reserve M for non empty MetrSpace,
        F,G for open Subset-Family of TopSpaceMetr M;
reserve L for Lebesgue_number of F;
reserve n,k for Nat,
        r for Real,
        X for set,
        M for Reflexive non empty MetrStruct,
        A for Subset of M,
        K for SimplicialComplexStr;
reserve V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve A for Subset of TOP-REAL n;
reserve A for affinely-independent Subset of TOP-REAL n;

theorem Th23:
  for A st card A = n+1
  for f be continuous Function of(TOP-REAL n)|conv A,(TOP-REAL n)|conv A
  ex p be Point of TOP-REAL n st p in dom f & f.p=p
  proof
  set TRn=TOP-REAL n;
  let A be affinely-independent Subset of TRn such that
   A1: card A=n+1;
  A2: A is non empty by A1;
  set cA=conv A;
  let f be continuous Function of TRn|cA,TRn|cA;
  reconsider cA as non empty Subset of TRn by A2;
  set T=TRn|cA;
  reconsider ff=f as continuous Function of T,T;
  set E=the Enumeration of A;
  deffunc P(object)
    ={v where v is Element of T: |--(A,E.$1).(f.v)<=|--(A,E.$1).v};
  consider F be FinSequence such that
   A3: len F=card A & for k st k in dom F holds F.k=P(k) from FINSEQ_1:sch 2;
  rng F c=bool the carrier of T
  proof
   let y be object;
   assume y in rng F;
   then consider x be object such that
    A4: x in dom F and
    A5: F.x=y by FUNCT_1:def 3;
   A6: P(x)c=the carrier of T
   proof
    let z be object;
    assume z in P(x);
    then ex v be Element of T st z=v & |--(A,E.x).(f.v)<=|--(A,E.x).v;
    hence thesis;
   end;
   F.x=P(x) by A3,A4;
   hence thesis by A5,A6;
  end;
  then reconsider F as FinSequence of bool the carrier of T by FINSEQ_1:def 4;
  A7: [#]T=cA by PRE_TOPC:def 5;
  A8: dom f=the carrier of T by FUNCT_2:def 1;
  now let W be Subset of T;
   reconsider Z=0 as Real;
   reconsider L=].-infty,Z.] as Subset of R^1 by TOPMETR:17;
   assume W in rng F;
   then consider i be object such that
    A9: i in dom F and
    A10: F.i=W by FUNCT_1:def 3;
   reconsider AEi=|--(A,E.i) as continuous Function of TRn,R^1 by A1,
RLAFFIN3:32;
   set AT=AEi|T;
   set Af=AT*ff;
   set AfT=Af-AT;
   A11: dom AfT=the carrier of T by FUNCT_2:def 1;
   reconsider AfT as Function of T,R^1 by TOPMETR:17;
   A12: dom AT=the carrier of T by FUNCT_2:def 1;
   A13: AT=AEi|the carrier of T by A7,TMAP_1:def 3;
   A14: dom Af=the carrier of T by FUNCT_2:def 1;
   A15: AfT"L c=P(i)
   proof
    let x be object;
    reconsider xx=x as set by TARSKI:1;
    assume A16: x in AfT"L;
    then reconsider v=x as Point of T;
    x in dom AfT by A16,FUNCT_1:def 7;
    then A17: AfT.x=Af.xx-AT.xx by VALUED_1:13;
    AfT.x in L & Af.x=AT.(f.x) by A14,A16,FUNCT_1:12,def 7;
    then AT.(f.xx)-AT.xx<=0 by A17,XXREAL_1:2;
    then A18: AT.(f.x)<=AT.xx by XREAL_1:50;
    f.x in dom AT by A14,A16,FUNCT_1:11;
    then A19: AT.(f.x)=AEi.(f.v) by A13,FUNCT_1:47;
    AT.x=AEi.v by A12,A13,FUNCT_1:47;
    hence thesis by A18,A19;
   end;
   A20: P(i)c=AfT"L
   proof
    let x be object;
    assume x in P(i);
    then consider v be Element of T such that
     A21: x=v and
     A22: AEi.(f.v)<=AEi.v;
    f.v in rng f by A8,FUNCT_1:def 3;
    then A23: AEi.(f.v)=AT.(f.v) by A12,A13,FUNCT_1:47;
    AEi.v=AT.v by A12,A13,FUNCT_1:47;
    then Af.v<=AT.v by A14,A22,A23,FUNCT_1:12;
    then Af.v-AT.v<=0 by XREAL_1:47;
    then AfT.v<=0 by A11,VALUED_1:13;
    then AfT.v in L by XXREAL_1:234;
    hence thesis by A11,A21,FUNCT_1:def 7;
   end;
   L is closed by BORSUK_5:41;
   then A24: AfT"L is closed by PRE_TOPC:def 6;
   F.i=P(i) by A3,A9;
   hence W is closed by A10,A15,A20,A24,XBOOLE_0:def 10;
  end;
  then A25: rng F is closed by TOPS_2:def 2;
  A26: dom E=Seg len E by FINSEQ_1:def 3;
  A27: conv A c=Affin A by RLAFFIN1:65;
  A28: rng E=A by RLAFFIN3:def 1;
  then len E=card A by FINSEQ_4:62;
  then A29: dom F=dom E by A3,FINSEQ_3:29;
  for S be Subset of dom F holds conv(E.:S)c=union(F.:S)
  proof
   let S be Subset of dom F;
   set ES=E.:S;
   per cases;
   suppose S is empty;
    then conv ES is empty;
    hence thesis;
   end;
   suppose A30: S is non empty;
    let x be object;
    set fx=f.x,xES=x|--ES,fxA=fx|--A,xA=x|--A;
    assume A31: x in conv ES;
    A32: conv ES c=conv A by A28,RELAT_1:111,RLAFFIN1:3;
    then reconsider v=x as Point of T by A7,A31;
    A33: len(fxA*E)=len E by FINSEQ_2:33;
    A34: len E=len(xES*E) by FINSEQ_2:33;
    then reconsider fxAE=fxA*E,xESE=xES*E as Element of len E-tuples_on REAL
by A33,FINSEQ_2:92;
    A35: dom fxAE=Seg len E by A33,FINSEQ_1:def 3;
    A36: conv ES c=Affin ES by RLAFFIN1:65;
    then A37: xA=xES by A28,A31,RELAT_1:111,RLAFFIN1:77;
    A38: ES c=A by A28,RELAT_1:111;
    A39: Carrier xES c=ES by RLVECT_2:def 6;
    ES is affinely-independent by A28,RELAT_1:111,RLAFFIN1:43;
    then sum xES=1 & Carrier xES c= A by A31,A36,A38,A39,RLAFFIN1:def 7;
    then A40: Carrier fxA c=A & 1=Sum(xES*E) by A28,RLAFFIN1:30,RLVECT_2:def 6;
    A41: fx in rng f by A7,A8,A31,A32,FUNCT_1:def 3;
    then A42: fx in conv A by A7;
    then sum fxA=1 by A27,RLAFFIN1:def 7;
    then A43: Sum fxAE=Sum xESE by A28,A40,RLAFFIN1:30;
    A44: dom xESE=Seg len E by A34,FINSEQ_1:def 3;
    per cases by A43,RVSUM_1:83;
    suppose ex j be Nat st j in Seg len E & fxAE.j<xESE.j;
     then consider j be Nat such that
      A45: j in Seg len E and
      A46: fxAE.j<xESE.j;
     A47: fxAE.j=fxA.(E.j) by A35,A45,FUNCT_1:12;
     A48: xESE.j=xES.(E.j) by A44,A45,FUNCT_1:12;
     then xESE.j=|--(A,E.j).x by A31,A37,RLAFFIN3:def 3;
     then A49: |--(A,E.j).(f.v)<=|--(A,E.j).v by A42,A46,A47,RLAFFIN3:def 3;
     A50: E.j in dom fxA by A35,A45,FUNCT_1:11;
     then 0<xES.(E.j) by A7,A41,A46,A47,A48,RLAFFIN1:71;
     then E.j in Carrier xES by A50,RLVECT_2:19;
     then consider i be object such that
      A51: i in dom E and
      A52: i in S and
      A53: E.i=E.j by A39,FUNCT_1:def 6;
     i=j by A26,A45,A51,A53,FUNCT_1:def 4;
     then A54: F.j in F.:S by A52,FUNCT_1:def 6;
     P(j)=F.j by A3,A26,A29,A45;
     then x in F.j by A49;
     hence thesis by A54,TARSKI:def 4;
    end;
    suppose A55: for j be Nat st j in Seg len E holds fxAE.j<=xESE.j;
     consider j be object such that
      A56: j in S by A30,XBOOLE_0:def 1;
     reconsider j as Nat by A56;
     A57: fxAE.j<=xESE.j by A26,A29,A55,A56;
     A58: P(j)=F.j by A3,A56;
     xESE.j=xES.(E.j) by A26,A29,A44,A56,FUNCT_1:12;
     then A59: xESE.j=|--(A,E.j).x by A31,A37,RLAFFIN3:def 3;
     fxAE.j=fxA.(E.j) by A26,A29,A35,A56,FUNCT_1:12;
     then |--(A,E.j).(f.v)<=|--(A,E.j).v by A42,A59,A57,RLAFFIN3:def 3;
     then A60: x in F.j by A58;
     F.j in F.:S by A56,FUNCT_1:def 6;
     hence thesis by A60,TARSKI:def 4;
    end;
   end;
  end;
  then meet rng F is non empty by A2,A3,A25,Th22;
  then consider v be object such that
   A61: v in meet rng F by XBOOLE_0:def 1;
  A62: v in cA by A7,A61;
  then reconsider v as Element of TRn;
  set fv=f.v,vA=v|--A,fvA=fv|--A;
  A63: len(fvA*E)=len E by FINSEQ_2:33;
  fv in rng f by A8,A61,FUNCT_1:def 3;
  then A64: fv in cA by A7;
  then Carrier fvA c=A & sum fvA=1 by A27,RLAFFIN1:def 7,RLVECT_2:def 6;
  then A65: Carrier vA c=A & 1=Sum(fvA*E) by A28,RLAFFIN1:30,RLVECT_2:def 6;
  A66: len E=len(vA*E) by FINSEQ_2:33;
  then reconsider fvAE=fvA*E,vAE=vA*E as Element of len E-tuples_on REAL by A63
,FINSEQ_2:92;
  A67: dom fvAE=Seg len E by A63,FINSEQ_1:def 3;
  A68: dom vAE=Seg len E by A66,FINSEQ_1:def 3;
  A69: for j be Nat st j in Seg len E holds fvAE.j<=vAE.j
  proof
   let j be Nat;
   assume A70: j in Seg len E;
   then F.j=P(j) & F.j in rng F by A3,A26,A29,FUNCT_1:def 3;
   then v in P(j) by A61,SETFAM_1:def 1;
   then A71: ex w be Element of T st w=v & |--(A,E.j).(f.w)<=|--(A,E.j).w;
   A72: |--(A,E.j).v=vA.(E.j) by RLAFFIN3:def 3
    .=vAE.j by A68,A70,FUNCT_1:12;
   |--(A,E.j).fv=fvA.(E.j) by A64,RLAFFIN3:def 3
    .=fvAE.j by A67,A70,FUNCT_1:12;
   hence thesis by A71,A72;
  end;
  A73: Carrier vA c=A by RLVECT_2:def 6;
  sum vA=1 by A27,A62,RLAFFIN1:def 7;
  then A74: Sum fvAE=Sum vAE by A28,A65,RLAFFIN1:30;
  A75: Carrier fvA c=A by RLVECT_2:def 6;
  A76: now let w be Element of TRn;
   per cases;
   suppose w in A;
    then consider j be object such that
     A77: j in dom E and
     A78: E.j=w by A28,FUNCT_1:def 3;
    A79: fvAE.j=fvA.w & vAE.j=vA.w by A77,A78,FUNCT_1:13;
    fvAE.j<=vAE.j & fvAE.j>=vAE.j by A26,A74,A69,A77,RVSUM_1:83;
    hence vA.w=fvA.w by A79,XXREAL_0:1;
   end;
   suppose A80: not w in A;
    then not w in Carrier vA by A73;
    then A81: vA.w=0 by RLVECT_2:19;
    not w in Carrier fvA by A75,A80;
    hence vA.w=fvA.w by A81,RLVECT_2:19;
   end;
  end;
  A82: Sum vA=v by A27,A62,RLAFFIN1:def 7;
  Sum fvA=fv by A27,A64,RLAFFIN1:def 7;
  then v=fv by A76,A82,RLVECT_2:def 9;
  hence thesis by A8,A61;
 end;
