reserve X for RealUnitarySpace;
reserve x, y, z, g, g1, g2 for Point of X;
reserve a, q, r for Real;
reserve seq, seq1, seq2, seq9 for sequence of X;
reserve k, n, m, m1, m2 for Nat;

theorem Th5:
  seq is convergent implies a * seq is convergent
proof
  assume seq is convergent;
  then consider g such that
A1: for r st r > 0 ex m st for n st n >= m holds dist((seq.n) , g) < r;
  take h = a * g;
A2: now
A3: 0/|.a.| = 0;
    assume
A4: a <> 0;
    then
A5: |.a.| > 0 by COMPLEX1:47;
    let r;
    assume r > 0;
    then consider m1 such that
A6: for n st n >= m1 holds dist(seq.n, g) < r/|.a.| by A1,A5,A3,XREAL_1:74;
    take k = m1;
    let n;
    assume n >= k;
    then
A7: dist((seq.n) , g) < r/|.a.| by A6;
A8: |.a.| <> 0 by A4,COMPLEX1:47;
A9: |.a.| * (r/|.a.|) = |.a.| * (|.a.|" * r) by XCMPLX_0:def 9
      .= |.a.| *|.a.|" * r
      .= 1 * r by A8,XCMPLX_0:def 7
      .= r;
    dist(a * (seq.n) , a * g) = ||.(a * (seq.n)) - (a * g).|| by BHSP_1:def 5
      .= ||.a * ((seq.n) - g).|| by RLVECT_1:34
      .= |.a.| * ||.(seq.n) - g.|| by BHSP_1:27
      .= |.a.| * dist((seq.n) , g) by BHSP_1:def 5;
    then dist((a *(seq.n)) , h) < r by A5,A7,A9,XREAL_1:68;
    hence dist((a * seq).n, h) < r by NORMSP_1:def 5;
  end;
  now
    assume
A10: a = 0;
    let r;
    assume r > 0;
    then consider m1 such that
A11: for n st n >= m1 holds dist((seq.n) , g) < r by A1;
    take k = m1;
    let n;
    assume n >= k;
    then
A12: dist((seq.n) , g) < r by A11;
    dist(a * (seq.n) , a * g) = dist(0 * (seq.n) , 09(X)) by A10,RLVECT_1:10
      .= dist(09(X) , 09(X)) by RLVECT_1:10
      .= 0 by BHSP_1:34;
    then dist(a * (seq.n) , h) < r by A12,BHSP_1:37;
    hence dist((a * seq).n, h) < r by NORMSP_1:def 5;
  end;
  hence thesis by A2;
end;
