A1: F is_naturally_transformable_to F2 by B1, B2, Th23;
A2: F1 is_transformable_to F2 by B2, Def7;
A3: for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t2 . b) * (F1 /. f) = (F2 /. f) * (t2 . a) by B2, Def8;
A4: for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds (t1 . b) * (F /. f) = (F1 /. f) * (t1 . a) by B1, Def8;
F is_transformable_to F1 by B1, Def7;
then for a, b being Object of A st Hom (a,b) <> {} holds
for f being Morphism of a,b holds ((t2 `*` t1) . b) * (F /. f) = (F2 /. f) * ((t2 `*` t1) . a) by A2, A4, A3, Lm2;
hence t2 `*` t1 is natural_transformation of F,F2 by A1, Def8; :: thesis: verum