
theorem
  for f be without-infty Function of [:NAT,NAT:],ExtREAL holds
  Partial_Sums f is convergent_in_cod1_to_-infty implies
   ex m be Element of NAT st
     ProjMap2(Partial_Sums_in_cod1 f,m) is convergent_to_-infty
proof
   let f be without-infty Function of [:NAT,NAT:],ExtREAL;
   assume
A1: Partial_Sums f is convergent_in_cod1_to_-infty;
A3:ProjMap2(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),0)
     = ProjMap2(Partial_Sums f,0)
    .= ProjMap2(Partial_Sums_in_cod1 f,0) by Th54;
   assume for m be Element of NAT holds
     not ProjMap2(Partial_Sums_in_cod1 f,m) is convergent_to_-infty;
   hence contradiction by A1,A3;
end;
