reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;
reserve F,F1,F2,F3 for Functor of A,B,
  G,G1,G2,G3 for Functor of B,C,
  H,H1,H2 for Functor of C,D,
  s for natural_transformation of F1,F2,
  s9 for natural_transformation of F2,F3,
  t for natural_transformation of G1,G2,
  t9 for natural_transformation of G2,G3,
  u for natural_transformation of H1,H2;

theorem
  F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
F3 & G1 is_naturally_transformable_to G2 & G2 is_naturally_transformable_to G3
  implies (t9`*`t)(#)(s9`*`s) = (t9(#)s9)`*`(t(#)s)
proof
  assume that
A1: F1 is_naturally_transformable_to F2 and
A2: F2 is_naturally_transformable_to F3 and
A3: G1 is_naturally_transformable_to G2 and
A4: G2 is_naturally_transformable_to G3;
A5: t9(#)s9 = (G3*s9)`*`(t9*F2) & t(#)s = (G2*s)`*`(t*F1) by A1,A2,A3,A4,Th34;
A6: G1*F1 is_naturally_transformable_to G2*F1 by A3,Th20;
A7: G2*F1 is_naturally_transformable_to G2*F2 by A1,Th20;
  then
A8: G1*F1 is_naturally_transformable_to G2*F2 by A6,NATTRA_1:23;
A9: G2*F2 is_naturally_transformable_to G3*F2 by A4,Th20;
A10: G1 is_naturally_transformable_to G3 by A3,A4,NATTRA_1:23;
  then
A11: G1*F1 is_naturally_transformable_to G3*F1 by Th20;
A12: G3*F1 is_naturally_transformable_to G3*F2 by A1,Th20;
A13: G2*F1 is_naturally_transformable_to G3*F1 by A4,Th20;
A14: G3*F2 is_naturally_transformable_to G3*F3 by A2,Th20;
  F1 is_naturally_transformable_to F3 by A1,A2,NATTRA_1:23;
  hence (t9`*`t)(#)(s9`*`s) = (G3*(s9`*`s))`*`((t9`*`t)*F1) by A10,Th34
    .= (G3*s9)`*`(G3*s)`*`((t9`*`t)*F1) by A1,A2,Th25
    .= (G3*s9)`*`(G3*s)`*`((t9*F1)`*`(t*F1)) by A3,A4,Th26
    .= (G3*s9)`*`((G3*s)`*`((t9*F1)`*`(t*F1))) by A14,A12,A11,NATTRA_1:26
    .= (G3*s9)`*`((G3*s)`*`(t9*F1)`*`(t*F1)) by A12,A6,A13,NATTRA_1:26
    .= (G3*s9)`*`((t9(#)s)`*`(t*F1)) by A1,A4,Th34
    .= (G3*s9)`*`((t9*F2)`*`((G2*s)`*`(t*F1))) by A6,A9,A7,NATTRA_1:26
    .= (t9(#)s9)`*`(t(#)s) by A14,A9,A8,A5,NATTRA_1:26;
end;
