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 Th32:
  |.Ka.| c= [#]Ka implies degree Ka = degree BCS(n,Ka)
 proof
  defpred P[Nat] means
   degree Ka=degree BCS($1,Ka) & BCS($1,Ka) is non void affinely-independent;
  assume A1: |.Ka.|c=[#]Ka;
  A2: for n st P[n] holds P[n+1]
  proof
   let n such that
    A3: P[n];
   A4: [#]BCS(n,Ka)=[#]Ka by A1,Th18;
   BCS(n+1,Ka)=BCS BCS(n,Ka) & |.BCS(n,Ka).|=|.Ka.| by A1,Th10,Th20;
   hence thesis by A1,A3,A4,Th28,Th31;
  end;
  A5: P[0 qua Nat] by A1,Th16;
  for n holds P[n] from NAT_1:sch 2(A5,A2);
  hence thesis;
 end;
