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;
reserve KX for SimplicialComplexStr of X,
        SX for SubSimplicialComplex of KX;
reserve SC for SimplicialComplex of X;
reserve K for non void subset-closed SimplicialComplexStr;

theorem
  for S be finite Simplex of K holds S is Simplex of card S - 1,K
  proof
  let S be finite Simplex of K;
  card S<=degree K+1 by Th24;
  then card S-1<=degree K+1-1 by XXREAL_3:37;
  then A1: card S-1<=degree K by XXREAL_3:24;
  card S-1>=0-1 & card S=card S-1+1 by XREAL_1:6;
  hence thesis by A1,Def18;
 end;
