reserve rseq, rseq1, rseq2 for Real_Sequence;
reserve seq, seq1, seq2 for Complex_Sequence;
reserve k, n, n1, n2, m for Nat;
reserve p, r for Real;
reserve z for Complex;
reserve Nseq,Nseq1 for increasing sequence of NAT;

theorem
  seq is summable implies for n holds Sum(seq) = Partial_Sums(seq).n +
  Sum(seq^\(n+1))
proof
  assume
A1: seq is summable;
  then Sum(seq)=Sum(Re(seq))+(Sum(Im(seq)))*<i> by Th53;
  then
A2: Re(Sum(seq))=Sum(Re(seq)) & Im(Sum(seq))=Sum(Im(seq)) by COMPLEX1:12;
  let n;
A3: Sum(seq^\(n+1)) = Sum(Re(seq^\(n+1)))+(Sum(Im(seq^\(n+1))))*<i> by A1,Th53;
A4: Sum(Im(seq)) =Partial_Sums(Im(seq)).n + Sum(Im(seq)^\(n+1)) by A1,
SERIES_1:15
    .=Im(Partial_Sums(seq)).n + Sum(Im(seq)^\(n+1)) by Th26
    .=Im(Partial_Sums(seq)).n + Sum(Im(seq^\(n+1))) by Th23
    .=Im(Partial_Sums(seq)).n + Im(Sum(seq^\(n+1))) by A3,COMPLEX1:12
    .=Im(Partial_Sums(seq).n) + Im(Sum(seq^\(n+1))) by Def6
    .=Im(Partial_Sums(seq).n+ Sum(seq^\(n+1))) by COMPLEX1:8;
  Sum(Re(seq)) =Partial_Sums(Re(seq)).n + Sum(Re(seq)^\(n+1)) by A1,SERIES_1:15
    .=Re(Partial_Sums(seq)).n + Sum(Re(seq)^\(n+1)) by Th26
    .=Re(Partial_Sums(seq)).n + Sum(Re(seq^\(n+1))) by Th23
    .=Re(Partial_Sums(seq)).n + Re(Sum(seq^\(n+1))) by A3,COMPLEX1:12
    .=Re(Partial_Sums(seq).n) + Re(Sum(seq^\(n+1))) by Def5
    .=Re(Partial_Sums(seq).n+ Sum(seq^\(n+1))) by COMPLEX1:8;
  hence thesis by A2,A4,COMPLEX1:def 3;
end;
