Sentence examples for auxiliary proposition from inspiring English sources

First, Let us prove the auxiliary proposition.

First, let us present the following auxiliary proposition.

Now let us give an auxiliary proposition which is used in the following arguments.

In Section 5 we provide some auxiliary propositions.

In Section 3, we prove some auxiliary propositions.

Sections 3 and 4 are devoted to auxiliary propositions and proofs of the main results, respectively.

Before formulating the results on a relative controllability of (1.6), we present some auxiliary propositions.

In this section we formulate auxiliary propositions on the existence and continuity of the Marchaud fractional derivative and its integral representation in terms of the higher-order derivative.

Strong verification required that the truth of a proposition be conclusively ascertainable; weak verification required only that an observation statement be deducible from the proposition together with other, auxiliary, propositions, provided that the observation statement was not deducible from these auxiliaries alone.

In this section, we will establish auxiliary results needed for the proofs of the main propositions.

Before further providing the proof of Proposition 2.2, we shall establish the following four auxiliary lemmas.