
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL, n,m be Nat
  st (for i be Nat st i <= m holds f.(n,i) is Real)
 holds (Partial_Sums_in_cod2 f).(n,m) < +infty
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL, n,m be Nat;
   assume
A2: for i be Nat st i<=m holds f.(n,i) is Real;
   defpred P[Nat] means $1<=m implies
     (Partial_Sums_in_cod2 f).(n,$1) < +infty;
   (Partial_Sums_in_cod2 f).(n,0) = f.(n,0) by DefCSM; then
   (Partial_Sums_in_cod2 f).(n,0) is Real by A2; then
A4:P[0] by XXREAL_0:9,XREAL_0:def 1;
A5:for k be Nat st P[k] holds P[k+1]
   proof
    let k be Nat;
    assume A6: P[k];
    now assume A7: k+1<=m; then
A8:  f.(n,k+1) is Real & f.(n,k+1) >= 0 by A2,SUPINF_2:51;
     (Partial_Sums_in_cod2 f).(n,k+1)
      = (Partial_Sums_in_cod2 f).(n,k) + f.(n,k+1) by DefCSM;
     hence (Partial_Sums_in_cod2 f).(n,k+1) < +infty
       by A6,A7,A8,NAT_1:13,XXREAL_3:16,XXREAL_0:4;
    end;
    hence P[k+1];
   end;
   for k be Nat holds P[k] from NAT_1:sch 2(A4,A5);
   hence thesis;
end;
