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
  seq is convergent & lim seq = g implies ||.seq - x.|| is convergent &
  lim ||.seq - x.|| = ||.g - x.||
proof
  assume
A1: seq is convergent & lim seq = g;
  then lim ||.seq + (-x).|| = ||.g + (-x).|| by Lm1;
  then
A2: lim ||.seq - x.|| = ||.g + (-x).|| by CSSPACE:71;
  ||.seq + (-x).|| is convergent by A1,Lm1;
  hence thesis by A2,CSSPACE:71,RLVECT_1:def 11;
end;
