nyuuzyou/wikis huggingface dataset:
Zorn's Lemma implies Kuratowski's Lemma

== Theorem ==
Let Zorn's Lemma be accepted as true.
Then Kuratowski's Lemma holds.
== Proof ==
Recall Zorn's Lemma:
Recall Kuratowski's Lemma:
Let $\struct {S, \preceq}$ be a non-empty ordered set.
Let $C$ be a chain in $\struct {S, \preceq}$.
Let $T$ be the set of all chains in $\struct {S, \preceq}$ that are supersets of $C$.
Let $\CC$ be a chain in the power set $\powerset T$ of $T$ under the ordering that is the subset relation.
Define $C' = \bigcup \CC$ be the union of $\CC$.
We have that the elements of $T$ are chains on $\struct {S, \preceq}$.
We have that $\CC$ is a subset of $T$.
Hence the elements of $\CC$ are also chains in $\struct {S, \preceq}$.
Thus $\bigcup \CC$ contains elements in $\struct {S, \preceq}$.
Hence:
:$C' \subseteq S$
We have that $C'$ is a chain in $\struct {S, \preceq}$.
Let $x, y \in C'$.
Then $x \in X$ and $y \in Y$ for some $X, Y \in \CC$.
As $\CC$ is a chain in $\powerset T$:
:either $X \subseteq Y$ or $Y \subseteq X$.
So $x$ and $y$ belong to the same chain in $\struct {S, \preceq}$.
Thus either $x \preceq y$ or $y \preceq x$.
Thus $C'$ is a chain on $\struct {S, \preceq}$. 
Now let $x \in C$.
Then:
:$\forall A \in T: x \in A$
Then because $\CC \subseteq T$:
:$\forall A \in \CC: x \in A$
So:
:$x \in \bigcup \CC$
and so:;
:$C \subseteq C'$
Thus:
:$C' \in T$
We now show that $C'$ is an upper bound on $\CC$.
Indeed, consider $x \in D \in \CC$.
This means:
:$x \in \bigcup \CC = C'$
so:
:$D\subseteq C'$
Thus $C'$ is an upper bound on $\CC$.
The chain in $T$ was arbitrary.
Hence every chain in $T$ has an upper bound.
Thus, by Zorn's Lemma, $T$ has a maximal element.
This must be a maximal chain containing $C$.
== Sources ==
* 
Category:Zorn's Lemma
Category:Kuratowski's Lemma
Zsigmondy's Theorem for Sums

== Theorem ==
Let $a > b > 0$ be coprime positive integers.
Let $n \ge 1$ be a (strictly) positive integer.
Then there is a prime number $p$ such that
:$p$ divides $a^n + b^n$
:$p$ does not divide $a^k + b^k$ for all $k < n$
with the following exception:
:$n = 3$, $a = 2$, $b = 1$
== Outline of Proof ==
We apply Zsigmondy's Theorem to $a^{2 n} - b^{2 n}$.
== Proof ==
By Zsigmondy's Theorem, there exists a prime divisor $p$ of $a^{2 n} - b^{2 n}$ which does not divide $a^k - b^k$ for all $k < 2 n$ unless:
:$n = 1$ and $a + b$ is a power of $2$
:$n = 3$, $a = 2$, $b = 1$
In particular, $p$ does not divide $a^{2 k} - b^{2 k} = \paren {a^k - b^k} \paren {a^k + b^k}$ for $k < n$.
It remains to check the case $n = 1$ and $a + b$ a power of $2$.
We have to show that $a^2 + b^2$ has an odd prime divisor.
Since $a$ and $b$ are coprime, both $a$ and $b$ are odd.
By Square Modulo 4, $a^2 + b^2 \equiv 2 \pmod 4$.
Because $a > b > 0$, $a^2 + b^2 > 2$.
But $4 \divides 2^k$ for $k > 1$.
Thus $a^2 + b^2$ is not a power of $2$.
Hence $a^2 + b^2$ has an odd prime divisor.
Category:Number Theory
Łoś-Vaught Test

== Theorem ==
Let $T$ be a satisfiable $\LL$-theory with no finite models.
Let $T$ be $\kappa$-categorical for some infinite cardinal $\kappa \ge \card \LL$.
Then $T$ is complete.
== Proof ==
We prove the contrapositive.
The main idea is that if such a theory $T$ is incomplete, we can construct size $\kappa$ models which disagree on a sentence.
Suppose $T$ is not complete.
By the definition of complete, this means that there is some sentence $\phi$ such that both $T \not \models \phi$ and $T \not \models \neg \phi$.
This in turn means that both $T \cup \set {\neg \phi}$ and $T \cup \set \phi$ have models.
Since $T$ has no finite models, this means that $T \cup \set {\neg \phi}$ and $T \cup \set \phi$ both have infinite models.
We have that $\kappa$ is infinite and greater than the cardinality of the language.
We also have that these theories have infinite models.
From the Upward Löwenheim-Skolem Theorem one can prove that there are size $\kappa$ models $\MM_{\neg \phi}$ and $\MM_\phi$ of $T \cup \set {\neg \phi}$ and $T \cup \set \phi$ respectively.
In particular, $\MM_{\neg \phi}$ and $\MM_\phi$ are models of $T$ which disagree about the sentence $\phi$.
Such models cannot be isomorphic since isomorphisms preserve the truth of sentences.
Thus, $T$ is not $\kappa$-categorical.
This result is also known as Vaught's Test.
Category:Model Theory for Predicate Logic
Łukasiewicz's Polish Notation/Examples/Arbitrary Example 1

== Example of Łukasiewicz's Polish Notation ==
The compound statement expresssed in infix notation as:
:$\paren {p \land \lnot q} \implies \paren {p \lor \lnot r}$
would be expressed in Łukasiewicz's Polish notation as:
:$\operatorname C \operatorname K p \operatorname N q \operatorname A p \operatorname N r$
== Sources ==
* 
Category:Examples of Łukasiewicz's Polish Notation
Main Page

Welcome to the Sand Box!
==Settings==
Crown Arginium ■ Socialist Arginium ■ Imperia
Pax Occidentalis ■ Fallout 69 ■ Worlmarch ■ Goldacre
Golbé ■ Aviria ■ SQ ■ UAC
==Franchises==
Armour Ace ■ Black Chamber ■ Commstar ■ Darkstorm
Dissident Realms ■ Project 81
==Gaming==
Systems & Secrets ■ Inter*Plan ■ Fallout London ■ Broken Banners
==Others==
DSRA ■ Aurilogne ■ Hiller's ■ Big MT ■ NRP
Hanwell ■ L&L Gang ■ Brighto Labs ■ Omega Black
Securis ■ TEC ■ APR ■ STRADA ■ Amity Insurance
FG ■ GCA ■ AWR ■ COA ■ COGM ■ INFRA ■ Novesperoj
Category:Content
Brighto Labs

Brighto Labs is a prominent US-based pharmaceutical research and development firm.
==Overview==
Brighto develops and produces a wide range of products from prescription and OTC medicines to healthcare products, first aid supplies, wellness products, and surgical-grade sanitation supplies.
==Divisions==
* Accounts
** Handles all vendor and supplier accounts, disbursements and payments.
* Facility Services
** Maintenance, moving services and supply.
* Payroll
** Bi-weekly paychecks, expense account disbursements, and all personnel related pay-outs.
* Personnel
** Hiring, training, and compliance certifications for all employees.
* Production
** Manufactures all products.
* Quality Assurance
** Develops and enforces internal quality standards.
* Quality Control
** Tests all products to ensure quality, purity, and safety.
* Regulatory Affairs
** Handles all applications for research, development, production and marketing of products.
* Research & Development
** Studies and develops all products.
* Security
** Maintains facility and grounds safety and security.
* Technical Services
** Provides support for all computer, communications, and electronic systems.
* Validation
** Assesses all equipment and processes to ensure proper operation, safety, and regulatory compliance.
==Facilities==
* Facility № 27
==Products==
* Stimpak - general purpose healing injectible
* Med-X - non-addictive morphine strength painkiller
* Serum A - cardiovascular restorative
* Serum G - diabetes corrective
* Serum K - muscular dystrophy corrective
* Serum Z - cancer reversal
* SPREADOUT - cure-all (code name)
Broken Banners

Broken Banners is a high fantasy massively multiplayer online roleplaying game (MMORPG) developed by Atomic Lion Studios.
==Synopsis==
===Plot===
===Characters===
==Setting==
Basinford - Dawncrest - Portsgrave - Goldacre - Oxgate - Roseminster - Castleside - Millstone - Hazelbridge - Fairhold - Saltwatch - Ambercrest - Embershire - Rusthaven - Cloudrest - Yewdark - Stormsbury - Highspire - Eastwick - Rogueham - Grimwell
===Aglar Heights===
* settlements - 
* points of interest - 
* natural features - 
===Dacla Hills===
* settlements - 
* points of interest - 
* natural features - 
===Drevanna===
RENAME
* settlements - Fairhold (capital), Ambercrest, 
* points of interest - 
* natural features - 
===Eslerene Plains===
* settlements - 
* points of interest - 
* natural features - 
===Haimore===
* settlements - 
* points of interest - 
* natural features - 
===Opria Valley===
* settlements - 
* points of interest - 
* natural features - 
===Ostinaweth===
* settlements - 
* points of interest - 
* natural features - 
===Qaldobyt===
* settlements - 
* points of interest - 
* natural features - 
===Patral Mountains===
* settlements - 
* points of interest - 
* natural features - 
===Ravaria===
* settlements - Basinford (capital), Portsgrave, Roseminster
* points of interest - 
* natural features - 
===Strole Mountains===
* settlements - 
* points of interest - 
* natural features - 
===Tagrana Desert===
* settlements - Saltwatch
* points of interest - 
* natural features - 
===Valdaria===
* settlements - 
* points of interest - 
* natural features - 
===Worlmarch===
* settlements - Castleside, 
* points of interest - 
* natural features -
==Gameplay==
===Classes===
* Archer
* Cleric
* Rogue
* Warrior
* Wizard
===Skills===
* Attack
* Defence
* Stregth
* Hits
* Ranged
* Prayer
* Magic
* Cooking
* Woodcutting
* Fatigue
* Fletching
* Fishing
* Firemaking
* Crafting
* Smithing
* Mining
* Herbology
* Agility
* Thieving
* Quest Points
===Equipment status===
* Armour
* Weapon aim
* Weapon power
* Magic
* Prayer
===Status conditions===
* Burn
* Dead Short
* Mark
* Slow
* Confusion
* High Threat
* Poison
* Stun
* Blind
===Abilities===
* TBD
Darkstorm

Darkstorm is the working title for a high fantasy MMORPG world building project currently being crafted by author Tom Brown. Work on the setting, and story, first began in 2009, and has proceeded gradually ever since. The entire project is dedicated to the memory of Tom's mother, Phyllis Brown, who passed away on 9 October 2020.
==Setting==
===Locations===
===Landmarks===
===Factions===
====Determined States====
The alliance of northern nations. Their strength is in a combination of technology, arcane magic, and firepower and they approach challenges with reason and logic. Their agnostic outlook is influenced by the religious wars of the Shattered Banners period which nearly tore the Northlands apart. The faction formulates collective policy through a body known as the Wise Council, where the leaders of the member nations meet four times a year. Special sessions are called in the event of emergencies.
====Radiant League====
The alliance of southern nations. They believe they are blessed by the gods of the Circle Celestial and their strength is a blend of holy weaponry, light magic, and disciplined tactics. Faith united the Southlands in the aftermath of the Shattered Banners period and is seen as their common cause, providing guidance and protection to their works.
==Gameplay==
===Player races===
* Cat-people (Whiskerfolk)
* Dwarf
* Elf
* Human
===Player classes===
* Warrior / Armiger
* Paladin / Partisan
* Rogue / Agent
* Arcanist
* Hunter
* Cleric / Vitalist
===Professions===
====Gathering====
* Fishing
* Herbalism
* Mining
====Crafting====
* Apothecary
* Armourcraft
* Artifice
* Tailoring
* Weaponscraft
Efram A. Grossman

Efram Alexander Grossman, D.Eng. - born 1 March 1978 - is an American-born engineer and senior member of the Inquest.
==Early Life==
Grossman was born in the Bay Ridge neighborhood of New York City, New York. He is the oldest of three, with a brother and a sister younger than him.
==Education==
Grossman is a 2002 valedictorian graduate of the Massachusetts Institute of Technology, and a 2005 valedictorian graduate of ETH Zurich.
Aside from his native English, he can fluently read and speak German, French, Russian, Mandarin Chinese, Vietnamese, and Japanese.
==Current Activity==
Grossman is a founding member of the Inquest, and is likewise a recognised lieutenant of the organisation's leader, Doctor Liam Gregory, the Duke of Kent. As a founder member and as a part of Gregory's inner circle, he is sits on the organisation's governing body, the Select Committee. He has top level access to the Inquest's databases and security protocols, and is involved in numerous initiatives. He directly oversees Research Facility Nº 3826, which is hidden in the forests southwest of Moscow. From this facility, he oversaw the successful completion of Project ARGUS, Project EPHEMERIS, and Project SEMELE. He is currently overseeing Project EUROPA and Project JUBILEE.
Doctor Grossman is also in consultation with one of his peers who is overseeing Project I71A at Research Facility Nº 2795.
==Personality==
According to the Myers-Briggs personality assessment, Doctor Grossman is classified as an "Architect. " In keeping with this, he is described as having a "quick, imaginative, and strategic mind", as well as "high self-confidence, and is independent & decisive". He is likewise described as "an open-minded jack-of-all-trades scientist". However, he is also believed by some to be "arrogant, judgmental, and overly analytical".
Golbé

Golbé is an ongoing high fantasy worldbuilding project being undertaken by Tom Brown.
==Overview==
==History==
===Major historical events===
===Ancient civilisations===
===Conflicts and wars===
===Important figures and legacies===
==Geography==
===Landmasses===
===Climate zones===
===Mountains===
===Bodies of water===
===Forests===
===Mountains===
==Ecology==
===Flora===
===Fauna===
===Ecosystems===
===Environmental concerns===
==Politics==
===Government systems===
===Political factions===
===Power dynamics===
==Society==
===Class structures===
===Family dynamics===
===Racial and ethnic groups===
Humans (2 or 3 ethic groups), elves (2 or 3 ethnic groups), dwarves, gnomes, orcs...
Trolls, goblins, dragons...
===Gender roles===
==Culture==
===Customs and traditions===
===Social hierarchies===
===Religious beliefs===
===Art and literature===
==Language==
===Unique languages===
===Dialects===
===Linguistic families===
==Magic==
===How magic works===
===Magic users and their roles===
===Magical creatures and phenomena===
==Technology==
===Level of advancement===
===Key inventions===
===Tools and techniques===
Goldacre

The Goldacre is a legendary island hidden somewhere in the southern Pacific Ocean.
==Overview==
==Government==
==Military==
==Economy==
Currency is known as “palmers”, abbreviated “P”. 100 copper palmers equal one silver palmer.
==Culture==
==Locations==
====Villages====
* Alnerstoke
* Basinford
* Bellstar
* Bronzewater
* Ellerstone
* Fairstrand
* Highcross
* Maplefair
* Millpost
* Oldgate
* Riverview
* Roseminster
* Stillwick
* Winvale
====Geographic Features====
* Amberwood
* Argent Coast
* Candle Ridge
* Elderwood
* Greenwall Mountain
* Mount Ernmore
* Parstead Basin
* Singing Shore
* Veridian Plain
====Unique Sites====
* Laventhorpe Hall
==People==
* Mister and Missus Pell, run the bakery in Highcross.
* Mister Tane, runs the tobacconist in Highcross.
==Organisations==
* Healer's League
* Lodge of Co-Operation
* Merchant's Guild
==Miscellaneous==
* Circulus
Hanwell

Hanwell Memorial Hospital - commonly referred to as "the Hanwell" or as "HMH" - is a Class One medical center located in Chattanooga, Tennessee.
==Overview==
Hanwell Memorial Hospital holds numerous distinctions. It is the single largest hospital in Tennessee, the United States, and the Western Hemisphere, and is also the single largest medical facility in the world. The facility is equipped for all medical specialties and can provide any and all forms of treatment and surgery.
Hanwell has accommodations for approximately 5,000 patients - 2,500 in single-patient rooms, 2,000 in dual-patient rooms, and 500 in ten-patient wards. The hospital has 50 major operating rooms, 125 minor operating rooms, and five operating theaters. The Critical Care Department has space for 500 patients, and the Emergency Services Department has spaces for 500 patients. The hospital has a staff of 2,500 doctors, 12,500 nurses, and 20,000 orderlies.
The Riverview Mill burned to the ground on 1 November 1869. Exactly seventy years to the day later, on 1 November 1939, Erlanger Hospital burned to the ground. The property sat unused for 71 years until Hanwell Memorial was constructed in 2010.
==Departments==
note - Heads of department marked with a "+" are also members of the Senior Staff.
====Medical Centre====
* Anesthetics (burgundy) - 
* Critical Care (red) - 
* Diagnostic Imaging (submarine) - Jennifer Strickland MD
* Emergency Services (regal blue) - J. Adam Colston MD+
* Laboratory (khaki) - Justin Luttrell MD+
* Nursing Administration (royal blue) - Sharon Largin MSN
* Orderly Services (white) - Chris Rodgers
* Pyschological Services (silver gray) - David Dancy MD+
* Surgical Services (surgical green) - Tom Brown MD+
====Medical Mall====
* Cardiology
* Dermatology
* Ear, Nose & Throat
* Elderly Services
* Gastroenterology
* Genitourinary Medicine
* Gynaecology
* Haematology
* Maternity
* Microbiology
* Neonatal Services
* Nephrology
* Neurology
* Nutrition & Dietetics
* Obstetrics
* Occupational Therapy
* Oncology
* Opthalmology
* Orthopaedics
* Pain Management
* Pediatrics
* Physiotherapy
* Pulmonology
* Radiotherapy
* Rheumatology
* Urology
====Auxiliary====
* Chaplaincy
* Discharge Lounge
* Pharmacy (hunter green)
* Transport Services
====Support====
* Catering Services
* General Administration
* Housekeeping
* Maintenance
* Records & Archives
* Technical Services
Hileen

Hileen is a planet located within the territory of the Galactic Empire. It is the sole planet in the Hylos system, orbiting the star of the same name. The planet has seven moons, and is accompanied in orbit around the star Hylos by two separate asteroid belts which intersect. The planet has a cool, dry climate with scattered bodies of water and few rivers. It has a diameter of 16,890 kilometers and a gravity of 1.15 standard. It has a standard atmosphere and water sources. There are no indigenous sentient life forms on the planet.
The Hylos system is located deep in Imperial territory, in Oversector Zeta, near the very peripheral edge of the Outer Rim Territories. It does not appear on any star charts, and buoys situated at the outskirts of the system ping any ships detected nearby and warn them away if they approach.
It is said that the Order of Varani was first founded on Hileen, and as such the planet plays a central role in the history and rituals of the Order. The planet is home to hundreds of crypts where centuries of powerful Shadow Lords are laid to rest, and in the intervening thousands of years since the Order's founding, it has become the Varani mausoleum world, primarily a planet of the dead. The most prestigious and powerful Varani were laid to rest in the oldest and most ominous area, known as the Valley of the Shadow Lords. The Valley is said to be guarded by ancient horrors created by Varani alchemy, as well as the vengeful spirits of the followers of several of the lords interred there.
Because of its Varani history, the planet is steeped - extremely so - in the Black Link of the Ambience. Any who visit that are not attuned to the Black Link, be they Pracado or mundane or otherwise, will feel the heavy influence of the dark energies that permeate the world. These energies grow exponentially stronger the closer one gets to the Valley of the Shadow Lords.
Category:Imperia
INFRA

INFRA is an exploration, adventure, and puzzle-solving alternative history game setting developed by Atomic Lion Studios.
==Setting==
The game takes place in Belokovia, a fictional nation which occupies the same territory as Finland. In this world, Belokovia was a part of the Russian Empire from the turn of the 19th century until 1920, when the country seceded in the height of the Russian Revolution. It would go on to resist a full-scale invasion by the Soviet Army in 1939. The country remained neutral during World War 2 while allowing refugees from war-torn states to find safety there. In the post-war era, it would build a prosperous society with a capitalist economy and strong social services.
Beginning in 1988, cracks began to show in the Eastern bloc as Moscow started losing control of the Baltic states and the Caucasus descended into civil war and chaos. The government of the period in Belokovia, which had been working towards closer ties with the USSR, saw its popularity dramatically decline as anti-Soviet sentiment among Belikovans grew stronger. When the 1991 general election came, the ruling party lost massively to the opposition, breaking a 12-year tenure for the left wing parties. In the following years, information came to light of a conspiracy throughout the mid-1970s to the mid-1980s where funding intended for maintenance of vital infrastructure and public transport was instead rerouted to officials in the government and unions. Ultimately several prominent politicians and union leaders were arrested and prosecuted for corruption, graft, and illegal patronage.
This is where you, the player, come in. The year is 2005. You and your colleagues are engineers with the National Consulting Group (Национальная Консалтинговая Группа or НКГ), an up and coming engineering and architectural consultation firm that has earned a reputation for honesty and efficiency as several of its older, more established competitors were brought down by the wide-ranging corruption investigation. NCG has recently received an expansive contract to survey and identify problems with a variety of systems and structures that need service. This includes water, public transport, and communications infrastructure, primarily around the nation's capital city of Cherenovsk.
==The Nation==
===History===
===Government===
Belokovia, formally the Republic of Belokovia (Республика Белоковия) is a unitary parliamentary republic.
===Military===
===Economy===
===Society===
The nation has a population as of 1 January 2005 is 11,207,702.
Imperial Ruling Council

The Imperial Ruling Council is the name given to the group of senior Imperial officials who constitute the most powerful collective body in the Galactic Empire, and the core of its political and military authority. Operating under the authority of Emperor Tuscano, it is the Council that manages the daily affairs of the Empire, advises the Emperor as needed, and carries out the Imperial will.
The Council consists of the Emperor and his five most senior officers - his Lord Privy Seal, Speaker of the Imperial Court, Minister President of the Imperial Government, and Supreme Commander of the Imperial Military. Providing technical and scientific counsel is the Master Engineer. These five men are considered his top lieutenants, and it is no secret throughout the Empire that they hold authority far above and beyond what most would even imagine. Theirs is the only counsel that directly reaches the Emperor's ear, though this aspect of their function is little more than a formality, as Tuscano takes advice from no one. To go against them is to risk drawing the direct wrath of the Emperor himself, a prospect none in their right mind finds desirable.
Image:Tuscano.png|Hieronymus Tuscano, Emperor
Image:Cavazerre.jpeg|Jordan Cavazerre, Speaker of the Imperial Court
Image:Eccari.jpeg|Theodore Eccari, Minister President of the Imperial Government
Image:Leovarus.jpg|Alexander Leovarus, Master Engineer
Image:Tavari.png|Hanseph Tavari, Lord Privy Seal
Image:Virini.jpeg|Dominic Virini, Supreme Commander of the Imperial Military
Category:Imperia
L&L Gang

The L&L Gang is a powerful Raider group operating in the Commonwealth.
==Overboss==
* Blackheart
** Liam Blackwood
** Adam Colston
** Corvega Assembly Plant
==Underbosses==
* Boomer
** Brendan Graf
** Jordan Dent
** Corvega Assembly Plant
* Cinder
** Raylan Collins
** Bryan Costley
** Saugus Ironworks
* Gage
** Taylor Wayne
** Justin Luttrell
** Kendall Hospital
* Hook
** Marcus Grant
** TBD
** Milton General Hospital
* Slab
** John Hammitt
** TBD
** Shamrock Taphouse
* Torque
** Patrick Lawson
** TBD
** Revere Beach Station
* Wire
** Lucas Dalton
** TBD
** Poseidon Energy
Category:Miscellaneous