Sentence examples for use in the proof from inspiring English sources

This is a useful lemma that we will use in the proof of the main result.

The "twisted tube" method we use in the proof is interesting in itself, as it provides other useful characterizations of Cp smoothness related to the existence of a certain kind of deleting diffeomorphisms, as well as to the failure of Brouwer's fixed point theorem even for smooth self-mappings of starlike bodies in all infinite-dimensional spaces.

There are four paths we will use in the proof.

In this section, we give some results which we shall use in the proof of theorems.

We present here the matrix inequalities that we will use in the proof of our main results.

Employing the technique that we use in the proof of Theorem 3.1, we get ∥ v n − s ∥ ≤ ∥ u n − s ∥.

This notation is deliberately similar to that of the transpose operator we use in the proofs.

Now we compile some identities to use in the proofs of the following theorems.

Before going in this assertion, let us show some definitions and theorems used in the proof.

Firstly, let us prove a conclusion which will be used in the proof of Theorem 3.2.

Let us begin with some lemmas, which will be used in the proof of Theorem 1.1.