** 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-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 regular ordinal is the initial ordinal of a cardinal.

Every well-ordered set is uniquely order isomorphic to a unique ordinal number, called the order type of the well-ordered set.

Every ordinal other than 0 is either a successor ordinal or a limit ordinal.

Every natural-born citizen of a foreign state who is also an American citizen and every natural-born American citizen who is a citizen of a foreign land owes a double allegiance, one to the United States, and one to his homeland ( in the event of an immigrant becoming a citizen of the US ), or to his adopted land ( in the event of an emigrant natural born citizen of the US becoming a citizen of another nation ).

* Every group G acts on G, i. e. in two natural but essentially different ways:, or.

Every man-made product is composed of natural resources ( at its fundamental level ).

* natural language — Every good boy deserves fruit ; and

Every smooth ( or differentiable ) map φ: M → N between smooth ( or differentiable ) manifolds induces natural linear maps between the corresponding tangent spaces:

Every twin prime pair except ( 3, 5 ) is of the form ( 6n − 1, 6n + 1 ) for some natural number n, and with the exception of < var > n </ var > = 1, < var > n </ var > must end in 0, 2, 3, 5, 7, or 8.

Every natural hollow is full of water, around the margin of which, long grasses, reeds and other aquatic plants grow in the greatest profusion, often making it difficult to say where the land ends and the water begins.

Every topological ring is a topological group ( with respect to addition ) and hence a uniform space in a natural manner.

Every smooth manifold defined in this way has a natural diffeology, for which the plots correspond to the smooth maps from open subsets of R < sup > n </ sup > to the manifold.

* Every natural number a has a successor, denoted by Sa.

Every step of the process is natural and healthy, it is only when a person gets stuck in one step for a long period of time then the grieving can become unhealthy, destructive and even dangerous.

* Every number of the form 2 < sup > m </ sup > p for a natural number m and a prime number p such that p < 2 < sup > m + 1 </ sup > is also semiperfect.

Every minimal pair of natural numbers has and, for otherwise the pair would belong to and be at least as small in both coordinates, contradicting the assumption that is minimal.

Every year Plum Creek replants approximately 85 million seedlings and plans for the natural regeneration of millions of trees.

Every infinitely divisible probability distribution corresponds in a natural way to a Lévy process, i. e., a stochastic process

Every classical theorem that applies to the natural numbers applies to the non-standard natural numbers.

Every C *- algebra has an approximate identity of positive elements of norm ≤ 1 ; indeed, the net of all positive elements of norm ≤ 1 ; in A with its natural order always suffices.

* Every individual will have a responsibility to contribute back to the greater natural and social community from his / her Brown School experience.

These rights are guaranteed to " Every individual ," that is, every natural person.

Every aspect of the ecosystem from the different species of plants to the diversity of animal species contributes to natural services like water purification, provision of food, fuel, fiber, and biochemicals, nutrient cycling, pollination and seed dispersal, and climate regulation, just to name a few.

Every August it is held the " Strawberry Festival " where the producers exhibit their natural production or their by-products elaborated from the fruit.

Every year two or more junior Bachelors of Arts students who had made the greatest progress in mathematics and natural philosophy were to be awarded a prize from the fund.