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 convergent & seq is convergent implies Rseq * seq is
  convergent & lim (Rseq * seq) = lim Rseq * lim seq
proof
  assume that
A1: Rseq is convergent and
A2: seq is convergent;
   consider b being Real such that
A3: b > 0 and
A4: for n being Nat holds |.Rseq.n.| < b by A1,SEQ_2:3,13;
  reconsider b as Real;
A5: b + ||.lim seq.|| > 0 + 0 by A3,BHSP_1:28,XREAL_1:8;
A6: ||.lim seq.|| >= 0 by BHSP_1:28;
A7: now
    let r;
    assume r > 0;
    then
A8: r/(b + ||.lim seq.||) > 0 by A5,XREAL_1:139;
    then consider m1 being Nat  such that
A9: for n being Nat st n >= m1
   holds |.Rseq.n - lim Rseq.| < r/(b + ||.lim seq
    .||) by A1,SEQ_2:def 7;
    consider m2 being Nat such that
A10: for n being Nat
  st n >= m2 holds dist(seq.n, lim seq) < r/(b + ||.lim seq
    .||) by A2,A8,BHSP_2:def 2;
    take m = m1 + m2;
    let n be Nat such that
A11: n >= m;
    m1 + m2 >= m1 by NAT_1:12;
    then n >= m1 by A11,XXREAL_0:2;
    then |.Rseq.n - lim Rseq.| <= r/(b + ||.lim seq.||) by A9;
    then
A12: ||.lim seq.|| * |.Rseq.n - lim Rseq.| <= ||.lim seq.|| * (r/(b + ||.
    lim seq.||)) by A6,XREAL_1:64;
A13: ||.seq.n - lim seq.|| >= 0 by BHSP_1:28;
    m >= m2 by NAT_1:12;
    then n >= m2 by A11,XXREAL_0:2;
    then dist(seq.n, lim seq) < r/(b + ||.lim seq.||) by A10;
    then
A14: ||.seq.n - lim seq.|| < r/(b + ||.lim seq.||) by BHSP_1:def 5;
    |.Rseq.n.| < b & |.Rseq.n.| >= 0 by A4,COMPLEX1:46;
    then |.Rseq.n.| * ||.seq.n - lim seq.|| < b * (r/(b + ||.lim seq.||)) by
A14,A13,XREAL_1:96;
    then |.Rseq.n.| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.Rseq.n -
lim Rseq.| <
  b * (r/(b + ||.lim seq.||)) + ||.lim seq.|| * (r/(b + ||.lim seq.||
    )) by A12,XREAL_1:8;
    then |.Rseq.n.| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.Rseq.n -
lim Rseq.| <
  (b * r)/(b + ||.lim seq.||) + ||.lim seq.|| * (r/(b + ||.lim seq.||
    )) by XCMPLX_1:74;
    then |.Rseq.n.| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.Rseq.n -
lim Rseq.| < (b * r)/(b + ||.lim seq.||) + (||.lim seq.|| * r)/(b + ||.lim seq
    .||) by XCMPLX_1:74;
    then |.Rseq.n.| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.Rseq.n -
    lim Rseq.| <
  (b * r + ||.lim seq.|| * r)/(b + ||.lim seq.||) by XCMPLX_1:62;
    then |.Rseq.n.| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.Rseq.n -
    lim Rseq.| < ((b + ||.lim seq.||) * r)/(b + ||.lim seq.||);
    then
A15: |.Rseq.n.| * ||.seq.n - lim seq.|| + ||.lim seq.|| * |.Rseq.n -
    lim Rseq.| < r by A5,XCMPLX_1:89;
    ||.(Rseq * seq).n - lim Rseq * lim seq.|| = ||.Rseq.n * seq.n - lim
    Rseq * lim seq.|| by Def7
      .= ||.(Rseq.n * seq.n - lim Rseq * lim seq) + 09(X).||
      .= ||.(Rseq.n * seq.n - lim Rseq * lim seq) + (Rseq.n * lim seq - Rseq
    .n * lim seq).|| by RLVECT_1:15
      .= ||.Rseq.n * seq.n - (lim Rseq * lim seq - (Rseq.n * lim seq - Rseq.
    n * lim seq)).|| by RLVECT_1:29
      .= ||.Rseq.n * seq.n - (Rseq.n * lim seq + (lim Rseq * lim seq - Rseq.
    n * lim seq)).|| by RLVECT_1:29
      .= ||.(Rseq.n * seq.n - Rseq.n * lim seq) - (lim Rseq * lim seq - Rseq
    .n * lim seq).|| by RLVECT_1:27
      .= ||.(Rseq.n * seq.n - Rseq.n * lim seq) + (Rseq.n * lim seq - lim
    Rseq * lim seq).|| by RLVECT_1:33;
    then
    ||.(Rseq * seq).n - lim Rseq * lim seq.|| <= ||.Rseq.n * seq.n - Rseq
    .n * lim seq.|| + ||.Rseq.n * lim seq - lim Rseq * lim seq.|| by BHSP_1:30;
    then
    ||.(Rseq * seq).n - lim Rseq * lim seq.|| <= ||.Rseq.n * (seq.n - lim
    seq).|| + ||.Rseq.n * lim seq - lim Rseq * lim seq.|| by RLVECT_1:34;
    then
    ||.(Rseq * seq).n - lim Rseq * lim seq.|| <= ||.Rseq.n * (seq.n - lim
    seq).|| + ||.(Rseq.n - lim Rseq) * lim seq.|| by RLVECT_1:35;
    then
    ||.(Rseq * seq).n - lim Rseq * lim seq.|| <= |.Rseq.n.| * ||.seq.n -
    lim seq.|| + ||.(Rseq.n - lim Rseq) * lim seq.|| by BHSP_1:27;
    then
    ||.(Rseq * seq).n - lim Rseq * lim seq.|| <= |.Rseq.n.| * ||.seq.n -
    lim seq.|| + ||.lim seq.|| * |.Rseq.n - lim Rseq.| by BHSP_1:27;
    then ||.(Rseq * seq).n - lim Rseq * lim seq.|| < r by A15,XXREAL_0:2;
    hence dist((Rseq * seq).n, (lim Rseq * lim seq)) < r by BHSP_1:def 5;
  end;
  Rseq * seq is convergent by A1,A2,Th48;
  hence thesis by A7,BHSP_2:def 2;
end;
