reserve n for Nat,
  i,j for Nat,
  r,s,r1,s1,r2,s2,r9,s9 for Real,
  p,q for Point of TOP-REAL 2,
  G for Go-board,
  x,y for set,
  v for Point of Euclid 2;

theorem
  1 <= j & j <= width G & p in Int h_strip(G,j) implies p`2 > G*(1,j)`2
proof
  assume that
A1: 1 <= j and
A2: j <= width G and
A3: p in Int h_strip(G,j);
  per cases by A2,XXREAL_0:1;
  suppose
    j = width G;
    then Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s } by Th16;
    then ex r,s st p = |[r,s]| & G*(1,j)`2 < s by A3;
    hence thesis by EUCLID:52;
  end;
  suppose
    j < width G;
    then
    Int h_strip(G,j) = { |[r,s]| : G*(1,j)`2 < s & s < G*(1,j+1)`2 } by A1,Th17
;
    then ex r,s st p = |[r,s]| & G*(1,j)`2 < s & s < G*(1,j+1)`2 by A3;
    hence thesis by EUCLID:52;
  end;
end;
