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
  seq is convergent & lim seq = g implies ||.a * seq.|| is convergent &
  lim ||.a * seq.|| = ||.a * g.||
proof
  assume seq is convergent & lim seq = g;
  then a * seq is convergent & lim (a * seq) = a * g by Th5,Th15;
  hence thesis by Th21;
end;
