Sentence examples for some preparatory lemma from inspiring English sources

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.

Theorem 3.3 will be proved after some preparatory lemmas.

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.

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

We need the following preparatory lemmas.

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

The proof is given at the end of the subsection after some preparatory lemmata.

Some preparatory work has been done.

For this purpose, some preparatory concepts are needed.

This section is devoted to some preparatory results.