We went on and on at length, strengthening our points, using anecdotes, pulling out the nonfiction stops to prove the other's reasoning fallacious, a veteran lawyer and a potential novice going at it.
To prove the necessity, we are reasoning as in the proof for parts (i - ii).
In this paper, we prove the soundness of the proposed reasoning method and demonstrate its effectiveness with several illustrative examples.
As before, replacing the section S with the alternative (tilde{S}), we can apply the same reasoning to prove the presence of a canard value along which there is a saddle-node homoclinic of ('jump-away') canard type.
These lines of reasoning cannot prove the dates given in the Chronicle, much less the existence of Ælle himself, but they do support the idea of an early conquest and the establishment of a settled kingdom.
Gaunilo's reply to Anselm centuries earlier was similar: by parity of reasoning, one can prove the existence in reality (and not just in the understanding) of the maximally perfect island (Anselm 102).
Johannes Caterus, for instance, objected to Descartes that by a precisely parallel form of reasoning we could prove the real existence of the object of an idea of an existent lion (AT 7.99).
Moreover, it is clear that, by using the same line of reasoning we can prove the existence of local extrema in many other objective functionals as, for example, for the case of Renyi entropy which was already proposed and studied for ICA[6, 7].
The piece of reasoning just carried out to prove the truth of KS should be available to any agent (person) with sufficient reasoning capabilities.
By a similar reasoning to that of Theorem 3.1, we can prove the following theorem.
