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 Th22:
  S is convergent implies a * S 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 = a * g;
A2: now
    assume
A3: a <> 0; then
A4: 0 < |.a.| by COMPLEX1:47;
    let r;
    assume 0 < r;
    then consider m1 such that
A5: for n st m1 <= n holds ||.(S.n) - g.|| < r/|.a.| by A1,A4;
    take k = m1;
    let n;
    assume k <= n; then
A6: ||.(S.n) - g.|| < r/|.a.| by A5;
A7: 0 <> |.a.| by A3,COMPLEX1:47;
A8: |.a.| * (r/|.a.|) = |.a.| * (|.a.|" * r) by XCMPLX_0:def 9
      .= |.a.| *|.a.|" * r
      .= 1 * r by A7,XCMPLX_0:def 7
      .= r;
    ||.(a * (S.n)) - (a * g).|| = ||.a * ((S.n) - g).|| by RLVECT_1:34
      .= |.a.| * ||.(S.n) - g.|| by Def1;
    then ||.(a *(S.n)) - h.|| < r by A4,A6,A8,XREAL_1:68;
    hence ||.(a * S).n - h.|| < r by Def5;
  end;
  now
    assume
A9: a = 0;
    let r;
    assume 0 < r;
    then consider m1 such that
A10: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1;
    take k = m1;
    let n;
    assume k <= n; then
A11: ||.(S.n) - g.|| < r by A10;
    ||.a * (S.n) - a * g.|| = ||.0 * (S.n) - 09(RNS).|| by A9,RLVECT_1:10
      .= ||.09(RNS) - 09(RNS).|| by RLVECT_1:10
      .= ||.09(RNS).||
      .= 0;
    then ||.a * (S.n) - h.|| < r by A11;
    hence ||.(a * S).n - h.|| < r by Def5;
  end;
  hence thesis by A2;
end;
