
theorem Th74:
for f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Nat holds
  (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).m = +infty
iff
  ex k be Element of NAT
   st k <= m & ProjMap1(Partial_Sums_in_cod2 f,k) is convergent_to_+infty
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Nat;
   hereby assume
A2: (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).m = +infty;
    now assume
A3:  for k be Element of NAT holds k > m or
      not ProjMap1(Partial_Sums_in_cod2 f,k) is convergent_to_+infty;
     defpred P[Nat] means $1 <= m implies
       (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).$1 <> +infty;
     (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).0
       = (lim_in_cod2(Partial_Sums_in_cod2 f)).0 by MESFUNC9:def 1
      .= lim ProjMap1(Partial_Sums_in_cod2 f,0) by D1DEF6; then
A5:  P[0] by A3,Th72;
A6:  for k be Nat st P[k] holds P[k+1]
     proof
      let k be Nat;
      assume A7: P[k];
      assume A8: k+1 <= m;
A10:  (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).(k+1)
       = (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).k
        + (lim_in_cod2(Partial_Sums_in_cod2 f)).(k+1) by MESFUNC9:def 1;
      now assume (lim_in_cod2(Partial_Sums_in_cod2 f)).(k+1) = +infty; then
       lim ProjMap1(Partial_Sums_in_cod2 f,k+1) = +infty by D1DEF6;
       hence contradiction by A3,A8,Th72;
      end;
      hence (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).(k+1)
              <> +infty by A7,A8,NAT_1:13,A10,XXREAL_3:16;
     end;
     for k be Nat holds P[k] from NAT_1:sch 2(A5,A6);
     hence contradiction by A2;
    end;
    hence
     ex k be Element of NAT st k <= m
      & ProjMap1(Partial_Sums_in_cod2 f,k) is convergent_to_+infty;
   end;
   given k be Element of NAT such that
B1: k <= m
  & ProjMap1(Partial_Sums_in_cod2 f,k) is convergent_to_+infty;
   lim ProjMap1(Partial_Sums_in_cod2 f,k) = +infty by B1,MESFUNC9:7; then
B2:(lim_in_cod2(Partial_Sums_in_cod2 f)).k = +infty by D1DEF6;
B3:Partial_Sums_in_cod2 f is convergent_in_cod2 by RINFSUP2:37; then
   (lim_in_cod2(Partial_Sums_in_cod2 f)) is nonnegative by Th65; then
B5:(Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).k
     >= +infty by B2,Th4;
   lim_in_cod2(Partial_Sums_in_cod2 f) is nonnegative by B3,Th65; then
   (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).k
    <= (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).m
        by B1,RINFSUP2:7,MESFUNC9:16;
   hence (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).m = +infty
     by B5,XXREAL_0:2,4;
end;
