Sentence examples for using the same proof from inspiring English sources

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

Using the same proof method as in Step 3, we know that y ∗ ∈ Γ.

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

Using the same proof as in Lemma 3.2, we get a contradiction.

Then, using the same proof as that of Lemma 2.3, it can be shown that (2.20).

Using the same proof method as in Theorem 1, we get assertion of Theorem 2.

Property 1 can be derived from the definition of the Z-parameter and we can use the same proof process of (24) that exploits the inequality relation between the arithmetic mean and the geometric mean.

end{aligned} Using the same idea of proof (7), we can obtain the proof of (8).

For (N=infty), we can prove the most results by using the same method of proof.

Using the same means, our proof method can easily carry over to the general finite case.

Moreover, we have p ( x ∗, x ∗ ) = 0. Proof Using the same notations as in the proof of Corollary 3.1, one can show easily from (16) that for all x, y ∈ X, we have.