Sentence examples for useful for the proof from inspiring English sources

To show this proposition, the following lemmas are useful for the proof.

The following lemmas will be useful for the proof of the main results.

The following lemmas are useful for the proof of the main results.

The following proposition is useful for the proof of the main result.

The following two theorems will be useful for the proof of our results in this article.

In this section, we first give Lemma 4.1 which is useful for the proof of global attractivity.

These results will be useful for the proofs of Theorem 1 in the next section.

The following lemma is useful for the proofs of our main results of this paper.

The following famous Lions vanishing lemma is useful for the proofs of Theorems 3.3 3.4.

In Section 2 some functional analytic preliminaries are stated and some inequalities are derived in Section 3, which are useful for the proofs of our main theorems.

First, we introduce some specifics of the graph theory that are useful for the proofs of asymptotic stability of an endemic equilibrium in this section and the next section.