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
  j < width G & p in Int h_strip(G,j) implies p`2 < G*(1,j+1)`2
proof
  assume that
A1: j < width G and
A2: p in Int h_strip(G,j);
  per cases by NAT_1:14;
  suppose
    j = 0;
    then Int h_strip(G,j) = { |[r,s]| : s < G*(1,j+1)`2 } by Th15;
    then ex r,s st p = |[r,s]| & G*(1,j+1)`2 > s by A2;
    hence thesis by EUCLID:52;
  end;
  suppose
    j >= 1;
    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 A2;
    hence thesis by EUCLID:52;
  end;
end;
