reserve I for set;

theorem
  for A being ManySortedSet of I, F being ManySortedFunction of I st A
  c= rngs F holds F.:.:(F""A) = A
proof
  let A be ManySortedSet of I, F be ManySortedFunction of I such that
A1: A c= rngs F;
  now
    let i be object;
    assume
A2: i in I;
    then A.i c= (rngs F).i by A1;
    then
A3: A.i c= rng (F.i) by A2,MSSUBFAM:13;
    thus (F.:.:(F""A)).i = (F.i).:((F""A).i) by A2,PBOOLE:def 20
      .= (F.i).:((F.i)"(A.i)) by A2,Def1
      .= A.i by A3,FUNCT_1:77;
  end;
  hence thesis;
end;
