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;

theorem Th12:
  for A be affinely-independent Subset of TOP-REAL n
  for E be Enumeration of A st dom E\X is non empty
    holds
  conv(E.:X) = meet{conv(A\{E.k}) where k is Element of NAT: k in dom E\X}
  proof
  let A be affinely-independent Subset of TOP-REAL n;
  let E be Enumeration of A such that
   A1: dom E\X is non empty;
  set d=dom E;
  defpred P[Nat] means
   $1<>0 implies for X st card((dom E)\X)=$1 holds conv(A\E.:(d\X))=meet{conv(A
\{E.k}) where k is Element of NAT:k in dom E\X};
  A2: rng E=A by RLAFFIN3:def 1;
  A3: for i be Nat st P[i] holds P[i+1]
  proof
   deffunc C(set)=conv(A\{E.$1});
   let i be Nat;
   assume A4: P[i];
   set i1=i+1;
   assume i1<>0;
   let X;
   set U={C(k) where k is Element of NAT:k in d\X};
   assume A5: card(d\X)=i1;
   then d\X is non empty;
   then consider m be object such that
    A6: m in d\X by XBOOLE_0:def 1;
   A7: m in d by A6,XBOOLE_0:def 5;
   reconsider m as Element of NAT by A6;
   A8: E.:{m}=Im(E,m) by RELAT_1:def 16
    .={E.m} by A7,FUNCT_1:59;
   per cases;
   suppose i=0;
    then consider x be object such that
     A9: d\X={x} by A5,CARD_2:42;
    A10: x=m by A6,A9,TARSKI:def 1;
    A11: U c={C(m)}
    proof
     let u be object;
     assume u in U;
     then ex k be Element of NAT st u=C(k) & k in d\X;
     then u=C(m) by A9,A10,TARSKI:def 1;
     hence thesis by TARSKI:def 1;
    end;
    C(m) in U by A6;
    then A12: U={C(m)} by A11,ZFMISC_1:33;
    E.:(d\X)={E.m} by A6,A8,A9,TARSKI:def 1;
    hence thesis by A12,SETFAM_1:10;
   end;
   suppose A13: i>0;
    set Xm=X\/{m};
    set Um={C(k) where k is Element of NAT:k in d\Xm};
    set t=the Element of(d\Xm);
    A14: d\X\{m}\/{m}=d\X by A6,ZFMISC_1:116;
    A15: d\X\{m}=d\Xm by XBOOLE_1:41;
    A16: Um c=U
    proof
     let x be object;
     assume x in Um;
     then consider k be Element of NAT such that
      A17: x=C(k) and
      A18: k in d\Xm;
     k in d\X by A15,A14,A18,XBOOLE_0:def 3;
     hence thesis by A17;
    end;
    m in {m} by TARSKI:def 1;
    then not m in d\X\{m} by XBOOLE_0:def 5;
    then A19: card(d\X\{m})+1=card(d\X) by A14,CARD_2:41;
    then d\Xm is non empty by A5,A13,A15;
    then t in d\Xm;
    then A20: C(t) in Um;
    set c =C(m),CC={c};
    set CA=Complex_of{A};
    A21: the topology of CA =bool A by SIMPLEX0:4;
    A22: U c=Um\/CC
    proof
     let x be object;
     assume x in U;
     then consider k be Element of NAT such that
      A23: x=C(k) and
      A24: k in d\X;
     k in d\Xm or k in {m} by A15,A14,A24,XBOOLE_0:def 3;
     then k in d\Xm or k=m by TARSKI:def 1;
     then x in Um or x in CC by A23,TARSKI:def 1;
     hence thesis by XBOOLE_0:def 3;
    end;
    C(m) in U by A6;
    then CC c=U by ZFMISC_1:31;
    then A25: Um\/CC c=U by A16,XBOOLE_1:8;
    reconsider A1=A\E.:(d\Xm),A2=A\{E.m} as Subset of CA;
    A\{E.m}c=A by XBOOLE_1:36;
    then A26: A2 is simplex-like by A21,PRE_TOPC:def 2;
    A\E.:(d\Xm)c=A by XBOOLE_1:36;
    then A1 is simplex-like by A21,PRE_TOPC:def 2;
    then A27: conv@A1/\conv@A2=conv@(A1/\A2) by A26,SIMPLEX1:def 8;
    E.:(d\Xm)\/{E.m} =E.:((d\Xm)\/{m}) by A8,RELAT_1:120
     .=E.:(d\X) by A14,XBOOLE_1:41;
    then A28: A1/\A2=A\E.:(d\X) by XBOOLE_1:53;
    conv(A\E.:(d\Xm))=meet Um by A4,A5,A13,A15,A19;
    then conv(A\E.:(d\X))=meet Um/\meet CC by A28,A27,SETFAM_1:10
     .=meet(Um\/CC) by A20,SETFAM_1:9;
    hence thesis by A25,A22,XBOOLE_0:def 10;
   end;
  end;
  A29: P[0];
  for i be Nat holds P[i] from NAT_1:sch 2(A29,A3);
  then A30: P[card(d\X)];
  d\(d\X)=d/\X by XBOOLE_1:48;
  then E.:X=E.:(d\(d\X)) by RELAT_1:112
   .=E.:d\E.:(d\X) by FUNCT_1:64
   .=A\E.:(d\X) by A2,RELAT_1:113;
  hence thesis by A1,A30;
 end;
