reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th11:
  for U1,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
proof
  let U1,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 the Sorts of Image F is OrderSortedSet of S1 by A2,MSUALG_3:def 12;
  hence thesis by OSALG_1:17;
end;
