** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

# Every open cover of A has a finite subcover.

* Countably compact: Every countable open cover has a finite subcover.

# Every finite and contingent being has a cause.

Every finite simple group is isomorphic to one of the following groups:

Hilbert's example: " the assertion that either there are only finitely many prime numbers or there are infinitely many " ( quoted in Davis 2000: 97 ); and Brouwer's: " Every mathematical species is either finite or infinite.

* 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 finite tree structure has a member that has no superior.

Every rational number / has two closely related expressions as a finite continued fraction, whose coefficients can be determined by applying the Euclidean algorithm to.

* Every finite tree with n vertices, with, has at least two terminal vertices ( leaves ).

Every finite group of exponent n with m generators is a homomorphic image of B < sub > 0 </ sub >( m, n ).

Every known Sierpinski number k has a small covering set, a finite set of primes with at least one dividing k · 2 < sup > n </ sup >+ 1 for each n > 0.

Every finite-dimensional Hausdorff topological vector space is reflexive, because J is bijective by linear algebra, and because there is a unique Hausdorff vector space topology on a finite dimensional vector space.

: Every oriented prime closed 3-manifold can be cut along tori, so that the interior of each of the resulting manifolds has a geometric structure with finite volume.

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 field of either type can be realized as the field of fractions of a Dedekind domain in which every non-zero ideal is of finite index.

Every process involving charged particles emits infinitely many coherent photons of infinite wavelength, and the amplitude for emitting any finite number of photons is zero.

Every finite group has a composition series, but not every infinite group has one.

* Every finite or cofinite subset of the natural numbers is computable.

* Every subset of may be covered by a finite set of positive orthants, whose apexes all belong to

* Every finite subextension of F / k is separable.

Every finite ordinal ( natural number ) is initial, but most infinite ordinals are not initial.

* Every finite-dimensional central simple algebra over a finite field must be a matrix ring over that field.

* Every commutative semisimple ring must be a finite direct product of fields.

Every movement, reflex, facial expression, and reaction is measured by this system, monitored by the Ministry of Love.

Every statement in F #, including statements and loops, is a composable expression with a definite return type as well.

Every attendee is expected to participate in some way for the event, with a performance, art piece, other form of creative expression, or volunteering to help.

Every butoh performance itself is an ultimate expression ; there are not and cannot be second or third places.

Every deputy enjoyed the freedom of expression at the Assembly and has immunity for the words uttered and for other acts he performed in his capacity as deputy.

Every lambda expression has a fixed point, and a fixed-point combinator is a " function " which takes as input a lambda expression and produces as output a fixed point of that expression.

* Every member of the organization performs the acts we assume to constitute an expression of preference among the scheduled alternatives, e. g., voting.

Professor Herman Bondi, who gave the first J. L. Synge Public Lecture in 1992, had this to say: " Every one of the other 88 % has been deeply influenced by his geometric vision and the clarity of his expression ".

Every line in a ScriptX program was an expression that returned a value, and a ScriptX program comprised a list of expressions.

For other theorists, there is no such thing as psychological dysfunction or being mentally ill. Every way of being is merely an expression of how one chooses to live one's life.

Following a series of screenings which he organised at the National Film Theatre of independently produced short films including his own Every Day Except Christmas ( about the Covent Garden fruit and vegetable market ), Karel Reisz's Momma Don't Allow and others, he developed a philosophy of cinema which found expression in what became known as the Free Cinema Movement in Britain by the late 1950s.

“ Every Brazilian, even the light skinned fair haired one carries about him on his soul, when not on soul and body alike, the shadow or at least the birthmark of the aborigine or the negro, in our affections, our excessive mimicry, our Catholicism which so delights the senses, our music, our gait, our speech, our cradle songs, in everything that is a sincere expression of our lives, we almost all of us bear the mark of that influence .”

In a review in The New York Times, rock critic Robert Palmer wrote of Nile ; " Every once in awhile the times seems to produce an artist who is at once an iconoclast and near-perfect expression of contemporary currents.

Every soldier in the army has, somewhere, relatives who are close to starvation.

Every woman has had the experience of saying no when she meant yes, and saying yes when she meant no.

Every detail in his interpretation has been beautifully thought out, and of these I would especially cite the delicious laendler touch the pianist brings to the fifth variation ( an obvious indication that he is playing with Viennese musicians ), and the gossamer shading throughout.

Every calculation has been made independently by two workers and checked by one of the editors.

Every retiring person has a different situation facing him.

Every family of Riviera Presbyterian Church has been asked to read the Bible and pray together daily during National Christian Family Week and to undertake one project in which all members of the family participate.

Every community, if it is alive has a spirit, and that spirit is the center of its unity and identity.

`` Every woman in the block has tried that ''.

: 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.

** Every surjective function has a right inverse.

** 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 vector space has a basis.

* Every small category has a skeleton.

* 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 field has an algebraic closure.

** Every field extension has a transcendence basis.

** Every Tychonoff space has a Stone – Čech compactification.

* Every rectangle R is in M. If the rectangle has length h and breadth k then a ( R ) =

Every unit of length has a corresponding unit of area, namely the area of a square with the given side length.

Every field has an algebraic extension which is algebraically closed ( called its algebraic closure ), but proving this in general requires some form of the axiom of choice.

Every ATM cell has an 8-or 12-bit Virtual Path Identifier ( VPI ) and 16-bit Virtual Channel Identifier ( VCI ) pair defined in its header.