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
  S is convergent implies lim (S - x) = (lim S) - x
proof
  set g = lim S;
  set h = g - x;
  assume
A1: S is convergent;
A2: now
    let r;
    assume 0 < r;
    then consider m1 such that
A3: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,Def7;
    take k = m1;
    let n;
    assume k <= n; then
A4: ||.(S.n) - g.|| < r by A3;
    ||.(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 ||.(S - x).n - h.|| < r by A4,Def4;
  end;
  S - x is convergent by A1,Th21;
  hence thesis by A2,Def7;
end;
