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The Euclidean Space

FORMALIZEDMATHEMATICS

Vol.2,No.4,September–October1991

Universit´eCatholiquedeLouvain

TheEuclideanSpace

AgataDarmochwa󰀗l

WarsawUniversity

Bia󰀗lystok

Summary.ThegeneraldefinitionofEuclideanSpace.

MMLIdentifier:EUCLID.

Thepapers[14],[6],[9],[8],[12],[1],[5],[10],[3],[13],[4],[15],[16],[7],[11],and[2]providethenotationandterminologyforthispaper.Inthesequelk,ndenotenaturalnumbersandrdenotesarealnumber.Letusconsidern.TheandisdefinedasfunctorRnyieldsanon-emptysetoffinitesequencesof

follows:

(Def.1)Rn=n.

Inthesequelxwilldenoteafinitesequenceofelementsof.Thefunction||fromintoisdefinedasfollows:

(Def.2)foreveryrholds||(r)=|r|.

Letusconsiderx.Thefunctor|x|yieldsafinitesequenceofelement
sof

Letusconsiderx.Thefunctor|x|yieldsafinitesequenceofelementsof

andisdefinedasfollows:

(Def.3)|x|=||·x.

Letusconsidern.Thefunctor󰀊0,...,0󰀋yieldsafinitesequenceofelementsofandisdefinedby:

(Def.4)󰀊0,...,0󰀋=n−→0quaarealnumber.󰀄n󰀆󰀄n󰀆

Letusconsidern.Then󰀊0,...,0󰀋isanelementofRn.

Inthesequelx,x1,x2,ydenoteelementsofRn.Onecanprovethefollowingproposition

(1)xisanelementofn.󰀄n󰀆

601c󰀁1991FondationPhilippeleHodey

ISSN0777–4028

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