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 Simplex of i,K st S is non empty holds i is natural
 proof
  let S be Simplex of i,K;
  assume S is non empty;
  then -1<=i & i<>-1 by Def18;
  then -1<i by XXREAL_0:1;
  then i>=0 by INT_1:8;
  then i in NAT by INT_1:3;
  hence thesis;
 end;
