reserve x, y for set;
reserve D for non empty set;
reserve UN for Universe;
reserve C for Category;
reserve O for non empty Subset of the carrier of C;
reserve G,H for AddGroup;
reserve V for Group_DOMAIN;

theorem Th33:
  for f,g being Morphism of GroupCat(UN) holds [g,f] in dom(the
  Comp of GroupCat(UN)) iff dom g = cod f
proof
  set C = GroupCat(UN), V = GroupObjects(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 Def25,Def26;
  hence thesis by Def27;
end;
