Sentence examples for proof proceeding from inspiring English sources

We start the proof proceeding the lines similar to the proof of Theorem 5 (resp. Theorem 6 or Theorem 7).

uniformly in all compact subsets in Ω. Proof Proceeding as in (2.4), we have lim t → T ∗ sup u ( 0, t ) G ( t ) ≤ a ( 0 ), (2.13).

The latter was proved by the program by asserting that there is no onto function from individuals to sets of individuals, with the proof proceeding by a diagonal argument.

Proof Proceeding exactly as in Theorem 2.2, we have that ( g x n ) and ( g y n ) are Cauchy sequences in the complete G-metric space ( g ( X ), G ).

Proof Proceeding exactly as in Theorem 3.1, we have that { g x n }, { g y n }, { g z n }, and { g w n } are Cauchy sequences in the complete metric space ( g ( X ), d ).

Proof Proceeding as in Theorem 3.1, we assume that (1) has a nonoscillatory solution, say x ( t ) > 0 for all t ≥ t 1, where t 1 is chosen so large that Lemma 2.1 and Lemma 2.3 hold.

In some proofs, however, universal instantiations are required in such large numbers as the proof proceeds that even the most powerful computers cannot produce them fast enough.

The proof proceeds in four steps.

The proof proceeds using arguments of contradiction.

The proof proceeds as that of Proposition 3.2.

The proof proceeds along the lines of the proof of Theorem 1 in [13].