theorem
  w in Sphere(x,r) iff dist(x,w) = r
proof
  thus w in Sphere(x,r) implies dist(x,w) = r
  proof
    assume w in Sphere(x,r);
    then ||.x - w.|| = r by Th51;
    hence thesis by CSSPACE:def 16;
  end;
  assume dist(x,w) = r;
  then ||.x - w.|| = r by CSSPACE:def 16;
  hence thesis;
end;
