reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;
reserve K for SimplicialComplexStr;

theorem Th24:
  for S be finite Subset of K st S is simplex-like holds card S <= degree K+1
 proof
  let S be finite Subset of K such that
   A1: S is simplex-like;
  S in the topology of K by A1;
  then A2: K is non void by PENCIL_1:def 4;
  per cases;
  suppose K is finite-degree;
   hence thesis by A1,A2,Def12;
  end;
  suppose K is non finite-degree;
   then degree K=+infty by A2,Def12;
   then degree K+1=+infty by XXREAL_3:def 2;
   hence thesis by XXREAL_0:3;
  end;
 end;
