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

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.

Let be an inverse of Then:

Let R be a domain and f a Euclidean function on R. Then:

Let a, b, and c be elements of G. Then:

Let t and s ( t > s ) be the sides of the two inscribed squares in a right triangle with hypotenuse c. Then s < sup > 2 </ sup > equals half the harmonic mean of c < sup > 2 </ sup > and t < sup > 2 </ sup >.

* 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.

Let the line of symmetry intersect the parabola at point Q, and denote the focus as point F and its distance from point Q as f. Let the perpendicular to the line of symmetry, through the focus, intersect the parabola at a point T. Then ( 1 ) the distance from F to T is 2f, and ( 2 ) a tangent to the parabola at point T intersects the line of symmetry at a 45 ° angle.

Let U be a functor from D to C, and let X be an object of C. Then the following statements are equivalent:

Let X be a normed topological vector space over F, compatible with the absolute value in F. Then in X *, the topological dual space X of continuous F-valued linear functionals on X, all norm-closed balls are compact in the weak -* topology.

Let S be a subgroup of G, and let N be a normal subgroup of G. Then:

Let R be a Noetherian ring and let I be an ideal of R. Then I may be written as the intersection of finitely many primary ideals with distinct radicals ; that is:

# Let p = ( p < sub > 1 </ sub >, p < sub > 2 </ sub >) be an element of W, that is, a point in the plane such that p < sub > 1 </ sub > = p < sub > 2 </ sub >, and let c be a scalar in R. Then cp = ( cp < sub > 1 </ sub >, cp < sub > 2 </ sub >); since p < sub > 1 </ sub > = p < sub > 2 </ sub >, then cp < sub > 1 </ sub > = cp < sub > 2 </ sub >, so cp is an element of W.

Let K be a closed subset of a compact set T in R < sup > n </ sup > and let C < sub > K </ sub > be an open cover of K. Then

Let us suppose that L is a complete lattice and let f be a monotonic function from L into L. Then, any x ′ such that f ′( x ′) ≤ x ′ is an abstraction of the least fixed-point of f, which exists, according to the Knaster – Tarski theorem.

Let be the mean of the values in associated with class c, and let be the variance of the values in associated with class c. Then, the probability of some value given a class,, can be computed by plugging into the equation for a Normal distribution parameterized by and.

' Then Útgarda-Loki answered, looking about him on the benches, and spake: ' I see no such man here within, who would not hold it a disgrace to wrestle with thee ;' and yet he said: ' Let us see first ; let the old woman my nurse be called hither, Elli, and let Thor wrestle with her if he will.

Let denote the number of kernels which have popped up to time t. Then this is a continuous time, non-homogenous Markov process.

Let p be an odd prime and a an integer coprime to p. Then

Let vectors and let denote the matrix with elements of a and b. Then the area of the parallelogram generated by a and b is equal to.

Let vectors and let Then the area of the parallelogram generated by a and b is equal to.

Theorem: Let R be a Dedekind domain with fraction field K. Let L be a finite degree field extension of K and denote by S the integral closure of R in L. Then S is itself a Dedekind domain.

Let me pass over the trip to Sante Fe with something of the same speed which made Mrs. Roebuck `` wonduh if the wahtahm speed limit '' ( 35 m.p.h. ) `` is still in ee-faket ''.

Let me quote him even more fully, for his analysis is important to my theme.

Let us not confuse the issue by labeling the objective or the method `` psychoanalytic '', for this is a well established term of art for the specific ideas and procedures initiated by Sigmund Freud and his followers for the study and treatment of disordered personalities.

Let us quote once more from R. G. Collingwood: `` History is properly concerned with the actions of human beings Regarded from the outside, an action is an event or series of events occurring in the physical world ; ;

Let us see just how typical Krim is.

Let us assume that Af is identical to the form of an occurrence Af which preceded Af in the text.

Let us also assume the existence of only one class or type of service, all of which is supplied at the same voltage, phase, etc. to residential, commercial, and industrial customers.

Let us therefore consider each of the three types of cost in turn, recognizing that this simplified classification is used only for illustrative purposes ; ;

Let Af where the maximization is over all admissible policies Af.

* Let Q be a set enclosed between two step regions S and T. A step region is formed from a finite union of adjacent rectangles resting on a common base, i. e. S ⊆ Q ⊆ T. If there is a unique number c such that a ( S ) ≤ c ≤ a ( T ) for all such step regions S and T, then a ( Q )

Let us suppose ( as is the practical case ) that the star is sufficiently distant that all light from the star travels in parallel paths to the Earth observer, regardless of where the Earth is in its orbit.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

Let us call this act a ' to be ,' in contradistinction to what is commonly called ' a being.

Let V, W and X be three vector spaces over the same base field F. A bilinear map is a function

Let be a Hilbert space and is a vector in.

Let A be a complex unital Banach algebra in which every non-zero element x is invertible ( a division algebra ).

Let A be a unital commutative Banach algebra over C. Since A is then a commutative ring with unit, every non-invertible element of A belongs to some maximal ideal of A.

Using terms from formal language theory, the precise mathematical definition of this concept is as follows: Let S and T be two finite sets, called the source and target alphabets, respectively.

:: Let no-one deceive you by any means, for that day will not come unless the falling away comes first, and the man of sin is revealed, the son of perdition, who opposes and exalts himself above all that is called God or that is worshiped, so that he sits as God in the temple of God, showing himself that he is God. Thess 2: 3-4