Sentence examples for details of proof from inspiring English sources

To avoid repetition, we omit the details of proof.

The following two lemmas can be derived from [[17], Theorem 1] (see also [18]), we here choose to omit the details of proof.

Details of proof are given in Additional file 1. DS is the synergism assessment factor derivative of the original system while DS' could be seen as that of a simplified system.

Details of proof are given in Additional file 4. We adapted numerical examples in [ 1] as shown in Figure 1 and Figure 2 respectively to verify the Corollary 1 and 2 above.

So the details of proofs are omitted.

In these cases we omit proofs or details of proofs making references to corresponding pages of the articles [3] or [13].

In the cases when we will use the results from other papers we will omit the proofs or details of proofs, but we will refer to them appropriately.

That is to say, By Corollary 1 and 2, whether the simplified systems could generate synergism under equivalent inhibition can help to determine whether the original systems can generate synergism effect under actual inhibition (see details of proofs in Additional file 1).

Such coins have been polished to the brilliance of proof coins, but they lack the detail of proof coins.

We give an outline focusing on the innovative parts of the proof, and the details of the proof are similar to those of Theorem 3.1.

Since the proof is similar to the proof of Theorem 3.2 in [17], we delete the details of the proof.