
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Nat holds
  (Partial_Sums(lim_in_cod1(Partial_Sums_in_cod1 f))).m = +infty
iff
  ex k be Element of NAT
   st k <= m & ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Nat;
   hereby assume
A1: (Partial_Sums(lim_in_cod1(Partial_Sums_in_cod1 f))).m = +infty;
    lim_in_cod1(Partial_Sums_in_cod1 f)
     = lim_in_cod2(~Partial_Sums_in_cod1 f) by Th38
    .= lim_in_cod2(Partial_Sums_in_cod2 ~f) by Th40; then
    consider k be Element of NAT such that
A2:  k <= m & ProjMap1(Partial_Sums_in_cod2 ~f,k) is convergent_to_+infty
       by A1,Th74;
    ProjMap1(Partial_Sums_in_cod2 ~f,k)
     = ProjMap2(~Partial_Sums_in_cod2 ~f,k) by Th32
    .= ProjMap2(Partial_Sums_in_cod1 ~(~f),k) by Th40
    .= ProjMap2(Partial_Sums_in_cod1 f,k) by DBLSEQ_2:7;
    hence ex k be Element of NAT
     st k <= m & ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty
      by A2;
   end;
   given k be Element of NAT such that
A3: k <= m & ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty;
A4:ProjMap2(Partial_Sums_in_cod1 f,k)
    = ProjMap2(Partial_Sums_in_cod1 ~(~f),k) by DBLSEQ_2:7
   .= ProjMap2(~Partial_Sums_in_cod2 ~f,k) by Th40
   .= ProjMap1(Partial_Sums_in_cod2 ~f,k) by Th32;
    lim_in_cod1(Partial_Sums_in_cod1 f)
     = lim_in_cod2(~Partial_Sums_in_cod1 f) by Th38
    .= lim_in_cod2(Partial_Sums_in_cod2 ~f) by Th40;
   hence (Partial_Sums(lim_in_cod1(Partial_Sums_in_cod1 f))).m = +infty
     by A3,A4,Th74;
end;
