If A admits a totally ordered cofinal subset, then we can find a subset B which is well-ordered and cofinal in A.

If every object X < sub > i </ sub > of C admits a initial morphism to U, then the assignment and defines a functor V from C to D. The maps φ < sub > i </ sub > then define a natural transformation from 1 < sub > C </ sub > ( the identity functor on C ) to UV.

If a set is well-ordered ( or even if it merely admits a wellfounded relation ), the proof technique of transfinite induction can be used to prove that a given statement is true for all elements of the set.

If the accused admits to having a motive consistent with the elements of foresight and desire, this will add to the level of probability that the actual outcome was intended ( it makes the prosecution case more credible ).

If the patient admits to hair pulling, diagnosis is not difficult ; if patients deny hair pulling, a differential diagnosis must be pursued.

If a given manifold admits a geometric structure, then it admits one whose model is maximal.

If such an operation admits an identity element o ( such that for all ), then this element is unique ( ).

In 1743 he published the paper On the structure and diseases of articulating cartilages – which is often cited – especially the following sentence: “ If we consult the standard Chirurgical Writers from Hippocrates down to the present Age, we shall find, that an ulcerated Cartilage is universally allowed to be a very troublesome Disease ; that it admits of a Cure with more Difficulty than carious Bone ; and that, when destroyed, it is not recovered ”.

If the accused admits to the offence, or there are reliable witness statements, the chair will call for proposals.

When Johnny admits himself into a hospital late at night, he is met by Mr. Carlson, a devoted Christian, who tells Johnny that there is nothing wrong with hearing God's voice, but doubted it was God, saying " If God had something He wanted to say, you'd hear it.

If the multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations.

If a differential operator L admits a set of eigenvectors ( i. e., a set of functions and scalars such that ) that is complete, then it is possible to construct a Green's function from these eigenvectors and eigenvalues.

If a bialgebra B admits an antipode S, then S is unique (" a bialgebra admits at most 1 Hopf algebra structure ").

Ure is a recovering alcoholic ; something he openly admits and discusses in his autobiography If I Was.

The length of a finite resolution is the subscript n such that P < sub > n </ sub > is nonzero and P < sub > i </ sub >= 0 for i greater than n. If M admits a finite projective resolution, the minimal length among all finite projective resolutions of M is called its projective dimension and denoted pd ( M ).

Let P be a principal H-bundle on M, equipped with a Cartan connection η: TP → g. If g is a reductive module for H, meaning that g admits an Ad ( H )- invariant splitting of vector spaces g = h ⊕ m, then the m-component of η generalizes the solder form for an affine connection.

If a vector field v admits a vector potential A, then from the equality

If ( and only if ) the victim is more than 1 – 2 hours away from a medical facility, it is recommended to place a lightly constricting band ( that admits one finger beneath it ) above the bitten area to prevent the systemic spread of the venom.

The length of a finite flat resolution is the first subscript n such that F < sub > n </ sub > is nonzero and F < sub > i </ sub >= 0 for i greater than n. If a module M admits a finite flat resolution, the minimal length among all finite flat resolutions of M is called its flat dimension and denoted fd ( M ).

If the restriction of this action to H is trivial, then H is said to be normal, and the quotient scheme admits a natural group law.

If M admits an almost complex structure, it must be even-dimensional.

If M admits local holomorphic coordinates for J around every point then these patch together to form a holomorphic atlas for M giving it a complex structure, which moreover induces J. J is then said to be ' integrable '.

Also, in Don Quixote by Cervantes, there are many stories within the story that influence the hero's actions ( there are others that even the author himself admits are purely digressive ).

Greg Gillis, in interview outtakes for the movie " RIP: A Remix Manifesto ", admits that Evolution Control Committee was a major influence and inspiration to create the mashups for his artistic persona Girl Talk.

Haushofer admits that after 1933 much of what he wrote was distorted under duress: his wife had to be protected by Hess's influence ( who managed to have her awarded ' honorary German ' status ); his son was implicated in the July 20 plot to assassinate Hitler and was executed by the Gestapo ; he himself was imprisoned in Dachau concentration camp for eight months ; and his son and grandson were imprisoned for two-and-a-half months.

SPS admits both male and female students who have already shown a great deal of interest in public service, as well as an ability to positively influence others.

( Despite his harsh view of her political influence, Saint-Simon admits that he personally liked and admired her.

For example, while Buxton notes the existence of a tradition that " Abyssinians invoked the aid of foreigners " to construct these monolithic churches, and admits that " there are clearly signs of Coptic influence in some decorative details " ( hardly surprising given the theological, ecclesiastical, and cultural links between the Ethiopian Orthodox and Coptic Orthodox Churches ), he is adamant about the native origins of these creations: " But the significant fact is remains that the rock-churches continue to follow the style of the local built-up prototypes, which themselves retain clear evidence of their basically Axumite origin.

However, he admits that the pottery also shows Roman and Sarmatian influence.

Haushofer admits that after 1933 much of what he wrote was distorted under duress: his wife had to be protected by Hess's influence ; his son was murdered by the Gestapo ; he himself was imprisoned in Dachau concentration camp for eight months ; and his son and grandson were imprisoned for two-and-a-half months.

Díaz portrays the executions as unjust and based on no evidence, and admits to having liked Cuauhtémoc personally.

The Banach fixed point theorem states that a contraction mapping on a complete metric space admits a fixed point.

In Victoria on 21 December 2011, legislation was passed by the Victorian Parliament allowing new trials where there is " fresh and compelling DNA evidence, where the person acquitted subsequently admits to the crime, or where it becomes clear that key witnesses have given false evidence ".

Dartmouth meets 100 % of students ' demonstrated financial need in order to attend the College, and currently admits all students, including internationals, on a need-blind basis.

When pressed on the issue of Idaho, he also admits that the Bene Tleilax have conditioned their own agenda into him.

On leaving Antioch he appointed Alexander of Heliopolis as governor, a violent and cruel man whom the Antiochene Libanius, a friend of the emperor, admits on first thought was a " dishonourable " appointment.

Daniel, though claiming to be a contemporary, admits to no knowledge of John's origins — any speculation on John's birth is the result of much later speculation, and is confined to references in the Menologion.

The madness that is a side effect of Venom takes hold, and during the ensuing fight with Superman and Batman, Luthor admits he had traded the creature Doomsday to Darkseid in return for weapons during the Our Worlds at War crisis ; in doing so, he inadvertently provides a confession which is captured on video by Batman.

In his autobiography, he admits that they left an indelible impression on his mind.

Marge admits that she " put up with a lot in marriage ," and has left Homer or thrown him out of the house on several occasions.

In his two part work The Age of Reason, the American revolutionary Thomas Paine wrote, " The study of theology, as it stands in Christian churches, is the study of nothing ; it is founded on nothing ; it rests on no principles ; it proceeds by no authorities ; it has no data ; it can demonstrate nothing ; and it admits of no conclusion.

The Temple admits members on all continents except Antarctica, though it is largely a U. S .- based organization.

* November 28 – Acting on the ordinance passed by the Jackson government, the Confederate Congress admits Missouri as the 12th Confederate state.

* March 20 – Australian Prime Minister Bob Hawke weeps on national television as he admits marital infidelity.

* The proof that every Haefliger structure on a manifold can be integrated to a foliation ( this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one ).

Further, the most recent edition of Ives's biography admits that Anne may very well have had a personal spiritual awakening in her youth which spurred her on, not just as catalyst but expediter for Henry's Reformation, though the process took a number of years.

Daniel Burston, however, has argued that overall the published works of Szasz and Laing demonstrate far more points of convergence and intellectual kinship than Szasz admits, despite the divergence on a number of issues related to Szasz being a libertarian and Laing an existentialist ; that Szasz employs a good deal of exaggeration and distortion in his criticism of Laing's personal character, and unfairly uses Laing's personal failings and family woes to discredit his work and ideas ; and that Szasz's " clear-cut, crystalline ethical principles are designed to spare us the agonizing and often inconclusive reflections that many clinicians face frequently in the course of their work ".

Producer and story editor Dennis Marks created the character and admits to basing him on Indiana Jones.

Trejo admits that Van Zant's men called in the tip on the robbery from information Waingro provided.

According to him it is obvious that the culture on Tierra del Fuego is at a lower level of development than the European culture, but he also admits that the conditions of life there are much more difficult and this gives people very little chance to develop a higher culture.

" She has confessed to having a longtime crush on Salma Hayek and admits to having fantasized about her.

Ruth admits that she was sent to spy on him.