Sentence examples for and in the proof from inspiring English sources

The act of prototyping has historical significance rooted in the process of invention and in the proof and protection of intellectual property.

Proof Applying the same method as in the proof of Corollary 3.3 and in the proof of Lemma 2.1, we can obtain Corollary 3.5.

By using the same argument of Saejung [8, Theorem, page 6] and in the proof of Theorem 3.1, we have and.

Indeed, notice that assumptions (2.2) and (2.5) of Theorem 2.3 have been used only in the choice of the constant C (see (2.11)) and in the proof of Step 1.

For the case π < θ < 2 π, applying the same method as above and in the proof of Lemma 1.1 and Lemma 2.2, we have 0 ≥ arg { f ( z ) z } ≥ arg { f ′ ( z ) z p − 1 } (2.7).

end{aligned} (9) Here, L is a positive constant, and in the proof of the last inequality, we have used Lemma 1 (i.e., for (forall tgeq 0), (x t)) and (y(t)) are bounded).

and of equivalent sea introduced, and in the proofs it emphasizes the geometric point of view using terms like triangle, triangle height and triangle base in place of e.t.s., Hsmax of the e.t.s, and duration of the e.t.s.s

A sliding mode controller is formulated through performance parameters which include the settling time and overshoots in the proof mass.

We use the same notations for and as in the proof of Theorem 2.4.

It is easy to show that for all and as in the proof of Theorem 2.2.

Asymptotic formulas then follow similarly as (10) and (11) in the proof of Theorem 2.