Sentence examples for through corollary from inspiring English sources

Further, our theorem has been validated through Corollary 6.3, which is a result of Bor [14].

So, upon replacing,,, respectively, by,,, and then dividing by, in each of the expressions of Theorem 4.1 through Corollary 4.15, we will get the corresponding symmetric identities for Type.  .

So, upon replacing,,, respectively, by,,, and then dividing by, in each of the expressions of Theorem 4.1 through Corollary 4.15, we will get the corresponding symmetric identities for Type.

Moreover, we prove that the equality of 4-WFRFT space and multi-WFRFT space through the corollary.

This region can be obtained through either Corollary 2 or Theorem 7 (by setting and eliminating redundant bounds) implying the consistency of the results.

We will only carry through the details for Corollary 3.

It was as if I could suddenly discern, in this contemporary vignette, the ancient corollary through which Plato and some of his contemporaries might have strolled; to wit, I was sensing the eternal in the ephemeral.

has the extremal solutions between the lower solution σ 1 ≡ − 1 and the upper solution σ 2 ≡ 1. Conditions (H1) through (H3) in Corollary 1.1 are obviously satisfied.

Similarly, we will prove Corollary 3.1 through the following two steps.

Corollary 2.2, through Lemma 3.1(i), leads directly to the subsequent result.

Without introducing auxiliary equation and the existence result of conjugate points as [2, 4], we can prove this corollary directly through the Sobolev inequality (the idea of the proof origins to Brown and Hinton [5, page 5]).