reserve e for set;

theorem
  for I being set, A,B,C being ManySortedSet of I st A
  is_transformable_to B for F being ManySortedFunction of A,B, G being
  ManySortedFunction of B,C holds G**F is ManySortedFunction of A,C
proof
  let I be set, A,B,C be ManySortedSet of I such that
A1: A is_transformable_to B;
  let F be ManySortedFunction of A,B, G be ManySortedFunction of B,C;
  reconsider GF = G**F as ManySortedFunction of I by MSSUBFAM:15;
  GF is ManySortedFunction of A,C
  proof
    let i be object;
    assume
A2: i in I;
    then reconsider Gi = G.i as Function of B.i,C.i by PBOOLE:def 15;
    reconsider Fi = F.i as Function of A.i,B.i by A2,PBOOLE:def 15;
    i in dom GF by A2,PARTFUN1:def 2;
    then
A3: (G**F).i = (Gi)*(Fi) by PBOOLE:def 19;
    B.i = {} implies A.i = {} by A1,A2,PZFMISC1:def 3;
    hence thesis by A3,FUNCT_2:13;
  end;
  hence thesis;
end;
