
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL,
    m be Element of NAT
  st not ProjMap2(Partial_Sums_in_cod1 f,m) is convergent_to_+infty
 holds for n be Nat holds f.(n,m) is Real
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL;
   let m be Element of NAT;
   assume
A2:not ProjMap2(Partial_Sums_in_cod1 f,m) is convergent_to_+infty;
   given N be Nat such that
A3: not f.(N,m) is Real;
   not f.(N,m) in REAL by A3; then
A4:f.(N,m) = +infty or f.(N,m) = -infty by XXREAL_0:14;
   reconsider N1=N as Element of NAT by ORDINAL1:def 12;
   now let g be Real;
    assume 0 < g;
    take N;
    hereby let k be Nat;
     assume A7: N<=k;
     per cases;
     suppose A8: N = 0;
      ProjMap2(Partial_Sums_in_cod1 f,m).N
       = (Partial_Sums_in_cod1 f).(N1,m) by MESFUNC9:def 7
      .= f.(N,m) by A8,DefRSM; then
A9:   g <= ProjMap2(Partial_Sums_in_cod1 f,m).N
         by A4,SUPINF_2:51,XXREAL_0:3;
      ProjMap2(Partial_Sums_in_cod1 f,m).N
        <= ProjMap2(Partial_Sums_in_cod1 f,m).k by A7,RINFSUP2:7;
      hence g <= ProjMap2(Partial_Sums_in_cod1 f,m).k by A9,XXREAL_0:2;
     end;
     suppose N <> 0; then
      consider N2 be Nat such that
A11:   N = N2 + 1 by NAT_1:6;
      reconsider N3=N2 as Element of NAT by ORDINAL1:def 12;
A12:  (Partial_Sums_in_cod1 f).(N3,m) >= 0 by SUPINF_2:51;
      ProjMap2(Partial_Sums_in_cod1 f,m).N1
       = (Partial_Sums_in_cod1 f).(N1,m) by MESFUNC9:def 7
      .= (Partial_Sums_in_cod1 f).(N2,m) + f.(N1,m) by A11,DefRSM; then
      ProjMap2(Partial_Sums_in_cod1 f,m).N1 = +infty
        by A4,SUPINF_2:51,XXREAL_0:4,A12,XXREAL_3:39; then
      ProjMap2(Partial_Sums_in_cod1 f,m).k = +infty
        by XXREAL_0:4,A7,RINFSUP2:7;
      hence g <= ProjMap2(Partial_Sums_in_cod1 f,m).k by XXREAL_0:3;
     end;
    end;
   end;
   hence contradiction by A2,MESFUNC5:def 9;
end;
