reserve x,y,X for set,
        r for Real,
        n,k for Nat;
reserve RLS for non empty RLSStruct,
        Kr,K1r,K2r for SimplicialComplexStr of RLS,
        V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;
reserve Ks for simplex-join-closed SimplicialComplex of V,
        As,Bs for Subset of Ks,
        Ka for non void affinely-independent SimplicialComplex of V,
        Kas for non void affinely-independent simplex-join-closed
                 SimplicialComplex of V,
        K for non void affinely-independent simplex-join-closed total
                 SimplicialComplex of V;
reserve Aff for finite affinely-independent Subset of V,
        Af,Bf for finite Subset of V,
        B for Subset of V,
        S,T for finite Subset-Family of V,
        Sf for c=-linear finite finite-membered Subset-Family of V,
        Sk,Tk for finite simplex-like Subset-Family of K,
        Ak for Simplex of K;

theorem Th44:
  for Af st K is Subdivision of Complex_of{Af} & card Af = n+1 & degree K = n &
  for S be Simplex of n-1,K,X st X = {S1 where S1 is Simplex of n,K: S c= S1}
      holds (conv @S meets  Int Af implies card X = 2) &
            (conv @S misses Int Af implies card X = 1)
  holds
    for S be Simplex of n-1,BCS K,X st
        X = {S1 where S1 is Simplex of n,BCS K:S c= S1}
    holds (conv @S meets  Int Af implies card X = 2) &
          (conv @S misses Int Af implies card X = 1)
  proof
  let A be finite Subset of V;
  assume that
   A1: K is Subdivision of Complex_of{A} and
   A2: card A=n+1 and
   A3: degree K=n and
   A4: for S be Simplex of n-1,K,X be set st X={S1 where S1 is Simplex of n,K:S
c=S1} holds(conv@S meets Int A implies card X=2) & (conv@S misses Int A implies
card X=1);
  |.Complex_of{A}.|=conv A by Th8;
  then A5: |.K.|=conv A by A1,Th10;
  A6: K is finite-degree by A3,SIMPLEX0:def 12;
  A7: A is affinely-independent
  proof
   consider a be Subset of K such that
    A8: a is simplex-like and
    A9: card a=degree K+1 by A6,SIMPLEX0:def 12;
   conv@a c=conv A by A5,A8,Th5;
   then A10: Affin@a c=Affin A by RLAFFIN1:68;
   card A=card a by A2,A3,A9,XXREAL_3:def 2;
   hence thesis by A8,A10,RLAFFIN1:80;
  end;
  set B=center_of_mass V;
  reconsider Z=0 as Nat;
  set TK=the TopStruct of K;
  reconsider n1=n-1 as ExtReal;
  let S be Simplex of n-1,BCS K,X be set such that
   A11: X={S1 where S1 is Simplex of n,BCS K:S c=S1};
  [#]K=the carrier of V by SIMPLEX0:def 10;
  then A12: |.K.|c=[#]K;
  then A13: degree K=degree BCS K by Th31;
  then A14: n+-1>=-1 & n-1<=degree BCS K by A3,XREAL_1:31,146;
  then A15: card S=n1+1 by SIMPLEX0:def 18;
  then A16: card S=(n-1)+1 by XXREAL_3:def 2;
  A17: BCS K=subdivision(B,K) by A12,Def5;
  per cases;
  suppose A18: n=0;
   then A19: TK=BCS K by A3,A12,Th21;
   then S in the topology of K by PRE_TOPC:def 2;
   then reconsider s=S as Simplex of K by PRE_TOPC:def 2;
   reconsider s as Simplex of n-1,K by A3,A15,A18,SIMPLEX0:def 18;
   set XX={W where W is Simplex of n,K:s c=W};
   A20: @S=@s;
   then A21: conv@S meets Int A implies card XX=2 by A4;
   A22: XX c=X
   proof
    let x be object;
    assume x in XX;
    then consider W be Simplex of n,K such that
     A23: x=W & S c=W;
    W in the topology of BCS K by A19,PRE_TOPC:def 2;
    then reconsider w=W as Simplex of BCS K by PRE_TOPC:def 2;
    card W=(degree K)+1 by A3,SIMPLEX0:def 18;
    then w is Simplex of n,BCS K by A3,A13,SIMPLEX0:def 18;
    hence thesis by A11,A23;
   end;
   A24: X c=XX
   proof
    let x be object;
    assume x in X;
    then consider W be Simplex of n,BCS K such that
     A25: x=W & S c=W by A11;
    W in the topology of K by A19,PRE_TOPC:def 2;
    then reconsider w=W as Simplex of K by PRE_TOPC:def 2;
    card W=(degree BCS K)+1 by A3,A13,SIMPLEX0:def 18;
    then w is Simplex of n,K by A3,A13,SIMPLEX0:def 18;
    hence thesis by A25;
   end;
   conv@S misses Int A implies card XX=1 by A4,A20;
   hence thesis by A22,A24,A21,XBOOLE_0:def 10;
  end;
  suppose A26: n>0;
   consider SS be c=-linear finite simplex-like Subset-Family of K such that
    A27: S=B.:SS by A17,SIMPLEX0:def 20;
   SS\{{}}c=SS by XBOOLE_1:36;
   then reconsider SS1=SS\{{}} as c=-linear finite simplex-like Subset-Family
of K by TOPS_2:11;
A28: SS1 c=bool@(union SS1) & bool@(union SS1)c=bool the carrier of V by
ZFMISC_1:67,82;
   A29: not{} in SS1 by ZFMISC_1:56;
   then A30: SS1 is with_non-empty_elements;
   A31: dom B=(bool the carrier of V)\{{}} by FUNCT_2:def 1;
   then A32: SS/\dom B=(SS/\(bool the carrier of V))\{{}} by XBOOLE_1:49
    .=SS1/\(bool the carrier of V) by XBOOLE_1:49
    .=SS1 by A28,XBOOLE_1:1,28;
   then A33: B.:SS=B.:SS1 by RELAT_1:112;
   then A34: card SS1=n by A16,A27,A30,Th33;
   A35: S=B.:SS1 by A27,A32,RELAT_1:112;
   A36: card SS1=card S by A27,A30,A33,Th33;
   then A37: SS1 is non empty by A16,A26;
   then A38: union SS1 in SS1 by SIMPLEX0:9;
   then reconsider U=union SS1 as Simplex of K by TOPS_2:def 1;
   Segm card SS1 c= Segm card U by A30,SIMPLEX0:10;
   then A39: n<=card U by A16,A36,NAT_1:39;
   card U<=degree K+1 by SIMPLEX0:24;
   then A40: card U<=n+1 by A3,XXREAL_3:def 2;
   A41: conv@U c=conv A by A5,Th5;
   SS1 c=bool the carrier of V by A28;
   then A42: conv@S c=conv@U by A35,CONVEX1:30,RLAFFIN2:17;
   per cases by A39,A40,NAT_1:9;
   suppose A43: card U=n;
    set XX={W where W is Simplex of n,K:U c=W};
    A44: U is Simplex of n-1,K by A13,A14,A15,A16,A43,SIMPLEX0:def 18;
    hereby assume conv@S meets Int A;
     then conv@U meets Int A by A42,XBOOLE_1:63;
     then A45: card XX=2 by A4,A44;
     consider w1,w2 be object such that
      A46: w1<>w2 and
      A47: XX={w1,w2} by A45,CARD_2:60;
     w2 in XX by A47,TARSKI:def 2;
     then consider W2 be Simplex of n,K such that
      A48: w2=W2 and
      A49: U c=W2;
A50: SS1 is with_non-empty_elements & S=B.:SS1 by A27,A29,A32,RELAT_1:112;
     w1 in XX by A47,TARSKI:def 2;
     then consider W1 be Simplex of n,K such that
      A51: w1=W1 and
      A52: U c=W1;
     A53: card W1=degree K+1 by A3,SIMPLEX0:def 18;
     then A54: card W1=n+1 by A3,XXREAL_3:def 2;
     then {W where W is Simplex of n,BCS K:S c=W & conv@W c=conv@W1}={S\/B.:{
W1}} by A16,A27,A30,A33,A36,A43,A52,Th42;
     then S\/B.:{W1} in {W where W is Simplex of n,BCS K:S c=W & conv@W c=conv
@W1} by TARSKI:def 1;
     then A55: ex R be Simplex of n,BCS K st R=S\/B.:{W1} & S c=R & conv@R c=
conv@W1;
     A56: S\/B.:{W1}<>S\/B.:{W2}
     proof
      for A be Subset of K st A in {W1} holds A is simplex-like by TARSKI:def 1
;
      then {W1} is simplex-like;
      then A57: SS1\/{W1} is simplex-like by TOPS_2:13;
      A58: S\/B.:{W1}=B.:(SS1\/{W1}) & S\/B.:{W2}=B.:(SS1\/{W2}) by A35,
RELAT_1:120;
      W1 in {W1} by TARSKI:def 1;
      then A59: W1 in SS1\/{W1} by XBOOLE_0:def 3;
      for A be Subset of K st A in {W2} holds A is simplex-like by TARSKI:def 1
;
      then {W2} is simplex-like;
      then A60: SS1\/{W2} is simplex-like by TOPS_2:13;
      assume A61: S\/B.:{W1}=S\/B.:{W2};
      W1 is non empty by A3,A53;
      then SS1\/{W1}c=SS1\/{W2} by A12,A30,A55,A60,A58,A57,A61,Th34;
      then W1 in SS1 or W1 in {W2} by A59,XBOOLE_0:def 3;
      then W1 c=U by A46,A48,A51,TARSKI:def 1,ZFMISC_1:74;
      then W1=U by A52;
      hence contradiction by A43,A54;
     end;
     A62: card SS1+Z<=degree K by A3,A16,A27,A30,A33,Th33;
     A63: X c={S\/B.:{W1},S\/B.:{W2}}
     proof
      let x be object;
      A64: n+1=degree K+1 & n=degree K+1-1 by A3,XXREAL_3:22,def 2;
      assume x in X;
      then consider W be Simplex of n,BCS K such that
       A65: x=W and
       A66: S c=W by A11;
      consider T be simplex-like finite Subset-Family of K such that
       A67: T misses SS1 and
       A68: T\/SS1 is c=-linear with_non-empty_elements and
       A69: card T=Z+1 and
       A70: @W=B.:SS1\/B.:T by A16,A36,A50,A62,A66,Th41;
      consider t be object such that
       A71: T={t} by A69,CARD_2:42;
      set TS=T\/SS1;
      A72: card TS=n+1 by A16,A36,A67,A69,CARD_2:40;
      A73: union TS in TS by A68,A71,SIMPLEX0:9;
      TS is simplex-like by TOPS_2:13;
      then reconsider UTS=union TS as Simplex of K by A73;
      Segm card TS c= Segm card UTS by A68,SIMPLEX0:10;
      then A74: card TS<=card UTS by NAT_1:39;
      UTS in T
      proof
       assume not UTS in T;
       then UTS in SS1 by A73,XBOOLE_0:def 3;
       then card UTS<=card U by NAT_1:43,ZFMISC_1:74;
       hence contradiction by A43,A72,A74,NAT_1:13;
      end;
      then A75: UTS=t by A71,TARSKI:def 1;
      card UTS<=degree K+1 by SIMPLEX0:24;
      then card UTS<=n+1 by A3,XXREAL_3:def 2;
      then card UTS=n+1 by A72,A74,XXREAL_0:1;
      then A76: UTS is Simplex of n,K by A64,SIMPLEX0:48;
      U c=UTS by XBOOLE_1:7,ZFMISC_1:77;
      then UTS in XX by A76;
      then W=B.:SS1\/B.:{W1} or W=B.:SS1\/B.:{W2} by A47,A48,A51,A70,A71,A75,
TARSKI:def 2;
      hence thesis by A35,A65,TARSKI:def 2;
     end;
     card W2=degree K+1 by A3,SIMPLEX0:def 18;
     then card W2=n+1 by A3,XXREAL_3:def 2;
     then {W where W is Simplex of n,BCS K:S c=W & conv@W c=conv@W2}={S\/B.:{
W2}} by A16,A30,A35,A36,A43,A49,Th42;
     then S\/B.:{W2} in {W where W is Simplex of n,BCS K:S c=W & conv@W c=conv
@W2} by TARSKI:def 1;
     then ex R be Simplex of n,BCS K st R=S\/B.:{W2} & S c=R & conv@R c=conv@
W2;
     then A77: S\/B.:{W2} in X by A11;
     S\/B.:{W1} in X by A11,A55;
     then {S\/B.:{W1},S\/B.:{W2}}c=X by A77,ZFMISC_1:32;
     then X={S\/B.:{W1},S\/B.:{W2}} by A63;
     hence card X=2 by A56,CARD_2:57;
    end;
    A78: conv@S c=conv A & A is non empty by A2,A41,A42;
    assume conv@S misses Int A;
    then consider BB be Subset of V such that
     A79: BB c<A and
     A80: conv@S c=conv BB by A7,A78,RLAFFIN2:23;
    A81: BB c=A by A79;
    then reconsider B1=BB as finite Subset of V;
    card B1<n+1 by A2,A79,CARD_2:48;
    then A82: card B1<=n by NAT_1:13;
    Affin@S c=Affin BB by A80,RLAFFIN1:68;
    then n<=card B1 by A16,RLAFFIN1:79;
    then card B1=n by A82,XXREAL_0:1;
    then card(A\BB)=n+1-n by A2,A81,CARD_2:44;
    then consider ab be object such that
     A83: A\BB={ab} by CARD_2:42;
    U is non empty by A26,A43;
    then @U in dom B by A31,ZFMISC_1:56;
    then A84: S c=conv@S & B.U in @S by A35,A38,FUNCT_1:def 6,RLAFFIN1:2;
    then B.U in conv@S;
    then A85: B.U in conv@U by A42;
    set BUU=B.U|--@U;
    @U c=conv@U by RLAFFIN1:2;
    then A86: @U c=conv A by A41;
    A87: ab in {ab} by TARSKI:def 1;
    then reconsider ab as Element of V by A83;
    A88: SS1 is with_non-empty_elements & S=B.:SS1 by A27,A29,A32,RELAT_1:112;
    A89: conv@U c=Affin@U by RLAFFIN1:65;
    then sum BUU=1 by A85,RLAFFIN1:def 7;
    then consider F be FinSequence of REAL,G be FinSequence of the carrier of V
such that
     A90: (Sum BUU|--A).ab=Sum F and
     A91: len G=len F and
     G is one-to-one and
     A92: rng G=Carrier BUU and
     A93: for n st n in dom F holds F.n=BUU.(G.n)*(G.n|--A).ab by A7,A86,
RLAFFIN2:3;
    A94: dom G=dom F by A91,FINSEQ_3:29;
    U c=conv B1
    proof
     A95: Carrier BUU c=U by RLVECT_2:def 6;
     A96: now let i be Nat such that
       A97: i in dom F;
      A98: F.i=BUU.(G.i)*(G.i|--A).ab by A93,A97;
      A99: G.i in rng G by A94,A97,FUNCT_1:def 3;
      then G.i in U by A92,A95;
      then A100: (G.i|--A).ab>=0 by A7,A86,RLAFFIN1:71;
      BUU.(G.i)=1/card U by A92,A95,A99,RLAFFIN2:18;
      hence 0<=F.i by A98,A100;
     end;
     B.U in conv@S by A84;
     then A101: B.U in conv BB by A80;
     assume not U c=conv B1;
     then consider t be object such that
      A102: t in U and
      A103: not t in conv B1;
    reconsider tt=t as set by TARSKI:1;
     A104: (t|--A).ab>0
     proof
      A\{ab}c=B1
      proof
       let x be object;
       assume x in A\{ab};
       then x in A & not x in {ab} by XBOOLE_0:def 5;
       hence thesis by A83,XBOOLE_0:def 5;
      end;
      then A105: conv(A\{ab})c=conv B1 by RLAFFIN1:3;
      assume A106: (t|--A).ab<=0;
      (t|--A).ab>=0 by A7,A86,A102,RLAFFIN1:71;
      then for x st x in {ab} holds(t|--A).x=0 by A106,TARSKI:def 1;
      then t in conv(A\{ab}) by A7,A86,A102,RLAFFIN1:76;
      hence contradiction by A103,A105;
     end;
     A107: BUU.t=1/card U by A102,RLAFFIN2:18;
     then t in Carrier BUU by A102;
     then consider n be object such that
      A108: n in dom G and
      A109: G.n=t by A92,FUNCT_1:def 3;
     reconsider n as Nat by A108;
     F.n=BUU.tt*(t|--A).ab by A93,A94,A108,A109;
     then 0<Sum F by A94,A96,A102,A104,A107,A108,RVSUM_1:85;
     then A110: (B.U|--A).ab>0 by A85,A89,A90,RLAFFIN1:def 7;
     Carrier(B.U|--BB)c=BB by RLVECT_2:def 6;
     then A111: not ab in Carrier(B.U|--BB) by A83,A87,XBOOLE_0:def 5;
     conv BB c=Affin BB by RLAFFIN1:65;
     then B.U|--A=B.U|--BB by A7,A81,A101,RLAFFIN1:77;
     hence contradiction by A111,A110;
    end;
    then conv@U c=conv B1 by CONVEX1:30;
    then conv@U misses Int A by A79,RLAFFIN2:7,XBOOLE_1:63;
    then card XX=1 by A4,A44;
    then consider w1 be object such that
     A112: XX={w1} by CARD_2:42;
    w1 in XX by A112,TARSKI:def 1;
    then consider W1 be Simplex of n,K such that
     A113: w1=W1 and
     A114: U c=W1;
    A115: card SS1+Z<=degree K by A3,A16,A27,A30,A33,Th33;
    A116: X c={S\/B.:{W1}}
    proof
     let x be object;
     A117: n+1=degree K+1 by A3,XXREAL_3:def 2;
     assume x in X;
     then consider W be Simplex of n,BCS K such that
      A118: x=W and
      A119: S c=W by A11;
     consider T be simplex-like finite Subset-Family of K such that
      A120: T misses SS1 and
      A121: T\/SS1 is c=-linear with_non-empty_elements and
      A122: card T=Z+1 and
      A123: @W=B.:SS1\/B.:T by A16,A36,A88,A115,A119,Th41;
     consider t be object such that
      A124: T={t} by A122,CARD_2:42;
     set TS=T\/SS1;
     A125: card TS=n+1 by A16,A36,A120,A122,CARD_2:40;
     A126: union TS in TS by A121,A124,SIMPLEX0:9;
     TS is simplex-like by TOPS_2:13;
     then reconsider UTS=union TS as Simplex of K by A126;
     Segm card TS c= Segm card UTS by A121,SIMPLEX0:10;
     then A127: card TS<=card UTS by NAT_1:39;
     UTS in T
     proof
      assume not UTS in T;
      then UTS in SS1 by A126,XBOOLE_0:def 3;
      then card UTS<=card U by NAT_1:43,ZFMISC_1:74;
      hence contradiction by A43,A125,A127,NAT_1:13;
     end;
     then A128: UTS=t by A124,TARSKI:def 1;
     card UTS<=degree K+1 by SIMPLEX0:24;
     then card UTS<=n+1 by A3,XXREAL_3:def 2;
     then card UTS=n+1 & SS1 c= TS by A125,A127,XBOOLE_1:7,XXREAL_0:1;
     then U c= UTS & UTS is Simplex of n,K by A3,A117,SIMPLEX0:def 18
,ZFMISC_1:77;
     then UTS in XX;
     then W=B.:SS1\/B.:{W1} by A112,A113,A123,A124,A128,TARSKI:def 1;
     hence thesis by A35,A118,TARSKI:def 1;
    end;
    card W1=degree K+1 by A3,SIMPLEX0:def 18;
    then card W1=n+1 by A3,XXREAL_3:def 2;
    then {W where W is Simplex of n,BCS K:S c=W & conv@W c=conv@W1}={S\/B.:{W1
}} by A16,A27,A30,A33,A36,A43,A114,Th42;
    then S\/B.:{W1} in {W where W is Simplex of n,BCS K:S c=W & conv@W c=conv@
W1} by TARSKI:def 1;
    then ex R be Simplex of n,BCS K st R=S\/B.:{W1} & S c=R & conv@R c=conv@W1;
    then S\/B.:{W1} in X by A11;
    then X={S\/B.:{W1}} by A116,ZFMISC_1:33;
    hence card X=1 by CARD_1:30;
   end;
   suppose A129: card U=n+1;
    A130: conv@S meets Int A
    proof
     U is non empty by A129;
     then @U in dom B by A31,ZFMISC_1:56;
     then A131: S c=conv@S & B.U in @S by A35,A38,FUNCT_1:def 6,RLAFFIN1:2;
     then B.U in conv@S;
     then A132: B.U in conv@U by A42;
     set BUU=B.U|--@U;
     assume A133: conv@S misses Int A;
     conv@S c=conv A & A is non empty by A2,A41,A42;
     then consider BB be Subset of V such that
      A134: BB c<A and
      A135: conv@S c=conv BB by A7,A133,RLAFFIN2:23;
     A136: BB c=A by A134;
     then reconsider B1=BB as finite Subset of V;
     Affin@S c=Affin BB by A135,RLAFFIN1:68;
     then A137: n<=card B1 by A16,RLAFFIN1:79;
     A138: card B1<n+1 by A2,A134,CARD_2:48;
     then card B1<=n by NAT_1:13;
     then card B1=n by A137,XXREAL_0:1;
     then card(A\BB)=n+1-n by A2,A136,CARD_2:44;
     then consider ab be object such that
      A139: A\BB={ab} by CARD_2:42;
     @U c=conv@U by RLAFFIN1:2;
     then A140: @U c=conv A by A41;
     A141: ab in {ab} by TARSKI:def 1;
     then reconsider ab as Element of V by A139;
     A142: conv@U c=Affin@U by RLAFFIN1:65;
     then sum BUU=1 by A132,RLAFFIN1:def 7;
     then consider F be FinSequence of REAL,G be FinSequence of the carrier of
V such that
      A143: (Sum BUU|--A).ab=Sum F and
      A144: len G=len F and
      G is one-to-one and
      A145: rng G=Carrier BUU and
      A146: for n st n in dom F holds F.n=BUU.(G.n)*(G.n|--A).ab by A7,A140,
RLAFFIN2:3;
     A147: dom G=dom F by A144,FINSEQ_3:29;
     A148: A\{ab}=A/\BB by A139,XBOOLE_1:48
      .=BB by A136,XBOOLE_1:28;
     U c=conv B1
     proof
      A149: Carrier BUU c=U by RLVECT_2:def 6;
      A150: now let i be Nat such that
        A151: i in dom F;
       A152: F.i=BUU.(G.i)*(G.i|--A).ab by A146,A151;
       A153: G.i in rng G by A147,A151,FUNCT_1:def 3;
       then G.i in U by A145,A149;
       then A154: (G.i|--A).ab>=0 by A7,A140,RLAFFIN1:71;
       BUU.(G.i)=1/card U by A145,A149,A153,RLAFFIN2:18;
       hence 0<=F.i by A152,A154;
      end;
      B.U in conv@S by A131;
      then A155: B.U in conv BB by A135;
      assume not U c=conv B1;
      then consider t be object such that
       A156: t in U and
       A157: not t in conv B1;
    reconsider tt=t as set by TARSKI:1;
      U c=conv@U by RLAFFIN1:2;
      then A158: t in conv@U by A156;
      A159: (t|--A).ab>0
      proof
       assume A160: (t|--A).ab<=0;
       (t|--A).ab>=0 by A7,A41,A158,RLAFFIN1:71;
       then for y st y in A\B1 holds(t|--A).y=0 by A139,A160,TARSKI:def 1;
       hence contradiction by A7,A41,A139,A148,A157,A158,RLAFFIN1:76;
      end;
      A161: BUU.t=1/card U by A156,RLAFFIN2:18;
      then t in Carrier BUU by A156;
      then consider n be object such that
       A162: n in dom G and
       A163: G.n=t by A145,FUNCT_1:def 3;
      reconsider n as Nat by A162;
      F.n=BUU.tt*(t|--A).ab by A146,A147,A162,A163;
      then 0<Sum F by A147,A150,A156,A159,A161,A162,RVSUM_1:85;
      then A164: (B.U|--A).ab>0 by A132,A142,A143,RLAFFIN1:def 7;
      Carrier(B.U|--BB)c=BB by RLVECT_2:def 6;
      then A165: not ab in Carrier(B.U|--BB) by A139,A141,XBOOLE_0:def 5;
      conv BB c=Affin BB by RLAFFIN1:65;
      then B.U|--A=B.U|--BB by A7,A136,A155,RLAFFIN1:77;
      hence contradiction by A165,A164;
     end;
     then conv@U c=conv B1 by CONVEX1:30;
     then Affin@U c=Affin B1 by RLAFFIN1:68;
     hence contradiction by A129,A138,RLAFFIN1:79;
    end;
    set XX={S1 where S1 is Simplex of n,BCS K:S c=S1 & conv@S1 c=conv@U};
    A166: card XX=2 by A16,A30,A35,A36,A129,Th43;
    consider w1,w2 be object such that
     w1<>w2 and
     A167: XX={w1,w2} by A166,CARD_2:60;
    w2 in XX by A167,TARSKI:def 2;
    then consider W2 be Simplex of n,BCS K such that
     A168: w2=W2 and
     A169: S c=W2 and
     conv@W2 c=conv@U;
    w1 in XX by A167,TARSKI:def 2;
    then consider W1 be Simplex of n,BCS K such that
     A170: w1=W1 and
     A171: S c=W1 and
     conv@W1 c=conv@U;
    A172: W1 in X by A11,A171;
    A173: X c=XX
    proof
     let w be object;
     assume w in X;
     then consider W be Simplex of n,BCS K such that
      A174: w=W and
      A175: S c=W by A11;
     card SS1+Z<=degree K by A3,A16,A27,A30,A33,Th33;
     then consider T be simplex-like finite Subset-Family of K such that
      T misses SS1 and
      A176: T\/SS1 is c=-linear with_non-empty_elements and
      card T=Z+1 and
      A177: @W=B.:SS1\/B.:T by A27,A30,A33,A34,A175,Th41;
     reconsider TS=T\/SS1 as finite simplex-like Subset-Family of K by
TOPS_2:13;
     A178: W=B.:@TS by A177,RELAT_1:120;
     union TS in TS by A37,A176,SIMPLEX0:9;
     then reconsider UTS=union TS as Simplex of K by TOPS_2:def 1;
     card UTS<=degree K+1 by SIMPLEX0:24;
     then A179: card UTS<=n+1 by A3,XXREAL_3:def 2;
     A180: U c=union TS by XBOOLE_1:7,ZFMISC_1:77;
     then n+1<=card UTS by A129,NAT_1:43;
     then UTS=U by A129,A179,A180,CARD_2:102,XXREAL_0:1;
     then conv@W c=conv@U by A178,CONVEX1:30,RLAFFIN2:17;
     hence thesis by A174,A175;
    end;
    W2 in X by A11,A169;
    then XX c=X by A167,A170,A168,A172,ZFMISC_1:32;
    hence thesis by A130,A166,A173,XBOOLE_0:def 10;
   end;
  end;
 end;
