reserve a, b, c, d, r, s for Real,
  n for Element of NAT,
  p, p1, p2 for Point of TOP-REAL 2,
  x, y for Point of TOP-REAL n,
  C for Simple_closed_curve,
  A, B, P for Subset of TOP-REAL 2,
  U, V for Subset of (TOP-REAL 2)|C`,
  D for compact with_the_max_arc Subset of TOP-REAL 2;

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;
