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
  seq is bounded implies z * seq is bounded
proof
  assume seq is bounded;
  then consider M such that
A1: M > 0 and
A2: for n holds ||.seq.n.|| <= M;
A3: z <> 0 implies z * seq is bounded
  proof
A4: now
      let n;
A5:   |.z.| >= 0 by COMPLEX1:46;
      ||.(z * seq).n.|| = ||.z * (seq.n).|| by CLVECT_1:def 14
        .= |.z.| * ||.seq.n.|| by CSSPACE:43;
      hence ||.(z * seq).n.|| <= |.z.| * M by A2,A5,XREAL_1:64;
    end;
    assume z <> 0;
    then |.z.| > 0 by COMPLEX1:47;
    then |.z.| * M > 0 by A1,XREAL_1:129;
    hence thesis by A4;
  end;
  z = 0 implies z * seq is bounded
  proof
    assume
A6: z = 0;
    now
      let n;
      ||.(z * seq).n.|| = ||.0c * (seq.n).|| by A6,CLVECT_1:def 14
        .= ||.09(X).|| by CLVECT_1:1
        .= 0 by CSSPACE:42;
      hence ||.(z * seq).n.|| <= M by A1;
    end;
    hence thesis by A1;
  end;
  hence thesis by A3;
end;
