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 (a * S) = a * (lim S)
proof
  set g = lim S;
  set h = a * g;
  assume
A1: S is convergent;
A2: now
    assume
A3: a = 0;
    let r;
    assume 0 < r;
    then consider m1 such that
A4: for n st m1 <= n holds ||.(S.n) - g.|| < r by A1,Def7;
    take k = m1;
    let n;
    assume k <= n; then
A5: ||.(S.n) - g.|| < r by A4;
    ||.a * (S.n) - a * g.|| = ||.0 * (S.n) - 09(RNS).|| by A3,RLVECT_1:10
      .= ||.09(RNS) - 09(RNS).|| by RLVECT_1:10
      .= ||.09(RNS).||
      .= 0;
    then ||.a * (S.n) - h.|| < r by A5;
    hence ||.(a * S).n - h.|| < r by Def5;
  end;
A6: now
    assume
A7: a <> 0; then
A8: 0 < |.a.| by COMPLEX1:47;
    let r;
    assume 0 < r;
    then 0 < r/|.a.| by A8;
    then consider m1 such that
A9: for n st m1 <= n holds ||.(S.n) - g.|| < r/|.a.| by A1,Def7;
    take k = m1;
    let n;
    assume k <= n; then
A10: ||.(S.n) - g.|| < r/|.a.| by A9;
A11: 0 <> |.a.| by A7,COMPLEX1:47;
A12: |.a.| * (r/|.a.|) = |.a.| * (|.a.|" * r) by XCMPLX_0:def 9
      .= |.a.| *|.a.|" * r
      .= 1 * r by A11,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 A8,A10,A12,XREAL_1:68;
    hence ||.(a * S).n - h.|| < r by Def5;
  end;
  a * S is convergent by A1,Th22;
  hence thesis by A2,A6,Def7;
end;
