
theorem
for f be nonnegative Function of [:NAT,NAT:],ExtREAL st
  Partial_Sums f is convergent_in_cod2_to_finite
 holds
  for m be Element of NAT holds
    (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).m
   = lim(ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m))
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL;
   assume Partial_Sums f is convergent_in_cod2_to_finite; then
   Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)
     is convergent_in_cod2_to_finite by Lm8; then
   ~Partial_Sums_in_cod1(Partial_Sums_in_cod2 f)
     is convergent_in_cod1_to_finite by Th36; then
   Partial_Sums_in_cod2~Partial_Sums_in_cod2 f
     is convergent_in_cod1_to_finite by Th40; then
A1:Partial_Sums (~f) is convergent_in_cod1_to_finite by Th40;
   hereby let m be Element of NAT;
    lim_in_cod2(Partial_Sums_in_cod2 f)
      = lim_in_cod1(~Partial_Sums_in_cod2 f) by Th38
     .= lim_in_cod1(Partial_Sums_in_cod1 (~f)) by Th40; then
    (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).m
     = lim(ProjMap2(Partial_Sums_in_cod1(Partial_Sums_in_cod2 (~f)),m))
         by A1,Th80
    .= lim(ProjMap2(Partial_Sums_in_cod1~Partial_Sums_in_cod1 f,m)) by Th40
    .= lim(ProjMap2(~Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m))
         by Th40;
    hence (Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 f))).m
     = lim(ProjMap1(Partial_Sums_in_cod2(Partial_Sums_in_cod1 f),m)) by Th32;
   end;
end;
