Sentence examples for the above proofs from inspiring English sources

In fact by modifying the above proofs one can obtain the regularity properties for a larger pointclass, namely, ∃ℝΓ.

Also, the above proofs may be similar to the method used in [2] and [3].

Combining the above proofs of both "if" and "only if" parts, Theorem 1 thus follows.

From the above proofs, we know that the assumptions of Lemma 1.4 are satisfied.

However, we find that both of the above proofs omit the verification of constraints of assurance regions, which makes them incredible.

The methods used in the above proofs can be modified to obtain the following results for unsaturated solutions.

According to the above proof.

Concluding the above proof, we obtain.

As the above proof, Tv = v.

(1 If, then the inequality (2.2) holds from the above proof.

If, then the assertion is true from the above proof.