{ localUrl: '../page/5nr.html', arbitalUrl: 'https://arbital.com/p/5nr', rawJsonUrl: '../raw/5nr.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: '5nr', edit: '1', editSummary: '', prevEdit: '0', currentEdit: '1', wasPublished: 'true', type: 'wiki', title: 'Without loss of generality', clickbait: '', textLength: '1041', alias: '5nr', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'JaimeSevillaMolina', editCreatedAt: '2016-07-30 03:38:47', pageCreatorId: 'JaimeSevillaMolina', pageCreatedAt: '2016-07-30 03:38:47', seeDomainId: '0', editDomainId: 'arbital_featured_project', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '3', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '17', text: '*Without loss of generality* (abbreviated as w.l.o.g.) is a common idiom in mathematics that remarks that we can introduce a new assumption reducing the proof to a special case, and the proof for the other cases either follows from the special case, can be reasoned in an [ analogous way], or is [ trivial].\n\nwlog is tightly related to [ case exhaustion].\n\n##Example with reduction to a special case\n\n##Example with analogous reasoning\n\n##Example with triviality\n\nTheorem: In every set of $5$ natural numbers there are three numbers which sum a multiple of $3$.\n\nProof: \n\nw.l.o.g. assume that there are no three numbers with the same residue modulo $3$ in the set. Otherwise, the sum of those three numbers is a multiple of $3$.\n\nNow, there are $5$ numbers and $3$ possible residues, so at least there is one number for each residue (otherwise, there could be a maximum of $2$ residue classes times a maximum of $2$ number per class, for a total of $4$ numbers). But $3a + (3b+1)+(3c+2) = 3 (a+b+c) + 3$, which is a multiple of $3$. Q.E.D.\n\n', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'JaimeSevillaMolina' ], childIds: [], parentIds: [], commentIds: [], questionIds: [], tagIds: [], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '3350', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '17782', pageId: '5nr', userId: 'JaimeSevillaMolina', edit: '1', type: 'newEdit', createdAt: '2016-07-30 03:38:47', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'false', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }