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 Th31:
  |.Ka.| c= [#]Ka implies degree Ka = degree BCS Ka
 proof
  A1: for n st n<=degree Ka ex S be Simplex of Ka st card S=n+1 & @S is
affinely-independent
  proof
   let n;
   reconsider N=n as ExtReal;
   set S=the Simplex of n,Ka;
   assume n<=degree Ka;
   then A2: card S=N+1 by SIMPLEX0:def 18;
   N+1=n+1 & @S is affinely-independent by XXREAL_3:def 2;
   hence thesis by A2;
  end;
  assume|.Ka.|c=[#]Ka;
  hence thesis by A1,Th30;
 end;
