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.|| 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 BHSP_1:56;
  ||.seq + (-x).|| is convergent by A1,Lm1;
  hence thesis by A2,BHSP_1:56,RLVECT_1:def 11;
end;
