Sentence examples for statements and proof from inspiring English sources

She was charged with carrying out a series of armed robberies, based on what the police said was a wealth of evidence, including credible witness statements and proof that her car was the getaway vehicle.

Applicants should bring documentation for: identification, proof of income for 30 days, bank statements and proof of expenses.

For details, exact statements and proofs, see Chapter 28 of [14].

Section 3 is devoted to the statements and proofs of our results.

We were not able to find these corrected statements and proofs for a bifurcation from infinity anywhere in the literature.

It is almost ready and is available online and we will often refer to it for details, exact statements and proofs.

Despite these apparent differences, it turns out that the statements and proofs of all the duality theorems for ((P) - (DI)) and ((P) - (tilde{D}I)) are almost identical and, therefore, we shall consider only the pair ((P) - (DI)).

The limits at ±∞ in (d) are taken to be the same in order to simplify the formulation, but the statements and proofs can readily be adapted to the case of different limits.

Given this condition we get (all formal statements and proofs are in Appendix A): Observation 1 If the consumer has the minor problem, then the expert strictly prefers providing the low over providing the high quality service.

In the above-mentioned works [ 23, 25, 45, 64, 79] only concise formulations are provided, while in this section we give mathematical statements and proofs to describe the two problems on the case when only gene duplication, loss, and divergence during speciation are the considered evolutionary events.

Section 2 is devoted to the statement and proof of our main result.