For brevity, we prove only part (1) of the lemma.
when (hleqmin{h_{1},-frac{varsigma}{p}}), (H_{1}subseteq S) if and only if j is even, when (hleq h_{2}), (H_{2}subseteq S) if and only if j is odd, when (hleqmin{h_{1},frac{eta}{p}}), (H_{3}subseteq S) if and only if k is odd, when (hleq h_{2}), (H_{4}subseteq S) if and only if k is even, For brevity, we only prove case 1 of the theorem; the others, which can be proved similarly, are omitted.
For brevity, we only prove the case of upward switching.
In this section, we shall, for brevity, state and prove our results only for triple sequences.
The "new director's cut" is an act of commercial jujitsu whereby a studio flatters a star director into adding back scenes once shed to satisfy the market's mindless bias for brevity, and the studio proves, by selling the ungainly result, that it despises money.
For brevity, we show only the former.
I need more practice at balancing the need for brevity with the requirements of proving I meet often long lists of essential and desirable criteria.
Gold assumed that the hypotheses, in the case of language learning, were generative grammars (or alternatively parsers; he proves results concerning both, but for brevity we follow most of the literature and neglect the very similar results on parsers).
For the sake of brevity, we only prove the uniqueness.
