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This is a misnomer.No, I don't think so. Fair Use isn't merely a matter of court precedent, it specifically is in the Copyright Act (17 U.S. Code § 107 [1]), the language of which indicates it's on the plaintiff, and subsequent case law does seem to have affirmed that unless you have something further to cite? What you cited right there was Lenz v. Universal Music Corp, a 9th Circuit decision about abuse of DMCA takedowns, and in turn considering the "under penalty of perjury" aspect of the DMCA not "Fair Use" as a defense in general. The quote you gave was in the context of §512, Judge Tallman wrote that §512 "unambiguously contemplates fair use as a use authorized by the law". But again that's specific to the DMCA, and even there while the 9th seemed to want to try stemming abuse a bit, they only required the plaintiff to show a purely subjective lack of belief in infringement. Which could be without any real consideration of fair use factors at all. As well as being circuit only, analysis at the time indicated that if anything it might encourage copyright holders specifically to do as little as possible to consider fair use. Lenz did appeal to SCOTUS on that question but certiorari was not granted. Harvard Law had a fairly in-depth looking analysis [2].
In contrast for the 9th Circuit specifically in Perfect 10 v. Amazon/A9.com/Google [3] they explicitly covered Fair Use as an affirmative defense where the burden was on the plaintiffs:
>C. Fair Use Defense
>Because Perfect 10 has succeeded in showing it would prevail in its prima facie case that Google’s thumbnail images infringe Perfect 10’s display rights, the burden shifts to Google to show that it will likely succeed in establishing an affirmative defense. Google contends that its use of thumbnails is a fair use of the images and therefore does not constitute an infringement of Perfect 10’s copyright. See 17 U.S.C. § 107.
Additionally, I can find modern SCOTUS opinions such as in Campbell v. Acuff-Rose Music which support Fair Use as an affirmative defense:
>The fair use factors thus reinforce the importance of keeping the definition of parody within proper limits. More than arguable parodic content should be required to deem a would-be parody a fair use. Fair use is an affirmative defense, so doubts about whether a given use is fair should not be resolved in favor of the self-proclaimed parodist.
It'd be nice if plaintiffs were required to demonstrate as part of a suit that there was not a fair use defense for the defendants, but I really don't think that's the case nationally right now.
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1: https://www.law.cornell.edu/uscode/text/17/107
2: https://harvardlawreview.org/2016/06/lenz-v-universal-music-...
3: http://cdn.ca9.uscourts.gov/datastore/opinions/2007/12/03/06...