: Every set has a choice function.

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Well-ordering theorem: Every set can be well-ordered.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

The restricted principle " Every partially ordered set has a maximal totally ordered subset " is also equivalent to AC over ZF.

** Antichain principle: Every partially ordered set has a maximal antichain.

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

: Every non-empty set A contains an element B which is disjoint from A.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

Every subset A of the vector space is contained within a smallest convex set ( called the convex hull of A ), namely the intersection of all convex sets containing A.

* Every cofinal subset of a partially ordered set must contain all maximal elements of that set.

Every corporation, whether financial or union, as well as every division of the administration, were set up as branches of the party, the CEOs, Union leaders, and division directors being sworn-in as section presidents of the party.

Every DNS zone must be assigned a set of authoritative name servers that are installed in NS records in the parent zone, and should be installed ( to be authoritative records ) as self-referential NS records on the authoritative name servers.

Group actions / representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a group action of G on a particular set, i. e. a G-set.

# " Personality " Argument: this argument is based on a quote from Hegel: " Every man has the right to turn his will upon a thing or make the thing an object of his will, that is to say, to set aside the mere thing and recreate it as his own ".

Every atom across this plane has an individual set of emission cones .</ p > < p > Drawing the billions of overlapping cones is impossible, so this is a simplified diagram showing the extents of all the emission cones combined.

* Every singleton set

Every processor or processor family has its own machine code instruction set.

Every set is a class, no matter which foundation is chosen.

Every non-empty totally ordered set is directed.

* Every preorder can be given a topology, the Alexandrov topology ; and indeed, every preorder on a set is in one-to-one correspondence with an Alexandrov topology on that set.

Every binary relation R on a set S can be extended to a preorder on S by taking the transitive closure and reflexive closure, R < sup >+=</ sup >.

* Every quadratic Bézier curve is also a cubic Bézier curve, and more generally, every degree n Bézier curve is also a degree m curve for any m > n. In detail, a degree n curve with control points P < sub > 0 </ sub >, …, P < sub > n </ sub > is equivalent ( including the parametrization ) to the degree n + 1 curve with control points P '< sub > 0 </ sub >, …, P '< sub > n + 1 </ sub >, where.

Cantor points out that his constructions prove more — namely, they provide a new proof of Liouville's theorem: Every interval contains infinitely many transcendental numbers.

Every space filling curve hits some points multiple times, and does not have a continuous inverse.

* Every finite topological space gives rise to a preorder on its points, in which x ≤ y if and only if x belongs to every neighborhood of y, and every finite preorder can be formed as the specialization preorder of a topological space in this way.

Every pair of points belongs to a unique line.

Every finite or bounded interval of the real numbers that contains an infinite number of points must have at least one point of accumulation.

* Every line contains at least two points.

Every point of PG ( 3, 2 ) is contained in 7 lines and every line contains three points.

He observed that his patients who used his mantra-like conscious suggestion, " Every day, in every way, I'm getting better and better ", ( French: Tous les jours à tous points de vue je vais de mieux en mieux ), replacing their " thought of illness " with a new " thought of cure ," could augment their medication plan.

Every time one levels up a class division, one is given one extra attribute point until it reaches division level 50, such that a non reincarnated char has a total of 236 points, 216 points the character can invest as there is always 5 Tactics, Clout, Education and Mechanical Aptitude to start with.

Every player is generally encouraged to invest points in Tactics, as Shattered Galaxy is a game that values overall quality over individual qualities: the individual units owned by a 120-Tactics player may have fewer hit points than a player with 120 Clout, or do less damage per second than a player with 120 Education, but his overall squad is undoubtedly stronger on both counts.

Every 12 points, the player is awarded a new promotion.

Every Jew who entered Palestine between 1918 and 1939 did so at the points of British bayonets.

Note: Every cyclic permutation of definition type 3 may be seen as an union of a cyclic permutation of definition type 2 and some fixed points.

Every cyclic permutation of definition type 2 may be seen ″ as a cyclic permutation of definition type 3 with zero fixed points.

Every fifth level, one additional character point is awarded for a total of 64 character points.

Every domain D can be embedded in an ω-complete domain completion ( called the ω-completion or simply the completion of D ) that is, in a precise sense, the smallest ω-complete domain containing D. The isolated elements of completion are precisely the isolated elements of D, but in general completion contains limit points that are not found in D. completion is uniquely determined up to isomorphism.

Every team plays every other team in its group once, receiving 2 points for a win, 1 point for a tie, or 0 points for a loss.

Every two extra points is with half a second head start.

Every year, the houses competed for the House Shield ; points were awarded for all forms of competitions, from sword dancing to vaulting.

The team adopted a motto – " Every Man a Tiger " – and went on to score 174 points while allowing its opponents just 29 points.