Sentence examples for needed for the proof from inspiring English sources

Only the former is needed for the proof of Frege's Theorem, however.

In Section 2, we introduce some lemmas needed for the proof of our main results.

The following lemma will be needed for the proof of our main result.

This section presents some technical lemmas needed for the proof of main Theorem 3.1.

In this section, we give some notations and material needed for the proof of our result.

We introduce the following lemma which is needed for the proof of the next theorem.

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

Then we state the critical point theorems needed for the proofs of our main results.

We next recall some definitions and basic facts about Orlicz spaces needed for the proofs of the main results.

We present several important lemmas in this section, which will be needed for the proofs of our main theorems.

The heat kernel analysis needed for the proofs of the Weyl asymptotic formulae for the class of operators under consideration is given in Sect.