Sentence examples for proved sufficiency from inspiring English sources

On the one hand, for the conditions (22), (23), (24) and (27), (28), (29) will be proved sufficiency, respectively, in Section 4.

Now, we prove sufficiency.

In the following, we will prove sufficiency of the equality.

Using the necessary result proved above in (iii), we conclude that the optimum (psi ^ t,varphi =psi _{1}^{foc} t,varphi)), which also contradicts the given assumption (psi ^ t,varphi =psi _{2}^{foc} t,varphi)) due to the inequality (34) in Lemma 5. Thus we have proven sufficiency in case (i).

Also, we have proved the sufficiency of the introduced G-Karush-Kuhn-Tucker necessary optimality conditions for (CVP) under the new nondifferentiable vector invexity assumption.

In 1981, Hanson [5] introduced the concept of invexity which is an extension of differentiable convex functions and proved the sufficiency of Kuhn-Tucker condition.

In Section 7, some important cases will be considered when the solvability of the main equation can be proved by sufficiency, namely, the selfadjoint case, the case of finite-dimensional perturbations of the spectral data and the case of small perturbations.

We prove the sufficiency.

This proves the sufficiency.

Finally, we prove the sufficiency.

Let us prove the sufficiency.