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;

theorem
  for H1, H2 being covariant Functor of C, D for t being transformation
  of F1, F2, s being transformation of G1, G2 for u being transformation of H1,
  H2 st F1 is_transformable_to F2 & G1 is_transformable_to G2 & H1
  is_transformable_to H2 holds u(#)s(#)t = u(#)(s(#)t)
proof
  let H1, H2 be covariant Functor of C, D, t be transformation of F1, F2, s be
  transformation of G1, G2, u be transformation of H1, H2;
  assume that
A1: F1 is_transformable_to F2 and
A2: G1 is_transformable_to G2 and
A3: H1 is_transformable_to H2;
A4: G1*F2 is_transformable_to G2*F2 & G1*F1 is_transformable_to G1*F2 by A1,A2
,Th10;
A5: H1*s*F2 = H1*(s*F2) & H1*G1*t = H1*(G1*t) by A1,A2,Th16,Th17;
A6: H1*G2*F2 = H1*(G2*F2) & H2*G2*F2 = H2*(G2*F2) by FUNCTOR0:32;
A7: H1*G1*F1 is_transformable_to H1*G1*F2 & u*G2*F2 = u*(G2*F2) by A1,A3,Th10
,Th15;
A8: H1*G1*F1 = H1*(G1*F1) & H1*G1*F2 = H1*(G1*F2) by FUNCTOR0:32;
A9: H1*G1 is_transformable_to H1*G2 by A2,Th10;
  then
A10: H1*G1*F2 is_transformable_to H1*G2*F2 by Th10;
A11: H1*G2 is_transformable_to H2*G2 by A3,Th10;
  then
A12: H1*G2*F2 is_transformable_to H2*G2*F2 by Th10;
  thus u(#)s(#)t = ((u*G2)*F2) `*` ((H1*s)*F2) `*` ((H1*G1)*t) by A11,A9,Th14
    .= (u*(G2*F2)) `*` ((H1*(s*F2)) `*` (H1*(G1*t))) by A12,A10,A7,A5,A8,A6,
FUNCTOR2:6
    .= u(#)(s(#)t) by A4,Th13;
end;
