
theorem
for f be without-infty Function of [:NAT,NAT:],ExtREAL holds
  Partial_Sums f is convergent_in_cod2_to_finite iff
    Partial_Sums_in_cod2 f is convergent_in_cod2_to_finite
proof
   let f be without-infty Function of [:NAT,NAT:],ExtREAL;
   hereby assume Partial_Sums f is convergent_in_cod2_to_finite; then
    ~Partial_Sums_in_cod2(Partial_Sums_in_cod1 f)
     is convergent_in_cod1_to_finite by Th36; then
    Partial_Sums_in_cod1~Partial_Sums_in_cod1 f
     is convergent_in_cod1_to_finite by Th40; then
    Partial_Sums_in_cod1(Partial_Sums_in_cod2 ~f)
     is convergent_in_cod1_to_finite by Th40; then
    Partial_Sums (~f) is convergent_in_cod1_to_finite by Lm8; then
    Partial_Sums_in_cod1 (~f) is convergent_in_cod1_to_finite
      by Th79; then
    ~Partial_Sums_in_cod2 f is convergent_in_cod1_to_finite by Th40;
    hence Partial_Sums_in_cod2 f is convergent_in_cod2_to_finite by Th36;
   end;
   assume Partial_Sums_in_cod2 f is convergent_in_cod2_to_finite; then
   ~Partial_Sums_in_cod2 f is convergent_in_cod1_to_finite by Th36; then
   Partial_Sums_in_cod1 (~f) is convergent_in_cod1_to_finite by Th40; then
   Partial_Sums (~f) is convergent_in_cod1_to_finite by Th79; then
   Partial_Sums_in_cod1(Partial_Sums_in_cod2 ~f)
     is convergent_in_cod1_to_finite by Lm8; then
   Partial_Sums_in_cod1~Partial_Sums_in_cod1 f
     is convergent_in_cod1_to_finite by Th40; then
   ~Partial_Sums_in_cod2(Partial_Sums_in_cod1 f)
     is convergent_in_cod1_to_finite by Th40;
   hence Partial_Sums f is convergent_in_cod2_to_finite by Th36;
end;
