Sentence examples for proceeding with the proof from inspiring English sources

Then, proceeding with the proof of the theorem similar to the proof of Theorem 3.1, we can derive the conclusion.

Before proceeding with the proof of Theorem 3.4, let us state some technical result about (mathcal{N}_{epsilon}^{pm}).

Before proceeding with the proof, it is important to note that Theorem 3.1 is a generalization of the special case (n = 2) in Hürlimann [21], Proposition II.2.

Thus, all we have to do is to show Propositions 1 and 2. Before proceeding with the proof of these propositions, we would like to show a corollary obtained from Proposition 1. Corollary 1 Suppose a non-trivial solution u of the non-linear equation ( − 1 ) M u ( x ) u ( 2 M ) ( x ) − r ( x ) ( u ( m ) ( x ) ) 2 = 0 (10).

Before proceeding with the proof of Theorem 2.3, which is the main result in this section, we shall need the following two lemmas: Lemma 2.1 If an odd mapping f : X → Y satisfies (1.2) for all x, y ∈ X, then f is additive.

Proceeding with the proof in much the same way as [9], the equivalent definition for polyconvexity is given by Iwaniec and Lutoborski.

Let us proceed with the proof of (ii).

Now we proceed with the proof of Theorem 3.1.

We will proceed with the proof in two steps.

We now proceed with the proof of (3.5).

In the following, we proceed with the proof of the "only if" part.