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 imperfect field is necessarily transcendental over its prime subfield ( the minimal subfield ), because the latter is perfect.

Every entity ( unless it is a weak entity ) must have a minimal set of uniquely identifying attributes, which is called the entity's primary key.

Every non-singular rational surface can be obtained by repeatedly blowing up a minimal rational surface.

: 2. Every object of can be decomposed into a sum of minimal objects.

Every projective variety is complete, but not vice versa.

* Every holomorphic vector bundle on a projective variety is induced by a unique algebraic vector bundle.

* Every holomorphic line bundle on a projective variety is a line bundle of a divisor.

* Every module over a field or skew field is projective ( even free ).

* Every projective module is flat.

Every module possesses a projective resolution.

* Every ( biregular ) algebraic automorphism of a projective space is projective linear.

Every rational variety, including the projective spaces, is rationally connected, but the converse is false.

Every compact, connected 2-manifold ( or surface ) is homeomorphic to the sphere, a connected sum of tori, or a connected sum of projective planes.

* Every projective set of reals -- and therefore every analytic set and every Borel set of reals -- is an element of L ( R ).

Every king from Penda until Ceolwulf, who was deposed in 823, was said to be a descendant of Pybba, either through Penda, Eowa, or Coenwalh ( perhaps excluding Beornrad, who ruled briefly and whose background is unknown ).

Every tribe within the confederation was ruled by a chief or cacique.

Every community retains its own traditions, and thus halakha should be ruled for each community according to its own tradition.

Every conic surface is ruled and developable.

Every month a consul, twelve senators and two censors were chosen from the pupils, and over all Trotzendorff ruled as dictator perpetuus.

President of the United States George W. Bush addressed the United Nations, stating, " Every civilized nation has a responsibility to stand up for people suffering under a brutal military regime like the one that has ruled Burma for so long.

Every form is the delimitation of a surface by another one ; it possesses an inner content, the effect it produces on one who looks at it attentively.

Every aspect of the store is mapped out and attention is paid to colour, wording and even surface texture.

Every gland is formed by an ingrowth from an epithelial surface.

Every antiseptic, however good, is more or less toxic and irritating to a wounded surface ; as a result, in surgery, the antiseptic method has been replaced by aseptic method, which is preventative in nature and relies on keeping free from the invasion of bacteria rather than destroying them when present.

Every Riemann surface is a two-dimensional real analytic manifold ( i. e., a surface ), but it contains more structure ( specifically a complex structure ) which is needed for the unambiguous definition of holomorphic functions.

Every Riemann surface is the quotient of a free, proper and holomorphic action of a discrete group on its universal covering and this universal covering is holomorphically isomorphic ( one also says: " conformally equivalent ") to one of the following:

# Every Riemannian metric on a Riemann surface is Kähler, since the condition for ω to be closed is trivial in 2 ( real ) dimensions.

# Every K3 surface is Kähler ( by a theorem of Y .- T. Siu ).

Every smooth surface S has a unique affine plane tangent to it at each point.

Every now and then, the surface water sloshes back across the ocean, bringing warm water temperatures along the eastern coasts of the pacific.

Every such black body emits from its surface with a spectral radiance that Kirchhoff labeled ( for specific intensity, the traditional name for spectral radiance ).

Every 17 years, Brood X cicadas tunnel en masse to the surface of the ground, lay eggs, and then die off in several weeks.

Every finite group is a subgroup of the mapping class group of a closed, orientable surface, moreover one can realize any finite group as the group of isometries of some compact Riemann surface.

Every vehicle function that changes direction or speed relies on friction between the tires and the road surface.

Every single object or point is dwarfed ; the valley of the Hudson is only a wrinkle in the earth's surface.

Every surface of the vaulted ceiling is occupied with preserved fishes, stuffed mammals and curious shells, with a stuffed crocodile suspended in the centre.

Every integer 2g ' 2g matrix with < sup >*</ sup > arises as the Seifert matrix of a knot with genus g Seifert surface.

Every morning before sunrise, the floor of the owners house, or where ever it may be, is cleaned with water and the muddy floor is swept well for an even surface.

Every hyperbolic Riemann surface has a non-trivial fundamental group.

According to Dale E. Dawkins, AMC's vice-president, " Every square inch of inner surface on exterior body panels is galvanized on our Spirit, Concord, and Eagle models.

Every particular finite line, surface, or solid which may possibly be the object of our thought is an idea existing only in the mind, and consequently each part of it must be perceived.