let A, B be Category; :: thesis: for F1, F2, F3 being Functor of A,B st F1 ~= F2 & F2 ~= F3 holds

F1 ~= F3

let F1, F2, F3 be Functor of A,B; :: thesis: ( F1 ~= F2 & F2 ~= F3 implies F1 ~= F3 )

assume A1: F1 is_naturally_transformable_to F2 ; :: according to NATTRA_1:def 11 :: thesis: ( for t being natural_transformation of F1,F2 holds not t is invertible or not F2 ~= F3 or F1 ~= F3 )

given t being natural_transformation of F1,F2 such that A2: t is invertible ; :: thesis: ( not F2 ~= F3 or F1 ~= F3 )

assume A3: F2 is_naturally_transformable_to F3 ; :: according to NATTRA_1:def 11 :: thesis: ( for t being natural_transformation of F2,F3 holds not t is invertible or F1 ~= F3 )

given t9 being natural_transformation of F2,F3 such that A4: t9 is invertible ; :: thesis: F1 ~= F3

thus F1 is_naturally_transformable_to F3 by A1, A3, Th19; :: according to NATTRA_1:def 11 :: thesis: ex t being natural_transformation of F1,F3 st t is invertible

take t9 `*` t ; :: thesis: t9 `*` t is invertible

let a be Object of A; :: according to NATTRA_1:def 10 :: thesis: (t9 `*` t) . a is invertible

A5: t9 . a is invertible by A4;

A6: t . a is invertible by A2;

(t9 `*` t) . a = (t9 . a) * (t . a) by A1, A3, Th21;

hence (t9 `*` t) . a is invertible by A5, A6, CAT_1:45; :: thesis: verum

F1 ~= F3

let F1, F2, F3 be Functor of A,B; :: thesis: ( F1 ~= F2 & F2 ~= F3 implies F1 ~= F3 )

assume A1: F1 is_naturally_transformable_to F2 ; :: according to NATTRA_1:def 11 :: thesis: ( for t being natural_transformation of F1,F2 holds not t is invertible or not F2 ~= F3 or F1 ~= F3 )

given t being natural_transformation of F1,F2 such that A2: t is invertible ; :: thesis: ( not F2 ~= F3 or F1 ~= F3 )

assume A3: F2 is_naturally_transformable_to F3 ; :: according to NATTRA_1:def 11 :: thesis: ( for t being natural_transformation of F2,F3 holds not t is invertible or F1 ~= F3 )

given t9 being natural_transformation of F2,F3 such that A4: t9 is invertible ; :: thesis: F1 ~= F3

thus F1 is_naturally_transformable_to F3 by A1, A3, Th19; :: according to NATTRA_1:def 11 :: thesis: ex t being natural_transformation of F1,F3 st t is invertible

take t9 `*` t ; :: thesis: t9 `*` t is invertible

let a be Object of A; :: according to NATTRA_1:def 10 :: thesis: (t9 `*` t) . a is invertible

A5: t9 . a is invertible by A4;

A6: t . a is invertible by A2;

(t9 `*` t) . a = (t9 . a) * (t . a) by A1, A3, Th21;

hence (t9 `*` t) . a is invertible by A5, A6, CAT_1:45; :: thesis: verum