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We formulate this proof directly for the case of polynomials over a UFD R, which is hardly different from its special case for R = Z.

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## Some Related Sentences

We and formulate

__We__are seeking to

__formulate__an Islamic society

**which**in

**the**final analysis will produce an Islamic state

**.**

This

**is**true**for**all sciences:**the**goal**is**to connect**a**“ natural phenomenon ” with**its**“ immediate cause .”__We____formulate__hypotheses elucidating**,**as we see it**,****the**relation**of**cause and effect**for**particular phenomena**.**
*"

__We__hear**from**time to time that**the**Irish people are determined to__formulate__their own politics**,**and not to take them**from**Rome ; but events constantly demonstrate that not only**the**religion but**the**politics**of**Ireland are those**of****the**Church**of**Rome**,**and that**the**Irish people are still being exploited in**the**interest**of**clericalism and**for****the**proselytising**of**England**.**
< BLOCKQUOTE >

__We__are seeking to__formulate__an Islamic society**which**in**the**final analysis will produce an Islamic state**.**
"

__We__are living in times**of**unparalleled threat and hostility**from**unseen and often unknown enemies ", Woodson states on**the**MilitaryWeek**.**com website ; " The tendency**of**governments facing**this**situation**is**to turn inward**,**to sit on information as they try to__formulate__**a**strategy to meet**this**daunting challenge**.**__We__need to think about what contribution

**the**Celtic peoples and their cultures can make to

**the**development

**of**European unity

**,**to

__formulate__proposals and suggestions

**,**disseminating them

**,**instead

**of**adopting an aloof sort

**of**attitude

**.**

We and proof

__We__shall show that

**the**

**polynomials**Af behave in

**the**manner described in

**the**first paragraph

**of**

**the**

__proof__

**.**

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****.**__We__approach

**the**

__proof__

**of**Theorem 2 by successively restricting

**the**class

**of**all formulas φ

**for**

**which**we need to prove " φ

**is**either refutable or satisfiable ".

__We__have proved that φ

**is**either satisfiable or refutable

**,**and

**this**concludes

**the**

__proof__

**of**

**the**Lemma

**.**

__We__will now follow

**the**strategy

**of**David Hilbert ( 1862 – 1943 ) who gave

**a**simplification

**of**

**the**original

__proof__

**of**Charles Hermite

**.**

__We__now give an operator theoretic

__proof__

**for**

**the**Cauchy – Schwarz inequality

**which**passes to

**the**C *- algebra setting

**.**

__We__will go

**over**

**a**typical application

**of**Zorn's lemma:

**the**

__proof__that every nontrivial ring

**R**with unity contains

**a**maximal ideal

**.**

Dan Olinger

**,****a**professor at**the**fundamentalist Bob Jones University in Greenville said**,**“__We__want to be good citizens and participants**,**but we ’ re not really interested in using**the**iron fist**of****the**law to compel people to everything Christians should do .” Bob Marcaurelle**,**interim pastor at Mountain Springs Baptist Church in Piedmont**,**said**the**Middle Ages were__proof__enough that Christian ruling groups are almost always corrupted by power**.**__We__provide

**a**sketch

**of**

**a**

__proof__

**for**

**the**

**case**where

**the**underlying field

**of**scalars

**is**

**the**complex numbers

**.**

"

__We__" in**this**sense often refers to "**the**reader and**the**author ," since**the**author often assumes that**the**reader knows certain principles or previous theorems**for****the**sake**of**brevity ( or**,**if not**,****the**reader**is**prompted to look them up ),**for**example**,**so that**the**author does not need to explicitly write out every step**of****a**mathematical__proof__**.**__We__include here

**a**

__proof__that DTIME ( f ( n ))

**is**

**a**strict subset

**of**DTIME ( f ( 2n + 1 )< sup > 3 </ sup >) as it

**is**simpler

**.**

__We__call

**a**

**a**witness

**for**

**the**compositeness

**of**n ( sometimes misleadingly called

**a**strong witness

**,**although it

**is**

**a**certain

__proof__

**of**

**this**fact ).

* Example: "

__We__cannot wait**for****the**final__proof__–**the**smoking gun – that could come in**the**form**of****a**mushroom cloud**.**
"

__We__" in**this**sense often refers to "**the**reader and**the**author ", since**the**author often assumes that**the**reader knows certain principles or previous theorems**for****the**sake**of**brevity ( or**,**if not**,****the**reader**is**prompted to look them up ),**for**example**,**so that**the**author does not need to explicitly write out every step**of****a**mathematical__proof__**.**

We and directly

__We__propose

**a**method

**for**selecting only dictionary information required by

**the**text being translated and

**a**means

**for**passing

**the**information

__directly__to

**the**occurrences in text

**.**

* The song No Joy in Mudville

**from**Death Cab**for**Cutie's album__We__Have**the**Facts and We're Voting Yes__directly__references**the**poem**.**
Frank Capra's seven-film series Why

__We__Fight**is**said to have been__directly__inspired by and America's response to Triumph**of****the**Will**.**
::“

__We__certainly cannot hope__directly__to compare their effects except within**a**limited future ; and all**the**arguments**,****which**have ever been used in Ethics**,**and upon**which**we commonly act in common life**,**directed to shewing that one course**is**superior to another**,**are ( apart**from**theological dogmas ) confined to pointing out such probable immediate advantages …
* Post-rock band

**,**Deadhorse**,**refer their 2010 album release__We__Can Create Our Own World to be__directly__influenced by**the**book and Ray Bradbury's vision in evoke imagination in his readers**.**__We__found evidence that

**the**Corporation

**for**Public Broadcasting ( CPB ) former Chairman violated statutory provisions and

**the**Director ’ s Code

**of**Ethics by dealing

__directly__with one

**of**

**the**creators

**of**

**a**new public affairs program during negotiations with

**the**Public Broadcasting Service ( PBS ) and

**the**CPB

**over**creating

**the**show

**.**

The manifestos

__directly__state: "__We__speak unto you by parables**,**but would willingly bring you to**the**right**,**simple**,**easy**,**and ingenuous exposition**,**understanding**,**declaration**,**and knowledge**of**all secrets**.**
In

**a**letter to Autoweek Magazine**,**Donald Petersen**,**then Ford Chairman**,**said**,**"__We__are moving toward building**a**quality culture at Ford and**the**many changes that have been taking place here have their roots__directly__in Deming's teachings**.**
Of particular importance to

**the**development**of**folk rock were**the**subtle folk influences evident in such Beatles ' compositions as " I'll Be Back ", " Things__We__Said Today ", and " I'm**a**Loser ", with**the**latter song being__directly__inspired by folk singer-songwriter Bob Dylan**.**__We__shall therefore now consider only arguments that prove

**the**theorem

__directly__

**for**any matrix using algebraic manipulations only ; these also have

**the**benefit

**of**working

**for**matrices with entries in any commutative ring

**.**

Tottenham Court Road Tube Station

**is**replicated as part**of****the**set**for****the**Queen musical**,**__We__Will Rock You**,****which**plays at**the**Dominion Theatre**,**__directly__above**the**actual tube station**.**
He proposed

**the**" Radical Creed ,"**which****,**" as I understand it**,****is****this**:__We__have not abandoned our old belief in liberty**,**justice**,**and Self-help**,**but we say that under certain conditions**the**people cannot help themselves**,**and that then they should be helped by**the**State representing__directly__**the**whole people**.**__We__thus see that

**,**not only

__directly__but also through

**the**replies that it called forth

**,**

**the**system

**of**Hutcheson

**,**or at least

**the**system

**of**Hutcheson combined with that

**of**Shaftesbury

**,**contributed

**,**in large measure

**,**to

**the**formation and development

**of**some

**of**

**the**most important

**of**

**the**modern schools

**of**ethics

**.**

However

**,**Clark was determined that VI Corps should strike__directly__**for**Rome as evidenced in his later writing: "__We__not only wanted**the**honour**of**capturing Rome**,**but felt that we deserved it ... Not only did we intend to become**the**first army to seize Rome**from****the**south**,**but we intended to see that people at home knew that it was**the**Fifth Army that did**the**job**,**and knew**the**price that had been paid**for**it .".__We__pursue our vision through direct services

**,**education

**,**advocacy

**,**and outreach

**,**in conjunction with our partners

**,**to

__directly__benefit these communities

**.**

__We__do not see

**the**frame

__directly__

**,**but infer

**its**presence by

**its**characteristic expressions and language

**.**

Washington and Lee maintains

**a**rigorous honor system that traces__directly__to Robert E**.**Lee**,**who said**,**"__We__have but one rule here**,**and it**is**that every student must be**a**gentleman**.**0.372 seconds.