let G be Go-board; :: thesis: for j being Element of NAT st j <= width G holds
not cell G,0 ,j is Bounded
let j be Element of NAT ; :: thesis: ( j <= width G implies not cell G,0 ,j is Bounded )
assume A1: j <= width G ; :: thesis: not cell G,0 ,j is Bounded
per cases ( j = 0 or ( j >= 1 & j < width G ) or j = width G ) by A1, NAT_1:14, XXREAL_0:1;
suppose j = 0 ; :: thesis: not cell G,0 ,j is Bounded
then A2: cell G,0 ,j = { |[r,s]| where r, s is Real : ( r <= (G * 1,1) `1 & s <= (G * 1,1) `2 ) } by GOBRD11:24;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,j & not |.q.| < r )
proof
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,j & not |.q.| < r )
take q = |[(min (- r),((G * 1,1) `1 )),(min (- r),((G * 1,1) `2 ))]|; :: thesis: ( q in cell G,0 ,j & not |.q.| < r )
A3: min (- r),((G * 1,1) `2 ) <= (G * 1,1) `2 by XXREAL_0:17;
min (- r),((G * 1,1) `1 ) <= (G * 1,1) `1 by XXREAL_0:17;
hence q in cell G,0 ,j by A2, A3; :: thesis: not |.q.| < r
A4: abs (q `1 ) <= |.q.| by JGRAPH_1:50;
per cases ( r <= 0 or r > 0 ) ;
suppose A5: r > 0 ; :: thesis: not |.q.| < r
q `1 = min (- r),((G * 1,1) `1 ) by EUCLID:56;
then A6: abs (- r) <= abs (q `1 ) by A5, TOPREAL6:8, XXREAL_0:17;
- (- r) > 0 by A5;
then - r < 0 ;
then - (- r) <= abs (q `1 ) by A6, ABSVALUE:def 1;
hence not |.q.| < r by A4, XXREAL_0:2; :: thesis: verum
end;
end;
end;
hence not cell G,0 ,j is Bounded by JORDAN2C:40; :: thesis: verum
end;
suppose A7: ( j >= 1 & j < width G ) ; :: thesis: not cell G,0 ,j is Bounded
then A8: cell G,0 ,j = { |[r,s]| where r, s is Element of REAL : ( r <= (G * 1,1) `1 & (G * 1,j) `2 <= s & s <= (G * 1,(j + 1)) `2 ) } by GOBRD11:26;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,j & not |.q.| < r )
proof
len G <> 0 by GOBOARD1:def 5;
then A9: 1 <= len G by NAT_1:14;
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,j & not |.q.| < r )
take q = |[(min (- r),((G * 1,1) `1 )),((G * 1,j) `2 )]|; :: thesis: ( q in cell G,0 ,j & not |.q.| < r )
A10: j < j + 1 by NAT_1:13;
A11: min (- r),((G * 1,1) `1 ) <= (G * 1,1) `1 by XXREAL_0:17;
j + 1 <= width G by A7, NAT_1:13;
then (G * 1,j) `2 <= (G * 1,(j + 1)) `2 by A7, A9, A10, GOBOARD5:5;
hence q in cell G,0 ,j by A8, A11; :: thesis: not |.q.| < r
A12: abs (q `1 ) <= |.q.| by JGRAPH_1:50;
per cases ( r <= 0 or r > 0 ) ;
suppose A13: r > 0 ; :: thesis: not |.q.| < r
q `1 = min (- r),((G * 1,1) `1 ) by EUCLID:56;
then A14: abs (- r) <= abs (q `1 ) by A13, TOPREAL6:8, XXREAL_0:17;
- (- r) > 0 by A13;
then - r < 0 ;
then - (- r) <= abs (q `1 ) by A14, ABSVALUE:def 1;
hence not |.q.| < r by A12, XXREAL_0:2; :: thesis: verum
end;
end;
end;
hence not cell G,0 ,j is Bounded by JORDAN2C:40; :: thesis: verum
end;
suppose j = width G ; :: thesis: not cell G,0 ,j is Bounded
then A15: cell G,0 ,j = { |[r,s]| where r, s is Element of REAL : ( r <= (G * 1,1) `1 & (G * 1,(width G)) `2 <= s ) } by GOBRD11:25;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,j & not |.q.| < r )
proof
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell G,0 ,j & not |.q.| < r )
take q = |[((G * 1,1) `1 ),(max r,((G * 1,(width G)) `2 ))]|; :: thesis: ( q in cell G,0 ,j & not |.q.| < r )
A16: abs (q `2 ) <= |.q.| by JGRAPH_1:50;
(G * 1,(width G)) `2 <= max r,((G * 1,(width G)) `2 ) by XXREAL_0:25;
hence q in cell G,0 ,j by A15; :: thesis: not |.q.| < r
end;
hence not cell G,0 ,j is Bounded by JORDAN2C:40; :: thesis: verum
end;
end;