reserve a, b for Real;
reserve RNS for RealNormSpace;
reserve x, y, z, g, g1, g2 for Point of RNS;
reserve S, S1, S2 for sequence of RNS;
reserve k, n, m, m1, m2 for Nat;
reserve r for Real;
reserve f for Function;
reserve d, s, t for set;

theorem Th21:
  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(RNS)) - g.||
    .= ||.((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,Def4;
end;
