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 Th8:
  for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= s2 & s2
  < s1 implies |[r,s+s2]| in Ball(u,s1)
proof
  let u be Point of Euclid 2 such that
A1: u = |[r,s]| and
A2: 0 <= s2 and
A3: s2 < s1;
  reconsider v = |[r,s+s2]| as Point of Euclid 2 by TOPREAL3:8;
  dist(u,v) = sqrt ((r - r)^2 + (s - (s+s2))^2) by A1,Th6
    .= sqrt ((-(s - (s+s2)))^2)
    .= s2 by A2,SQUARE_1:22;
  hence thesis by A3,METRIC_1:11;
end;
