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 Th48:
  Rseq is convergent & seq is convergent implies Rseq * seq is convergent
proof
  assume that
A1: Rseq is convergent and
A2: seq is convergent;
  consider p being Real such that
A3: for r being Real st r > 0
  ex m being Nat  st for n being Nat st n >= m holds |.
  Rseq.n - p.| < r by A1,SEQ_2:def 6;
  consider g such that
A4: for r st r > 0
  ex m being Nat st
   for n being Nat st n >= m holds ||.seq.n - g.|| < r by A2,
BHSP_2:9;
  reconsider p as Real;
  now
    take h = p * g;
    let r;
    consider b being Real such that
A5: b > 0 and
A6: for n being Nat  holds |.Rseq.n.| < b by A1,SEQ_2:3,13;
    reconsider b as Real;
A7: b + ||.g.|| > 0 + 0 by A5,BHSP_1:28,XREAL_1:8;
    assume
A8: r > 0;
    then consider m1 being Nat such that
A9: for n being Nat st n >= m1
        holds |.Rseq.n - p.| < r/(b + ||.g.||) by A3,A7,
XREAL_1:139;
    consider m2 being Nat such that
A10: for n being Nat
     st n >= m2 holds ||.seq.n - g.|| < r/(b + ||.g.||) by A4,A7,A8,
XREAL_1:139;
    reconsider m = m1 + m2 as Nat;
    take m;
    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 ||.g.|| >= 0 & |.Rseq.n - p.| <= r/(b + ||.g.||) by A9,BHSP_1:28;
    then
A12: ||.g.|| * |.Rseq.n - p.| <= ||.g.|| * (r/(b + ||.g.||)) by XREAL_1:64;
A13: |.Rseq.n.| >= 0 & ||.seq.n - g.|| >= 0 by BHSP_1:28,COMPLEX1:46;
    m >= m2 by NAT_1:12;
    then n >= m2 by A11,XXREAL_0:2;
    then
A14: ||.seq.n - g.|| < r/(b + ||.g.||) by A10;
    |.Rseq.n.| < b by A6;
    then |.Rseq.n.| * ||.seq.n - g.|| < b * (r/(b + ||.g.||)) by A14,A13,
XREAL_1:96;
    then
    |.Rseq.n.| * ||.seq.n - g.|| + ||.g.|| * |.Rseq.n - p.| < b * (r/(b
    + ||.g.||)) + ||.g.|| * (r/(b + ||.g.||)) by A12,XREAL_1:8;
    then
    |.Rseq.n.| * ||.seq.n - g.|| + ||.g.|| * |.Rseq.n - p.| < (b * r)/(
    b + ||.g.||) + ||.g.|| * (r/(b + ||.g.||)) by XCMPLX_1:74;
    then
    |.Rseq.n.| * ||.seq.n - g.|| + ||.g.|| * |.Rseq.n - p.| < (b * r)/(
    b + ||.g.||) + (||.g.|| * r)/(b + ||.g.||) by XCMPLX_1:74;
    then |.Rseq.n.| * ||.seq.n - g.|| + ||.g.|| * |.Rseq.n - p.| < (b * r +
    ||.g.|| * r)/(b + ||.g.||) by XCMPLX_1:62;
    then
    |.Rseq.n.| * ||.seq.n - g.|| + ||.g.|| * |.Rseq.n - p.| < ((b + ||.
    g.||) * r)/(b + ||.g.||);
    then
A15: |.Rseq.n.| * ||.seq.n - g.|| + ||.g.|| * |.Rseq.n - p.| < r by A7,
XCMPLX_1:89;
    ||.(Rseq * seq).n - p * g.|| = ||.Rseq.n * seq.n - p * g.|| by Def7
      .= ||.(Rseq.n * seq.n - p * g) + 09(X).||
      .= ||.(Rseq.n * seq.n - p * g) + (Rseq.n * g - Rseq.n * g).|| by
RLVECT_1:15
      .= ||.Rseq.n * seq.n - (p * g - (Rseq.n * g - Rseq.n * g)).|| by
RLVECT_1:29
      .= ||.Rseq.n * seq.n - (Rseq.n * g + (p * g - Rseq.n * g)).|| by
RLVECT_1:29
      .= ||.(Rseq.n * seq.n - Rseq.n * g) - (p * g - Rseq.n * g).|| by
RLVECT_1:27
      .= ||.(Rseq.n * seq.n - Rseq.n * g) + (Rseq.n * g - p * g).|| by
RLVECT_1:33;
    then ||.(Rseq * seq).n - p * g.|| <= ||.Rseq.n * seq.n - Rseq.n * g.|| +
    ||.Rseq.n * g - p * g.|| by BHSP_1:30;
    then
    ||.(Rseq * seq).n - p * g.|| <= ||.Rseq.n * (seq.n - g).|| + ||.Rseq.
    n * g - p * g.|| by RLVECT_1:34;
    then
    ||.(Rseq * seq).n - p * g.|| <= ||.Rseq.n * (seq.n - g).|| + ||.(Rseq
    .n - p) * g.|| by RLVECT_1:35;
    then ||.(Rseq * seq).n - p * g.|| <= |.Rseq.n.| * ||.seq.n - g.|| + ||.(
    Rseq.n - p) * g.|| by BHSP_1:27;
    then ||.(Rseq * seq).n - p * g.|| <= |.Rseq.n.| * ||.seq.n - g.|| + ||.g
    .|| * |.Rseq.n - p.| by BHSP_1:27;
    hence ||.(Rseq * seq).n - h.|| < r by A15,XXREAL_0:2;
  end;
  hence thesis by BHSP_2:9;
end;
