theorem Th27:
  H1 is_naturally_transformable_to H2 implies u*G*F = u*(G*F)
proof
  assume
A1: H1 is_naturally_transformable_to H2;
A2: H1*(G*F) = H1*G*F by RELAT_1:36;
  then reconsider v = u*(G*F) as natural_transformation of H1*G*F, H2*G*F by
RELAT_1:36;
A3: H2*(G*F) = H2*G*F by RELAT_1:36;
A4: now
    let a be Object of A;
    thus (u*G*F).a = (u*G).(F.a) by A1,Th20,Th22
      .= u.(G.(F.a)) by A1,Th22
      .= u.((G*F).a) by CAT_1:76
      .= v.a by A1,A2,A3,Th22;
  end;
  H1*G is_naturally_transformable_to H2*G by A1,Th20;
  hence thesis by A4,Th20,Th24;
end;
