reserve x,y,z for set;

theorem
  for I being set, A being ManySortedSet of I for F being
  ManySortedFunction of I st F is "1-1" & A c= doms F holds F""(F.:.:A) = A
proof
  let I be set, A be ManySortedSet of I;
  let F be ManySortedFunction of I such that
A1: F is "1-1" and
A2: A c= doms F;
A3: dom F = I by PARTFUN1:def 2;
  now
    let i be object;
    assume
A4: i in I;
    then
A5: F.i is one-to-one by A1,A3,MSUALG_3:def 2;
    A.i c= (doms F).i by A2,A4;
    then
A6: A.i c= dom (F.i) by A3,A4,FUNCT_6:22;
    thus (F""(F.:.:A)).i = (F.i)"((F.:.:A).i) by A4,EQUATION:def 1
      .= (F.i)"((F.i).:(A.i)) by A4,PBOOLE:def 20
      .= A.i by A6,A5,FUNCT_1:94;
  end;
  hence thesis;
end;
