reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem
  seq1 is convergent & lim seq1 = g1 & seq2 is convergent & lim seq2 =
g2 implies ||.seq1 + seq2.|| is convergent & lim ||.seq1 + seq2.|| = ||.g1 + g2
  .||
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 Th21;
end;
