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 Th47:
  z in cl_Ball(x,r) iff ||.x - z.|| <= r
proof
  thus z in cl_Ball(x,r) implies ||.x - z.|| <= r
  proof
    assume z in cl_Ball(x,r);
    then ex y be Point of X st z = y & ||.x - y.|| <= r;
    hence thesis;
  end;
  assume ||.x - z.|| <= r;
  hence thesis;
end;
