reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;
reserve I for set,
  A, B, C for ManySortedSet of I;

theorem Th12:
  for A, B be non-empty ManySortedSet of I for F be
  ManySortedFunction of A, B st F is "1-1" "onto" holds F"" is "1-1" "onto"
proof
  let A, B be non-empty ManySortedSet of I;
  let F be ManySortedFunction of A, B;
  assume
A1: F is "1-1" "onto";
  now
    let i be set;
    assume
A2: i in I;
    then reconsider g = F.i as Function of A.i, B.i by PBOOLE:def 15;
    g is one-to-one by A1,A2,MSUALG_3:1;
    then g" is one-to-one;
    hence (F"".i) is one-to-one by A1,A2,MSUALG_3:def 4;
  end;
  hence F"" is "1-1" by MSUALG_3:1;
  thus F"" is "onto"
  proof
    let i be set;
    assume
A3: i in I;
    then reconsider g = F.i as Function of A.i, B.i by PBOOLE:def 15;
A4: g is one-to-one by A1,A3,MSUALG_3:1;
    A.i = dom g by A3,FUNCT_2:def 1
      .= rng (g") by A4,FUNCT_1:33;
    hence thesis by A1,A3,MSUALG_3:def 4;
  end;
end;
