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

theorem Th15:
  for O being non empty Subset of the carrier of C holds union{Hom
  (a,b) where a is Object of C,b is Object of C: a in O & b in O} is non empty
  Subset of the carrier' of C
proof
  let O be non empty Subset of the carrier of C;
  set H = {Hom(a,b) where a is Object of C, b is Object of C: a in O & b in O};
  set M = union H;
A1: M c= the carrier' of C
  proof
    let x be object;
    assume x in M;
    then consider X being set such that
A2: x in X and
A3: X in H by TARSKI:def 4;
    ex a,b being Object of C st X = Hom(a,b) & a in O & b in O by A3;
    hence thesis by A2;
  end;
  now
    set a = the Element of O;
    reconsider a as Object of C;
    id a in Hom(a,a) & Hom(a,a) in H by CAT_1:27;
    then id a in M by TARSKI:def 4;
    hence ex f being set st f in M;
  end;
  hence thesis by A1;
end;
