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
  for X being RealHilbertSpace, seq being sequence of X
   holds seq is Cauchy & Rseq is Cauchy implies Rseq *
  seq is Cauchy
proof
 let X be RealHilbertSpace, seq be sequence of X;
  assume that
A1: seq is Cauchy and
A2: Rseq is Cauchy;
  now
    let r be Real;
    assume
A3: r > 0;
    consider k such that
A4: for n, m st n >= k & m >= k holds |.(Rseq.n - Rseq.m).| < r by A2,A3;
     reconsider k as Nat;
    take k;
    let n be Nat;
    thus n >= k implies |.(Rseq.n - Rseq.k).| < r by A4;
  end;
  then
A5: Rseq is convergent by SEQ_4:41;
   Rseq * seq is convergent by A5,Th48,A1,BHSP_3:def 4;
  hence thesis;
end;
