reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;
reserve t for natural_transformation of F,F1,
  t1 for natural_transformation of F1,F2;

theorem Th21:
  F is_naturally_transformable_to F1 & F1
is_naturally_transformable_to F2 implies for t1 being natural_transformation of
F,F1 for t2 being natural_transformation of F1,F2 for a being Object of A holds
  (t2`*`t1).a = (t2.a)*(t1.a)
proof
  assume that
A1: F is_naturally_transformable_to F1 and
A2: F1 is_naturally_transformable_to F2;
A3: F1 is_transformable_to F2 by A2;
  let t1 be natural_transformation of F,F1;
  let t2 be natural_transformation of F1,F2;
  let a be Object of A;
  reconsider t19 = t1 as transformation of F,F1;
  reconsider t29 = t2 as transformation of F1,F2;
A4: F is_transformable_to F1 by A1;
  thus (t2`*`t1).a = (t29`*`t19).a by A1,A2,Def8
    .= (t2.a)*(t1.a) by A4,A3,Def5;
end;
