Sentence examples for in their proof from inspiring English sources

Safeguard against malfunction is obtained by duplication or triplication; parallel computer systems are arranged to examine electronic route-setting commands in different ways, and only if automatic comparison shows no discrepancy in their proof that conflicting routes have been secured will the apparatus set the required route.

However, we think there is a mistake in their proof.

Doob's maximal inequality has played an important role in their proof.

We shall remark that Eastham et al. employed continuous eigenvalue branch which studied in [2], in their proof.

But we will remark that in [8, Chapter 8] the authors employed the Riccati equation and a positive definite quadratic functional in their proof.

Firstly, we point out that Dalal et al. [31] used the completeness of X in their proof but failed to mention the same in the hypotheses.

However, there exists some problem in their proofs.

In their proofs, Chen and Zhong also used the definition of the discrete form for modulation spaces.

The following notations and observations may help to understand the next results, and they will be used in their proofs.

Unlike many previous related results, we do not assume the condition (inf_{tinmathbb{R}}a(t)>0), which is a key assumption in their proofs.

In this note, we would like to supplement several conditions, which are used in their proofs but not referred to in the conditions of the main results, to achieve our claim.