Although flawed, Kempe's original purported proof of the four color theorem provided some of the basic tools later used to prove it.

Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic.

Although the proof is necessary to produce a theorem, it is not usually considered part of the theorem.

Although the virial theorem depends on averaging the total kinetic and potential energies, the presentation here postpones the averaging to the last step.

Although this at first appears to be a stronger statement, it is a direct consequence of the other form of the theorem, through the use of successive polynomial division by linear factors.

Although the issue of existence and uniqueness of solutions of ordinary differential equations has a very satisfactory answer with the Picard – Lindelöf theorem, that is far from the case for partial differential equations.

Although useful in its own right, the version of her theorem just given was a special case of the general version she derived in 1915.

Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.

Although there is no proof that a smooth group order will be found in the Hasse-interval, by using heuristic probabilistic methods, the Canfield-Erdős-Pomerance theorem with suitably optimized parameter choices, and the L-notation, we can expect to try L < nowiki >/ 2, √ 2 < nowiki ></ nowiki > curves before getting a smooth group order.

Although he didn't publish much in the field, Dirichlet lectured on probability theory and least squares, introducing some original methods and results, in particular for limit theorems and an improvement of Laplace's method of approximation related to the central limit theorem.

Although that is strictly speaking a question about a real vector bundle ( the " hairs " on a ball are actually copies of the real line ), there are generalizations in which the hairs are complex ( see the example of the complex hairy ball theorem below ), or for 1-dimensional projective spaces over many other fields.

Although apparently a rather special case of the above theorem, the general result can be reduced to this simpler case.

Although in general these two can be different there is a theorem that, under very mild conditions, gives the equality of the two.

Although Desargues ' theorem chooses different roles for these ten lines and points, the Desargues configuration itself is more symmetric: any of the ten points may be chosen to be the center of perspectivity, and that choice determines which six points will be the vertices of triangles and which line will be the axis of perspectivity.

" Although this conjecture is true, most of its known proofs depend on the theory of separable and purely inseparable extensions ; for instance, in the case corresponding to the extension being separable, one known proof involves the use of the primitive element theorem in the context of Galois extensions.

Although the deduction theorem could be taken as primitive rule of inference in such systems, this approach is not generally followed ; instead, the deduction theorem is obtained as an admissible rule using the other logical axioms and modus ponens.

Although Pappus's Theorem usually refers to Pappus's hexagon theorem, it may also refer to Pappus's centroid theorem.

Although the equipartition theorem makes very accurate predictions in certain conditions, it becomes inaccurate when quantum effects are significant, such as at low temperatures.

Although in general such spaces are not necessarily normable, the existence of a convex local base for the zero vector is strong enough for the Hahn – Banach theorem to hold, yielding a sufficiently rich theory of continuous linear functionals.

Although the full description of the splitting of every prime p in a general Galois extension is a major unsolved problem, the Chebotarev density theorem says that the frequency of the occurrence of a given pattern, for all primes p less than a large integer N, tends to a certain limit as N goes to infinity.

Although the possession of a common language shows that its speakers have lived together and have a common history, peoples also change languages.

Although the perception by the public may be that the " central bank " controls some or all interest rates and currency rates, economic theory ( and substantial empirical evidence ) shows that it is impossible to do both at once in an open economy.

Although the current term " register file " shows the early concept of files, it has largely disappeared.

Although there is a downward population curve, explained by a larger death than birth rate, as well as a larger number of emigrants than immigrants, the line graph of the natural population increase shows the rate of population decrease was slowly diminishing.

Although the statement of the theory in terms of graphs may imply perturbation theory, use of graphical methods in the many-body problem shows that this formalism is flexible enough to deal with phenomena of nonperturbative characters ...

Although ABC does not air any national game shows, its syndication wing Disney-ABC Domestic Television distributes Who Wants to Be a Millionaire ?, and many of their local affiliates air it in syndication.

" Although such similarities have given rise to arguments for the existence of a standardized system of urban layout and planning, the similarities are largely due to the presence of a semi-orthogonal type of civic layout, and a comparison of the layouts of Mohenjo-daro and Harappa shows that they are in fact, arranged in a quite dissimilar fashion.

Although this vaccine is likely not safe for human use, it still shows that attacking Aβ has efficacy in mice against IBM.

Although it has previously been thought that morphine was contraindicated in acute pancreatitis, a review of the literature shows no evidence for this.

Although Hammerstein's father managed the Victoria Theatre for his father and was a producer of vaudeville shows ( he is generally credited with inventing the " pie-in-the-face " routine ), he was opposed to his son's desire to participate in the arts.

Although the first arrivals were hunters and gatherers, early evidence shows that people managed the forest environment to provide food.

Although often written as Paint Shop Pro, Corel's website shows the name for the product as PaintShop Pro.

" GameSpot stated " Although Red Faction rarely outstrips the games it draws inspiration from, the fact that there are times when it shows them up at all is pretty impressive.

Although the site shows signs of continuous occupation from before 3000 BC, its power grew and reached its apogee in the second half of the following millennium.

Although he did not perform as much in the early shows Milligan eventually became a lead performer in almost all of the Goon Show episodes, portraying a wide range of characters including Eccles, Minnie Bannister, Jim Spriggs and the nefarious Count Moriarty.

Although the shows involve a traditional talent search, the shows follow the reality-competition conventions of removing one or more contestants per episode and allowing the public to vote on who is removed ; the Idol series also require the contestants to live together during the run of the show ( though their daily life is never shown onscreen ).

Although other bands, including The Beatles, had made short promotional films or videos of songs prior to this, generally, those were specifically made to be aired on specific television shows.

Although the story follows the same general storyline, there are some changes: the film shows Dr. No to be an operative of SPECTRE and his island fortress is nuclear powered ; No is killed not by the mountain of guano, but by drowning in reactor coolant.

Although this diagram shows discrete models and events, the continuous case may be visualized similarly using probability densities.

Although earlier BBC radio shows such as Much Binding in the Marsh, Take It From Here and Beyond Our Ken had conditioned listeners to a mix of music, sketches and jokes within a 30 minute show, and Round the Horne was also doing this, ISIRTA ( as it was known to its friends ) accelerated the transitions, and it certainly seemed more improvised.

Although there are twelve Clue shows broadcast per year these are the result of just six recording sessions each of two programmes recorded back-to-back.

Although the three musicians chose not to speak publicly about the shows, Clapton would later state that he had become more " generous " in regard to his past, and that the physical health of Bruce and Baker was a major factor: Bruce had recently undergone a liver transplant for liver cancer, and had almost lost his life, while Baker had severe arthritis.

Although she enjoys the company of the boys, she often shows disdain, naivity or exasperation toward some of their more boyish behaviour, though more often than not she is in the thick of such situations.

Although the science fiction film Destination Moon is generally described as being based on Heinlein's novel Rocket Ship Galileo, the story in fact bears a much closer resemblance to The Man Who Sold the Moon, whose copyright date shows that it was written in 1949, although it wasn't published until 1951, the year after Destination Moon came out.