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 ||.(seq - x) - (g - x).|| is
  convergent & lim ||.(seq - x) - (g - x).|| = 0
proof
  assume
A1: seq is convergent & lim seq = g;
  then lim ||.(seq + (-x)) - (g + (-x)).|| = 0 by Th33;
  then
A2: lim ||.(seq - x) - (g + (-x)).|| = 0 by BHSP_1:56;
  ||.(seq + (-x)) - (g + (-x)).|| is convergent by A1,Th33;
  then ||.(seq - x) - (g + (-x)).|| is convergent by BHSP_1:56;
  hence thesis by A2,RLVECT_1:def 11;
end;
