:: The Geometric Interior in Real Linear Spaces
:: by Karol P\c{a}k
::
:: Received February 9, 2010
:: Copyright (c) 2010 Association of Mizar Users
environ
vocabularies ARYTM_1, ARYTM_3, XBOOLE_0, CARD_1, CONVEX1, CONVEX2, CONVEX3,
FINSEQ_1, FINSEQ_2, FINSEQ_4, FINSET_1, FUNCOP_1, FUNCT_1, FUNCT_2,
FUNCT_4, MEMBERED, ORDERS_1, RELAT_1, RLVECT_1, RLVECT_2, RUSUB_4,
SEMI_AF1, SETFAM_1, TARSKI, TOPS_1, RLAFFIN1, RLAFFIN2, ZFMISC_1, REAL_1,
CARD_3, XXREAL_0, NAT_1, SUBSET_1, NUMBERS, ORDINAL1, STRUCT_0, SUPINF_2,
ORDINAL4, VALUED_1, XREAL_0, PARTFUN1;
notations TARSKI, XBOOLE_0, SUBSET_1, ZFMISC_1, ORDINAL1, NUMBERS, NAT_1,
XCMPLX_0, XXREAL_0, ORDERS_1, CARD_1, XREAL_0, REAL_1, FINSET_1,
SETFAM_1, DOMAIN_1, RELAT_1, FUNCT_1, PARTFUN1, FUNCT_2, RELSET_1,
FINSEQ_1, STRUCT_0, FINSEQ_2, SEQ_1, FINSEQOP, RVSUM_1, RLVECT_1,
RLVECT_2, RUSUB_4, CONVEX1, CONVEX2, CONVEX3, RLAFFIN1, FUNCT_4,
FUNCOP_1, MEMBERED, XXREAL_2;
constructors BINOP_2, CONVEX1, CONVEX3, DOMAIN_1, FINSEQOP, REAL_1, RLVECT_3,
RVSUM_1, RUSUB_5, SEQ_1, SETFAM_1, RLAFFIN1, SIMPLEX0, XXREAL_2, FUNCT_4,
RELSET_1, PARTFUN1;
registrations CARD_1, FINSET_1, FINSEQ_2, FUNCT_2, MEMBERED, NAT_1, NUMBERS,
RELAT_1, RLVECT_1, RLVECT_3, STRUCT_0, SUBSET_1, VALUED_0, XCMPLX_0,
XREAL_0, XXREAL_0, XBOOLE_0, RLAFFIN1, SIMPLEX0, FUNCOP_1, XXREAL_3,
REALSET1, SETFAM_1, XXREAL_2, ABIAN, PARTFUN1, RVSUM_1, FINSEQ_1,
RELSET_1, DILWORTH, RLVECT_2;
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM;
definitions ORDERS_1, RLVECT_2, TARSKI, XBOOLE_0;
theorems BORSUK_4, CARD_1, CARD_2, CARD_FIN, CONVEX1, CONVEX3, FINSEQ_1,
FINSEQ_2, FINSEQ_3, FINSEQ_4, FUNCOP_1, FUNCT_1, FUNCT_2, FUNCT_4, NAT_1,
MEMBERED, PARTFUN1, RELAT_1, RLAFFIN1, RLTOPSP1, RUSUB_4, RLVECT_2,
RLVECT_3, RLVECT_4, RVSUM_1, STIRL2_1, SIMPLEX0, TARSKI, WELLORD2,
XBOOLE_0, XBOOLE_1, XCMPLX_1, XREAL_0, XREAL_1, XXREAL_0, XXREAL_2,
ZFMISC_1, RLVECT_1;
schemes FRAENKEL, FUNCT_2, NAT_1, RLVECT_4;
begin :: Preliminaries
reserve x,y for set,
r,s for Real,
n for Nat,
V for RealLinearSpace,
v,u,w,p for VECTOR of V,
A,B for Subset of V,
Af for finite Subset of V,
I for affinely-independent Subset of V,
If for finite affinely-independent Subset of V,
F for Subset-Family of V,
L1,L2 for Linear_Combination of V;
Lm1: r<>1 & r*u+(1-r)*w=v implies w=(1/(1-r)*v)+(1-1/(1-r))*u
proof
assume that
A1: r<>1 and
A2: r*u+(1-r)*w=v;
A3: (1-r)*w = v-r*u by A2,RLVECT_4:1;
A4: 1-r<>1-1 by A1;
then A5: (1/(1-r))*(1-r)=1 by XCMPLX_1:107;
A6: 1/(1-r)*(r*u)=(1/(1-r)*r)*u by RLVECT_1:def 10
.=((1-(1-r))/(1-r))*u by XCMPLX_1:100
.=(1/(1-r)-(1-r)/(1-r))*u by XCMPLX_1:121
.=(1/(1-r)-1)*u by A4,XCMPLX_1:60;
thus w=1*w by RLVECT_1:def 11
.=(1/(1-r))*((1-r)*w) by A5,RLVECT_1:def 10
.=(1/(1-r)*v)-1/(1-r)*(r*u) by A3,RLVECT_1:48
.=(1/(1-r)*v)+-(1/(1-r)-1)*u by A6,RLVECT_1:def 14
.=(1/(1-r)*v)+(1/(1-r)-1)*(-u) by RLVECT_1:39
.=(1/(1-r)*v)+(-(1/(1-r)-1))*u by RLVECT_1:38
.=(1/(1-r)*v)+(1-1/(1-r))*u;
end;
theorem Th1:
for L be Linear_Combination of A st L is convex & v <> Sum L & L.v <> 0
ex p st p in conv(A\{v}) & Sum L = L.v*v + (1-L.v)*p &
1/L.v*(Sum L) + (1-1/L.v)*p = v
proof
let L be Linear_Combination of A such that
A1: L is convex and
A2: v<>Sum L and
A3: L.v<>0;
set Lv=L.v,1Lv=1-L.v;
A4: Carrier L c=A by RLVECT_2:def 8;
Carrier L<>{} by A1,CONVEX1:21;
then reconsider A1=A as non empty Subset of V by A4;
consider K be Linear_Combination of{v} such that
A5: K.v=Lv by RLVECT_4:40;
A6: Lv<>1
proof
assume A7: Lv=1;
then Carrier L={v} by A1,RLAFFIN1:64;
then Sum L=1*v by A7,RLVECT_2:53
.=v by RLVECT_1:def 11;
hence contradiction by A2;
end;
Lv<=1 by A1,RLAFFIN1:63;
then Lv<1 by A6,XXREAL_0:1;
then A8: 1Lv>1-1 by XREAL_1:12;
v in Carrier L by A3;
then {v}c=A1 by A4,ZFMISC_1:37;
then K is Linear_Combination of A1 by RLVECT_2:33;
then reconsider LK=L-K as Linear_Combination of A1 by RLVECT_2:81;
1/1Lv*LK is Linear_Combination of A by RLVECT_2:67;
then A9: Carrier(1/1Lv*LK)c=A1 by RLVECT_2:def 8;
LK.v=Lv-Lv by A5,RLVECT_2:79;
then A10: (1/1Lv*LK).v=1/1Lv*(Lv-Lv) by RLVECT_2:def 13;
then not v in Carrier(1/1Lv*LK) by RLVECT_2:28;
then A11: Carrier(1/1Lv*LK)c=A\{v} by A9,ZFMISC_1:40;
A12: Carrier K c={v} by RLVECT_2:def 8;
A13: for w be Element of V holds(1/1Lv*LK).w>=0
proof
let w be Element of V;
A14: (1/1Lv*LK).w=(1/1Lv)*(LK.w) by RLVECT_2:def 13
.=(1/1Lv)*(L.w-K.w) by RLVECT_2:79;
per cases;
suppose w=v;
hence thesis by A10;
end;
suppose w<>v;
then not w in Carrier K by A12,TARSKI:def 1;
then A15: K.w=0;
L.w>=0 by A1,RLAFFIN1:62;
hence thesis by A8,A14,A15;
end;
end;
sum LK=sum L-sum K by RLAFFIN1:36
.=sum L-Lv by A5,A12,RLAFFIN1:32
.=1Lv by A1,RLAFFIN1:62;
then A16: sum(1/1Lv*LK)=1/1Lv*1Lv by RLAFFIN1:35
.=1 by A8,XCMPLX_1:107;
then 1/1Lv*LK is convex by A13,RLAFFIN1:62;
then Carrier(1/1Lv*LK)<>{} by CONVEX1:21;
then reconsider Av=A\{v} as non empty Subset of V by A11;
reconsider 1LK=1/1Lv*LK as Convex_Combination of Av
by A11,A13,A16,RLAFFIN1:62,RLVECT_2:def 8;
take p=Sum 1LK;
1LK in ConvexComb(V) by CONVEX3:def 1;
then p in {Sum(M) where M is Convex_Combination of Av:M in ConvexComb(V)};
hence p in conv(A\{v}) by CONVEX3:5;
A17: Sum(LK)=Sum L-Sum K by RLVECT_3:4
.=Sum L-Lv*v by A5,RLVECT_2:50;
then 1Lv*p=1Lv*(1/1Lv*(Sum L-Lv*v)) by RLVECT_3:2
.=(1Lv*(1/1Lv))*(Sum L-Lv*v) by RLVECT_1:def 10
.=1*(Sum L-Lv*v) by A8,XCMPLX_1:107
.=Sum L-Lv*v by RLVECT_1:def 11;
hence Sum L=Lv*v+1Lv*p by RLVECT_4:1;
1-1/Lv=Lv/Lv-1/Lv by A3,XCMPLX_1:60
.=(Lv-1)/Lv by XCMPLX_1:121
.=(-1Lv)/Lv
.=-1Lv/Lv by XCMPLX_1:188;
then (1-1/Lv)*Sum 1LK=(-1Lv/Lv)*(1/1Lv*(Sum L-Lv*v)) by A17,RLVECT_3:2
.=((-1Lv/Lv)*(1/1Lv))*(Sum L-Lv*v) by RLVECT_1:
def 10
.=(-(1Lv/Lv)*(1/1Lv))*(Sum L-Lv*v)
.=(-(1Lv/1Lv)*(1/Lv))*(Sum L-Lv*v) by XCMPLX_1:86
.=(-1*(1/Lv))*(Sum L-Lv*v) by A8,XCMPLX_1:60
.=(-(1/Lv))*(Sum L)-(-(1/Lv))*(Lv*v) by RLVECT_1:48
.=(-(1/Lv))*(Sum L)+-(-(1/Lv))*(Lv*v)
by RLVECT_1:def 14
.=(-(1/Lv))*(Sum L)+(-(-(1/Lv)))*(Lv*v) by RLVECT_4:6
.=(-(1/Lv))*(Sum L)+(1/Lv*Lv)*v by RLVECT_1:
def 10
.=(-(1/Lv))*(Sum L)+1*v by A3,XCMPLX_1:107
.=(-(1/Lv))*(Sum L)+v by RLVECT_1:def 11
.=-((1/Lv)*(Sum L))+v by RLVECT_4:6;
hence 1/Lv*(Sum L)+(1-1/Lv)*p =(1/Lv*Sum L+-(1/Lv)*(Sum L))+v
by RLVECT_1:def 6
.=(1/Lv*Sum L-(1/Lv)*(Sum L))+v by RLVECT_1:def 14
.=v by RLVECT_4:1;
end;
theorem
for p1,p2,w1,w2 be Element of V st v in conv I & u in conv I &
not u in conv(I\{p1}) & not u in conv(I\{p2}) &
w1 in conv(I\{p1}) & w2 in conv(I\{p2}) &
r*u+(1-r)*w1 = v & s*u + (1-s)*w2 = v & r < 1 & s < 1
holds w1 = w2 & r = s
proof
let p1,p2,w1,w2 be Element of V;
assume that
A1: v in conv I and
A2: u in conv I and
A3: not u in conv(I\{p1}) and
A4: not u in conv(I\{p2}) and
A5: w1 in conv(I\{p1}) and
A6: w2 in conv(I\{p2}) and
A7: r*u+(1-r)*w1=v and
A8: s*u+(1-s)*w2=v and
A9: r<1 and
A10: s<1;
set Lu=u|--I,Lv=v|--I,Lw1=w1|--I,Lw2=w2|--I;
A11: conv I c=Affin I by RLAFFIN1:65;
A12: Lu.p2>0
proof
assume A13: Lu.p2<=0;
Lu.p2>=0 by A2,RLAFFIN1:71;
then for y st y in {p2} holds Lu.y=0 by A13,TARSKI:def 1;
hence contradiction by A2,A4,RLAFFIN1:76;
end;
conv(I\{p2})c=Affin(I\{p2}) by RLAFFIN1:65;
then Lw2=w2|--(I\{p2}) by A6,RLAFFIN1:77,XBOOLE_1:36;
then Carrier Lw2 c=I\{p2} by RLVECT_2:def 8;
then not p2 in Carrier Lw2 by ZFMISC_1:64;
then A14: Lw2.p2=0;
A15: Lu.p1>0
proof
assume A16: Lu.p1<=0;
Lu.p1>=0 by A2,RLAFFIN1:71;
then for y st y in {p1} holds Lu.y=0 by A16,TARSKI:def 1;
hence contradiction by A2,A3,RLAFFIN1:76;
end;
conv(I\{p1})c=Affin(I\{p1}) by RLAFFIN1:65;
then Lw1=w1|--(I\{p1}) by A5,RLAFFIN1:77,XBOOLE_1:36;
then Carrier Lw1 c=I\{p1} by RLVECT_2:def 8;
then not p1 in Carrier Lw1 by ZFMISC_1:64;
then A17: Lw1.p1=0;
A18: conv(I\{p2})c=conv I by RLAFFIN1:3,XBOOLE_1:36;
then w2 in conv I by A6;
then s*Lu+(1-s)*Lw2=Lv by A2,A8,A11,RLAFFIN1:70;
then Lv.p2=(s*Lu).p2+((1-s)*Lw2).p2 by RLVECT_2:def 12
.=s*(Lu.p2)+((1-s)*Lw2).p2 by RLVECT_2:def 13
.=s*(Lu.p2)+(1-s)*(Lw2.p2) by RLVECT_2:def 13
.=s*(Lu.p2) by A14;
then A19: Lv.p2<1*Lu.p2 by A10,A12,XREAL_1:70;
then A20: Lu.p2-Lv.p2>=Lv.p2-Lv.p2 by XREAL_1:11;
A21: conv(I\{p1})c=conv I by RLAFFIN1:3,XBOOLE_1:36;
then Lw1.p2>=0 by A5,RLAFFIN1:71;
then A22: 1/(1-s)-0>=1/(1-s)-Lw1.p2/(Lu.p2-Lv.p2) by A20,XREAL_1:12;
w1 in conv I by A5,A21;
then Lv=r*Lu+(1-r)*Lw1 by A2,A7,A11,RLAFFIN1:70;
then Lv.p1=(r*Lu).p1+((1-r)*Lw1).p1 by RLVECT_2:def 12
.=r*(Lu.p1)+((1-r)*Lw1).p1 by RLVECT_2:def 13
.=r*(Lu.p1)+(1-r)*(Lw1.p1) by RLVECT_2:def 13
.=r*(Lu.p1) by A17;
then A23: Lv.p1<1*Lu.p1 by A9,A15,XREAL_1:70;
then A24: Lu.p1-Lv.p1>=Lv.p1-Lv.p1 by XREAL_1:11;
w2=(1/(1-s)*v)+(1-1/(1-s))*u by A8,A10,Lm1;
then A25: Lw2=1/(1-s)*Lv+(1-1/(1-s))*Lu by A1,A2,A11,RLAFFIN1:70;
then A26: 1/(1-s)=(Lu.p2-0)/(Lu.p2-Lv.p2) by A14,A19,RLAFFIN1:81;
A27: w1=(1/(1-r)*v)+(1-1/(1-r))*u by A7,A9,Lm1;
then A28: Lw1=1/(1-r)*Lv+(1-1/(1-r))*Lu by A1,A2,A11,RLAFFIN1:70;
then A29: 1/(1-r)=(Lu.p2-Lw1.p2)/(Lu.p2-Lv.p2) by A19,RLAFFIN1:81
.=1/(1-s)-Lw1.p2/(Lu.p2-Lv.p2) by A26,XCMPLX_1:121;
Lw2.p1>=0 by A6,A18,RLAFFIN1:71;
then A30: 1/(1-r)-0>=1/(1-r)-Lw2.p1/(Lu.p1-Lv.p1) by A24,XREAL_1:12;
A31: 1/(1-r)=(Lu.p1-0)/(Lu.p1-Lv.p1) by A17,A23,A28,RLAFFIN1:81;
1/(1-s)=(Lu.p1-Lw2.p1)/(Lu.p1-Lv.p1) by A23,A25,RLAFFIN1:81
.=1/(1-r)-Lw2.p1/(Lu.p1-Lv.p1) by A31,XCMPLX_1:121;
then 1-r=1-s by A30,A22,A29,XCMPLX_1:59,XXREAL_0:1;
hence thesis by A8,A10,A27,Lm1;
end;
theorem Th3:
for L be Linear_Combination of Af st Af c=conv If & sum L=1 holds
Sum L in Affin If
& for x be Element of V
ex F be FinSequence of REAL,G be FinSequence of V st
(Sum L|--If).x = Sum F & len G = len F & G is one-to-one &
rng G = Carrier L &
for n st n in dom F holds F.n = L.(G.n)*(G.n|--If).x
proof
defpred P[Nat] means
for B be finite Subset of V st card B=$1 & B c=conv If for L be
Linear_Combination of B st Carrier L=B & sum L=1 holds Sum L in Affin If &
for x be Element of V
ex F be FinSequence of REAL,G be FinSequence of V st
(Sum L|--If).x=Sum F & len G=len F & G is one-to-one & rng G=Carrier L &
for n st n in dom F holds F.n=L.(G.n)*(G.n|--If).x;
A1: for m be Nat st P[m] holds P[m+1]
proof
let m be Nat such that
A2: P[m];
let B be finite Subset of V such that
A3: card B=m+1 and
A4: B c=conv If;
conv If c=Affin If by RLAFFIN1:65;
then A5: B c=Affin If by A4,XBOOLE_1:1;
then A6: Affin B c=Affin If by RLAFFIN1:51;
let L be Linear_Combination of B such that
A7: Carrier L=B and
A8: sum L=1;
Sum L in {Sum K where K is Linear_Combination of B:sum K=1} by A8;
then Sum L in Affin B by RLAFFIN1:59;
hence Sum L in Affin If by A6;
per cases;
suppose A9: m=0;
let x be Element of V;
consider b be set such that
A10: B={b} by A3,A9,CARD_2:60;
b in B by A10,TARSKI:def 1;
then reconsider b as Element of V;
A11: sum L=L.b by A7,A10,RLAFFIN1:32;
set F=<*(b|--If).x*>,G=<*b*>;
take F,G;
Sum L=L.b*b by A10,RLVECT_2:50;
then len G=1 & Sum L=b by A8,A11,FINSEQ_1:56,RLVECT_1:def 11;
hence (Sum L|--If).x=Sum F & len G=len F & G is one-to-one & rng G=
Carrier L by A7,A10,FINSEQ_1:56,FINSEQ_3:102,RVSUM_1:103;
let n;
assume n in dom F;
then n in Seg 1 by FINSEQ_1:55;
then A12: n=1 by FINSEQ_1:4,TARSKI:def 1;
then G.n=b by FINSEQ_1:57;
hence thesis by A8,A11,A12,FINSEQ_1:57;
end;
suppose A13: m>0;
ex v be Element of V st L.v<>1 & v in Carrier L
proof
consider F be FinSequence of V such that
A14: F is one-to-one and
A15: rng F=Carrier L and
A16: 1=Sum(L*F) by A8,RLAFFIN1:def 3;
dom F,B are_equipotent by A7,A14,A15,WELLORD2:def 4;
then A17: card B=card dom F by CARD_1:21
.=card Seg len F by FINSEQ_1:def 3
.=len F by FINSEQ_1:78;
A18: len F=len(L*F) & len(len F|->1)=len F
by FINSEQ_1:def 18,FINSEQ_2:37;
Sum(len F|->1)=len F*1 by RVSUM_1:110;
then len F|->1<>L*F by A3,A13,A16,A17;
then consider k be Nat such that
A19: 1<=k & k<=len F and
A20: (len F|->1).k<>(L*F).k by A18,FINSEQ_1:18;
A21: k in Seg len F by A19,FINSEQ_1:3;
A22: k in dom F by A19,FINSEQ_3:27;
then A23: F.k in Carrier L by A15,FUNCT_1:def 5;
L.(F.k)=(L*F).k by A22,FUNCT_1:23;
then L.(F.k)<>1 by A20,A21,FINSEQ_2:71;
hence thesis by A23;
end;
then consider v be Element of V such that
A24: L.v<>1 and
A25: v in Carrier L;
set 1Lv=1-L.v;
consider K be Linear_Combination of{v} such that
A26: K.v=L.v by RLVECT_4:40;
set LK=L-K;
A27: 1Lv<>0 by A24;
set 1LK=1/1Lv*LK;
A28: Carrier K c={v} by RLVECT_2:def 8;
then sum K=K.v by RLAFFIN1:32;
then sum LK=1Lv by A8,A26,RLAFFIN1:36;
then A29: sum 1LK=1/1Lv*1Lv by RLAFFIN1:35;
LK.v=L.v-K.v by RLVECT_2:79;
then A30: not v in Carrier LK by A26,RLVECT_2:28;
A31: card(B\{v})=m by A3,A7,A25,STIRL2_1:65;
A32: Sum K=L.v*v by A26,RLVECT_2:50;
B\{v}c=B by XBOOLE_1:36;
then A33: B\{v}c=conv If by A4,XBOOLE_1:1;
L.v<>0 by A25,RLVECT_2:28;
then v in Carrier K by A26;
then A34: Carrier K={v} by A28,ZFMISC_1:39;
A35: B\{v}c=Carrier LK
proof
let x;
assume A36: x in B\{v};
then reconsider u=x as Element of V;
u in B by A36,ZFMISC_1:64;
then A37: L.u<>0 by A7,RLVECT_2:28;
LK.u=L.u-K.u & not u in {v} by A36,RLVECT_2:79,XBOOLE_0:def 5;
then LK.u<>0 by A34,A37;
hence thesis;
end;
let x be Element of V;
A38: 1/1Lv*1Lv=(1*1Lv)/1Lv by XCMPLX_1:75
.=1 by A27,XCMPLX_1:60;
Sum 1LK=1/1Lv*Sum LK by RLVECT_3:2;
then 1Lv*Sum 1LK=(1Lv*(1/1Lv))*Sum LK by RLVECT_1:def 10
.=Sum LK by A38,RLVECT_1:def 11;
then A39: 1Lv*Sum 1LK+L.v*v=Sum L-L.v*v+L.v*v by A32,RLVECT_3:4
.=Sum L by RLVECT_4:1;
B\/{v}=B by A7,A25,ZFMISC_1:46;
then Carrier LK c=B\{v} by A7,A34,A30,RLVECT_2:80,ZFMISC_1:40;
then B\{v}=Carrier LK by A35,XBOOLE_0:def 10;
then A40: Carrier 1LK=B\{v} by A27,RLVECT_2:65;
then A41: 1LK is Linear_Combination of B\{v} by RLVECT_2:def 8;
then consider F be FinSequence of REAL,
G be FinSequence of V such that
A42: (Sum 1LK|--If).x=Sum F and
A43: len G=len F and
A44: G is one-to-one and
A45: rng G=Carrier 1LK and
A46: for n st n in dom F holds F.n=1LK.(G.n)*(G.n|--If).x
by A2,A29,A38,A31,A33,A40;
Sum 1LK in Affin If by A2,A29,A38,A31,A33,A40,A41;
then A47: Sum L|--If=1Lv*(Sum 1LK|--If)+L.v*(v|--If)
by A5,A7,A25,A39,RLAFFIN1:70;
take F1=(1Lv*F)^<*L.v*(v|--If).x*>,G1=G^<*v*>;
thus Sum F1=Sum(1Lv*F)+L.v*(v|--If).x by RVSUM_1:104
.=1Lv*(Sum 1LK|--If).x+L.v*(v|--If).x by A42,RVSUM_1:117
.=(1Lv*(Sum 1LK|--If)).x+L.v*(v|--If).x by RLVECT_2:def 13
.=(1Lv*(Sum 1LK|--If)).x+(L.v*(v|--If)).x by RLVECT_2:def 13
.=(Sum L|--If).x by A47,RLVECT_2:def 12;
reconsider f=F as Element of len F-tuples_on REAL by FINSEQ_2:110;
A48: len F=len(1Lv*f) by FINSEQ_1:def 18;
then A49: len F1=len F+1 by FINSEQ_2:19;
hence len F1=len G1 by A43,FINSEQ_2:19;
A50: rng<*v*>={v} by FINSEQ_1:55;
then <*v*> is one-to-one & rng G misses rng<*v*>
by A40,A45,FINSEQ_3:102,XBOOLE_1:79;
hence G1 is one-to-one by A44,FINSEQ_3:98;
thus rng G1=B\{v}\/{v} by A40,A45,A50,FINSEQ_1:44
.=Carrier L by A7,A25,ZFMISC_1:140;
let n;
assume A51: n in dom F1;
then A52: n<=len F1 by FINSEQ_3:27;
per cases by A49,A51,A52,FINSEQ_3:27,NAT_1:8;
suppose A53: 1<=n & n<=len F;
then n in dom F by FINSEQ_3:27;
then A54: (1Lv*f).n=1Lv*f.n & F.n=1LK.(G.n)*(G.n|--If).x
by A46,RVSUM_1:67;
A55: n in dom G by A43,A53,FINSEQ_3:27;
then A56: G1.n=G.n by FINSEQ_1:def 7;
A57: G.n in B\{v} by A40,A45,A55,FUNCT_1:def 5;
then not G.n in {v} by XBOOLE_0:def 5;
then K.(G.n)=0 by A34,A57;
then LK.(G.n)=L.(G.n)-0 by A57,RLVECT_2:79;
then 1LK.(G.n)=(1/1Lv)*(L.(G.n)) by A57,RLVECT_2:def 13;
then 1Lv*1LK.(G1.n)=1Lv*(1/1Lv)*(L.(G.n)) by A56;
then A58: 1Lv*1LK.(G1.n)=L.(G.n) by A38;
n in dom(1Lv*F) by A48,A53,FINSEQ_3:27;
hence thesis by A54,A56,A58,FINSEQ_1:def 7;
end;
suppose A59: n=len F+1;
then G1.n=v by A43,FINSEQ_1:59;
hence thesis by A48,A59,FINSEQ_1:59;
end;
end;
end;
let L be Linear_Combination of Af such that
A60: Af c=conv If and
A61: sum L=1;
set C=Carrier L;
C c=Af by RLVECT_2:def 8;
then A62: C c=conv If by A60,XBOOLE_1:1;
reconsider L1=L as Linear_Combination of C by RLVECT_2:def 8;
A63: P[0 qua Nat]
proof
let B be finite Subset of V such that
A64: card B=0 and
B c=conv If;
let L be Linear_Combination of B such that
Carrier L=B and
A65: sum L=1;
B={}the carrier of V by A64;
then L=ZeroLC(V) by RLVECT_2:35;
hence thesis by A65,RLAFFIN1:31;
end;
for m be Nat holds P[m] from NAT_1:sch 2(A63,A1);
then sum L=sum L1 & P[card C];
hence thesis by A61,A62;
end;
theorem
for Aff be Subset of V st Aff is Affine & conv A /\ conv B c= Aff &
conv(A\{v}) c= Aff & not v in Aff
holds conv (A\{v}) /\ conv B = conv A /\ conv B
proof
let Aff be Subset of V;
assume that
A1: Aff is Affine and
A2: conv A/\conv B c=Aff and
A3: conv(A\{v})c=Aff and
A4: not v in Aff;
conv(A\{v})c=conv A by RLTOPSP1:21,XBOOLE_1:36;
hence conv(A\{v})/\conv B c=conv A/\conv B by XBOOLE_1:26;
let x;
assume A5: x in conv A/\conv B;
then reconsider A1=A as non empty Subset of V by XBOOLE_0:def 4;
A6: x in Aff by A2,A5;
conv A={Sum(L) where L is Convex_Combination of A1:L in ConvexComb(V)} &
x in conv A by A5,CONVEX3:5,XBOOLE_0:def 4;
then consider L be Convex_Combination of A1 such that
A7: x=Sum L and
L in ConvexComb(V);
set Lv=L.v;
A8: Carrier L c=A by RLVECT_2:def 8;
A9: x in conv B by A5,XBOOLE_0:def 4;
per cases;
suppose Lv=0;
then not v in Carrier L by RLVECT_2:28;
then A10: Carrier L c=A\{v} by A8,ZFMISC_1:40;
then reconsider K=L as Linear_Combination of A\{v} by RLVECT_2:def 8;
Carrier L<>{} by CONVEX1:21;
then reconsider Av=A\{v} as non empty Subset of V by A10;
K in ConvexComb(V) by CONVEX3:def 1;
then Sum K in {Sum(M) where M is Convex_Combination of Av:
M in ConvexComb(V)};
then x in conv Av by A7,CONVEX3:5;
hence thesis by A9,XBOOLE_0:def 4;
end;
suppose Lv<>0;
then ex p be Element of V st p in conv(A\{v}) & Sum L=L.v*v+(1-L.v)*p &
1/L.v*(Sum L)+(1-1/L.v)*p=v by A4,A6,A7,Th1;
hence thesis by A1,A2,A3,A4,A5,A7,RUSUB_4:def 5;
end;
end;
begin :: The Geometric Interior
definition
let V be non empty RLSStruct;
let A be Subset of V;
func Int A -> Subset of V means :Def1:
x in it iff x in conv A & not ex B be Subset of V st B c< A & x in conv B;
existence
proof
set I={x where x is Element of V:x in conv A & not ex B be Subset of V st
B c< A & x in conv B};
now let x;
assume x in I;
then ex y be Element of V st x=y & y in conv A & not ex B be Subset of V
st B c empty;
coherence
proof
Int A c=conv A by Lm2;
hence thesis;
end;
end;
theorem
for V be non empty RLSStruct for A be Subset of V holds Int A c= conv A
by Lm2;
theorem
for V be vector-distributive scalar-distributive scalar-associative
scalar-unital(non empty RLSStruct)
for A be Subset of V holds Int A = A iff A is trivial
proof
let V be vector-distributive scalar-distributive scalar-associative
scalar-unital(non empty RLSStruct);
let A be Subset of V;
per cases;
suppose A is empty;
hence thesis;
end;
suppose A1: A is non empty;
hereby assume A2: Int A=A;
now let x,y;
assume that
A3: x in A and
A4: y in A;
A\{x}c=A & A\{x}<>A by A3,XBOOLE_1:36,ZFMISC_1:64;
then A\{x}c=conv(A\{x}) & A\{x}cA;
then B cA;
then Carrier L cv by A16,ZFMISC_1:64;
then not w in Carrier(v|--{v}) by A8,TARSKI:def 1;
then A18: (v|--I).w=0 by A12;
(u|--I).w=((1-r)*(p|--I)).w+(r*(v|--I)).w by A7,RLVECT_2:def 12
.=((1-r)*(p|--I)).w+r*((v|--I).w) by RLVECT_2:def 13
.=(1-r)*((p|--I).w) by A18,RLVECT_2:def 13;
then (p|--I).w<>0 by A10,A17,RLVECT_2:28;
hence thesis by A14;
end;
Carrier(p|--(I\{v}))c=I\{v} by RLVECT_2:def 8;
then A19: I\{v}=Carrier(p|--(I\{v})) by A15,XBOOLE_0:def 10;
A20: I\{v} is affinely-independent by RLAFFIN1:43,XBOOLE_1:36;
then Sum(p|--(I\{v}))=p by A3,A13,RLAFFIN1:def 7;
hence thesis by A3,A19,A20,Th12,RLAFFIN1:71;
end;
begin :: The Center of Mass
definition let V;
func center_of_mass V -> Function of BOOL the carrier of V,the carrier of V
means :Def2:
(for A be non empty finite Subset of V holds it.A = 1/card A * Sum(A)) &
for A st A is infinite holds it.A = 0.V;
existence
proof
defpred P[set,set] means
(for A be non empty finite Subset of V st A=$1 holds$2=1/card A*Sum(A)) &
for A be Subset of V st A is infinite & A=$1 holds$2=0.V;
set cV=the carrier of V;
A1: for x st x in BOOL cV ex y st y in cV & P[x,y]
proof
let x;
assume x in BOOL cV;
then reconsider A=x as Subset of V;
per cases;
suppose A is finite;
then reconsider A1=A as finite Subset of V;
take 1/card A1*Sum A1;
thus thesis;
end;
suppose A2: A is infinite;
take 0.V;
thus thesis by A2;
end;
end;
consider f be Function of BOOL cV,cV such that
A3: for x st x in BOOL cV holds P[x,f.x] from FUNCT_2:sch 1(A1);
take f;
hereby let A be non empty finite Subset of V;
A in BOOL cV by ZFMISC_1:64;
hence f.A=1/card A*Sum(A) by A3;
end;
let A be Subset of V;
assume A4: A is infinite;
then A in bool cV\{{}} by ZFMISC_1:64;
hence thesis by A3,A4;
end;
uniqueness
proof
set cV=the carrier of V;
let F1,F2 be Function of BOOL cV,cV such that
A5: for A be non empty finite Subset of V holds F1.A=1/card A*Sum(A) and
A6: for A be Subset of V st A is infinite holds F1.A=0.V and
A7: for A be non empty finite Subset of V holds F2.A=1/card A*Sum(A) and
A8: for A be Subset of V st A is infinite holds F2.A=0.V;
for x st x in BOOL cV holds F1.x=F2.x
proof
let x;
assume x in BOOL cV;
then reconsider A=x as non empty Subset of V by ZFMISC_1:64;
per cases;
suppose A is finite;
then reconsider A1=A as non empty finite Subset of V;
thus F1.x=1/card A1*Sum(A1) by A5
.=F2.x by A7;
end;
suppose A9: A is infinite;
hence F1.x=0.V by A6
.=F2.x by A8,A9;
end;
end;
hence thesis by FUNCT_2:18;
end;
end;
theorem Th15:
ex L be Linear_Combination of Af st Sum L = r*Sum Af & sum L = r * card Af &
L = (ZeroLC V) +* (Af-->r)
proof
set cV=the carrier of V;
set Ar=ZeroLC(V)+*(Af-->r);
A1: dom(Af-->r)=Af by FUNCOP_1:19;
dom ZeroLC(V)=cV by FUNCT_2:def 1;
then dom Ar=cV\/Af by A1,FUNCT_4:def 1;
then rng Ar c=rng(Af-->r)\/rng ZeroLC(V) & dom Ar=cV
by FUNCT_4:18,XBOOLE_1:12;
then Ar is Function of the carrier of V,REAL by FUNCT_2:4;
then reconsider Ar as Element of Funcs(the carrier of V,REAL)by FUNCT_2:11;
now take Af;
let v;
assume not v in Af;
hence Ar.v=ZeroLC(V).v by A1,FUNCT_4:12
.=0 by RLVECT_2:30;
end;
then reconsider Ar as Linear_Combination of V by RLVECT_2:def 5;
Carrier Ar c=Af
proof
let x be set;
assume A2: x in Carrier Ar;
then reconsider v=x as Element of V;
assume not x in Af;
then Ar.x=ZeroLC(V).v by A1,FUNCT_4:12
.=0 by RLVECT_2:30;
hence thesis by A2,RLVECT_2:28;
end;
then reconsider Ar=(ZeroLC V)+*(Af-->r) as Linear_Combination of Af
by RLVECT_2:def 8;
A3: Carrier Ar c=Af by RLVECT_2:def 8;
per cases;
suppose A4: r=0;
Carrier Ar={}
proof
assume Carrier Ar<>{};
then consider x such that
A5: x in Carrier Ar by XBOOLE_0:def 1;
Ar.x=(Af-->r).x & (Af-->r).x=0 by A1,A3,A4,A5,FUNCOP_1:13,FUNCT_4:14;
hence contradiction by A5,RLVECT_2:28;
end;
then Ar=ZeroLC(V) by RLVECT_2:def 7;
then A6: Sum Ar=0.V & sum Ar=0 by RLAFFIN1:31,RLVECT_2:48;
r*Sum Af=0.V by A4,RLVECT_1:23;
hence thesis by A4,A6;
end;
suppose A7: r<>0;
consider F be FinSequence of V such that
A8: F is one-to-one and
A9: rng F=Carrier(Ar) and
A10: Sum Ar=Sum(Ar(#)F) by RLVECT_2:def 10;
Af c=Carrier Ar
proof
let x;
assume A11: x in Af;
then Ar.x=(Af-->r).x by A1,FUNCT_4:14;
hence thesis by A7,A11;
end;
then A12: Af=Carrier Ar by A3,XBOOLE_0:def 10;
then dom F,Af are_equipotent by A8,A9,WELLORD2:def 4;
then A13: card Af=card dom F by CARD_1:21
.=card Seg len F by FINSEQ_1:def 3
.=len F by FINSEQ_1:78;
set L=len F|->r;
A14: len(Ar*F)=len F by FINSEQ_2:37;
then reconsider ArF=Ar*F as Element of len F-tuples_on REAL
by FINSEQ_2:110;
now let i be Nat;
assume A15: i in Seg len F;
then i in dom F by FINSEQ_1:def 3;
then A16: F.i in rng F by FUNCT_1:def 5;
then A17: (Af-->r).(F.i)=r by A3,A9,FUNCOP_1:13;
i in dom ArF by A14,A15,FINSEQ_1:def 3;
then ArF.i=Ar.(F.i) by FUNCT_1:22;
then ArF.i=(Af-->r).(F.i) by A1,A3,A9,A16,FUNCT_4:14;
hence ArF.i=L.i by A15,A17,FINSEQ_2:71;
end;
then ArF=L by FINSEQ_2:139;
then A18: sum Ar=Sum L by A8,A9,RLAFFIN1:def 3
.=(len F)*r by RVSUM_1:110;
set AF=Ar(#)F;
A19: len AF=len F by RLVECT_2:def 9;
then A20: dom AF=dom F by FINSEQ_3:31;
now let i be Element of NAT;
assume A21: i in dom F;
then F/.i=F.i & F.i in rng F by FUNCT_1:def 5,PARTFUN1:def 8;
then Ar.(F/.i)=(Af-->r).(F/.i) & (Af-->r).(F/.i)=r
by A1,A3,A9,FUNCOP_1:13,FUNCT_4:14;
hence AF.i=r*F/.i by A20,A21,RLVECT_2:def 9;
end;
then Sum Ar=r*Sum F by A10,A19,RLVECT_2:5
.=r*Sum Af by A8,A9,A12,RLVECT_2:def 4;
hence thesis by A13,A18;
end;
end;
theorem Th16:
Af is non empty implies (center_of_mass V).Af in conv Af
proof
assume Af is non empty;
then reconsider a=Af as non empty finite Subset of V;
consider L be Linear_Combination of Af such that
A1: Sum L=1/card a*Sum a and
A2: sum L=1/card a*card a and
A3: L=(ZeroLC V)+*(Af-->1/card a) by Th15;
A4: dom(Af-->1/card a)=Af by FUNCOP_1:19;
A5: 0<=L.v
proof
per cases;
suppose A6: v in Af;
then L.v=(Af-->1/card a).v by A3,A4,FUNCT_4:14
.=1/card a by A6,FUNCOP_1:13;
hence thesis;
end;
suppose not v in Af;
then L.v=(ZeroLC V).v by A3,A4,FUNCT_4:12
.=0 by RLVECT_2:30;
hence thesis;
end;
end;
sum L=1 by A2,XCMPLX_1:88;
then A7: L is convex by A5,RLAFFIN1:62;
then L in ConvexComb(V) by CONVEX3:def 1;
then Sum L in {Sum K where K is Convex_Combination of a:K in ConvexComb(V)}
by A7;
then Sum L in conv Af by CONVEX3:5;
hence thesis by A1,Def2;
end;
theorem Th17:
union F is finite implies (center_of_mass V).:F c= conv union F
proof
set B=center_of_mass V;
assume A1: union F is finite;
let y;
assume y in B.:F;
then consider x such that
A2: x in dom B and
A3: x in F and
A4: B.x=y by FUNCT_1:def 12;
reconsider x as non empty Subset of V by A2,ZFMISC_1:64;
x c=union F by A3,ZFMISC_1:92;
then A5: y in conv x by A1,A4,Th16;
conv x c=conv union F by A3,RLTOPSP1:21,ZFMISC_1:92;
hence thesis by A5;
end;
theorem Th18:
v in If implies ((center_of_mass V).If |-- If).v = 1/card If
proof
consider L be Linear_Combination of If such that
A1: Sum L=1/card If*Sum If & sum L=1/card If*card If and
A2: L=(ZeroLC V)+*(If-->1/card If) by Th15;
assume A3: v in If;
then A4: conv If c=Affin If & (center_of_mass V).If in conv If
by Th16,RLAFFIN1:65;
(center_of_mass V).If=Sum L & sum L=1 by A1,A3,Def2,XCMPLX_1:88;
then dom(If-->1/card If)=If & L=(center_of_mass V).If|--If
by A4,FUNCOP_1:19,RLAFFIN1:def 7;
hence ((center_of_mass V).If|--If).v=(If-->1/card If).v by A2,A3,FUNCT_4:14
.=1/card If by A3,FUNCOP_1:13;
end;
theorem Th19:
(center_of_mass V).If in If iff card If=1
proof
set B=center_of_mass V;
hereby assume A1: B.If in If;
then reconsider BA=B.If as Element of V;
B.If in conv If by A1,Th16;
then 1=(BA|--If).BA by A1,RLAFFIN1:72
.=1/card If by A1,Th18;
hence 1=card If by XCMPLX_1:58;
end;
assume A2: card If=1;
then consider x such that
A3: {x}=If by CARD_2:60;
x in If by A3,TARSKI:def 1;
then reconsider x as Element of V;
(center_of_mass V).If=1/card If*Sum If by A3,Def2
.=1/1*x by A2,A3,RLVECT_2:15
.=x by RLVECT_1:def 11;
hence thesis by A3,TARSKI:def 1;
end;
theorem Th20:
If is non empty implies (center_of_mass V).If in Int If
proof
set BA=(center_of_mass V).If|--If;
A1: If c=Carrier BA
proof
let x;
assume A2: x in If;
then BA.x=1/card If by Th18;
hence thesis by A2;
end;
assume If is non empty;
then A3: (center_of_mass V).If in conv If by Th16;
Carrier BA c=If by RLVECT_2:def 8;
then Carrier BA=If & BA is convex by A1,A3,RLAFFIN1:71,XBOOLE_0:def 10;
then conv If c=Affin If & Sum BA in Int If by Th12,RLAFFIN1:65;
hence thesis by A3,RLAFFIN1:def 7;
end;
theorem
A c= If & (center_of_mass V).If in Affin A implies If = A
proof
set B=center_of_mass V;
assume that
A1: A c=If and
A2: B.If in Affin A;
A3: B.If|--If=B.If|--A by A1,A2,RLAFFIN1:77;
reconsider i=If as finite set;
assume If<>A;
then not If c=A by A1,XBOOLE_0:def 10;
then consider x such that
A4: x in If and
A5: not x in A by TARSKI:def 3;
reconsider x as Element of V by A4;
A6: (B.If|--If).x=1/(card i) by A4,Th18;
Carrier(B.If|--A)c=A by RLVECT_2:def 8;
then not x in Carrier(B.If|--A) by A5;
hence contradiction by A3,A4,A6;
end;
theorem Th22:
v in Af & Af\{v} is non empty implies (center_of_mass V).Af =
(1-1/card Af) * (center_of_mass V)/.(Af\{v}) + 1/card Af*v
proof
set Av=Af\{v};
assume that
A1: v in Af and
A2: Av is non empty;
consider L3 be Linear_Combination of{v} such that
A3: L3.v=1/card Af by RLVECT_4:40;
consider L1 be Linear_Combination of Av such that
A4: Sum L1=1/card Av*Sum Av and
sum L1=1/card Av*card Av and
A5: L1=(ZeroLC V)+*(Av-->1/card Av) by Th15;
consider L2 be Linear_Combination of Af such that
A6: Sum L2=1/card Af*Sum Af and
sum L2=1/card Af*card Af and
A7: L2=(ZeroLC V)+*(Af-->1/card Af) by Th15;
A8: Sum L1=(center_of_mass V).Av by A2,A4,Def2;
for u be Element of V holds L2.u=((1-1/card Af)*L1+L3).u
proof
let u be Element of V;
A9: ((1-1/card Af)*L1+L3).u=((1-1/card Af)*L1).u+L3.u &
((1-1/card Af)*L1).u=(1-1/card Af)*(L1.u) by RLVECT_2:def 12,def 13;
A10: ZeroLC(V).u=0 by RLVECT_2:30;
A11: Carrier L3 c={v} by RLVECT_2:def 8;
A12: dom(Af-->1/card Af)=Af by FUNCOP_1:19;
A13: dom(Av-->1/card Av)=Av by FUNCOP_1:19;
per cases;
suppose A14: not u in Af;
then not u in Carrier L3 by A1,A11,TARSKI:def 1;
then A15: L3.u=0;
not u in Av by A14,ZFMISC_1:64;
then L1.u=0 by A5,A10,A13,FUNCT_4:12;
hence thesis by A7,A9,A10,A12,A14,A15,FUNCT_4:12;
end;
suppose A16: v=u;
then not u in Av by ZFMISC_1:64;
then A17: L1.u=0 by A5,A10,A13,FUNCT_4:12;
L2.u=(Af-->1/card Af).v by A1,A7,A12,A16,FUNCT_4:14;
hence thesis by A1,A3,A9,A16,A17,FUNCOP_1:13;
end;
suppose A18: u in Af & u<>v;
then L2.u=(Af-->1/card Af).u by A7,A12,FUNCT_4:14;
then A19: L2.u=1/card Af by A18,FUNCOP_1:13;
not u in Carrier L3 by A11,A18,TARSKI:def 1;
then A20: L3.u=0;
(not v in Av) & Av\/{v}=Af by A1,ZFMISC_1:64,140;
then A21: card Af=card Av+1 by CARD_2:54;
1-1/card Af=card Af/card Af-1/card Af by A1,XCMPLX_1:60
.=(card Af-1)/card Af by XCMPLX_1:121
.=card Av/card Af by A21;
then A22: (1-1/card Af)*(1/card Av) = card Av/card Af/card Av
by XCMPLX_1:100
.=card Av/card Av/card Af by XCMPLX_1:48
.=1/card Af by A2,XCMPLX_1:60;
A23: u in Av by A18,ZFMISC_1:64;
then L1.u=(Av-->1/card Av).u by A5,A13,FUNCT_4:14;
hence thesis by A9,A19,A20,A22,A23,FUNCOP_1:13;
end;
end;
then A24: L2=(1-1/card Af)*L1+L3 by RLVECT_2:def 11;
dom(center_of_mass V)=BOOL the carrier of V by FUNCT_2:def 1;
then A25: Av in dom(center_of_mass V) by A2,ZFMISC_1:64;
Sum L2=(center_of_mass V).Af by A1,A6,Def2;
hence (center_of_mass V).Af=Sum((1-1/card Af)*L1)+Sum L3 by A24,RLVECT_3:1
.=(1-1/card Af)*Sum L1+Sum L3 by RLVECT_3:2
.=(1-1/card Af)*Sum L1+1/card Af*v by A3,RLVECT_2:50
.=(1-1/card Af)*(center_of_mass V)/.Av+1/card Af*v
by A8,A25,PARTFUN1:def 8;
end;
theorem
conv A c=conv If & If is non empty & conv A misses Int If implies
ex B be Subset of V st B c< If & conv A c= conv B
proof
assume that
A1: conv A c=conv If and
A2: If is non empty and
A3: conv A misses Int If;
reconsider If as non empty finite affinely-independent Subset of V by A2;
set YY={B where B is Subset of V:B c=If & ex x st x in conv A &
x in Int B};
YY c=bool the carrier of V
proof
let x;
assume x in YY;
then ex B be Subset of V st B=x & B c=If &
ex y st y in conv A & y in Int B;
hence thesis;
end;
then reconsider YY as Subset-Family of V;
take U=union YY;
A4: conv A c=conv U
proof
let v be set;
assume A5: v in conv A;
then v in conv If by A1;
then v in union{Int B where B is Subset of V:B c=If} by Th8;
then consider IB be set such that
A6: v in IB and
A7: IB in {Int B where B is Subset of V:B c=If} by TARSKI:def 4;
consider B be Subset of V such that
A8: IB=Int B and
A9: B c=If by A7;
Int B c=conv B by Lm2;
then A10: v in conv B by A6,A8;
B in YY by A5,A6,A8,A9;
then conv B c=conv U by RLAFFIN1:3,ZFMISC_1:92;
hence thesis by A10;
end;
A11: U c=If
proof
let x;
assume x in U;
then consider b be set such that
A12: x in b and
A13: b in YY by TARSKI:def 4;
ex B be Subset of V st b=B & B c=If & ex y st y in conv A &
y in Int B by A13;
hence thesis by A12;
end;
U<>If
proof
defpred P[set,set] means
for B be Subset of V st B=$2 holds$1 in B & ex x st x in conv A &
x in Int B;
assume A14: U=If;
A15: for x st x in If ex y st y in bool If & P[x,y]
proof
let x;
assume x in If;
then consider b be set such that
A16: x in b and
A17: b in YY by A14,TARSKI:def 4;
consider B be Subset of V such that
A18: b=B & B c=If & ex y st y in conv A & y in Int B by A17;
take B;
thus thesis by A16,A18;
end;
consider p be Function of If,bool If such that
A19: for x st x in If holds P[x,p.x] from FUNCT_2:sch 1(A15);
defpred Q[set,set] means
for B be Subset of V st B=p.$1 holds$2 in conv A & $2 in Int B;
A20: dom p=If by FUNCT_2:def 1;
A21: for x st x in If ex y st y in the carrier of V & Q[x,y]
proof
let x;
assume A22: x in If;
then p.x in rng p by A20,FUNCT_1:def 5;
then reconsider px=p.x as Subset of V by XBOOLE_1:1;
consider y such that
A23: y in conv A & y in Int px by A19,A22;
take y;
thus thesis by A23;
end;
consider q be Function of If,V such that
A24: for x st x in If holds Q[x,q.x] from FUNCT_2:sch 1(A21);
reconsider R=rng q as non empty finite Subset of V;
A25: R c=conv A
proof
let y;
assume y in R;
then consider x such that
A26: x in dom q and
A27: y=q.x by FUNCT_1:def 5;
p.x in rng p by A20,A26,FUNCT_1:def 5;
then reconsider px=p.x as Subset of V by XBOOLE_1:1;
px=p.x;
hence thesis by A24,A26,A27;
end;
then A28: R c=conv U by A4,XBOOLE_1:1;
A29: conv R c=conv A by A25,CONVEX1:30;
A30: dom q=If by FUNCT_2:def 1;
A31: 1/card R*card R=card R/card R by XCMPLX_1:100
.=1 by XCMPLX_1:60;
consider L be Linear_Combination of R such that
A32: Sum L=1/card R*Sum R and
A33: sum L=1/card R*card R and
A34: L=(ZeroLC V)+*(R-->1/card R) by Th15;
set SL=Sum L;
set SLIf=SL|--If;
Sum L=(center_of_mass V).R by A32,Def2;
then A35: Sum L in conv R by Th16;
A36: dom(R-->1/card R)=R by FUNCOP_1:19;
A37: now let x;
assume A38: x in R;
hence L.x=(R-->1/card R).x by A34,A36,FUNCT_4:14
.=1/card R by A38,FUNCOP_1:13;
end;
A39: R c=Carrier L
proof
let x;
assume A40: x in R;
then L.x<>0 by A37;
hence thesis by A40;
end;
A41: conv U c=conv If by A11,RLAFFIN1:3;
then A42: R c=conv If by A28,XBOOLE_1:1;
then A43: conv R c=conv If by CONVEX1:30;
then A44: SL in conv If by A35;
A45: R c=conv If by A28,A41,XBOOLE_1:1;
Carrier L c=R by RLVECT_2:def 8;
then A46: R=Carrier L by A39,XBOOLE_0:def 10;
A47: If c=Carrier SLIf
proof
let x;
assume A48: x in If;
then consider F be FinSequence of REAL,
G be FinSequence of V such that
A49: SLIf.x=Sum F and
A50: len G=len F and
G is one-to-one and
A51: rng G=Carrier L and
A52: for n st n in dom F holds F.n=L.(G.n)*(G.n|--If).x
by A31,A33,A42,Th3;
A53: p.x in rng p by A20,A48,FUNCT_1:def 5;
then reconsider px=p.x as Subset of V by XBOOLE_1:1;
A54: Int px c=conv px by Lm2;
A55: q.x in Int px by A24,A48;
then A56: q.x in conv px by A54;
A57: x in px by A19,A48;
A58: conv px c=Affin px by RLAFFIN1:65;
A59: px is affinely-independent by A53,RLAFFIN1:43;
then Sum(q.x|--px)=q.x by A56,A58,RLAFFIN1:def 7;
then A60: Carrier(q.x|--px)=px by A54,A55,A59,Th11,RLAFFIN1:71;
q.x|--px=q.x|--If by A53,A56,A58,RLAFFIN1:77;
then A61: (q.x|--If).x<>0 by A57,A60,RLVECT_2:28;
conv px c=conv If by A53,RLAFFIN1:3;
then A62: (q.x|--If).x>=0 by A48,A56,RLAFFIN1:71;
A63: q.x in R by A30,A48,FUNCT_1:def 5;
then A64: L.(q.x)=1/card R by A37;
A65: dom G=dom F by A50,FINSEQ_3:31;
A66: now let m be Nat;
assume A67: m in dom F;
then G.m in R by A46,A51,A65,FUNCT_1:def 5;
then A68: L.(G.m)>0 & (G.m|--If).x>=0
by A37,A45,A48,RLAFFIN1:71;
F.m=L.(G.m)*(G.m|--If).x by A52,A67;
hence 0<=F.m by A68;
end;
consider n be set such that
A69: n in dom G and
A70: G.n=q.x by A39,A51,A63,FUNCT_1:def 5;
F.n=L.(q.x)*(q.x|--If).x by A52,A65,A69,A70;
then SLIf.x>0 by A49,A61,A62,A64,A65,A66,A69,RVSUM_1:115;
hence thesis by A48;
end;
Carrier SLIf c=If by RLVECT_2:def 8;
then Carrier SLIf=If & SLIf is convex
by A47,A35,A43,RLAFFIN1:71,XBOOLE_0:def 10;
then conv If c=Affin If & Sum SLIf in Int If by Th12,RLAFFIN1:65;
then SL in Int If by A44,RLAFFIN1:def 7;
hence contradiction by A3,A29,A35,XBOOLE_0:3;
end;
hence thesis by A4,A11,XBOOLE_0:def 8;
end;
theorem Th24:
Sum L1 <> Sum L2 & sum L1 = sum L2 implies ex v st L1.v > L2.v
proof
assume that
A1: Sum L1<>Sum L2 and
A2: sum L1=sum L2;
consider F be FinSequence such that
A3: rng F=Carrier(L1)\/Carrier(L2) and
A4: F is one-to-one by FINSEQ_4:73;
reconsider F as FinSequence of V by A3,FINSEQ_1:def 4;
A5: len(L2*F)=len F by FINSEQ_2:37;
A6: len(L1*F)=len F by FINSEQ_2:37;
then reconsider L1F=L1*F,L2F=L2*F as Element of len F-tuples_on REAL
by A5,FINSEQ_2:110;
A7: len(L2F-L1F)=len F by FINSEQ_1:def 18;
assume A8: for v be Element of V holds L1.v<=L2.v;
A9: for i be Nat st i in dom(L2F-L1F) holds 0<=(L2F-L1F).i
proof
let i be Nat;
L2.(F/.i)>=L1.(F/.i) by A8;
then A10: L2.(F/.i)-L1.(F/.i)>=L1.(F/.i)-L1.(F/.i) by XREAL_1:11;
assume A11: i in dom(L2F-L1F);
then i in dom F by A7,FINSEQ_3:31;
then A12: F/.i=F.i by PARTFUN1:def 8;
i in dom L2F by A5,A7,A11,FINSEQ_3:31;
then A13: L2F.i=L2.(F.i) by FUNCT_1:22;
i in dom L1F by A6,A7,A11,FINSEQ_3:31;
then L1F.i=L1.(F.i) by FUNCT_1:22;
hence thesis by A10,A12,A13,RVSUM_1:48;
end;
A14: Sum(L2F-L1F)=Sum L2F-Sum L1F by RVSUM_1:120
.=sum L2-Sum L1F by A3,A4,RLAFFIN1:30,XBOOLE_1:7
.=sum L2-sum L1 by A3,A4,RLAFFIN1:30,XBOOLE_1:7
.=0 by A2;
now let v be Element of V;
now per cases by A3,XBOOLE_0:def 3;
suppose A15: not v in Carrier(L1) & not v in Carrier(L2);
then L1.v=0;
hence L1.v=L2.v by A15;
end;
suppose v in rng F;
then consider i be set such that
A16: i in dom F and
A17: F.i=v by FUNCT_1:def 5;
reconsider i as Nat by A16;
i in dom L2F by A5,A16,FINSEQ_3:31;
then A18: (L2F-L1F).i=L2F.i-L1F.i & L2F.i=L2.v
by A17,FUNCT_1:22,RVSUM_1:48;
i in dom L1F by A6,A16,FINSEQ_3:31;
then A19: L1F.i=L1.v by A17,FUNCT_1:22;
A20: i in dom(L2F-L1F) by A7,A16,FINSEQ_3:31;
then L2.v-L1.v<=0 by A9,A14,A18,A19,RVSUM_1:115;
then L2.v-L1.v=0 by A9,A18,A19,A20;
hence L1.v=L2.v;
end;
end;
hence L1.v=L2.v;
end;
hence contradiction by A1,RLVECT_2:def 11;
end;
Lm3: L1.v<>L2.v implies ((r*L1+(1-r)*L2).v = s iff r = (L2.v-s)/(L2.v-L1.v))
proof
set u1=L1.v,u2=L2.v;
A1: (r*L1+(1-r)*L2).v=(r*L1).v+((1-r)*L2).v by RLVECT_2:def 12
.=r*u1+((1-r)*L2).v by RLVECT_2:def 13
.=r*u1+(-r+1)*u2 by RLVECT_2:def 13
.=r*(u1-u2)+u2;
assume A2: u1<>u2;
then A3: u1-u2<>0;
A4: u2-u1<>0 by A2;
hereby assume(r*L1+(1-r)*L2).v=s;
then r*(u2-u1)=(u2-s)*1 by A1;
then r/1=(u2-s)/(u2-u1) by A4,XCMPLX_1:95;
hence r=(u2-s)/(u2-u1);
end;
assume r=(u2-s)/(u2-u1);
hence (r*L1+(1-r)*L2).v=(u2-s)/(-(u1-u2))*(u1-u2)+u2 by A1
.=(-(u2-s))/(u1-u2)*(u1-u2)+u2 by XCMPLX_1:193
.=-(u2-s)+u2 by A3,XCMPLX_1:88
.=s;
end;
theorem Th25:
for p be Real st (r*L1+(1-r)*L2).v <= p & p <= (s*L1+(1-s)*L2).v
ex rs be Real st (rs*L1+(1-rs)*L2).v = p &
(r <= s implies r <= rs & rs <= s) &
(s <= r implies s <= rs & rs <= r)
proof
let p be Real;
set rv=(r*L1+(1-r)*L2).v,sv=(s*L1+(1-s)*L2).v;
set v1=L1.v,v2=L2.v;
A1: rv=(r*L1).v+((1-r)*L2).v by RLVECT_2:def 12
.=r*v1+((1-r)*L2).v by RLVECT_2:def 13
.=r*v1+(1-r)*v2 by RLVECT_2:def 13;
A2: sv=(s*L1).v+((1-s)*L2).v by RLVECT_2:def 12
.=s*v1+((1-s)*L2).v by RLVECT_2:def 13
.=s*v1+(1-s)*v2 by RLVECT_2:def 13;
assume that
A3: rv<=p and
A4: p<=sv;
per cases;
suppose A5: rv=sv;
take r;
thus thesis by A3,A4,A5,XXREAL_0:1;
end;
suppose rv<>sv;
then A6: sv-rv<>0;
set y=(p-rv)/(sv-rv);
set x=(sv-p)/(sv-rv);
take rs=r*x+s*y;
A7: r*x+r*y=r*(x+y) & s*x+s*y=s*(x+y);
A8: x+y=((sv-p)+(p-rv))/(sv-rv) by XCMPLX_1:63
.=1 by A6,XCMPLX_1:60;
A9: p-rv>=rv-rv by A3,XREAL_1:11;
thus p=p*(sv-rv)/(sv-rv) by A6,XCMPLX_1:90
.=(rv*(sv-p)+sv*(p-rv))/(sv-rv)
.=(rv*(sv-p))/(sv-rv)+(sv*(p-rv))/(sv-rv) by XCMPLX_1:63
.=(rv*(sv-p))/(sv-rv)+(p-rv)/(sv-rv)*sv by XCMPLX_1:75
.=x*(r*v1+(1-r)*v2)+y*(s*v1+(1-s)*v2) by A1,A2,XCMPLX_1:75
.=rs*v1+(x+y)*v2-rs*v2
.=rs*v1+1*v2-rs*v2 by A8
.=rs*v1+(1-rs)*v2
.=rs*v1+((1-rs)*L2).v by RLVECT_2:def 13
.=(rs*L1).v+((1-rs)*L2).v by RLVECT_2:def 13
.=(rs*L1+(1-rs)*L2).v by RLVECT_2:def 12;
A10: sv-rv>=sv-p & sv-p>=p-p by A3,A4,XREAL_1:11,12;
hereby assume r<=s;
then r*x<=s*x & r*y<=s*y by A9,A10,XREAL_1:66;
hence r<=rs & rs<=s by A7,A8,XREAL_1:8;
end;
assume A11: s<=r;
then A12: r*x>=s*x by A10,XREAL_1:66;
sv-rv>=p-rv by A4,XREAL_1:11;
then r*y>=s*y by A9,A11,XREAL_1:66;
hence thesis by A7,A8,A12,XREAL_1:8;
end;
end;
theorem
v in conv A & u in conv A & v <> u implies
ex p,w,r st p in A & w in conv(A\{p}) & 0<=r & r<1 & r*u+(1-r)*w = v
proof
reconsider Z=0 as Real by XREAL_0:def 1;
assume that
A1: v in conv A and
A2: u in conv A and
A3: v<>u;
reconsider A1=A as non empty Subset of V by A1;
A4: conv A1={Sum(L) where L is Convex_Combination of A1:L in ConvexComb(V)}
by CONVEX3:5;
then consider Lv be Convex_Combination of A1 such that
A5: v=Sum Lv and
A6: Lv in ConvexComb(V) by A1;
set Cv=Carrier(Lv);
A7: Cv c=A by RLVECT_2:def 8;
consider Lu be Convex_Combination of A1 such that
A8: u=Sum Lu and
Lu in ConvexComb(V) by A2,A4;
set Cu=Carrier(Lu);
A9: Cu c=A by RLVECT_2:def 8;
then A10: Cv\/Cu c=A by A7,XBOOLE_1:8;
per cases;
suppose not Cu c=Cv;
then consider p be set such that
A11: p in Cu and
A12: not p in Cv by TARSKI:def 3;
reconsider p as Element of V by A11;
Carrier Lv<>{} & Carrier(Lv)c=A\{p} by A7,A12,CONVEX1:21,ZFMISC_1:40;
then reconsider Ap=A\{p} as non empty Subset of V;
Carrier(Lv)c=Ap by A7,A12,ZFMISC_1:40;
then reconsider LV=Lv as Linear_Combination of Ap by RLVECT_2:def 8;
take p,w=v,Z;
A13: Z*u+(1-Z)*w=0.V+1*w by RLVECT_1:23
.=0.V+w by RLVECT_1:def 11
.=v by RLVECT_1:10;
Sum(LV) in {Sum(K) where K is Convex_Combination of Ap:
K in ConvexComb(V)} by A6;
hence thesis by A5,A9,A11,A13,CONVEX3:5;
end;
suppose A14: Cu c=Cv;
defpred P[set,set] means
for r for p be Element of V st r=$2 & p=$1 holds r<0 & Lv.p<>Lu.p &
(r*Lu+(1-r)*Lv).p=0;
set P = {r where r is Element of REAL:ex p be Element of V st
P[p,r] & p in Cv\/Cu};
A15: now let x;
assume x in P;
then ex r be Element of REAL st r=x & ex p be Element of V st
P[p,r] & p in Cv\/Cu;
hence x is real;
end;
A16: for p being Element of V,r,s being Element of REAL st
P[p,r] & P[p,s] holds r=s
proof
let p be Element of V,r,s be Element of REAL;
assume A17: P[p,r];
then A18: (r*Lu+(1-r)*Lv).p=0;
A19: Lv.p<>Lu.p by A17;
assume P[p,s];
then (s*Lu+(1-s)*Lv).p=0;
then s=(Lv.p-0)/(Lv.p-Lu.p) by A19,Lm3
.=r by A18,A19,Lm3;
hence thesis;
end;
sum Lu=1 & sum Lv=1 by RLAFFIN1:62;
then consider p be Element of V such that
A20: Lu.p>Lv.p by A3,A5,A8,Th24;
A21: Lv.p<>0
proof
assume A22: Lv.p=0;
then not p in Cu by A14,RLVECT_2:28;
hence contradiction by A20,A22;
end;
then p in Cv;
then A23: p in Cv\/Cu by XBOOLE_0:def 3;
set r=Lv.p/(Lv.p-Lu.p);
A24: r=(Lv.p-Z)/(Lv.p-Lu.p) & Lv.p-Lu.p0 by A21,RLAFFIN1:62;
then P[p,r] by A24,Lm3;
then A25: r in P by A23;
A26: Cv\/Cu is finite;
P is finite from FRAENKEL:sch 28(A26,A16);
then reconsider P as finite non empty real-membered set
by A15,A25,MEMBERED:def 3;
set M=max P;
M in P by XXREAL_2:def 8;
then consider r be Element of REAL such that
A27: M=r and
A28: ex p be Element of V st P[p,r] & p in Cv\/Cu;
set Lw=r*Lu+(1-r)*Lv;
consider p be Element of V such that
A29: P[p,r] and
A30: p in Cv\/Cu by A28;
set w=r*u+(1-r)*v,R=(-r)/(1-r);
A31: Sum Lw=Sum(r*Lu)+Sum((1-r)*Lv) by RLVECT_3:1
.=r*u+Sum((1-r)*Lv) by A8,RLVECT_3:2
.=w by A5,RLVECT_3:2;
A32: for z be Element of V holds 0<=Lw.z
proof
let z be Element of V;
A33: (Z*Lu+(1-Z)*Lv).z=(Z*Lu).z+((1-Z)*Lv).z by RLVECT_2:def 12
.=Z*(Lu.z)+((1-Z)*Lv).z by RLVECT_2:def 13
.=Z*(Lu.z)+(1-0)*(Lv.z) by RLVECT_2:def 13
.=Lv.z;
assume A34: 0>Lw.z;
A35: Lw.z=(r*Lu).z+((1-r)*Lv).z by RLVECT_2:def 12
.=r*(Lu.z)+((1-r)*Lv).z by RLVECT_2:def 13
.=r*(Lu.z)+(1-r)*(Lv.z) by RLVECT_2:def 13;
A36: Lv.z<>0
proof
assume A37: Lv.z=0;
then not z in Cu by A14,RLVECT_2:28;
then Lu.z=0;
hence contradiction by A34,A35,A37;
end;
then z in Cv;
then A38: z in Cv\/Cu by XBOOLE_0:def 3;
Lv.z>=0 by RLAFFIN1:62;
then consider rs be Real such that
A39: (rs*Lu+(1-rs)*Lv).z=0 and
A40: r<=0 implies r<=rs & rs<=0 and
0<=r implies 0<=rs & rs<=r by A33,A34,Th25;
rs<>0 by A33,A36,A39;
then P[z,rs] by A29,A34,A35,A39,A40,RLAFFIN1:62;
then rs in P by A38;
then rs<=r by A27,XXREAL_2:def 8;
then rs=r by A28,A40,XXREAL_0:1;
hence contradiction by A34,A39;
end;
r*Lu is Linear_Combination of A & (1-r)*Lv is Linear_Combination of A
by RLVECT_2:67;
then Lw is Linear_Combination of A by RLVECT_2:59;
then A41: Carrier Lw c=A by RLVECT_2:def 8;
Lw.p=0 by A29;
then not p in Carrier Lw by RLVECT_2:28;
then A42: Carrier Lw c=A\{p} by A41,ZFMISC_1:40;
A43: sum Lw=sum(r*Lu)+sum((1-r)*Lv) by RLAFFIN1:34
.=r*sum Lu+sum((1-r)*Lv) by RLAFFIN1:35
.=r*1+sum((1-r)*Lv) by RLAFFIN1:62
.=r*1+(1-r)*sum(Lv) by RLAFFIN1:35
.=r*1+(1-r)*1 by RLAFFIN1:62
.=1;
then Lw is convex by A32,RLAFFIN1:62;
then Carrier Lw<>{} by CONVEX1:21;
then reconsider Ap=A\{p} as non empty Subset of V by A42;
reconsider LW=Lw as Linear_Combination of Ap by A42,RLVECT_2:def 8;
A44: LW is convex by A32,A43,RLAFFIN1:62;
then LW in ConvexComb(V) by CONVEX3:def 1;
then A45: Sum(LW) in {Sum(K) where K is Convex_Combination of Ap:K in
ConvexComb(V)} by A44;
take p,w,R;
A46: 0>r by A29;
then A47: 1+-r>0+-r & (1+-r)/(1+-r)=1 by XCMPLX_1:60,XREAL_1:8;
R*u+(1-R)*w=(-r)/(1-r)*u+((1-r)/(1-r)-(-r)/(1-r))*w by A46,XCMPLX_1:60
.=(-r)/(1-r)*u+((1-r-(-r))/(1-r))*w by XCMPLX_1:121
.=(-r)/(1-r)*u+(1/(1-r)*(r*u)+(1/(1-r))*((1-r)*v)) by RLVECT_1:def 8
.=(-r)/(1-r)*u+((1/(1-r)*r)*u+(1/(1-r))*((1-r)*v)) by RLVECT_1:
def 10
.=(-r)/(1-r)*u+((1/(1-r)*r)*u+(1/(1-r)*(1-r))*v) by RLVECT_1:
def 10
.=(-r)/(1-r)*u+((r/(1-r)*1)*u+(1/(1-r)*(1-r))*v) by XCMPLX_1:76
.=(-r)/(1-r)*u+(r/(1-r)*u+1*v) by A46,XCMPLX_1:88
.=((-r)/(1-r)*u+r/(1-r)*u)+1*v by RLVECT_1:def 6
.=((-r)/(1-r)+r/(1-r))*u+1*v by RLVECT_1:def 9
.=(-r+r)/(1-r)*u+1*v by XCMPLX_1:63
.=0/(1-r)*u+v by RLVECT_1:def 11
.=0.V+v by RLVECT_1:23
.=v by RLVECT_1:def 7;
hence thesis by A10,A30,A31,A45,A46,A47,CONVEX3:5,XREAL_1:76;
end;
end;
theorem Th27:
A\/{v} is affinely-independent iff
A is affinely-independent & (v in A or not v in Affin A) by RLAFFIN1:82;
theorem Th28:
Af c= I & v in Af implies
I\{v}\/{(center_of_mass V).Af} is affinely-independent Subset of V
proof
assume that
A1: Af c=I and
A2: v in Af;
set Iv=I\{v},Av=Af\{v};
A3: Iv\/{v}=I by A1,A2,ZFMISC_1:140;
set BA=(center_of_mass V)/.Af;
A4: (center_of_mass V).Af=1/card Af*Sum Af by A2,Def2;
A5: dom(center_of_mass V)=BOOL the carrier of V by FUNCT_2:def 1;
then Af in dom(center_of_mass V) by A2,ZFMISC_1:64;
then A6: (center_of_mass V).Af=(center_of_mass V)/.Af by PARTFUN1:def 8;
per cases;
suppose Af={v};
then Sum Af=v & card Af=1 by CARD_1:50,RLVECT_2:15;
hence thesis by A3,A4,RLVECT_1:def 11;
end;
suppose A7: Af<>{v};
A8: not BA in Affin Iv
proof
A9: Av is non empty
proof
assume Av is empty;
then Af c={v} by XBOOLE_1:37;
hence contradiction by A2,A7,ZFMISC_1:39;
end;
then Av in dom(center_of_mass V) by A5,ZFMISC_1:64;
then A10: (center_of_mass V).Av=(center_of_mass V)/.Av
by PARTFUN1:def 8;
Av c=Iv by A1,XBOOLE_1:33;
then A11: Affin Av c=Affin Iv by RLAFFIN1:52;
reconsider c =card Af as Real by XREAL_0:def 1;
A12: c/c =c*(1/c) by XCMPLX_1:100;
conv Av c=Affin Av & (center_of_mass V).Av in conv Av
by A9,Th16,RLAFFIN1:65;
then A13: (center_of_mass V).Av in Affin Av;
assume BA in Affin Iv;
then A14:not v in Iv &
(1-c)*(center_of_mass V)/.Av+c*((center_of_mass V)/.Af)
in Affin Iv by A10,A11,A13,RUSUB_4:def 5,ZFMISC_1:64;
(center_of_mass V)/.Af-(1-1/c)*(center_of_mass V)/.Av =
(1-1/c)*(center_of_mass V)/.Av+1/c*v-(1-1/c)*(center_of_mass V)/.Av
by A2,A6,A9,Th22;
then A15: 1/c*v=(center_of_mass V)/.Af-(1-1/c)*(center_of_mass V)/.Av
by RLVECT_4:1
.=(center_of_mass V)/.Af+-(1-1/c)*(center_of_mass V)/.Av
by RLVECT_1:def 14
.=(-(1-1/c))*(center_of_mass V)/.Av+(center_of_mass V)/.Af
by RLVECT_4:6;
A16: 1=c/c by A2,XCMPLX_1:60;
(1-c)*(center_of_mass V)/.Av+c*((center_of_mass V)/.Af)
=1*((1-c)*(center_of_mass V)/.Av+c*((center_of_mass V)/.Af))
by RLVECT_1:def 11
.=c*(1/c*((1-c)*(center_of_mass V)/.Av+c*(center_of_mass V)/.Af))
by A12,A16,RLVECT_1:def 10
.=c*(1/c*((1-c)*(center_of_mass V)/.Av)+
1/c*(c*(center_of_mass V)/.Af))
by RLVECT_1:def 8
.=c*(1/c*((1-c)*(center_of_mass V)/.Av)+1*(center_of_mass V)/.Af)
by A12,A16,RLVECT_1:def 10
.=c*(1/c*((1-c)*(center_of_mass V)/.Av)+(center_of_mass V)/.Af)
by RLVECT_1:def 11
.=c*(1/c*(1-c)*(center_of_mass V)/.Av+(center_of_mass V)/.Af)
by RLVECT_1:def 10
.=1*v by A16,A15,A12,RLVECT_1:def 10
.=v by RLVECT_1:def 11;
hence contradiction by A3,A14,Th27;
end;
Iv is affinely-independent by RLAFFIN1:43,XBOOLE_1:36;
hence thesis by A6,A8,Th27;
end;
end;
theorem Th29:
for F be c=-linear Subset-Family of V st
union F is finite affinely-independent
holds (center_of_mass V).:F is affinely-independent Subset of V
proof
set B=center_of_mass V;
defpred P[Nat] means
for k be Nat st k<=$1for S be c=-linear Subset-Family of V st
card union S=k & union S is finite affinely-independent
holds B.:S is affinely-independent Subset of V;
let S be c=-linear Subset-Family of V;
A1: dom B=BOOL the carrier of V by FUNCT_2:def 1;
A2: for n be Nat st P[n] holds P[n+1]
proof
let n be Nat such that
A3: P[n];
let k be Nat such that
A4: k<=n+1;
per cases by A4,NAT_1:8;
suppose k<=n;
hence thesis by A3;
end;
suppose A5: k=n+1;
let S be c=-linear Subset-Family of V such that
A6: card union S=k and
A7: union S is finite affinely-independent;
set U=union S;
A8: S c=bool U by ZFMISC_1:100;
set SU=S\{U};
A9: union SU c=U by XBOOLE_1:36,ZFMISC_1:95;
then reconsider Usu=union SU as finite set by A7;
A10: SU c=S by XBOOLE_1:36;
A11: union SU<>U
proof
assume A12: union SU=U;
then SU is non empty by A5,A6,ZFMISC_1:2;
then union SU in SU by A7,A10,A8,SIMPLEX0:9;
hence contradiction by A12,ZFMISC_1:64;
end;
then union SU c__{};
then consider x such that
A21: x in SU/\dom B by XBOOLE_0:def 1;
x in SU by A21,XBOOLE_0:def 4;
then A22: x c=union SU by ZFMISC_1:92;
then x c=U by A9,XBOOLE_1:1;
then A23: x=U or x={} by A20,ZFMISC_1:39;
not v in x by A14,A22;
hence thesis by A20,A21,A23,TARSKI:def 1,ZFMISC_1:64;
end;
then B.:SU=B.:{} by RELAT_1:145
.={} by RELAT_1:149;
hence thesis by A17;
end;
suppose A24: n>0;
reconsider u=U as finite set by A7;
A25: Usu c=u by XBOOLE_1:36,ZFMISC_1:95;
then card Usu<=n+1 by A5,A6,NAT_1:44;
then card Usu1 by A5,A6,A24;
then not BU in U by A7,Th19;
then not BU in U\{v} by XBOOLE_0:def 5;
then not BU in Affin BSU by A15,A33,Th27;
hence thesis by A17,Th27;
end;
end;
end;
A34: P[0 qua Nat]
proof
let k be Nat;
assume A35: k<=0;
let S be c=-linear Subset-Family of V such that
A36: card union S=k and
union S is finite affinely-independent;
A37: union S={} by A35,A36;
S/\dom B={}
proof
assume S/\dom B<>{};
then consider x such that
A38: x in S/\dom B by XBOOLE_0:def 1;
x in S by A38,XBOOLE_0:def 4;
then A39: x c={} by A37,ZFMISC_1:92;
x<>{} by A38,ZFMISC_1:64;
hence contradiction by A39;
end;
then B.:S=B.:{} by RELAT_1:145
.={} by RELAT_1:149;
hence thesis;
end;
A40: for n be Nat holds P[n] from NAT_1:sch 2(A34,A2);
assume A41: union S is finite affinely-independent;
then card union S is Nat;
hence thesis by A40,A41;
end;
theorem
for F be c=-linear Subset-Family of V st
union F is affinely-independent finite
holds Int ((center_of_mass V).:F) c= Int union F
proof
set B=center_of_mass V;
let S be c=-linear Subset-Family of V such that
A1: union S is affinely-independent finite;
reconsider BS=B.:S as affinely-independent Subset of V by A1,Th29;
set U=union S;
let x such that
A2: x in Int(B.:S);
BS is non empty by A2;
then consider y such that
A3: y in BS by XBOOLE_0:def 1;
consider X be set such that
A4: X in dom B and
A5: X in S and
B.X=y by A3,FUNCT_1:def 12;
X c=U & X is non empty by A4,A5,ZFMISC_1:64,92;
then reconsider U as non empty finite Subset of V by A1;
set xBS=x|--BS;
A6: Int BS c=conv BS by Lm2;
then A7: xBS is convex by A2,RLAFFIN1:71;
S c=bool U
proof
let s be set;
assume s in S;
then s c=U by ZFMISC_1:92;
hence thesis;
end;
then A8: U in S by A5,SIMPLEX0:9;
dom B=(bool the carrier of V)\{{}} by FUNCT_2:def 1;
then U in dom B by ZFMISC_1:64;
then A9: B.U in BS by A8,FUNCT_1:def 12;
then reconsider BU=B.U as Element of V;
conv BS c=Affin BS by RLAFFIN1:65;
then A10: Int BS c=Affin BS by A6,XBOOLE_1:1;
then A11: Sum xBS=x by A2,RLAFFIN1:def 7;
then Carrier xBS=BS by A2,A6,Th11,RLAFFIN1:71;
then A12: xBS.BU<>0 by A9,RLVECT_2:28;
then A13: xBS.BU>0 by A2,A6,RLAFFIN1:71;
set xU=x|--U;
A14: conv U c=Affin U by RLAFFIN1:65;
A15: conv(B.:S)c=conv U by Th17,CONVEX1:30;
then A16: Int BS c=conv U by A6,XBOOLE_1:1;
then Int BS c=Affin U by A14,XBOOLE_1:1;
then A17: Sum xU=x by A1,A2,RLAFFIN1:def 7;
BS c=conv BS by RLAFFIN1:2;
then A18: B.U in conv BS by A9;
per cases;
suppose x=B.U;
hence thesis by A1,Th20;
end;
suppose x<>BU;
then consider p be Element of V such that
A19: p in conv(BS\{BU}) and
A20: Sum xBS=xBS.BU*BU+(1-xBS.BU)*p and
1/xBS.BU*(Sum xBS)+(1-1/xBS.BU)*p=BU by A7,A11,A12,Th1;
A21: x=(1-xBS.BU)*p+xBS.BU*BU by A2,A10,A20,RLAFFIN1:def 7;
xBS.BU<=1 by A2,A6,RLAFFIN1:71;
then A22: 1-xBS.BU>=1-1 by XREAL_1:12;
A23: BU in conv U by A15,A18;
conv(BS\{BU})c=conv BS by RLAFFIN1:3,XBOOLE_1:36;
then A24: p in conv BS by A19;
then p in conv U by A15;
then A25: xU=(1-xBS.BU)*(p|--U)+xBS.BU*(BU|--U)
by A1,A14,A21,A23,RLAFFIN1:70;
A26: U c=Carrier xU
proof
let u be set;
assume A27: u in U;
then A28: xU.u=((1-xBS.BU)*(p|--U)).u+(xBS.BU*(BU|--U)).u
by A25,RLVECT_2:def 12
.=((1-xBS.BU)*(p|--U)).u+xBS.BU*((BU|--U).u) by A27,RLVECT_2:def 13
.=(1-xBS.BU)*((p|--U).u)+xBS.BU*((BU|--U).u) by A27,RLVECT_2:def 13;
(BU|--U).u=1/card U & (p|--U).u>=0 by A1,A15,A24,A27,Th18,RLAFFIN1:71;
hence thesis by A13,A22,A27,A28;
end;
Carrier xU c=U by RLVECT_2:def 8;
then Carrier xU=U by A26,XBOOLE_0:def 10;
hence thesis by A1,A2,A16,A17,Th12,RLAFFIN1:71;
end;
end;
__