 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem th101:
  for rseq being Real_Sequence, m being Nat st
    rseq is nonnegative holds
      rseq.m <= (Partial_Sums rseq).m
proof
   let rseq be Real_Sequence, m be Nat;
   assume A1: rseq is nonnegative;
   defpred P[Nat] means rseq.$1 <= (Partial_Sums rseq).$1;
a3:P[0] by SERIES_1:def 1;
a4:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume P[k];
    (Partial_Sums rseq).(k+1) = (Partial_Sums rseq).k + rseq.(k+1)
       by SERIES_1:def 1;
    hence P[k+1] by XREAL_1:31,SERIES_3:34,A1;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(a3,a4);
   hence rseq.m <= Partial_Sums(rseq).m;
end;
