reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th22:
  for n being Element of NAT, a,b being Element of TOP-REAL n, r1,
r2 being positive Real st |.a-b.| <= r1-r2 holds Ball(b,r2) c= Ball(a,
  r1)
proof
  let n be Element of NAT, a,b be Element of TOP-REAL n, r1,r2 be positive
  Real;
  assume |.a-b.| <= r1-r2;
  then
A1: |.b-a.| <= r1-r2 by TOPRNS_1:27;
  let x be object;
  assume
A2: x in Ball(b,r2);
  then reconsider x as Element of TOP-REAL n;
  |.x-b.| < r2 by A2,TOPREAL9:7;
  then
A3: |.x-b.|+|.b-a.| < r2+(r1-r2) by A1,XREAL_1:8;
  |.x-a.| <= |.x-b.|+|.b-a.| by TOPRNS_1:34;
  then |.x-a.| < r2+(r1-r2) by A3,XXREAL_0:2;
  hence thesis by TOPREAL9:7;
end;
