let A, D, B, C be non empty transitive with_units AltCatStr ; :: thesis: for F1 being covariant Functor of A,B
for G1, G2 being covariant Functor of B,C
for H1 being covariant Functor of C,D
for q being transformation of G1,G2 st G1 is_transformable_to G2 holds
(H1 * q) * F1 = H1 * (q * F1)
let F1 be covariant Functor of A,B; :: thesis: for G1, G2 being covariant Functor of B,C
for H1 being covariant Functor of C,D
for q being transformation of G1,G2 st G1 is_transformable_to G2 holds
(H1 * q) * F1 = H1 * (q * F1)
let G1, G2 be covariant Functor of B,C; :: thesis: for H1 being covariant Functor of C,D
for q being transformation of G1,G2 st G1 is_transformable_to G2 holds
(H1 * q) * F1 = H1 * (q * F1)
let H1 be covariant Functor of C,D; :: thesis: for q being transformation of G1,G2 st G1 is_transformable_to G2 holds
(H1 * q) * F1 = H1 * (q * F1)
let q be transformation of G1,G2; :: thesis: ( G1 is_transformable_to G2 implies (H1 * q) * F1 = H1 * (q * F1) )
assume A1: G1 is_transformable_to G2 ; :: thesis: (H1 * q) * F1 = H1 * (q * F1)
A2: (H1 * G2) * F1 = H1 * (G2 * F1) by FUNCTOR0:33;
then reconsider m = H1 * (q * F1) as transformation of (H1 * G1) * F1,(H1 * G2) * F1 by FUNCTOR0:33;
H1 * G1 is_transformable_to H1 * G2 by A1, Th10;
then A3: (H1 * G1) * F1 is_transformable_to (H1 * G2) * F1 by Th10;

now

let a be object of A; :: thesis: ((H1 * q) * F1) ! a = m ! a
A4: ( (G1 * F1) . a = G1 . (F1 . a) & (G2 * F1) . a = G2 . (F1 . a) ) by FUNCTOR0:34;
thus ((H1 * q) * F1) ! a = (H1 * q) ! (F1 . a) by A1, Th10, Th12
.= H1 . (q ! (F1 . a)) by A1, Th11
.= H1 . ((q * F1) ! a) by A1, A4, Th12
.= (H1 * (q * F1)) ! a by A1, Th10, Th11
.= m ! a by A2, FUNCTOR0:33 ; :: thesis: verum

end;

hence (H1 * q) * F1 = H1 * (q * F1) by A3, FUNCTOR2:5; :: thesis: verum