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) & r <= q implies y in Ball(x,q)
proof
  assume that
A1: y in Ball(x,r) and
A2: r <= q;
  ||.x - y.|| < r by A1,Th40;
  then ||.x - y.|| < q by A2,XXREAL_0:2;
  hence thesis;
end;
