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|yieldsafinitesequenceofelementsofLetusconsiderx.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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