reserve X for ComplexUnitarySpace;
reserve x, y, w, g, g1, g2 for Point of X;
reserve z for Complex;
reserve p, q, r, M, M1, M2 for Real;
reserve seq, seq1, seq2, seq3 for sequence of X;
reserve k,n,m for Nat;
reserve Nseq for increasing sequence of NAT;

theorem Th15:
  seq is convergent implies lim (z * seq) = z * (lim seq)
proof
  set g = lim seq;
  set h = z * g;
A1: now
    set m1 = the Nat;
    assume
A2: z = 0;
    let r;
    assume
A3: r > 0;
    take k = m1;
    let n;
    assume n >= k;
    dist(z*(seq.n), z*g) = dist(0c*(seq.n), 09(X)) by A2,CLVECT_1:1
      .= dist(09(X), 09(X)) by CLVECT_1:1
      .= 0 by CSSPACE:50;
    hence dist((z * seq).n, h) < r by A3,CLVECT_1:def 14;
  end;
  assume
A4: seq is convergent;
A5: now
A6: 0/|.z.| =0;
    assume
A7: z <> 0;
    then
A8: |.z.| > 0 by COMPLEX1:47;
    let r;
    assume r > 0;
    then r/|.z.| > 0 by A8,A6,XREAL_1:74;
    then consider m1 be Nat such that
A9: for n st n >= m1 holds dist((seq.n),g) < r/|.z.| by A4,Def2;
    take k = m1;
    let n;
    assume n >= k;
    then
A10: dist((seq.n) , g) < r/|.z.| by A9;
A11: |.z.| <> 0 by A7,COMPLEX1:47;
A12: |.z.| * (r/|.z.|) = |.z.| * (|.z.|" * r) by XCMPLX_0:def 9
      .= |.z.| *|.z.|" * r
      .= 1 * r by A11,XCMPLX_0:def 7
      .= r;
    dist(z*(seq.n), z*g) = ||.(z*(seq.n)) - (z*g).|| by CSSPACE:def 16
      .= ||.z * ((seq.n) - g).|| by CLVECT_1:9
      .= |.z.| * ||.(seq.n) - g.|| by CSSPACE:43
      .= |.z.| * dist((seq.n) , g) by CSSPACE:def 16;
    then dist(z * (seq.n) , h) < r by A8,A10,A12,XREAL_1:68;
    hence dist((z * seq).n, h) < r by CLVECT_1:def 14;
  end;
  z * seq is convergent by A4,Th5;
  hence thesis by A1,A5,Def2;
end;
