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;

theorem Th22:
  for A be non empty affinely-independent Subset of TOP-REAL n
  for E be Enumeration of A
  for F be FinSequence of bool the carrier of ((TOP-REAL n)|(conv A)) st
    len F = card A & rng F is closed &
    for S be Subset of dom F holds conv(E.:S) c= union(F.:S)
      holds meet rng F is non empty
 proof
  set TRn=TOP-REAL n;
  let A be non empty affinely-independent Subset of TRn;
  set cA=conv A;
  let E be Enumeration of A;
  let F be FinSequence of bool the carrier of(TRn|cA) such that
   A1: len F=card A and
   A2: rng F is closed and
   A3: for S be Subset of dom F holds conv(E.:S)c=union(F.:S);
  A4: F<>{} by A1;
  A5: rng E=A by RLAFFIN3:def 1;
  then len E=card A by FINSEQ_4:62;
  then A6: dom E=dom F by A1,FINSEQ_3:29;
  set En=Euclid n;
  set Comp=Complex_of{A};
  defpred P[object,object] means
   $1 in F.(E".$2) & for B be Subset of TRn st B c=A & $1 in conv B holds$2 in
B;
  A7: TopSpaceMetr En=the TopStruct of TRn by EUCLID:def 8;
  then reconsider CA=cA as non empty Subset of TopSpaceMetr En;
  reconsider ca=cA as non empty Subset of En by A7;
  A8: (TopSpaceMetr En)|CA=TopSpaceMetr(En|ca) by HAUSDORF:16;
  then reconsider CrF=COMPLEMENT(rng F) as Subset-Family of TopSpaceMetr(En|ca)
by A7,PRE_TOPC:36;
  CA is compact by A7,COMPTS_1:23;
  then A9: TopSpaceMetr(En|ca) is compact by A8,COMPTS_1:3;
  A10: (TopSpaceMetr En)|CA=TRn|cA by A7,PRE_TOPC:36;
  assume meet rng F is empty;
  then [#](TRn|cA)=(meet rng F)`
   .=union CrF by A4,TOPS_2:7;
  then A11: CrF is Cover of TopSpaceMetr(En|ca) by A8,A10,SETFAM_1:45;
  set L=the Lebesgue_number of CrF;
  A12: |.Comp.|c=[#]Comp;
  then consider k be Nat such that
   A13: diameter BCS(k,Comp)<L by Th18;
  set Bcs=BCS(k,Comp);
  A14: |.Bcs.|=|.Comp.| by A12,SIMPLEX1:19;
  A15: the topology of Comp=bool A by SIMPLEX0:4;
  A16: for x be object st x in Vertices Bcs
ex y be object st y in A & P[x,y]
  proof
   let x be object;
   assume A17: x in Vertices Bcs;
   then reconsider v=x as Element of Bcs;
   v is vertex-like by A17,SIMPLEX0:def 4;
   then consider S be Subset of Bcs such that
    A18: S is simplex-like and
    A19: v in S;
   @S c=conv@S by RLAFFIN1:2;
   then A20: v in conv@S by A19;
   conv@S c=|.Comp.| by A14,A18,SIMPLEX1:5;
   then consider W be Subset of Comp such that
    A21: W is simplex-like and
    A22: v in Int@W by A20,SIMPLEX1:6;
   A23: v in conv@W by A22,RLAFFIN2:def 1;
   A24: W in the topology of Comp by A21,PRE_TOPC:def 2;
   then E"W c=dom E & E.:(E"W)=W by A5,A15,FUNCT_1:77,RELAT_1:132;
   then conv@W c=union(F.:(E"W)) by A3,A6;
   then consider Y be set such that
    A25: v in Y and
    A26: Y in F.:(E"W) by A23,TARSKI:def 4;
   consider i be object such that
    i in dom F and
    A27: i in E"W and
    A28: F.i=Y by A26,FUNCT_1:def 6;
   take y=E.i;
   A29: y in W by A27,FUNCT_1:def 7;
   i in dom E by A27,FUNCT_1:def 7;
   hence y in A & x in F.(E".y) by A15,A24,A25,A28,A29,FUNCT_1:34;
   let B be Subset of TRn;
   assume that
    A30: B c=A and
    A31: x in conv B;
   reconsider b=B as Simplex of Comp by A15,A30,PRE_TOPC:def 2;
   conv@b meets Int@W by A22,A31,XBOOLE_0:3;
   then W c=b by A21,SIMPLEX1:26;
   hence thesis by A29;
  end;
  consider G be Function of Vertices Bcs,A such that
   A32: for x be object st x in Vertices Bcs holds P[x,G.x]
from FUNCT_2:sch 1(A16
);
  A33: |.Comp.|=conv A by SIMPLEX1:8;
  then Bcs is with_non-empty_element by A14,SIMPLEX1:7;
  then for v be Vertex of Bcs,B be Subset of TRn st B c=A & v in conv B holds G
.v in B by A32;
  then consider S be Simplex of card A-1,Bcs such that
   A34: G.:S=A by SIMPLEX1:47;
  A35: [#]Bcs=[#]Comp by A12,SIMPLEX1:18;
  then reconsider SS=S as Subset of En by A7;
  S is non empty by A34;
  then consider s be object such that
   A36: s in S by XBOOLE_0:def 1;
  A37: conv@S c=cA by A14,A33,SIMPLEX1:5;
  reconsider s as Point of En by A7,A35,A36;
  A38: S c=conv@S by RLAFFIN1:2;
  then s in conv@S by A36;
  then reconsider ss=s as Point of En|ca by A37,TOPMETR:def 2;
  A39: SS is bounded;
  CrF is open by A2,A8,A10,TOPS_2:9;
  then consider CRF be Subset of TopSpaceMetr(En|ca) such that
   A40: CRF in CrF and
   A41: Ball(ss,L)c=CRF by A9,A11,Def1;
  CRF` in rng F by A8,A10,A40,SETFAM_1:def 7;
  then consider i be object such that
   A42: i in dom F and
   A43: F.i=CRF` by FUNCT_1:def 3;
  E.i in A by A5,A6,A42,FUNCT_1:def 3;
  then consider w be object such that
   A44: w in dom G and
   A45: w in S and
   A46: G.w=E.i by A34,FUNCT_1:def 6;
  A47: w in conv@S by A38,A45;
  A48: conv@S c=cA by A14,A33,SIMPLEX1:5;
  then A49: [#](En|ca)=ca & w in cA by A47,TOPMETR:def 2;
  reconsider SS=S as bounded Subset of En by A39;
  diameter SS<=diameter BCS(k,Comp) by Def4;
  then A50: diameter SS<L by A13,XXREAL_0:2;
  w in F.(E".(G.w)) by A32,A44;
  then A51: w in F.i by A6,A42,A46,FUNCT_1:34;
  reconsider w as Point of En by A49;
  reconsider ww=w as Point of En|ca by A47,A48,TOPMETR:def 2;
  conv@S c=cl_Ball(s,diameter SS) by A36,A38,Th13;
  then dist(s,w)=dist(ss,ww) & dist(s,w)<=diameter SS by A47,METRIC_1:12
,TOPMETR:def 1;
  then dist(ss,ww)<L by A50,XXREAL_0:2;
  then ww in Ball(ss,L) by METRIC_1:11;
  hence contradiction by A41,A43,A51,XBOOLE_0:def 5;
 end;
