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 Th12:
  for f,g being Morphism of LModCat(UN,R) holds [g,f] in dom(the
  Comp of LModCat(UN,R)) iff dom g = cod f
proof
  set C = LModCat(UN,R), V = LModObjects(UN,R);
  let f,g be Morphism of C;
  reconsider f9 = f as Element of Morphs(V);
  reconsider g9 = g as Element of Morphs(V);
A1: cod f = cod'(f9) by Def12
    .= cod (f9);
A2: dom g = dom'(g9) by Def11
    .= dom (g9);
A3: now
    assume dom g = cod f;
    then dom' g9 = cod' f9 by A2,A1;
    hence [g,f] in dom(the Comp of C) by Def13;
  end;
  now
    assume [g,f] in dom(the Comp of C);
    then dom' g9 = cod' f9 by Def13
      .= cod f9;
    hence dom g = cod f by A2,A1;
  end;
  hence thesis by A3;
end;
