let A, B be Category; :: thesis: for F1, F2, F3 being Functor of A,B st F1 ~= F2 & F2 ~= F3 holds
for t being natural_equivalence of F1,F2
for t9 being natural_equivalence of F2,F3 holds t9 `*` t is natural_equivalence of F1,F3
let F1, F2, F3 be Functor of A,B; :: thesis: ( F1 ~= F2 & F2 ~= F3 implies for t being natural_equivalence of F1,F2
for t9 being natural_equivalence of F2,F3 holds t9 `*` t is natural_equivalence of F1,F3 )
assume that
A1: F1,F2 are_naturally_equivalent and
A2: F2,F3 are_naturally_equivalent ; :: thesis: for t being natural_equivalence of F1,F2
for t9 being natural_equivalence of F2,F3 holds t9 `*` t is natural_equivalence of F1,F3
let t be natural_equivalence of F1,F2; :: thesis: for t9 being natural_equivalence of F2,F3 holds t9 `*` t is natural_equivalence of F1,F3
let t9 be natural_equivalence of F2,F3; :: thesis: t9 `*` t is natural_equivalence of F1,F3
thus F1,F3 are_naturally_equivalent by A1, A2, Th32; :: according to NATTRA_1:def 14 :: thesis: t9 `*` t is invertible
let a be Object of A; :: according to NATTRA_1:def 10 :: thesis: (t9 `*` t) . a is invertible
t9 is invertible by A2, Def14;
then A3: t9 . a is invertible by Def10;
t is invertible by A1, Def14;
then A4: t . a is invertible by Def10;
A5: F1 is_naturally_transformable_to F2 by A1, Def11;
then F1 is_transformable_to F2 by Def7;
then A6: Hom ((F1 . a),(F2 . a)) <> {} by Def2;
A7: F2 is_naturally_transformable_to F3 by A2, Def11;
then F2 is_transformable_to F3 by Def7;
then A8: Hom ((F2 . a),(F3 . a)) <> {} by Def2;
(t9 `*` t) . a = (t9 . a) * (t . a) by A5, A7, Th27;
hence (t9 `*` t) . a is invertible by A6, A8, A3, A4, CAT_1:75; :: thesis: verum