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;

theorem Th24:
  |.Kv.| c= [#]Kv implies Vertices Kv c= Vertices BCS(n,Kv)
 proof
  set S=Skeleton_of(Kv,0);
  assume A1: |.Kv.|c=[#]Kv;
  per cases;
  suppose n=0;
   hence thesis by A1,Th16;
  end;
  suppose A2: n>0;
   the topology of S c=the topology of Kv by SIMPLEX0:def 13;
   then |.S.|c=|.Kv.| by Th4;
   then A3: |.S.|c=[#]S by A1;
   then degree S<=0 & BCS(n,S) is SubSimplicialComplex of BCS(n,Kv) by A1,Th23,
SIMPLEX0:44;
   then S is SubSimplicialComplex of BCS(n,Kv) by A2,A3,Th22;
   then A4: Vertices S c=Vertices BCS(n,Kv) by SIMPLEX0:31;
   let x be object;
   assume A5: x in Vertices Kv;
   then reconsider v=x as Element of Kv;
   v is vertex-like by A5,SIMPLEX0:def 4;
   then consider A be Subset of Kv such that
    A6: A is simplex-like and
    A7: v in A;
   reconsider vv={v} as Subset of Kv by A7,ZFMISC_1:31;
   {v}c=A by A7,ZFMISC_1:31;
   then vv is simplex-like by A6,MATROID0:1;
   then A8: vv in the topology of Kv;
   card vv=1 & card 1=1 by CARD_1:30;
   then vv in the_subsets_with_limited_card(1,the topology of Kv) by A8,
SIMPLEX0:def 2;
   then vv in the topology of S by SIMPLEX0:2;
   then reconsider vv as Simplex of S by PRE_TOPC:def 2;
   A9: v in vv by TARSKI:def 1;
   reconsider v as Element of S;
   v is vertex-like by A9;
   then v in Vertices S by SIMPLEX0:def 4;
   hence thesis by A4;
  end;
 end;
