reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem
  y in Ball(x,r) implies (y - x) in Ball (0.(X),r)
proof
  assume y in Ball(x,r);
  then ||.x - y.|| < r by Th40;
  then ||.(-y) + x.|| < r by RLVECT_1:def 11;
  then ||.-(y - x).|| < r by RLVECT_1:33;
  then ||.09(X) - (y - x).|| < r by RLVECT_1:14;
  hence thesis;
end;
