let A, B be category; :: thesis: for F1, F2, F3 being covariant Functor of A,B
for e being natural_equivalence of F1,F2
for e1 being natural_equivalence of F2,F3 st F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent holds
e1 `*` e is natural_equivalence of F1,F3
let F1, F2, F3 be covariant Functor of A,B; :: thesis: for e being natural_equivalence of F1,F2
for e1 being natural_equivalence of F2,F3 st F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent holds
e1 `*` e is natural_equivalence of F1,F3
let e be natural_equivalence of F1,F2; :: thesis: for e1 being natural_equivalence of F2,F3 st F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent holds
e1 `*` e is natural_equivalence of F1,F3
let e1 be natural_equivalence of F2,F3; :: thesis: ( F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent implies e1 `*` e is natural_equivalence of F1,F3 )
assume A1: ( F1,F2 are_naturally_equivalent & F2,F3 are_naturally_equivalent ) ; :: thesis: e1 `*` e is natural_equivalence of F1,F3
hence A2: F1,F3 are_naturally_equivalent by Th33; :: according to FUNCTOR3:def 5 :: thesis: for a being object of A holds (e1 `*` e) ! a is iso
let a be object of A; :: thesis: (e1 `*` e) ! a is iso
A3: ( F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F3 ) by A1, Def4;
A4: ( F1 is_transformable_to F2 & F2 is_transformable_to F3 & F3 is_transformable_to F1 ) by A1, A2, Def4;
A5: (e1 `*` e) ! a = (e1 `*` e) ! a by A3, FUNCTOR2:def 8
.= (e1 ! a) * (e ! a) by A4, FUNCTOR2:def 5 ;
A6: ( <^(F1 . a),(F2 . a)^> <> {} & <^(F2 . a),(F3 . a)^> <> {} & <^(F3 . a),(F1 . a)^> <> {} ) by A4, FUNCTOR2:def 1;
( e1 ! a is iso & e ! a is iso ) by A1, Def5;
hence (e1 `*` e) ! a is iso by A5, A6, ALTCAT_3:7; :: thesis: verum