theorem
  F1 is_naturally_transformable_to F2 & G1 is_naturally_transformable_to
  G2 & H1 is_naturally_transformable_to H2 implies u(#)(t(#)s) = (u(#)t)(#)s
proof
  assume that
A1: F1 is_naturally_transformable_to F2 and
A2: G1 is_naturally_transformable_to G2 and
A3: H1 is_naturally_transformable_to H2;
A4: u*(G2*F2) = u*G2*F2 & H1*(t*F2) = H1*t*F2 by A2,A3,Th27,Th28;
A5: H1*G1 is_naturally_transformable_to H1*G2 by A2,Th20;
  then
A6: H1*G1*F2 is_naturally_transformable_to H1*G2*F2 by Th20;
A7: H1*(G1*s) = H1*G1*s & H1*G1*F1 is_naturally_transformable_to H1*G1*F2 by A1
,Th20,Th29;
A8: H1*G1*F1 = H1*(G1*F1) & H1*G1*F2 = H1*(G1*F2) by RELAT_1:36;
A9: H1*G2 is_naturally_transformable_to H2*G2 by A3,Th20;
  then
A10: H1*G2*F2 is_naturally_transformable_to H2*G2*F2 by Th20;
A11: H1*G2*F2 = H1*(G2*F2) & H2*G2*F2 = H2*(G2*F2) by RELAT_1:36;
  G1*F2 is_naturally_transformable_to G2*F2 & G1*F1
  is_naturally_transformable_to G1*F2 by A1,A2,Th20;
  hence u(#)(t(#)s) = (u*(G2*F2))`*`((H1*(t*F2))`*`(H1*(G1*s))) by Th25
    .= (u*G2*F2)`*`(H1*t*F2)`*`((H1*G1)*s) by A8,A11,A4,A7,A6,A10,NATTRA_1:26
    .= u(#)t(#)s by A5,A9,Th26;
end;
