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, FUNCTOR2:4, FUNCTOR2:10; :: 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: ( F1 is_transformable_to F2 & F2 is_transformable_to F3 & F3 is_transformable_to F1 ) by A1, A2, A4, A5, FUNCTOR2:4, FUNCTOR2:def 6;
A8: (t1 `*` t) ! a = (t1 `*` t) ! a by A1, A4, FUNCTOR2:def 8
.= (t1 ! a) * (t ! a) by A7, FUNCTOR2:def 5 ;
A9: ( <^(F1 . a),(F2 . a)^> <> {} & <^(F2 . a),(F3 . a)^> <> {} & <^(F3 . a),(F1 . a)^> <> {} ) by A7, FUNCTOR2:def 1;
( t1 ! a is iso & t ! a is iso ) by A3, A6;
hence (t1 `*` t) ! a is iso by A8, A9, ALTCAT_3:7; :: thesis: verum