theorem Th115:
  S is convergent implies S - x is convergent
proof
  assume S is convergent;
  then consider g such that
A1: for r st 0 < r ex m st for n st m <= n holds ||.(S.n) - g.|| < r;
  take h = g - x;
  let r;
  assume 0 < r;
  then consider m1 such that
A2: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1;
  take k = m1;
  let n;
  assume k <= n;
  then
A3: ||.(S.n) - g.|| < r by A2;
  ||.(S.n) - g.|| = ||.((S.n) - 09(CNS)) - g.|| by RLVECT_1:13
    .= ||.((S.n) - (x - x)) - g.|| by RLVECT_1:15
    .= ||.(((S.n) - x) + x) - g.|| by RLVECT_1:29
    .= ||.((S.n) - x) + ((-g) + x).|| by RLVECT_1:def 3
    .= ||.((S.n) - x) - h.|| by RLVECT_1:33;
  hence thesis by A3,NORMSP_1:def 4;
end;
