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
  F1 ~= F2 & F2 ~= F3 implies for t being natural_equivalence of F1,F2,
t9 being natural_equivalence of F2,F3 holds t9`*`t is natural_equivalence of F1
  ,F3
proof
  assume that
A1: F1,F2 are_naturally_equivalent and
A2: F2,F3 are_naturally_equivalent;
  let t be natural_equivalence of F1,F2, t9 be natural_equivalence of F2,F3;
  thus F1,F3 are_naturally_equivalent by A1,A2,Th25;
  let a be Object of A;
  t9 is invertible by A2,Def13;
  then
A3: t9.a is invertible;
  t is invertible by A1,Def13;
  then
A4: t.a is invertible;
A5: F1 is_naturally_transformable_to F2 by A1;
A6: F2 is_naturally_transformable_to F3 by A2;
  (t9`*`t).a = (t9.a)*(t.a) by A5,A6,Th21;
  hence thesis by A3,A4,CAT_1:45;
end;
