Sentence examples for by the same proof from inspiring English sources

By the same proof as that of (2) in case (i), we have (4.5).

By the same proof of (4.21) we can show that (4.32).

By the same proof as the one above, we have q ˆ ∈ F ( S ) ∩ MEP.

By the same proof as in Case 1, we obtain that (3.65).

Then, by the same proof as in the first part of Lemma 2.2, one gets.

By the same proof as the one above, we have x ˆ ∈ F ( T ) ∩ ( A + B ) − 1 0.

By using the same proof as Theorem 3.2, we can easily obtain the following conclusions.

By using the same proof of Theorem 1, we show the existence of optimal controls.

The converse of Theorem 1.4 is also valid by following the same proof of Theorem 1.2.

By using the same proof, we obtain that and are nonexpansive.

In addition, the stability of the pinning-based consensus algorithm defined in (4) also can be ensured by using the same proof.