theorem Th47: :: DBLSEQ_3:47

for f being without-infty Function of [:NAT,NAT:],ExtREAL
for n, m being Nat holds
( (Partial_Sums f) . ((n + 1),m) = ((Partial_Sums_in_cod2 f) . ((n + 1),m)) + ((Partial_Sums f) . (n,m)) & (Partial_Sums_in_cod1 (Partial_Sums_in_cod2 f)) . (n,(m + 1)) = ((Partial_Sums_in_cod1 f) . (n,(m + 1))) + ((Partial_Sums_in_cod1 (Partial_Sums_in_cod2 f)) . (n,m)) )