theorem Th7:
  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
    .= sqrt ((-(r - (r+r2)))^2)
    .= r2 by A2,SQUARE_1:22;
  hence thesis by A3,METRIC_1:11;
end;
