reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;

theorem
  for F, G be ManySortedFunction of I holds G ** F is ManySortedFunction of I
proof
  let F, G be ManySortedFunction of I;
  dom (G ** F) = (dom F) /\ (dom G) by PBOOLE:def 19
    .= I /\ (dom G) by PARTFUN1:def 2
    .= I /\ I by PARTFUN1:def 2
    .= I;
  hence thesis by PARTFUN1:def 2,RELAT_1:def 18;
end;
