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

theorem Th23:
  for F1,F2,F3 being Functor of [:A,B:],C st F1
  is_naturally_transformable_to F2 & F2 is_naturally_transformable_to F3 for t1
being natural_transformation of F1,F2, t2 being natural_transformation of F2,F3
  holds export(t2`*`t1) = (export t2)`*`(export t1)
proof
  let F1,F2,F3 be Functor of [:A,B:],C such that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_naturally_transformable_to F3;
A3: F2 is_transformable_to F3 by A2;
  let t1 be natural_transformation of F1,F2, t2 be natural_transformation of
  F2,F3;
A4: F1 is_transformable_to F2 by A1;
A5: export F1 is_naturally_transformable_to export F2 by A1,Th21;
  then
A6: export F1 is_transformable_to export F2;
A7: export F2 is_naturally_transformable_to export F3 by A2,Th21;
  then
A8: export F2 is_transformable_to export F3;
A9: F1 is_naturally_transformable_to F3 by A1,A2,NATTRA_1:23;
  then
A10: F1 is_transformable_to F3;
  now
    let a be Object of A;
    reconsider S1 = (export F1).a, S2 = (export F2).a, S3 = (export F3).a as
    Functor of B,C by Th5;
    reconsider
    s1 = t1, s2 = t2, s3 = t2`*`t1 as Function of [:the carrier of
    A, the carrier of B:], the carrier' of C;
A11: S2 = F2?-a by Th18;
A12: S3 = F3?-a by Th18;
    then reconsider
    T2 = (curry s2).a as natural_transformation of S2,S3 by A2,A11,Th11;
A13: S2 is_naturally_transformable_to S3 by A2,A11,A12,Th11;
    then
A14: S2 is_transformable_to S3;
A15: S1 = F1?-a by Th18;
    then reconsider
    T1 = (curry s1).a as natural_transformation of S1,S2 by A1,A11,Th11;
A16: (export t2).a = [[S2,S3],T2] & (export t1).a = [[S1,S2],T1] by A1,A2,Def5;
A17: S1 is_naturally_transformable_to S2 by A1,A15,A11,Th11;
    then S1 is_naturally_transformable_to S3 by A13,NATTRA_1:23;
    then
A18: S1 is_transformable_to S3;
    reconsider T3 = (curry s3).a as natural_transformation of S1,S3 by A9,A15
,A12,Th11;
A19: Hom((export F1).a,(export F2).a) <> {} & Hom((export F2).a,(export F3
    ).a) <> {} by A6,A8;
A20: S1 is_transformable_to S2 by A17;
    now
      let b be Object of B;
A21:  Hom(F1.[a,b],F2.[a,b]) <> {} & Hom(F2.[a,b],F3.[a,b]) <> {} by A4,A3;
A22:  Hom(S1.b,S2.b) <> {} & Hom(S2.b,S3.b) <> {} by A20,A14;
A23:  T1.b = (T1 qua Function of the carrier of B, the carrier' of C).b
      by A20,NATTRA_1:def 5
        .= s1.(a,b) by FUNCT_5:69
        .= t1.[a,b] by A4,NATTRA_1:def 5;
A24:  T2.b = (T2 qua Function of the carrier of B, the carrier' of C).b
      by A14,NATTRA_1:def 5
        .= s2.(a,b) by FUNCT_5:69
        .= t2.[a,b] by A3,NATTRA_1:def 5;
      thus T3.b = (T3 qua Function of the carrier of B, the carrier' of C).b
      by A18,NATTRA_1:def 5
        .= s3.(a,b) by FUNCT_5:69
        .= (t2`*`t1).[a,b] by A10,NATTRA_1:def 5
        .= t2.[a,b]*t1.[a,b] by A1,A2,NATTRA_1:25
        .= (T2.b)(*)(T1.b qua Morphism of C) by A21,A24,A23,CAT_1:def 13
        .= T2.b*T1.b by A22,CAT_1:def 13
        .= (T2`*`T1).b by A17,A13,NATTRA_1:25;
    end;
    then
A25: T3 = T2`*`T1 by A18,NATTRA_1:19;
    thus (export(t2`*`t1)).a = [[S1,S3],T3] by A9,Def5
      .= ((export t2).a)(*)((export t1).a qua Morphism of Functors(B,C))
             by A16,A25,NATTRA_1:36
      .= (export t2).a*(export t1).a by A19,CAT_1:def 13
      .= ((export t2)`*`(export t1)).a by A5,A7,NATTRA_1:25;
  end;
  hence thesis by A6,A8,NATTRA_1:18,19;
end;
