reserve x, y for set;

theorem Th17:
  for s being finite set, t being non empty finite set
  st card s = 23 & card t = 365
  holds 2 * card (not-one-to-one (s, t)) > card Funcs (s, t)
  proof
    let s be finite set, t be non empty finite set;
    assume A1: card s = 23;
    assume A2: card t = 365;
    then card (not-one-to-one (s, t)) = (365 |^ 23) - (365! / ((365 -' 23)!))
                                                               by Th16,A1;
    hence 2 * card (not-one-to-one (s, t)) > card Funcs (s, t)
      by Lm1,A1,A2,CARD_FIN:4;
  end;
