Sentence examples for the proofs given from inspiring English sources

The development uses only elementarily algebraic ideas, mainly the bounded real lemma, thus the proofs given are simple and clear.

In this paper, the proofs given by Malis are discussed and it will be shown that there is a limitation to the validity of the ESM.

Our proofs are different from the proofs given by Elcrat [1].

The proofs given to Caristi's result vary and use different techniques (see [3, 6 8]).

The concept of minimal element is crucial in the proofs given for Caristi's fixed point theorem.

But, from the proofs given there, it is obvious that the statements hold under the more general setting given here.

The proof given relies upon properties of the medial axis, which is explained in this paper.

In one instance, a proof devised by the program was more elegant than the proof given in the books.

The proof given here is a simple application of the notion of quasicentral approximate unit.

The proof given there structurally closely follows the proof of the original PCP theorem by Dinur (2007).

The proof given by Hiai and Petz is quite different.