let A, B be Category; :: thesis: for F, F1, F2, F3 being Functor of A,B st F is_transformable_to F1 & F1 is_transformable_to F2 & F2 is_transformable_to F3 holds
for t1 being transformation of F,F1
for t2 being transformation of F1,F2
for t3 being transformation of F2,F3 holds (t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)
let F, F1, F2, F3 be Functor of A,B; :: thesis: ( F is_transformable_to F1 & F1 is_transformable_to F2 & F2 is_transformable_to F3 implies for t1 being transformation of F,F1
for t2 being transformation of F1,F2
for t3 being transformation of F2,F3 holds (t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1) )
assume A1:
( F is_transformable_to F1 & F1 is_transformable_to F2 & F2 is_transformable_to F3 )
; :: thesis: for t1 being transformation of F,F1
for t2 being transformation of F1,F2
for t3 being transformation of F2,F3 holds (t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)
let t1 be transformation of F,F1; :: thesis: for t2 being transformation of F1,F2
for t3 being transformation of F2,F3 holds (t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)
let t2 be transformation of F1,F2; :: thesis: for t3 being transformation of F2,F3 holds (t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)
let t3 be transformation of F2,F3; :: thesis: (t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)
A2:
( F is_transformable_to F2 & F1 is_transformable_to F3 )
by A1, Th19;
then A3:
F is_transformable_to F3
by A1, Th19;
now let a be
Object of
A;
:: thesis: ((t3 `*` t2) `*` t1) . a = (t3 `*` (t2 `*` t1)) . aA4:
(
Hom (F . a),
(F1 . a) <> {} &
Hom (F1 . a),
(F2 . a) <> {} &
Hom (F2 . a),
(F3 . a) <> {} )
by A1, Def2;
thus ((t3 `*` t2) `*` t1) . a =
((t3 `*` t2) . a) * (t1 . a)
by A1, A2, Def6
.=
((t3 . a) * (t2 . a)) * (t1 . a)
by A1, Def6
.=
(t3 . a) * ((t2 . a) * (t1 . a))
by A4, CAT_1:54
.=
(t3 . a) * ((t2 `*` t1) . a)
by A1, Def6
.=
(t3 `*` (t2 `*` t1)) . a
by A1, A2, Def6
;
:: thesis: verum end;
hence
(t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)
by A3, Th20; :: thesis: verum