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 Th18:
  for n holds for seq, sq st len(sq)=n & (for k st k<n holds seq.k
=sq.(k+1)) & (for k st k>=n holds seq.k=0) holds seq is summable & Sum(seq)=Sum
  (sq)
proof
  defpred P[Nat] means
for seq, sq st len(sq)=$1 & (for k st k<$1
holds seq.k=sq.(k+1)) & (for k st k>=$1 holds seq.k=0) holds seq is summable &
  Sum(seq)=Sum(sq);
  now
    let n;
    assume
A1: for seq, sq st len(sq)=n & (for k st k<n holds seq.k=sq.(k+1)) &
    (for k st k>=n holds seq.k=0) holds seq is summable & Sum(seq)=Sum(sq);
    let seq, sq;
    assume that
A2: len(sq)=n+1 and
A3: for k st k<n+1 holds seq.k=sq.(k+1) and
A4: for k st k>=n+1 holds seq.k=0;
A5: now
      let k;
A6:   k+1>=0+1 by XREAL_1:6;
      assume k<n;
      then
A7:   k+1<n+1 by XREAL_1:6;
      thus (seq^\1).k = seq.(k+1) by NAT_1:def 3
        .= sq.((k+1)+1) by A3,A7
        .= (sq/^1).(k+1) by A2,A7,A6,Th16;
    end;
A8: now
      let k;
      assume k>=n;
      then
A9:   k+1>=n+1 by XREAL_1:6;
      thus (seq^\1).k = seq.(k+1) by NAT_1:def 3
        .= 0 by A4,A9;
    end;
    n+1>=0+1 by XREAL_1:6;
    then
A10: len(sq/^1) = len(sq)-1 by A2,RFINSEQ:def 1
      .= n by A2;
    then
A11: Sum(seq^\1)=Sum(sq/^1) by A1,A5,A8;
A12: seq^\1 is summable by A1,A10,A5,A8;
    hence seq is summable by Th15;
    thus Sum(seq) = (seq.0)+Sum(seq^\1) by A12,Th15
      .= (sq.(0+1))+Sum(seq^\1) by A3
      .= Sum(sq) by A2,A11,Th17;
  end;
  then
A13: P[k] implies P[k+1];
  now
    let seq, sq;
    assume that
A14: len(sq)=0 and
    for k st k<0 holds seq.k=sq.(k+1) and
A15: for k st k>=0 holds seq.k=0;
    sq is Element of 0-tuples_on REAL by A14,FINSEQ_2:92;
    then
A16: Sum(sq)=0 by RVSUM_1:79;
    defpred P[Nat] means Partial_Sums(seq).$1=0;
A17: now
      let k be Nat;
A18:  Partial_Sums(seq).(k+1) = (Partial_Sums(seq).k)+(seq.(k+1)) by
SERIES_1:def 1;
      assume P[k];
      hence P[k+1] by A15,A18;
    end;
    seq.0=0 by A15;
    then
A19: P[0] by SERIES_1:def 1;
A20: for k being Nat holds P[k] from NAT_1:sch 2(A19,A17);
    then Partial_Sums(seq) is convergent by Th9;
    hence seq is summable by SERIES_1:def 2;
    lim(Partial_Sums(seq))=0 by A20,Th9;
    hence Sum(seq) = Sum(sq) by A16,SERIES_1:def 3;
  end;
  then
A21: P[0];
  thus P[n] from NAT_1:sch 2(A21,A13);
end;
