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 Th48:
  for seq1, seq2, n ex Fr st Partial_Sums(seq1 (##) seq2).n = Sum
Fr & dom Fr=n+1 & for i st i in n+1 holds Fr.i=seq1.i * Partial_Sums(seq2).(n-'
  i)
proof
  let seq1,seq2,n;
  set S=seq1 (##) seq2;
  set P=Partial_Sums(seq2);
  defpred P[Nat] means ex Fr st Partial_Sums(S).$1 = Sum Fr & dom Fr=$1+1 &
  for i st i in $1+1 holds Fr.i=seq1.i * P.($1-'i);
A1: for n st P[n] holds P[n+1]
  proof
    set A=addreal;
    let n;
    set n1=n+1;
    defpred Q[set,set] means for i st i=$1 holds $2=seq1.i * P.(n1-'i);
A2: n1-'n1=0 & P.0=seq2.0 by SERIES_1:def 1,XREAL_1:232;
A3: for i st i in Segm(n1+1) ex x be Element of REAL st Q[i,x]
    proof
      let i such that
      i in Segm(n1+1);
       reconsider ss = seq1.i * Partial_Sums(seq2).(n1-'i) as Element of REAL;
      take ss;
      thus thesis;
    end;
    consider Fr2 such that
A4: dom Fr2 = Segm(n1+1) and
A5: for i st i in Segm(n1+1) holds Q[i,Fr2.i] from STIRL2_1:sch 5(A3);
    assume P[n];
    then consider Fr such that
A6: Partial_Sums(S).n = Sum Fr and
A7: dom Fr=n1 and
A8: for i st i in n+1 holds Fr.i=seq1.i * P.(n-'i);
    consider Fr1 such that
A9: dom Fr1 = n1+1 and
A10: for i st i in n1+1 holds Fr1.i = seq1.i * seq2.(n1-'i) and
A11: Sum Fr1 = S.n1 by Def4;
A12: Fr1|(n1+1)=Fr1 by A9;
A13: for i be Nat
    st i in dom (Fr2|n1) holds (Fr2|n1).i=addreal.(Fr.i,(Fr1|n1).i)
    proof
      let i be Nat such that
A14:  i in dom (Fr2|n1);
A15:  i in dom Fr2 /\ n1 by A14,RELAT_1:61;
      then i in dom(Fr1|n1) by A9,A4,RELAT_1:61;
      then
A16:  Fr1.i=(Fr1|n1).i by FUNCT_1:47;
A17:  i in Segm n1 by A15,XBOOLE_0:def 4;
      then
A18:  i < n1 by NAT_1:44;
      then i <=n by NAT_1:13;
      then
A19:  n-'i=n-i by XREAL_1:233;
      i in n1+1 & i in NAT by A4,A15,XBOOLE_0:def 4;
      then
A20:  Fr1.i = seq1.i * seq2.(n1-'i) & Fr2.i=seq1.i * P.(n1-'i) by A10,A5;
A21:  Fr2.i=(Fr2|n1).i by A14,FUNCT_1:47;
      n1-'i=n1-i by A18,XREAL_1:233;
      then (n-'i)+1=n1-'i by A19;
      then
A22:  P.(n1-'i)=P.(n-'i)+seq2.(n1-'i) by SERIES_1:def 1;
      Fr.i=seq1.i * P.(n-'i) by A8,A17;
      then Fr2.i=Fr.i+Fr1.i by A20,A22;
      hence thesis by A16,A21,BINOP_2:def 9;
    end;
    n1<=n1+1 by NAT_1:11;
    then
A23: Segm n1 c= Segm(n1+1) by NAT_1:39;
    then
A24: len (Fr1|n1)=len Fr by A7,A9,RELAT_1:62;
    n1 < n1+1 by NAT_1:13;
    then
A25: n1 in Segm(n1+1) by NAT_1:44;
    then
A26: Fr1.n1=seq1.n1 * seq2.(n1-'n1) & Sum(Fr1|(n1+1)) = Fr1.n1 + Sum(Fr1|
    n1) by A9,A10,AFINSQ_2:65;
    len (Fr2|n1)=len Fr by A7,A4,A23,RELAT_1:62;
    then A"**"(Fr2|n1)=A"**"(Fr^(Fr1|n1)) by A13,A24,AFINSQ_2:46
 .= Sum(Fr^(Fr1|n1)) by AFINSQ_2:48
.= Sum(Fr)+Sum(Fr1|n1) by AFINSQ_2:55;
    then
A27: Sum(Fr2|n1)=Sum Fr+Sum(Fr1|n1) by AFINSQ_2:48;
    take Fr2;
    Fr2.n1=seq1.n1 * P.(n1-'n1) & Sum(Fr2|(n1+1)) = Fr2.n1 + Sum(Fr2|n1)
    by A4,A5,A25,AFINSQ_2:65;
    then Sum Fr2=Partial_Sums(S).n + S.n1 & n in NAT & n1 in NAT
      by A6,A11,A4,A27,A2,A26,A12,ORDINAL1:def 12;
    hence thesis by A4,A5,SERIES_1:def 1;
  end;
A28: P[0]
  proof
     reconsider rr = seq1.0*seq2.0 as Element of REAL;
    set Fr=1--> rr;
    reconsider Fr as XFinSequence of REAL;
    take Fr;
A29: dom Fr=1;
A30: Sum(Fr|zz) = 0;
A31: 0 in Segm 1 by NAT_1:44;
    then
A32: Fr.zz=seq1.0*seq2.0 by FUNCOP_1:7;
A33:  Sum(Fr|zz)+Fr.zz=Sum(Fr|(zz+1)) by A29,A31,AFINSQ_2:65;
    Sum Fr = Sum(Fr|((0 qua Nat)+1))
        .= Sum(Fr|zz)+Fr.zz by A33
        .= Fr.zz by A30
        .=S.0 by Th47,A32;
    hence Partial_Sums(S).0 = Sum Fr & dom Fr=(0 qua Nat)+1
         by SERIES_1:def 1;
    let i such that
A34: i in (0 qua Nat)+1;
    i in Segm 1 by A34;
    then i<1 by NAT_1:44;
    then
A35: i=0 by NAT_1:14;
    then (0-'i)=0 by XREAL_1:232;
    then P.(0-'i)= seq2.0 by SERIES_1:def 1;
    hence thesis by A34,A35,FUNCOP_1:7;
  end;
  for i holds P[i] from NAT_1:sch 2(A28,A1);
  hence thesis;
end;
