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;

theorem Th19:
  F is_naturally_transformable_to F1 & F1
  is_naturally_transformable_to F2 implies F is_naturally_transformable_to F2
proof
  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 Hom(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 Hom(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,Th14;
  take t2`*`t1;
  thus thesis by A1,A2,A3,A4,Lm2;
end;
