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 Th14:
  for f,g being (Morphism of LModCat(UN,R)), f9,g9 being Element
of Morphs(LModObjects(UN,R)) st f = f9 & g = g9 holds (dom g = cod f iff dom g9
= cod f9) & (dom g = cod f iff [g9,f9] in dom comp(LModObjects(UN,R))) & (dom g
  = cod f implies g(*)f = g9*f9) & (dom f = dom g iff dom f9 = dom g9) &
(cod f =
  cod g iff cod f9 = cod g9)
proof
  set C = LModCat(UN,R), V = LModObjects(UN,R);
  set X = Morphs(V);
  let f,g be Morphism of C;
  let f9,g9 be Element of X;
  assume that
A1: f = f9 and
A2: g = g9;
A3: cod f = cod f9 by A1,Th13;
  hence dom g = cod f iff dom g9 = cod f9 by A2,Th13;
  dom g = dom g9 by A2,Th13;
  hence
A4: dom g = cod f iff [g9,f9] in dom comp(V) by A3,Th11;
  thus dom g = cod f implies g(*)f = g9*f9
  proof
    assume
A5: dom g = cod f;
    then [g,f] in dom (the Comp of C) by Th12;
    hence g(*)f = (comp(V)).(g9,f9) by A1,A2,CAT_1:def 1
      .= g9*f9 by A4,A5,Def13;
  end;
  dom f = dom f9 by A1,Th13;
  hence dom f = dom g iff dom f9 = dom g9 by A2,Th13;
  cod g = cod g9 by A2,Th13;
  hence thesis by A1,Th13;
end;
