reserve a, b, r, M2 for Real;
reserve Rseq,Rseq1,Rseq2 for Real_Sequence;
reserve k, n, m, m1, m2 for Nat;
reserve X for RealUnitarySpace;
reserve g for Point of X;
reserve seq, seq1, seq2 for sequence of X;

theorem
  Rseq is bounded & seq is bounded implies Rseq * seq is bounded
proof
  assume that
A1: Rseq is bounded and
A2: seq is bounded;
  consider M1 being Real such that
A3: M1 > 0 and
A4: for n being Nat holds |.Rseq.n.| < M1 by A1,SEQ_2:3;
  consider M2 such that
A5: M2 > 0 and
A6: for n holds ||.seq.n.|| <= M2 by A2,BHSP_3:def 3;
  now
    reconsider M = M1 * M2 as Real;
    take M;
    now
      let n;
      |.Rseq.n.| >= 0 by COMPLEX1:46;
      then
A7:   |.Rseq.n.| * ||.seq.n.|| <= |.Rseq.n.| * M2 by A6,XREAL_1:64;
      |.Rseq.n.| <= M1 by A4;
      then
A8:   |.Rseq.n.| * M2 <= M1 * M2 by A5,XREAL_1:64;
      ||.(Rseq * seq).n.|| = ||.Rseq.n * seq.n.|| by Def7
        .= |.Rseq.n.| * ||.seq.n.|| by BHSP_1:27;
      hence ||.(Rseq * seq).n.|| <= M by A7,A8,XXREAL_0:2;
    end;
    hence M > 0 & for n holds ||.(Rseq * seq).n.|| <= M by A3,A5,XREAL_1:129;
  end;
  hence thesis by BHSP_3:def 3;
end;
