Sentence examples for proofs were provided from inspiring English sources

A more important result, now known as Fermat's lesser theorem, asserts that if p is a prime number and if a is any positive integer, then ap - a is divisible by p. Fermat seldom gave demonstrations of his results, and in this case proofs were provided by Gottfried Leibniz, the 17th-century German mathematician and philosopher, and Leonhard Euler, the 18th-century Swiss mathematician.

The first well-established proofs were provided by comparing recognition processes involved in faces and objects.

All proofs are provided in the section "Proofs".

Further, all proofs are provided in Appendix.

All proofs are provided in the Appendix.

Some proofs are provided in the appendices.

In Section 3, main results and their proofs are provided.

For purposes explained above, however, the paper is kept self-contained and full proofs are provided.

In order to verify the functionality of the proposed device, some physical proofs are provided.

An important normal form theorem for the structure of cut-free proofs was provided by Andreoli (1992).

Main results and their proofs are provided in Section 3 and Section 4, respectively.