Then two members of the Cartesian product are equivalent precisely if they are equal almost everywhere on the index set.

Then one loads the molecular structures ' Cartesian coordinates ( e. g. from PDB files ).

Then the curve has Cartesian equation

Suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a tubular neighborhood of c. Then A is an annulus and so is homeomorphic to the Cartesian product of

Then the previous expression is equal to the product of two factors:

* Let H be a group, and let G be the direct product H × H. Then the subgroups

Then most pure part of the condensate may be extracted as product.

Then, in the early 550s, two monks succeeded in smuggling eggs of silk worms from Central Asia back to Constantinople, and silk became an indigenous product.

Then to define multiplication, it suffices by the distributive law to describe the product of any two such terms, which is given by the rule

Then the assignment extends uniquely to an algebra homomorphism by sending the monomial in the Clifford algebra to the product of matrices and extending linearly.

Then each element in G is represented in some way by a product

Then, by linearity of the inner product in its first argument, one has

Then there are three basic constructions in universal algebra: homomorphic image, subalgebra, and product.

Then by making use of the scalar product,

Then R (( G )) is the ring of formal power series on G ; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same.

Then the unit prices could be grouped in terms of the product type, say " clothing " and " food ".

Then Edwin Binney, working with his wife, Alice Stead Binney, developed his own famous product line of wax crayons beginning on 10 June 1903, which it sold under the brand name " Crayola.

Then the unrestricted wreath product A Wr < sub > Ω </ sub > H of A by H is the semidirect product K ⋊ H. The subgroup K of A Wr < sub > Ω </ sub > H is called the base of the wreath product.

Then the total number of intersection points of X and Y with coordinates in an algebraically closed field E which contains F, counted with their multiplicities, is equal to the product of the degrees of X and Y.

Then it can be shown ( see next subsection ) that the quaternion product

Then the contravariant coordinates of any vector v can be obtained by the dot product of v with the contravariant basis vectors:

Then that container is refilled from the next oldest container, and that one in succession from the second-oldest, down to the youngest container, which is refilled with new product.

Then there is his resuscitation of the method of " water-culture ," and the application of it to the investigation of the problems of nutrition ; and further, his discovery that the starch-grains to be found in chloroplasts are the first visible product of their assimilatory activity.

Then we obtain the definition of a product.

Then, as the matrix product is associative,

Then, on July 2, there occurred another incident which set tongues to wagging at a furious clip.

Then O'Banion was presented with a platinum watch set with rubies and diamonds.

`` Then maybe it was the shockwave we set up.

Then our choice function can choose the least element of every set under our unusual ordering.

Then, taking in on the first cable as the boat is motored into the wind and letting slack while drifting back, a second anchor is set approximately a half-scope away from the first on a line perpendicular to the wind.

Then those small compound bodies that are least removed from the impetus of the atoms are set in motion by the impact of their invisible blows and in turn cannon against slightly larger bodies.

Then codenamed " Stella ," the machine was also set to utilize cartridges ; after seeing the Channel F, Atari realized they needed to release it before the market was flooded with cartridge-based machines.

Indeed, following, suppose ƒ is a complex function defined in an open set Ω ⊂ C. Then, writing for every z ∈ Ω, one can also regard Ω as an open subset of R < sup > 2 </ sup >, and ƒ as a function of two real variables x and y, which maps Ω ⊂ R < sup > 2 </ sup > to C. We consider the Cauchy – Riemann equations at z = 0 assuming ƒ ( z ) = 0, just for notational simplicity – the proof is identical in general case.

Then Asmund set himself to wait the agreed upon three days:

Let G be a set and let "~" denote an equivalence relation over G. Then we can form a groupoid representing this equivalence relation as follows.

Then the equivalence class of the pair can be identified with the rational number, and this equivalence relation and its equivalence classes can be used to give a formal definition of the set of rational numbers.

Then an " estimator " is a function that maps the sample space to a set of sample estimates.

Then, in the grand final against Melbourne, Essendon set a grand final record score of 22. 18 ( 150 ) to Melbourne 13. 9 ( 87 ), with 7 goal centre half forward Gordon Lane.

Then one " flows " the charge by systematically redistributing the charge from a vertex to its neighboring vertices according to a set of rules, the discharging procedure.

is partially ordered by set inclusion, therefore it contains a maximal totally ordered subset P. Then the set satisfies the desired properties.

Then the image set f ( I ) is also an interval, and either it contains f ( b ), or it contains f ( a ); that is,

* Let the index set I of an inverse system ( X < sub > i </ sub >, f < sub > ij </ sub >) have a greatest element m. Then the natural projection π < sub > m </ sub >: X → X < sub > m </ sub > is an isomorphism.

Then Lothair finally set out with a large Lombard army, but Louis had promised his sons Louis the German and Pepin of Aquitaine greater shares of the inheritance, prompting them to shift loyalties in favour of their father.

Then worship Allah and do not associate anything with Him, perform the Hajj and the ‘ Umrah, be righteous, and all affairs will be set right for you.

Then the null set automatically has measure zero because of countable additivity, because and is finite if and only if the empty set has measure zero.

Then N < sub > x </ sub > is a directed set, where the direction is given by reverse inclusion, so that S ≥ T if and only if S is contained in T. For S in N < sub > x </ sub >, let x < sub > S </ sub > be a point in S. Then ( x < sub > S </ sub >) is a net.