reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve R for Ring;
reserve G,H for LeftMod of R;
reserve V for LeftMod_DOMAIN of R;

theorem Th13:
  for f being Morphism of LModCat(UN,R) for f9 being Element of
  Morphs(LModObjects(UN,R)) st f = f9 holds dom f = dom f9 & cod f = cod f9
proof
  set C = LModCat(UN,R), V = LModObjects(UN,R);
  set X = Morphs(V);
  let f be (Morphism of C), f9 be Element of X;
  assume
A1: f = f9;
  hence dom f = dom'(f9) by Def11
    .= dom (f9);
  thus cod f = cod' f9 by A1,Def12
    .= cod f9;
end;
