Sentence examples for conclusion of the proof from inspiring English sources

end{aligned} (4.33)Note that the conclusion of the proof will follow immediately from (4.33).

Third we state a technical lemma on hypergeometric functions, the proof being given in Lemma F.0.4 of Appendix F. The last step is devoted to the conclusion of the proof.

In this case the function \[\mathcal{P} y) = \begin{cases} \phi & \text{if \ \phi\) is the conclusion of the proof coded} \\ & \quad\quad\text{by } y \text{ and } R(\ulcorner \phi \urcorner,y) \\ \top & \text{otherwise} \end{cases} \] where \ \top\) is some fixed tautology would be a polynomial bounded proof system.

After doing so, we would easily reach the conclusion of the proof.

We can actually obtain a stronger conclusion from the last part of the proof, that is, the convergence can be assured if (D.14).

Remark 3.4 Both the conclusion and the proof methods of Theorem 3.1 are different from those previous related results in the literature (see, e.g., [2, 3]).

If a connective does not appear in the conclusion of a proof, it will not appear in the proof at all, since connectives cannot be eliminated.

end{aligned} Thus, as the above wrong conclusion, the proof of the inequality of Lemma 3.3 is not correct.

The above discussion guarantees that the trivial solution of equation (1) is uniformly asymptotically stable, and the same conclusion as in the proof of the theorem can be drawn for square integrability of solutions of equation (1).

Thus the conclusion of the theorem holds and the proof is complete.

This implies that the conclusion of the theorem holds and the proof is complete.