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;

theorem
  (for n holds 0 <= rseq.n) implies (for n, m st n <= m holds |.
Partial_Sums(rseq).m-Partial_Sums(rseq).n.| =Partial_Sums(rseq).m-Partial_Sums(
  rseq).n) & for n holds |.Partial_Sums(rseq).n.|=Partial_Sums(rseq).n
proof
  assume
A1: for n holds 0 <= rseq.n;
  then
A2: Partial_Sums(rseq) is non-decreasing by SERIES_1:16;
A3: now
    let n,m;
    assume n <= m;
    then Partial_Sums(rseq).n <= Partial_Sums(rseq).m by A2,SEQM_3:6;
    then Partial_Sums(rseq).n-Partial_Sums(rseq).n <= Partial_Sums(rseq).m-
    Partial_Sums(rseq).n by XREAL_1:9;
    hence
    |.Partial_Sums(rseq).m-Partial_Sums(rseq).n.| =Partial_Sums(rseq).m-
    Partial_Sums(rseq).n by ABSVALUE:def 1;
  end;
  now
    let n;
A4: Partial_Sums(rseq).0 <= Partial_Sums(rseq).n by A2,SEQM_3:6;
    Partial_Sums(rseq).0 = rseq.0 & 0 <= rseq.0 by A1,SERIES_1:def 1;
    hence |.Partial_Sums(rseq).n.|=Partial_Sums(rseq).n by A4,ABSVALUE:def 1;
  end;
  hence thesis by A3;
end;
