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, [|B,C|] st M in doms F holds for
  i be set st i in I holds (F..M).i = [((Mpr1 F)..M).i, ((Mpr2 F)..M).i]
proof
  let F be ManySortedFunction of A, [|B,C|] such that
A1: M in doms F;
  let i be set;
  assume
A2: i in I;
  then M.i in (doms F).i by A1;
  then
A3: M.i in dom (F.i) by A2,MSSUBFAM:14;
  A is_transformable_to [|B,C|];
  then M in A by A1,MSSUBFAM:17;
  then F..M in [|B,C|] by CLOSURE1:3;
  then (F..M).i in [|B,C|].i by A2;
  then (F..M).i in [:B.i,C.i:] by A2,PBOOLE:def 16;
  then
A4: (F.i).(M.i) in [:B.i,C.i:] by A2,PRALG_1:def 20;
  set z = (F.i).(M.i);
  (Mpr2 F).i = pr2 (F.i) by A2,Def2;
  then
A5: ((Mpr2 F)..M).i = (pr2 (F.i)).(M.i) by A2,PRALG_1:def 20
    .= z`2 by A3,MCART_1:def 13;
  (Mpr1 F).i = pr1 (F.i) by A2,Def1;
  then ((Mpr1 F)..M).i = (pr1 (F.i)).(M.i) by A2,PRALG_1:def 20
    .= z`1 by A3,MCART_1:def 12;
  then z = [((Mpr1 F)..M).i, ((Mpr2 F)..M).i] by A4,A5,MCART_1:21;
  hence thesis by A2,PRALG_1:def 20;
end;
