Sentence examples for a brief proof from inspiring English sources

We provide a brief proof here.

Below, we provide a brief proof of the previous result.

In Appendix 1, a brief proof is given.

For the sake of completeness, we provide a brief proof.

In here, we only provide a brief proof.

We next state their results and give a brief proof.

We also use fitness graphs in order to give a new brief proof of the equivalent characterizations of fitness landscapes lacking genetic constraints on accessible mutational trajectories.

A brief first proof of this statement may be found in Tasković [6].

We can see other brief proofs for this in Tasković [7, 8], and [13].

By a similar method in [[14], Lemma 2.8], it is not difficult to verify that f 0 ∗ = f 0. In the rest of this paper, we shall study the existence of positive solutions of (1.3) by using the fixed point index theory to give a brief and clear proof for the existence of positive solutions of (1.1).

Although the proof is in "Appendix", we give a brief explanation of the proof.