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 Th61:
  seq is Cauchy implies z * seq is Cauchy
proof
  assume
A1: seq is Cauchy;
A2: now
A3: 0/|.z.| = 0;
    assume
A4: z <> 0;
    then
A5: |.z.| > 0 by COMPLEX1:47;
    let r;
    assume r > 0;
    then r/|.z.| > 0 by A5,A3,XREAL_1:74;
    then consider m1 be Nat such that
A6: for n, m st n >= m1 & m >= m1 holds dist((seq.n), (seq.m)) < r/|.
    z.| by A1;
    take k = m1;
    let n, m;
    assume n >= k & m >= k;
    then
A7: dist((seq.n), (seq.m)) < r/|.z.| by A6;
A8: |.z.| <> 0 by A4,COMPLEX1:47;
A9: |.z.| * (r/|.z.|) = |.z.| * (|.z.|" * r) by XCMPLX_0:def 9
      .= |.z.| *|.z.|" * r
      .= 1 * r by A8,XCMPLX_0:def 7
      .= r;
    dist(z * (seq.n), z * (seq.m)) = ||.(z * (seq.n)) - (z * (seq.m)).||
    by CSSPACE:def 16
      .= ||.z * ((seq.n) - (seq.m)).|| by CLVECT_1:9
      .= |.z.| * ||.(seq.n) - (seq.m).|| by CSSPACE:43
      .= |.z.| * dist((seq.n), (seq.m)) by CSSPACE:def 16;
    then dist((z * (seq.n)), (z * (seq.m))) < r by A5,A7,A9,XREAL_1:68;
    then dist((z * seq).n, (z * (seq.m))) < r by CLVECT_1:def 14;
    hence dist((z * seq).n, (z * seq).m) < r by CLVECT_1:def 14;
  end;
  now
    assume
A10: z = 0;
    let r;
    assume r > 0;
    then consider m1 be Nat such that
A11: for n, m st n >= m1 & m >= m1 holds dist((seq.n), (seq.m)) < r by A1;
    take k = m1;
    let n, m;
    assume n >= k & m >= k;
    then
A12: dist((seq.n), (seq.m)) < r by A11;
    dist(z * (seq.n), z * (seq.m)) = dist(09(X), 0c * (seq.m)) by A10,
CLVECT_1:1
      .= dist(09(X), 09(X)) by CLVECT_1:1
      .= 0 by CSSPACE:50;
    then dist(z * (seq.n), z * (seq.m)) < r by A12,CSSPACE:53;
    then dist((z * seq).n, z * (seq.m)) < r by CLVECT_1:def 14;
    hence dist((z * seq).n, (z * seq).m) < r by CLVECT_1:def 14;
  end;
  hence thesis by A2;
end;
