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If P.Ferma knew the proof of the following lemma

"Lemma. If numbers a, b, c, have no common factor, numbers c-a and c-b are also mutually-prime and n is odd, then the numbers (c^n-a^n)/(c-a) and (c^n-b^n)/(c-b) are also mutually-prime",

then with its aid it is possible to briefly and simply prove Fermat's last theorem.

Actually, in the Fermat’s equality (where numbers a, b, c have no common factor and n is odd) numbers c-a and c-b, obviously, mutually-prime. And then from the Fermat's little theorem it follows that with prime q>2c the numbers c^(q-1)-a^(q-1) and c^(q-1)-b^(q-1) are multiple by q.

And since, according to lemma, the numbers (c^(q-1)-a^(q-1))/(c-a) and (c^(q-1)-b^(q-1))/(c-b) are mutually-prime (i.e. have no common factor), then one of the numbers c-a and c-b is divided by q (>2c>c-b>c-a), i.e., the solution of the Fermat’s equation is not integer.

It remains to learn, who and when proved lemma.