theorem Th21:
  r < s implies cl_Ball(x,r) c= Ball(x,s)
proof
  assume
A1: r < s;
  let a be object;
  assume
A2: a in cl_Ball(x,r);
  then reconsider a as Point of TOP-REAL n;
  |. a-x .| <= r by A2,TOPREAL9:8;
  then |. a-x .| < s by A1,XXREAL_0:2;
  hence thesis by TOPREAL9:7;
end;
