reserve x, y for set;

theorem Th9:
  for s being non empty finite set
  holds |. card(derangements s) - (((card s)!) / number_e) .| < 1/2
  proof
    let s be non empty finite set;
    set n = card s;
    consider c being Real such that
A1:  c in ]. -1, 0 .[ and
A2:  card(derangements s) - (((n)!) / number_e) =
          -(n)! * (exp_R c * (-1) |^ ((n)+1) / (((n)+1)!)) by Th8;
    c < 0 by A1,XXREAL_1:4;
    hence thesis by A2,Th6;
  end;
