reserve k, m, n, p, K, N for Nat;
reserve i for Integer;
reserve x, y, eps for Real;
reserve seq, seq1, seq2 for Real_Sequence;
reserve sq for FinSequence of REAL;

theorem Th24:
  for K holds for dseqK being Real_Sequence st for n holds dseqK.n
  =Partial_Sums(cseq(n)).K holds dseqK is convergent & lim(dseqK)=Partial_Sums(
  eseq).K
proof
  defpred P[Nat] means
for dseqK being Real_Sequence st for n holds
  dseqK.n=Partial_Sums(cseq(n)).$1 holds dseqK is convergent & lim(dseqK)=
  Partial_Sums(eseq).$1;
  now
    let K;
    deffunc F(Nat)=Partial_Sums(cseq($1)).K;
    consider dseqK being Real_Sequence such that
A1: for n holds dseqK.n=F(n) from SEQ_1:sch 1;
    assume
A2: for dseqK being Real_Sequence st for n holds dseqK.n=Partial_Sums(
    cseq(n)).K holds dseqK is convergent & lim(dseqK)=Partial_Sums(eseq).K;
    then
A3: dseqK is convergent by A1;
    let dseqK1 be Real_Sequence;
    assume
A4: for n holds dseqK1.n=Partial_Sums(cseq(n)).(K+1);
    now
      let n be Element of NAT;
      thus dseqK1.n = Partial_Sums(cseq(n)).(K+1) by A4
        .= Partial_Sums(cseq(n)).K+cseq(n).(K+1) by SERIES_1:def 1
        .= dseqK.n+cseq(n).(K+1) by A1
        .= dseqK.n+bseq(K+1).n by Th3
        .= (dseqK+bseq(K+1)).n by SEQ_1:7;
    end;
    then
A5: dseqK1=dseqK+bseq(K+1) by FUNCT_2:63;
A6: lim(dseqK)=Partial_Sums(eseq).K by A2,A1;
A7: bseq(K+1) is convergent by Th12;
    hence dseqK1 is convergent by A3,A5;
    thus lim(dseqK1) = lim(dseqK)+lim(bseq(K+1)) by A3,A5,A7,SEQ_2:6
      .= Partial_Sums(eseq).K+eseq.(K+1) by A6,Th12
      .= Partial_Sums(eseq).(K+1) by SERIES_1:def 1;
  end;
  then
A8: P[n] implies P[n+1];
  now
    let dseq0 be Real_Sequence;
    assume
A9: for n holds dseq0.n=Partial_Sums(cseq(n)).0;
    now
      let n be Element of NAT;
      thus dseq0.n = Partial_Sums(cseq(n)).0 by A9
        .= cseq(n).0 by SERIES_1:def 1
        .= bseq(0).n by Th3;
    end;
    then
A10: dseq0 = bseq(0) by FUNCT_2:63;
    hence dseq0 is convergent by Th12;
    thus lim(dseq0) = eseq.0 by A10,Th12
      .= Partial_Sums(eseq).0 by SERIES_1:def 1;
  end;
  then
A11: P[0];
  thus P[n] from NAT_1:sch 2(A11,A8);
end;
