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 - z in Ball(x - z,r)
proof
  assume y in Ball(x,r);
  then
A1: dist(x,y) < r by Th41;
  dist(x - z,y - z) = dist(x,y) by BHSP_1:42;
  hence thesis by A1,Th41;
end;
