reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem Th69:
  for C,D being Category holds
  NatTrans(C,D) c= [: [:bool [:(the carrier' of C), (the carrier' of D):],
                        bool [:(the carrier' of C), (the carrier' of D):]:],
                      bool [:(the carrier of C), (the carrier' of D):] :]
  proof
    let C,D be Category;
    set CD = [:(the carrier' of C), (the carrier' of D):],
        CDp = [:(the carrier of C), (the carrier' of D):];
    now
      let x be set;
      assume x in NatTrans(C,D);
      then consider F1, F2 be Functor of C,D such that
A1: ex t be natural_transformation of F1,F2 st
      ( x = [[F1,F2],t] & F1 is_naturally_transformable_to F2)
        by NATTRA_1:def 16;
      consider t be natural_transformation of F1,F2 such that
A2:   x = [[F1,F2],t] & F1 is_naturally_transformable_to F2 by A1;
      [F1,F2] in [:bool CD, bool CD:] by ZFMISC_1:def 2;
      hence x in [: [:bool CD,bool CD:],bool CDp :] by A2,ZFMISC_1:def 2;
    end;
    hence thesis;
  end;
