
theorem Th10:
  for C1,C2,C3,C4 being category, F being Functor of C1,C2,
  G being Functor of C2,C3, H being Functor of C3,C4 st
  F is covariant & G is covariant & H is covariant
  holds H (*) (G (*) F) = (H (*) G) (*) F
  proof
    let C1,C2,C3,C4 be category;
    let F be Functor of C1,C2;
    let G be Functor of C2,C3;
    let H be Functor of C3,C4;
    assume
A1: F is covariant;
    assume
A2: G is covariant;
    assume
A3: H is covariant;
    set GF = G (*) F, HG = H (*) G;
A4: GF is covariant by A1,A2,CAT_6:35;
A5: HG is covariant by A2,A3,CAT_6:35;
    thus H (*) (G (*) F) = GF * H by A4,A3,CAT_6:def 27
    .= (F * G) * H by A1,A2,CAT_6:def 27
    .= F * (G * H) by RELAT_1:36
    .= F * HG by A2,A3,CAT_6:def 27
    .= (H (*) G) (*) F by A5,A1,CAT_6:def 27;
  end;
