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 Th33:
  for S be simplex-like Subset-Family of Kas st S is with_non-empty_elements
    holds card S = card ((center_of_mass V).:S)
 proof
  set B=center_of_mass V;
  set T=the topology of Kas;
  let S be simplex-like Subset-Family of Kas such that
   A1: S is with_non-empty_elements;
  A2: not{} in S by A1;
  [#]Kas c=the carrier of V by SIMPLEX0:def 9;
  then bool the carrier of Kas c=bool the carrier of V by ZFMISC_1:67;
  then dom B=(bool the carrier of V)\{{}} & S c=bool the carrier of V by
FUNCT_2:def 1;
  then A3: dom(B|S)=S by A2,RELAT_1:62,ZFMISC_1:34;
  S c=T
  proof
   let x be object;
   assume x in S;
   then x is Simplex of Kas by TOPS_2:def 1;
   hence thesis by PRE_TOPC:def 2;
  end;
  then B|T|S=B|S by RELAT_1:74;
  then A4: B|S is one-to-one by FUNCT_1:52;
  B.:S=rng(B|S) by RELAT_1:115;
  then S,B.:S are_equipotent by A3,A4,WELLORD2:def 4;
  hence thesis by CARD_1:5;
 end;
