theorem Th15:
  H1 is_transformable_to H2 implies r*G1*F1 = r*(G1*F1)
proof
A1: H2*G1*F1 = H2*(G1*F1) by FUNCTOR0:32;
  then reconsider m = r*(G1*F1) as transformation of H1*G1*F1, H2*G1*F1 by
FUNCTOR0:32;
  assume
A2: H1 is_transformable_to H2;
A3: now
    let a be Object of A;
    thus (r*G1*F1)!a = (r*G1)!(F1.a) by A2,Th10,Th12
      .= r!(G1.(F1.a)) by A2,Th12
      .= r!((G1*F1).a) by FUNCTOR0:33
      .= (r*(G1*F1))!a by A2,Th12
      .= m!a by A1,FUNCTOR0:32;
  end;
  H1*G1 is_transformable_to H2*G1 by A2,Th10;
  hence thesis by A3,Th10,FUNCTOR2:3;
end;
