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

theorem Th3:
  the carrier' of E c= the carrier' of C
proof
  let x be object;
  assume x in the carrier' of E;
  then reconsider f = x as Morphism of E;
  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;
  then f in Hom(a9,b9) & Hom(a9,b9) <> {};
  hence thesis;
end;
