 reserve i,j,k,k0,m,n,N for Nat;
 reserve x,y for Real;
 reserve p for Prime;
 reserve s for Real_Sequence;

theorem SumMono:
  s is summable & (for n holds 0 <= s.n) & i <= j implies
  Sum(s^\j) <= Sum(s^\i)
  proof
    assume A1: s is summable & for n holds 0<=s.n;
    assume i <= j; then
    per cases by XXREAL_0:1;
    suppose i = j;
      hence thesis;
    end;
    suppose i < j; then
      i+1 <= j by NAT_1:13; then
      0 <= j-(i+1) by XREAL_1:48; then
      reconsider k = j-(i+1) as Nat;
      s^\i is summable by A1,SERIES_1:12; then
A2:   Sum(s^\i) = Partial_Sums(s^\i).k + Sum(s^\i^\(k+1)) by SERIES_1:15
        .= Partial_Sums(s^\i).k + Sum(s^\(i+(k+1))) by NAT_1:48
        .= Partial_Sums(s^\i).k + Sum(s^\j);
      for n holds 0 <= (s^\i).n
      proof
        let n;
        0 <= s.(i+n) by A1;
        hence thesis by NAT_1:def 3;
      end;
      hence thesis by A2,PartialNonneg,XREAL_1:31;
    end;
  end;
