
theorem Th4:
  for seq be nonnegative ExtREAL_sequence, m be Nat holds
    seq.m <= (Partial_Sums seq).m
proof
   let seq be nonnegative ExtREAL_sequence, m be Nat;
   defpred P[Nat] means seq.$1 <= (Partial_Sums seq).$1;
A1:P[0] by MESFUNC9:def 1;
A2:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume P[k];
    (Partial_Sums seq).(k+1) = (Partial_Sums seq).k + seq.(k+1)
       by MESFUNC9:def 1;
    hence P[k+1] by XXREAL_3:39,SUPINF_2:51;
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A1,A2);
   hence seq.m <= Partial_Sums(seq).m;
end;
