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) - (a * g).|| is
  convergent & lim ||.(a * seq) - (a * g).|| = 0
proof
  assume seq is convergent & lim seq = g;
  then a * seq is convergent & lim (a * seq) = a * g by Th5,Th15;
  hence thesis by Th22;
end;
