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On the Bourbaki's fixed point theorem and the axiom of choice

In this note we generalize the Moroianu's fixed point theorem. We propose a very elegant common proof of the Bourbaki's fixed point theorem and our result. We apply our result to give a very elegant proof of the fact that, in the Zermelo-Fraenkel system, the axiom of choice is equivalent to each of the following statements: H. Kneser's Lemma, Zorn's Lemma, Zermelo's Lemma