Sentence examples for using the above lemma from inspiring English sources

Further, by using the above lemma, we can make the following useful comparisons.

Using the above lemma, one can obtain the following result.

By using the above lemma, we obtain Theorem 4.3.

Using the above lemma, we can prove our next result.

By using the above lemma, we present some typical examples of a probabilistic b-metric space.

By using the above Lemma 2, we now give the following theorem.

The proof is clear using the above lemmas.

In this section, we will prove our assertions by using the above lemmas.

Using the above Lemmas and Theorem 1, we can get the following Corollaries describing the uniqueness results.

By using the above lemmas and Theorem 1.2, as in [2], we can prove the following results.

Using the above lemmas and following the same lines as the proof of Theorem 4.1 in [35], we obtain the main result concerning the approximation errors.