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
  seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies ||.(seq1 + seq2) - (g1 + g2).|| is convergent & lim ||.(seq1 + seq2)
  - (g1 + g2).|| = 0
proof
  assume
  seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 = g2;
  then seq1 + seq2 is convergent & lim (seq1 + seq2) = g1 + g2 by Th3,Th13;
  hence thesis by Th22;
end;
