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 Th7:
  seq is convergent implies seq + x is convergent
proof
  assume seq is convergent;
  then consider g such that
A1: for r st r > 0 ex m st for n st n >= m holds dist((seq.n) , g) < r;
  take g + x;
  let r;
  assume r > 0;
  then consider m1 such that
A2: for n st n >= m1 holds dist((seq.n) , g) < r by A1;
  take k = m1;
  let n;
  dist((seq.n) + x, g + x) <= dist((seq.n) , g) + dist(x, x) by BHSP_1:40;
  then
A3: dist((seq.n) + x, g + x) <= dist((seq.n) , g) + 0 by BHSP_1:34;
  assume n >= k;
  then dist((seq.n) , g) < r by A2;
  then dist((seq.n) + x, g + x) < r by A3,XXREAL_0:2;
  hence thesis by BHSP_1:def 6;
end;
