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

theorem Th23:
  for a being Real, r1,r2 being positive Real st
  r1 <= r2 holds Ball(|[a,r1]|,r1) c= Ball(|[a,r2]|,r2)
proof
  let a be Real;
  let r1,r2 be positive Real;
  assume r1 <= r2;
  then
A1: r2-r1 >= 0 by XREAL_1:48;
  let x be object;
  assume
A2: x in Ball(|[a,r1]|,r1);
  then reconsider x as Element of TOP-REAL 2;
A3: |.x-|[a,r1]|.| < r1 by A2,TOPREAL9:7;
  |[a,r1]|-|[a,r2]| = |[a-a,r1-r2]| by EUCLID:62;
  then |.|[a,r1]|-|[a,r2]|.| = |.r1-r2.| by TOPREAL6:23;
  then |.|[a,r1]|-|[a,r2]|.| = |.r2-r1.| by COMPLEX1:60;
  then |.|[a,r1]|-|[a,r2]|.| = r2-r1 by A1,ABSVALUE:def 1;
  then
A4: |.x-|[a,r1]|.|+|.|[a,r1]|-|[a,r2]|.| < r1+(r2-r1) by A3,XREAL_1:8;
  |.x-|[a,r2]|.| <= |.x-|[a,r1]|.|+|.|[a,r1]|-|[a,r2]|.| by TOPRNS_1:34;
  then |.x-|[a,r2]|.| < r2 by A4,XXREAL_0:2;
  hence thesis by TOPREAL9:7;
end;
