07:03 Jan 12 

As I understand it, a company (let's call it company Y) has a registered trademark YYY. Company Y has filed suit against company X for trademark infringement, alleging that company X's trademark XXX is not registered.

The "presuntos infractores" are company X. Company X cannot prove that XXX IS a registered trademark, because it isn't ("dado que es falso que así sea"): so says company Y.

Company Y, however, cannot prove the "hecho negativo" that XXX is NOT a trademark, which is what they are alleging, because it is impossible to prove that a trademark is NOT registered. Nor, apparently, are they required to do so by article 82.

This makes sense provided that "es IMPOSIBLE de comprobar para (company)" refers to company Y.

I think "hecho negativo" is much more likely to mean a negative fact or proposition than a proposition that someone has denied (or negated). Company Y denies that XXX is registered. Company X (presumably) denies that it is NOT registered.

23:45 Jan 11 

Desde luego que LOS PRESUNTOS INFRACTORES, se verán imposibilitados para acreditar fehacientemente que el diseño , es una marca registrada ante la autoridad marcaria competente y que lo está amparando servicios como los que ofrece y presta mediante el uso de dicho diseño, a saber, servicios de seguridad, protección y de diferentes tipos de alarma, entre otros, como son servicios de estaciones centrales de protección y monitoreo y control de alarmas y sistemas de seguridad, DADO QUE ES FALSO QUE ASÍ SEA.

23:13 Jan 11 

I'm still confused to tell you the truth. I don't see what makes it a negative fact, which is why I was considering using “negated fact”, as the owner of the (YYY) registered trademark, does deny that the (XXX) trademark is registered. I guess I've been thinking about it too long. %-(
Thanks at lot, Charles

22:43 Jan 11 

I'm not going to post an answer, because I think your proposal is correct: "the negative fact that the (XXX) design is not...".

It would not be right to omit the second "not"; it's got to be there. The negative fact that cannot be proved is that the (XXX) design is not a trademark. It is indeed very difficult, in principle, to think of a way you might prove that something is not a trademark. You'd have to examine all the trademarks ever registered in order to prove that this design is not among them.

If you leave out "not", you are saying that they cannot prove the negative fact that the design is a trademark, which is nonsensical (the proposition that it is a trademark is not a negative fact but a positive one).

Seriously, I think you're OK. These negatives are confusing, but they work here.

On proving negatives, see http://en.wikipedia.org/wiki/Proving_a_negative