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;
