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 Th22:
  r < s implies Sphere(x,r) c= Ball(x,s)
proof
  assume r < s;
  then
A1: cl_Ball(x,r) c= Ball(x,s) by Th21;
  Sphere(x,r) c= cl_Ball(x,r) by TOPREAL9:17;
  hence thesis by A1;
end;
