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

theorem
  for a,b being Object of E, a9,b9 being Object of C,f being Morphism of
  a, b st a = a9 & b = b9 & Hom(a,b)<>{} holds f is Morphism of a9,b9
proof
  let a,b be Object of E, a9,b9 be Object of C, f be Morphism of a,b;
  assume a = a9 & b = b9 & Hom(a,b)<>{};
  then f in Hom(a,b) & Hom(a,b) c= Hom(a9,b9) by Def4,CAT_1:def 5;
  then f in Hom(a9,b9) & Hom(a9,b9) <> {};
  hence thesis by CAT_1:def 5;
end;
