reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th13:
  for U1 being monotone non-empty OSAlgebra of S1, U2 being
  non-empty OSAlgebra of S1 for F being ManySortedFunction of U1,U2 st F is
order-sorted & F is_homomorphism U1,U2 holds Image F is order-sorted & Image F
  is monotone OSAlgebra of S1
proof
  let U1 be monotone non-empty OSAlgebra of S1, U2 be non-empty OSAlgebra of
  S1;
  let F be ManySortedFunction of U1,U2 such that
A1: F is order-sorted and
A2: F is_homomorphism U1,U2;
  reconsider O1 = the Sorts of U1 as OrderSortedSet of S1 by OSALG_1:17;
  F.:.:O1 is OrderSortedSet of S1 by A1,Th7;
  then
A3: the Sorts of Image F is OrderSortedSet of S1 by A2,MSUALG_3:def 12;
  then reconsider I = Image F as non-empty OSAlgebra of S1 by OSALG_1:17;
  thus Image F is order-sorted by A3,OSALG_1:17;
  consider G being ManySortedFunction of U1,I such that
A4: F = G and
A5: G is_epimorphism U1,I by A2,MSUALG_3:21;
A6: G is_homomorphism U1,I by A5,MSUALG_3:def 8;
A7: G is "onto" by A5,MSUALG_3:def 8;
  for o1,o2 being OperSymbol of S1 st o1 <= o2 holds Den(o1,I) c= Den(o2,I )
  proof
    let o1,o2 be OperSymbol of S1 such that
A8: o1 <= o2;
A9: Args(o1,I) c= Args(o2,I) by A8,OSALG_1:26;
A10: Args(o1,U1) c= Args(o2,U1) by A8,OSALG_1:26;
A11: dom Den(o2,I) = Args(o2,I) by FUNCT_2:def 1;
A12: Den(o2,U1)|Args(o1,U1) = Den(o1,U1) by A8,OSALG_1:def 21;
A13: the_result_sort_of o1 <= the_result_sort_of o2 by A8;
    for a,b being object holds [a,b] in Den(o1,I) implies [a,b] in Den(o2,I)
    proof
      set s1 = the_result_sort_of o1, s2 = the_result_sort_of o2;
      o1 in the carrier' of S1;
      then
A14:  o1 in dom the ResultSort of S1 by FUNCT_2:def 1;
      let a,b be object such that
A15:  [a,b] in Den(o1,I);
A16:  a in Args(o1,I) by A15,ZFMISC_1:87;
      then consider y being Element of Args(o1,U1) such that
A17:  G # y = a by A7,MSUALG_9:17;
      reconsider y1 = y as Element of Args(o2,U1) by A10;
A18:  G # y1 = G # y & Den(o2,U1).y = Den(o1,U1).y by A1,A4,A8,A12,Th12,
FUNCT_1:49;
      set x = Den(o1,U1).y;
      (G.s1).x = Den(o1,I).a by A6,A17,MSUALG_3:def 7;
      then
A19:  b = (G.s1).x by A15,FUNCT_1:1;
      Result(o1,U1) = (O1 * (the ResultSort of S1)).o1 by MSUALG_1:def 5
        .= O1.((the ResultSort of S1).o1) by A14,FUNCT_1:13
        .= O1.s1 by MSUALG_1:def 2
        .= dom (G.s1) by FUNCT_2:def 1;
      then (G.s1).x = (G.s2).x by A1,A4,A13;
      then b = Den(o2,I).a by A6,A17,A19,A18,MSUALG_3:def 7;
      hence thesis by A9,A11,A16,FUNCT_1:1;
    end;
    hence thesis by RELAT_1:def 3;
  end;
  hence thesis by OSALG_1:27;
end;
