stronger A theorem is said to be strong if it deduces restrictive results from general hypotheses. One celebrated example is Donaldson's theorem, which puts tight restraints on what would otherwise appear to be a large class of manifolds. This (informal) usage reflects the opinion of the mathematical community: not only should such a theorem be strong in the descriptive sense (below) but it should also be definitive in its area. A theorem, result, or condition is further called stronger than another one if a proof of the second can be easily obtained from the first but not conversely. An example is the sequence of theorems: Fermat's little theorem, Euler's theorem, Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (see sharp above) is a stronger result than a non-sharp one. Finally, the adjective strong or the adverb strongly may be added to a mathematical notion to indicate a related stronger notion; for example, a strong antichain is an antichain satisfying certain additional conditions, and likewise a strongly regular graph is a regular graph meeting stronger conditions. When used in this way, the stronger notion (such as "strong antichain") is a technical term with a precisely defined meaning; the nature of the extra conditions cannot be derived from the definition of the weaker notion (such as "antichain"). sufficiently large, suitably small, sufficiently close In the context of limits, these terms refer to some (unspecified, even unknown) point at which a phenomenon prevails as the limit is approached. A statement such as that predicate P holds for sufficiently large values, can be expressed in more formal notation by ∃x : ∀y ≥ x : P(y). See also eventually. upstairs, downstairs A descriptive term referring to notation in which two objects are written one above the other; the upper one is upstairs and the lower, downstairs. For example, in a fiber bundle, the total space is often said to be upstairs, with the base space downstairs. In a fraction, the numerator is occasionally referred to as upstairs and the denominator downstairs, as in "bringing a term upstairs". up to, modulo, mod out by An extension to mathematical discourse of the notions of modular arithmetic. A statement is true up to a condition if the establishment of that condition is the only impediment to the truth of the statement. Also used when working with members of