reserve R for non empty Poset,
  S1 for OrderSortedSign;

theorem Th15:
  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 G
  be ManySortedFunction of U1,Image F st F = G & G is order-sorted & G
  is_epimorphism U1,Image F
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;
  consider G being ManySortedFunction of U1,Image F such that
A3: F = G & G is_epimorphism U1,Image F by A1,MSUALG_3:21;
  take G;
  thus thesis by A2,A3;
end;
