Thus, the goal of a conditional proof is to demonstrate that if the CPA were true, then the desired conclusion necessarily follows.

Thus this code number of a proof of contradition of T must be a non-standard number.

If a statement P is provable, then it is certainly impossible to prove that there is no proof of P. But even if it can be shown that no disproof of P is possible, we cannot conclude from this absence that there is a proof of P. Thus P is a stronger statement than not-not-P.

Thus the only thing we don't have is a formal proof of consistency of whatever version of set theory we may prefer, such as ZF.

Thus the claim is true and we have completed the proof for 2 colours.

Thus, assuming the Axiom of Choice, we can infer that κ is a strong limit cardinal, which completes the proof of its inaccessibility.

Thus, given a proof of the conclusion in the special case, it is trivial to adapt it to prove the conclusion in all other cases.

Thus, though hard proof is lacking but given the extent of possible habitat and the precautionary principle, it is believed to be extant for the time being.

Thus, the proof shown above is not entirely rigorous when.

Thus the conjecture of Pólya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.

Thus, the perfect graph theorem can be used to prove Dilworth's theorem from the ( much easier ) proof of Mirsky's theorem, or vice versa.

Thus, there is a lower burden of proof, making it easier for the DOJ to pursue infringers.

Thus, the proof is completed.

Thus shall be done for all grievances or thievery, if the accuser cannot provide proof, and the grievance is for less than half of gold grivna, then give him a trial by iron in captivity ; if the grievance is for less than that, but more than two silver grivnas, then trial by water ; if it is even less, then he must make an oath.

Thus, the complementation property of perfect graphs can provide an alternative proof of Dilworth's theorem.

Thus, like the typical small claims court, the plaintiff has the burden of proof and must argue his / her case before the Judge and answer any questions directed to him / her.

Thus, loosely speaking, a proof calculus is a template or design pattern, characterized by a certain style of formal inference, that may be specialized to produce specific formal systems, namely by specifying the actual inference rules for such a system.

Thus, we have the condition that ƒ is a ( Hurwitz ) stable polynomial if and only if p − q = n ( the proof is given below ).

Thus, an act is illegal without extrinsic proof of any surrounding circumstances such as lack of scienter ( knowledge ) or other defenses.

Thus they are easier to prove in some circumstances, but conversely the burden of proof is on the defendant.

Thus the proof might, after all, beg the question against a kind of scepticism radical enough to put in doubt the rule of non-contradiction.

He is callous to supplicants in need .” Steblin ( 2007, p. 169 ) also discovered a document headed " Table of Rules " and dated 5 – 11 July with a list of ethical categories in the hand-writing of Christoph von Stackelberg: " Thus this whole document, dated at the time when ... he ... was deliberating about his future, is surely further proof that Josephine was left alone ... in June and July 1812.

Thus, the scheme serves mainly as a proof of concept, and more efficient provably-secure schemes such as Elgamal have been developed since.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus, besides the training provided to youth in school, the existence of the school program can have supplementary benefits to industry which make it an asset to industrial development efforts.

Thus it too has an objective, observer-independent existence.

Thus, he reasoned that existence must be due to an agent cause that necessitates, imparts, gives, or adds existence to an essence.

Thus, theory has its roots in the materiality of social existence.

Thus, in the sentence " There exists a man ", the term " man " is asserted to be part of existence.

Thus Ockham argued that " Socrates has wisdom ", which apparently asserts the existence of a reference for " wisdom ", can be rewritten as " Socrates is wise ", which contains only the referring phrase " Socrates ".

Thus, Heracles ' very existence proved at least one of Zeus ' many illicit affairs, and Hera often conspired against Zeus ' mortal offspring as revenge for her husband's infidelities.

Thus accounts such as that of the miraculous " Image Not Made by Hands ", and the weeping and moving " Mother of God of the Sign " of Novgorod are accepted as fact: " Church Tradition tells us, for example, of the existence of an Icon of the Savior during His lifetime ( the " Icon-Made-Without-Hands ") and of Icons of the Most-Holy Theotokos immediately after Him.

Thus, many typical household applications already in existence like word processing and email were designed specifically to avoid using floating point operations.

Thus: " existence precedes essence ".

The book takes the term from Friedrich Nietzsche's Thus Spoke Zarathustra, where it is used in the context of the often nauseating quality of existence.

Thus, while the probability might be extremely small that any particular universe would have the requisite conditions for life ( as we understand life ) to emerge and evolve, this does not require intelligent design per the teleological argument as the only explanation for the conditions in the Universe that promote our existence in it.

Thus, not only the existence of the crime depends on there being a previous legal provision declaring it to be a penal offense ( nullum crimen sine praevia lege ), but also, for a specific penalty to be imposed in a certain case, it is also necessary that the penal legislation in force at the time when the crime was committed ranked the penalty to be imposed as one of the possible sanctions to that crime ( nulla poena sine praevia lege ).

Thus, reasoned Oort, a comet could not have formed while in its current orbit, and must have been held in an outer reservoir for almost all of its existence.

Thus the paradox of creating new information, information itself, or creating existence itself and an information system such as a cognitive system into existence from a position of infinitely knowing, becomes a self-refuting argument and defies the listed " omni " attributes.

Thus the Peloponnesian League was not an " alliance " in the strictest sense of the word ( nor was it wholly Peloponnesian for the entirety of its existence ).

Thus by inspiring the game designers to create the games, he insured his own future existence.

Thus ended the legal existence of the Saudi – Iraqi neutral zone.

Thus, they have no legal existence independent from the trustee and his or her ownership of the subject matter of the trust.

Thus several psychologists such as Howe have argued against the existence of innate talent, while in neuroscience there have been debates that the brain functions with Mass action, rather than by a series of interacting mechanisms – for instance by Karl Lashley and others.