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 Th40:
  Aff is Simplex of K implies (B is Simplex of BCS Complex_of{Aff} iff
     B is Simplex of BCS K & conv B c= conv Aff)
 proof
  set Bag=center_of_mass V;
  set C=Complex_of{Aff};
  A1: the topology of C=bool Aff by SIMPLEX0:4;
  assume Aff is Simplex of K;
  then reconsider s=Aff as Simplex of K;
  A2: [#]K=the carrier of V by SIMPLEX0:def 10;
  then |.K.|c=[#]K;
  then A3: subdivision(Bag,K)=BCS K by Def5;
  @s is affinely-independent;
  then A4: Complex_of{Aff} is SubSimplicialComplex of K by Th3;
  then the topology of C c=the topology of K by SIMPLEX0:def 13;
  then A5: |.C.|c=|.K.| by Th4;
  [#]C=[#]V;
  then A6: subdivision(Bag,C)=BCS C by A5,Def5;
  then BCS C is SubSimplicialComplex of BCS K by A3,A4,SIMPLEX0:58;
  then A7: the topology of BCS C c=the topology of BCS K by SIMPLEX0:def 13;
  hereby assume B is Simplex of BCS C;
   then reconsider A=B as Simplex of BCS C;
   A in the topology of BCS C by PRE_TOPC:def 2;
   then A in the topology of BCS K by A7;
   then reconsider a=A as Simplex of BCS K by PRE_TOPC:def 2;
   |.BCS C.|=|.C.| & conv@A c=|.BCS C.| by Th5,Th10;
   then conv@a c=conv Aff by Th8;
   hence B is Simplex of BCS K & conv B c=conv Aff;
  end;
  assume that
   A8: B is Simplex of BCS K and
   A9: conv B c=conv Aff;
  reconsider A=B as Simplex of BCS K by A8;
  consider SS be c=-linear finite simplex-like Subset-Family of K such that
   A10: B=Bag.:SS by A3,A8,SIMPLEX0:def 20;
  reconsider ss=SS as c=-linear finite Subset-Family of C by A2;
  [#]subdivision(Bag,C)=[#]C by SIMPLEX0:def 20;
  then reconsider Bss=Bag.:ss as Subset of BCS C by A5,Def5;
  A11: dom Bag=(bool the carrier of V)\{{}} by FUNCT_2:def 1;
  ss is simplex-like
  proof
   let a be Subset of C such that
    A12: a in ss;
   reconsider aK=a as Simplex of K by A12,TOPS_2:def 1;
   per cases;
   suppose aK is empty;
    hence thesis;
   end;
   suppose A13: aK is non empty;
    then aK in dom Bag by A11,ZFMISC_1:56;
    then A14: Bag.aK in A by A10,A12,FUNCT_1:def 6;
    A15: Bag.aK in Int@aK by A13,RLAFFIN2:20;
    A c=conv@A by RLAFFIN1:2;
    then Bag.aK in conv@A by A14;
    then Int@aK meets conv@s by A9,A15,XBOOLE_0:3;
    then aK c=Aff by Th26;
    hence thesis by A1;
   end;
  end;
  then Bss is simplex-like by A6,SIMPLEX0:def 20;
  hence thesis by A10;
 end;
