reserve x,y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve f for RingMorphismStr;
reserve G,H,G1,G2,G3,G4 for Ring;
reserve F for RingMorphism;
reserve V for Ring_DOMAIN;

theorem Th21:
  for f,g being Morphism of RingCat(UN) holds [g,f] in dom(the
  Comp of RingCat(UN)) iff dom g = cod f
proof
  set C = RingCat(UN), V = RingObjects(UN);
  let f,g be Morphism of C;
  reconsider f9 = f as Element of Morphs(V);
  reconsider g9 = g as Element of Morphs(V);
  dom g = dom(g9) & cod f = cod(f9) by Def19,Def20;
  hence thesis by Def21;
end;
