reserve B,C,D,C9,D9 for Category;
reserve E for Subcategory of C;

theorem Th5:
  for f being (Morphism of E), f9 being Morphism of C st f = f9
  holds dom f = dom f9 & cod f = cod f9
proof
  let f be (Morphism of E), f9 be Morphism of C such that
A1: f = f9;
  set a = dom f, b = cod f;
  reconsider a9 = a, b9 = b as Object of C by Th2;
  f in Hom(a,b) & Hom(a,b) c= Hom(a9,b9) by Def4;
  hence thesis by A1,CAT_1:1;
end;
