
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Element of NAT
 st (ex k be Element of NAT st k <= m
       & ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty)
holds
  ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 f),m)
   is convergent_to_+infty
& lim(ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 f),m)) = +infty
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Element of NAT;
   given k be Element of NAT such that
A1: k <= m & ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty;
A3: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;
   ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 ~f),m)
    = ProjMap2(~Partial_Sums_in_cod2(Partial_Sums_in_cod1 ~f),m) by Th32
   .= ProjMap2(Partial_Sums_in_cod1(~Partial_Sums_in_cod1 ~f),m) by Th40
   .= ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 ~(~f)),m) by Th40
   .= ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 f),m) by DBLSEQ_2:7;
   hence thesis by A3,A1,Th77;
end;
