reserve a, I for set,
  S for non empty non void ManySortedSign;
reserve A, M for ManySortedSet of I,
  B, C for non-empty ManySortedSet of I;

theorem
  for F being ManySortedFunction of A, [| I-->{a} , I-->{a} |] holds
  Mpr1 F = Mpr2 F
proof
  let F be ManySortedFunction of A, [| I-->{a} , I-->{a} |];
  now
    let i be object;
A1: dom (pr2 (F.i)) = dom (F.i) by MCART_1:def 13;
    assume
A2: i in I;
A3: now
      let y be object such that
A4:   y in dom (F.i);
A5:   (F.i).y in rng (F.i) by A4,FUNCT_1:def 3;
      (F.i) is Function of A.i, [| I-->{a} , I-->{a} |].i by A2,PBOOLE:def 15;
      then rng (F.i) c= [| I-->{a} , I-->{a} |].i by RELAT_1:def 19;
      then
A6:   rng (F.i) c= [: (I-->{a}).i, (I-->{a}).i :] by A2,PBOOLE:def 16;
      then ((F.i).y)`1 in (I-->{a}).i by A5,MCART_1:10;
      then
A7:   ((F.i).y)`1 in {a} by A2,FUNCOP_1:7;
      ((F.i).y)`2 in (I-->{a}).i by A5,A6,MCART_1:10;
      then
A8:   ((F.i).y)`2 in {a} by A2,FUNCOP_1:7;
      thus (pr2 (F.i)).y = ((F.i).y)`2 by A4,MCART_1:def 13
        .= a by A8,TARSKI:def 1
        .= ((F.i).y)`1 by A7,TARSKI:def 1;
    end;
    (Mpr1 F).i = pr1 (F.i) & (Mpr2 F).i = pr2 (F.i) by A2,Def1,Def2;
    hence (Mpr1 F).i = (Mpr2 F).i by A1,A3,MCART_1:def 12;
  end;
  hence thesis;
end;
