reserve x, y for set;

theorem Th12:
 for x being object holds  derangements { x } = {}
  proof let x be object;
A1: card { x } = 1 by CARD_1:30;
    1 / number_e < 1/2 by TAYLOR_1:11,XREAL_1:76;
    then -(1/2) < -1/number_e by XREAL_1:24;
    then |. (0 qua Nat) - 1 / number_e .| < 1/2 by SEQ_2:1;
    then round(1 / number_e) = 0 by Th4;
    then card(derangements { x }) = 0 by Th10,A1,NEWTON:13;
    hence thesis;
  end;
