
theorem Th77:
for f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Element of NAT
 st (ex k be Element of NAT st k <= m
       & ProjMap1(Partial_Sums_in_cod2 f,k) is convergent_to_+infty)
holds
  ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m)
   is convergent_to_+infty
& lim(ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 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
A2:  k<=m & ProjMap1(Partial_Sums_in_cod2 f,k) is convergent_to_+infty;
A5: for g be Real st 0 < g
     ex N be Nat st for n be Nat st N<=n holds
      g <= ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m).n
    proof
     let g be Real;
     assume 0<g; then
     consider N be Nat such that
A4:   for n be Nat st N<=n holds
       g <= ProjMap1(Partial_Sums_in_cod2 f,k).n by A2,MESFUNC5:def 9;
     now let n be Nat;
      reconsider n1=n as Element of NAT by ORDINAL1:def 12;
      assume N<=n; then
      g <= ProjMap1(Partial_Sums_in_cod2 f,k).n by A4; then
A7:   g <= (Partial_Sums_in_cod2 f).(k,n1) by MESFUNC9:def 6;
      ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 f),n1).k
       <= ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 f),n1).m
         by A2,RINFSUP2:7; then
      (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(k,n1)
       <= ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 f),n1).m
         by MESFUNC9:def 7; then
A10:  (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(k,n1)
       <= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(m,n1)
         by MESFUNC9:def 7;
      (Partial_Sums_in_cod2 f).(k,n1)
       <= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(k,n1)
         by Th76; then
A9:   (Partial_Sums_in_cod2 f).(k,n1)
       <= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(m,n)
         by A10,XXREAL_0:2;
      ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m).n1
       = (Partial_Sums f).(m,n) by MESFUNC9:def 6
      .= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)).(m,n)
           by Lm8;
      hence g <= ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m).n
        by A7,A9,XXREAL_0:2;
     end;
     hence thesis;
    end;
    hence ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m)
            is convergent_to_+infty by MESFUNC5:def 9;
    thus
    lim(ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m)) = +infty
      by A5,MESFUNC5:def 9,MESFUNC9:7;
end;
