reserve x, x1, x2, y, X, D for set,
  i, j, k, l, m, n, N for Nat,
  p, q for XFinSequence of NAT,
  q9 for XFinSequence,
  pd, qd for XFinSequence of D;
reserve pN, qN for Element of NAT^omega;
reserve seq1,seq2,seq3,seq4 for Real_Sequence,
  r,s,e for Real,
  Fr,Fr1, Fr2 for XFinSequence of REAL;

theorem Th49:
  for seq1, seq2, n st seq2 is summable ex Fr st Partial_Sums(seq1
(##) seq2).n= Sum seq2 * Partial_Sums(seq1).n- Sum Fr & dom Fr=n+1 & for i st i
  in n+1 holds Fr.i=seq1.i * Sum (seq2^\(n-'i+1))
proof
  let seq1, seq2, n such that
A1: seq2 is summable;
  defpred Q[set,set] means for i st i=$1 holds $2=seq1.i * Sum (seq2^\(n-'i+1)
  );
  set P2=Partial_Sums(seq2);
  set P1=Partial_Sums(seq1);
  set S=seq1 (##) seq2;
A2: for i st i in Segm(n+1) ex x be Element of REAL st Q[i,x]
  proof
    let i such that
    i in Segm(n+1);
     reconsider ss = seq1.i * Sum (seq2^\(n-'i+1)) as Element of REAL
      by XREAL_0:def 1;
    take ss;
    thus thesis;
  end;
  consider Fr such that
A3: dom Fr = Segm(n+1) and
A4: for i st i in Segm(n+1) holds Q[i,Fr.i] from STIRL2_1:sch 5(A2);
  consider Fr1 such that
A5: Partial_Sums(S).n = Sum Fr1 and
A6: dom Fr1=n+1 and
A7: for i st i in n+1 holds Fr1.i=seq1.i * P2.(n-'i) by Th48;
A8: 0 in Segm(n+1) by NAT_1:44;
  then
A9: Fr1.0=seq1.0 * P2.(n-'0) &
    Sum (Fr1|(zz+1))=Fr1.0 + Sum (Fr1|zz) by A6,A7,AFINSQ_2:65;
  defpred P[Nat] means $1+1 <= n+1 implies Sum (Fr1|($1+1)) + Sum (
  Fr|($1+1))= Sum seq2 * P1.$1;
A10: for k st P[k] holds P[k+1]
  proof
    let k such that
A11: P[k];
    reconsider k1=k+1 as Nat;
    assume
 A12:    k+1+1<=n+1;
    then k1<n+1 by NAT_1:13;
    then
A13: k1 in Segm(n+1) by NAT_1:44;
    then
A14: Fr.k1=seq1.k1*Sum(seq2^\(n-'k1+1)) & Sum(Fr1|(k1+1))=Fr1.k1+Sum (Fr1|
    k1) by A4,A6,AFINSQ_2:65;
A15:Sum (Fr1|k1) + Sum (Fr|k1)= Sum seq2 * P1.k by A12,A11,NAT_1:13;
    Sum(Fr|(k1+1))=Fr.k1 + Sum (Fr|k1) & Fr1.k1=seq1.k1 * P2.(n-'k1) by A3,A7
,A13,AFINSQ_2:65;
    then Sum(Fr|(k1+1)) + Sum(Fr1|(k1+1))
       =seq1.k1 * ( Sum(seq2^\(n-'k1+1)) + P2.(n-'k1) )
      + Sum seq2 * P1.k by A15,A14
      .=seq1.k1*Sum seq2 +Sum seq2 * P1.k by A1,SERIES_1:15
      .=Sum seq2*(P1.k+seq1.k1)
      .=P1.k1*Sum seq2 by SERIES_1:def 1;
    hence thesis;
  end;
A16: Sum (Fr|zz) = 0;
A17: Sum (Fr1|zz) = 0;
A18: Sum (Fr|(zz+1))=Fr.0 + Sum (Fr|zz) &
  Fr.0= seq1.0 * Sum (seq2^\(n-'0+1)) by A3,A4,A8,AFINSQ_2:65;
  then Sum (Fr|(zz+1))+Sum (Fr1|(zz+1))
       = Fr.0 + Sum (Fr|zz)+Sum (Fr1|(zz+1))
      .= Fr.0 + Sum (Fr1|(zz+1)) by A16
      .= seq1.0 * Sum (seq2^\(n-'0+1)) + Sum (Fr1|(zz+1)) by A18
      .= seq1.0 * Sum (seq2^\(n-'0+1)) +
              Fr1.0 + Sum (Fr1|zz) by A9
      .= seq1.0 * Sum (seq2^\(n-'0+1)) +
              seq1.0 * P2.(n-'0) + Sum (Fr1|zz) by A9
      .= seq1.0*(Sum(seq2^\(n-'0+1))+P2.(n-'0)) by A17
    .=seq1.0*Sum seq2 by A1,SERIES_1:15;
  then
A19: P[0] by SERIES_1:def 1;
A20: for k holds P[k] from NAT_1:sch 2(A19,A10);
  take Fr;
A21: Fr1|(n+1)=Fr1 by A6;
  Fr|(n+1)=Fr by A3;
  then Sum Fr1 + Sum Fr= Sum seq2 * P1.n by A20,A21;
  hence thesis by A3,A4,A5;
end;
