 2 · (u#v − u) = v − u and 2 · (v − u#v) = v − u.

(13) u, u#v \begin{matrix} \\ \\ ∥ \end{matrix} u#v, v.

(14) u, v \begin{matrix} \\ \\ ∥ \end{matrix} u, u#v and u, v \begin{matrix} \\ \\ ∥ \end{matrix} u#v, v.

(15) If u, y \begin{matrix} \\ \\ ∥ \end{matrix} y, v, then u#y, y \begin{matrix} \\ \\ ∥ \end{matrix} y, y#v.

(16) If u, v \begin{matrix} \\ \\ ∥ \end{matrix} u_{1}, v_{1}, then u, v \begin{matrix} \\ \\ ∥ \end{matrix} u#u_{1}, v#v_{1}.

Let us consider V, w, y, u, u_{1}, v, v_{1}. We say that u, u_{1} and v, v_{1} form a directed trapezium w.r.t. w, y if and only if: (Def.3) u, u_{1} \begin{matrix} \\ \\ ∥ \end{matrix} v, v_{1} and u, u_{1}, u#u_{1} and v#v_{1} are orthogonal w.r.t. w, y and

v, v_{1}, u#u_{1} and v#v_{1} are orthogonal w.r.t. w, y. We now state a number of propositions:

(17) If w, y span the space, then u, u and v, v form a directed trapezium w.r.t. w, y.

(18) If w, y span the space, then u, v and u, v form a directed trapezium w.r.t. w, y.

(19) If u, v and v, u form a directed trapezium w.r.t. w, y, then u = v.

(20) If w, y span the space and v_{1}, u and u, v_{2} form a directed trapezium w.r.t. w, y, then v_{1} = u and u = v_{2}.

(21) If w, y span the space and u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y and u, v and u_{2}, v_{2} form a directed trapezium w.r.t. w, y and u ≠ v, then u_{1}, v_{1} and u_{2}, v_{2} form a directed trapezium w.r.t. w, y.

(22) If w, y span the space, then there exists a vector t of V such that u, v and u_{1}, t form a directed trapezium w.r.t. w, y or u, v and t, u_{1} form a directed trapezium w.r.t. w, y.

(23) If w, y span the space and u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y, then u_{1}, v_{1} and u, v form a directed trapezium w.r.t. w, y.

(24) If w, y span the space and u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y, then v, u and v_{1}, u_{1} form a directed trapezium w.r.t. w, y.

(25) If w, y span the space and v, u_{1} and v, u_{2} form a directed trapezium w.r.t. w, y, then u_{1} = u_{2}.

(26) If w, y span the space and u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y and u, v and u_{1}, v_{2} form a directed trapezium w.r.t. w, y, then u = v or v_{1} = v_{2}.

(27) If w, y span the space and u ≠ u_{1} and u, u_{1} and v, v_{1} form a directed trapezium w.r.t. w, y but u, u_{1} and v, v_{2} form a directed trapezium w.r.t. w, y or u, u_{1} and v_{2}, v form a directed trapezium w.r.t. w, y, then v_{1} = v_{2}.

(28) If w, y span the space and u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y, then u, v and u#u_{1}, v#v_{1} form a directed trapezium w.r.t. w, y.

(29) If w, y span the space and u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y, then u, v and u#v_{1}, v#u_{1} form a directed trapezium w.r.t. w, y or u, v and v#u_{1}, u#v_{1} form a directed trapezium w.r.t. w, y.

(30) Let u, u_{1}, u_{2}, v_{1}, v_{2}, t_{1}, t_{2}, w_{1}, w_{2} be vectors of V. Then if w, y span the space and u = u_{1}#t_{1} and u = u_{2}#t_{2} and u = v_{1}#w_{1} and u = v_{2}#w_{2} and u_{1}, u_{2} and v_{1}, v_{2} form a directed trapezium w.r.t. w, y, then t_{1}, t_{2} and w_{1}, w_{2} form a directed trapezium w.r.t. w, y.

Let us consider V, w, y, u. Let us assume that w, y span the space. The functor π_{w,y}^{1}(u) yielding a real number is defined as follows: (Def.4) there exists b such that u = π_{w,y}^{1}(u) · w + b · y.

Let us consider V, w, y, u. Let us assume that w, y span the space. The functor π_{w,y}^{2}(u) yields a real number and is defined by: (Def.5) there exists a such that u = a · w + π_{w,y}^{2}(u) · y.

Let us consider V, w, y, u, v. Let us assume that w, y span the space. The functor u ·_{w,y} v yields a real number and is defined as follows: (Def.6) u ·_{w,y} v = π_{w,y}^{1}(u) · π_{w,y}^{1}(v) + π_{w,y}^{2}(u) · π_{w,y}^{2}(v).

We now state a number of propositions:

(31) If w, y span the space, then for all u, v holds u ·_{w,y} v = v ·_{w,y} u.

(32) Suppose w, y span the space. Given u, v, v_{1}. Then (i) u ·_{w,y} (v + v_{1}) = u ·_{w,y} v + u ·_{w,y} v_{1}, (ii) u ·_{w,y} (v − v_{1}) = u ·_{w,y} v − u ·_{w,y} v_{1}, (iii) (v − v_{1}) ·_{w,y} u = v ·_{w,y} u − v_{1} ·_{w,y} u, (iv) (v + v_{1}) ·_{w,y} u = v ·_{w,y} u + v_{1} ·_{w,y} u.

(33) Suppose w, y span the space. Let u, v be vectors of V. Let a be a real

number. Then

(i) (a · u) ·_{w,y} v = a · u ·_{w,y} v, (ii) u ·_{w,y} (a · v) = a · u ·_{w,y} v, (iii) (a · u) ·_{w,y} v = u ·_{w,y} v · a, (iv) u ·_{w,y} (a · v) = u ·_{w,y} v · a.

(34) If w, y span the space, then for all vectors u, v of V holds u, v are orthogonal w.r.t. w, y if and only if u ·_{w,y} v = 0.

(35) If w, y span the space, then for all vectors u, v, u_{1}, v_{1} of V holds u, v, u_{1} and v_{1} are orthogonal w.r.t. w, y if and only if (v −u)·_{w,y} (v_{1} −u_{1}) = 0.

(36) If w, y span the space, then for all vectors u, v, v_{1} of V holds 2 · u ·_{w,y} (v#v_{1}) = u ·_{w,y} v + u ·_{w,y} v_{1}.

(37) If w, y span the space, then for all vectors u, v of V such that u ≠ v holds (u − v) ·_{w,y} (u − v) ≠ 0.

(38) Suppose w, y span the space. Let p, q, u, v, v′ be vectors of V. Let A

be a real number. Suppose that

(i) p, q and u, v form a directed trapezium w.r.t. w, y, (ii) p ≠ q, (iii) A = ((p − q) ·_{w,y} (p + q) − 2 · (p − q) ·_{w,y} u) · (p − q) ·_{w,y} (p − q)^{−1}, (iv) v′ = u + A · (p − q).

Then v = v′.

(39) Suppose w, y span the space. Let u, u′, u_{1}, u_{2}, v_{1}, v_{2}, t_{1}, t_{2}, w_{1}, w_{2} be vectors of V. Then if u ≠ u′ and u, u′ and u_{1}, t_{1} form a directed trapezium w.r.t. w, y and u, u′ and u_{2}, t_{2} form a directed trapezium w.r.t. w, y and u, u′ and v_{1}, w_{1} form a directed trapezium w.r.t. w, y and u, u′ and v_{2}, w_{2} form a directed trapezium w.r.t. w, y and u_{1}, u_{2} \begin{matrix} \\ \\ ∥ \end{matrix} v_{1}, v_{2}, then

t_{1}, t_{2} \begin{matrix} \\ \\ ∥ \end{matrix} w_{1}, w_{2}.

(40) Suppose w, y span the space. Then for all vectors u, u′, u_{1}, u_{2}, v_{1}, t_{1}, t_{2}, w_{1} of V such that u ≠ u′ and u, u′ and u_{1}, t_{1} form a directed trapezium w.r.t. w, y and u, u′ and u_{2}, t_{2} form a directed trapezium w.r.t. w, y and u, u′ and v_{1}, w_{1} form a directed trapezium w.r.t. w, y and v_{1} = u_{1}#u_{2} holds w_{1} = t_{1}#t_{2}.

(41) If w, y span the space, then for all vectors u, u′, u_{1}, u_{2}, t_{1}, t_{2} of V such that u ≠ u′ and u, u′ and u_{1}, t_{1} form a directed trapezium w.r.t. w, y and u, u′ and u_{2}, t_{2} form a directed trapezium w.r.t. w, y holds u, u′ and u_{1}#u_{2}, t_{1}#t_{2} form a directed trapezium w.r.t. w, y.

(42) Suppose w, y span the space. Let u, u′, u_{1}, u_{2}, v_{1}, v_{2}, t_{1}, t_{2}, w_{1}, w_{2} be vectors of V. Suppose u ≠ u′ and u, u′ and u_{1}, t_{1} form a directed trapezium w.r.t. w, y and u, u′ and u_{2}, t_{2} form a directed trapezium w.r.t. w, y and u, u′ and v_{1}, w_{1} form a directed trapezium w.r.t. w, y and u, u′ and v_{2}, w_{2} form a directed trapezium w.r.t. w, y and u_{1}, u_{2}, v_{1} and v_{2} are orthogonal w.r.t. w, y. Then t_{1}, t_{2}, w_{1} and w_{2} are orthogonal w.r.t. w, y.

(43) Let u, u′, u_{1}, u_{2}, v_{1}, v_{2}, t_{1}, t_{2}, w_{1}, w_{2} be vectors of V. Suppose w, y span the space and u ≠ u′ and u, u′ and u_{1}, t_{1} form a directed trapezium w.r.t. w, y and u, u′ and u_{2}, t_{2} form a directed trapezium w.r.t. w, y and u, u′ and v_{1}, w_{1} form a directed trapezium w.r.t. w, y and u, u′ and v_{2}, w_{2} form a directed trapezium w.r.t. w, y and u_{1}, u_{2} and v_{1}, v_{2} form a directed trapezium w.r.t. w, y. Then t_{1}, t_{2} and w_{1}, w_{2} form a directed trapezium w.r.t. w, y.

Let us consider V, w, y. The directed trapezium relation defined over V in the basis w, y yielding a binary relation on [: the vectors of V, the vectors of V :] is defined

as follows: (Def.7) ⟨⟨x, z⟩⟩ ∈ the directed trapezium relation defined over V in the basis w, y

if and only if there exist u, u_{1}, v, v_{1} such that x = ⟨⟨u, u_{1}⟩⟩ and z = ⟨⟨v, v_{1}⟩⟩ and u, u_{1} and v, v_{1} form a directed trapezium w.r.t. w, y.

The following proposition is true

(44) If w, y span the space, then ⟨⟨⟨⟨u, v⟩⟩, ⟨⟨u_{1}, v_{1}⟩⟩⟩⟩ ∈ the directed trapezium relation defined over V in the basis w, y if and only if u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y.

Let us consider V. The midpoint operation in V yields a binary operation on the vectors of V and is defined as follows: (Def.8) for all u, v holds (the midpoint operation in V)(u, v) = u#v.

We consider affine midpoint structures which are systems ⟨points, a midpoint operation, a congruence⟩,

where the points constitute a non-empty set, the midpoint operation is a binary operation on the points, and the congruence is a binary relation on [: the points, the points :].

Let us consider V, w, y. Let us assume that w, y span the space. The directed trapezium space defined over V in the basis w, y yielding a affine midpoint structure is defined as follows: (Def.9) the directed trapezium space defined over V in the basis w, y = ⟨the vec-

tors of V, the midpoint operation in V, the directed trapezium relation defined over V in the basis w, y⟩.

The following proposition is true

(45) For all V, w, y such that w, y span the space holds the directed trapezium space defined over V in the basis w, y = ⟨the vectors of V, the midpoint operation in V, the directed trapezium relation defined over V in the basis w, y⟩.

Let A_{1} be a affine midpoint structure. The affine reduct of A_{1} yielding an affine structure is defined by:

(Def.10) the affine reduct of A_{1} = ⟨ the points of A_{1}, the congruence of A_{1}⟩.

Let A_{1} be a affine midpoint structure, and let a, b, c, d be elements of the points of A_{1}. The predicate a, b⊤^{>}c, d is defined by:

(Def.11) ⟨⟨⟨⟨a, b⟩⟩, ⟨⟨c, d⟩⟩⟩⟩ ∈ the congruence of A_{1}.

Let A_{1} be a affine midpoint structure, and let a, b be elements of the points of A_{1}. The functor a#b yielding an element of the points of A_{1} is defined by:

(Def.12) a#b = (the midpoint operation of A_{1})(a, b).

In the sequel a, b, a_{1}, b_{1} denote elements of the points of the directed trapezium space defined over V in the basis w, y. We now state three propositions:

(46) If w, y span the space, then for an arbitrary x holds x is an element of the points of the directed trapezium space defined over V in the basisw, y if and only if x is a vector of V.

(47) If w, y span the space and u = a and v = b and u_{1} = a_{1} and v_{1} = b_{1}, then a, b⊤^{>}a_{1}, b_{1} if and only if u, v and u_{1}, v_{1} form a directed trapezium w.r.t. w, y.

(48) If w, y span the space and u = a and v = b, then u#v = a#b. A affine midpoint structure is called an ordered midpoint trapezium space if

it satisfies the condition (Def.13). (Def.13) Let a, b, c, d, a_{1}, b_{1}, c_{1}, d_{1}, p, q be elements of the points of it . Then

(i) a#b = b#a, (ii) a#a = a, (iii) a#b#(c#d) = a#c#(b#d), (iv) there exists an element p of the points of it such that p#a = b,

(v) if a#b = a#c, then b = c, (vi) if a, b⊤^{>}c, d, then a, b⊤^{>}a#c, b#d, (vii) if a, b⊤^{>}c, d, then a, b⊤^{>}a#d, b#c or a, b⊤^{>}b#c, a#d, (viii) if a, b⊤^{>}c, d and a#a_{1} = p and b#b_{1} = p and c#c_{1} = p and d#d_{1} = p,

then a_{1}, b_{1}⊤^{>}c_{1}, d_{1}, (ix) if p ≠ q and p, q⊤^{>}a, a_{1} and p, q⊤^{>}b, b_{1} and p, q⊤^{>}c, c_{1} and p, q⊤^{>}d, d_{1}

and a, b⊤^{>}c, d, then a_{1}, b_{1}⊤^{>}c_{1}, d_{1},

(x) if a, b⊤^{>}b, c, then a = b and b = c, (xi) if a, b⊤^{>}a_{1}, b_{1} and a, b⊤^{>}c_{1}, d_{1} and a ≠ b, then a_{1}, b_{1}⊤^{>}c_{1}, d_{1}, (xii) if a, b⊤^{>}c, d, then c, d⊤^{>}a, b and b, a⊤^{>}d, c, (xiii) there exists an element d of the points of it such that a, b⊤^{>}c, d or

a, b⊤^{>}d, c, (xiv) if a, b⊤^{>}c, p and a, b⊤^{>}c, q, then a = b or p = q.

One can prove the following proposition

(49) If w, y span the space, then the directed trapezium space defined over V in the basis w, y is an ordered midpoint trapezium space.

An affine structure is called an ordered trapezium space if it satisfies the condition (Def.14).

(Def.14) Let a, b, c, d, a_{1}, b_{1}, c_{1}, d_{1}, p, q be elements of the points of it . Then (i) if a, b ∥ b, c, then a = b and b = c, (ii) if a, b ∥ a_{1}, b_{1} and a, b ∥ c_{1}, d_{1} and a ≠ b, then a_{1}, b_{1} ∥ c_{1}, d_{1}, (iii) if a, b ∥ c, d, then c, d ∥ a, b and b, a ∥ d, c, (iv) there exists an element d of the points of it such that a, b ∥ c, d or

a, b ∥ d, c, (v) if a, b ∥ c, p and a, b ∥ c, q, then a = b or p = q.

Let M_{1} be an ordered midpoint trapezium space. Then the affine reduct of M_{1} is an ordered trapezium space.

We follow a convention: O_{1} denotes an ordered trapezium space, a, b, c, d denote elements of the points of O_{1}, and a′, b′, c′, d′ denote elements of the points of Λ(O_{1}). We now state two propositions:

(50) For an arbitrary x holds x is an element of the points of O_{1} if and only if x is an element of the points of Λ(O_{1}).

(51) If a = a′ and b = b′ and c = c′ and d = d′, then a′, b′ ∥ c′, d′ if and only

if a, b ∥ c, d or a, b ∥ d, c. An affine structure is called a trapezium space if it satisfies the condition

(Def.15). (Def.15) Let a′, b′, c′, d′, p′, q′ be elements of the points of it . Then

(i) a′, b′ ∥ b′, a′, (ii) if a′, b′ ∥ c′, d′ and a′, b′ ∥ c′, q′, then a′ = b′ or d′ = q′, (iii) if p′ ≠ q′ and p′, q′ ∥ a′, b′ and p′, q′ ∥ c′, d′, then a′, b′ ∥ c′, d′, (iv) if a′, b′ ∥ c′, d′, then c′, d′ ∥ a′, b′,

(v) there exists an element x′ of the points of it such that a′, b′ ∥ c′, x′.

Let O_{1} be an ordered trapezium space. Then Λ(O_{1}) is a trapezium space. An affine structure is regular if it satisfies the condition (Def.16).

(Def.16) Let p, q, a, a_{1}, b, b_{1}, c, c_{1}, d, d_{1} be elements of the points of it . Then if p ≠ q and p, q ∥ a, a_{1} and p, q ∥ b, b_{1} and p, q ∥ c, c_{1} and p, q ∥ d, d_{1} and a, b ∥ c, d, then a_{1}, b_{1} ∥ c_{1}, d_{1}.

Let M_{1} be an ordered midpoint trapezium space. Then the affine reduct of M_{1} is an regular ordered trapezium space.

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On Projections in Projective Planes. Part II ^{1}

Eugeniusz Kusak Warsaw University

Białystok Wojciech Leończuk Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. We study in greater datail projectivities on Desarguesian projective planes. We are particularly interested in the situation when the composition of given two projectivities can be replaced by another two, with a given axis or centre of one of them.

MML Identifier: PROJRED2.

The articles [7], [9], [6], [8], [10], [11], [5], [4], [1], [2], and [3] provide the notation and terminology for this paper. In the sequel I_{1} will denote a projective space defined in terms of incidence and z will denote an element of the points of I_{1}. Let us consider I_{1}, and let A, B, C be elements of the lines of I_{1}. We say that A, B, C are concurrent if and only if: (Def.1) there exists an element o of the points of I_{1} such that o | A and o | B

and o | C.

Let us consider I_{1}, and let Z be an element of the lines of I_{1}. The functor chain(Z) yields a subset of the points of I_{1} and is defined by: (Def.2) chain(Z) = {z : z | Z}.

We adopt the following rules: I_{2} will denote an Desarguesian 2-dimensional projective space defined in terms of incidence, a, b, c, d, p, p′_{1}, q, o, o′, o′′, o′_{1}, r, s, x, y, o_{1}, o_{2} will denote elements of the points of I_{2}, and O_{1}, O_{2}, O_{3}, A, B, C, O, Q, R, S will denote elements of the lines of I_{2}. Let us consider I_{2}. A partial function from the points of I_{2} to the points of I_{2} is said to be a projection of I_{2} if: (Def.3) there exist a, A, B such that a A and a B and it = π_{a}(A → B).

The following propositions are true:

1Supported by RPBP.III-24.C6

(1) If A = B or B = C or C = A, then A, B, C are concurrent.

(2) If A, B, C are concurrent, then A, C, B are concurrent and B, A, C are concurrent and B, C, A are concurrent and C, A, B are concurrent and C, B, A are concurrent.

(3) If o A and o B and y | B, then there exists x such that x | A and π_{o}(A → B)(x) = y.

(4) If o A and o B, then rng π_{o}(A → B) ⊆ the points of I_{2}.

(5) If o A and o B, then dom π_{o}(A → B) = chain(A).

(6) If o A and o B, then rng π_{o}(A → B) = chain(B).

(7) For an arbitrary x holds x ∈ chain(A) if and only if there exists a such that x = a and a | A.

(8) If o A and o B, then π_{o}(A → B) is one-to-one.

(9) If o A and o B, then π_{o}(A → B)^{−1} = π_{o}(B → A).

(10) For every projection f of I_{2} holds f^{−1} is a projection of I_{2}.

(11) If o A, then π_{o}(A → A) = id_{chain(A)}.

(12) id_{chain(A)} is a projection of I_{2}.

(13) If o A and o B and o C, then π_{o}(C → B) · π_{o}(A → C) = π_{o}(A → B).

(14) Suppose o_{1} O_{1} and o_{1} O_{2} and o_{2} O_{2} and o_{2} O_{3} and O_{1}, O_{2}, O_{3} are concurrent and O_{1} ≠ O_{3}. Then there exists o such that o O_{1} and o O_{3} and π_{o_{2}}(O_{2} → O_{3}) · π_{o_{1}}(O_{1} → O_{2}) = π_{o}(O_{1} → O_{3}).

(15) Suppose that (i) a A, (ii) b B, (iii) a C, (iv) b C,

(v) A, B, C are not concurrent, (vi) c | A, (vii) c | C, (viii) c | Q, (ix) b Q,

(x) A ≠ Q, (xi) a ≠ b, (xii) b ≠ q, (xiii) a | O, (xiv) b | O, (xv) B, C, O are not concurrent, (xvi) d | C, (xvii) d | B, (xviii) a | O_{1}, (xix) d | O_{1}, (xx) p | A, (xxi) p | O_{1}, (xxii) q | O, (xxiii) q | O_{2}, (xxiv) p | O_{2}, (xxv) p′_{1} | O_{2}, (xxvi) d | O_{3}, (xxvii) b | O_{3}, (xxviii) p′_{1} | O_{3}, (xxix) p′_{1} | Q, (xxx) Q ≠ C, (xxxi) q ≠ a, (xxxii) q A, (xxxiii) q Q.

Then π_{b}(C → B) · π_{a}(A → C) = π_{b}(Q → B) · π_{q}(A → Q).

(16) Suppose that (i) a A, (ii) a C, (iii) b B, (iv) b C,

(v) b Q, (vi) A, B, C are not concurrent, (vii) a ≠ b, (viii) b ≠ q, (ix) A ≠ Q,

(x) c, o | A, (xi) o, o′′, d | B, (xii) c, d, o′ | C, (xiii) a, b, d | O, (xiv) c, o′_{1} | Q, (xv) a, o, o′ | O_{1}, (xvi) b, o′, o′_{1} | O_{2}, (xvii) o, o′_{1}, q | O_{3}, (xviii) q | O.

Then π_{b}(C → B) · π_{a}(A → C) = π_{b}(Q → B) · π_{q}(A → Q).

(17) Suppose that (i) a A, (ii) a C, (iii) b B, (iv) b C,

(v) b Q, (vi) A, B, C are not concurrent, (vii) B, C, O are not concurrent, (viii) A ≠ Q, (ix) Q ≠ C,

(x) a ≠ b, (xi) c, p | A, (xii) d | B, (xiii) c, d | C, (xiv) a, b, q | O, (xv) c, p′_{1} | Q, (xvi) a, d, p | O_{1}, (xvii) q, p, p′_{1} | O_{2}, (xviii) b, d, p′_{1} | O_{3}.

Then q ≠ a and q ≠ b and q A and q Q.

(18) Suppose that (i) a A, (ii) a C, (iii) b B, (iv) b C,

(v) b Q, (vi) A, B, C are not concurrent, (vii) a ≠ b, (viii) A ≠ Q, (ix) c, o | A,

(x) o, o′′, d | B, (xi) c, d, o′ | C, (xii) a, b, d | O, (xiii) c, o′_{1} | Q, (xiv) a, o, o′ | O_{1}, (xv) b, o′, o′_{1} | O_{2}, (xvi) o, o′_{1}, q | O_{3}, (xvii) q | O.

Then q A and q Q and b ≠ q.

(19) Suppose that (i) a A, (ii) a C, (iii) b B, (iv) b C,

(v) q A, (vi) A, B, C are not concurrent, (vii) B, C, O are not concurrent, (viii) a ≠ b, (ix) b ≠ q,

(x) q ≠ a, (xi) c, p | A, (xii) d | B, (xiii) c, d | C, (xiv) a, b, q | O, (xv) c, p′_{1} | Q, (xvi) a, d, p | O_{1}, (xvii) q, p, p′_{1} | O_{2}, (xviii) b, d, p′_{1} | O_{3}.

Then Q ≠ A and Q ≠ C and q Q and b Q.

(20) Suppose that (i) a A, (ii) a C, (iii) b B, (iv) b C,

(v) q A, (vi) A, B, C are not concurrent, (vii) a ≠ b, (viii) b ≠ q, (ix) c, o | A,

(x) o, o′′, d | B, (xi) c, d, o′ | C, (xii) a, b, d | O, (xiii) c, o′_{1} | Q, (xiv) a, o, o′ | O_{1}, (xv) b, o′, o′_{1} | O_{2}, (xvi) o, o′_{1}, q | O_{3}, (xvii) q | O.

Then b Q and q Q and A ≠ Q.

(21) Suppose that (i) a A, (ii) b B, (iii) a C, (iv) b C,

(v) A, B, C are not concurrent, (vi) A, C, Q are concurrent, (vii) b Q, (viii) A ≠ Q, (ix) a ≠ b,

(x) a | O, (xi) b | O.

Then there exists q such that q | O and q A and q Q and π_{b}(C → B) · π_{a}(A → C) = π_{b}(Q → B) · π_{q}(A → Q).

(22) Suppose that (i) a A, (ii) b B, (iii) a C, (iv) b C,

(v) A, B, C are not concurrent, (vi) B, C, Q are concurrent, (vii) a Q, (viii) B ≠ Q, (ix) a ≠ b,

(x) a | O, (xi) b | O.

Then there exists q such that q | O and q B and q Q and π_{b}(C → B) · π_{a}(A → C) = π_{q}(Q → B) · π_{a}(A → Q).

(23) Suppose that (i) a A, (ii) b B, (iii) a C, (iv) b C,

(v) a B, (vi) b A, (vii) c | A, (viii) c | C, (ix) d | B,

(x) d | C, (xi) a | S, (xii) d | S, (xiii) c | R, (xiv) b | R, (xv) s | A, (xvi) s | S, (xvii) r | B, (xviii) r | R, (xix) s | Q, (xx) r | Q, (xxi) A, B, C are not concurrent.

Then π_{b}(C → B) · π_{a}(A → C) = π_{a}(Q → B) · π_{b}(A → Q).

(24) Suppose a A and b B and a C and b C and a ≠ b and a | O and b | O and q | O and q A and q ≠ b and A, B, C are not concurrent. Then there exists Q such that A, C, Q are concurrent and b Q and q Q and π_{b}(C → B) · π_{a}(A → C) = π_{b}(Q → B) · π_{q}(A → Q).

(25) Suppose a A and b B and a C and b C and a ≠ b and a | O and b | O and q | O and q B and q ≠ a and A, B, C are not concurrent. Then there exists Q such that B, C, Q are concurrent and a Q and q Q and π_{b}(C → B) · π_{a}(A → C) = π_{q}(Q → B) · π_{a}(A → Q).

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[2] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

On Projections in Projective Planes. … 329

[3] Eugeniusz Kusak and Wojciech Leończuk. Incidence projective space (a reduction theorem in a plane). Formalized Mathematics, 2(2):271–274, 1991.

[4] Wojciech Leończuk and Krzysztof Prażmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225–232, 1991.

[5] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[6] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[7] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[8] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[9] Zinaida Trybulec and Halina Świe‘czkowska. Boolean properties of sets. Formalized Mathematics, 1(1):17–23, 1990.

[10] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[11] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

330

Metric-Affine Configurations in Metric Affine Planes - Part I

Jolanta Świerzyńska Warsaw University Białystok Bogdan Świerzyński Warsaw University Białystok

Summary. We introduce several configurational axioms for metric affine planes such as theorem on three perpendiculars, orthogonalization of major Desargues Axiom, orthogonalization of the trapezium variant of Desargues Axiom, axiom on parallel projection together with its indirect forms. For convenience we also consider affine Major Desargues Axiom. The aim is to prove logical relationships which hold between the introduced statements.

MML Identifier: CONAFFM.

The notation and terminology used here have been introduced in the following papers: [7], [8], [6], [3], [5], [4], [1], and [2]. We adopt the following rules: X will denote a metric affine plane and o, a, a_{1}, b, b_{1}, c, c_{1} will denote elements of the points of X. Let us consider X. We say that Desargues Axiom holds in X if and only if the condition (Def.1) is satisfied. (Def.1) Given o, a, a_{1}, b, b_{1}, c, c_{1}. Suppose that

(i) o ≠ a, (ii) o ≠ a_{1}, (iii) o ≠ b, (iv) o ≠ b_{1},

(v) o ≠ c,

(vi) o ≠ c_{1},

(vii) not L(b, b_{1}, a), (viii) not L(a, a_{1}, c), (ix) L(o, a, a_{1}),

(x) L(o, b, b_{1}), (xi) L(o, c, c_{1}), (xii) a, b ∥ a_{1}, b_{1}, (xiii) a, c ∥ a_{1}, c_{1}.

Then b, c ∥ b_{1}, c_{1}.

Let us consider X. We say that AH holds in X if and only if the condition (Def.2) is satisfied. (Def.2) Given o, a, a_{1}, b, b_{1}, c, c_{1}. Suppose o, a ⊥ o, a_{1} and o, b ⊥ o, b_{1} and

o, c ⊥ o, c_{1} and a, b ⊥ a_{1}, b_{1} and o, a ∥ b, c and a, c ⊥ a_{1}, c_{1} and o, c o, a and o, a o, b. Then b, c ⊥ b_{1}, c_{1}.

Let us consider X. We say that theorem on three perpendiculars holds in X if and only if: (Def.3) for all a, b, c such that not L(a, b, c) there exists an element d of the

points of X such that d, a ⊥ b, c and d, b ⊥ a, c and d, c ⊥ a, b.

Let us consider X. We say that othogonal verion of Desargues Axiom holds in X if and only if the condition (Def.4) is satisfied. (Def.4) Given o, a, a_{1}, b, b_{1}, c, c_{1}. Then if o, a ⊥ o, a_{1} and o, b ⊥ o, b_{1} and

o, c ⊥ o, c_{1} and a, b ⊥ a_{1}, b_{1} and a, c ⊥ a_{1}, c_{1} and o, c o, a and o, a o, b, then b, c ⊥ b_{1}, c_{1}.

Let us consider X. We say that LIN holds in X if and only if the condition (Def.5) is satisfied. (Def.5) Given o, a, a_{1}, b, b_{1}, c, c_{1}. Suppose that

(i) o ≠ a, (ii) o ≠ a_{1}, (iii) o ≠ b, (iv) o ≠ b_{1},

(v) o ≠ c, (vi) o ≠ c_{1}, (vii) a ≠ b, (viii) o, c ⊥ o, c_{1}, (ix) o, a ⊥ o, a_{1},

(x) o, b ⊥ o, b_{1}, (xi) not L(o, c, a), (xii) L(o, a, b), (xiii) L(o, a_{1}, b_{1}), (xiv) a, c ⊥ a_{1}, c_{1}, (xv) b, c ⊥ b_{1}, c_{1}.

Then a, a_{1} ∥ b, b_{1}.

Let us consider X. We say that first indirect form of LIN holds in X if and only if the condition (Def.6) is satisfied. (Def.6) Given o, a, a_{1}, b, b_{1}, c, c_{1}. Suppose that

(i) o ≠ a, (ii) o ≠ a_{1}, (iii) o ≠ b, (iv) o ≠ b_{1},

(v) o ≠ c, (vi) o ≠ c_{1}, (vii) a ≠ b, (viii) o, c ⊥ o, c_{1}, (ix) o, a ⊥ o, a_{1},

(x) o, b ⊥ o, b_{1}, (xi) not L(o, c, a), (xii) L(o, a, b), (xiii) L(o, a_{1}, b_{1}), (xiv) a, c ⊥ a_{1}, c_{1}, (xv) a, a_{1} ∥ b, b_{1}.

Then b, c ⊥ b_{1}, c_{1}.

Let us consider X. We say that second indirect form of LIN holds in X if and only if the condition (Def.7) is satisfied. (Def.7) Given o, a, a_{1}, b, b_{1}, c, c_{1}. Suppose that

(i) o ≠ a, (ii) o ≠ a_{1}, (iii) o ≠ b, (iv) o ≠ b_{1},

(v) o ≠ c, (vi) o ≠ c_{1}, (vii) a ≠ b, (viii) a, a_{1} ∥ b, b_{1}, (ix) o, a ⊥ o, a_{1},

(x) o, b ⊥ o, b_{1}, (xi) not L(o, c, a), (xii) L(o, a, b), (xiii) L(o, a_{1}, b_{1}), (xiv) a, c ⊥ a_{1}, c_{1}, (xv) b, c ⊥ b_{1}, c_{1}.

Then o, c ⊥ o, c_{1}.

We now state several propositions:

(1) If othogonal verion of Desargues Axiom holds in X, then Desargues Axiom holds in X.

(2) If othogonal verion of Desargues Axiom holds in X, then AH holds in X.

(3) If LIN holds in X, then first indirect form of LIN holds in X.

(4) If first indirect form of LIN holds in X, then second indirect form of LIN holds in X.

(5) If LIN holds in X, then othogonal verion of Desargues Axiom holds in X.

(6) If LIN holds in X, then theorem on three perpendiculars holds in X.

334 Jolanta Świerzyńska et al.

References

[1] Eugeniusz Kusak, Wojciech Leończuk, and Michał Muzalewski. Construction of a bilinear symmetric form in orthogonal vector space. Formalized Mathematics, 1(2):353–356, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical metric affine spaces and planes. Formalized Mathematics, 1(5):891–899, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[6] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[7] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[8] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

Metric-Affine Configurations in Metric Affine Planes - Part II

Jolanta Świerzyńska Warsaw University Białystok Bogdan Świerzyński Warsaw University Białystok

Summary. A continuation of [5]. We introduce more configurational axioms i.e. orthogonalizations of ”scherungssatzes” (direct and indirect), ”Scherungssatz” with orthogonal axes, Pappus axiom with orthogonal axes; we also consider the affine Major Pappus Axiom and affine minor Desargues Axiom. We prove a number of implications which hold between the above axioms.

MML Identifier: CONMETR.

The articles [2], [4], [1], [3], and [5] provide the notation and terminology for this paper. We adopt the following rules: X will denote a metric affine plane, o, a, a_{1}, a_{2}, a_{3}, a_{4}, b, b_{1}, b_{2}, b_{3}, b_{4}, c, c_{1}, d will denote elements of the points of X, and A, K, M, N will denote subsets of the points of X. Let us consider X. We say that Pappos Axiom with orthogonal axes holds in X if and only if the condition (Def.1) is satisfied. (Def.1) Given o, a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, M, N. Suppose that

(i) o ∈ M, (ii) a_{1} ∈ M, (iii) a_{2} ∈ M, (iv) a_{3} ∈ M,

(v) o ∈ N, (vi) b_{1} ∈ N, (vii) b_{2} ∈ N, (viii) b_{3} ∈ N, (ix) b_{2} ∉ M,

(x) a_{3} ∉ N, (xi) M ⊥ N, (xii) o ≠ a_{1}, (xiii) o ≠ a_{2}, (xiv) o ≠ a_{3}, (xv) o ≠ b_{1}, (xvi) o ≠ b_{2}, (xvii) o ≠ b_{3}, (xviii) a_{3}, b_{2} ∥ a_{2}, b_{1}, (xix) a_{3}, b_{3} ∥ a_{1}, b_{1}.

Then a_{1}, b_{2} ∥ a_{2}, b_{3}.

Let us consider X. We say that Pappos Axiom holds in X if and only if the condition (Def.2) is satisfied. (Def.2) Given o, a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}, M, N. Suppose that

(i) M is a line, (ii) N is a line, (iii) o ∈ M, (iv) a_{1} ∈ M,

(v) a_{2} ∈ M, (vi) a_{3} ∈ M, (vii) o ∈ N, (viii) b_{1} ∈ N, (ix) b_{2} ∈ N,

(x) b_{3} ∈ N, (xi) b_{2} ∉ M, (xii) a_{3} ∉ N, (xiii) o ≠ a_{1}, (xiv) o ≠ a_{2}, (xv) o ≠ a_{3}, (xvi) o ≠ b_{1}, (xvii) o ≠ b_{2}, (xviii) o ≠ b_{3}, (xix) a_{3}, b_{2} ∥ a_{2}, b_{1}, (xx) a_{3}, b_{3} ∥ a_{1}, b_{1}.

Then a_{1}, b_{2} ∥ a_{2}, b_{3}.

Let us consider X. We say that MH1 holds in X if and only if the condition (Def.3) is satisfied. (Def.3) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M ⊥ N, (ii) a_{1} ∈ M, (iii) a_{3} ∈ M, (iv) b_{1} ∈ M,

(v) b_{3} ∈ M, (vi) a_{2} ∈ N, (vii) a_{4} ∈ N, (viii) b_{2} ∈ N, (ix) b_{4} ∈ N, (x) a_{2} ∉ M, (xi) a_{4} ∉ M, (xii) a_{1}, a_{2} ⊥ b_{1}, b_{2}, (xiii) a_{2}, a_{3} ⊥ b_{2}, b_{3}, (xiv) a_{3}, a_{4} ⊥ b_{3}, b_{4}.

Then a_{1}, a_{4} ⊥ b_{1}, b_{4}.

Let us consider X. We say that MH2 holds in X if and only if the condition (Def.4) is satisfied. (Def.4) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M ⊥ N, (ii) a_{1} ∈ M, (iii) a_{3} ∈ M, (iv) b_{2} ∈ M,

(v) b_{4} ∈ M, (vi) a_{2} ∈ N, (vii) a_{4} ∈ N, (viii) b_{1} ∈ N, (ix) b_{3} ∈ N,

(x) a_{2} ∉ M, (xi) a_{4} ∉ M, (xii) a_{1}, a_{2} ⊥ b_{1}, b_{2}, (xiii) a_{2}, a_{3} ⊥ b_{2}, b_{3}, (xiv) a_{3}, a_{4} ⊥ b_{3}, b_{4}.

Then a_{1}, a_{4} ⊥ b_{1}, b_{4}.

Let us consider X. We say that trapezium variant of Desargues Axiom holds in X if and only if the condition (Def.5) is satisfied. (Def.5) Given o, a, a_{1}, b, b_{1}, c, c_{1}. Suppose that

(i) o ≠ a, (ii) o ≠ a_{1}, (iii) o ≠ b, (iv) o ≠ b_{1},

(v) o ≠ c, (vi) o ≠ c_{1}, (vii) not L(b, b_{1}, a), (viii) not L(b, b_{1}, c), (ix) L(o, a, a_{1}),

(x) L(o, b, b_{1}), (xi) L(o, c, c_{1}), (xii) a, b ∥ a_{1}, b_{1}, (xiii) a, b ∥ o, c, (xiv) b, c ∥ b_{1}, c_{1}.

Then a, c ∥ a_{1}, c_{1}.

Let us consider X. We say that Scherungssatz holds in X if and only if the condition (Def.6) is satisfied. (Def.6) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M is a line, (ii) N is a line, (iii) a_{1} ∈ M, (iv) a_{3} ∈ M,

(v) b_{1} ∈ M, (vi) b_{3} ∈ M, (vii) a_{2} ∈ N, (viii) a_{4} ∈ N, (ix) b_{2} ∈ N,

(x) b_{4} ∈ N, (xi) a_{4} ∉ M, (xii) a_{2} ∉ M, (xiii) b_{2} ∉ M, (xiv) b_{4} ∉ M, (xv) a_{1} ∉ N, (xvi) a_{3} ∉ N, (xvii) b_{1} ∉ N, (xviii) b_{3} ∉ N, (xix) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xx) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xxi) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}.

Let us consider X. We say that Scherungssatz with orthogonal axes holds in X if and only if the condition (Def.7) is satisfied. (Def.7) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M ⊥ N, (ii) a_{1} ∈ M, (iii) a_{3} ∈ M, (iv) b_{1} ∈ M,

(v) b_{3} ∈ M, (vi) a_{2} ∈ N, (vii) a_{4} ∈ N, (viii) b_{2} ∈ N, (ix) b_{4} ∈ N,

(x) a_{4} ∉ M, (xi) a_{2} ∉ M, (xii) b_{2} ∉ M, (xiii) b_{4} ∉ M, (xiv) a_{1} ∉ N, (xv) a_{3} ∉ N, (xvi) b_{1} ∉ N, (xvii) b_{3} ∉ N, (xviii) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xix) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xx) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}.

Let us consider X. We say that minor Desargues Axiom holds in X if and only if: (Def.8) for all a, a_{1}, b, b_{1}, c, c_{1} such that not L(a, a_{1}, b) and not L(a, a_{1}, c)

and a, a_{1} ∥ b, b_{1} and a, a_{1} ∥ c, c_{1} and a, b ∥ a_{1}, b_{1} and a, c ∥ a_{1}, c_{1} holds b, c ∥ b_{1}, c_{1}.

One can prove the following propositions:

(1) There exist a, b, c such that L(a, b, c) and a ≠ b and b ≠ c and c ≠ a.

(2) For all a, b such that a ≠ b there exists c such that L(a, b, c) and a ≠ c and b ≠ c.

(3) For all A, a such that A is a line there exists K such that a ∈ K and A ⊥ K.

(4) If A is a line and a ∈ A and b ∈ A and c ∈ A, then L(a, b, c).

(5) If A is a line and M is a line and a ∈ A and b ∈ A and a ∈ M and b ∈ M, then a = b or A = M.

(6) For all a, b, c, d, M and for every subset M′ of the points of the affine reduct of X and for all elements c′, d′ of the points of the affine reduct of X such that c = c′ and d = d′ and M = M′ and a ∈ M and b ∈ M and c′, d′ ∥ M′ holds c, d ∥ a, b.

(7) If trapezium variant of Desargues Axiom holds in X, then the affine reduct of X satisfies TDES.

(8) If the affine reduct of X satisfies des, then minor Desargues Axiom holds in X.

(9) If MH1 holds in X, then Scherungssatz with orthogonal axes holds in X.

(10) If MH2 holds in X, then Scherungssatz with orthogonal axes holds in X.

(11) If AH holds in X, then trapezium variant of Desargues Axiom holds in X.

(12) If Scherungssatz with orthogonal axes holds in X and trapezium variant of Desargues Axiom holds in X, then Scherungssatz holds in X.

(13) If Pappos Axiom with orthogonal axes holds in X and Desargues Axiom holds in X, then Pappos Axiom holds in X.

(14) If MH1 holds in X and MH2 holds in X, then Pappos Axiom with orthogonal axes holds in X.

(15) If theorem on three perpendiculars holds in X, then Pappos Axiom with orthogonal axes holds in X.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical metric affine spaces and planes. Formalized Mathematics, 1(5):891–899, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Classical configurations in affine planes. Formalized Mathematics, 1(4):625–633, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[5] Jolanta Świerzyńska and Bogdan Świerzyński. Metric-affine configurations in metric affine planes - Part I. Formalized Mathematics, 2(3):331–334, 1991.

Fanoian, Pappian and Desarguesian Affine Spaces

Krzysztof Prażmowski^{1} Warsaw University Białystok

Summary. We introduce basic types of affine spaces such as Desarguesian, Fanoian, Pappian, and translation affine and ordered affine spaces and we prove that suitably chosen analytically defined affine structures satify the required properties.

MML Identifier: PAPDESAF.

The articles [6], [1], [4], [5], [2], and [3] provide the notation and terminology for this paper. Let O_{1} be an ordered affine space. Then Λ(O_{1}) is an affine space.

Let O_{1} be an ordered affine plane. Then Λ(O_{1}) is an affine plane. We now state several propositions:

(1) There exists a real linear space V and there exist vectors u, v of V such that for all real numbers a, b such that a · u + b · v = 0_{V} holds a = 0 and b = 0.

(2) For every ordered affine space O_{1} and for an arbitrary x holds x is an element of the points of O_{1} if and only if x is an element of the points of Λ(O_{1}) but x is a subset of the points of O_{1} if and only if x is a subset of the points of Λ(O_{1}).

(3) For every ordered affine space O_{1} and for all elements a, b, c of the points of O_{1} and for all elements a′, b′, c′ of the points of Λ(O_{1}) such that a = a′ and b = b′ and c = c′ holds L(a, b, c) if and only if L(a′, b′, c′).

(4) For every real linear space V and for an arbitrary x holds x is an element of the points of OASpace V if and only if x is a vector of V.

(5) Let V be a real linear space. Then for every ordered affine space O_{1} such that O_{1} = OASpace V for all elements a, b, c, d of the points of O_{1} and for all vectors u, v, w, y of V such that a = u and b = v and c = w and d = y holds a, b |||| c, d if and only if u, v |||| w, y.

1Supported by RPBP.III-24.C2

(6) For every real linear space V and for every ordered affine space O_{1} such that O_{1} = OASpace V there exist vectors u, v of V such that for all real numbers a, b such that a · u + b · v = 0_{V} holds a = 0 and b = 0.

Let A_{1} be an affine space. We say that A_{1} satisfies PAP’ if and only if the condition (Def.1) is satisfied. (Def.1) Let M, N be subsets of the points of A_{1}. Let o, a, b, c, a′, b′, c′ be

elements of the points of A_{1}. Suppose that

(i) M is a line, (ii) N is a line, (iii) M ≠ N, (iv) o ∈ M,

(v) o ∈ N, (vi) o ≠ a, (vii) o ≠ a′, (viii) o ≠ b, (ix) o ≠ b′,

(x) o ≠ c, (xi) o ≠ c′, (xii) a ∈ M, (xiii) b ∈ M, (xiv) c ∈ M, (xv) a′ ∈ N, (xvi) b′ ∈ N, (xvii) c′ ∈ N, (xviii) a, b′ ∥ b, a′, (xix) b, c′ ∥ c, b′.

Then a, c′ ∥ c, a′.

Let A_{1} be an affine space. We say that A_{1} satisfies DES’ if and only if the condition (Def.2) is satisfied. (Def.2) Let A, P, C be subsets of the points of A_{1}. Let o, a, b, c, a′, b′, c′ be

elements of the points of A_{1}. Suppose that

(i) o ∈ A, (ii) o ∈ P, (iii) o ∈ C, (iv) o ≠ a,

(v) o ≠ b, (vi) o ≠ c, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) c′ ∈ C, (xiii) A is a line, (xiv) P is a line, (xv) C is a line, (xvi) A ≠ P, (xvii) A ≠ C, (xviii) a, b ∥ a′, b′, (xix) a, c ∥ a′, c′.

Then b, c ∥ b′, c′.

Let A_{1} be an affine space. We say that A_{1} satisfies TDES’ if and only if the condition (Def.3) is satisfied. (Def.3) Let K be a subset of the points of A_{1}. Let o, a, b, c, a′, b′, c′ be elements

of the points of A_{1}. Suppose that

(i) K is a line, (ii) o ∈ K, (iii) c ∈ K, (iv) c′ ∈ K,

(v) a ∉ K, (vi) o ≠ c, (vii) a ≠ b, (viii) L(o, a, a′), (ix) L(o, b, b′),

(x) a, b ∥ a′, b′, (xi) a, c ∥ a′, c′, (xii) a, b ∥ K.

Then b, c ∥ b′, c′.

Let A_{1} be an affine space. We say that A_{1} satisfies des’ if and only if the condition (Def.4) is satisfied. (Def.4) Let A, P, C be subsets of the points of A_{1}. Let a, b, c, a′, b′, c′ be

elements of the points of A_{1}. Suppose that

(i) A ∥ P, (ii) A ∥ C, (iii) a ∈ A, (iv) a′ ∈ A,

(v) b ∈ P, (vi) b′ ∈ P, (vii) c ∈ C, (viii) c′ ∈ C, (ix) A is a line,

(x) P is a line, (xi) C is a line, (xii) A ≠ P, (xiii) A ≠ C, (xiv) a, b ∥ a′, b′, (xv) a, c ∥ a′, c′.

Then b, c ∥ b′, c′. Let A_{1} be an affine space. We say that A_{1} satisfies Fano Axiom if and only

if: (Def.5) for all elements a, b, c, d of the points of A_{1} such that a, b ∥ c, d and

a, c ∥ b, d and a, d ∥ b, c holds a, b ∥ a, c. One can prove the following propositions:

(7) For every affine plane A_{1} holds A_{1} satisfies PAP if and only if A_{1} satisfies PAP’.

(8) For every affine plane A_{1} holds A_{1} satisfies DES if and only if A_{1} satisfies DES’.

(9) For every affine plane A_{1} holds A_{1} satisfies TDES if and only if A_{1} satisfies TDES’.

(10) For every affine plane A_{1} holds A_{1} satisfies des if and only if A_{1} satisfies

des’. An affine space is Pappian if:

(Def.6) it satisfies PAP’.

An affine space is Desarguesian if: (Def.7) it satisfies DES’.

An affine space is Moufangian if: (Def.8) it satisfies TDES’.

An affine space is translation if: (Def.9) it satisfies des’.

An affine space is Fanoian if: (Def.10) it satisfies Fano Axiom.

An ordered affine space is Pappian if: (Def.11) Λ(it) satisfies PAP’.

An ordered affine space is Desarguesian if: (Def.12) Λ(it) satisfies DES’.

An ordered affine space is Moufangian if: (Def.13) Λ(it) satisfies TDES’.

An ordered affine space is translation if: (Def.14) Λ(it) satisfies des’.

Let O_{1} be an ordered affine space. We say that O_{1} satisfies DES if and only if the condition (Def.15) is satisfied.

(Def.15) Let o, a, b, c, a_{1}, b_{1}, c_{1} be elements of the points of O_{1}. Then if o, a \begin{matrix} \\ \\ ∥ \end{matrix}

o, a_{1} and o, b \begin{matrix} \\ \\ ∥ \end{matrix} o, b_{1} and o, c \begin{matrix} \\ \\ ∥ \end{matrix} o, c_{1} and not L(o, a, b) and not L(o, a, c)

and a, b \begin{matrix} \\ \\ ∥ \end{matrix} a_{1}, b_{1} and a, c \begin{matrix} \\ \\ ∥ \end{matrix} a_{1}, c_{1}, then b, c \begin{matrix} \\ \\ ∥ \end{matrix} b_{1}, c_{1}.

Let O_{1} be an ordered affine space. We say that O_{1} satisfies DES_{1} if and only if the condition (Def.16) is satisfied.

(Def.16) Let o, a, b, c, a_{1}, b_{1}, c_{1} be elements of the points of O_{1}. Then if a, o \begin{matrix} \\ \\ ∥ \end{matrix}

o, a_{1} and b, o \begin{matrix} \\ \\ ∥ \end{matrix} o, b_{1} and c, o \begin{matrix} \\ \\ ∥ \end{matrix} o, c_{1} and not L(o, a, b) and not L(o, a, c)

and a, b \begin{matrix} \\ \\ ∥ \end{matrix} b_{1}, a_{1} and a, c \begin{matrix} \\ \\ ∥ \end{matrix} c_{1}, a_{1}, then b, c \begin{matrix} \\ \\ ∥ \end{matrix} c_{1}, b_{1}.

One can prove the following propositions:

(11) For every ordered affine space O_{1} such that O_{1} satisfies DES_{1} holds O_{1} satisfies DES.

(12) For every ordered affine space O_{1} and for all elements o, a, b, a′, b′ of the points of O_{1} such that not L(o, a, b) and a, o \begin{matrix} \\ \\ ∥ \end{matrix} o, a′ and L(o, b, b′) and

a, b |||| a′, b′ holds b, o \begin{matrix} \\ \\ ∥ \end{matrix} o, b′ and a, b \begin{matrix} \\ \\ ∥ \end{matrix} b′, a′.

(13) For every ordered affine space O_{1} and for all elements o, a, b, a′, b′ of the points of O_{1} such that not L(o, a, b) and o, a \begin{matrix} \\ \\ ∥ \end{matrix} o, a′ and L(o, b, b′) and

a, b |||| a′, b′ holds o, b \begin{matrix} \\ \\ ∥ \end{matrix} o, b′ and a, b \begin{matrix} \\ \\ ∥ \end{matrix} a′, b′.

(14) For every ordered affine space O_{2} such that O_{2} satisfies DES_{1} holds Λ(O_{2}) satisfies DES’.

(15) Let V be a real linear space. Let o, u, v, u_{1}, v_{1} be vectors of V. Let r be a real number. Suppose o − u = r · (u_{1} − o) and r ≠ 0 and o, v |||| o, v_{1} and o, u |||| o, v and u, v |||| u_{1}, v_{1}. Then v_{1} = u_{1} + (−r)^{−1} · (v − u) and v_{1} = o + (−r)^{−1} · (v − o) and v − u = (−r) · (v_{1} − u_{1}).

(16) For every real number r such that r ≠ 0 holds (−r)^{−1} = −r^{−1}.

(17) For every real linear space V and for every ordered affine space O_{1} such that O_{1} = OASpace V holds O_{1} satisfies DES_{1}.

(18) For every real linear space V and for every ordered affine space O_{1} such that O_{1} = OASpace V holds O_{1} satisfies DES_{1} and O_{1} satisfies DES.

(19) For every real linear space V and for every ordered affine space O_{1} such that O_{1} = OASpace V holds Λ(O_{1}) satisfies PAP’.

(20) For every real linear space V and for every ordered affine space O_{1} such that O_{1} = OASpace V holds Λ(O_{1}) satisfies DES’.

(21) For every affine space A_{1} such that A_{1} satisfies DES’ holds A_{1} satisfies TDES’.

(22) For every real linear space V and for every ordered affine space O_{1} such that O_{1} = OASpace V holds Λ(O_{1}) satisfies TDES’.

(23) For every real linear space V and for every ordered affine space O_{1} such that O_{1} = OASpace V holds Λ(O_{1}) satisfies des’.

(24) For every ordered affine space O_{1} holds Λ(O_{1}) satisfies Fano Axiom.

Let O_{1} be an ordered affine space. Then Λ(O_{1}) is an Fanoian affine space. Let O_{1} be a Pappian ordered affine space. Then Λ(O_{1}) is a Pappian Fanoian

affine space.

Let O_{1} be a Desarguesian ordered affine space. Then Λ(O_{1}) is an Desarguesian Fanoian affine space.

346 Krzysztof Prażmowski

Let O_{1} be a Moufangian ordered affine space. Then Λ(O_{1}) is an Moufangian Fanoian affine space.

Let O_{1} be a translation ordered affine space. Then Λ(O_{1}) is a translation Fanoian affine space.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Classical configurations in affine planes. Formalized Mathematics, 1(4):625–633, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. A construction of analytical ordered trapezium spaces. Formalized Mathematics, 2(3):315–322, 1991.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[6] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291– 296, 1990.

Elementary Variants of Affine Configurational Theorems ^{1}

Krzysztof Prażmowski Warsaw University Białystok Krzysztof Radziszewski Gdańsk University

Summary. We present elementary versions of Pappus, Major Desargues and Minor Desargues Axioms (i.e. statements formulated entirely in the language of points and parallelism of segments). Evidently they are consequences of appropriate configurational axioms introduced in the article [2]. In particular it follows that there exists an affine plane satisfying all of them.

MML Identifier: PARDEPAP.

The terminology and notation used in this paper have been introduced in the following papers: [1], [3], [2], and [4]. In the sequel S_{1} will be an affine plane. The following propositions are true:

(1) If S_{1} satisfies PAP, then for all elements a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} of the points of S_{1} such that a_{1}, a_{2} ∥ a_{1}, a_{3} and b_{1}, b_{2} ∥ b_{1}, b_{3} and a_{1}, b_{2} ∥ a_{2}, b_{1} and a_{2}, b_{3} ∥ a_{3}, b_{2} holds a_{3}, b_{1} ∥ a_{1}, b_{3}.

(2) Suppose S_{1} satisfies DES. Let o, a, a′, b, b′, c, c′ be elements of the points of S_{1}. Then if o, a o, b and o, a o, c and o, a ∥ o, a′ and o, b ∥ o, b′ and o, c ∥ o, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′, then b, c ∥ b′, c′.

(3) Suppose S_{1} satisfies des. Let a, a′, b, b′, c, c′ be elements of the points of S_{1}. Then if a, a′ a, b and a, a′ a, c and a, a′ ∥ b, b′ and a, a′ ∥ c, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′, then b, c ∥ b′, c′.

(4) If S_{1} satisfies Fano Axiom, then for all elements a, b, c, d of the points of S_{1} such that a, b a, c and a, b ∥ c, d and a, c ∥ b, d holds a, d b, c.

(5) There exists S_{1} such that for all elements o, a, a′, b, b′, c, c′ of the points of S_{1} such that o, a o, b and o, a o, c and o, a ∥ o, a′ and o, b ∥ o, b′ and o, c ∥ o, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′ holds b, c ∥ b′, c′ and for all elements a, a′, b, b′, c, c′ of the points of S_{1} such that a, a′ a, b and a, a′ a, c and a, a′ ∥ b, b′ and a, a′ ∥ c, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′ holds b, c ∥ b′, c′ and for all elements a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} of the points of S_{1} such that a_{1}, a_{2} ∥ a_{1}, a_{3} and b_{1}, b_{2} ∥ b_{1}, b_{3} and a_{1}, b_{2} ∥ a_{2}, b_{1} and a_{2}, b_{3} ∥ a_{3}, b_{2} holds a_{3}, b_{1} ∥ a_{1}, b_{3} and for all elements a, b, c, d of the points of S_{1} such that a, b a, c and a, b ∥ c, d and a, c ∥ b, d holds a, d b, c.

1Supported by RPBP.III-24.C2

(6) For every elements o, a of the points of S_{1} there exists an element p of the points of S_{1} such that for all elements b, c of the points of S_{1} holds o, a ∥ o, p and there exists an element d of the points of S_{1} such that if o, p ∥ o, b, then o, c ∥ o, d and p, c ∥ b, d.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Classical configurations in affine planes. Formalized Mathematics, 1(4):625–633, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[4] Krzysztof Prażmowski. Fanoian, Pappian and Desarguesian affine spaces. Formalized Mathematics, 2(3):341–346, 1991.

Semi-Affine Space ^{1}

Eugeniusz Kusak Warsaw University Białystok Krzysztof Radziszewski Gdańsk University

Summary. A brief survey on semi-affine geometry, which results from the classical Pappian and Desarguesian affine (dimension free) geometry by weakening the so called trapezium axiom. With the help of the relation of parallelogram in every semi-affine space we define the operation of ”addition” of ”vectors”. Next we investigate in greater details the relation of (affine) trapezium in such spaces.

MML Identifier: SEMI AF1.

The papers [3], [2], and [1] provide the notation and terminology for this paper. An affine structure is called a semi affine space if it satisfies the conditions (Def.1). (Def.1) (i) For all elements a, b of the points of it holds a, b ∥ b, a,

(ii) for all elements a, b, c of the points of it holds a, b ∥ c, c, (iii) for all elements a, b, p, q, r, s of the points of it such that a ≠ b and

a, b ∥ p, q and a, b ∥ r, s holds p, q ∥ r, s, (iv) for all elements a, b, c of the points of it such that a, b ∥ a, c holds

b, a ∥ b, c,

(v) there exist elements a, b, c of the points of it such that a, b a, c, (vi) for every elements a, b, p of the points of it there exists an element q

of the points of it such that a, b ∥ p, q and a, p ∥ b, q, (vii) for every elements o, a of the points of it there exists an element p of

the points of it such that for all elements b, c of the points of it holds o, a ∥ o, p and there exists an element d of the points of it such that if o, p ∥ o, b, then o, c ∥ o, d and p, c ∥ b, d,

(viii) for all elements o, a, a′, b, b′, c, c′ of the points of it such that o, a o, b and o, a o, c and o, a ∥ o, a′ and o, b ∥ o, b′ and o, c ∥ o, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′ holds b, c ∥ b′, c′, (ix) for all elements a, a′, b, b′, c, c′ of the points of it such that a, a′ a, b

1Supported by RPBP.III-24.C2

and a, a′ a, c and a, a′ ∥ b, b′ and a, a′ ∥ c, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′ holds b, c ∥ b′, c′,

(x) for all elements a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3} of the points of it such that a_{1}, a_{2} ∥ a_{1}, a_{3} and b_{1}, b_{2} ∥ b_{1}, b_{3} and a_{1}, b_{2} ∥ a_{2}, b_{1} and a_{2}, b_{3} ∥ a_{3}, b_{2} holds a_{3}, b_{1} ∥ a_{1}, b_{3},

(xi) for all elements a, b, c, d of the points of it such that a, b a, c and

a, b ∥ c, d and a, c ∥ b, d holds a, d b, c. We adopt the following convention: S_{1} will be a semi affine space and a, a′,

a_{1}, a_{2}, a_{3}, a_{4}, b, b′, b_{1}, b_{2}, b_{3}, c, c′, d, d′, d_{1}, d_{2}, o, p, p_{1}, p_{2}, q, r, r_{1}, r_{2}, s, x, y, z will be elements of the points of S_{1}. The following propositions are true:

(1) a, b ∥ b, a.

(2) a, b ∥ c, c.

(3) If a ≠ b and a, b ∥ p, q and a, b ∥ r, s, then p, q ∥ r, s.

(4) If a, b ∥ a, c, then b, a ∥ b, c.

(5) There exist a, b, c such that a, b a, c.

(6) There exists q such that a, b ∥ p, q and a, p ∥ b, q.

(7) For every o, a there exists p such that for all b, c holds o, a ∥ o, p and there exists d such that if o, p ∥ o, b, then o, c ∥ o, d and p, c ∥ b, d.

(8) If o, a o, b and o, a o, c and o, a ∥ o, a′ and o, b ∥ o, b′ and o, c ∥ o, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′, then b, c ∥ b′, c′.

(9) If a, a′ a, b and a, a′ a, c and a, a′ ∥ b, b′ and a, a′ ∥ c, c′ and a, b ∥ a′, b′ and a, c ∥ a′, c′, then b, c ∥ b′, c′.

(10) If a_{1}, a_{2} ∥ a_{1}, a_{3} and b_{1}, b_{2} ∥ b_{1}, b_{3} and a_{1}, b_{2} ∥ a_{2}, b_{1} and a_{2}, b_{3} ∥ a_{3}, b_{2}, then a_{3}, b_{1} ∥ a_{1}, b_{3}.

(11) If a, b a, c and a, b ∥ c, d and a, c ∥ b, d, then a, d b, c.

(12) a, b ∥ a, b.

(13) If a, b ∥ c, d, then c, d ∥ a, b.

(14) a, a ∥ b, c.

(15) If a, b ∥ c, d, then b, a ∥ c, d.

(16) If a, b ∥ c, d, then a, b ∥ d, c.

(17) If a, b ∥ c, d, then b, a ∥ c, d and a, b ∥ d, c and b, a ∥ d, c and c, d ∥ a, b and d, c ∥ a, b and c, d ∥ b, a and d, c ∥ b, a.

(18) Suppose a, b ∥ a, c. Then a, c ∥ a, b and b, a ∥ a, c and a, b ∥ c, a and a, c ∥ b, a and b, a ∥ c, a and c, a ∥ a, b and c, a ∥ b, a and b, a ∥ b, c and a, b ∥ b, c and b, a ∥ c, b and b, c ∥ b, a and a, b ∥ c, b and c, b ∥ b, a and b, c ∥ a, b and c, b ∥ a, b and c, a ∥ c, b and a, c ∥ c, b and c, a ∥ b, c and a, c ∥ b, c and c, b ∥ c, a and b, c ∥ c, a and c, b ∥ a, c and b, c ∥ a, c.

(19) If a, b ∥ p, q and a, b ∥ r, s, then a = b or p, q ∥ r, s.

(20) If a ≠ b and p, q ∥ a, b and a, b ∥ r, s, then p, q ∥ r, s.

(21) If a, b a, d, then a ≠ b and b ≠ d and d ≠ a.

(22) If a, b p, q, then a ≠ b and p ≠ q.

(23) If a, b ∥ a, x and b, c ∥ b, x and c, a ∥ c, x, then a, b ∥ a, c.

(24) If a, b a, c, then a, b a, x or b, c b, x or c, a c, x.

(25) If a, b a, c and p ≠ q, then p, q p, a or p, q p, b or p, q p, c.

(26) If p ≠ q, then there exists r such that p, q p, r.

(27) Suppose a, b c, d. Then a, b d, c and b, a c, d and b, a d, c and c, d a, b and c, d b, a and d, c a, b and d, c b, a.

(28) Suppose a, b a, c. Then a, b c, a and b, a a, c and b, a c, a and a, c a, b and a, c b, a and c, a a, b and c, a b, a and b, a b, c and b, a c, b and a, b b, c and a, b c, b and b, c b, a and b, c a, b and c, b a, b and c, b b, a and c, b c, a and c, b a, c and b, c c, a and b, c a, c and c, a c, b and c, a b, c and a, c b, c and a, c c, b.

(29) If a, b c, d and a, b ∥ p, q and c, d ∥ r, s and p ≠ q and r ≠ s, then p, q r, s.

(30) If a, b a, c and a, b ∥ p, q and a, c ∥ p, r and b, c ∥ q, r and p ≠ q, then p, q p, r.

(31) If a, b a, c and a, c ∥ p, r and b, c ∥ p, r, then p = r.

We now state four propositions:

(32) If p, q p, r_{1} and p, r_{1} ∥ p, r_{2} and q, r_{1} ∥ q, r_{2}, then r_{1} = r_{2}.

(33) If a, b a, c and a, b ∥ p, q and a, c ∥ p, r_{1} and a, c ∥ p, r_{2} and b, c ∥ q, r_{1} and b, c ∥ q, r_{2}, then r_{1} = r_{2}.

(34) If a = b or c = d or a = c and b = d or a = d and b = c, then a, b ∥ c, d.

(35) If a = b or a = c or b = c, then a, b ∥ a, c.

Let us consider S_{1}, a, b, c. We say that a, b and c are collinear if and only if: (Def.2) a, b ∥ a, c.

We now state a number of propositions: (37)^{2} If a_{1}, a_{2} and a_{3} are collinear, then a_{1}, a_{3} and a_{2} are collinear and a_{2},

a_{1} and a_{3} are collinear and a_{2}, a_{3} and a_{1} are collinear and a_{3}, a_{1} and a_{2} are collinear and a_{3}, a_{2} and a_{1} are collinear.

(38) If a_{1}, a_{2} and a_{3} are not collinear, then a_{1}, a_{3} and a_{2} are not collinear and a_{2}, a_{1} and a_{3} are not collinear and a_{2}, a_{3} and a_{1} are not collinear and a_{3}, a_{1} and a_{2} are not collinear and a_{3}, a_{2} and a_{1} are not collinear.

(39) If a, b and c are not collinear and a, b ∥ p, q and a, c ∥ p, r and p ≠ q and p ≠ r, then p, q and r are not collinear.

(40) If a = b or b = c or c = a, then a, b and c are collinear.

(41) If p ≠ q, then there exists r such that p, q and r are not collinear.

(42) If a, b and c are collinear and a, b and d are collinear, then a, b ∥ c, d.

(43) If a, b and c are not collinear and a, b ∥ c, d, then a, b and d are not collinear.

2The proposition (36) was either repeated or obvious.

(44) If a, b and c are not collinear and a, b ∥ c, d and c ≠ d and c, d and x are collinear, then a, b and x are not collinear.

(45) If o, a and b are not collinear and o, a and x are collinear and o, b and x are collinear, then o = x.

(46) If o ≠ a and o ≠ b and o, a and b are collinear and o, a and a′ are collinear and o, b and b′ are collinear, then a, b ∥ a′, b′.

(48)^{3} If a, b c, d and a, b and p_{1} are collinear and a, b and p_{2} are collinear

and c, d and p_{1} are collinear and c, d and p_{2} are collinear, then p_{1} = p_{2}.

(49) If a ≠ b and a, b and c are collinear and a, b ∥ c, d, then a, c ∥ b, d.

(50) If a ≠ b and a, b and c are collinear and a, b ∥ c, d, then c, b ∥ c, d.

(51) If o, a and c are not collinear and o, a and b are collinear and o, c and d_{1} are collinear and o, c and d_{2} are collinear and a, c ∥ b, d_{1} and a, c ∥ b, d_{1} and a, c ∥ b, d_{2}, then d_{1} = d_{2}.

(52) If a ≠ b and a, b and c are collinear and a, b and d are collinear, then

a, c and d are collinear. Let us consider S_{1}, a, b, c, d. We say that a, b, c, d form a parallelogram if

and only if: (Def.3) a, b and c are not collinear and a, b ∥ c, d and a, c ∥ b, d.

We now state a number of propositions: (54)^{4} If a, b, c, d form a parallelogram, then a ≠ b and a ≠ c and c ≠ b and

a ≠ d and b ≠ d and c ≠ d.

(55) If a, b, c, d form a parallelogram, then a, b and c are not collinear and b, a and d are not collinear and c, d and a are not collinear and d, c and b are not collinear.

(56) Suppose a_{1}, a_{2}, a_{3}, a_{4} form a parallelogram. Then a_{1}, a_{2} and a_{3} are not collinear and a_{1}, a_{3} and a_{2} are not collinear and a_{1}, a_{2} and a_{4} are not collinear and a_{1}, a_{4} and a_{2} are not collinear and a_{1}, a_{3} and a_{4} are not collinear and a_{1}, a_{4} and a_{3} are not collinear and a_{2}, a_{1} and a_{3} are not collinear and a_{2}, a_{3} and a_{1} are not collinear and a_{2}, a_{1} and a_{4} are not collinear and a_{2}, a_{4} and a_{1} are not collinear and a_{2}, a_{3} and a_{4} are not collinear and a_{2}, a_{4} and a_{3} are not collinear and a_{3}, a_{1} and a_{2} are not collinear and a_{3}, a_{2} and a_{1} are not collinear and a_{3}, a_{1} and a_{4} are not collinear and a_{3}, a_{4} and a_{1} are not collinear and a_{3}, a_{2} and a_{4} are not collinear and a_{3}, a_{4} and a_{2} are not collinear and a_{4}, a_{1} and a_{2} are not collinear and a_{4}, a_{2} and a_{1} are not collinear and a_{4}, a_{1} and a_{3} are not collinear and a_{4}, a_{3} and a_{1} are not collinear and a_{4}, a_{2} and a_{3} are not collinear and a_{4}, a_{3} and a_{2} are not collinear.

(57) If a, b, c, d form a parallelogram, then a, b and x are not collinear or c, d and x are not collinear.

(58) If a, b, c, d form a parallelogram, then a, c, b, d form a parallelogram.

3The proposition (47) was either repeated or obvious.

4The proposition (53) was either repeated or obvious.

(59) If a, b, c, d form a parallelogram, then c, d, a, b form a parallelogram.

(60) If a, b, c, d form a parallelogram, then b, a, d, c form a parallelogram.

(61) If a, b, c, d form a parallelogram, then a, c, b, d form a parallelogram and c, d, a, b form a parallelogram and b, a, d, c form a parallelogram and c, a, d, b form a parallelogram and d, b, c, a form a parallelogram and b, d, a, c form a parallelogram.

(62) If a, b and c are not collinear, then there exists d such that a, b, c, d form a parallelogram.

(63) If a, b, c, d_{1} form a parallelogram and a, b, c, d_{2} form a parallelogram, then d_{1} = d_{2}.

(64) If a, b, c, d form a parallelogram, then a, d b, c.

(65) If a, b, c, d form a parallelogram, then a, b, d, c do not form a parallelogram.

(66) If a ≠ b, then there exists c such that a, b and c are collinear and c ≠ a and c ≠ b.

(67) If a, a′, b, b′ form a parallelogram and a, a′, c, c′ form a parallelogram, then b, c ∥ b′, c′.

(68) If b, b′ and c are not collinear and a, a′, b, b′ form a parallelogram and a, a′, c, c′ form a parallelogram, then b, b′, c, c′ form a parallelogram.

(69) If a, b and c are collinear and b ≠ c and a, a′, b, b′ form a parallelogram and a, a′, c, c′ form a parallelogram, then b, b′, c, c′ form a parallelogram.

(70) If a, a′, b, b′ form a parallelogram and a, a′, c, c′ form a parallelogram and b, b′, d, d′ form a parallelogram, then c, d ∥ c′, d′.

(71) If a ≠ d, then there exist b, c such that a, b, c, d form a parallelogram. Let us consider S_{1}, a, b, r, s. We say that a, b are congruent to r, s if and

only if: (Def.4) a = b and r = s or there exist p, q such that p, q, a, b form a parallelo-

gram and p, q, r, s form a parallelogram. Next we state a number of propositions:

(73)^{5} If a, a are congruent to b, c, then b = c.

(74) If a, b are congruent to c, c, then a = b.

(75) If a, b are congruent to b, a, then a = b.

(76) If a, b are congruent to c, d, then a, b ∥ c, d.

(77) If a, b are congruent to c, d, then a, c ∥ b, d.

(78) If a, b are congruent to c, d and a, b and c are not collinear, then a, b, c, d form a parallelogram.

(79) If a, b, c, d form a parallelogram, then a, b are congruent to c, d.

(80) If a, b are congruent to c, d and a, b and c are collinear and r, s, a, b form a parallelogram, then r, s, c, d form a parallelogram.

5The proposition (72) was either repeated or obvious.

(81) If a, b are congruent to c, x and a, b are congruent to c, y, then x = y.

(82) There exists d such that a, b are congruent to c, d.

(83) a, a are congruent to b, b.

(84) a, b are congruent to a, b.

(85) If r, s are congruent to a, b and r, s are congruent to c, d, then a, b are congruent to c, d.

(86) If a, b are congruent to c, d, then c, d are congruent to a, b.

(87) If a, b are congruent to c, d, then b, a are congruent to d, c.

(88) If a, b are congruent to c, d, then a, c are congruent to b, d.

(89) If a, b are congruent to c, d, then c, d are congruent to a, b and b, a are congruent to d, c and a, c are congruent to b, d and d, c are congruent to b, a and b, d are congruent to a, c and c, a are congruent to d, b and d, b are congruent to c, a.

(90) If a, b are congruent to p, q and b, c are congruent to q, s, then a, c are congruent to p, s.

(91) If b, a are congruent to p, q and c, a are congruent to p, r, then b, c are congruent to r, q.

(92) If a, o are congruent to o, p and b, o are congruent to o, q, then a, b are congruent to q, p.

(93) If b, a are congruent to p, q and c, a are congruent to p, r, then b, c ∥ q, r.

(94) If a, o are congruent to o, p and b, o are congruent to o, q, then a, b ∥ p, q. Let us consider S_{1}, a, b, o. The functor sum_{o}(a, b) yielding an element of the

points of S_{1} is defined as follows: (Def.5) o, a are congruent to b, sum_{o}(a, b).

Next we state the proposition

(95) sum_{o}(a, b) = c if and only if o, a are congruent to b, c.

Let us consider S_{1}, a, o. The functor opposite_{o}(a) yields an element of the points of S_{1} and is defined as follows: (Def.6) sum_{o}(a, opposite_{o}(a)) = o.

We now state the proposition

(96) opposite_{o}(a) = b if and only if sum_{o}(a, b) = o.

Let us consider S_{1}, a, b, o. The functor diff_{o}(a, b) yielding an element of the points of S_{1} is defined as follows: (Def.7) diff_{o}(a, b) = sum_{o}(a, opposite_{o}(b)).

Next we state a number of propositions:

(97) diff_{o}(a, b) = sum_{o}(a, opposite_{o}(b)).

(98) o, a are congruent to b, sum_{o}(a, b).

(99) sum_{o}(a, o) = a.

(100) There exists x such that sum_{o}(a, x) = o.

(101) sum_{o}(sum_{o}(a, b), c) = sum_{o}(a, sum_{o}(b, c)).

(102) sum_{o}(a, b) = sum_{o}(b, a).

(103) If sum_{o}(a, a) = o, then a = o.

(104) If sum_{o}(a, x) = sum_{o}(a, y), then x = y.

(105) sum_{o}(a, opposite_{o}(a)) = o.

(106) a, o are congruent to o, opposite_{o}(a).

(107) If opposite_{o}(a) = opposite_{o}(b), then a = b.

(108) a, b ∥ opposite_{o}(a), opposite_{o}(b).

(109) opposite_{o}(o) = o.

(110) p, q ∥ sum_{o}(p, r), sum_{o}(q, r).

(111) If p, q ∥ r, s, then p, q ∥ sum_{o}(p, r), sum_{o}(q, s). (113)^{6} diff_{o}(a, b) = o if and only if a = b.

(114) o, diff_{o}(b, a) ∥ a, b.

(115) o, diff_{o}(b, a) and diff_{o}(d, c) are collinear if and only if a, b ∥ c, d. Let us consider S_{1}, a, b, c, d, o. We say that a, b, c, d form a trapezium with

vertex o if and only if: (Def.8) o, a and c are not collinear and o, a and b are collinear and o, c and d

are collinear and a, c ∥ b, d. Let us consider S_{1}, o, p. We say that there are trapeziums through p with

vertex o if and only if: (Def.9) for every b, c there exists d such that if o, p and b are collinear, then o,

c and d are collinear and p, c ∥ b, d. One can prove the following propositions:

(118)^{7} If a, b, c, d form a trapezium with vertex o, then o ≠ a and a ≠ c and

c ≠ o.

(119) If a, b, c, x form a trapezium with vertex o and a, b, c, y form a trapezium with vertex o, then x = y.

(120) If o, a and b are not collinear, then a, o, b, o form a trapezium with vertex o.

(121) If a, b, c, d form a trapezium with vertex o, then c, d, a, b form a trapezium with vertex o.

(122) If o ≠ b and a, b, c, d form a trapezium with vertex o, then o ≠ d.

(123) If o ≠ b and a, b, c, d form a trapezium with vertex o, then o, b and d are not collinear.

(124) If o ≠ b and a, b, c, d form a trapezium with vertex o, then b, a, d, c form a trapezium with vertex o.

(125) If o = b or o = d but a, b, c, d form a trapezium with vertex o, then o = b and o = d.

6The proposition (112) was either repeated or obvious.

7The propositions (116)–(117) were either repeated or obvious.

(126) If a, p, b, q form a trapezium with vertex o and a, p, c, r form a trapezium with vertex o, then b, c ∥ q, r.

(127) If a, p, b, q form a trapezium with vertex o and a, p, c, r form a trapezium with vertex o and o, b and c are not collinear, then b, q, c, r form a trapezium with vertex o.

(128) If a, p, b, q form a trapezium with vertex o and a, p, c, r form a trapezium with vertex o and b, q, d, s form a trapezium with vertex o, then c, d ∥ r, s.

(129) For every o, a there exists p such that o, a and p are collinear and there are trapeziums through p with vertex o.

(130) There exist x, y, z such that x ≠ y and y ≠ z and z ≠ x.

(131) If there are trapeziums through p with vertex o, then o ≠ p.

(132) If there are trapeziums through p with vertex o, then there exists q such that o, p and q are not collinear and there are trapeziums through q with vertex o.

(133) If o, p and c are not collinear and o, p and b are collinear and there are trapeziums through p with vertex o, then there exists d such that p, b, c, d form a trapezium with vertex o.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[3] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

Planes in Affine Spaces ^{1}

Wojciech Leończuk Warsaw University Białystok Henryk Oryszczyszyn Warsaw University Białystok

Krzysztof Prażmowski Warsaw University Białystok

Summary. We introduce the notion of plane in affine space and investigate fundamental properties of them. Further we introduce the relation of parallelism defined for arbitrary subsets. In particular we are concerned with parallelisms which hold between lines and planes and between planes. We also define a function which assigns to every line and every point the unique line passing through the point and parallel to the given line. With the help of the introduced notions we prove that every at least 3-dimensional affine space is Desarguesian and translation.

MML Identifier: AFF 4.

The articles [5], [1], [2], [3], and [4] provide the notation and terminology for this paper. We follow a convention: A_{1} will be an affine space, a, b, c, d, a′, b′, c′, p, q will be elements of the points of A_{1}, and A, C, K, M, N, P, Q, X, Y, Z will be subsets of the points of A_{1}. Let us consider A_{1}, X, Y. Then X ∩ Y is a subset of the points of A_{1}.

The following propositions are true:

(1) If L(p, a, a′) or L(p, a′, a) but p ≠ a, then there exists b′ such that L(p, b, b′) and a, b ∥ a′, b′.

(2) If a, b ∥ A or b, a ∥ A but a ∈ A, then b ∈ A.

(3) If a, b ∥ A or b, a ∥ A but A ∥ K or K ∥ A, then a, b ∥ K and b, a ∥ K.

(4) If a, b ∥ A or b, a ∥ A but a, b ∥ c, d or c, d ∥ a, b and a ≠ b, then c, d ∥ A and d, c ∥ A.

(5) If a, b ∥ M or b, a ∥ M but a, b ∥ N or b, a ∥ N and a ≠ b, then M ∥ N and N ∥ M.

1Supported by RPBP.III-24.C2

(6) If a, b ∥ M or b, a ∥ M but c, d ∥ M or d, c ∥ M, then a, b ∥ c, d and a, b ∥ d, c.

(7) If A ∥ C or C ∥ A but a ≠ b but a, b ∥ c, d or c, d ∥ a, b and a ∈ A and b ∈ A and c ∈ C, then d ∈ C.

(8) Suppose that

(i) q ∈ M, (ii) q ∈ N, (iii) a ∈ M, (iv) a′ ∈ M,

(v) b ∈ N, (vi) b′ ∈ N, (vii) q ≠ a, (viii) q ≠ b, (ix) M ≠ N,

(x) a, b ∥ a′, b′ or b, a ∥ b′, a′, (xi) M is a line, (xii) N is a line, (xiii) q = a′.

Then q = b′.

(9) Suppose that

(i) q ∈ M, (ii) q ∈ N, (iii) a ∈ M, (iv) a′ ∈ M,

(v) b ∈ N, (vi) b′ ∈ N, (vii) q ≠ a, (viii) q ≠ b, (ix) M ≠ N,

(x) a, b ∥ a′, b′ or b, a ∥ b′, a′, (xi) M is a line, (xii) N is a line, (xiii) a = a′.

Then b = b′.

(10) If M ∥ N or N ∥ M but a ∈ M and a′ ∈ M and b ∈ N and b′ ∈ N and M ≠ N but a, b ∥ a′, b′ or b, a ∥ b′, a′ and a = a′, then b = b′.

(11) There exists A such that a ∈ A and b ∈ A and A is a line.

(12) If A is a line, then there exists q such that q ∉ A.

Let us consider A_{1}, K, P. The functor Plane(K, P) yielding a subset of the points of A_{1} is defined by: (Def.1) Plane(K, P) = {a : ⋁_{b}[a, b ∥ K ∧ b ∈ P]}.

Let us consider A_{1}, X. We say that X is a plane if and only if:

(Def.2) there exist K, P such that K is a line and P is a line and K P and X = Plane(K, P).

We now state a number of propositions:

(13) If K is not a line, then Plane(K, P) = ∅.

(14) If K is a line, then P ⊆ Plane(K, P).

(15) If K ∥ P, then Plane(K, P) = P.

(16) If K ∥ M, then Plane(K, P) = Plane(M, P).

(17) Suppose that (i) p ∈ M, (ii) a ∈ M, (iii) b ∈ M, (iv) p ∈ N,

(v) a′ ∈ N, (vi) b′ ∈ N, (vii) p ∉ P, (viii) p ∉ Q, (ix) M ≠ N,

(x) a ∈ P, (xi) a′ ∈ P, (xii) b ∈ Q, (xiii) b′ ∈ Q, (xiv) M is a line, (xv) N is a line, (xvi) P is a line, (xvii) Q is a line.

Then P ∥ Q or there exists q such that q ∈ P and q ∈ Q.

(18) Suppose a ∈ M and b ∈ M and a′ ∈ N and b′ ∈ N and a ∈ P and a′ ∈ P and b ∈ Q and b′ ∈ Q and M ≠ N and M ∥ N and P is a line and Q is a line. Then P ∥ Q or there exists q such that q ∈ P and q ∈ Q.

(19) If X is a plane and a ∈ X and b ∈ X and a ≠ b, then Line(a, b) ⊆ X.

(20) If K is a line and P is a line and Q is a line and K P and K Q and Q ⊆ Plane(K, P), then Plane(K, Q) = Plane(K, P).

(21) If K is a line and P is a line and Q is a line and K P and Q ⊆ Plane(K, P), then P ∥ Q or there exists q such that q ∈ P and q ∈ Q.

(22) If X is a plane and M is a line and N is a line and M ⊆ X and N ⊆ X, then M ∥ N or there exists q such that q ∈ M and q ∈ N.

(23) If X is a plane and a ∈ X and M ⊆ X and a ∈ N but M ∥ N or N ∥ M, then N ⊆ X.

(24) If X is a plane and Y is a plane and a ∈ X and b ∈ X and a ∈ Y and b ∈ Y and X ≠ Y and a ≠ b, then X ∩ Y is a line.

(25) If X is a plane and Y is a plane and a ∈ X and b ∈ X and c ∈ X and a ∈ Y and b ∈ Y and c ∈ Y and not L(a, b, c), then X = Y.

(26) If X is a plane and Y is a plane and M is a line and N is a line and

M ⊆ X and N ⊆ X and M ⊆ Y and N ⊆ Y and M ≠ N, then X = Y. Let us consider A_{1}, a, K. Let us assume that K is a line. The functor a · K

yields a subset of the points of A_{1} and is defined by: (Def.3) a ∈ a · K and K ∥ a · K.

We now state several propositions:

(27) If A is a line, then a · A is a line.

(28) If X is a plane and M is a line and a ∈ X and M ⊆ X, then a · M ⊆ X.

(29) If X is a plane and a ∈ X and b ∈ X and c ∈ X and a, b ∥ c, d and a ≠ b, then d ∈ X.

(30) If A is a line, then a ∈ A if and only if a · A = A.

(31) If A is a line, then a · A = a · (q · A).

(32) If K ∥ M, then a · K = a · M.

Let us consider A_{1}, X, Y. The predicate X||Y is defined by: (Def.4) for all a, A such that a ∈ Y and A is a line and A ⊆ X holds a · A ⊆ Y.

Next we state a number of propositions:

(33) If X ⊆ Y but X is a line and Y is a line or X is a plane and Y is a plane, then X = Y.

(34) If X is a plane, then there exist a, b, c such that a ∈ X and b ∈ X and c ∈ X and not L(a, b, c).

(35) If M is a line and X is a plane and M ⊆ X, then there exists q such that q ∈ X and q ∉ M.

(36) For all a, A such that A is a line there exists X such that a ∈ X and A ⊆ X and X is a plane.

(37) There exists X such that a ∈ X and b ∈ X and c ∈ X and X is a plane.

(38) If q ∈ M and q ∈ N and M is a line and N is a line, then there exists X such that M ⊆ X and N ⊆ X and X is a plane.

(39) If M ∥ N, then there exists X such that M ⊆ X and N ⊆ X and X is a plane.

(40) If M is a line and N is a line, then M ∥ N if and only if M||N.

(41) If M is a line and X is a plane, then M||X if and only if there exists N such that N ⊆ X but M ∥ N or N ∥ M.

(42) If M is a line and X is a plane and M ⊆ X, then M||X.

(43) If A is a line and X is a plane and a ∈ A and a ∈ X and A||X, then

A ⊆ X. Let us consider A_{1}, K, M, N. We say that K, M, N are coplanar if and

only if: (Def.5) there exists X such that K ⊆ X and M ⊆ X and N ⊆ X and X is a

plane. The following propositions are true:

(44) If K, M, N are coplanar, then K, N, M are coplanar and M, K, N are coplanar and M, N, K are coplanar and N, K, M are coplanar and N, M, K are coplanar.

(45) If A is a line and K is a line and M is a line and N is a line and M, N, K are coplanar and M, N, A are coplanar and M ≠ N, then M, K, A are coplanar.

(46) If K is a line and M is a line and X is a plane and K ⊆ X and M ⊆ X and K ≠ M, then K, M, A are coplanar if and only if A ⊆ X.

(47) If q ∈ K and q ∈ M and K is a line and M is a line, then K, M, M are coplanar and M, K, M are coplanar and M, M, K are coplanar.

(48) If A_{1} is not an affine plane and X is a plane, then there exists q such that q ∉ X.

(49) Suppose that (i) A_{1} is not an affine plane, (ii) q ∈ A, (iii) q ∈ P, (iv) q ∈ C,

(v) q ≠ a, (vi) q ≠ b, (vii) q ≠ c, (viii) a ∈ A, (ix) a′ ∈ A,

(x) b ∈ P, (xi) b′ ∈ P, (xii) c ∈ C, (xiii) c′ ∈ C, (xiv) A is a line, (xv) P is a line, (xvi) C is a line, (xvii) A ≠ P, (xviii) A ≠ C, (xix) a, b ∥ a′, b′, (xx) a, c ∥ a′, c′.

Then b, c ∥ b′, c′.

(50) If A_{1} is not an affine plane, then A_{1} is Desarguesian.

(51) Suppose that (i) A_{1} is not an affine plane, (ii) A ∥ P, (iii) A ∥ C, (iv) a ∈ A,

(v) a′ ∈ A, (vi) b ∈ P, (vii) b′ ∈ P, (viii) c ∈ C, (ix) c′ ∈ C,

(x) A is a line, (xi) P is a line, (xii) C is a line, (xiii) A ≠ P, (xiv) A ≠ C, (xv) a, b ∥ a′, b′, (xvi) a, c ∥ a′, c′.

Then b, c ∥ b′, c′.

(52) If A_{1} is not an affine plane, then A_{1} is translation.

(53) If A_{1} is an affine plane and not L(a, b, c), then there exists c′ such that a, c ∥ a′, c′ and b, c ∥ b′, c′.

(54) If not L(a, b, c) and a′ ≠ b′ and a, b ∥ a′, b′, then there exists c′ such that a, c ∥ a′, c′ and b, c ∥ b′, c′.

(55) Suppose X is a plane and Y is a plane. Then X||Y if and only if there exist A, P, M, N such that A P and A ⊆ X and P ⊆ X and M ⊆ Y and N ⊆ Y but A ∥ M or M ∥ A but P ∥ N or N ∥ P.

(56) If A ∥ M and M||X, then A||X.

(57) If X is a plane, then X||X.

(58) If X is a plane and Y is a plane and X||Y, then Y ||X.

(59) If X is a plane, then X ≠ ∅.

(60) If X||Y and Y ||Z and Y ≠ ∅, then X||Z.

(61) If X is a plane and Y is a plane and Z is a plane but X||Y and Y ||Z or X||Y and Z||Y or Y ||X and Y ||Z or Y ||X and Z||Y, then X||Z and Z||X.

(62) If X is a plane and Y is a plane and a ∈ X and a ∈ Y and X||Y, then X = Y.

(63) If X is a plane and Y is a plane and Z is a plane and X||Y and X ≠ Y and a ∈ X ∩Z and b ∈ X ∩Z and c ∈ Y ∩Z and d ∈ Y ∩Z, then a, b ∥ c, d.

(64) Suppose X is a plane and Y is a plane and Z is a plane and X||Y and a ∈ X ∩ Z and b ∈ X ∩ Z and c ∈ Y ∩ Z and d ∈ Y ∩ Z and X ≠ Y and a ≠ b and c ≠ d. Then X ∩ Z ∥ Y ∩ Z.

(65) For all a, X such that X is a plane there exists Y such that a ∈ Y and

X||Y and Y is a plane. Let us consider A_{1}, a, X. Let us assume that X is a plane. The functor

a + X yields a subset of the points of A_{1} and is defined as follows: (Def.6) a ∈ a + X and X||a + X and a + X is a plane.

Next we state four propositions:

(66) If X is a plane, then a ∈ X if and only if a + X = X.

(67) If X is a plane, then a + X = a + (q + X).

(68) If A is a line and X is a plane and A||X, then a · A ⊆ a + X.

Planes in Affine Spaces 363

(69) If X is a plane and Y is a plane and X||Y, then a + X = a + Y.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[4] Krzysztof Prażmowski. Fanoian, Pappian and Desarguesian affine spaces. Formalized Mathematics, 2(3):341–346, 1991.

[5] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

364

Graphs

Krzysztof Hryniewiecki Warsaw University

Summary. Definitions of graphs are introduced and their basic properties are proved. The following notions related to graph theory are introduced: Subgraph, Finite graph, Chain and oriented chain - as a finite sequence of edges, Path and oriented path - as a finite sequence of different edges, Cycle and oriented cycle, Incidency of graph’s vertices, A sum of two graphs, A degree of a vertice, A set of all subgraphs of a graph. Many ideas in this article have been taken from [12].

MML Identifier: GRAPH 1.

The terminology and notation used in this paper are introduced in the following papers: [10], [4], [5], [3], [9], [7], [6], [1], [8], [2], and [11]. We adopt the following convention: x, y, v will be arbitrary and n, m will be natural numbers. We consider multi graph structures which are systems

⟨vertices, edges, a source, a target⟩, where the vertices, the edges constitute a set and the source, the target are a function from the edges into the vertices.

A multi graph structure is said to be a graph if: (Def.1) the vertices of it is a non-empty set.

In the sequel G, G_{1}, G_{2}, G_{3} are graphs. Let us consider G_{1}, G_{2}. Let us assume that the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2}. The functor G_{1} ∪ G_{2} yielding a graph is defined by the conditions (Def.2). (Def.2) (i) The vertices of G_{1} ∪ G_{2} = (the vertices of G_{1})∪ the vertices of G_{2},

(ii) the edges of G_{1} ∪ G_{2} = (the edges of G_{1})∪ the edges of G_{2}, (iii) for every v such that v ∈ the edges of G_{1} holds (the source of G_{1} ∪

G_{2})(v) = (the source of G_{1})(v) and (the target of G_{1} ∪ G_{2})(v) = (the target of G_{1})(v),

(iv) for every v such that v ∈ the edges of G_{2} holds (the source of G_{1} ∪ G_{2})(v) = (the source of G_{2})(v) and (the target of G_{1} ∪ G_{2})(v) = (the target of G_{2})(v).

Let G, G_{1}, G_{2} be graphs. We say that G is a sum of G_{1} and G_{2} if and only if: (Def.3) the target of G_{1} ≈ the target of G_{2} and the source of G_{1} ≈ the source

of G_{2} and G = G_{1} ∪ G_{2}.

We now define five new attributes. A graph is oriented if: (Def.4) for all x, y such that x ∈ the edges of it and y ∈ the edges of it and

(the source of it)(x) = (the source of it)(y) and (the target of it)(x) = (the target of it)(y) holds x = y.

A graph is non-multi if it satisfies the condition (Def.5). (Def.5) Given x, y. Suppose x ∈ the edges of it and y ∈ the edges of it but

(the source of it)(x) = (the source of it)(y) and (the target of it)(x) = (the target of it)(y) or (the source of it)(x) = (the target of it)(y) and (the source of it)(y) = (the target of it)(x). Then x = y.

A graph is simple if: (Def.6) for no x holds x ∈ the edges of it and (the source of it)(x) = (the target

of it)(x). A graph is connected if: (Def.7) for no graphs G_{1}, G_{2} holds (the vertices of G_{1})∩ the vertices of G_{2} = ∅

and it is a sum of G_{1} and G_{2}. A multi graph structure is finite if: (Def.8) the vertices of it is finite and the edges of it is finite.

In the sequel x, y will denote elements of the vertices of G. Let us consider G, x, y, v. We say that v joins x with y if and only if: (Def.9) (the source of G)(v) = x and (the target of G)(v) = y or (the source

of G)(v) = y and (the target of G)(v) = x.

Let us consider G, and let x, y be elements of the vertices of G. We say that x and y are incydent if and only if:

(Def.10) there exists arbitrary v such that v ∈ the edges of G and v joins x with y.

Let G be a graph. A finite sequence is called a chain of G if it satisfies the conditions (Def.11).

(Def.11) (i) For every n such that 1 ≤ n and n ≤ len it holds it(n) ∈ the edges

of G, (ii) there exists a finite sequence p such that len p = len it+1 and for every

n such that 1 ≤ n and n ≤ len p holds p(n) ∈ the vertices of G and for every n such that 1 ≤ n and n ≤ len it there exist elements x′, y′ of the vertices of G such that x′ = p(n) and y′ = p(n + 1) and it(n) joins x′ with y′.

Let G be a graph. A chain of G is said to be an oriented chain of G if: (Def.12) for every n such that 1 ≤ n and n < len it holds (the source of G)(it(n+

1)) = (the target of G)(it(n)).

Let G be a graph. A chain of G is said to be a path of G if: (Def.13) for all n, m such that 1 ≤ n and n < m and m ≤ len it holds it(n) ≠

it(m).

Let G be a graph. An oriented chain of G is said to be an oriented path of G if:

(Def.14) it is a path of G.

Let G be a graph. A path of G is said to be a cycle of G if it satisfies the condition (Def.15).

(Def.15) There exists a finite sequence p such that len p = len it+1 and for every n such that 1 ≤ n and n ≤ len p holds p(n) ∈ the vertices of G and for every n such that 1 ≤ n and n ≤ len it there exist elements x′, y′ of the vertices of G such that x′ = p(n) and y′ = p(n + 1) and it(n) joins x′ with y′ and p(1) = p(len p).

Let G be a graph. An oriented path of G is called an oriented cycle of G if: (Def.16) it is a cycle of G.

Let G be a graph. A graph is said to be a subgraph of G if it satisfies the conditions (Def.17).

(Def.17) (i) The vertices of it ⊆ the vertices of G, (ii) the edges of it ⊆ the edges of G, (iii) for every v such that v ∈ the edges of it holds (the source of it)(v) =

(the source of G)(v) and (the target of it)(v) = (the target of G)(v) and (the source of G)(v) ∈ the vertices of it and (the target of G)(v) ∈ the vertices of it.

We now define two new functors. Let G be an finite graph. The number of vertices of G yielding a natural number is defined by:

(Def.18) the number of vertices of G = card (the vertices of G).

The number of edges of G yielding a natural number is defined by: (Def.19) the number of edges of G = card (the edges of G).

We now define two new functors. Let G be an finite graph, and let x be an element of the vertices of G. The functor EdgIn(x) yields a natural number and is defined as follows:

(Def.20) there exists a set X such that for an arbitrary z holds z ∈ X if and only

if z ∈ the edges of G and (the target of G)(z) = x and EdgIn(x) = card X. The functor EdgOut(x) yielding a natural number is defined by:

(Def.21) there exists a set X such that for an arbitrary z holds z ∈ X if and only if z ∈ the edges of G and (the source of G)(z) = x and EdgOut(x) = card X.

Let G be an finite graph, and let x be an element of the vertices of G. The degree of x yields a natural number and is defined by:

(Def.22) the degree of x = EdgIn(x) + EdgOut(x).

Let G_{1}, G_{2} be graphs. The predicate G_{1} ⊆ G_{2} is defined by: (Def.23) G_{1} is a subgraph of G_{2}.

Let G be a graph. The functor 2^{G} yields a set and is defined by: (Def.24) for an arbitrary x holds x ∈ 2^{G} if and only if x is a subgraph of G.

The scheme GraphSeparation deals with a graph A, and a unary predicate P, and states that:

there exists a set X such that for an arbitrary x holds x ∈ X if and only if x is a subgraph of A and P[x] for all values of the parameters.

Next we state a number of propositions:

(1) For every graph G holds dom (the source of G) = the edges of G and dom (the target of G) = the edges of G and rng (the source of G) ⊆ the vertices of G and rng (the target of G) ⊆ the vertices of G.

(2) For every element x of the vertices of G holds x ∈ the vertices of G.

(3) For an arbitrary v such that v ∈ the edges of G holds (the source of G)(v) ∈ the vertices of G and (the target of G)(v) ∈ the vertices of G.

(4) For every chain p of G holds p Seg n is a chain of G.

(5) If G_{1} ⊆ G, then graph (the source of G_{1}) ⊆ graph (the source of G) and graph (the target of G_{1}) ⊆ graph (the target of G).

(6) If the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2}, then graph (the source of G_{1} ∪ G_{2}) = graph (the source of G_{1}) ∪ graph (the source of G_{2}) and graph (the target of G_{1} ∪ G_{2}) = graph (the target of G_{1}) ∪ graph (the target of G_{2}).

(7) G = G ∪ G.

(8) If the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2}, then G_{1} ∪ G_{2} = G_{2} ∪ G_{1}.

(9) If the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2} and the source of G_{1} ≈ the source of G_{3} and the target of G_{1} ≈ the target of G_{3} and the source of G_{2} ≈ the source of G_{3} and the target of G_{2} ≈ the target of G_{3}, then G_{1} ∪ G_{2} ∪ G_{3} = G_{1} ∪ (G_{2} ∪ G_{3}).

(10) If G is a sum of G_{1} and G_{2}, then G is a sum of G_{2} and G_{1}.

(11) G is a sum of G and G.

(12) If there exists G such that G_{1} ⊆ G and G_{2} ⊆ G, then G_{1}∪G_{2} = G_{2}∪G_{1}.

(13) If there exists G such that G_{1} ⊆ G and G_{2} ⊆ G and G_{3} ⊆ G, then G_{1} ∪ G_{2} ∪ G_{3} = G_{1} ∪ (G_{2} ∪ G_{3}).

(14) G ⊆ G.

(15) For all subgraphs H_{1}, H_{2} of G such that the vertices of H_{1} = the vertices of H_{2} and the edges of H_{1} = the edges of H_{2} holds H_{1} = H_{2}.

(16) If G_{1} ⊆ G_{2} and G_{2} ⊆ G_{1}, then G_{1} = G_{2}.

(17) If G_{1} ⊆ G_{2} and G_{2} ⊆ G_{3}, then G_{1} ⊆ G_{3}.

(18) If G is a sum of G_{1} and G_{2}, then G_{1} ⊆ G and G_{2} ⊆ G.

(19) If the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2}, then G_{1} ⊆ G_{1} ∪ G_{2} and G_{2} ⊆ G_{1} ∪ G_{2}.

(20) If there exists G such that G_{1} ⊆ G and G_{2} ⊆ G, then G_{1} ⊆ G_{1} ∪ G_{2} and G_{2} ⊆ G_{1} ∪ G_{2}.

(21) If G_{1} ⊆ G_{3} and G_{2} ⊆ G_{3} and G is a sum of G_{1} and G_{2}, then G ⊆ G_{3}.

(22) If G_{1} ⊆ G and G_{2} ⊆ G, then G_{1} ∪ G_{2} ⊆ G.

(23) If G_{1} ⊆ G_{2}, then G_{1} ∪ G_{2} = G_{2} and G_{2} ∪ G_{1} = G_{2}.

(24) If the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2} but G_{1} ∪ G_{2} = G_{2} or G_{2} ∪ G_{1} = G_{2}, then G_{1} ⊆ G_{2}.

(25) If G_{2} is a sum of G_{1} and G_{2} or G_{2} is a sum of G_{2} and G_{1}, then G_{1} ⊆ G_{2}.

(26) If there exists G such that G_{1} ⊆ G and G_{2} ⊆ G but G_{2} = G_{1} ∪ G_{2} or G_{2} = G_{2} ∪ G_{1}, then G_{1} ⊆ G_{2}.

(27) For every oriented graph G such that G_{1} ⊆ G holds G_{1} is oriented.

(28) For every non-multi graph G such that G_{1} ⊆ G holds G_{1} is non-multi.

(29) For every simple graph G such that G_{1} ⊆ G holds G_{1} is simple.

(30) G_{1} ∈ 2^{G} if and only if G_{1} ⊆ G.

(31) G ∈ 2^{G}.

We now state several propositions:

(32) G_{1} ⊆ G_{2} if and only if 2^{G_{1}} ⊆ 2^{G_{2}}.

(33) 2^{G} ≠ ∅.

(34) {G} ⊆ 2^{G}.

(35) If the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2} and 2^{G_{1}∪G_{2}} ⊆ 2^{G_{1}} ∪ 2^{G_{2}}, then G_{1} ⊆ G_{2} or G_{2} ⊆ G_{1}.

(36) If the source of G_{1} ≈ the source of G_{2} and the target of G_{1} ≈ the target of G_{2}, then 2^{G_{1}} ∪ 2^{G_{2}} ⊆ 2^{G_{1}∪G_{2}}.

(37) If G_{1} ∈ 2^{G} and G_{2} ∈ 2^{G}, then G_{1} ∪ G_{2} ∈ 2^{G}.

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[7] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[8] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[9] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[10] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11,

370 Krzysztof Hryniewiecki

[11] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[12] Robin Wilson. Wprowadzenie do teorii grafów. PWN, 1985.

Mostowski’s Fundamental Operations - Part I

Andrzej Kondracki Warsaw University

Summary. In the chapter II.4 of his book [17] A.Mostowski introduces what he calls fundamental operations:

A_{1}(a, b) = {{⟨0, x⟩, ⟨1, y⟩} : x ∈ y ∧ x ∈ a ∧ y ∈ a},

A_{2}(a, b) = {a, b},
A_{3}(a, b) = ⋃ a,
A_{4}(a, b) = {{⟨x, y⟩} : x ∈ a ∧ y ∈ b},
A_{5}(a, b) = {x ∪ y : x ∈ a ∧ y ∈ b},
A_{6}(a, b) = {x \ y : x ∈ a ∧ y ∈ b},
A_{7}(a, b) = {x ∘ y : x ∈ a ∧ y ∈ b}.

He proves that if a non-void class is closed under these operations then it is predicatively closed. Then he formulates sufficient criteria for a class to be a model of ZF set theory (theorem 4.12).

The article includes the translation of this part of Mostowski’s book. The fundamental operations are defined (to be precise, not these operations, but the notions of closure of a class with respect to them). Some properties of classes closed under these operations are proved. At last it is proved that if a non-void class X is closed under the operations A_{1} −A_{7} then D_{H}(a) ∈ X for every a in X and every H being formula of ZF language (D_{H}(a) consists of all finite sequences with terms belonging to a which satisfy H in a).

MML Identifier: ZF FUND1.

The articles [20], [12], [7], [10], [4], [11], [13], [18], [2], [1], [24], [19], [8], [5], [9], [6], [16], [21], [14], [22], [15], [3], and [23] provide the notation and terminology for this paper. For simplicity we follow the rules: V will be a universal class, a, b, x, y will be elements of V, X will be a subclass of V, o, p, q, r, s, t, u will be arbitrary, A, B will be sets, n will be an element of ω, f_{1} will be a finite subset of ω, E will be a non-empty set, f will be a function from VAR into E, k will be a natural number, v_{1}, v_{2} will be elements of VAR, and H, H′ will be ZF-formulae. Let us consider A, B. The functor AB yielding a set is defined as follows:

(Def.1) p ∈ AB if and only if there exist q, r, s such that p = ⟨⟨q, s⟩⟩ and

⟨⟨q, r⟩⟩ ∈ A and ⟨⟨r, s⟩⟩ ∈ B. Let us consider V, x, y. Then xy is an element of V. The function decode from ω into VAR is defined by:

(Def.2) for every p such that p ∈ ω holds decode(p) = x_{card p}.

Let us consider v_{1}. The functor ^{v_{1}}x yielding a natural number is defined by: (Def.3) x_{^{v_{1}}x} = v_{1}.

Let A be a finite subset of VAR. The functor code(A) yielding a finite subset of ω is defined as follows: (Def.4) code(A) = (decode^{−1})^{∘} A.

Let us consider H. Then Free H is a finite subset of VAR. Let us consider v_{1}. Then {v_{1}} is a finite subset of VAR. Let us consider v_{2}.

Then {v_{1}, v_{2}} is a finite subset of VAR.

Let us consider H, E. The functor D_{E}(H) yielding a set is defined by: (Def.5) p ∈ D_{E}(H) if and only if there exists f such that p = (f · decode)

code(Free H) and f ∈ St_{E}(H). Let us consider n. Then {n} is a finite subset of ω. We now define several new predicates. Let us consider V, X. We say that

X is closed w.r.t. A1 if and only if: (Def.6) for every a such that a ∈ X holds {{⟨⟨0_{V} , x⟩⟩, ⟨⟨1_{V} , y⟩⟩} : x ∈ y ∧ x ∈

a ∧ y ∈ a} ∈ X. We say that X is closed w.r.t. A2 if and only if: (Def.7) for all a, b such that a ∈ X and b ∈ X holds {a, b} ∈ X. We say that X is closed w.r.t. A3 if and only if: (Def.8) for every a such that a ∈ X holds ⋃ a ∈ X. We say that X is closed w.r.t. A4 if and only if: (Def.9) for all a, b such that a ∈ X and b ∈ X holds {{⟨⟨x, y⟩⟩} : x ∈ a ∧ y ∈

b} ∈ X. We say that X is closed w.r.t. A5 if and only if:

(Def.10) for all a, b such that a ∈ X and b ∈ X holds {x∪y : x ∈ a∧y ∈ b} ∈ X.

We say that X is closed w.r.t. A6 if and only if: (Def.11) for all a, b such that a ∈ X and b ∈ X holds {x \ y : x ∈ a ∧ y ∈ b} ∈ X.

We say that X is closed w.r.t. A7 if and only if: (Def.12) for all a, b such that a ∈ X and b ∈ X holds {xy : x ∈ a ∧ y ∈ b} ∈ X.

Let us consider V, X. We say that X is closed w.r.t. A1-A7 if and only if: (Def.13) X is closed w.r.t. A1 and X is closed w.r.t. A2 and X is closed w.r.t.

A3 and X is closed w.r.t. A4 and X is closed w.r.t. A5 and X is closed w.r.t. A6 and X is closed w.r.t. A7.

We now state a number of propositions:

(1) X ⊆ V but if o ∈ X, then o is an element of V but if o ∈ A and A ∈ X, then o is an element of V.

(2) If X is closed w.r.t. A1-A7, then o ∈ X if and only if {o} ∈ X but if A ∈ X, then ⋃ A ∈ X.

(3) If X is closed w.r.t. A1-A7, then ∅ ∈ X and 0 ∈ X.

(4) If X is closed w.r.t. A1-A7 and A ∈ X and B ∈ X, then A ∪ B ∈ X and A \ B ∈ X and AB ∈ X.

(5) If X is closed w.r.t. A1-A7 and A ∈ X and B ∈ X, then A ∩ B ∈ X.

(6) If X is closed w.r.t. A1-A7 and o ∈ X and p ∈ X, then {o, p} ∈ X and ⟨⟨o, p⟩⟩ ∈ X.

(7) If X is closed w.r.t. A1-A7, then ω ⊆ X.

(8) If X is closed w.r.t. A1-A7, then ω^{f_{1}} ⊆ X.

(9) If X is closed w.r.t. A1-A7 and a ∈ X, then a^{f_{1}} ∈ X.

(10) If X is closed w.r.t. A1-A7 and a ∈ ω^{f_{1}} and b ∈ X, then {ax : x ∈ b} ∈ X.

(11) If X is closed w.r.t. A1-A7 and n ∈ f_{1} and a ∈ X and b ∈ X and b ⊆ a^{f_{1}}, then {x : x ∈ a^{f_{1}\{n}} ∧ ⋁_{u}{⟨⟨n, u⟩⟩} ∪ x ∈ b} ∈ X.

(12) If X is closed w.r.t. A1-A7 and n ∉ f_{1} and a ∈ X and b ∈ X and b ⊆ a^{f_{1}}, then {{⟨⟨n, x⟩⟩} ∪ y : x ∈ a ∧ y ∈ b} ∈ X.

(13) If X is closed w.r.t. A1-A7 and B is finite and for every o such that o ∈ B holds o ∈ X, then B ∈ X.

(14) If X is closed w.r.t. A1-A7 and A ⊆ X and y ∈ A^{f_{1}}, then y ∈ X.

(15) If X is closed w.r.t. A1-A7 and n ∉ f_{1} and a ∈ X and a ⊆ X and y ∈ a^{f_{1}}, then {{⟨⟨n, x⟩⟩} ∪ y : x ∈ a} ∈ X.

(16) Suppose X is closed w.r.t. A1-A7 and n ∉ f_{1} and a ∈ X and a ⊆ X and y ∈ a^{f_{1}} and b ⊆ a^{f_{1}∪{n}} and b ∈ X. Then {x : x ∈ a ∧ {⟨⟨n, x⟩⟩} ∪ y ∈ b} ∈ X.

(17) If X is closed w.r.t. A1-A7 and a ∈ X, then {{⟨⟨0_{V} , x⟩⟩, ⟨⟨1_{V} , x⟩⟩} : x ∈ a} ∈ X.

(18) If X is closed w.r.t. A1-A7 and E ∈ X, then for all v_{1}, v_{2} holds D_{E}(v_{1}=v_{2}) ∈ X and D_{E}(v_{1}ϵv_{2}) ∈ X.

(19) If X is closed w.r.t. A1-A7 and E ∈ X, then for every H such that D_{E}(H) ∈ X holds D_{E}(¬H) ∈ X.

(20) If X is closed w.r.t. A1-A7 and E ∈ X, then for all H, H′ such that D_{E}(H) ∈ X and D_{E}(H′) ∈ X holds D_{E}(H ∧ H′) ∈ X.

(21) If X is closed w.r.t. A1-A7 and E ∈ X, then for all H, v_{1} such that D_{E}(H) ∈ X holds D_{E}(∀_{v_{1}}H) ∈ X.

(22) If X is closed w.r.t. A1-A7 and E ∈ X, then D_{E}(H) ∈ X.

(23) If X is closed w.r.t. A1-A7, then n ∈ X and 0_{V} ∈ X and 1_{V} ∈ X.

(24) {⟨⟨o, p⟩⟩, ⟨⟨p, p⟩⟩}{⟨⟨p, q⟩⟩} = {⟨⟨o, q⟩⟩, ⟨⟨p, q⟩⟩}.

(25) If p ≠ r, then {⟨⟨o, p⟩⟩, ⟨⟨q, r⟩⟩}{⟨⟨p, s⟩⟩, ⟨⟨r, t⟩⟩} = {⟨⟨o, s⟩⟩, ⟨⟨q, t⟩⟩}.

374 Andrzej Kondracki

(26) ^{x_{k}}x = k.

(27) code({v_{1}}) = {ord(^{v_{1}}x)} and code({v_{1}, v_{2}}) = {ord(^{v_{1}}x), ord(^{v_{2}}x)}.

(28) dom f = {o, q} if and only if graph f = {⟨⟨o, f(o)⟩⟩, ⟨⟨q, f(q)⟩⟩}.

(29) dom decode = ω and rng decode = VAR and decode is one-to-one and decode^{−1} is one-to-one and dom(decode^{−1}) = VAR and rng(decode^{−1}) = ω.

(30) For every finite subset A of VAR holds A ≈ code(A).

(31) If A ∈ ω, then A = ord(card A) and A = ord(^{x_{card A}}x).

One can prove the following propositions:

(32) dom((f · decode) f_{1}) = f_{1} and rng((f · decode) f_{1}) ⊆ E and (f · decode) f_{1} ∈ E^{f_{1}} and dom(f · decode) = ω and rng(f · decode) ⊆ E.

(33) decode(ord(^{v_{1}}x)) = v_{1} and decode^{−1}(v_{1}) = ord(^{v_{1}}x) and (f · decode)(ord(^{v_{1}}x)) = f(v_{1}).

(34) For every finite subset A of VAR holds p ∈ code(A) if and only if there exists v_{1} such that v_{1} ∈ A and p = ord(^{v_{1}}x).

(35) For all finite subsets A, B of VAR holds code(A ∪ B) = code(A) ∪ code(B) and code(A \ B) = code(A) \ code(B).

(36) If v_{1} ∈ Free H, then ((f · decode) code(Free H))(ord(^{v_{1}}x)) = f(v_{1}).

(37) For all functions f, g from VAR into E such that (f · decode) code(Free H) = (g · decode) code(Free H) and f ∈ St_{E}(H) holds g ∈ St_{E}(H).

(38) If p ∈ E^{f_{1}}, then there exists f such that p = (f · decode) f_{1}.

References

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376

A Projective Closure and Projective Horizon of an Affine Space

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. With every affine space A we correlate two incidence structures. The first, called Inc-ProjSp(A), is the usual projective closure of A, i.e. the structure obtained from A by adding directions of lines and planes of A. The second, called projective horizon of A, is the structure build from directions. We prove that Inc-ProjSp(A) is always a projective space, and projective horizon of A is a projective space provided A is at least 3-dimensional. Some evident relationships between projective and affine configurational axioms that may hold in A and in Inc-ProjSp(A) are established.

MML Identifier: AFPROJ.

The notation and terminology used in this paper have been introduced in the following articles: [9], [11], [12], [8], [6], [13], [10], [3], [4], [5], [1], [7], and [2]. We adopt the following rules: A_{1} will denote an affine space, A, K, M, X, Y will denote subsets of the points of A_{1}, and x, y will be arbitrary. Next we state several propositions:

(1) If A_{1} is an affine plane and X = the points of A_{1}, then X is a plane.

(2) If A_{1} is an affine plane and X is a plane, then X = the points of A_{1}.

(3) If A_{1} is an affine plane and X is a plane and Y is a plane, then X = Y.

(4) If X = the points of A_{1} and X is a plane, then A_{1} is an affine plane.

(5) If A K and A||X and A||Y and K||X and K||Y and A is a line and K is a line and X is a plane and Y is a plane, then X||Y.

(6) If A is a line and X is a plane and Y is a plane and A||X and X||Y, then A||Y.

Let D be a non-empty set, and let X be a set. Then D ∪ X is a non-empty set.

Let us consider A_{1}. The lines of A_{1} yields a family of subsets of the points of A_{1} and is defined as follows:

(Def.1) the lines of A_{1} = {A : A is a line }.

Let us consider A_{1}. The planes of A_{1} yielding a family of subsets of the points of A_{1} is defined as follows: (Def.2) the planes of A_{1} = {A : A is a plane }.

The following two propositions are true:

(7) For every x holds x ∈ the lines of A_{1} if and only if there exists X such that x = X and X is a line.

(8) For every x holds x ∈ the planes of A_{1} if and only if there exists X such that x = X and X is a plane.

Let us consider A_{1}. The parallelity of lines of A_{1} yields an equivalence relation of the lines of A_{1} and is defined by: (Def.3) the parallelity of lines of A_{1} = {⟨⟨K, M⟩⟩ : K is a line ∧ M is a line

∧K||M}.

Let us consider A_{1}. The parallelity of planes of A_{1} yielding an equivalence relation of the planes of A_{1} is defined as follows: (Def.4) the parallelity of planes of A_{1} = {⟨⟨X, Y ⟩⟩ : X is a plane ∧ Y is a plane

∧X||Y }.

Let us consider A_{1}, X. Let us assume that X is a line. The direction of X yields a subset of the lines of A_{1} and is defined by: (Def.5) the direction of X = [X]_{the parallelity of lines of A_{1}}.

Let us consider A_{1}, X. Let us assume that X is a plane. The direction of X yielding a subset of the planes of A_{1} is defined as follows: (Def.6) the direction of X = [X]_{the parallelity of planes of A_{1}}.

Next we state several propositions:

(9) If X is a line, then for every x holds x ∈ the direction of X if and only if there exists Y such that x = Y and Y is a line and X||Y.

(10) If X is a plane, then for every x holds x ∈ the direction of X if and only if there exists Y such that x = Y and Y is a plane and X||Y.

(11) If X is a line and Y is a line, then the direction of X = the direction of Y if and only if X ∥ Y.

(12) If X is a line and Y is a line, then the direction of X = the direction of Y if and only if X||Y.

(13) If X is a plane and Y is a plane, then the direction of X = the direction of Y if and only if X||Y.

Let us consider A_{1}. The directions of lines of A_{1} yields a non-empty set and is defined as follows: (Def.7) the directions of lines of A_{1} = Classes(the parallelity of lines of A_{1}).

Let us consider A_{1}. The directions of planes of A_{1} yielding a non-empty set is defined by:

(Def.8) the directions of planes of A_{1} = Classes(the parallelity of planes of A_{1}).

One can prove the following propositions:

(14) For every x holds x ∈ the directions of lines of A_{1} if and only if there exists X such that x = the direction of X and X is a line.

(15) For every x holds x ∈ the directions of planes of A_{1} if and only if there exists X such that x = the direction of X and X is a plane.

(16) (the points of A_{1}) ∩ the directions of lines of A_{1} = ∅.

(17) If A_{1} is an affine plane, then (the lines of A_{1}) ∩ the directions of planes of A_{1} = ∅.

(18) For every x holds x ∈ [: the lines of A_{1}, {1}:] if and only if there exists X such that x = ⟨⟨X, 1⟩⟩ and X is a line.

(19) For every x holds x ∈ [: the directions of planes of A_{1}, {2}:] if and only

if there exists X such that x = ⟨⟨ the direction of X, 2⟩⟩ and X is a plane. Let us consider A_{1}. The projective points over A_{1} yielding a non-empty set is

defined as follows: (Def.9) the projective points over A_{1} = (the points of

A_{1}) ∪ the directions of lines of A_{1}.

Let us consider A_{1}. The functor L(A_{1}) yielding a non-empty set is defined as follows:

(Def.10) L(A_{1}) = [: the lines of A_{1}, {1}:] ∪ [: the directions of planes of A_{1}, {2}:].

Let us consider A_{1}. The functor I_{A_{1}} yielding a relation between the projective points over A_{1} and L(A_{1}) is defined by the condition (Def.11).

(Def.11) Given x, y. Then ⟨⟨x, y⟩⟩ ∈ I_{A_{1}} if and only if there exists K such that K is a line and y = ⟨⟨K, 1⟩⟩ but x ∈ the points of A_{1} and x ∈ K or x = the direction of K or there exist K, X such that K is a line and X is a plane and x = the direction of K and y = ⟨⟨ the direction of X, 2⟩⟩ and K||X.

Let us consider A_{1}. The incidence of directions of A_{1} yields a relation between the directions of lines of A_{1} and the directions of planes of A_{1} and is defined as follows:

(Def.12) for all x, y holds ⟨⟨x, y⟩⟩ ∈ the incidence of directions of A_{1} if and only if there exist A, X such that x = the direction of A and y = the direction of X and A is a line and X is a plane and A||X.

Let us consider A_{1}. The functor Inc-ProjSp(A_{1}) yielding a projective incidence structure is defined as follows:

(Def.13) Inc-ProjSp(A_{1}) = ⟨the projective points over A_{1}, L(A_{1}), I_{A_{1}}⟩.

Let us consider A_{1}. The projective horizon of A_{1} yielding a projective incidence structure is defined as follows:

(Def.14) the projective horizon of A_{1} = ⟨the directions of lines of A_{1}, the directions of planes of A_{1}, the incidence of directions of A_{1}⟩.

We now state several propositions:

(20) For every x holds x is an element of the points of Inc-ProjSp(A_{1}) if and only if x is an element of the points of A_{1} or there exists X such that x = the direction of X and X is a line.

(21) x is an element of the points of the projective horizon of A_{1} if and only if there exists X such that x = the direction of X and X is a line.

(22) If x is an element of the points of the projective horizon of A_{1}, then x is an element of the points of Inc-ProjSp(A_{1}).

(23) For every x holds x is an element of the lines of Inc-ProjSp(A_{1}) if and only if there exists X such that x = ⟨⟨X, 1⟩⟩ and X is a line or x = ⟨⟨ the direction of X, 2⟩⟩ and X is a plane.

(24) x is an element of the lines of the projective horizon of A_{1} if and only if there exists X such that x = the direction of X and X is a plane.

(25) If x is an element of the lines of the projective horizon of A_{1}, then ⟨⟨x, 2⟩⟩ is an element of the lines of Inc-ProjSp(A_{1}).

For simplicity we adopt the following rules: x will denote an element of the points of A_{1}, X, Y, X′ will denote subsets of the points of A_{1}, a, p, q will denote elements of the points of Inc-ProjSp(A_{1}), and A will denote an element of the lines of Inc-ProjSp(A_{1}). We now state a number of propositions:

(26) If x = a and ⟨⟨X, 1⟩⟩ = A, then a | A if and only if X is a line and x ∈ X.

(27) If x = a and ⟨⟨ the direction of X, 2⟩⟩ = A and X is a plane, then a A.

(28) If a = the direction of Y and ⟨⟨X, 1⟩⟩ = A and Y is a line and X is a line, then a | A if and only if Y ||X.

(29) If a = the direction of Y and A = ⟨⟨ the direction of X, 2⟩⟩ and Y is a line and X is a plane, then a | A if and only if Y ||X.

(30) If X is a line and a = the direction of X and A = ⟨⟨X, 1⟩⟩, then a | A.

(31) If X is a line and Y is a plane and X ⊆ Y and a = the direction of X and A = ⟨⟨ the direction of Y, 2⟩⟩, then a | A.

(32) If Y is a plane and X ⊆ Y and X′ ∥ X and a = the direction of X′ and A = ⟨⟨ the direction of Y, 2⟩⟩, then a | A.

(33) If A = ⟨⟨ the direction of X, 2⟩⟩ and X is a plane and a | A, then a is not an element of the points of A_{1}.

(34) If A = ⟨⟨X, 1⟩⟩ and X is a line and p | A and p is not an element of the points of A_{1}, then p = the direction of X.

(35) If A = ⟨⟨X, 1⟩⟩ and X is a line and p | A and a | A and a ≠ p and p is not an element of the points of A_{1}, then a is an element of the points of A_{1}.

(36) For every element a of the points of the projective horizon of A_{1} and for every element A of the lines of the projective horizon of A_{1} such that a = the direction of X and A = the direction of Y and X is a line and Y is a plane holds a | A if and only if X||Y.

(37) For every element a of the points of the projective horizon of A_{1} and for every element a′ of the points of Inc-ProjSp(A_{1}) and for every element A of the lines of the projective horizon of A_{1} and for every element A′ of the lines of Inc-ProjSp(A_{1}) such that a′ = a and A′ = ⟨⟨A, 2⟩⟩ holds a | A if and only if a′ | A′.

In the sequel P, Q denote elements of the lines of Inc-ProjSp(A_{1}). We now state several propositions:

(38) For all elements a, b of the points of the projective horizon of A_{1} and for all elements A, K of the lines of the projective horizon of A_{1} such that a | A and a | K and b | A and b | K holds a = b or A = K.

(39) For every element A of the lines of the projective horizon of A_{1} there exist elements a, b, c of the points of the projective horizon of A_{1} such that a | A and b | A and c | A and a ≠ b and b ≠ c and c ≠ a.

(40) For every elements a, b of the points of the projective horizon of A_{1} there exists an element A of the lines of the projective horizon of A_{1} such that a | A and b | A.

(41) For all elements x, y of the points of the projective horizon of A_{1} and for every element X of the lines of Inc-ProjSp(A_{1}) such that x ≠ y and ⟨⟨x, X⟩⟩ ∈ the incidence of Inc-ProjSp(A_{1}) and ⟨⟨y, X⟩⟩ ∈ the incidence of Inc-ProjSp(A_{1}) there exists an element Y of the lines of the projective horizon of A_{1} such that X = ⟨⟨Y, 2⟩⟩.

(42) For every element x of the points of Inc-ProjSp(A_{1}) and for every element X of the lines of the projective horizon of A_{1} such that ⟨⟨x, ⟨⟨X, 2⟩⟩⟩⟩ ∈ the incidence of Inc-ProjSp(A_{1}) holds x is an element of the points of the projective horizon of A_{1}.

(43) If Y is a plane and X is a line and X′ is a line and X ⊆ Y and X′ ⊆ Y and P = ⟨⟨X, 1⟩⟩ and Q = ⟨⟨X′, 1⟩⟩, then there exists q such that q | P and q | Q.

(44) Let a, b, c, d, p be elements of the points of the projective horizon of A_{1}. Let M, N, P, Q be elements of the lines of the projective horizon of A_{1}. Suppose that

(i) a | M, (ii) b | M, (iii) c | N, (iv) d | N,

(v) p | M, (vi) p | N, (vii) a | P, (viii) c | P, (ix) b | Q,

(x) d | Q, (xi) p P, (xii) p Q, (xiii) M ≠ N.

Then there exists an element q of the points of the projective horizon of A_{1} such that q | P and q | Q.

Let us consider A_{1}. Then Inc-ProjSp(A_{1}) is a projective space defined in terms of incidence.

Let A_{1} be an affine plane. Then Inc-ProjSp(A_{1}) is a 2-dimensional projective space defined in terms of incidence.

The following propositions are true:

(45) If Inc-ProjSp(A_{1}) is 2-dimensional, then A_{1} is an affine plane.

(46) If A_{1} is not an affine plane, then the projective horizon of A_{1} is a projective space defined in terms of incidence.

(47) If the projective horizon of A_{1} is a projective space defined in terms of incidence, then A_{1} is not an affine plane.

(48) Let M, N be subsets of the points of A_{1}. Let o, a, b, c, a′, b′, c′ be

elements of the points of A_{1}. Suppose that

(i) M is a line, (ii) N is a line, (iii) M ≠ N, (iv) o ∈ M,

(v) o ∈ N, (vi) o ≠ a, (vii) o ≠ a′, (viii) o ≠ b, (ix) o ≠ b′,

(x) o ≠ c, (xi) o ≠ c′, (xii) a ∈ M, (xiii) b ∈ M, (xiv) c ∈ M, (xv) a′ ∈ N, (xvi) b′ ∈ N, (xvii) c′ ∈ N, (xviii) a, b′ ∥ b, a′, (xix) b, c′ ∥ c, b′, (xx) a = b or b = c or a = c.

Then a, c′ ∥ c, a′.

(49) If Inc-ProjSp(A_{1}) is Pappian, then A_{1} is Pappian.

(50) Let A, P, C be subsets of the points of A_{1}. Let o, a, b, c, a′, b′, c′ be

elements of the points of A_{1}. Suppose that

(i) o ∈ A, (ii) o ∈ P, (iii) o ∈ C, (iv) o ≠ a,

A Projective Closure and Projective Horizon … 383

(v) o ≠ b, (vi) o ≠ c, (vii) a ∈ A, (viii) a′ ∈ A, (ix) b ∈ P,

(x) b′ ∈ P, (xi) c ∈ C, (xii) c′ ∈ C, (xiii) A is a line, (xiv) P is a line, (xv) C is a line, (xvi) A ≠ P, (xvii) A ≠ C, (xviii) a, b ∥ a′, b′, (xix) a, c ∥ a′, c′, (xx) o = a′ or a = a′.

Then b, c ∥ b′, c′.

(51) If Inc-ProjSp(A_{1}) is Desarguesian, then A_{1} is Desarguesian.

(52) If Inc-ProjSp(A_{1}) is Fanoian, then A_{1} is Fanoian.

References

[1] Wojciech Leoṅczuk, Henryk Oryszczyszyn, and Krzysztof Prażmowski. Planes in affine spaces. Formalized Mathematics, 2(3):357–363, 1991.

[2] Wojciech Leończuk and Krzysztof Prażmowski. Incidence projective spaces. Formalized Mathematics, 2(2):225–232, 1991.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[6] Beata Padlewska. Families of sets. Formalized Mathematics, 1(1):147–152, 1990.

[7] Krzysztof Prażmowski. Fanoian, Pappian and Desarguesian affine spaces. Formalized Mathematics, 2(3):341–346, 1991.

[8] Konrad Raczkowski and Paweł Sadowski. Equivalence relations and classes of abstraction. Formalized Mathematics, 1(3):441–444, 1990.

[9] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[10] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[11] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[12] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186,

384 Henryk Oryszczyszyn et al.

[13] Edmund Woronowicz and Anna Zalewska. Properties of binary relations. Formalized

Mathematics, 1(1):85–89, 1990.

Received December 17, 1990

Schemes ^{1}

Stanisław T. Czuba Warsaw University Białystok

Summary. Some basic schemes of quantifier calculus are proved.

MML Identifier: SCHEMS 1.

In the sequel a, b will be arbitrary. In this article we present several logical schemes. The scheme Schemat0 concerns a unary predicate P, and states that:

there exists a such that P[a] provided the parameter meets the following requirement:

• for every a holds P[a].

The scheme Schemat1a deals with Q, and a unary predicate P, and states that:

for every a holds P[a] and Q[] provided the parameters meet the following requirement:

• for every a holds P[a] and Q[].

The scheme Schemat1b concerns Q, and a unary predicate P, and states that:

for every a holds P[a] and Q[] provided the parameters have the following property:

• for every a holds P[a] and Q[].

The scheme Schemat2a concerns Q, and a unary predicate P, and states that:

there exists a such that P[a] or Q[] provided the parameters meet the following requirement:

• there exists a such that P[a] or Q[].

The scheme Schemat2b deals with Q, and a unary predicate P, and states that:

there exists a such that P[a] or Q[] provided the following condition is met:

1Supported by RPBP.III-24.C10

• there exists a such that P[a] or Q[]. The scheme Schemat3 concerns a binary predicate P, and states that: for every b there exists a such that P[a, b]

provided the parameter has the following property: • there exists a such that for every b holds P[a, b]. The scheme Schemat4a concerns two unary predicates P and Q, and states

that:

there exists a such that P[a] or there exists a such that Q[a] provided the following condition is satisfied:

• there exists a such that P[a] or Q[a]. The scheme Schemat4b deals with two unary predicates P and Q, and states

that:

there exists a such that P[a] or Q[a] provided the parameters meet the following requirement:

• there exists a such that P[a] or there exists a such that Q[a]. The scheme Schemat5 concerns two unary predicates P and Q, and states

that:

there exists a such that P[a] and there exists a such that Q[a] provided the following condition is met:

• there exists a such that P[a] and Q[a]. The scheme Schemat6a concerns two unary predicates P and Q, and states

that:

for every a holds P[a] and for every a holds Q[a] provided the parameters satisfy the following condition:

• for every a holds P[a] and Q[a]. The scheme Schemat6b deals with two unary predicates P and Q, and states

that:

for every a holds P[a] and Q[a] provided the following requirement is met:

• for every a holds P[a] and for every a holds Q[a]. The scheme Schemat7 deals with two unary predicates P and Q, and states

that:

for every a holds P[a] or Q[a] provided the following condition is satisfied:

• for every a holds P[a] or for every a holds Q[a]. The scheme Schemat8 concerns two unary predicates P and Q, and states

that:

if for every a holds P[a], then for every a holds Q[a] provided the parameters satisfy the following condition:

• for every a such that P[a] holds Q[a]. The scheme Schemat9 concerns two unary predicates P and Q, and states

that:

for every a holds P[a] if and only if for every a holds Q[a] provided the parameters have the following property:

• for every a holds P[a] if and only if Q[a].

The scheme Schemat10a concerns P and states that: P[]

provided the parameter satisfies the following condition: • for every a holds P[]. The scheme Schemat10b concerns P and states that: for every a holds P[]

provided the parameter satisfies the following condition: • P[]. The scheme Schemat11a concerns Q, and a unary predicate P, and states

that:

for every a holds P[a] or Q[] provided the following requirement is met:

• for every a holds P[a] or Q[]. The scheme Schemat11b deals with Q, and a unary predicate P, and states

that:

for every a holds P[a] or Q[] provided the parameters satisfy the following condition:

• for every a holds P[a] or Q[]. The scheme Schemat12a concerns Q, and a unary predicate P, and states

that:

there exists a such that Q[] and P[a] provided the following condition is satisfied:

• Q[] and there exists a such that P[a]. The scheme Schemat12b concerns Q, and a unary predicate P, and states

that:

Q[] and there exists a such that P[a] provided the following condition is satisfied:

• there exists a such that Q[] and P[a]. The scheme Schemat13a concerns Q, and a unary predicate P, and states

that:

for every a such that Q[] holds P[a] provided the parameters satisfy the following condition:

• if Q[], then for every a holds P[a]. The scheme Schemat13b deals with Q, and a unary predicate P, and states

that:

if Q[], then for every a holds P[a] provided the parameters satisfy the following condition:

• for every a such that Q[] holds P[a]. The scheme Schemat14 concerns Q, and a unary predicate P, and states

that:

there exists a such that if Q[], then P[a] provided the parameters meet the following requirement:

• if Q[], then there exists a such that P[a]. The scheme Schemat15 deals with Q, and a unary predicate P, and states

that:

for every a such that P[a] holds Q[] provided the following condition is met:

• if there exists a such that P[a], then Q[]. The scheme Schemat16 deals with Q, and a unary predicate P, and states

that:

there exists a such that if P[a], then Q[] provided the parameters meet the following requirement:

• if for every a holds P[a], then Q[]. The scheme Schemat17 concerns Q, and a unary predicate P, and states

that:

if for every a holds P[a], then Q[] provided the parameters meet the following requirement:

• for every a such that P[a] holds Q[]. The scheme Schemat18a deals with two unary predicates P and Q, and states

that:

there exists a such that for every b holds P[a] or Q[b] provided the following condition is satisfied:

• there exists a such that P[a] or for every b holds Q[b]. The scheme Schemat18b deals with two unary predicates P and Q, and states

that:

there exists a such that P[a] or for every b holds Q[b] provided the parameters meet the following condition:

• there exists a such that for every b holds P[a] or Q[b]. The scheme Schemat19a concerns two unary predicates P and Q, and states

that:

for every b there exists a such that P[a] or Q[b] provided the following condition is met:

• there exists a such that P[a] or for every b holds Q[b]. The scheme Schemat19b concerns two unary predicates P and Q, and states

that:

there exists a such that P[a] or for every b holds Q[b] provided the following condition is met:

• for every b there exists a such that P[a] or Q[b]. The scheme Schemat20a deals with two unary predicates P and Q, and states

that:

for every b there exists a such that P[a] or Q[b] provided the following condition is met:

• there exists a such that for every b holds P[a] or Q[b]. The scheme Schemat20b concerns two unary predicates P and Q, and states

that:

there exists a such that for every b holds P[a] or Q[b] provided the following requirement is met:

• for every b there exists a such that P[a] or Q[b]. The scheme Schemat21a deals with two unary predicates P and Q, and states

that:

there exists a such that for every b holds P[a] and Q[b] provided the following condition is satisfied:

• there exists a such that P[a] and for every b holds Q[b]. The scheme Schemat21b deals with two unary predicates P and Q, and states

that:

there exists a such that P[a] and for every b holds Q[b] provided the following condition is satisfied:

• there exists a such that for every b holds P[a] and Q[b]. The scheme Schemat22a deals with two unary predicates P and Q, and states

that:

for every b there exists a such that P[a] and Q[b] provided the parameters meet the following condition:

• there exists a such that P[a] and for every b holds Q[b]. The scheme Schemat22b deals with two unary predicates P and Q, and states

that:

there exists a such that P[a] and for every b holds Q[b] provided the following requirement is met:

• for every b there exists a such that P[a] and Q[b]. The scheme Schemat23a deals with two unary predicates P and Q, and states

that:

for every b there exists a such that P[a] and Q[b] provided the following requirement is met:

• there exists a such that for every b holds P[a] and Q[b]. The scheme Schemat23b deals with two unary predicates P and Q, and states

that:

there exists a such that for every b holds P[a] and Q[b] provided the parameters satisfy the following condition:

• for every b there exists a such that P[a] and Q[b]. The scheme Schemat24a concerns a unary predicate Q, and a binary predi-

cate P, and states that:

for every a there exists b such that if P[a, b], then Q[a] provided the parameters satisfy the following condition:

• for every a such that for every b holds P[a, b] holds Q[a]. The scheme Schemat24b deals with a unary predicate Q, and a binary pred-

icate P, and states that:

for every a such that for every b holds P[a, b] holds Q[a] provided the following requirement is met:

• for every a there exists b such that if P[a, b], then Q[a]. The scheme Schemat25a concerns a unary predicate Q, and a binary predi-

cate P, and states that:

for all a, b such that P[a, b] holds Q[a] provided the parameters have the following property:

• for every a such that there exists b such that P[a, b] holds Q[a]. The scheme Schemat25b concerns a unary predicate Q, and a binary predi-

cate P, and states that:

for every a such that there exists b such that P[a, b] holds Q[a] provided the following condition is met:

• for all a, b such that P[a, b] holds Q[a]. The scheme Schemat26 deals with a binary predicate P, and states that: there exists a such that for every b holds P[a, b]

provided the following condition is met:

• for all a, b holds P[a, b]. The scheme Schemat27 deals with a binary predicate P, and states that: for every a holds P[a, a]

provided the parameter meets the following condition: • for all a, b holds P[a, b]. The scheme Schemat28 concerns a binary predicate P, and states that: there exists b such that for every a holds P[a, b]

provided the following requirement is met:

• for all a, b holds P[a, b]. The scheme Schemat29 deals with a binary predicate P, and states that: for every b there exists a such that P[a, b]

provided the parameter has the following property: • there exists a such that for every b holds P[a, b]. The scheme Schemat30 deals with a binary predicate P, and states that: there exists a such that P[a, a]

provided the parameter meets the following requirement: • there exists a such that for every b holds P[a, b]. The scheme Schemat31 concerns a binary predicate P, and states that: for every a there exists b such that P[b, a]

provided the following condition is satisfied:

• for every a holds P[a, a]. The scheme Schemat32 concerns a binary predicate P, and states that: there exists a such that P[a, a]

provided the parameter meets the following condition: • for every a holds P[a, a]. The scheme Schemat33 deals with a binary predicate P, and states that: for every a there exists b such that P[a, b]

provided the following condition is satisfied:

• for every a holds P[a, a]. The scheme Schemat34 concerns a binary predicate P, and states that: there exists b such that P[b, b]

provided the parameter meets the following requirement: • there exists b such that for every a holds P[a, b]. The scheme Schemat35 deals with a binary predicate P, and states that: for every a there exists b such that P[a, b]

provided the parameter meets the following condition: • there exists b such that for every a holds P[a, b]. The scheme Schemat36 deals with a binary predicate P, and states that: there exist a, b such that P[a, b] provided the parameter meets the following requirement:

• for every b there exists a such that P[a, b]. The scheme Schemat37 deals with a binary predicate P, and states that: there exist a, b such that P[a, b]

provided the following condition is satisfied:

• there exists a such that P[a, a]. The scheme Schemat38 concerns a binary predicate P, and states that: there exist a, b such that P[a, b]

provided the parameter satisfies the following condition: • for every a there exists b such that P[a, b].

392

Algebra of Normal Forms Is a Heyting Algebra ^{1}

Andrzej Trybulec Warsaw University Białystok

Summary. We prove that the lattice of normal forms over an arbitrary set, introduced in [7], is an implicative lattice. The relative psedo-complement α ⇒ β is defined as ⨆_{α_{1}∪α_{2}=α} −α_{1} ⊓ α_{2} β, where −α is the pseudo-complement of α and α β is a rather strong implication introduced in this paper.

MML Identifier: HEYTING1.

The articles [13], [4], [5], [2], [14], [3], [8], [6], [15], [9], [16], [10], [11], [12], [7], and [1] provide the notation and terminology for this paper. One can prove the following proposition

(1) For all non-empty sets A, B, C and for every function f from A into B such that for every element x of A holds f(x) ∈ C holds f is a function from A into C.

In the sequel A will be a non-empty set and a will be an element of A. Let us consider A, and let B, C be elements of Fin A. Let us note that one can characterize the predicate B ⊆ C by the following (equivalent) condition: (Def.1) for every a such that a ∈ B holds a ∈ C.

Let A be a non-empty set, and let B be a non-empty subset of A. Then {B \atop ↪} is a function from B into A.

The following proposition is true

(2) For every non-empty set A and for every non-empty subset B of A and for every element x of B holds ({B \atop ↪})(x) = x.

In the sequel A denotes a set. Let us consider A. Let us assume that A is non-empty. The functor [A] yielding an non-empty set is defined by:

1Partially supported by RPBP.III-24.B1

(Def.2) [A] = A.

We follow the rules: B, C will denote elements of Fin DP(A), a, b, c, s, t_{1}, t_{2} will denote elements of DP(A), and u, v, w will denote elements of the carrier of the lattice of normal forms over A. The following propositions are true:

(3) If B = ∅, then μB = ∅.

(4) For an arbitrary x such that x ∈ B holds x is an element of DP(A). Let us consider A, a. Then {a} is an element of the normal forms over A. Let us consider A, and let u be an element of the carrier of the lattice of normal forms over A. The functor ^{@}u yields an element of the normal forms over A and is defined

as follows: (Def.3) ^{@}u = u.

One can prove the following two propositions:

(5) ⊓_{A}(^{@}u, ^{@}v) = (the meet operation of the lattice of normal forms over A)(u, v).

(6) ⊔_{A}(^{@}u, ^{@}v) = (the join operation of the lattice of normal forms over A)(u, v).

In the sequel K, L will denote elements of the normal forms over A. One can prove the following propositions:

(7) μ(K ^{}K) = K.

(8) For every set X such that X ⊆ K holds X ∈ the normal forms over A.

(9) ∅ is an element of the normal forms over A.

(10) For every set X such that X ⊆ u holds X is an element of the carrier

of the lattice of normal forms over A. Let us consider A. The functor { }_{A} yields a function from DP(A) into the

carrier of the lattice of normal forms over A and is defined by: (Def.4) { }_{A}(a) = {a}.

The following propositions are true:

(11) If c ∈ { }_{A}(a), then c = a.

(12) a ∈ { }_{A}(a).

(13) { }_{A}(a) = singleton_{DP(A)}(a).

(14) ⨆_{K}^{f}({ }_{A}) = FinUnion(K, singleton_{DP(A)}).

(15) u = ⨆_{(^{@}u)}^{f}({ }_{A}).

In the sequel f will denote an element of [: Fin A, Fin A:]^{DP(A)} and g will denote an element of [A]^{DP(A)}. Let A be a set. The functor \_{A} yielding a binary operation on [: Fin A, Fin A:] is defined as follows: (Def.5) for all elements a, b of [: Fin A, Fin A:] holds \_{A} (a, b) = a \ b.

We now define two new functors. Let us consider A, B. The functor −B yielding an element of Fin DP(A) is defined by:

(Def.6) −B = DP(A) ∩ {⟨⟨{g(t_{1}) : g(t_{1}) ∈ t_{1}_{2} ∧ t_{1} ∈ B},

{g(t_{2}) : g(t_{2}) ∈ t_{2}_{1} ∧ t_{2} ∈ B}⟩⟩ : s ∈ B ⇒ g(s) ∈ s_{1} ∪ s_{2}}. Let us consider C. The functor B C yielding an element of Fin DP(A) is defined by: (Def.7) B C = DP(A) ∩ \{FinUnion(B, \_{A} ^{∘}(f, {DP(A) \atop ↪})) : f ^{∘} B ⊆ C\} .

The following propositions are true:

(16) Suppose c ∈ −B. Then there exists g such that for every s such that s ∈ B holds g(s) ∈ s_{1} ∪ s_{2} and c = ⟨⟨ {g(t_{1}) : g(t_{1}) ∈ t_{1}_{2} ∧ t_{1} ∈ B} , {g(t_{2}) : g(t_{2}) ∈ t_{2}_{1} ∧ t_{2} ∈ B} ⟩⟩.

(17) ⟨⟨∅, ∅⟩⟩ is an element of DP(A).

(18) For every K such that K = ∅ holds −K = {⟨⟨∅, ∅⟩⟩}.

(19) For all K, L such that K = ∅ and L = ∅ holds K L = {⟨⟨∅, ∅⟩⟩}.

(20) For every element a of DP(∅) holds a = ⟨⟨∅, ∅⟩⟩.

(21) DP(∅) = {⟨⟨∅, ∅⟩⟩}.

(22) {⟨⟨∅, ∅⟩⟩} is an element of the normal forms over A.

(23) If c ∈ B C, then there exists f such that f ^{∘} B ⊆ C and c = FinUnion(B, \_{A} ^{∘}(f, {DP(A) \atop ↪})).

(24) If K ^{}{a} = ∅, then there exists b such that b ∈ −K and b ⊆ a.

(25) If for every b such that b ∈ u holds b ∪ a ∈ DP(A) and for every c such that c ∈ u there exists b such that b ∈ v and b ⊆ c ∪ a, then there exists b such that b ∈ (^{@}u) ^{@}v and b ⊆ a.

(26) K ^{}−K = ∅.

We now define four new functors. Let us consider A. The functor ^{c}_{A} yielding a unary operation on the carrier of the lattice of normal forms over A is defined by: (Def.8) ^{c}_{A}(u) = μ(−^{@}u). The functor _{A} yields a binary operation on the carrier of the

lattice of normal forms over A

and is defined by: (Def.9) ( _{A} )(u, v) = μ((^{@}u) ^{@}v). Let us consider u. The functor 2^{u} yielding an element of Fin (the carrier of the lattice of normal forms over A) is defined by:

(Def.10) 2^{u} = 2^{u}.

The functor \_{u} yielding a unary operation on the carrier of the lattice of normal forms over A is defined as follows:

(Def.11) ( \_{u} )(v) = u \ v.

We now state several propositions:

(27) ( \_{u} )(v) ⊑ u.

(28) u ⊓ ^{c}_{A}(u) = ⊥_{the lattice of normal forms over A}.

396 Andrzej Trybulec

(29) u ⊓ ( _{A} )(u, v) ⊑ v.

(30) If (^{@}u)^{}{a} = ∅, then { }_{A}(a) ⊑ ^{c}_{A}(u).

(31) If for every b such that b ∈ u holds b∪a ∈ DP(A) and u⊓{ }_{A}(a) ⊑ w, then { }_{A}(a) ⊑ ( _{A} )(u, w).

(32) The lattice of normal forms over A is an implicative lattice.

(33) u ⇒ v = ⨆_{2^{u}}^{f}( (the meet operation of the lattice of normal forms over A)^{∘}( ^{c}_{A}, ( _{A} )^{∘}( \_{u} , v))).

(34) ⊤_{The lattice of normal forms over A} = {⟨⟨∅, ∅⟩⟩}.

References

[1] Grzegorz Bancerek. Filters - part I. Formalized Mathematics, 1(5):813–819, 1990.

[2] Czesław Byliński. Basic functions and operations on functions. Formalized Mathematics, 1(1):245–254, 1990.

[3] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[4] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[5] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[6] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[7] Andrzej Trybulec. Algebra of normal forms. Formalized Mathematics, 2(2):237–242, 1991.

[8] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[9] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115–122, 1990.

[10] Andrzej Trybulec. Finite join and finite meet and dual lattices. Formalized Mathematics, 1(5):983–988, 1990.

[11] Andrzej Trybulec. Function domains and Frænkel operator. Formalized Mathematics, 1(3):495–500, 1990.

[12] Andrzej Trybulec. Semilattice operations on finite subsets. Formalized Mathematics, 1(2):369–376, 1990.

[13] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[14] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[15] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

[16] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

König’s Lemma ^{1}

Grzegorz Bancerek Warsaw University Białystok

Summary. A contiuation of [5]. The notions of finite-order trees, succesors of an element of a tree, and chains, levels and branches of a tree are introduced. Those notions are used to formalize König’s Lemma which claims that there is a infinite branch of a finite-order tree if the tree has arbitrary long finite chains. Besides, the concept of decorated trees is introduced and some concepts dealing with trees are applied to decorated trees.

MML Identifier: TREES 2.

The articles [12], [7], [10], [4], [6], [9], [2], [1], [3], [8], [11], [13], and [5] provide the notation and terminology for this paper. For simplicity we adopt the following rules: x, y are arbitrary, W, W_{1}, W_{2} denote trees, w denotes an element of W, X denotes a set, f, f_{1}, f_{2} denote functions, D, D′ denote non-empty sets, k, k_{1}, k_{2}, m, n denote natural numbers, v, v_{1}, v_{2} denote finite sequences, and p, q, r denote finite sequences of elements of . The following propositions are true:

(1) For all v_{1}, v_{2}, v such that v_{1} ⪯ v and v_{2} ⪯ v holds v_{1} and v_{2} are comparable.

(2) For all v_{1}, v_{2}, v such that v_{1} ≺ v and v_{2} ⪯ v holds v_{1} and v_{2} are comparable and v_{2} and v_{1} are comparable.

(4)^{2} If len v_{1} = k + 1, then there exist v_{2}, x such that v_{1} = v_{2} ^{}⟨x⟩ and

len v_{2} = k.

(5) (v_{1} ^{}v_{2}) Seg len v_{1} = v_{1}.

(6) Seg_{⪯}(v^{}⟨x⟩) = Seg_{⪯}(v) ∪ {v}.

The scheme TreeStruct Ind concerns a tree A, and a unary predicate P, and states that:

for every element t of A holds P[t] provided the following requirements are met:

1Partially supported by RPBP.III-24.C1

2The proposition (3) was either repeated or obvious.

• P[ε],

• for every element t of A and for every n such that P[t] and t^{}⟨n⟩ ∈ A holds P[t^{}⟨n⟩].

We now state the proposition

(7) If for every p holds p ∈ W_{1} if and only if p ∈ W_{2}, then W_{1} = W_{2}. Let us consider W_{1}, W_{2}. Let us note that one can characterize the predicate

W_{1} = W_{2} by the following (equivalent) condition: (Def.1) for every p holds p ∈ W_{1} if and only if p ∈ W_{2}.

One can prove the following propositions:

(8) If p ∈ W, then W = W (p/(W p)).

(9) If p ∈ W and q ∈ W and p q, then q ∈ W (p/W_{1}).

(10) If p ∈ W and q ∈ W and p and q are not comparable, then

W (p/W_{1})(q/W_{2}) = W (q/W_{2})(p/W_{1}). A tree is finite-order if:

(Def.2) there exists n such that for every element t of it holds t^{}⟨n⟩ ∉ it.

We now define three new constructions. Let us consider W. A subset of W is said to be a chain of W if: (Def.3) for all p, q such that p ∈ it and q ∈ it holds p and q are comparable. A subset of W is called a level of W if: (Def.4) there exists n such that it = {w : len w = n}. Let us consider w. The functor succ w yielding a subset of W is defined by: (Def.5) succ w = {w^{}⟨n⟩ : w^{}⟨n⟩ ∈ W }.

One can prove the following propositions:

(11) For every level L of W holds L is an antichain of prefixes of W.

(12) succ w is an antichain of prefixes of W.

(13) For every antichain A of prefixes of W and for every chain C of W there

exists w such that A ∩ C ⊆ {w}. Let us consider W, n. The functor n_{W} yielding a level of W is defined by:

(Def.6) n_{W} = {w : len w = n}.

We now state several propositions:

(14) w^{}⟨n⟩ ∈ succ w if and only if w^{}⟨n⟩ ∈ W.

(15) If w = ε, then 1_{W} = succ w.

(16) W = ⋃{n_{W} }.

(17) For every finite tree W holds W = ⋃{n_{W} : n ≤ height W }.

(18) For every level L of W there exists n such that L = n_{W}.

Now we present three schemes. The scheme AuxSch concerns a tree A, and a unary predicate P, and states that:

{w : P[w]}, where w ranges over elements of A, is a subset of A for all values of the parameters.

The scheme FraenkelCard concerns a non-empty set A, a set B, and a unary functor F and states that:

\overline{\overline{\{F(w) : w ∈ B\}}} ≤ \overline{\overline{B}}, where w ranges over elements of A for all values of the parameters.

The scheme FraenkelFinCard concerns a non-empty set A, a set B, and a unary functor F and states that:

card{F(w) : w ∈ B} ≤ card B, where w ranges over elements of A provided the parameters meet the following requirement:

• B is finite.

The following four propositions are true:

(19) If W is finite-order, then there exists n such that for every w holds succ w is finite and card succ w ≤ n.

(20) If W is finite-order, then succ w is finite.

(21) ∅ is a chain of W.

(22) {ε} is a chain of W.

Let us consider W. A chain of W is said to be a branch of W if: (Def.7) for every p such that p ∈ it holds Seg_{⪯}(p) ⊆ it and for no p holds p ∈ W

and for every q such that q ∈ it holds q ≺ p.

Let us consider W. We see that the branch of W is an non-empty chain of W.

In the sequel C will be a chain of W and B will be a branch of W. The following propositions are true:

(23) If v_{1} ∈ C and v_{2} ∈ C, then v_{1} ∈ Seg_{⪯}(v_{2}) or v_{2} ⪯ v_{1}.

(24) If v_{1} ∈ C and v_{2} ∈ C and v ⪯ v_{2}, then v_{1} ∈ Seg_{⪯}(v) or v ⪯ v_{1}.

(25) If C is finite and card C > n, then there exists p such that p ∈ C and len p ≥ n.

(26) For every C holds {w : ⋁_{p}[p ∈ C ∧ w ⪯ p]} is a chain of W.

(27) If p ⪯ q and q ∈ B, then p ∈ B.

(28) ε ∈ B.

(29) If p ∈ C and q ∈ C and len p ≤ len q, then p ⪯ q.

(30) There exists B such that C ⊆ B.

Now we present two schemes. The scheme FuncExOfMinNat concerns a set A, and a binary predicate P, and states that:

there exists f such that dom f = A and for every x such that x ∈ A there exists n such that f(x) = n and P[x, n] and for every m such that P[x, m] holds n ≤ m provided the following condition is met:

• for every x such that x ∈ A there exists n such that P[x, n].

The scheme InfiniteChain concerns a set A, a constant B, a unary predicate P, and a binary predicate Q, and states that:

there exists f such that dom f = and rng f ⊆ A and f(0) = B and for every k holds Q[f(k), f(k + 1)] and P[f(k)] provided the parameters meet the following conditions:

• B ∈ A and P[B],

• for every x such that x ∈ A and P[x] there exists y such that y ∈ A and Q[x, y] and P[y].

The following two propositions are true:

(31) For every tree T such that for every n there exists a chain C of T such that C is finite and card C = n and for every element t of T holds succ t is finite there exists a chain B of T such that B is not finite.

(32) For every finite-order tree T such that for every n there exists a chain C of T such that C is finite and card C = n there exists a chain B of T such that B is not finite.

A function is said to be a decorated tree if: (Def.8) dom it is a tree.

In the sequel T, T_{1}, T_{2} are decorated trees. Let us consider T. Then dom T is a tree.

Let us consider D. A decorated tree is said to be a tree decorated by D if: (Def.9) rng it ⊆ D.

Let D be a non-empty set, and let T be a tree decorated by D, and let t be an element of dom T. Then T (t) is an element of D.

One can prove the following proposition

(33) If dom T_{1} = dom T_{2} and for every p such that p ∈ dom T_{1} holds T_{1}(p) =

T_{2}(p), then T_{1} = T_{2}. Now we present two schemes. The scheme DTreeEx concerns a tree A, and

a binary predicate P, and states that:

there exists T such that dom T = A and for every p such that p ∈ A holds P[p, T (p)] provided the following condition is satisfied:

• for every p such that p ∈ A there exists x such that P[p, x]. The scheme DTreeLambda deals with a tree A and a unary functor F and

states that:

there exists T such that dom T = A and for every p such that p ∈ A holds T (p) = F(p) for all values of the parameters.

We now define two new functors. Let us consider T. The functor Leaves T yielding a set is defined by:

(Def.10) Leaves T = T ^{∘} Leaves dom T.

Let us consider p. The functor T p yielding a decorated tree is defined by: (Def.11) dom(T p) = dom T p and for every q such that q ∈ dom T p holds

(T p)(q) = T (p^{}q).

The following proposition is true

(34) If p ∈ dom T, then rng(T p) ⊆ rng T.

Let us consider D, and let T be a tree decorated by D. Then Leaves T is a subset of D. Let p be an element of dom T. Then T p is a tree decorated by D.

Let us consider T, p, T_{1}. Let us assume that p ∈ dom T. The functor T (p/T_{1}) yielding a decorated tree is defined by the conditions (Def.12).

(Def.12) (i) dom(T (p/T_{1})) = (dom T)(p/ dom T_{1}),

(ii) for every q such that q ∈ (dom T)(p/ dom T_{1}) holds p q and T (p/T_{1})(q) = T (q) or there exists r such that r ∈ dom T_{1} and q = p^{}r and T (p/T_{1})(q) = T_{1}(r).

Let us consider W, x. Then W ⟼ x is a decorated tree.

Let D be a non-empty set, and let us consider W, and let d be an element of D. Then W ⟼ d is a tree decorated by D.

Next we state four propositions:

(35) If for every x such that x ∈ D holds x is a tree, then ⋃ D is a tree.

(36) If for every x such that x ∈ X holds x is a function and for all f_{1}, f_{2} such that f_{1} ∈ X and f_{2} ∈ X holds graph f_{1} ⊆ graph f_{2} or graph f_{2} ⊆ graph f_{1}, then ⋃ X is a function.

(37) If for every x such that x ∈ D holds x is a decorated tree and for all T_{1}, T_{2} such that T_{1} ∈ D and T_{2} ∈ D holds graph T_{1} ⊆ graph T_{2} or graph T_{2} ⊆ graph T_{1}, then ⋃ D is a decorated tree.

(38) If for every x such that x ∈ D′ holds x is a tree decorated by D and for all T_{1}, T_{2} such that T_{1} ∈ D′ and T_{2} ∈ D′ holds graph T_{1} ⊆ graph T_{2} or graph T_{2} ⊆ graph T_{1}, then ⋃ D′ is a tree decorated by D.

Now we present two schemes. The scheme DTreeStructEx deals with a nonempty set A, an element B of A, a unary functor F yielding a set, and a function C from [: A, :] into A and states that:

there exists a tree T decorated by A such that T (ε) = B and for every element t of dom T holds succ t = {t^{}⟨k⟩ : k ∈ F(T (t))} and for all n, x such that x = T (t) and n ∈ F(x) holds T (t^{}⟨n⟩) = C(⟨⟨x, n⟩⟩) provided the following condition is satisfied:

• for every element d of A and for all k_{1}, k_{2} such that k_{1} ≤ k_{2} and k_{2} ∈ F(d) holds k_{1} ∈ F(d).

The scheme DTreeStructFinEx deals with a non-empty set A, an element B of A, a unary functor F yielding a natural number, and a function C from [: A, :] into A and states that:

there exists a tree T decorated by A such that T (ε) = B and for every element t of dom T holds succ t = {t^{}⟨k⟩ : k < F(T (t))} and for all n, x such that x = T (t) and n < F(x) holds T (t^{}⟨n⟩) = C(⟨⟨x, n⟩⟩) for all values of the parameters.

402 Grzegorz Bancerek

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[3] Grzegorz Bancerek. Countable sets and Hessenberg’s theorem. Formalized Mathematics, 2(1):65–69, 1991.

[4] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[5] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[8] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[10] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[11] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[12] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[13] Wojciech A. Trybulec and Grzegorz Bancerek. Kuratowski - Zorn lemma. Formalized Mathematics, 1(2):387–393, 1990.

Monotonic and Continuous Real Function

Jarosław Kotowicz Warsaw University Białystok

Summary. A continuation of [16] and [13]. We prove a few theorems about real functions monotonic and continuous on interval, on halfline and on the set of real numbers and continuity of the inverse function. At the begining of the paper we show some facts about topological properties of the set of real numbers, halflines and intervals which rather belong to [17]

MML Identifier: FCONT 3.

The notation and terminology used in this paper are introduced in the following articles: [18], [5], [1], [2], [3], [20], [12], [6], [8], [15], [14], [4], [19], [9], [10], [17], [11], [16], and [7]. For simplicity we follow the rules: X will denote a set, x_{0}, r, r_{1}, g, p will denote real numbers, n will denote a natural number, a will denote a sequence of real numbers, and f will denote a partial function from to . Next we state several propositions:

(1) Ω_{}is closed.

(2) ∅_{}is open.

(3) ∅_{}is closed.

(4) Ω_{}is open.

(5) [r, +∞[ is closed.

(6) ]−∞, r] is closed.

(7) ]r, +∞[ is open.

(8) ]−∞, r[ is open. Let us consider r. Then ]r, +∞[ is a real open subset. Then HL(r) is a real

open subset.

Let us consider p, g. Then ]p, g[ is a real open subset. Next we state a number of propositions:

(9) 0 < r and g ∈ ]x_{0} − r, x_{0} + r[ if and only if there exists r_{1} such that g = x_{0} + r_{1} and |r_{1}| < r.

(10) 0 < r and g ∈ ]x_{0} − r, x_{0} + r[ if and only if g − x_{0} ∈ ]−r, r[.

(11) ]−∞, p] = {p} ∪]−∞, p[.

(12) [p, +∞[ = {p} ∪]p, +∞[.

(13) If for every n holds a(n) = x_{0}− \frac{p}{n+1}, then a is convergent and lim a = x_{0}.

(14) If for every n holds a(n) = x_{0}+ \frac{p}{n+1}, then a is convergent and lim a = x_{0}.

(15) If f is continuous in x_{0} and f(x_{0}) ≠ r and there exists a neighbourhood N of x_{0} such that N ⊆ dom f, then there exists a neighbourhood N of x_{0} such that N ⊆ dom f and for every g such that g ∈ N holds f(g) ≠ r.

(16) If f is increasing on X or f is decreasing on X, then f X is one-to-one.

(17) If f is increasing on X, then (f X)^{−1} is increasing on f ^{∘} X.

(18) If f is decreasing on X, then (f X)^{−1} is decreasing on f ^{∘} X.

(19) If X ⊆ dom f and f is monotone on X and there exists p such that f ^{∘} X = ]−∞, p[, then f is continuous on X.

(20) If X ⊆ dom f and f is monotone on X and there exists p such that f ^{∘} X = ]p, +∞[, then f is continuous on X.

(21) If X ⊆ dom f and f is monotone on X and there exists p such that f ^{∘} X = ]−∞, p], then f is continuous on X.

(22) If X ⊆ dom f and f is monotone on X and there exists p such that f ^{∘} X = [p, +∞[, then f is continuous on X.

(23) If X ⊆ dom f and f is monotone on X and there exist p, g such that f ^{∘} X = ]p, g[, then f is continuous on X.

(24) If X ⊆ dom f and f is monotone on X and f ^{∘} X = , then f is continuous on X.

(25) If f is increasing on ]p, g[ or f is decreasing on ]p, g[ but ]p, g[ ⊆ dom f, then (f ]p, g[)^{−1} is continuous on f ^{∘} ]p, g[.

(26) If f is increasing on ]−∞, p[ or f is decreasing on ]−∞, p[ but ]−∞, p[ ⊆ dom f, then (f ]−∞, p[)^{−1} is continuous on f ^{∘} ]−∞, p[.

(27) If f is increasing on ]p, +∞[ or f is decreasing on ]p, +∞[ but ]p, +∞[ ⊆ dom f, then (f ]p, +∞[)^{−1} is continuous on f ^{∘} ]p, +∞[.

(28) If f is increasing on ]−∞, p] or f is decreasing on ]−∞, p] but ]−∞, p] ⊆ dom f, then (f ]−∞, p])^{−1} is continuous on f ^{∘} ]−∞, p].

(29) If f is increasing on [p, +∞[ or f is decreasing on [p, +∞[ but [p, +∞[ ⊆ dom f, then (f [p, +∞[)^{−1} is continuous on f ^{∘} [p, +∞[.

(30) If f is increasing on Ω_{}or f is decreasing on Ω_{}but f is total, then f^{−1} is continuous on rng f.

(31) If f is continuous on ]p, g[ but f is increasing on ]p, g[ or f is decreasing on ]p, g[, then rng(f ]p, g[) is open.

(32) If f is continuous on ]−∞, p[ but f is increasing on ]−∞, p[ or f is decreasing on ]−∞, p[, then rng(f ]−∞, p[) is open.

(33) If f is continuous on ]p, +∞[ but f is increasing on ]p, +∞[ or f is decreasing on ]p, +∞[, then rng(f ]p, +∞[) is open.

Monotonic and Continuous Real Function 405

(34) If f is continuous on Ω_{}but f is increasing on Ω_{}or f is decreasing on Ω_{}, then rng f is open.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[2] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[5] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[6] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[7] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

[8] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[9] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[10] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

[11] Jarosław Kotowicz. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

[12] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[13] Jarosław Kotowicz and Konrad Raczkowski. Real function uniform continuity. Formalized Mathematics, 1(4):793–795, 1990.

[14] Andrzej Ne‘dzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

[15] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263–264, 1990.

[16] Konrad Raczkowski and Paweł Sadowski. Real function continuity. Formalized Mathematics, 1(4):787–791, 1990.

[17] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[19] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[20] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

406

Real Function Differentiability - Part II

Jarosław Kotowicz Warsaw University Białystok Konrad Raczkowski Warsaw University Białystok

Summary. A continuation of [18]. We prove an equivalent definition of the derivative of the real function at the point and theorems about derivative of composite functions, inverse function and derivative of quotient of two functions. At the begining of the paper a few facts which rather belong to [8], [10], [7] are proved.

MML Identifier: FDIFF 2.

The terminology and notation used in this paper have been introduced in the following papers: [20], [5], [1], [2], [3], [22], [14], [8], [10], [16], [15], [4], [21], [11], [12], [19], [13], [17], [18], [9], and [6]. For simplicity we adopt the following convention: x_{0}, r, r_{1}, r_{2}, g, p will be real numbers, n, m will be natural numbers, a, b, d will be sequences of real numbers, h, h_{1}, h_{2} will be real sequences convergent to 0, c will be a constant real sequence, A will be a real open subset, and f, f_{1}, f_{2} will be partial functions from to . Let us consider h. Then −h is a real sequence convergent to 0.

The following propositions are true:

(1) If a is convergent and b is convergent and lim a = lim b and for every n holds d(2 · n) = a(n) and d(2 · n + 1) = b(n), then d is convergent and lim d = lim a.

(2) If for every n holds a(n) = 2 · n, then a is an increasing sequence of naturals.

(3) If for every n holds a(n) = 2 · n + 1, then a is an increasing sequence of naturals.

(4) If rng c = {x_{0}}, then c is convergent and lim c = x_{0} and h + c is convergent and lim(h + c) = x_{0}.

(5) If rng a = {r} and rng b = {r}, then a = b.

(6) If a is a subsequence of h, then a is a real sequence convergent to 0.

(7) Suppose for all h, c such that rng c = {g} and rng(h + c) ⊆ dom f and {g} ⊆ dom f holds h^{−1}(f · (h + c) − f · c) is convergent. Given h_{1}, h_{2}, c. Suppose rng c = {g} and rng(h_{1} + c) ⊆ dom f and rng(h_{2} + c) ⊆ dom f and {g} ⊆ dom f. Then lim(h_{1}^{−1}(f · (h_{1} + c) − f · c)) = lim(h_{2}^{−1}(f · (h_{2} + c) − f · c)).

(8) If there exists a neighbourhood N of r such that N ⊆ dom f, then there exist h, c such that rng c = {r} and rng(h + c) ⊆ dom f and {r} ⊆ dom f.

(9) If rng a ⊆ dom(f_{2} · f_{1}), then rng a ⊆ dom f_{1} and rng(f_{1} · a) ⊆ dom f_{2}. The scheme ExInc Seq of Nat concerns a sequence of real numbers A, and a

unary predicate P, and states that:

there exists an increasing sequence q of naturals such that for every n holds P[(A · q)(n)] and for every n such that for every r such that r = A(n) holds P[r] there exists m such that n = q(m) provided the following requirement is met:

• for every n there exists m such that n ≤ m and P[A(m)]. One can prove the following propositions:

(10) If f(x_{0}) ≠ r and f is differentiable in x_{0}, then there exists a neighbourhood N of x_{0} such that N ⊆ dom f and for every g such that g ∈ N holds f(g) ≠ r.

(11) f is differentiable in x_{0} if and only if there exists a neighbourhood N of x_{0} such that N ⊆ dom f and for all h, c such that rng c = {x_{0}} and rng(h + c) ⊆ dom f holds h^{−1}(f · (h + c) − f · c) is convergent.

(12) f is differentiable in x_{0} and f′(x_{0}) = g if and only if the following

conditions are satisfied:

(i) there exists a neighbourhood N of x_{0} such that N ⊆ dom f, (ii) for all h, c such that rng c = {x_{0}} and rng(h + c) ⊆ dom f holds

h^{−1}(f · (h + c) − f · c) is convergent and lim(h^{−1}(f · (h + c) − f · c)) = g.

(13) If f_{1} is differentiable in x_{0} and f_{2} is differentiable in f_{1}(x_{0}), then f_{2} · f_{1} is differentiable in x_{0} and (f_{2} · f_{1})′(x_{0}) = f_{2}′(f_{1}(x_{0})) · f_{1}′(x_{0}).

(14) If f_{2}(x_{0}) ≠ 0 and f_{1} is differentiable in x_{0} and f_{2} is differentiable in x_{0}, then \frac{f_{1}}{f_{2}} is differentiable in x_{0} and (\frac{f_{1}}{f_{2}} )′(x_{0}) = \frac{f_{1}′(x_{0})·f_{2}(x_{0})−f_{2}′(x_{0})·f_{1}(x_{0})}{f_{2}(x_{0})2}.

(15) If f(x_{0}) ≠ 0 and f is differentiable in x_{0}, then \frac{1}{f} is differentiable in x_{0}

and (\frac{1}{f})′(x_{0}) = −\frac{f′(x_{0})}{f(x_{0})2} .

(16) If f is differentiable on A, then f A is differentiable on A and f′_{A} = (f A)′_{A}.

(17) If f_{1} is differentiable on A and f_{2} is differentiable on A, then f_{1} + f_{2} is differentiable on A and (f_{1} + f_{2})′_{A} = f_{1}′_{A} + f_{2}′_{A}.

(18) If f_{1} is differentiable on A and f_{2} is differentiable on A, then f_{1} − f_{2} is differentiable on A and (f_{1} − f_{2})′_{A} = f_{1}′_{A} − f_{2}′_{A}.

(19) If f is differentiable on A, then rf is differentiable on A and (rf)′_{A} = rf′_{A}.

(20) If f_{1} is differentiable on A and f_{2} is differentiable on A, then f_{1}f_{2} is differentiable on A and (f_{1}f_{2})′_{A} = f_{1}′_{A}f_{2} + f_{1}f_{2}′_{A}.

(21) If f_{1} is differentiable on A and f_{2} is differentiable on A and for every x_{0} such that x_{0} ∈ A holds f_{2}(x_{0}) ≠ 0, then \frac{f_{1}}{f_{2}} is differentiable on A and

(\frac{f_{1}}{f_{2}} )′_{A} = \frac{f_{1}′_{A} f_{2}−f_{2}′_{A} f_{1}}{f_{2}f_{2}} .

(22) If f is differentiable on A and for every x_{0} such that x_{0} ∈ A holds f(x_{0}) ≠ 0, then \frac{1}{f} is differentiable on A and (\frac{1}{f})′_{A} = −\frac{f′_{A}}{ff} .

(23) If f_{1} is differentiable on A and f_{1} ^{∘} A is a real open subset and f_{2} is differentiable on f_{1} ^{∘} A, then f_{2} · f_{1} is differentiable on A and (f_{2} · f_{1})′_{A} = (f_{2}′_{f_{1}^{∘}A} · f_{1})f_{1}′_{A}.

(24) If A ⊆ dom f and for all r, p such that r ∈ A and p ∈ A holds |f(r) − f(p)| ≤ (r − p)^{2}, then f is differentiable on A and for every x_{0} such that x_{0} ∈ A holds f′(x_{0}) = 0.

(25) Suppose for all r_{1}, r_{2} such that r_{1} ∈ ]p, g[ and r_{2} ∈ ]p, g[ holds |f(r_{1}) − f(r_{2})| ≤ (r_{1} − r_{2})^{2} and p < g and ]p, g[ ⊆ dom f. Then f is differentiable on ]p, g[ and f is a constant on ]p, g[.

(26) If ]−∞, r[ ⊆ dom f and for all r_{1}, r_{2} such that r_{1} ∈ ]−∞, r[ and r_{2} ∈ ]−∞, r[ holds |f(r_{1}) − f(r_{2})| ≤ (r_{1} − r_{2})^{2}, then f is differentiable on ]−∞, r[ and f is a constant on ]−∞, r[.

(27) If ]r, +∞[ ⊆ dom f and for all r_{1}, r_{2} such that r_{1} ∈ ]r, +∞[ and r_{2} ∈ ]r, +∞[ holds |f(r_{1}) − f(r_{2})| ≤ (r_{1} − r_{2})^{2}, then f is differentiable on ]r, +∞[ and f is a constant on ]r, +∞[.

(28) If f is total and for all r_{1}, r_{2} holds |f(r_{1}) − f(r_{2})| ≤ (r_{1} − r_{2})^{2}, then f is differentiable on Ω_{}and f is a constant on Ω_{}.

(29) If f is differentiable on ]−∞, r[ and for every x_{0} such that x_{0} ∈ ]−∞, r[ holds 0 < f′(x_{0}), then f is increasing on ]−∞, r[ and f ]−∞, r[ is oneto-one.

(30) If f is differentiable on ]−∞, r[ and for every x_{0} such that x_{0} ∈ ]−∞, r[ holds f′(x_{0}) < 0, then f is decreasing on ]−∞, r[ and f ]−∞, r[ is oneto-one.

(31) If f is differentiable on ]−∞, r[ and for every x_{0} such that x_{0} ∈ ]−∞, r[ holds 0 ≤ f′(x_{0}), then f is non-decreasing on ]−∞, r[.

(32) If f is differentiable on ]−∞, r[ and for every x_{0} such that x_{0} ∈ ]−∞, r[ holds f′(x_{0}) ≤ 0, then f is non-increasing on ]−∞, r[.

(33) If f is differentiable on ]r, +∞[ and for every x_{0} such that x_{0} ∈ ]r, +∞[ holds 0 < f′(x_{0}), then f is increasing on ]r, +∞[ and f ]r, +∞[ is oneto-one.

(34) If f is differentiable on ]r, +∞[ and for every x_{0} such that x_{0} ∈ ]r, +∞[ holds f′(x_{0}) < 0, then f is decreasing on ]r, +∞[ and f ]r, +∞[ is oneto-one.

(35) If f is differentiable on ]r, +∞[ and for every x_{0} such that x_{0} ∈ ]r, +∞[

holds 0 ≤ f′(x_{0}), then f is non-decreasing on ]r, +∞[.

(36) If f is differentiable on ]r, +∞[ and for every x_{0} such that x_{0} ∈ ]r, +∞[ holds f′(x_{0}) ≤ 0, then f is non-increasing on ]r, +∞[.

(37) If f is differentiable on Ω_{}and for every x_{0} holds 0 < f′(x_{0}), then f is increasing on Ω_{}and f is one-to-one.

(38) If f is differentiable on Ω_{}and for every x_{0} holds f′(x_{0}) < 0, then f is decreasing on Ω_{}and f is one-to-one.

(39) If f is differentiable on Ω_{}and for every x_{0} holds 0 ≤ f′(x_{0}), then f is non-decreasing on Ω_{}.

(40) If f is differentiable on Ω_{}and for every x_{0} holds f′(x_{0}) ≤ 0, then f is

non-increasing on Ω_{}. One can prove the following propositions:

(41) If f is differentiable on ]p, g[ but for every x_{0} such that x_{0} ∈ ]p, g[ holds 0 < f′(x_{0}) or for every x_{0} such that x_{0} ∈ ]p, g[ holds f′(x_{0}) < 0, then rng(f ]p, g[) is open.

(42) If f is differentiable on ]−∞, p[ but for every x_{0} such that x_{0} ∈ ]−∞, p[ holds 0 < f′(x_{0}) or for every x_{0} such that x_{0} ∈ ]−∞, p[ holds f′(x_{0}) < 0, then rng(f ]−∞, p[) is open.

(43) If f is differentiable on ]p, +∞[ but for every x_{0} such that x_{0} ∈ ]p, +∞[ holds 0 < f′(x_{0}) or for every x_{0} such that x_{0} ∈ ]p, +∞[ holds f′(x_{0}) < 0, then rng(f ]p, +∞[) is open.

(44) If f is differentiable on Ω_{}but for every x_{0} holds 0 < f′(x_{0}) or for every x_{0} holds f′(x_{0}) < 0, then rng f is open.

(45) Suppose f is differentiable on Ω_{}but for every x_{0} holds 0 < f′(x_{0}) or for every x_{0} holds f′(x_{0}) < 0. Then f is one-to-one and f^{−1} is differentiable on dom(f^{−1}) and for every x_{0} such that x_{0} ∈ dom(f^{−1}) holds (f^{−1})′(x_{0}) = \frac{1}{f′(f^{−1}(x_{0}))}.

(46) Suppose f is differentiable on ]−∞, p[ but for every x_{0} such that x_{0} ∈ ]−∞, p[ holds 0 < f′(x_{0}) or for every x_{0} such that x_{0} ∈ ]−∞, p[ holds f′(x_{0}) < 0. Then f ]−∞, p[ is one-to-one and (f ]−∞, p[)^{−1} is differentiable on dom((f ]−∞, p[)^{−1}) and for every x_{0} such that x_{0} ∈ dom((f ]−∞, p[)^{−1}) holds ((f ]−∞, p[)^{−1})′(x_{0}) = \frac{1}{f′((f ]−∞,p[)^{−1}(x_{0}))}.

(47) Suppose f is differentiable on ]p, +∞[ but for every x_{0} such that x_{0} ∈ ]p, +∞[ holds 0 < f′(x_{0}) or for every x_{0} such that x_{0} ∈ ]p, +∞[ holds f′(x_{0}) < 0. Then f ]p, +∞[ is one-to-one and (f ]p, +∞[)^{−1} is differentiable on dom((f ]p, +∞[)^{−1}) and for every x_{0} such that x_{0} ∈ dom((f ]p, +∞[)^{−1}) holds ((f ]p, +∞[)^{−1})′(x_{0}) = \frac{1}{f′((f ]p,+∞[)^{−1}(x_{0}))}.

(48) Suppose f is differentiable on ]p, g[ but for every x_{0} such that x_{0} ∈ ]p, g[ holds 0 < f′(x_{0}) or for every x_{0} such that x_{0} ∈ ]p, g[ holds f′(x_{0}) < 0. Then

(i) f ]p, g[ is one-to-one, (ii) (f ]p, g[)^{−1} is differentiable on dom((f ]p, g[)^{−1}),

Real Function Differentiability - Part II 411

(iii) for every x_{0} such that x_{0} ∈ dom((f ]p, g[)^{−1}) holds ((f ]p, g[)^{−1})′(x_{0}) = \frac{1}{f′((f ]p,g[)^{−1}(x_{0}))}.

(49) Suppose f is differentiable in x_{0}. Given h, c. Suppose rng c = {x_{0}} and rng(h + c) ⊆ dom f and rng(−h + c) ⊆ dom f. Then (2h)^{−1}(f · (c + h) − f · (c − h)) is convergent and lim((2h)^{−1}(f · (c + h) − f · (c − h))) = f′(x_{0}).

References

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[3] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[4] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[5] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[6] Jarosł aw Kotowicz. Monotonic and continuous real function. Formalized Mathematics, 2(3):403–405, 1991.

[7] Jarosław Kotowicz. Convergent real sequences. Upper and lower bound of sets of real numbers. Formalized Mathematics, 1(3):477–481, 1990.

[8] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273–275, 1990.

[9] Jarosław Kotowicz. The limit of a real function at infinity. Formalized Mathematics, 2(1):17–28, 1991.

[10] Jarosław Kotowicz. Monotone real sequences. Subsequences. Formalized Mathematics, 1(3):471–475, 1990.

[11] Jarosław Kotowicz. Partial functions from a domain to a domain. Formalized Mathematics, 1(4):697–702, 1990.

[12] Jarosław Kotowicz. Partial functions from a domain to the set of real numbers. Formalized Mathematics, 1(4):703–709, 1990.

[13] Jarosław Kotowicz. Properties of real functions. Formalized Mathematics, 1(4):781–786, 1990.

[14] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269–272, 1990.

[15] Andrzej Ne‘dzusiak. σ-fields and probability. Formalized Mathematics, 1(2):401–407, 1990.

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[18] Konrad Raczkowski and Paweł Sadowski. Real function differentiability. Formalized Mathematics, 1(4):797–801, 1990.

[19] Konrad Raczkowski and Paweł Sadowski. Topological properties of subsets in real numbers. Formalized Mathematics, 1(4):777–780, 1990.

[20] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[21] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.

[22] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

412

Preliminaries to the Lambek Calculus

Wojciech Zielonka^{1} Adam Mickiewicz University Poznań

Summary. Some preliminary facts concerning completeness and decidability problems for the Lambek calculus [13] are proved as well as some theses and derived rules of the calculus itself.

MML Identifier: PRELAMB.

The articles [16], [7], [9], [10], [18], [6], [8], [12], [17], [15], [14], [5], [1], [11], [2], [3], and [4] provide the terminology and notation for this paper. We consider structures of the type algebra which are systems

⟨types, a left quotient, a right quotient, a inner product⟩, where the types constitute a non-empty set and the left quotient, the right quotient, the inner product are a binary operation on the types.

Let s be a structure of the type algebra. A type of s is an element of the types of s.

We adopt the following rules: s will denote a structure of the type algebra, T, X, Y will denote finite sequences of elements of the types of s, and x, y, z will denote types of s. We now define three new functors. Let us consider s, x, y. The functor x \ y yields a type of s and is defined by: (Def.1) x \ y = (the left quotient of s)(x, y). The functor x/y yields a type of s and is defined as follows: (Def.2) x/y = (the right quotient of s)(x, y). The functor x · y yields a type of s and is defined by: (Def.3) x · y = (the inner product of s)(x, y).

Let T_{1} be a tree, and let v be an element of T_{1}. The branch degree of v is defined by: (Def.4) the branch degree of v = card succ v.

1This paper was written during author’s visit at the Warsaw University (Białystok) in Winter 1991.

Let us consider s. A preproof of s is a tree decorated by [: [: (the types of s)^{∗}, the types of s:], :].

In the sequel T_{1} is a preproof of s. Let us consider s, T_{1}, and let v be an element of dom T_{1}. We say that v is correct if and only if: (Def.5) (i) the branch degree of v = 0 and there exists x such that T_{1}(v)_{1} =

⟨⟨⟨x⟩, x⟩⟩ if T_{1}(v)_{2} = 0, (ii) the branch degree of v = 1 and there exist T, x, y such that T_{1}(v)_{1} =

⟨⟨T, x/y⟩⟩ and T_{1}(v^{}⟨0⟩)_{1} = ⟨⟨T ^{}⟨y⟩, x⟩⟩ if T_{1}(v)_{2} = 1, (iii) the branch degree of v = 1 and there exist T, x, y such that T_{1}(v)_{1} =

⟨⟨T, y \ x⟩⟩ and T_{1}(v^{}⟨0⟩)_{1} = ⟨⟨⟨y⟩^{}T, x⟩⟩ if T_{1}(v)_{2} = 2, (iv) the branch degree of v = 2 and there exist T, X, Y, x, y, z such that

T_{1}(v)_{1} = ⟨⟨X^{}⟨x/y⟩^{}T^{}Y, z⟩⟩ and T_{1}(v^{}⟨0⟩)_{1} = ⟨⟨T, y⟩⟩ and T_{1}(v^{}⟨1⟩)_{1} = ⟨⟨X ^{}⟨x⟩^{}Y, z⟩⟩ if T_{1}(v)_{2} = 3,

(v) the branch degree of v = 2 and there exist T, X, Y, x, y, z such that T_{1}(v)_{1} = ⟨⟨X^{}T^{}⟨y\x⟩^{}Y, z⟩⟩ and T_{1}(v^{}⟨0⟩)_{1} = ⟨⟨T, y⟩⟩ and T_{1}(v^{}⟨1⟩)_{1} = ⟨⟨X ^{}⟨x⟩^{}Y, z⟩⟩ if T_{1}(v)_{2} = 4,

(vi) the branch degree of v = 1 and there exist X, x, y, Y such that T_{1}(v)_{1} =

⟨⟨X ^{}⟨x · y⟩^{}Y, z⟩⟩ and T_{1}(v^{}⟨0⟩)_{1} = ⟨⟨X ^{}⟨x⟩^{}⟨y⟩^{}Y, z⟩⟩ if T_{1}(v)_{2} = 5, (vii) the branch degree of v = 2 and there exist X, Y, x, y such that T_{1}(v)_{1} =

⟨⟨X ^{}Y, x · y⟩⟩ and T_{1}(v^{}⟨0⟩)_{1} = ⟨⟨X, x⟩⟩ and T_{1}(v^{}⟨1⟩)_{1} = ⟨⟨Y, y⟩⟩ if T_{1}(v)_{2} = 6,

(viii) the branch degree of v = 2 and there exist T, X, Y, y, z such that T_{1}(v)_{1} = ⟨⟨X ^{}T ^{}Y, z⟩⟩ and T_{1}(v^{}⟨0⟩)_{1} = ⟨⟨T, y⟩⟩ and T_{1}(v^{}⟨1⟩)_{1} = ⟨⟨X ^{}⟨y⟩^{}Y, z⟩⟩ if T_{1}(v)_{2} = 7.

We now define three new attributes. Let us consider s. A type of s is left if: (Def.6) there exist x, y such that it = x \ y. A type of s is right if: (Def.7) there exist x, y such that it = x/y. A type of s is middle if: (Def.8) there exist x, y such that it = x · y.

Let us consider s. A type of s is primitive if: (Def.9) neither it is left nor it is right nor it is middle.

Let us consider s, and let T_{1} be a tree decorated by the types of s, and let us consider x. We say that T_{1} represents x if and only if the conditions (Def.10) is satisfied.

(Def.10) (i) dom T_{1} is finite,

(ii) for every element v of dom T_{1} holds the branch degree of v = 0 or the branch degree of v = 2 but if the branch degree of v = 0, then T_{1}(v) is primitive but if the branch degree of v = 2, then there exist y, z such that T_{1}(v) = y/z or T_{1}(v) = y \ z or T_{1}(v) = y · z but T_{1}(v^{}⟨0⟩) = y and T_{1}(v^{}⟨1⟩) = z.

A structure of the type algebra is free if:

(Def.11) for no type x of it holds x is left right or x is left middle or x is right middle and for every type x of it there exists a tree T_{1} decorated by the types of it such that for every tree T_{2} decorated by the types of it holds T_{2} represents x if and only if T_{1} = T_{2}.

Let us consider s, x. Let us assume that s is free. The representation of x yields a tree decorated by the types of s and is defined by:

(Def.12) for every tree T_{1} decorated by the types of s holds T_{1} represents x if and only if the representation of x = T_{1}.

Let us consider s, and let f be a finite sequence of elements of the types of s, and let t be a type of s. Then ⟨⟨f, t⟩⟩ is an element of [: (the types of s)^{∗}, the types of s:].

Let us consider s. A preproof of s is called a proof of s if: (Def.13) dom it is a finite tree and for every element v of dom it holds v is correct.

In the sequel p is a proof of s and v is an element of dom p. The following propositions are true:

(1) If the branch degree of v = 1, then v^{}⟨0⟩ ∈ dom p.

(2) If the branch degree of v = 2, then v^{}⟨0⟩ ∈ dom p and v^{}⟨1⟩ ∈ dom p.

(3) If p(v)_{2} = 0, then there exists x such that p(v)_{1} = ⟨⟨⟨x⟩, x⟩⟩.

(4) If p(v)_{2} = 1, then there exists an element w of dom p and there exist T, x, y such that w = v^{}⟨0⟩ and p(v)_{1} = ⟨⟨T, x/y⟩⟩ and p(w)_{1} = ⟨⟨T ^{}⟨y⟩, x⟩⟩.

(5) If p(v)_{2} = 2, then there exists an element w of dom p and there exist T, x, y such that w = v^{}⟨0⟩ and p(v)_{1} = ⟨⟨T, y \ x⟩⟩ and p(w)_{1} = ⟨⟨⟨y⟩^{}T, x⟩⟩.

(6) Suppose p(v)_{2} = 3. Then there exist elements w, u of dom p and there exist T, X, Y, x, y, z such that w = v^{}⟨0⟩ and u = v^{}⟨1⟩ and p(v)_{1} = ⟨⟨X ^{}⟨x/y⟩^{}T ^{}Y, z⟩⟩ and p(w)_{1} = ⟨⟨T, y⟩⟩ and p(u)_{1} = ⟨⟨X ^{}⟨x⟩^{}Y, z⟩⟩.

(7) Suppose p(v)_{2} = 4. Then there exist elements w, u of dom p and there exist T, X, Y, x, y, z such that w = v^{}⟨0⟩ and u = v^{}⟨1⟩ and p(v)_{1} = ⟨⟨X ^{}T ^{}⟨y \ x⟩^{}Y, z⟩⟩ and p(w)_{1} = ⟨⟨T, y⟩⟩ and p(u)_{1} = ⟨⟨X ^{}⟨x⟩^{}Y, z⟩⟩.

(8) Suppose p(v)_{2} = 5. Then there exists an element w of dom p and there exist X, x, y, Y such that w = v^{}⟨0⟩ and p(v)_{1} = ⟨⟨X^{}⟨x · y⟩^{}Y, z⟩⟩ and p(w)_{1} = ⟨⟨X ^{}⟨x⟩^{}⟨y⟩^{}Y, z⟩⟩.

(9) Suppose p(v)_{2} = 6. Then there exist elements w, u of dom p and there exist X, Y, x, y such that w = v ^{}⟨0⟩ and u = v ^{}⟨1⟩ and p(v)_{1} = ⟨⟨X ^{}Y, x · y⟩⟩ and p(w)_{1} = ⟨⟨X, x⟩⟩ and p(u)_{1} = ⟨⟨Y, y⟩⟩.

(10) Suppose p(v)_{2} = 7. Then there exist elements w, u of dom p and there exist T, X, Y, y, z such that w = v ^{}⟨0⟩ and u = v ^{}⟨1⟩ and p(v)_{1} = ⟨⟨X ^{}T ^{}Y, z⟩⟩ and p(w)_{1} = ⟨⟨T, y⟩⟩ and p(u)_{1} = ⟨⟨X ^{}⟨y⟩^{}Y, z⟩⟩.

(11) (i) p(v)_{2} = 0, or (ii) p(v)_{2} = 1, or (iii) p(v)_{2} = 2, or (iv) p(v)_{2} = 3, or

(v) p(v)_{2} = 4, or (vi) p(v)_{2} = 5, or (vii) p(v)_{2} = 6, or (viii) p(v)_{2} = 7. We now define two new constructions. Let us consider s. A preproof of s is

cut-free if: (Def.14) for every element v of dom it holds it(v)_{2} ≠ 7.

The size w.r.t. s yielding a function from the types of s into is defined by: (Def.15) for every x holds

(the size w.r.t. s)(x) = card dom(the representation of x).

Let D be a non-empty set, and let T be a finite sequence of elements of D, and let f be a function from D into . Then f ·T is a finite sequence of elements of .

Let D be a non-empty set, and let f be a function from D into , and let d be an element of D. Then f(d) is a natural number.

Let us consider s, and let p be a proof of s. Let us assume that s is free. The cut degree of p yields a natural number and is defined by:

(Def.16) (i) there exist T, X, Y, y, z such that p(ε)_{1} = ⟨⟨X ^{}T ^{}Y, z⟩⟩ and p(⟨0⟩)_{1} = ⟨⟨T, y⟩⟩ and p(⟨1⟩)_{1} = ⟨⟨X ^{}⟨y⟩^{}Y, z⟩⟩ and the cut degree of p = (the size w.r.t. s)(y) + (the size w.r.t. s)(z) + ∑((the size w.r.t. s) · (X^{}T ^{} Y)) if p(ε)_{2} = 7,

(ii) the cut degree of p = 0, otherwise.

We adopt the following convention: A denotes an non-empty set and a, a_{1}, a_{2}, b denote elements of A^{∗}. Let us consider s, A. A function from the types of s into 2^{A^{∗}} is said to be a model of s if it satisfies the condition (Def.17).

(Def.17) Given x, y. Then (i) it(x · y) = {a^{}b : a ∈ it(x) ∧ b ∈ it(y)}, (ii) it(x/y) = {a_{1} : ⋀_{b}[b ∈ it(y) ⇒ a_{1} ^{}b ∈ it(x)]}, (iii) it(y \ x) = {a_{2} : ⋀_{b}[b ∈ it(y) ⇒ b^{}a_{2} ∈ it(x)]}.

We consider type structures which are systems

⟨structures of the type algebra; a derivability⟩, where the derivability is a non-empty relation between

(the types of the structure of the type algebra)^{∗}

and the types of the structure of the type algebra.

In the sequel s will denote a type structure and x will denote a type of s. Let us consider s, and let f be a finite sequence of elements of the types of s, and let us consider x. The predicate f ⟶ x is defined by:

(Def.18) ⟨⟨f, x⟩⟩ ∈ the derivability of s.

A type structure is called a calculus of syntactic types if it satisfies the conditions (Def.19).

(Def.19) (i) For every type x of it holds ⟨x⟩ ⟶ x,

(ii) for every finite sequence T of elements of the types of it and for all types x, y of it such that T ^{}⟨y⟩ ⟶ x holds T ⟶ x/y, (iii) for every finite sequence T of elements of the types of it and for all

types x, y of it such that ⟨y⟩^{}T ⟶ x holds T ⟶ y \ x, (iv) for all finite sequences T, X, Y of elements of the types of it and for

all types x, y, z of it such that T ⟶ y and X ^{}⟨x⟩ ^{}Y ⟶ z holds X ^{}⟨x/y⟩^{}T ^{}Y ⟶ z,

(v) for all finite sequences T, X, Y of elements of the types of it and for all types x, y, z of it such that T ⟶ y and X ^{}⟨x⟩ ^{}Y ⟶ z holds X ^{}T ^{}⟨y \ x⟩^{}Y ⟶ z,

(vi) for all finite sequences X, Y of elements of the types of it and for all

types x, y, z of it such that X^{}⟨x⟩^{}⟨y⟩^{}Y ⟶ z holds X^{}⟨x·y⟩^{}Y ⟶ z, (vii) for all finite sequences X, Y of elements of the types of it and for all

types x, y of it such that X ⟶ x and Y ⟶ y holds X ^{}Y ⟶ x · y. In the sequel s will be a calculus of syntactic types and x, y, z will be types

of s. The following propositions are true:

(12) ⟨x/y⟩^{}⟨y⟩ ⟶ x and ⟨y⟩^{}⟨y \ x⟩ ⟶ x.

(13) ⟨x⟩ ⟶ y/(x \ y) and ⟨x⟩ ⟶ y/x \ y.

(14) ⟨x/y⟩ ⟶ x/z/(y/z).

(15) ⟨y \ x⟩ ⟶ z \ y \ (z \ x).

(16) If ⟨x⟩ ⟶ y, then ⟨x/z⟩ ⟶ y/z and ⟨z \ x⟩ ⟶ z \ y.

(17) If ⟨x⟩ ⟶ y, then ⟨z/y⟩ ⟶ z/x and ⟨y \ z⟩ ⟶ x \ z.

(18) ⟨y/(y/x \ y)⟩ ⟶ y/x.

(19) If ⟨x⟩ ⟶ y, then ε_{(the types of s)} ⟶ y/x and ε_{(the types of s)} ⟶ x \ y.

(20) ε_{(the types of s)} ⟶ x/x and ε_{(the types of s)} ⟶ x \ x.

(21) ε_{(the types of s)} ⟶ y/(x \ y)/x and ε_{(the types of s)} ⟶ x \ (y/x \ y).

(22) ε_{(the types of s)} ⟶ x/z/(y/z)/(x/y) and ε_{(the types of s)} ⟶ y \ x \ (z \ y \ (z \ x)).

(23) If ε_{(the types of s)} ⟶ x, then ε_{(the types of s)} ⟶ y/(y/x) and ε_{(the types of s)} ⟶ x \ y \ y.

(24) ⟨x/(y/y)⟩ ⟶ x.

Let us consider s, x, y. The predicate x ⟷ y is defined as follows: (Def.20) ⟨x⟩ ⟶ y and ⟨y⟩ ⟶ x.

Next we state several propositions:

(25) x ⟷ x.

(26) x/y ⟷ x/(x/y \ x).

(27) x/(z · y) ⟷ x/y/z.

(28) ⟨x · (y/z)⟩ ⟶ (x · y)/z.

(29) ⟨x⟩ ⟶ (x · y)/y and ⟨x⟩ ⟶ y \ y · x.

(30) x · y · z ⟷ x · (y · z).

418 Wojciech Zielonka

References

[1] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[3] Grzegorz Bancerek. Introduction to trees. Formalized Mathematics, 1(2):421–427, 1990.

[4] Grzegorz Bancerek. König’s lemma. Formalized Mathematics, 2(3):397–402, 1991.

[5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[6] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107–114, 1990.

[7] Czesław Byliński. Binary operations. Formalized Mathematics, 1(1):175–180, 1990.

[8] Czesław Byliński. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529–536, 1990.

[9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[11] Czesław Byliński. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661–668, 1990.

[12] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[13] Joachim Lambek. The mathematics of sentence structure. American Mathetical Monthly, (65):154–170, 1958.

[14] Michał Muzalewski. Midpoint algebras. Formalized Mathematics, 1(3):483–488, 1990.

[15] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1(2):329–334, 1990.

[16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[17] Andrzej Trybulec. Tuples, projections and Cartesian products. Formalized Mathematics, 1(1):97–105, 1990.

[18] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181–186, 1990.

Opposite Categories and Contravariant Functors

Czesław Byliński Warsaw University Białystok

Summary. The opposite category of a category, contravariant functors and duality functors are defined.

MML Identifier: OPPCAT 1.

The articles [6], [1], [2], [5], [4], and [3] provide the notation and terminology for this paper. In the sequel B, C, D will be categories. Let X be a set, and let C, D be non-empty sets, and let f be a function from X into C, and let g be a function from C into D. Then g · f is a function from X into D.

Let X, Y, Z be non-empty sets, and let f be a partial function from [: X, Y :] to Z. Then f is a partial function from [: Y, X:] to Z.

The following proposition is true

(1) ⟨The objects of C, the morphisms of C, the cod-map of C, the dom-map of C, (the composition of C), the id-map of C⟩ is a category.

Let us consider C. The functor C^{op} yielding a category is defined as follows: (Def.1) C^{op} = ⟨ the objects of C, the morphisms of C, the cod-map of C, the

dom-map of C, (the composition of C), the id-map of C⟩.

One can prove the following proposition

(2) (C^{op})^{op} = C.

Let us consider C, and let c be an object of C. The functor c^{op} yields an object of C^{op} and is defined by: (Def.2) c^{op} = c.

Let us consider C, and let c be an object of C^{op}. The functor ^{op}c yielding an object of C is defined by: (Def.3) ^{op}c = c^{op}.

One can prove the following three propositions:

(3) For every object c of C holds (c^{op})^{op} = c.

(4) For every object c of C holds ^{op}(c^{op}) = c.

(5) For every object c of C^{op} holds (^{op}c)^{op} = c. Let us consider C, and let f be a morphism of C. The functor f^{op} yields a

morphism of C^{op} and is defined as follows: (Def.4) f^{op} = f.

Let us consider C, and let f be a morphism of C^{op}. The functor ^{op}f yields a morphism of C and is defined by: (Def.5) ^{op}f = f^{op}.

One can prove the following propositions:

(6) For every morphism f of C holds (f^{op})^{op} = f.

(7) For every morphism f of C holds ^{op}(f^{op}) = f.

(8) For every morphism f of C^{op} holds (^{op}f)^{op} = f.

(9) For every morphism f of C holds dom(f^{op}) = cod f and cod(f^{op}) = dom f.

(10) For every morphism f of C^{op} holds dom^{op}f = cod f and cod^{op}f = dom f.

(11) For every morphism f of C holds (dom f)^{op} = cod(f^{op}) and (cod f)^{op} = dom(f^{op}).

(12) For every morphism f of C^{op} holds ^{op} dom f = cod^{op}f and ^{op} cod f = dom^{op}f.

(13) For all objects a, b of C and for every morphism f of C holds f ∈ hom(a, b) if and only if f^{op} ∈ hom(b^{op}, a^{op}).

(14) For all objects a, b of C^{op} and for every morphism f of C^{op} holds f ∈ hom(a, b) if and only if ^{op}f ∈ hom(^{op}b,^{op}a).

(15) For all objects a, b of C and for every morphism f from a to b such that hom(a, b) ≠ ∅ holds f^{op} is a morphism from b^{op} to a^{op}.

(16) For all objects a, b of C^{op} and for every morphism f from a to b such that hom(a, b) ≠ ∅ holds ^{op}f is a morphism from ^{op}b to ^{op}a.

(17) For all morphisms f, g of C such that dom g = cod f holds (g · f)^{op} = f^{op} · g^{op}.

(18) For all morphisms f, g of C such that cod(g^{op}) = dom(f^{op}) holds (g · f)^{op} = f^{op} · g^{op}.

(19) For all morphisms f, g of C^{op} such that dom g = cod f holds ^{op}(g · f) = opf ·^{op}g.

(20) For all objects a, b, c of C and for every morphism f from a to b and for every morphism g from b to c such that hom(a, b) ≠ ∅ and hom(b, c) ≠ ∅ holds (g · f)^{op} = f^{op} · g^{op}.

(21) For every object a of C holds id_{a}^{op} = id_{a^{op}}.

(22) For every object a of C^{op} holds ^{op}(id_{a}) = id_{(^{op}a)}.

(23) For every morphism f of C holds f^{op} is monic if and only if f is epi.

(24) For every morphism f of C holds f^{op} is epi if and only if f is monic.

(25) For every morphism f of C holds f^{op} is invertible if and only if f is invertible.

(26) For every object c of C holds c is an initial object if and only if c^{op} is a terminal object.

(27) For every object c of C holds c^{op} is an initial object if and only if c is

a terminal object. Let us consider C, B, and let S be a function from the morphisms of C^{op}

into the morphisms of B. The functor _{∗}S yields a function from the morphisms of C into the morphisms of B and is defined by:

(Def.6) for every morphism f of C holds (_{∗}S)(f) = S(f^{op}).

One can prove the following propositions:

(28) For every function S from the morphisms of C^{op} into the morphisms of B and for every morphism f of C^{op} holds (_{∗}S)(^{op}f) = S(f).

(29) For every functor S from C^{op} to B and for every object c of C holds (Obj_{∗}S)(c) = (Obj S)(c^{op}).

(30) For every functor S from C^{op} to B and for every object c of C^{op} holds

(Obj_{∗}S)(^{op}c) = (Obj S)(c). Let us consider C, D. A function from the morphisms of C into the mor-

phisms of D is called a contravariant functor from C into D if it satisfies the conditions (Def.7).

(Def.7) (i) For every object c of C there exists an object d of D such that

it(id_{c}) = id_{d}, (ii) for every morphism f of C holds it(id_{dom f}) = id_{cod(it(f))} and it(id_{cod f}) =

id_{dom(it(f))}, (iii) for all morphisms f, g of C such that dom g = cod f holds it(g · f) =

it(f) · it(g).

The following propositions are true:

(31) For every contravariant functor S from C into D and for every object c of C and for every object d of D such that S(id_{c}) = id_{d} holds (Obj S)(c) = d.

(32) For every contravariant functor S from C into D and for every object c of C holds S(id_{c}) = id_{(Obj S)(c)}.

(33) For every contravariant functor S from C into D and for every morphism f of C holds (Obj S)(dom f) = cod(S(f)) and (Obj S)(cod f) = dom(S(f)).

(34) For every contravariant functor S from C into D and for all morphisms f, g of C such that dom g = cod f holds dom(S(f)) = cod(S(g)).

(35) For every functor S from C^{op} to B holds _{∗}S is a contravariant functor from C into B.

(36) For every contravariant functor S_{1} from C into B and for every contravariant functor S_{2} from B into D holds S_{2} · S_{1} is a functor from C to D.

(37) For every contravariant functor S from C^{op} into B and for every object c of C holds (Obj_{∗}S)(c) = (Obj S)(c^{op}).

(38) For every contravariant functor S from C^{op} into B and for every object c of C^{op} holds (Obj_{∗}S)(^{op}c) = (Obj S)(c).

(39) For every contravariant functor S from C^{op} into B holds _{∗}S is a functor from C to B.

We now define two new functors. Let us consider C, B, and let S be a function from the morphisms of C into the morphisms of B. The functor ^{∗}S yielding a function from the morphisms of C^{op} into the morphisms of B is defined as follows:

(Def.8) for every morphism f of C^{op} holds (^{∗}S)(f) = S(^{op}f). The functor S^{∗} yields a function from the morphisms of C into the morphisms of B^{op} and is defined by:

(Def.9) for every morphism f of C holds S^{∗}(f) = S(f)^{op}.

The following propositions are true:

(40) For every function S from the morphisms of C into the morphisms of B and for every morphism f of C holds (^{∗}S)(f^{op}) = S(f).

(41) For every functor S from C to B and for every object c of C^{op} holds (Obj^{∗}S)(c) = (Obj S)(^{op}c).

(42) For every functor S from C to B and for every object c of C holds (Obj^{∗}S)(c^{op}) = (Obj S)(c).

(43) For every functor S from C to B and for every object c of C holds (Obj(S^{∗}))(c) = (Obj S)(c)^{op}.

(44) For every contravariant functor S from C into B and for every object c of C^{op} holds (Obj^{∗}S)(c) = (Obj S)(^{op}c).

(45) For every contravariant functor S from C into B and for every object c of C holds (Obj^{∗}S)(c^{op}) = (Obj S)(c).

(46) For every contravariant functor S from C into B and for every object c of C holds (Obj(S^{∗}))(c) = (Obj S)(c)^{op}.

(47) For every function F from the morphisms of C into the morphisms of D and for every morphism f of C holds (^{∗}F)^{∗}(f^{op}) = F (f)^{op}.

(48) For every function S from the morphisms of C into the morphisms of D holds _{∗}^{∗}S = S.

(49) For every function S from the morphisms of C^{op} into the morphisms of D holds ^{∗}_{∗}S = S.

(50) For every function S from the morphisms of C into the morphisms of D holds (^{∗}S)^{∗} = ^{∗}(S^{∗}).

(51) For every function S from the morphisms of C into the morphisms of D holds (S^{∗})^{∗} = S.

(52) For every function S from the morphisms of C into the morphisms of D holds ^{∗}(^{∗}S) = S.

(53) For every function S from the morphisms of C into the morphisms of B and for every function T from the morphisms of B into the morphisms of D holds ^{∗}(T · S) = T ·^{∗}S.

(54) For every function S from the morphisms of C into the morphisms of B and for every function T from the morphisms of B into the morphisms of D holds (T · S)^{∗} = T^{∗} · S.

(55) For every function F_{1} from the morphisms of C into the morphisms of B and for every function F_{2} from the morphisms of B into the morphisms of D holds (^{∗}(F_{2} · F_{1}))^{∗} = (^{∗}F_{2})^{∗} · (^{∗}F_{1})^{∗}.

(56) For every contravariant functor S from C into D holds ^{∗}S is a functor from C^{op} to D.

(57) For every contravariant functor S from C into D holds S^{∗} is a functor from C to D^{op}.

(58) For every functor S from C to D holds ^{∗}S is a contravariant functor from C^{op} into D.

(59) For every functor S from C to D holds S^{∗} is a contravariant functor from C into D^{op}.

(60) For every contravariant functor S_{1} from C into B and for every functor S_{2} from B to D holds S_{2} · S_{1} is a contravariant functor from C into D.

(61) For every functor S_{1} from C to B and for every contravariant functor S_{2} from B into D holds S_{2} · S_{1} is a contravariant functor from C into D.

(62) For every functor F from C to D and for every object c of C holds (Obj((^{∗}F)^{∗}))(c^{op}) = (Obj F)(c)^{op}.

(63) For every contravariant functor F from C into D and for every object c of C holds (Obj((^{∗}F)^{∗}))(c^{op}) = (Obj F)(c)^{op}.

(64) For every functor F from C to D holds (^{∗}F)^{∗} is a functor from C^{op} to D^{op}.

(65) For every contravariant functor F from C into D holds (^{∗}F)^{∗} is a con-

travariant functor from C^{op} into D^{op}. We now define two new functors. Let us consider C. The functor id^{op}(C)

yielding a contravariant functor from C into C^{op} is defined as follows: (Def.10) id^{op}(C) = id_{C}^{∗}.

The functor ^{op}id(C) yielding a contravariant functor from C^{op} into C is defined as follows:

(Def.11) ^{op}id(C) = ^{∗}(id_{C}).

One can prove the following propositions:

(66) For every morphism f of C holds id^{op}(C)(f) = f^{op}.

424 Czesław Byliński

(67) For every object c of C holds (Obj id^{op}(C))(c) = c^{op}.

(68) For every morphism f of C^{op} holds (^{op}id(C))(f) = ^{op}f.

(69) For every object c of C^{op} holds (Obj^{op}id(C))(c) = ^{op}c.

(70) For every function S from the morphisms of C into the morphisms of D holds ^{∗}S = S ·^{op}id(C) and S^{∗} = id^{op}(D) · S.

References

[1] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55– 65, 1990.

[2] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[3] Czesław Byliński. Introduction to categories and functors. Formalized Mathematics, 1(2):409–420, 1990.

[4] Czesław Byliński. The modification of a function by a function and the iteration of the composition of a function. Formalized Mathematics, 1(3):521–527, 1990.

[5] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357–367, 1990.

[6] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

Mostowski’s Fundamental Operations - Part II

Grzegorz Bancerek Warsaw University Białystok Andrzej Kondracki Warsaw University

Summary. The article consists of two parts. The first part is translation of chapter II.3 of [18]. A section of D_{H}(a) determined by f (symbolically S_{H}(a, f)) and a notion of predicative closure of a class are defined. It is proved that if following assumptions are satisfied: (o) A = ⋃_{ξ} A_{ξ}, (i) A_{ξ} ⊂ A_{η} for ξ < η, (ii) A_{λ} = ⋃_{ξ<λ} A_{λ} (λ is a limit number), (iii) A_{ξ} ∈ A, (iv) A_{ξ} is transitive, (v) (x, y ∈ A) → (x ∩ y ∈ A), (vi) A is predicatively closed, then the axiom of power sets and the axiom of substitution are valid in A. The second part is continuation of [17]. It is proved that if a non-void transitive class is closed under the operations A_{1} − A_{7} then it is predicatively closed. At last sufficient criteria for a class to be a model of ZF-theory are formulated: if A_{ξ} satisfies o – iv and A is closed under the operations A_{1} − A_{7} then A is a model of ZF.

MML Identifier: ZF FUND2.

The papers [21], [20], [3], [14], [15], [16], [8], [6], [7], [9], [12], [2], [1], [5], [11], [13], [19], [4], [10], [22], and [17] provide the terminology and notation for this paper. For simplicity we adopt the following rules: H will denote a ZF-formula, M, E will denote non-empty sets, e will denote an element of E, m will denote an element of M, v will denote a function from VAR into M, and f will denote a function from VAR into E. Let us consider H, M, v. The functor S_{v}(H) yields a subset of M and is defined by: (Def.1) (i) S_{v}(H) = \{m : M, v(\frac{x_{0}}{m}) ⊧ H\} if x_{0} ∈ Free H,

(ii) S_{v}(H) = ∅, otherwise.

Let us consider M. We say that M is predicatively closed if and only if: (Def.2) for all H, E, f such that E ∈ M holds S_{f}(H) ∈ M.

We now state the proposition

(1) If E is transitive, then S_{f( \frac{x_{1}}{e})}(∀_{x_{2}}(x_{2}ϵ(x_{0}) ⇒ x_{2}ϵ(x_{1}))) = E ∩ 2^{e}.

For simplicity we adopt the following convention: W denotes a universal class, Y denotes a subclass of W, a, b denote ordinals of W, and L denotes a transfinite sequence of non-empty sets from W. We now state several propositions:

(2) If for all a, b such that a ∈ b holds L(a) ⊆ L(b) and for every a holds L(a) ∈ ⋃ L and L(a) is transitive and ⋃ L is predicatively closed, then ⋃ L ⊧ the axiom of power sets.

(3) Suppose that

(i) ω ∈ W, (ii) for all a, b such that a ∈ b holds L(a) ⊆ L(b), (iii) for every a such that a ≠ 0 and a is a limit ordinal number holds

L(a) = ⋃(L a), (iv) for every a holds L(a) ∈ ⋃ L and L(a) is transitive,

(v) ⋃ L is predicatively closed. Then for every H such that {x_{0}, x_{1}, x_{2}} misses Free H holds ⋃ L ⊧ the axiom of substitution for H.

(4) S_{v}(H) = {m : {⟨⟨0, m⟩⟩} ∪ (v · decode) (code(Free H) \ {0}) ∈ D_{M}(H)}.

(5) If Y is closed w.r.t. A1-A7 and Y is transitive, then Y is predicatively closed.

(6) Suppose that

(i) ω ∈ W, (ii) for all a, b such that a ∈ b holds L(a) ⊆ L(b), (iii) for every a such that a ≠ 0 and a is a limit ordinal number holds

L(a) = ⋃(L a), (iv) for every a holds L(a) ∈ ⋃ L and L(a) is transitive,

(v) ⋃ L is closed w.r.t. A1-A7. Then ⋃ L is a model of ZF.

References

[1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543–547, 1990.

[2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377–382, 1990.

[3] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.

[4] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711–714, 1990.

[5] Grzegorz Bancerek. König’s theorem. Formalized Mathematics, 1(3):589–593, 1990.

[6] Grzegorz Bancerek. A model of ZF set theory language. Formalized Mathematics, 1(1):131–145, 1990.

[7] Grzegorz Bancerek. Models and satisfiability. Formalized Mathematics, 1(1):191–199, 1990.

[8] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.

[9] Grzegorz Bancerek. Properties of ZF models. Formalized Mathematics, 1(2):277–280, 1990.

[10] Grzegorz Bancerek. The reflection theorem. Formalized Mathematics, 1(5):973–977, 1990.

[11] Grzegorz Bancerek. Replacing of variables in formulas of ZF theory. Formalized Mathematics, 1(5):963–972, 1990.

Mostowski’s Fundamental Operations - Part … 427

[12] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281– 290, 1990.

[13] Grzegorz Bancerek. Tarski’s classes and ranks. Formalized Mathematics, 1(3):563–567, 1990.

[14] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55–65, 1990.

[15] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153–164, 1990.

[16] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165–167, 1990.

[17] Andrzej Kondracki. Mostowski’s fundamental operations - Part I. Formalized Mathematics, 2(3):371–375, 1991.

[18] Andrzej Mostowski. Constructible Sets with Applications. North Holland, 1969.

[19] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595–600, 1990.

[20] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25–34, 1990.

[21] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[22] Andrzej Trybulec and Agata Darmochwał. Boolean domains. Formalized Mathematics, 1(1):187–190, 1990.

428

Fundamental Types of Metric Affine Spaces

Henryk Oryszczyszyn Warsaw University Białystok Krzysztof Prażmowski Warsaw University Białystok

Summary. We distinguish in the class of metric affine spaces some fundamental types of them. First we can assume the underlying affine space to satisfy classical affine configurational axiom; thus we come to Pappian, Desarguesian, Moufangian, and translation spaces. Next we distinguish the spaces satisfying theorem on three perpendiculars and the homogeneous spaces; these properties directly refer to some axioms involving orthogonality. Some known relationships between the introduced classes of structures are established. We also show that the commonly investigated models of metric affine geometry constructed in a real linear space with the help of a symmetric bilinear form belong to all the classes introduced in the paper.

MML Identifier: EUCLMETR.

The papers [1], [3], [5], [6], [2], [4], [7], [8], and [9] provide the notation and terminology for this paper. A metric affine space is Euclidean if: (Def.1) for all elements a, b, c, d of the points of it such that a, b ⊥ c, d and

b, c ⊥ a, d holds b, d ⊥ a, c. A metric affine space is Pappian if:

(Def.2) the affine reduct of it is Pappian.

A metric affine space is Desarguesian if: (Def.3) the affine reduct of it is Desarguesian.

A metric affine space is Fanoian if: (Def.4) the affine reduct of it is Fanoian.

A metric affine space is Moufangian if: (Def.5) the affine reduct of it is Moufangian.

A metric affine space is translation if: (Def.6) the affine reduct of it is translation.

A metric affine space is homogeneous if it satisfies the condition (Def.7).

(Def.7) Let o, a, a_{1}, b, b_{1}, c, c_{1} be elements of the points of it . Then if o, a ⊥ o, a_{1} and o, b ⊥ o, b_{1} and o, c ⊥ o, c_{1} and a, b ⊥ a_{1}, b_{1} and a, c ⊥ a_{1}, c_{1} and o, c o, a and o, a o, b, then b, c ⊥ b_{1}, c_{1}.

In the sequel M_{1} denotes a metric affine plane and M_{2} denotes a metric affine space. The following propositions are true:

(1) For all elements a, b, c of the points of M_{2} such that not L(a, b, c) holds a ≠ b and b ≠ c and a ≠ c.

(2) For all elements a, b, c, d of the points of M_{1} and for every subset K of the points of M_{1} such that a, b ⊥ K and c, d ⊥ K holds a, b ∥ c, d and a, b ∥ d, c.

(3) For all elements a, b of the points of M_{1} and for all subsets A, K of the points of M_{1} such that a ≠ b but a, b ⊥ K or b, a ⊥ K but a, b ⊥ A or b, a ⊥ A holds K ∥ A.

(4) For all elements x, y, z of the points of M_{2} such that L(x, y, z) holds L(x, z, y) and L(y, x, z) and L(y, z, x) and L(z, x, y) and L(z, y, x).

(5) For all elements a, b, c of the points of M_{1} such that not L(a, b, c) there exists an element d of the points of M_{1} such that d, a ⊥ b, c and d, b ⊥ a, c.

(6) For all elements a, b, c, d_{1}, d_{2} of the points of M_{1} such that not L(a, b, c) and d_{1}, a ⊥ b, c and d_{1}, b ⊥ a, c and d_{2}, a ⊥ b, c and d_{2}, b ⊥ a, c holds d_{1} = d_{2}.

(7) For all elements a, b, c, d of the points of M_{1} such that a, b ⊥ c, d and b, c ⊥ a, d and L(a, b, c) holds a = c or a = b or b = c.

(8) M_{1} is Euclidean if and only if theorem on three perpendiculars holds in M_{1}.

(9) M_{1} is homogeneous if and only if othogonal verion of Desargues Axiom holds in M_{1}.

(10) M_{1} is Pappian if and only if Pappos Axiom holds in M_{1}.

(11) M_{1} is Desarguesian if and only if Desargues Axiom holds in M_{1}.

(12) M_{1} is Moufangian if and only if trapezium variant of Desargues Axiom holds in M_{1}.

(13) M_{1} is translation if and only if minor Desargues Axiom holds in M_{1}.

(14) If M_{1} is homogeneous, then M_{1} is Desarguesian.

(15) If M_{1} is Euclidean Desarguesian, then M_{1} is Pappian.

We adopt the following rules: V will denote a real linear space and w, y, u, v will denote vectors of V. The following propositions are true:

(16) Let o, c, c_{1}, a, a_{1}, a_{2} be elements of the points of M_{1}. Then if not L(o, c, a) and o ≠ c_{1} and o, c ⊥ o, c_{1} and o, a ⊥ o, a_{1} and o, a ⊥ o, a_{2} and c, a ⊥ c_{1}, a_{1} and c, a ⊥ c_{1}, a_{2}, then a_{1} = a_{2}.

(17) For all elements o, c, c_{1}, a of the points of M_{1} such that not L(o, c, a) and o ≠ c_{1} and o, c ⊥ o, c_{1} there exists an element a_{1} of the points of M_{1} such that o, a ⊥ o, a_{1} and c, a ⊥ c_{1}, a_{1}.

(18) Let a, b be real numbers. Suppose w, y span the space and 0_{V} ≠ u and 0_{V} ≠ v and u, v are orthogonal w.r.t. w, y and u = a · w + b · y. Then there exists a real number c such that c ≠ 0 and v = c · b · w + (−c · a) · y.

(19) Suppose w, y span the space and 0_{V} ≠ u and 0_{V} ≠ v and u, v are orthogonal w.r.t. w, y. Then there exists a real number c such that for all real numbers a, b holds a · w + b · y, c · b · w + (−c · a) · y are orthogonal w.r.t. w, y and (a · w + b · y) − u, (c · b · w + (−c · a) · y) − v are orthogonal w.r.t. w, y.

(20) If w, y span the space and M_{1} = AMSp(V, w, y), then for an arbitrary x holds x is a vector of V if and only if x is an element of the points of M_{1}.

(21) If w, y span the space and M_{1} = AMSp(V, w, y), then LIN holds in M_{1}.

(22) Suppose w, y span the space and M_{1} = AMSp(V, w, y). Let o, a, a_{1}, b, b_{1}, c, c_{1} be elements of the points of M_{1}. Suppose o, a ⊥ o, a_{1} and o, b ⊥ o, b_{1} and o, c ⊥ o, c_{1} and a, b ⊥ a_{1}, b_{1} and a, c ⊥ a_{1}, c_{1} and o, c o, a and o, a o, b and o = a_{1}. Then b, c ⊥ b_{1}, c_{1}.

(23) If w, y span the space and M_{1} = AMSp(V, w, y), then M_{1} is homogeneous.

The following proposition is true

(24) If w, y span the space and M_{1} = AMSp(V, w, y), then M_{1} is a metric

affine plane. Let M_{1} be an Pappian metric affine plane. Then the affine reduct of M_{1} is a

Pappian affine plane.

Let M_{1} be a Desarguesian metric affine plane. Then the affine reduct of M_{1} is a Desarguesian affine plane.

Let M_{1} be a Moufangian metric affine plane. Then the affine reduct of M_{1} is a Moufangian affine plane.

Let M_{1} be a translation metric affine plane. Then the affine reduct of M_{1} is an translation affine plane.

Let M_{1} be an Fanoian metric affine plane. Then the affine reduct of M_{1} is a Fanoian affine plane.

Let M_{1} be a homogeneous metric affine plane. Then the affine reduct of M_{1} is an Desarguesian affine plane.

Let M_{1} be a Euclidean Desarguesian metric affine plane. Then

the affine reduct of M_{1}

is a Pappian affine plane. Let M_{1} be an Pappian metric affine space. Then the affine reduct of M_{1} is a

Pappian affine space.

Let M_{1} be a Desarguesian metric affine space. Then the affine reduct of M_{1} is a Desarguesian affine space.

432 Henryk Oryszczyszyn et al.

Let M_{1} be an Moufangian metric affine space. Then the affine reduct of M_{1} is an Moufangian affine space.

Let M_{1} be a translation metric affine space. Then the affine reduct of M_{1} is a translation affine space.

Let M_{1} be a Fanoian metric affine space. Then the affine reduct of M_{1} is a Fanoian affine space.

References

[1] Krzysztof Hryniewiecki. Basic properties of real numbers. Formalized Mathematics, 1(1):35–40, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical metric affine spaces and planes. Formalized Mathematics, 1(5):891–899, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. A construction of analytical ordered trapezium spaces. Formalized Mathematics, 2(3):315–322, 1991.

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[6] Henryk Oryszczyszyn and Krzysztof Prażmowski. Parallelity and lines in affine spaces. Formalized Mathematics, 1(3):617–621, 1990.

[7] Krzysztof Prażmowski. Fanoian, Pappian and Desarguesian affine spaces. Formalized Mathematics, 2(3):341–346, 1991.

[8] Jolanta Świerzyńska and Bogdan Świerzyński. Metric-affine configurations in metric affine planes - Part II. Formalized Mathematics, 2(3):335–340, 1991.

[9] Jolanta Świerzyńska and Bogdan Świerzyński. Metric-affine configurations in metric affine planes - Part I. Formalized Mathematics, 2(3):331–334, 1991.

Filters - Part II.

Quotient Lattices Modulo Filters and Direct Product of Two Lattices

Grzegorz Bancerek Warsaw University Białystok

Summary. Binary and unary operation preserving binary relations and quotients of those operations modulo equivalence relations are introduced. It is shown that the quotients inherit some important properties (commutativity, associativity, distributivity, ect.). Based on it the quotient (also called factor) lattice modulo filter (ie. modulo the equivalence relation w.r.t the filter) is introduced. Similarly, some properties of the direct product of two binary (unary) operations are presented and then the direct product of two lattices is introduced. Besides, the heredity of distributivity, modularity, completeness, etc., for the product of lattices is also shown. Finally, the concept of isomorphic lattices is introduced, and it is shown that every Boolean lattice B is isomorphic with the direct product of the factor lattice B/[a] and the lattice latt[a], where a is an element of B.

MML Identifier: FILTER 1.

The notation and terminology used in this paper are introduced in the following papers: [11], [5], [6], [13], [4], [8], [12], [9], [2], [3], [7], [14], [1], and [10]. Let L be a lattice structure. An element of L is an element of the carrier of L.

For simplicity we adopt the following convention: L, L_{1}, L_{2} denote lattices, F_{1}, F_{2} denote filters of L, p, q denote elements of L, p_{1}, q_{1} denote elements of L_{1}, p_{2}, q_{2} denote elements of L_{2}, x, x_{1}, y, y_{1} are arbitrary, D, D_{1}, D_{2} denote non-empty sets, R denotes a binary relation, R_{1} denotes an equivalence relation of D, a, b, d denote elements of D, a_{1}, b_{1} denote elements of D_{1}, a_{2}, b_{2} denote elements of D_{2}, B denotes a boolean lattice, F_{3} denotes a filter of B, I denotes an implicative lattice, F_{4} denotes a filter of I, i, i_{1}, i_{2}, j, j_{1}, j_{2}, k denote elements of I, f_{1}, g_{1} denote binary operations on D_{1}, and f_{2}, g_{2} denote binary operations on D_{2}. One can prove the following two propositions:

(1) F_{1} ∩ F_{2} is a filter of L.

(2) If [p] = [q], then p = q. Let us consider L, F_{1}, F_{2}. Then F_{1} ∩ F_{2} is a filter of L. We now define two new modes. Let us consider D, R. A unary operation on

D is called a unary R-congruent operation on D if: (Def.1) for all elements x, y of D such that ⟨⟨x, y⟩⟩ ∈ R holds ⟨⟨it(x), it(y)⟩⟩ ∈ R. A binary operation on D is called a binary R-congruent operation on D if: (Def.2) for all elements x_{1}, y_{1}, x_{2}, y_{2} of D such that ⟨⟨x_{1}, y_{1}⟩⟩ ∈ R and ⟨⟨x_{2}, y_{2}⟩⟩ ∈

R holds ⟨⟨it(x_{1}, x_{2}), it(y_{1}, y_{2})⟩⟩ ∈ R. In the sequel F, G denote binary R_{1}-congruent operations on D. We now

define two new modes. Let us consider D, and let R be an equivalence relation of D. A unary operation on R is a unary R-congruent operation on D.

A binary operation on R is a binary R-congruent operation on D. Then Classes R is an non-empty subset of 2^{D}. Let X be a set, and let S be a non-empty subset of 2^{X}. We see that the

element of S is a subset of X.

Let us consider D, and let R be an equivalence relation of D, and let d be an element of D. Then [d]_{R} is an element of Classes R.

Let us consider D, and let R be an equivalence relation of D, and let u be a unary operation on D. Let us assume that u is a unary R-congruent operation on D. The functor u_{/R} yielding a unary operation on Classes R is defined as follows: (Def.3) for all x, y such that x ∈ Classes R and y ∈ x holds u_{/R}(x) = [u(y)]_{R}.

Let us consider D, and let R be an equivalence relation of D, and let b be a binary operation on D. Let us assume that b is a binary R-congruent operation on D. The functor b_{/R} yields a binary operation on Classes R and is defined by: (Def.4) for all x, y, x_{1}, y_{1} such that x ∈ Classes R and y ∈ Classes R and x_{1} ∈ x

and y_{1} ∈ y holds b_{/R}(x, y) = [b(x_{1}, y_{1})]_{R}. We now state the proposition

(3) F_{/R_{1}}([a]_{R_{1}}, [b]_{R_{1}}) = [F (a, b)]_{R_{1}}. The following propositions are true:

(4) If F is commutative, then F_{/R_{1}} is commutative.

(5) If F is associative, then F_{/R_{1}} is associative.

(6) If d is a left unity w.r.t. F, then [d]_{R_{1}} is a left unity w.r.t. F_{/R_{1}}.

(7) If d is a right unity w.r.t. F, then [d]_{R_{1}} is a right unity w.r.t. F_{/R_{1}}.

(8) If d is a unity w.r.t. F, then [d]_{R_{1}} is a unity w.r.t. F_{/R_{1}}.

(9) If F is left distributive w.r.t. G, then F_{/R_{1}} is left distributive w.r.t. G_{/R_{1}}.

(10) If F is right distributive w.r.t. G, then F_{/R_{1}} is right distributive w.r.t. G_{/R_{1}}.

(11) If F is distributive w.r.t. G, then F_{/R_{1}} is distributive w.r.t. G_{/R_{1}}.

(12) If F absorbs G, then F_{/R_{1}} absorbs G_{/R_{1}}.

(13) The join operation of I is a binary ≡_{F_{4}}-congruent operation on the carrier of I.

(14) The meet operation of I is a binary ≡_{F_{4}}-congruent operation on the

carrier of I. Let L be a lattice, and let F be a filter of L. Let us assume that L is an

implicative lattice. The functor L_{/F} yields a lattice and is defined as follows: (Def.5) for every equivalence relation R of the carrier of L such that R = ≡_{F}

holds L_{/F} = ⟨Classes R, (the join operation of L)_{/R}, (the meet operation of L)_{/R}⟩.

Let L be a lattice, and let F be a filter of L, and let a be an element of L. Let us assume that L is an implicative lattice. The functor a_{/F} yielding an element of L_{/F} is defined as follows: (Def.6) for every equivalence relation R of the carrier of L such that R = ≡_{F}

holds a_{/F} = [a]_{R}. Next we state several propositions:

(15) i_{/F_{4}} ⊔ j_{/F_{4}} = (i ⊔ j)_{/F_{4}} and i_{/F_{4}} ⊓ j_{/F_{4}} = (i ⊓ j)_{/F_{4}}.

(16) i_{/F_{4}} ⊑ j_{/F_{4}} if and only if i ⇒ j ∈ F_{4}.

(17) i ⊓ j ⇒ k = i ⇒ (j ⇒ k).

(18) If I is a lower bound lattice, then I_{/F_{4}} is a lower bound lattice and ⊥_{I_{/F_{4}}} = (⊥_{I})_{/F_{4}}.

(19) I_{/F_{4}} is an upper bound lattice and ⊤_{I_{/F_{4}}} = (⊤_{I})_{/F_{4}}.

(20) I_{/F_{4}} is an implicative lattice.

(21) B_{/F_{3}} is a boolean lattice.

Let D_{1}, D_{2} be non-empty sets, and let f_{1} be a binary operation on D_{1}, and let f_{2} be a binary operation on D_{2}. Then |:f_{1}, f_{2}:| is a binary operation on [: D_{1}, D_{2} :].

We now state the proposition

(22) |:f_{1}, f_{2}:|(⟨⟨a_{1}, a_{2}⟩⟩, ⟨⟨b_{1}, b_{2}⟩⟩) = ⟨⟨f_{1}(a_{1}, b_{1}), f_{2}(a_{2}, b_{2})⟩⟩.

One can prove the following propositions:

(23) f_{1} is commutative and f_{2} is commutative if and only if |:f_{1}, f_{2}:| is commutative.

(24) f_{1} is associative and f_{2} is associative if and only if |:f_{1}, f_{2}:| is associative.

(25) a_{1} is a left unity w.r.t. f_{1} and a_{2} is a left unity w.r.t. f_{2} if and only if ⟨⟨a_{1}, a_{2}⟩⟩ is a left unity w.r.t. |:f_{1}, f_{2}:|.

(26) a_{1} is a right unity w.r.t. f_{1} and a_{2} is a right unity w.r.t. f_{2} if and only if ⟨⟨a_{1}, a_{2}⟩⟩ is a right unity w.r.t. |:f_{1}, f_{2}:|.

(27) a_{1} is a unity w.r.t. f_{1} and a_{2} is a unity w.r.t. f_{2} if and only if ⟨⟨a_{1}, a_{2}⟩⟩ is a unity w.r.t. |:f_{1}, f_{2}:|.

(28) f_{1} is left distributive w.r.t. g_{1} and f_{2} is left distributive w.r.t. g_{2} if and only if |:f_{1}, f_{2}:| is left distributive w.r.t. |:g_{1}, g_{2}:|.

(29) f_{1} is right distributive w.r.t. g_{1} and f_{2} is right distributive w.r.t. g_{2} if and only if |:f_{1}, f_{2}:| is right distributive w.r.t. |:g_{1}, g_{2}:|.

(30) f_{1} is distributive w.r.t. g_{1} and f_{2} is distributive w.r.t. g_{2} if and only if |:f_{1}, f_{2}:| is distributive w.r.t. |:g_{1}, g_{2}:|.

(31) f_{1} absorbs g_{1} and f_{2} absorbs g_{2} if and only if |:f_{1}, f_{2}:| absorbs |:g_{1}, g_{2}:|. Let L_{1}, L_{2} be lattice structures. The functor [: L_{1}, L_{2} :] yielding a lattice

structure is defined by: (Def.7) [: L_{1}, L_{2} :] = ⟨[: the carrier of L_{1}, the carrier of L_{2} :], |: the join operation

of L_{1}, the join operation of L_{2}:|, |: the meet operation of L_{1}, the meet operation of L_{2}:|⟩.

Let L be a lattice. The functor LattRel(L) yields a binary relation and is defined as follows: (Def.8) LattRel(L) = {⟨⟨p, q⟩⟩ : p ⊑ q}, where p ranges over elements of the

carrier of L, and q ranges over elements of the carrier of L. We now state two propositions:

(32) ⟨⟨p, q⟩⟩ ∈ LattRel(L) if and only if p ⊑ q.

(33) dom LattRel(L) = the carrier of L and rng LattRel(L) = the carrier of

L and field LattRel(L) = the carrier of L. Let L_{1}, L_{2} be lattices. We say that L_{1} and L_{2} are isomorphic if and only if:

(Def.9) LattRel(L_{1}) and LattRel(L_{2}) are isomorphic. Let us notice that the predicate introduced above is reflexive and symmetric. Then [: L_{1}, L_{2} :] is a lattice.

Next we state two propositions:

(34) For all lattices L_{1}, L_{2}, L_{3} such that L_{1} and L_{2} are isomorphic and L_{2} and L_{3} are isomorphic holds L_{1} and L_{3} are isomorphic.

(35) For all L_{1}, L_{2} being lattice structures such that [: L_{1}, L_{2} :] is a lattice

holds L_{1} is a lattice and L_{2} is a lattice. Let L_{1}, L_{2} be lattices, and let a be an element of L_{1}, and let b be an element

of L_{2}. Then ⟨⟨a, b⟩⟩ is an element of [: L_{1}, L_{2} :].

The following propositions are true:

(36) ⟨⟨p_{1}, p_{2}⟩⟩⊔⟨⟨q_{1}, q_{2}⟩⟩ = ⟨⟨p_{1} ⊔q_{1}, p_{2} ⊔q_{2}⟩⟩ and ⟨⟨p_{1}, p_{2}⟩⟩⊓⟨⟨q_{1}, q_{2}⟩⟩ = ⟨⟨p_{1} ⊓q_{1}, p_{2} ⊓ q_{2}⟩⟩.

(37) ⟨⟨p_{1}, p_{2}⟩⟩ ⊑ ⟨⟨q_{1}, q_{2}⟩⟩ if and only if p_{1} ⊑ q_{1} and p_{2} ⊑ q_{2}.

(38) L_{1} is a modular lattice and L_{2} is a modular lattice if and only if [: L_{1}, L_{2} :] is a modular lattice.

(39) L_{1} is a distributive lattice and L_{2} is a distributive lattice if and only if [: L_{1}, L_{2} :] is a distributive lattice.

(40) L_{1} is a lower bound lattice and L_{2} is a lower bound lattice if and only if [: L_{1}, L_{2} :] is a lower bound lattice.

(41) L_{1} is an upper bound lattice and L_{2} is an upper bound lattice if and only if [: L_{1}, L_{2} :] is an upper bound lattice.

(42) L_{1} is a bound lattice and L_{2} is a bound lattice if and only if [: L_{1}, L_{2} :] is a bound lattice.

(43) If L_{1} is a lower bound lattice and L_{2} is a lower bound lattice, then ⊥_{[: L_{1}, L_{2} :]} = ⟨⟨⊥_{L_{1}}, ⊥_{L_{2}} ⟩⟩.

(44) If L_{1} is an upper bound lattice and L_{2} is an upper bound lattice, then ⊤_{[: L_{1}, L_{2} :]} = ⟨⟨⊤_{L_{1}}, ⊤_{L_{2}} ⟩⟩.

(45) If L_{1} is a bound lattice and L_{2} is a bound lattice, then p_{1} is a complement of q_{1} and p_{2} is a complement of q_{2} if and only if ⟨⟨p_{1}, p_{2}⟩⟩ is a complement of ⟨⟨q_{1}, q_{2}⟩⟩.

(46) L_{1} is a complemented lattice and L_{2} is a complemented lattice if and only if [: L_{1}, L_{2} :] is a complemented lattice.

(47) L_{1} is a boolean lattice and L_{2} is a boolean lattice if and only if [: L_{1}, L_{2} :] is a boolean lattice.

(48) L_{1} is an implicative lattice and L_{2} is an implicative lattice if and only if [: L_{1}, L_{2} :] is an implicative lattice.

(49) [: L_{1}, L_{2} :]^{∘} = [: L_{1}^{∘}, L_{2}^{∘}:].

(50) [: L_{1}, L_{2} :] and [: L_{2}, L_{1} :] are isomorphic.

We follow the rules: B will be a boolean lattice and a, b, c, d will be elements of B. One can prove the following propositions:

(51) a ⇔ b = a ⊓ b ⊔ a^{c} ⊓ b^{c}.

(52) (a ⇒ b)^{c} = a ⊓ b^{c} and (a ⇔ b)^{c} = a ⊓ b^{c} ⊔ a^{c} ⊓ b and (a ⇔ b)^{c} = a ⇔ b^{c} and (a ⇔ b)^{c} = a^{c} ⇔ b.

(53) If a ⇔ b = a ⇔ c, then b = c.

(54) a ⇔ (a ⇔ b) = b.

(55) i ⊔ j ⇒ i = j ⇒ i and i ⇒ i ⊓ j = i ⇒ j.

(56) i ⇒ j ⊑ i ⇒ j ⊔ k and i ⇒ j ⊑ i ⊓ k ⇒ j and i ⇒ j ⊑ i ⇒ k ⊔ j and i ⇒ j ⊑ k ⊓ i ⇒ j.

(57) (i ⇒ k) ⊓ (j ⇒ k) ⊑ i ⊔ j ⇒ k.

(58) (i ⇒ j) ⊓ (i ⇒ k) ⊑ i ⇒ j ⊓ k.

(59) If i_{1} ⇔ i_{2} ∈ F_{4} and j_{1} ⇔ j_{2} ∈ F_{4}, then i_{1} ⊔ j_{1} ⇔ i_{2} ⊔ j_{2} ∈ F_{4} and i_{1} ⊓ j_{1} ⇔ i_{2} ⊓ j_{2} ∈ F_{4}.

(60) If i ∈ [k]_{≡_{F_{4}}} and j ∈ [k]_{≡_{F_{4}}} , then i ⊔ j ∈ [k]_{≡_{F_{4}}} and i ⊓ j ∈ [k]_{≡_{F_{4}}} .

(61) c ⊔ (c ⇔ d) ∈ [c]_{≡_{[d]}} and for every b such that b ∈ [c]_{≡_{[d]}} holds b ⊑ c ⊔ (c ⇔ d).

(62) B and [: B_{/[a]}, _{[a]} :] are isomorphic.

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[11] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9–11, 1990.

[12] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.

[13] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73–83, 1990.

[14] Stanisław Żukowski. Introduction to lattice theory. Formalized Mathematics, 1(1):215– 222, 1990.

Shear Theorems and Their Role in Affine Geometry

Jolanta Świerzyńska Warsaw University Białystok Bogdan Świerzyński Warsaw University Białystok

Summary. Investigations on affine shear theorems, major and minor, direct and indirect. We prove logical relationships which hold between these statements and between them and other classical affine configurational axioms (eg. minor and major Pappus Axiom, Desargues Axioms et al.). For the shear, Desargues and Pappus Axioms formulated in terms of metric affine spaces we prove that they are equivalent to corresponding statements formulated in terms of affine reduct of the given space.

MML Identifier: CONMETR1.

The terminology and notation used in this paper have been introduced in the following papers: [2], [4], [1], [3], [6], [7], and [5]. We follow a convention: X will be an affine plane, o, a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4} will be elements of the points of X, and M, N will be subsets of the points of X. Let us consider X. We say that X satisfies minor Scherungssatz if and only if the condition (Def.1) is satisfied. (Def.1) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M ∥ N, (ii) a_{1} ∈ M, (iii) a_{3} ∈ M, (iv) b_{1} ∈ M,

(v) b_{3} ∈ M, (vi) a_{2} ∈ N, (vii) a_{4} ∈ N, (viii) b_{2} ∈ N, (ix) b_{4} ∈ N,

(x) a_{4} ∉ M, (xi) a_{2} ∉ M, (xii) b_{2} ∉ M, (xiii) b_{4} ∉ M, (xiv) a_{1} ∉ N, (xv) a_{3} ∉ N, (xvi) b_{1} ∉ N, (xvii) b_{3} ∉ N, (xviii) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xix) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xx) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}.

Let us consider X. We say that X satisfies major Scherungssatz if and only if the condition (Def.2) is satisfied. (Def.2) Given o, a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M is a line, (ii) N is a line, (iii) o ∈ M, (iv) o ∈ N,

(v) a_{1} ∈ M, (vi) a_{3} ∈ M, (vii) b_{1} ∈ M, (viii) b_{3} ∈ M, (ix) a_{2} ∈ N,

(x) a_{4} ∈ N, (xi) b_{2} ∈ N, (xii) b_{4} ∈ N, (xiii) a_{4} ∉ M, (xiv) a_{2} ∉ M, (xv) b_{2} ∉ M, (xvi) b_{4} ∉ M, (xvii) a_{1} ∉ N, (xviii) a_{3} ∉ N, (xix) b_{1} ∉ N, (xx) b_{3} ∉ N, (xxi) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xxii) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xxiii) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}.

Let us consider X. We say that X satisfies Scherungssatz if and only if the condition (Def.3) is satisfied. (Def.3) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M is a line, (ii) N is a line, (iii) a_{1} ∈ M, (iv) a_{3} ∈ M,

(v) b_{1} ∈ M, (vi) b_{3} ∈ M, (vii) a_{2} ∈ N, (viii) a_{4} ∈ N, (ix) b_{2} ∈ N,

(x) b_{4} ∈ N, (xi) a_{4} ∉ M, (xii) a_{2} ∉ M, (xiii) b_{2} ∉ M, (xiv) b_{4} ∉ M, (xv) a_{1} ∉ N, (xvi) a_{3} ∉ N, (xvii) b_{1} ∉ N, (xviii) b_{3} ∉ N, (xix) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xx) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xxi) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}.

Let us consider X. We say that X satisfies Scherungssatz^{∗} if and only if the condition (Def.4) is satisfied. (Def.4) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M is a line, (ii) N is a line, (iii) a_{1} ∈ M, (iv) a_{3} ∈ M,

(v) b_{2} ∈ M, (vi) b_{4} ∈ M, (vii) a_{2} ∈ N, (viii) a_{4} ∈ N, (ix) b_{1} ∈ N,

(x) b_{3} ∈ N, (xi) a_{4} ∉ M, (xii) a_{2} ∉ M, (xiii) b_{1} ∉ M, (xiv) b_{3} ∉ M, (xv) a_{1} ∉ N, (xvi) a_{3} ∉ N, (xvii) b_{2} ∉ N, (xviii) b_{4} ∉ N, (xix) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xx) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xxi) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}.

Let us consider X. We say that X satisfies minor Scherungssatz^{∗} if and only if the condition (Def.5) is satisfied. (Def.5) Given a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M ∥ N, (ii) a_{1} ∈ M, (iii) a_{3} ∈ M, (iv) b_{2} ∈ M,

(v) b_{4} ∈ M, (vi) a_{2} ∈ N, (vii) a_{4} ∈ N, (viii) b_{1} ∈ N, (ix) b_{3} ∈ N,

(x) a_{4} ∉ M, (xi) a_{2} ∉ M, (xii) b_{1} ∉ M, (xiii) b_{3} ∉ M, (xiv) a_{1} ∉ N, (xv) a_{3} ∉ N, (xvi) b_{2} ∉ N, (xvii) b_{4} ∉ N, (xviii) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xix) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xx) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}.

Let us consider X. We say that X satisfies major Scherungssatz^{∗} if and only if the condition (Def.6) is satisfied. (Def.6) Given o, a_{1}, a_{2}, a_{3}, a_{4}, b_{1}, b_{2}, b_{3}, b_{4}, M, N. Suppose that

(i) M is a line, (ii) N is a line, (iii) o ∈ M, (iv) o ∈ N,

(v) a_{1} ∈ M, (vi) a_{3} ∈ M, (vii) b_{2} ∈ M, (viii) b_{4} ∈ M, (ix) a_{2} ∈ N,

(x) a_{4} ∈ N, (xi) b_{1} ∈ N, (xii) b_{3} ∈ N, (xiii) a_{4} ∉ M, (xiv) a_{2} ∉ M, (xv) b_{1} ∉ M, (xvi) b_{3} ∉ M, (xvii) a_{1} ∉ N, (xviii) a_{3} ∉ N, (xix) b_{2} ∉ N, (xx) b_{4} ∉ N, (xxi) a_{3}, a_{2} ∥ b_{3}, b_{2}, (xxii) a_{2}, a_{1} ∥ b_{2}, b_{1}, (xxiii) a_{1}, a_{4} ∥ b_{1}, b_{4}.

Then a_{3}, a_{4} ∥ b_{3}, b_{4}. Next we state a number of propositions:

(1) X satisfies Scherungssatz^{∗} if and only if X satisfies minor Scherungssatz^{∗} and X satisfies major Scherungssatz^{∗}.

(2) X satisfies Scherungssatz if and only if X satisfies minor Scherungssatz and X satisfies major Scherungssatz.

(3) If X satisfies minor Scherungssatz^{∗}, then X satisfies minor Scherungssatz.

(4) If X satisfies major Scherungssatz^{∗}, then X satisfies major Scherungssatz.

(5) If X satisfies Scherungssatz^{∗}, then X satisfies Scherungssatz.

(6) If X satisfies des, then X satisfies minor Scherungssatz.

(7) If X satisfies DES, then X satisfies major Scherungssatz.

(8) X satisfies DES if and only if X satisfies Scherungssatz.

(9) X satisfies pap if and only if X satisfies minor Scherungssatz^{∗}.

(10) X satisfies PAP if and only if X satisfies major Scherungssatz^{∗}.

(11) X satisfies PPAP if and only if X satisfies Scherungssatz^{∗}.

(12) If X satisfies major Scherungssatz^{∗}, then X satisfies minor Scherungssatz

∗. In the sequel X denotes a metric affine plane. We now state several propo-

sitions:

(13) The affine reduct of X satisfies Scherungssatz if and only if Scherungssatz holds in X.

(14) trapezium variant of Desargues Axiom holds in X if and only if the affine reduct of X satisfies TDES.

(15) The affine reduct of X satisfies des if and only if minor Desargues Axiom holds in X.

(16) Pappos Axiom holds in X if and only if the affine reduct of X satisfies PAP.

(17) Desargues Axiom holds in X if and only if the affine reduct of X satisfies DES.

References

[1] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical metric affine spaces and planes. Formalized Mathematics, 1(5):891–899, 1990.

[2] Henryk Oryszczyszyn and Krzysztof Prażmowski. Analytical ordered affine spaces. Formalized Mathematics, 1(3):601–605, 1990.

[3] Henryk Oryszczyszyn and Krzysztof Prażmowski. Classical configurations in affine planes. Formalized Mathematics, 1(4):625–633, 1990.

[4] Henryk Oryszczyszyn and Krzysztof Prażmowski. Ordered affine spaces defined in terms of directed parallelity - part I. Formalized Mathematics, 1(3):611–615, 1990.

[5] Henryk Oryszczyszyn and Krzysztof Prażmowski. Translations in affine planes. Formalized Mathematics, 1(4):751–753, 1990.

[6] Jolanta Świerzyńska and Bogdan Świerzyński. Metric-affine configurations in metric affine planes - Part I. Formalized Mathematics, 2(3):331–334, 1991.

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Shear Theorems and Their Role in Affine … 445

alalalalalal alalalalalalal

Index of MML Identifiers

AFF 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .357 AFPROJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 ANALTRAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 CONAFFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 CONMETR1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 CONMETR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 EUCLMETR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 FCONT 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403 FDIFF 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 FILTER 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 GRAPH 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 HEYTING1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 OPPCAT 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 PAPDESAF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 PARDEPAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 PRELAMB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413 PROJRED2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 SCHEMS 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 SEMI AF1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 TREES 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 ZF FUND1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 ZF FUND2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Contents Form. Math. 2 (3)

A Construction of Analytical Ordered Trapezium Spaces

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

On Projections in Projective Planes. Part II

By Eugeniusz Kusak et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Metric-Affine Configurations in Metric Affine Planes - Part I

By Jolanta Świerzyńska et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

Metric-Affine Configurations in Metric Affine Planes - Part II

By Jolanta Świerzyńska et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Fanoian, Pappian and Desarguesian Affine Spaces

By Krzysztof Prażmowski . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341

Elementary Variants of Affine Configurational Theorems

By Krzysztof Prażmowski et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347

Semi-Affine Space

By Eugeniusz Kusak and Krzysztof Radziszewski . . . . . . 349

Planes in Affine Spaces

By Wojciech Leończuk et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357

Graphs

By Krzysztof Hryniewiecki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

Mostowski’s Fundamental Operations - Part I

By Andrzej Kondracki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371

A Projective Closure and Projective Horizon of an Affine Space

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

Schemes

By Stanisław T. Czuba . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

Continued on inside back cover

Algebra of Normal Forms Is a Heyting Algebra

By Andrzej Trybulec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

König’s Lemma

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

Monotonic and Continuous Real Function

By Jarosław Kotowicz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

Real Function Differentiability - Part II

By Jarosław Kotowicz and Konrad Raczkowski . . . . . . . 407

Preliminaries to the Lambek Calculus

By Wojciech Zielonka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

Opposite Categories and Contravariant Functors

By Czesław Byliński . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419

Mostowski’s Fundamental Operations - Part II

By Grzegorz Bancerek and Andrzej Kondracki . . . . . . . 425

Fundamental Types of Metric Affine Spaces

By Henryk Oryszczyszyn et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Filters - Part II. Quotient Lattices Modulo Filters and Direct Product of Two Lattices

By Grzegorz Bancerek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Shear Theorems and Their Role in Affine Geometry

By Jolanta Świerzyńska et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

Index of MML Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446