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 <= i & i <= len G & p in Int v_strip(G,i) implies p`1 > G*(i,1)`1
proof
  assume that
A1: 1 <= i and
A2: i <= len G and
A3: p in Int v_strip(G,i);
  per cases by A2,XXREAL_0:1;
  suppose
    i = len G;
    then Int v_strip(G,i) = { |[r,s]| : G*(i,1)`1 < r } by Th13;
    then ex r,s st p = |[r,s]| & G*(i,1)`1 < r by A3;
    hence thesis by EUCLID:52;
  end;
  suppose
    i < len G;
    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 A3;
    hence thesis by EUCLID:52;
  end;
end;
