
theorem Th4:
  for C being Category, D being Subcategory of C, E being Subcategory of D
  holds E is Subcategory of C
proof
  let C be Category, D be Subcategory of C, E be Subcategory of D;
A1: the carrier of D c= the carrier of C by CAT_2:def 4;
A2: the Comp of D c= the Comp of C by CAT_2:def 4;
A3: the carrier of E c= the carrier of D by CAT_2:def 4;
A4: the Comp of E c= the Comp of D by CAT_2:def 4;
  thus the carrier of E c= the carrier of C by A1,A3;
  hereby
    let a,b be Object of E, a9,b9 be Object of C;
    reconsider a1 = a, b1 = b as Object of D by CAT_2:6;
    assume that
A5: a = a9 and
A6: b = b9;
A7: Hom(a,b) c= Hom(a1,b1) by CAT_2:def 4;
    Hom(a1,b1) c= Hom(a9,b9) by A5,A6,CAT_2:def 4;
    hence Hom(a,b) c= Hom(a9,b9) by A7;
  end;
  thus the Comp of E c= the Comp of C by A2,A4;
  let a be Object of E, a9 be Object of C;
  reconsider a1 = a as Object of D by CAT_2:6;
  assume
A8: a = a9;
  id a = id a1 by CAT_2:def 4;
  hence thesis by A8,CAT_2:def 4;
end;
