Sentence examples for separate proofs from inspiring English sources

In this sense of conversion, the passage from a proposition to its converse is not, in general, a valid inference; and though often a mathematical proposition and its converse may both hold, separate proofs must be given for each case.

The correct way of dealing with this duality is to provide separate proofs.

But the Department of Housing and Urban Development, which provides housing vouchers and other crucial services, has required separate proofs that a family with children, or an unaccompanied minor, is actually homeless.

Supplementarily we now give a separate proof without using the topological equivalence of ({d_{L^p}}) and ({e_{L^p}}) specified in Corollary 15 (see [54, Section 5.1] or [53, Section 5.1] for the corresponding metrics on (L^2) with different arguments).

Our study is limited by the fact that many of our predicted miRNA:mRNA interactions remain in silico only, and require a separate proof of regulation on an individual basis for any given interaction.

We separate the proof into four steps.

We separate the proof into a sequence of subcases.

We separate the proof of Theorem 2.1 into two parts.

(3.17) For simplicity, we separate the proof into four steps.

In order to prove this result, we separate the proof in several subsections.

The systematic derivation of the generalization algorithms from the modal truth criterion obviates the need for carrying out a separate formal proof of correctness of the EBG algorithms.