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 Th33:
  F1, F2 are_naturally_equivalent & F2, F3
  are_naturally_equivalent implies F1, F3 are_naturally_equivalent
proof
  assume that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_transformable_to F1;
  given t being natural_transformation of F1, F2 such that
A3: for a being Object of A holds t!a is iso;
  assume that
A4: F2 is_naturally_transformable_to F3 and
A5: F3 is_transformable_to F2;
  given t1 being natural_transformation of F2, F3 such that
A6: for a being Object of A holds t1!a is iso;
  thus F1 is_naturally_transformable_to F3 & F3 is_transformable_to F1 by A1,A2
,A4,A5,FUNCTOR2:2,8;
  take t1 `*` t;
  let a be Object of A;
A7: t1!a is iso by A6;
  F3 is_transformable_to F1 by A2,A5,FUNCTOR2:2;
  then
A8: <^F3.a,F1.a^> <> {};
A9: t!a is iso by A3;
A10: F2 is_transformable_to F3 by A4;
  then
A11: <^F2.a,F3.a^> <> {};
A12: F1 is_transformable_to F2 by A1;
  then
A13: <^F1.a,F2.a^> <> {};
  (t1 `*` t)!a = ((t1 qua transformation of F2, F3) `*` t)!a by A1,A4,
FUNCTOR2:def 8
    .= (t1!a)*(t!a) by A12,A10,FUNCTOR2:def 5;
  hence thesis by A13,A11,A8,A7,A9,ALTCAT_3:7;
end;
