reserve x, y for set;

theorem
  for s, t being non empty finite set st card s = 23 & card t = 365
  holds prob (not-one-to-one (s, t)) > 1/2
  proof
    let s, t be non empty finite set;
    assume A1: card s = 23;
    assume A2: card t = 365;
    set E = not-one-to-one (s, t);
    set comega = card Funcs (s, t);
    2 * card E / 2 > comega / 2 by Th17,A1,A2,XREAL_1:74;
    then card E / comega > comega / 2 / comega by XREAL_1:74;
    then card E / comega > comega / comega / 2;
    then card E / comega > 1 / 2 by XCMPLX_0:def 7;
    hence prob E > 1/2 by RPR_1:def 1;
  end;
