let A, B be category; :: thesis: for F1, F2, F3 being covariant Functor of A,B st F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent holds
F1,F3 are_naturally_equivalent
let F1, F2, F3 be covariant Functor of A,B; :: thesis: ( F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent implies F1,F3 are_naturally_equivalent )
assume that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_transformable_to F1 ; :: according to FUNCTOR3:def 4 :: thesis: ( for t being natural_transformation of F1,F2 holds
not for a being Object of A holds t ! a is iso or not F2,F3 are_naturally_equivalent or F1,F3 are_naturally_equivalent )
given t being natural_transformation of F1,F2 such that A3: for a being Object of A holds t ! a is iso ; :: thesis: ( not F2,F3 are_naturally_equivalent or F1,F3 are_naturally_equivalent )
assume that
A4: F2 is_naturally_transformable_to F3 and
A5: F3 is_transformable_to F2 ; :: according to FUNCTOR3:def 4 :: thesis: ( for t being natural_transformation of F2,F3 holds
not for a being Object of A holds t ! a is iso or F1,F3 are_naturally_equivalent )
given t1 being natural_transformation of F2,F3 such that A6: for a being Object of A holds t1 ! a is iso ; :: thesis: F1,F3 are_naturally_equivalent
thus ( F1 is_naturally_transformable_to F3 & F3 is_transformable_to F1 ) by A1, A2, A4, A5, FUNCTOR2:2, FUNCTOR2:8; :: according to FUNCTOR3:def 4 :: thesis: ex t being natural_transformation of F1,F3 st
for a being Object of A holds t ! a is iso
take t1 `*` t ; :: thesis: for a being Object of A holds (t1 `*` t) ! a is iso
let a be Object of A; :: thesis: (t1 `*` t) ! a is iso
A7: t1 ! a is iso by A6;
F3 is_transformable_to F1 by A2, A5, FUNCTOR2:2;
then A8: <^(F3 . a),(F1 . a)^> <> {} ;
A9: t ! a is iso by A3;
A10: F2 is_transformable_to F3 by A4;
then A11: <^(F2 . a),(F3 . a)^> <> {} ;
A12: F1 is_transformable_to F2 by A1;
then A13: <^(F1 . a),(F2 . a)^> <> {} ;
(t1 `*` t) ! a = (t1 `*` t) ! a by A1, A4, FUNCTOR2:def 8
.= (t1 ! a) * (t ! a) by A12, A10, FUNCTOR2:def 5 ;
hence (t1 `*` t) ! a is iso by A13, A11, A8, A7, A9, ALTCAT_3:7; :: thesis: verum