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The proof above applies with almost no change to showing that any compact subset S of a Hausdorff topological space X is closed in X.
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proof and above
As the problem of P ≟ PSPACE has not yet been solved, the proof of inequality between BQP and classes mentioned above is supposed to be difficult.
The higher-dimensional chain rule can be proved using a technique similar to the second proof given above.
The above proof suggests another interpretation of the Cauchy – Riemann equations.
The very first geometric proof in the Elements, shown in the figure above, is that any line segment is part of a triangle ; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex.
The above proof is an example of a non-constructive proof disallowed by intuitionists:
Here is a sketch of the proof referred to in Tao's lecture mentioned above.
( The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, " The above proposition is occasionally useful.
Depending on the value of n, we specify a sufficiently large positive integer k ( to meet our needs later ), and multiply both sides of the above equation by, where the notation will be used in this proof as shorthand for the integral:
The uncountability of the real numbers was already established by Cantor's first uncountability proof, but it also follows from the above result.
The above proof fails for W. V. Quine's " New Foundations " set theory ( NF ).
The fundamental property of used in the proof above is that there cannot be a chain of positive integers where each integer in the chain is strictly less than its predecessor ; in other words, the ring of integers is not " too large " since it cannot sustain such a " large chain ".
A first glance at the second proof of Tychnoff may suggest that the proof uses no more than ( BPIT ), in contradiction to the above.
It is actually a more comprehensible proof than the above ( probably because it does not involve Zorn's Lemma, which is quite opaque to most mathematicians as far as intuition is concerned !).
The proof is a relatively straightforward generalization of the proof given above that a 60-degree angle is not trisectible.
Using the above rule, we can complete the proof of Fermat's little theorem quite easily, as follows.
There are two steps in the above proof that we need to justify:
Our use of this cancellation law in the above proof of Fermat's little theorem was valid, because the numbers 1, 2, ..., p − 1 are certainly not divisible by p ( indeed they are smaller than p ).
Alcohol concentrations above about 50 % alcohol by volume ( 100 proof ) are flammable and therefore dangerous to handle.
and it follows from the proof above that
The existence of proofs like the one above shows that such a task is not so simple, because at least one of the deduction rules used in the proof above must be omitted or restricted.
proof and applies
The original completeness proof applies to all classical models, not some special proper subclass of intended ones.
An alternate view takes qe-ra-si-ja and qe-ra-si-jo as proof of androgyny, and applies this name by similar arguments to the legendary seer, Tiresias, but these views are not mutually exclusive of one another.
In fact, it is possible to give a proof that is a Noetherian ring without appealing to its order structure and this proof applies more generally to principal ideal rings ( i. e., rings in which every ideal is generated by a single element ).
This is the highest standard used as the burden of proof in Anglo-American jurisprudence and typically only applies in criminal proceedings.
However, this equation only applies if the source resistance cannot be adjusted, e. g., with antennas ( see the first line in the proof stating " fixed source resistance ").
A simpler more illustrative proof is as follows, although it applies only in the discrete case.
In that case the more complicated proof of the product rule applies.
After the image is finished, or at least ready to proof, the artist applies ink to the plate with a dauber.
No oracle machine is capable of solving its own halting problem ( a variation of Turing's proof applies ).
The result used for Liouville numbers in the proof is effective in the way it applies the mean value theorem: but improvements ( to what is now the Thue-Siegel-Roth theorem ) were not.
In the first episode of the TV series Lois & Clark: The New Adventures of Superman Clark Kent applies for a job at the Daily Planet newspaper, producing an article on Knob-Tailed Geckos as proof of his writing skills.
In 1902, Hurwitz published a short proof using the Fourier series that applies to arbitrary rectifiable curves ( not assumed to be smooth ).
Kurt Gödel wrote the first paper on provability logic, which applies modal logic — the logic of necessity and possibility — to the theory of mathematical proof, but Gödel never developed the subject to any significant extent.
Similar proof also applies to many multi-dimensional generalizations.
For the same reasons, though, a lack of such proof applies to almost all long-range planning techniques.
Social proof naturally also applies to products and is used
*" The proof that Q is universal relative to the set of all 3-bit gates applies step by step, mutatis mutandis, to Q < sub > 4 </ sub >.
A proof by double counting due to Jim Pitman applies to trees.
The proof also applies to a generalization of CCM for any size block cipher, and for any size cryptographically strong pseudo-random function ( since in both counter mode and CBC-MAC, the block cipher is only ever used in one direction ).
While convicting someone of the crime of murder requires proof beyond a reasonable doubt, the slayer rule applies to civil law, not criminal law, so it is only necessary to prove the wrongful killing by a preponderance of the evidence, as in a wrongful death claim.
Plantinga defends Anselm's proof by averring that it applies exclusively to Him, a viewpoint that Anselm himself had stated but failed to elaborate.
This proof only applies in Case 2 since if ∠ BAC > 120 ° A lies inside the circumcircle of BPC which switches the relative positions of A and F. However it is easily modified to cover Case 1.
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