theorem Th48:
  w in cl_Ball(x,r) iff dist(x,w) <= r
proof
  thus w in cl_Ball(x,r) implies dist(x,w) <= r
  proof
    assume w in cl_Ball(x,r);
    then ||.x - w.|| <= r by Th47;
    hence thesis by CSSPACE:def 16;
  end;
  assume dist(x,w) <= r;
  then ||.x - w.|| <= r by CSSPACE:def 16;
  hence thesis;
end;
