A2: F is_naturally_transformable_to F2 by A1, Th8;
A3: ( ( for a, b being Object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds (t1 ! b) * (F . f) = (F1 . f) * (t1 ! a) ) & ( for a, b being Object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds (t2 ! b) * (F1 . f) = (F2 . f) * (t2 ! a) ) ) by A1, Def7;
( F is_transformable_to F1 & F1 is_transformable_to F2 ) by A1;
then for a, b being Object of A st <^a,b^> <> {} holds
for f being Morphism of a,b holds ((t2 `*` t1) ! b) * (F . f) = (F2 . f) * ((t2 `*` t1) ! a) by A3, Lm2;
then t2 `*` t1 is natural_transformation of F,F2 by A2, Def7;
hence ex b₁ being natural_transformation of F,F2 st b₁ = t2 `*` t1 ; :: thesis: verum