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 Th15:
  seq is convergent implies lim (a * seq) = a * (lim seq)
proof
  set g = lim seq;
  set h = a * g;
  assume
A1: seq is convergent;
A2: now
    assume
A3: a = 0;
    let r;
    assume r > 0;
    then consider m1 such that
A4: for n st n >= m1 holds dist((seq.n) , g) < r by A1,Def2;
    take k = m1;
    let n;
    assume n >= k;
    then
A5: dist((seq.n) , g) < r by A4;
    dist(a * (seq.n) , a * g) = dist(0 * (seq.n) , 09(X)) by A3,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 A5,BHSP_1:37;
    hence dist((a * seq).n, h) < r by NORMSP_1:def 5;
  end;
A6: now
A7: 0/|.a.| =0;
    assume
A8: a <> 0;
    then
A9: |.a.| > 0 by COMPLEX1:47;
    let r;
    assume r > 0;
    then r/|.a.| > 0 by A9,A7,XREAL_1:74;
    then consider m1 such that
A10: for n st n >= m1 holds dist((seq.n) , g) < r/|.a.| by A1,Def2;
    take k = m1;
    let n;
    assume n >= k;
    then
A11: dist((seq.n) , g) < r/|.a.| by A10;
A12: |.a.| <> 0 by A8,COMPLEX1:47;
A13: |.a.| * (r/|.a.|) = |.a.| * (|.a.|" * r) by XCMPLX_0:def 9
      .= |.a.| *|.a.|" * r
      .= 1 * r by A12,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 A9,A11,A13,XREAL_1:68;
    hence dist((a * seq).n, h) < r by NORMSP_1:def 5;
  end;
  a * seq is convergent by A1,Th5;
  hence thesis by A2,A6,Def2;
end;
