Sentence examples for are needed for the proof from inspiring English sources

The following results are needed for the proof of Theorem 3.2.

The following lemmas are needed for the proof of our main results in next section.

We also investigate conditions such that I a, b ; c f ∈ R t ( A, B ) ( UST, UCV, S λ ∗ and C λ ), whenever f ∈ R t ( A, B ). Now we introduce several lemmas which are needed for the proof of our main results.

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

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

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

We recall the fixed point theorem for a single-valued Reich-type operator, which is needed for the proof of our first main result.

The above lemma is Corollary 6 to Theorem 2 of [2] and only the special case d = c is needed for the proof of Theorem 1.3.

Remark 2.8 The condition (A1) is needed for the proof of Lemma 2.7, see details in [27] and the references therein.

In this section, we recall some concepts including the assumption which will be needed for the proof of our main result.

We added a condition p p ∈O(γ −(t+1+ε)) compared to the statement in [12], but as far as we could tell, it is needed for the proof to be complete.