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

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;

end;

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 )

end;for r being Real ex q being Point of (TOP-REAL 2) st

( q in cell (G,(len G),0) & not |.q.| < r )

proof

hence
not cell (G,(len G),i) is bounded
by A2, JORDAN2C:34; :: thesis: verum
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

end;( 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

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 )

end;for r being Real ex q being Point of (TOP-REAL 2) st

( q in cell (G,(len G),i) & not |.q.| < r )

proof

hence
not cell (G,(len G),i) is bounded
by JORDAN2C:34; :: thesis: verum
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;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;

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 )

end;for r being Real ex q being Point of (TOP-REAL 2) st

( q in cell (G,(len G),i) & not |.q.| < r )

proof

hence
not cell (G,(len G),i) is bounded
by JORDAN2C:34; :: thesis: verum
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;( 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