reserve I for set;

theorem
  for A, B being ManySortedSet of I, F being ManySortedFunction of A, B
  st A is_transformable_to B holds F""B = A
proof
  let A, B be ManySortedSet of I, F be ManySortedFunction of A, B such that
A1: A is_transformable_to B;
  now
    let i be object;
    assume
A2: i in I;
    then
A3: F.i is Function of A.i, B.i by PBOOLE:def 15;
A4: B.i = {} implies A.i = {} by A1,A2;
    thus (F""B).i = (F.i)"(B.i) by A2,Def1
      .= A.i by A4,A3,FUNCT_2:40;
  end;
  hence thesis;
end;
