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

Every non-empty totally ordered set is directed.

Every subfield of an ordered field is also an ordered field in the inherited order.

Every ordered field contains an ordered subfield that is isomorphic to the rational numbers.

Every ordered field is a formally real field.

Every subfield of an ordered field is also an ordered field ( inheriting the induced ordering ).

Every ordered field can be embedded into the surreal numbers.

Every ordered field is a formally real field, i. e., 0 cannot be written as a sum of nonzero squares.

* Every totally ordered set with the order topology is Tychonoff.

* Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element with respect to set inclusion.

* Every totally ordered set that is a bounded lattice is also a Heyting algebra, where is equal to when, and 1 otherwise.

* Every totally ordered set is a distributive lattice with max as join and min as meet.

Every relatively atomic partially ordered set with a least element is atomic.

Every time she entered, song typical of the Brazilian northeast would play, stopping only when ordered by Cirene herself.

Every partially ordered set is cofinal in itself.

** Every vector space has a basis.

** Every field extension has a transcendence basis.

Every module over a division ring has a basis ; linear maps between finite-dimensional modules over a division ring can be described by matrices, and the Gaussian elimination algorithm remains applicable.

Every vector space has a basis, and all bases of a vector space have the same number of elements, called the dimension of the vector space.

Every finite-dimensional vector space is isomorphic to its dual space, but this isomorphism relies on an arbitrary choice of isomorphism ( for example, via choosing a basis and then taking the isomorphism sending this basis to the corresponding dual basis ).

Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.

Every symmetric matrix is thus, up to choice of an orthonormal basis, a diagonal matrix.

Every rotation maps an orthonormal basis of R < sup > 3 </ sup > to another orthonormal basis.

Every finite-dimensional inner product space has an orthonormal basis, which may be obtained from an arbitrary basis using the Gram – Schmidt process.

Every free abelian group has a rank defined as the cardinality of a basis.

Every building has a canonical length metric inherited from the geometric realisation obtained by identifying the vertices with an orthonormal basis of a Hilbert space.

For example, second-order arithmetic can express the principle " Every countable vector space has a basis " but it cannot express the principle " Every vector space has a basis ".

Many principles that imply the axiom of choice in their general form ( such as " Every vector space has a basis ") become provable in weak subsystems of second-order arithmetic when they are restricted.

Every vector a in three dimensions is a linear combination of the standard basis vectors i, j, and k.

Every lattice in can be generated from a basis for the vector space by forming all linear combinations with integer coefficients.

Every software development methodology approach acts as a basis for applying specific frameworks to develop and maintain software.

Every faculty, school and department has its own administrative body, the members of which are democratically elected on the basis of collective processes.

Every day throughout the year commemorates some saint or some event in the lives of Christ or the Theotokos.

Every Wake Forest undergraduate student who lives on campus is required to sign up for some form of a meal plan in coordination with Aramark.

Every new Cercle Commander might well bring with him vast projects for development and the restructuring of the people's lives he governed.

Every year tens of millions of people around the world, including around four million cancer patients and 0. 8 million HIV / AIDS patients at the end of their lives suffer from such pain without treatment.

Every year on the battle's anniversary, a US ship in the area cruises into the waters and drops a wreath to commemorate those who lost their lives.

Every ten weeks, the producers would select two Lofters whose hosting skills were sub-par or who didn't share enough of their lives, and nominate them for eviction.

Every 10, 000 points is worth an extra life ; Baldric can have up to 9 lives in hand at once.

Every year the ICP holds an international conference to promote 45 categories within psychological science and raise awareness about the applications they can bring to our lives.

Every version of Dax before him lives in Daxia, including Darius Duck, Daxor, Daxian, Doomsdax, Mad Nazi Scientist Dax, and " Grim " Serial-Killer Transvestite 80's Dax.

Every time the main character was injured he would lose one of his limited lives.

" Parker J. Palmer wrote, " Every thoughtful adult I know is concerned that many American youth seem to lack a sense of purpose in their lives.

In New York, Ted Forstmann explained his goal for the scholarships, saying, " Every child, regardless of their parents ' income, should have access to a quality education – an education that will not only prepare them for successful private lives, but help them to build cohesive communities and a strong democracy.

Every person aboard was badly shaken or injured and five lives were lost.

Every day thousands of flight crews and their passengers risked their lives making this passage in C-46 and C-47 aircraft.

Faculty member Barbara Breitman says, “ Every spiritual tradition has within it the qualities of soul that people need to cultivate in their lives so that they can live according to a higher sense of purpose: generosity, patience, gratitude, truthfulness.

“ 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 .”

Her 1997 memoir, Every Secret Thing: My Family, My Country, is an account of her childhood in South Africa and her relationship with her parents Joe Slovo and Ruth First — both famous South Africans and major figures in the anti-apartheid struggle who lived perilous lives of exile, armed resistance, and occasional imprisonment, which culminated in her mother's murder by South African forces in 1982.

Every day, they saved one of the two men and put the brain of a sheep instead of his into the food, but they could not save the lives of both men.