Sentence examples for Preparatory lemmas from inspiring English sources

We need the following preparatory lemmas.

Theorem 3.3 will be proved after some preparatory lemmas.

In Section 3, the reader can find preparatory lemmas, results, and corollaries.

In order to render the main results as transparently as possible, we also give some preparatory lemmas and theorems.

If (M=M_{1}times_{f}M_{2}) is a warped product manifold then (M_{1}) is a totally geodesic and (M_{2}) is a totally umbilical submanifold of M. First of all we give some preparatory lemmas.

There exist M 1, M 2, M 3 > 0, for ∀ α ∈ X, such that ∥ α ∥ ≤ M 1 ∥ α ∥ 1 2 ≤ M 2 ∥ α ∥ 1 ≤ M 3 ∥ α ∥ 2. In order to prove the corresponding convergence conclusions, some preparatory lemma, which can simplify the proof procedure, has to be proved firstly.

Next we shall do some preparatory work from Lemma 3.1 to Lemma 3.7.

The case when I is a finite set is straightforward, but the case when I is infinite, although it has a similar idea, requires a preparatory result, namely Lemma 4.6.

Below, we give some fundamental definitions, lemmas, and preparatory facts from functional analysis, fractional calculus theory, and measures of non-compactness which will be employed in the course of this manuscript.

Section 2 is a review of preparatory facts, addressing both theoretical aspects and Lemma 2.3.

See also preparatory school.