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;

theorem Th28:
  |.Ka.| c= [#]Ka implies BCS Ka is affinely-independent
 proof
  set P=BCS Ka,B=center_of_mass V;
  assume|.Ka.|c=[#]Ka;
  then A1: P=subdivision(B,Ka) by Def5;
  let A be Subset of P;
  assume A is simplex-like;
  then consider S be c=-linear finite simplex-like Subset-Family of Ka such
that
   A2: A=B.:S by A1,SIMPLEX0:def 20;
  per cases;
  suppose S is empty;
   then A={} by A2;
   hence thesis;
  end;
  suppose A3: S is non empty;
   S c=bool union S & bool@union S c=bool the carrier of V by ZFMISC_1:67,82;
   then reconsider s=S as c=-linear finite Subset-Family of V by XBOOLE_1:1;
   union S in S by A3,SIMPLEX0:9;
   then union S is simplex-like by TOPS_2:def 1;
   then @union S is affinely-independent;
   then union s is affinely-independent;
   hence thesis by A2,RLAFFIN2:29;
  end;
 end;
