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
  i < len G & p in Int v_strip(G,i) implies p`1 < G*(i+1,1)`1
proof
  assume that
A1: i < len G and
A2: p in Int v_strip(G,i);
  per cases by NAT_1:14;
  suppose
    i = 0;
    then Int v_strip(G,i) = { |[r,s]| : r < G*(i+1,1)`1 } by Th12;
    then ex r,s st p = |[r,s]| & G*(i+1,1)`1 > r by A2;
    hence thesis by EUCLID:52;
  end;
  suppose
    i >= 1;
    then
    Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 } by A1,Th14
;
    then ex r,s st p = |[r,s]| & G*(i,1)`1 < r & r < G*(i+1,1)`1 by A2;
    hence thesis by EUCLID:52;
  end;
end;
