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;
