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 Th17:
  seq is convergent implies lim (seq + x) = (lim seq) + x
proof
  set g = lim seq;
  assume
A1: seq is convergent;
A2: now
    let r;
    assume r > 0;
    then consider m1 such that
A3: for n st n >= m1 holds dist((seq.n) , g) < r by A1,Def2;
    take k = m1;
    let n;
    assume n >= k;
    then
A4: dist((seq.n) , g) < r by A3;
    dist((seq.n) , g) = dist((seq.n) - (-x) , g - (-x)) by BHSP_1:42
      .= dist((seq.n) + (-(-x)) , g - (-x)) by RLVECT_1:def 11
      .= dist((seq.n) + x, g - (-x)) by RLVECT_1:17
      .= dist((seq.n) + x, g + (-(-x))) by RLVECT_1:def 11
      .= dist((seq.n) + x, g + x) by RLVECT_1:17;
    hence dist((seq + x).n, (g + x)) < r by A4,BHSP_1:def 6;
  end;
  seq + x is convergent by A1,Th7;
  hence thesis by A2,Def2;
end;
