
theorem Th53: :: TOPREAL6:65
  for V being RealUnitarySpace, M being Subset of TopUnitSpace V,
  v being VECTOR of V, r being Real st M = cl_Ball(v,r) holds M is closed
proof
  let V be RealUnitarySpace;
  let M be Subset of TopUnitSpace V;
  let v be VECTOR of V;
  let r be Real;
  assume
A1: M = cl_Ball(v,r);
  for x being set holds x in M` iff ex Q being Subset of TopUnitSpace V st
  Q is open & Q c= M` & x in Q
  proof
    let x be set;
    thus x in M` implies ex Q being Subset of TopUnitSpace V st Q is open & Q
    c= M` & x in Q
    proof
      assume
A2:   x in M`;
      then reconsider e = x as Point of V;
      reconsider Q = Ball(e,dist(e,v)-r) as Subset of TopUnitSpace V;
      take Q;
      thus Q is open by Th50;
      thus Q c= M`
      proof
        let q be object;
        assume
A3:     q in Q;
        then reconsider f = q as Point of V;
        dist(e,v) <= dist(e,f) + dist(f,v) by BHSP_1:35;
        then
A4:     dist(e,v) - r <= dist(e,f) + dist(f,v) - r by XREAL_1:9;
        dist(e,f) < dist(e,v)-r by A3,BHSP_2:41;
        then dist(e,f) < dist(e,f) + dist(f,v) - r by A4,XXREAL_0:2;
        then dist(e,f) - dist(e,f) < dist(e,f) + dist(f,v) - r - dist(e,f) by
XREAL_1:9;
        then 0 < dist(e,f) - dist(e,f) + dist(f,v) - r;
        then dist(f,v) > r by XREAL_1:47;
        then not q in M by A1,BHSP_2:48;
        hence thesis by A3,SUBSET_1:29;
      end;
      not x in M by A2,XBOOLE_0:def 5;
      then dist(v,e) > r by A1,BHSP_2:48;
      then dist(e,v)-r > r-r by XREAL_1:9;
      hence thesis by BHSP_2:42;
    end;
    thus thesis;
  end;
  then M` is open by TOPS_1:25;
  hence thesis;
end;
