reserve D,D9 for non empty set;
reserve R for Ring;
reserve G,H,S for non empty ModuleStr over R;
reserve UN for Universe;
reserve R for Ring;
reserve G, H for LeftMod of R;
reserve G1, G2, G3 for LeftMod of R;
reserve f for LModMorphismStr over R;

theorem Th5:
  for f being LModMorphismStr over R holds dom(f) = G & cod(f) = H
  & fun(f) is additive homogeneous implies f is Morphism of G,H
proof
  let f be LModMorphismStr over R;
  assume that
A1: dom(f) = G & cod(f) = H and
A2: fun(f) is additive homogeneous;
  reconsider f9 = f as LModMorphism of R by A2,Def7;
  f9 is Morphism of G,H by A1,Def8;
  hence thesis;
end;
