
theorem Th16:
  for X be non empty set holds card(product <*X*>) = card X
proof
  let X be non empty set;
  consider I be Function of X,product <*X*> such that
A1: I is one-to-one & I is onto &
 for x be object st x in X holds I.x = <*x*>
  by PRVECT_3:4;
  not {} in rng <*X*>
  proof
    assume not not {} in rng <*X*>;
    then {} in { X } by FINSEQ_1:39;
    hence contradiction by TARSKI:def 1;
  end;
  then product (<*X*>) <> {} by CARD_3:26;
  then
A2: dom I = X by FUNCT_2:def 1;
  rng I = product (<*X*>) by A1,FUNCT_2:def 3;
  hence card(X) = card(product(<*X*>)) by CARD_1:5,A1,A2,WELLORD2:def 4;
end;
