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
  S1 is convergent & S2 is convergent implies lim (S1 + S2) = (lim S1) +
  (lim S2)
proof
  assume that
A1: S1 is convergent and
A2: S2 is convergent;
  set g2 = lim S2;
  set g1 = lim S1;
  set g = g1 + g2;
A3: now
    let r;
    assume 0 < r; then
A4: 0< r/2;
    then consider m1 such that
A5: for n st m1 <= n holds ||.(S1.n) - g1.|| < r/2 by A1,Def7;
    consider m2 such that
A6: for n st m2 <= n holds ||.(S2.n) - g2.|| < r/2 by A2,A4,Def7;
    take k = m1 + m2;
    let n such that
A7: k <= n;
    m2 <= k by NAT_1:12;
    then m2 <= n by A7,XXREAL_0:2;
    then
A8: ||.(S2.n) - g2.|| < r/2 by A6;
A9: ||.(S1 + S2).n - g.|| = ||.-(g1+g2) + ((S1.n) + (S2.n)).|| by Def2
      .= ||.((-g1)+(-g2)) + ((S1.n) + (S2.n)).|| by RLVECT_1:31
      .= ||.(S1.n) + ((-g1)+(-g2)) + (S2.n).|| by RLVECT_1:def 3
      .= ||.(S1.n) - g1 + (-g2) + (S2.n).|| by RLVECT_1:def 3
      .= ||.(S1.n) - g1 + ((S2.n) - g2).|| by RLVECT_1:def 3;
A10: ||.(S1.n) - g1 + ((S2.n) - g2).|| <= ||.(S1.n) - g1.|| + ||.(S2.n) -
    g2.|| by Def1;
    m1 <= m1 + m2 by NAT_1:12;
    then m1 <= n by A7,XXREAL_0:2;
    then ||.(S1.n) - g1.|| < r/2 by A5;
    then ||.(S1.n) - g1.|| + ||.(S2.n) - g2.|| < r/2 + r/2 by A8,XREAL_1:8;
    hence ||.(S1 + S2).n - g.|| < r by A9,A10,XXREAL_0:2;
  end;
  S1 + S2 is convergent by A1,A2,Th19;
  hence thesis by A3,Def7;
end;
