reserve X for ComplexUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;
reserve Rseq for Real_Sequence;
reserve Cseq,Cseq1,Cseq2 for Complex_Sequence;
reserve z,z1,z2 for Complex;
reserve r for Real;
reserve k,n,m for Nat;

theorem
  Cseq is bounded & seq is bounded implies Cseq * seq is bounded
proof
  assume that
A1: Cseq is bounded and
A2: seq is bounded;
  consider M1 being Real such that
A3: M1 > 0 and
A4: for n holds |.(Cseq.n).| < M1 by A1,COMSEQ_2:8;
  consider M2 be Real such that
A5: M2 > 0 and
A6: for n holds ||.seq.n.|| <= M2 by A2,CLVECT_2:def 10;
  now
    set M = M1 * M2;
    take M;
    now
      let n;
      |.(Cseq.n).| >= 0 by COMPLEX1:46;
      then
A7:   |.(Cseq.n).| * ||.seq.n.|| <= |.(Cseq.n).| * M2 by A6,XREAL_1:64;
      |.(Cseq.n).| <= M1 by A4;
      then
A8:   |.(Cseq.n).| * M2 <= M1 * M2 by A5,XREAL_1:64;
      ||.(Cseq * seq).n.|| = ||.Cseq.n * seq.n.|| by Def8
        .= |.(Cseq.n).| * ||.seq.n.|| by CSSPACE:43;
      hence ||.(Cseq * seq).n.|| <= M by A7,A8,XXREAL_0:2;
    end;
    hence M > 0 & for n holds ||.(Cseq * seq).n.|| <= M by A3,A5;
  end;
  hence thesis by CLVECT_2:def 10;
end;
