
theorem Th8:
  for A,B be category, F,F1,F2 be covariant Functor of A,B holds F
is_naturally_transformable_to F1 & F1 is_naturally_transformable_to F2 implies
  F is_naturally_transformable_to F2
proof
  let A,B be category, F,F1,F2 be covariant Functor of A,B;
  assume
A1: F is_transformable_to F1;
  given t1 being transformation of F,F1 such that
A2: for a,b being Object of A st <^a,b^> <> {} for f being Morphism of a
  ,b holds t1!b*F.f = F1.f*(t1!a);
  assume
A3: F1 is_transformable_to F2;
  given t2 being transformation of F1,F2 such that
A4: for a,b being Object of A st <^a,b^> <> {} for f being Morphism of a
  ,b holds t2!b*F1.f = F2.f*(t2!a);
  thus F is_transformable_to F2 by A1,A3,Th2;
  take t2`*`t1;
  thus thesis by A1,A2,A3,A4,Lm2;
end;
