FERMAT’S LAST THEOREM PROVED BY INDUCTION (accompanied by a philosophical comment and a small perfection)

Abstract. A proof of Fermat’s last theorem is demonstrated. It is very brief, simple, elementary, and absolutely arithmetical. The necessary premises for the proof are only: the three definitive properties of the relation of equality (identity, symmetry, and transitivity), modus tollens, axiom of induction, the proof of Fermat’s last theorem in the case of 𝑛𝑛=3 as well as the premises necessary for the formulation of the theorem itself. It involves a modification of Fermat’s approach of infinite descent. The infinite descent is linked to induction starting from 𝑛𝑛=3 by modus tollens. An inductive series of modus tollens is constructed. The proof of the series by induction is equivalent to Fermat’s last theorem. As far as Fermat had been proved the theorem for 𝑛𝑛=4, one can suggest that the proof for 𝑛𝑛≥4 was accessible to him.
An idea for an elementary arithmetical proof of Fermat’s last theorem (FLT) by induction is suggested. It would be accessible to Fermat unlike Wiles’s proof (1995), and would justify Fermat’s claim (1637) for its proof. The inspiration for a simple proof would contradict to Descartes’s dualism for appealing to merge “mind” and “body”, “words” and “things”, “terms” and “propositions”, all orders of logic. A counterfactual course of history of mathematics and philosophy may be admitted. The bifurcation happened in Descartes and Fermat’s age. FLT is exceptionally difficult to be proved in our real branch rather than in the counterfactual one

Only the proof
The proof accompanied by a philosophical comment

A small perfection to the elementary proof of Fermat’s last theorem by induction published in PhilSci Archive is demonstrated. Only the property of identity is necessary in this second version of the proof. “Symmetry” and “transitivity” of the relation of equality are not necessary in it. This allows for simplifying and shorthening the proof. The refusal of a frequent objection to the proof is explicated. The utilized format is suitable for presenting the proof to wider audience

A small perfection of the proof

The last version (07.052020) as a PDF or @ EasyChair or @ FrenXiv, or @ SSRN, or @ PhilPapers