reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem
  for U1,U2 being non-empty OSAlgebra of S1 for F be ManySortedFunction
  of U1,U2 st F is_homomorphism U1,U2 & F is order-sorted ex F1 be
ManySortedFunction of U1,Image F, F2 be ManySortedFunction of Image F,U2 st F1
is_epimorphism U1,Image F & F2 is_monomorphism Image F,U2 & F = F2**F1 & F1 is
  order-sorted & F2 is order-sorted
proof
  let U1,U2 be non-empty OSAlgebra of S1;
  let F be ManySortedFunction of U1,U2;
  assume that
A1: F is_homomorphism U1,U2 and
A2: F is order-sorted;
  for H be ManySortedFunction of Image F,Image F holds H is
  ManySortedFunction of Image F,U2
  proof
    let H be ManySortedFunction of Image F,Image F;
    for i be object st i in the carrier of S1 holds H.i is Function of (the
    Sorts of Image F).i,(the Sorts of U2).i
    proof
      let i be object;
      assume
A3:   i in the carrier of S1;
      then reconsider
      f = F.i as Function of (the Sorts of U1).i,(the Sorts of U2).
      i by PBOOLE:def 15;
      reconsider h = H.i as Function of (the Sorts of Image F).i,(the Sorts of
      Image F).i by A3,PBOOLE:def 15;
A4:   dom f = (the Sorts of U1).i by A3,FUNCT_2:def 1;
      the Sorts of Image F = F.:.:(the Sorts of U1) by A1,MSUALG_3:def 12;
      then (the Sorts of Image F).i = f.:((the Sorts of U1).i) by A3,
PBOOLE:def 20
        .= rng f by A4,RELAT_1:113;
      then h is Function of (the Sorts of Image F).i,(the Sorts of U2).i by
FUNCT_2:7;
      hence thesis;
    end;
    hence thesis by PBOOLE:def 15;
  end;
  then reconsider
  F2 = id (the Sorts of Image F) as ManySortedFunction of Image F,
  U2;
  consider F1 being ManySortedFunction of U1,Image F such that
A5: F1 = F & F1 is order-sorted and
A6: F1 is_epimorphism U1,Image F by A1,A2,Th15;
  take F1,F2;
  thus F1 is_epimorphism U1,Image F by A6;
  thus F2 is_monomorphism Image F,U2 by MSUALG_3:22;
  thus F = F2**F1 & F1 is order-sorted by A5,MSUALG_3:4;
  Image F is order-sorted by A1,A2,Th11;
  then (the Sorts of Image F) is OrderSortedSet of S1 by OSALG_1:17;
  hence thesis;
end;
