reserve
  x for object, X for set,
  i, n, m for Nat,
  r, s for Real,
  c, c1, c2, d for Complex,
  f, g for complex-valued Function,
  g1 for n-element complex-valued FinSequence,
  f1 for n-element real-valued FinSequence,
  T for non empty TopSpace,
  p for Element of TOP-REAL n;

theorem Th23:
  for p, q being Point of TOP-REAL m holds p in Ball(q,r) iff -p in Ball(-q,r)
  proof
    let p, q be Point of TOP-REAL m;
A1: now
      let a, b be Point of TOP-REAL m;
      assume a in Ball(b,r);
      then
A2:   |.a-b.| < r by TOPREAL9:7;
      -a--b = (-a)+-(-b) .= -(a-b) by RLVECT_1:31;
      then |.-a--b.| = |.a-b.| by EUCLID:71;
      hence -a in Ball(-b,r) by A2,TOPREAL9:7;
    end;
    --p = p & --q = q;
    hence thesis by A1;
  end;
