
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Nat holds
  (for k be Element of NAT st k<=m holds
     not ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty)
iff
  (for k be Element of NAT st k<=m holds
     lim ProjMap2(Partial_Sums_in_cod1 f,k) < +infty)
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Nat;
   hereby assume
A1: for k be Element of NAT st k<=m holds
     not ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty;
    hereby let k be Element of NAT;
     assume k<=m; then
     not ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty
      by A1; then
     ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_finite_number
       by Th62; then
     ex g be Real st
      lim ProjMap2(Partial_Sums_in_cod1 f,k) = g
    & for p be Real st 0<p ex n be Nat st for m be Nat st n<=m holds
        |.(ProjMap2(Partial_Sums_in_cod1 f,k)).m
            - lim ProjMap2(Partial_Sums_in_cod1 f,k).| < p
                 by MESFUNC9:7;
     hence lim ProjMap2(Partial_Sums_in_cod1 f,k) < +infty
       by XREAL_0:def 1,XXREAL_0:9;
    end;
   end;
   assume
A2: for k be Element of NAT st k <= m holds
     lim ProjMap2(Partial_Sums_in_cod1 f,k) < +infty;
   now let k be Element of NAT;
    assume k <= m; then
    lim ProjMap2(Partial_Sums_in_cod1 f,k) < +infty by A2;
    hence not ProjMap2(Partial_Sums_in_cod1 f,k) is convergent_to_+infty
      by MESFUNC9:7;
   end;
   hence thesis;
end;
