reserve I for set;

theorem
  for A being ManySortedSet of I, B, C being non-empty ManySortedSet of
I for f being ManySortedFunction of A, B, g being ManySortedFunction of B, C st
  g ** f is "onto" holds g is "onto"
proof
  let A be ManySortedSet of I, B, C be non-empty ManySortedSet of I, f be
  ManySortedFunction of A, B, g be ManySortedFunction of B, C such that
A1: g ** f is "onto";
  let i be set;
  assume
A2: i in I;
  then
A3: f.i is Function of A.i, B.i & g.i is Function of B.i, C.i by PBOOLE:def 15;
  rng (g.i * f.i) = rng ((g ** f).i) by A2,MSUALG_3:2
    .= C.i by A1,A2;
  hence thesis by A2,A3,Th1;
end;
