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 that

A1: F is_transformable_to F1 and

A2: F1 is_transformable_to F2 and

A3: 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)

A4: F1 is_transformable_to F3 by A2, A3, Th14;

A5: F is_transformable_to F2 by A1, A2, Th14;

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 that

A1: F is_transformable_to F1 and

A2: F1 is_transformable_to F2 and

A3: 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)

A4: F1 is_transformable_to F3 by A2, A3, Th14;

A5: F is_transformable_to F2 by A1, A2, Th14;

now :: thesis: for a being Object of A holds ((t3 `*` t2) `*` t1) . a = (t3 `*` (t2 `*` t1)) . a

hence
(t3 `*` t2) `*` t1 = t3 `*` (t2 `*` t1)
by A1, A4, Th14, Th15; :: thesis: verumlet a be Object of A; :: thesis: ((t3 `*` t2) `*` t1) . a = (t3 `*` (t2 `*` t1)) . a

A6: Hom ((F . a),(F1 . a)) <> {} by A1;

A7: Hom ((F1 . a),(F2 . a)) <> {} by A2;

A8: Hom ((F2 . a),(F3 . a)) <> {} by A3;

thus ((t3 `*` t2) `*` t1) . a = ((t3 `*` t2) . a) * (t1 . a) by A1, A4, Def5

.= ((t3 . a) * (t2 . a)) * (t1 . a) by A2, A3, Def5

.= (t3 . a) * ((t2 . a) * (t1 . a)) by A6, A7, A8, CAT_1:25

.= (t3 . a) * ((t2 `*` t1) . a) by A1, A2, Def5

.= (t3 `*` (t2 `*` t1)) . a by A3, A5, Def5 ; :: thesis: verum

end;A6: Hom ((F . a),(F1 . a)) <> {} by A1;

A7: Hom ((F1 . a),(F2 . a)) <> {} by A2;

A8: Hom ((F2 . a),(F3 . a)) <> {} by A3;

thus ((t3 `*` t2) `*` t1) . a = ((t3 `*` t2) . a) * (t1 . a) by A1, A4, Def5

.= ((t3 . a) * (t2 . a)) * (t1 . a) by A2, A3, Def5

.= (t3 . a) * ((t2 . a) * (t1 . a)) by A6, A7, A8, CAT_1:25

.= (t3 . a) * ((t2 `*` t1) . a) by A1, A2, Def5

.= (t3 `*` (t2 `*` t1)) . a by A3, A5, Def5 ; :: thesis: verum