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Every element has a successor ( there is no largest element ).

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## Some Related Sentences

Every and element

__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.

** 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__.
** Tukey's lemma:

__Every__non-empty collection of finite character**has****a**maximal__element__with respect to inclusion.__Every__lattice

__element__of the structure

**is**in its proper place, whether it

**is**

**a**single atom or

**a**molecular grouping.

__Every__Boolean algebra

**(**A, ∧, ∨) gives rise to

**a**ring

**(**A, +, ·) by defining

**a**+ b :=

**(**

**a**∧ ¬ b ) ∨

**(**b ∧ ¬

**a**) =

**(**

**a**∨ b ) ∧ ¬(

**a**∧ b )

**(**this operation

**is**called symmetric difference in the case of sets and XOR in the case of logic ) and

**a**· b :=

**a**∧ b. The zero

__element__of this ring coincides with the 0 of the Boolean algebra ; the multiplicative identity

__element__of the ring

**is**the 1 of the Boolean algebra.

__Every__repetition of insertion sort removes an

__element__from the input data, inserting it into the correct position in the already-sorted list, until

**no**input elements remain.

__Every__singleton

**is**

**a**terminal object, with the functions mapping all elements of the source sets to the single target

__element__as morphisms.

__Every__

__element__s, except

**a**possible greatest

__element__,

**has**

**a**unique

**successor**

**(**next

__element__), namely the least

__element__of the subset of all elements greater than s.

__Every__subset which

**has**an upper bound

**has**

**a**least upper bound.

*

__Every__non-empty set of left ideals of R, partially ordered by inclusion,**has****a**maximal__element__with respect to set inclusion.__Every__time

**a**pixel on

**a**triangle

**is**rendered, the corresponding texel

**(**or texture

__element__) in the texture must be found.

__Every__non-inner automorphism yields

**a**non-trivial

__element__of Out

**(**G ), but different non-inner automorphisms may yield the same

__element__of Out

**(**G

**).**

Every and has

__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__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.

The restricted principle "

__Every__partially ordered set__has__**a**maximal totally ordered subset "**is**also equivalent to AC over ZF.
*

__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__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.

Every and successor

__Every__strong limit cardinal

**is**also

**a**weak limit cardinal, because κ < sup >+</ sup > ≤ 2 < sup > κ </ sup > for every cardinal κ, where κ < sup >+</ sup > denotes the

__successor__cardinal of κ.

__Every__branch of the Central Statistical Administration and its

__successor__the State Statistics Committee

**(**Goskomstat ) also had

**a**First Department to control access, distribution, and publication of official economic, population, and social statistics.

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