let G be Go-board; :: thesis: for i being Nat st i <= width G holds
not cell (G,(len G),i) is bounded
let i be Nat; :: thesis: ( i <= width G implies not cell (G,(len G),i) is bounded )
assume A1: i <= width G ; :: thesis: not cell (G,(len G),i) is bounded
per cases ( i = 0 or ( i >= 1 & i < width G ) or i = width G ) by A1, NAT_1:14, XXREAL_0:1;
suppose A2: i = 0 ; :: thesis: not cell (G,(len G),i) is bounded
A3: cell (G,(len G),0) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & s <= (G * (1,1)) `2 ) } by GOBRD11:27;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,(len G),0) & not |.q.| < r )
proof
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,(len G),0) & not |.q.| < r )
take q = |[((G * ((len G),1)) `1),(min ((- r),((G * (1,1)) `2)))]|; :: thesis: ( q in cell (G,(len G),0) & not |.q.| < r )
A4: |.(q `2).| <= |.q.| by JGRAPH_1:33;
min ((- r),((G * (1,1)) `2)) <= (G * (1,1)) `2 by XXREAL_0:17;
hence q in cell (G,(len G),0) by A3; :: thesis: not |.q.| < r
per cases ( r <= 0 or r > 0 ) ;
suppose A5: r > 0 ; :: thesis: not |.q.| < r
q `2 = min ((- r),((G * (1,1)) `2)) by EUCLID:52;
then A6: |.(- r).| <= |.(q `2).| by A5, TOPREAL6:3, XXREAL_0:17;
- (- r) > 0 by A5;
then - r < 0 ;
then - (- r) <= |.(q `2).| by A6, ABSVALUE:def 1;
hence not |.q.| < r by A4, XXREAL_0:2; :: thesis: verum
end;
end;
end;
hence not cell (G,(len G),i) is bounded by A2, JORDAN2C:34; :: thesis: verum
end;
suppose A7: ( i >= 1 & i < width G ) ; :: thesis: not cell (G,(len G),i) is bounded
then A8: cell (G,(len G),i) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,i)) `2 <= s & s <= (G * (1,(i + 1))) `2 ) } by GOBRD11:29;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,(len G),i) & not |.q.| < r )
proof
len G <> 0 by MATRIX_0:def 10;
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,(len G),i) & not |.q.| < r )
take q = |[(max (r,((G * ((len G),1)) `1))),((G * (1,i)) `2)]|; :: thesis: ( q in cell (G,(len G),i) & not |.q.| < r )
A10: i < i + 1 by NAT_1:13;
A11: max (r,((G * ((len G),1)) `1)) >= (G * ((len G),1)) `1 by XXREAL_0:25;
i + 1 <= width G by A7, NAT_1:13;
then (G * (1,i)) `2 <= (G * (1,(i + 1))) `2 by A7, A9, A10, GOBOARD5:4;
hence q in cell (G,(len G),i) by A8, A11; :: thesis: not |.q.| < r
A12: |.(q `1).| <= |.q.| by JGRAPH_1:33;
end;
hence not cell (G,(len G),i) is bounded by JORDAN2C:34; :: thesis: verum
end;
suppose A14: i = width G ; :: thesis: not cell (G,(len G),i) is bounded
A15: cell (G,(len G),(width G)) = { |[r,s]| where r, s is Real : ( (G * ((len G),1)) `1 <= r & (G * (1,(width G))) `2 <= s ) } by GOBRD11:28;
for r being Real ex q being Point of (TOP-REAL 2) st
( q in cell (G,(len G),i) & not |.q.| < r )
proof
let r be Real; :: thesis: ex q being Point of (TOP-REAL 2) st
( q in cell (G,(len G),i) & not |.q.| < r )
take q = |[((G * ((len G),1)) `1),(max (r,((G * (1,(width G))) `2)))]|; :: thesis: ( q in cell (G,(len G),i) & not |.q.| < r )
A16: |.(q `2).| <= |.q.| by JGRAPH_1:33;
(G * (1,(width G))) `2 <= max (r,((G * (1,(width G))) `2)) by XXREAL_0:25;
hence q in cell (G,(len G),i) by A14, A15; :: thesis: not |.q.| < r
end;
hence not cell (G,(len G),i) is bounded by JORDAN2C:34; :: thesis: verum
end;
end;