reserve A for non empty AltCatStr,
  B, C for non empty reflexive AltCatStr,
  F for feasible Covariant FunctorStr over A, B,
  G for feasible Covariant FunctorStr over B, C,
  M for feasible Contravariant FunctorStr over A, B,
  N for feasible Contravariant FunctorStr over B, C,
  o1, o2 for Object of A,
  m for Morphism of o1, o2;
reserve A, B, C, D for transitive with_units non empty AltCatStr,
  F1, F2, F3 for covariant Functor of A, B,
  G1, G2, G3 for covariant Functor of B, C,
  H1, H2 for covariant Functor of C, D,
  p for transformation of F1, F2,
  p1 for transformation of F2, F3,
  q for transformation of G1, G2,
  q1 for transformation of G2, G3,
  r for transformation of H1, H2;
reserve A, B, C, D for category,
  F1, F2, F3 for covariant Functor of A, B,
  G1, G2, G3 for covariant Functor of B, C;
reserve t for natural_transformation of F1, F2,
  s for natural_transformation of G1, G2,
  s1 for natural_transformation of G2, G3;
reserve e for natural_equivalence of F1, F2,
  e1 for natural_equivalence of F2, F3,
  f for natural_equivalence of G1, G2;

theorem
  for k being natural_equivalence of F1, F3 st k = e1 `*` e & F1, F2
are_naturally_equivalent & F2, F3 are_naturally_equivalent holds k" = e" `*` e1
  "
proof
  let k be natural_equivalence of F1, F3 such that
A1: k = e1 `*` e and
A2: F1, F2 are_naturally_equivalent and
A3: F2, F3 are_naturally_equivalent;
A4: F3 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
  F1 by A2,A3,Def4;
A5: F1 is_transformable_to F2 & F2 is_transformable_to F3 by A2,A3,Def4;
A6: F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
  F3 by A2,A3;
A7: F3 is_transformable_to F2 & F2 is_transformable_to F1 by A2,A3;
  then
A8: F3 is_transformable_to F1 by FUNCTOR2:2;
  now
    let a be Object of A;
A9: <^F1.a,F2.a^> <> {} & <^F2.a,F3.a^> <> {} by A5;
A10: <^F3.a,F1.a^> <> {} by A8;
A11: e!a is iso & e1!a is iso by A2,A3,Def5;
    thus k"!a = ((e1 `*` e)!a)" by A1,A2,A3,Th33,Th38
      .= (((e1 qua transformation of F2, F3)`*` e)!a)" by A6,FUNCTOR2:def 8
      .= ((e1!a)*(e!a))" by A5,FUNCTOR2:def 5
      .= ((e!a)")*((e1!a)") by A11,A9,A10,ALTCAT_3:7
      .= ((e!a)")*(e1"!a) by A3,Th38
      .= (e"!a)*(e1"!a) by A2,Th38
      .= ((e" qua transformation of F2, F1)`*` e1")!a by A7,FUNCTOR2:def 5
      .= (e" `*` e1")!a by A4,FUNCTOR2:def 8;
  end;
  hence thesis by A7,FUNCTOR2:2,3;
end;
