It depends on which meaning of "shown" you are using in English. "I had already proven that X equals Y." and "I had already shown that X equals Y" mean the same thing. "I had already shown the new house" is a different meaning of "shown" and not the same as either the English "proven" or the Spanish "demostrado" (that would be "mostrado", I think).
The hints list 3 options (shown, proven, demonstrated). If 'shown' is wrong, then it should not be listed. I chose 'shown' because it is the first one listed, and it was marked wrong (March 2015).
lo había demostrado
(I/he/she/it/you) had showed/shown (it/him)
(I/he/she/it/you) had proved/proven (it/him)
(I/he/she/it/you) had demonstrated (it/him)
Yes, proven for some reason just doesn't sound right to me, though I am uncertain why. It isn't that it is incorrect, I think, but that it could be tweaked with wording? I think the suggested wording of "shown" in this instance is fine! The two forms seem to translate well to me.
The hover-suggestion shows 3 possible interpretations of había demostrado: "had showed/shown", "had demonstrated", and "had proved/proven". Apparently "showed" and "proved" are grammatically acceptable in this conjugation, although perhaps not preferred (according to English dictionaries). Also, the object pronoun could be equally well translated as "it" or "him". There is nothing about the other 2 words of this sentence that would rule out any of these interpretations. (I put "I had already shown him", mostly out of confusion regarding direct vs indirect objects, but it should have been accepted nonetheless, along with the rest of the answers mentioned here. I will report mine as well.
Most math books use show or prove interchangably, and very rarely or never use demonstrate when reffering to math proofs being shown. Your first sentance would normally be transalted as "A mathematical rule can be shown or proved." Translatng it as "A mathematical rule can be demonstrated or proved." would be unusual since demonstrate kinda implies it being actively shown by someone.
I reported The use of Shown to be accepted it was not when I tried it but it may be now IDK. But you cannot use showed because it is a past tense for show, and not the past particple which is shown. I think this exercise was for perfect past participles hence the need for the helping verb haber.
An object pronoun follows and is attached to the affirmative command. It may either follow the infinitive and present participle, or precede the 'conjugated' verb. For example : Voy a tomarlo OR Lo voy a tomar. & Ella esta' tomandolo OR Ella lo esta' tomando. The object pronoun is not attached to the past participle.