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 Th9:
  for u being Point of Euclid 2 st u = |[r,s]| holds 0 <= r2 & r2
  < r1 implies |[r-r2,s]| in Ball(u,r1)
proof
  let u be Point of Euclid 2 such that
A1: u = |[r,s]| and
A2: 0 <= r2 and
A3: r2 < r1;
  reconsider v = |[r-r2,s]| as Point of Euclid 2 by TOPREAL3:8;
  dist(u,v) = sqrt ((r - (r-r2))^2 + (s - s)^2) by A1,Th6
    .= r2 by A2,SQUARE_1:22;
  hence thesis by A3,METRIC_1:11;
end;
