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 Th25:
  F1 ~= F2 & F2 ~= F3 implies F1 ~= F3
proof
  assume
A1: F1 is_naturally_transformable_to F2;
  given t being natural_transformation of F1,F2 such that
A2: t is invertible;
  assume
A3: F2 is_naturally_transformable_to F3;
  given t9 being natural_transformation of F2,F3 such that
A4: t9 is invertible;
  thus F1 is_naturally_transformable_to F3 by A1,A3,Th19;
  take t9`*`t;
  let a be Object of A;
A5: t9.a is invertible by A4;
A6: t.a is invertible by A2;
  (t9`*`t).a = (t9.a)*(t.a) by A1,A3,Th21;
  hence thesis by A5,A6,CAT_1:45;
end;
