reserve E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  i, j, m, n for Nat,
  p for Point of TOP-REAL 2;

theorem Th33:
  j <= width G implies cell(G,0,j) is not bounded
proof
  assume
A1: j <= width G;
  per cases by A1,NAT_1:14,XXREAL_0:1;
  suppose
    j = 0;
    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;
    not ex r being Real st
    for q being Point of TOP-REAL 2 st q in cell(G,0,j) holds |.q.| < r
    proof
      let r be Real;
      take q = |[min(-r,G*(1,1)`1),min(-r,G*(1,1)`2)]|;
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;
A4:   |.q`1.|<=|.q.| by JGRAPH_1:33;
      per cases;
      suppose
        r <= 0;
        hence thesis;
      end;
      suppose
A5:     r > 0;
        q`1 = min(-r,G*(1,1)`1) by EUCLID:52;
        then
A6:     |.-r.| <= |.q`1.| by A5,TOPREAL6:3,XXREAL_0:17;
        --r > 0 by A5;
        then -r < 0;
        then --r <= |.q`1.| by A6,ABSVALUE:def 1;
        hence thesis by A4,XXREAL_0:2;
      end;
    end;
    hence thesis by JORDAN2C:34;
  end;
  suppose
A7: j >= 1 & j < width G;
    then
A8: cell(G,0,j) = { |[r,s]| where r is Real, s is Real:
     r <= G*(1,1)`1 & G*(1,j)`2 <= s & s <= G* (1,j+1)`2 } by GOBRD11:26;
    not ex r being Real st
     for q being Point of TOP-REAL 2 st q in cell(G,0,j) holds |.q.| < r
    proof
      len G <> 0 by MATRIX_0:def 10;
      then
A9:   1 <= len G by NAT_1:14;
      let r be Real;
      take q = |[min(-r,G*(1,1)`1),G*(1,j)`2]|;
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,GOBOARD5:4;
      hence q in cell(G,0,j) by A8,A11;
A12:  |.q`1.|<=|.q.| by JGRAPH_1:33;
      per cases;
      suppose
        r <= 0;
        hence thesis;
      end;
      suppose
A13:    r > 0;
        q`1 = min(-r,G*(1,1)`1) by EUCLID:52;
        then
A14:    |.-r.| <= |.q`1.| by A13,TOPREAL6:3,XXREAL_0:17;
        --r > 0 by A13;
        then -r < 0;
        then --r <= |.q`1.| by A14,ABSVALUE:def 1;
        hence thesis by A12,XXREAL_0:2;
      end;
    end;
    hence thesis by JORDAN2C:34;
  end;
  suppose
    j = width G;
    then
A15: cell(G,0,j) = { |[r,s]| where r is Real, s is Real:
     r <= G*(1,1)`1 & G*(1,width G)`2 <= s } by GOBRD11:25;
    not ex r being Real st
     for q being Point of TOP-REAL 2 st q in cell(G,0,j) holds |.q.| < r
    proof
      let r be Real;
      take q = |[G*(1,1)`1,max(r,G*(1,width G)`2)]|;
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,0,j) by A15;
      per cases;
      suppose
        r <= 0;
        hence thesis;
      end;
      suppose
A17:    r > 0;
        q`2 = max(r,G*(1,width G)`2) by EUCLID:52;
        then r <= q`2 by XXREAL_0:25;
        then r <= |.q`2.| by A17,ABSVALUE:def 1;
        hence thesis by A16,XXREAL_0:2;
      end;
    end;
    hence thesis by JORDAN2C:34;
  end;
end;
