The necessity of this condition follows easily from the properties of an inner product.

Given a groupoid G, the vertex groups or isotropy groups or object groups in G are the subsets of the form G ( x, x ), where x is any object of G. It follows easily from the axioms above that these are indeed groups, as every pair of elements is composable and inverses are in the same vertex group.

Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily.

This is easily proven with elementary mathematics using a counting argument, as follows:

The second part can be easily deduced and follows a similar result of Apollonius of Perga for the circle.

The well ordering theorem follows easily from Zorn's Lemma.

However, one could easily modify the classic statement as follows: " An omnipotent being creates a universe which follows the laws of Aristotelian physics.

Auxiliary verbs in English are easily identified, since only auxiliary verbs allow subject-auxiliary inversion and only finite auxiliary verbs can take not as a postdependent ( a dependent that follows its head ).

The general solutions of the equation can easily be seen as follows.

According to Mueller's analysis, the duet that follows – easily the most famous of all the Astaire-Rogers partnered dances – reflects the complexity of the emotional situation in which the pair find themselves.

Using the above rule, we can complete the proof of Fermat's little theorem quite easily, as follows.

This follows easily by induction ; we have just proven this to be true for n

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

It follows easily from the definition that the determinant of any symplectic matrix is ± 1.

The formula for escape velocity is easily derived as follows.

He is a Republican and a devout Evangelical Christian who strictly follows the Bible literally and is easily shocked when challenged on any point of dogma.

The degree bound follows from this and the fact ( easily proved by induction ) that for all integers, where.

It follows easily from the chain rule that.

In pre-Roman times, the Chiltern ridge provided a relatively safe and easily negotiable route across southern Iron Age England, thus the Icknield Way ( one of England's ancient prehistoric trackways ) follows the line of the hills.

This follows easily by checking that the expression commutes with exterior derivative, is a derivation ( being an anticommutator of graded derivations ) and does the right thing on functions.

In fact, if G is a matrix group, then the result follows easily from the Stone-Weierstrass theorem.

The first equation is solved by using an integrating factor and the second follows easily ; the solutions are exactly the same as those obtained via Laplace transforms.

Courage the Cowardly Dog follows a dog named Courage, an easily frightened canine who lives in a farmhouse with Muriel and Eustace Bagge near the fictional town of Nowhere, Kansas.

These publications may be secured as follows: The Annual Catalog from the Director of Admissions and other issues from the Publications Office.

In type 1 the pulmonary vein closely follows the course of the bronchus and the pulmonary artery from the periphery to the hilum.

The specificity of staining with Af was established as follows: Af specifically stained tumor sections but not sections of healthy sweet clover stems or of crown gall tumor tissue from sweet clover.

It was assumed that the shift in autonomic hypothalamic balance occurring spontaneously in neuropsychiatric patients from the application of certain therapeutic procedures follows the pattern known from the sleep-wakefulness cycle.

It then follows that Af is large enough, but this is not obvious from the above expression.

The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and Af are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point.

Since the complex of singular lines is of order K and since there is no complex of invariant lines, it follows from the formula Af that the order of the involution is Af.

A final class of exceptional lines is identifiable from the following considerations: Since no two generators of Af can intersect, it follows that their image curves can have no free intersections.

Consider what follows from the positivist view.

Our staff man cabled from Turin as follows --

The configuration of these electrons follows from the principles of quantum mechanics.

For finite sets X, the axiom of choice follows from the other axioms of set theory.

And the axiom of global choice follows from the axiom of limitation of size.

This belief follows the theology of St. Paul: For it is by grace you have been saved, through faith — and this not from yourselves, it is the gift of God, not by works, so that no one can boast.

Although Dirac did not himself use the term antimatter, its use follows on naturally enough from antielectrons, antiprotons, etc.

For Hume, every effect only follows its cause arbitrarily — they are entirely distinct from one another.

The book of Acts follows just the opposite motion, taking the scene from Jerusalem ( chs.

The distance formula on the plane follows from the Pythagorean theorem.

Today, Methodism follows this ancient Alexandrian practice as bishops are elected from and by the order of the presbyterate: the Discipline of the Methodist Church, in ¶ 303, affirms that " ordination to this ministry is a gift from God to the Church.

This conservation law mathematically follows from continuous directional symmetry of space ( no direction in space is any different from any other direction ).

The debate might be framed starkly as follows: on the one hand, we can hypothesise a poem put together from various tales concerning the hero ( the Grendel episode, the Grendel's mother story, and the firedrake narrative ).