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 Th43:
  Sk is c=-linear with_non-empty_elements & card Sk+1 = card union Sk
  implies card {S1 where S1 is Simplex of card Sk,BCS K:
       (center_of_mass V).:Sk c= S1 & conv @S1 c= conv @union Sk} = 2
 proof
  set B=center_of_mass V;
  assume that
   A1: Sk is c=-linear with_non-empty_elements and
   A2: card Sk+1=card union Sk;
  Sk is non empty by A2,ZFMISC_1:2;
  then union Sk in Sk by A1,SIMPLEX0:9;
  then reconsider U=union Sk as Simplex of K by TOPS_2:def 1;
  reconsider Sk1=@Sk as c=-linear finite finite-membered Subset-Family of V by
A1;
  reconsider c =card U as ExtReal;
  @U=union Sk1;
  then reconsider U1=union Sk1 as finite affinely-independent Subset of V;
  set C=Complex_of{U1};
  A3: degree C=c-1 by SIMPLEX0:26
   .=card U +-1 by XXREAL_3:def 2
   .=card Sk by A2;
  set YY={W where W is Simplex of card Sk,BCS K:B.:Sk c=W & conv@W c=conv@
union Sk};
  [#]K=the carrier of V by SIMPLEX0:def 10;
  then |.K.|c=[#]K;
  then A4: subdivision(B,K)=BCS K by Def5;
  set XX={W where W is Simplex of card Sk,BCS C:B.:Sk c=W};
  A5: @U=U1;
  then A6: C is SubSimplicialComplex of K by Th3;
  then the topology of C c=the topology of K by SIMPLEX0:def 13;
  then A7: |.C.|c=|.K.| by Th4;
  A8: [#]C=[#]V;
  then A9: degree C=degree BCS C by A7,Th31;
  subdivision(B,C)=BCS C by A7,A8,Def5;
  then BCS C is SubSimplicialComplex of BCS K by A4,A6,SIMPLEX0:58;
  then A10: degree BCS C<=degree BCS K by SIMPLEX0:32;
  A11: XX c=YY
  proof
   let x be object;
   assume x in XX;
   then consider W be Simplex of card Sk,BCS C such that
    A12: x=W & B.:Sk1 c=W;
   W=@W;
   then reconsider w=W as Simplex of BCS K by A5,Th40;
   card W=degree BCS C+1 by A3,A9,SIMPLEX0:def 18;
   then A13: w is Simplex of card Sk,BCS K by A3,A9,A10,SIMPLEX0:def 18;
   conv@W c=conv@U & @w=w by Th40;
   hence thesis by A12,A13;
  end;
  A14: [#]subdivision(B,C)=[#]C by SIMPLEX0:def 20;
  A15: YY c=XX
  proof
   let x be object;
   reconsider c1=card Sk as ExtReal;

   assume x in YY;
   then consider W be Simplex of card Sk,BCS K such that
    A16: W=x & B.:Sk c=W and
    A17: conv@W c=conv@U;
   reconsider w=@W as Subset of BCS C by A7,A14,Def5;
   reconsider cW=card W as ExtReal;
   card W=c1+1 by A3,A9,A10,SIMPLEX0:def 18
    .=card Sk+1 by XXREAL_3:def 2;
   then card Sk=card W+-1;
   then A18: card Sk=cW-1 by XXREAL_3:def 2;
   w is simplex-like by A17,Th40;
   then w is Simplex of card Sk,BCS C by A18,SIMPLEX0:48;
   hence thesis by A16;
  end;
  card XX=2 by A1,A2,Th39;
  hence thesis by A11,A15,XBOOLE_0:def 10;
 end;
