PROPERTIES OF THE UPPER AND LOWER SEQUENCE... 2

for this paper. In this paper n is a natural number. We now state a number of propositions: (1) For all subsets A, B of E 2 T such that A meets B holds proj1 ? A meets proj1 ? B. (2) Let A, B be subsets of E 2 T and s be a real number. If A misses B and A

JOURNALOFFORMALIZEDMATHEMATICS

Volume14,Released2002,Published2003

Inst.ofComputerScience,Univ.ofBia ystok

PropertiesoftheUpperandLowerSequenceontheCage1RobertMilewskiUniversityofBia ystokMMLIdenti er:JORDAN15.WWW:doc.xuehai.net[25],[29],[2],[27],[1],[4],[5],[3],[15],[13],[26],[23],[24],[12],[22],[9],[10],[7],

[28],[16],[11],[19],[18],[6],[21],[20],[8],[14],and[17]providethenotationandterminologyforthispaper.

Inthispapernisanaturalnumber.

Wenowstateanumberofpropositions:

(1)2suchthatAmeetsBholdsproj1 Ameetsproj1 B.ForallsubsetsA,BofET

2andsbearealnumber.IfAmissesBandA HorizontalLine(s)(2)LetA,BbesubsetsofETandB HorizontalLine(s),thenproj1 Amissesproj1 B.

(3)

(4)

(5)2suchthatSisBoundedholdsproj1 Sisclosed.ForeveryclosedsubsetSofET2holdsproj1 Siscompact.ForeverycompactsubsetSofET2.SupposeL(p,q)isverticalandL(p,q)isverticalandLetp,q,p1,q1bepointsofET11

p1=(p1)1andp2≤(p1)2and(p1)2≤(q1)2and(q1)2≤q2.ThenL(p1,q1) L(p,q).

2.SupposeL(p,q)ishorizontalandL(p,q)ishorizontal(6)Letp,q,p1,q1bepointsofET11

andp2=(p1)2andp1≤(p1)1and(p1)1≤(q1)1and(q1)1≤q1.ThenL(p1,q1) L(p,q).

(7)LetGbeaGo-boardandi,j,k,j1,k1benaturalnumbers.Suppose1≤iandi≤lenGand

1≤jandj≤j1andj1≤k1andk1≤kandk≤widthG.ThenL(G (i,j1),G (i,k1)) L(G (i,j),G (i,k)).

(8)LetGbeaGo-boardandi,j,k,j1,k1benaturalnumbers.Suppose1≤iandi≤widthG

and1≤jandj≤j1andj1≤k1andk1≤kandk≤lenG.ThenL(G (j1,i),G (k1,i)) L(G (j,i),G (k,i)).

(9)LetGbeaGo-boardandj,k,j1,k1benaturalnumbers.Suppose1≤jandj≤j1

andj1≤k1andk1≤kandk≤widthG.ThenL(G (CenterG,j1),G (CenterG,k1)) L(G (CenterG,j),G (CenterG,k)).

(10)LetGbeaGo-board.SupposelenG=widthG.Letj,k,j1,k1benaturalnumbers.Sup-

pose1≤jandj≤j1andj1≤k1andk1≤kandk≤lenG.ThenL(G (j1,CenterG),G

(k1,CenterG)) L(G (j,CenterG),G (k,CenterG)).

1ThisworkhasbeenpartiallysupportedbyCALCULEMUSgrantHPRN-CT-2000-00102.

1cAssociationofMizarUsers

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