let Rseq be Function of [:NAT,NAT:],REAL; :: thesis: ( Partial_Sums Rseq is convergent_in_cod1 implies lim_in_cod1 (Partial_Sums Rseq) = Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq)) )
assume AS: Partial_Sums Rseq is convergent_in_cod1 ; :: thesis: lim_in_cod1 (Partial_Sums Rseq) = Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq))

now :: thesis: for m being Nat holds (lim_in_cod1 (Partial_Sums Rseq)) . m = (Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq))) . m

let m be Nat; :: thesis: (lim_in_cod1 (Partial_Sums Rseq)) . m = (Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq))) . m
reconsider m1 = m as Element of NAT by ORDINAL1:def 12;
defpred S₁[ Nat] means (Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq))) . $1 = (lim_in_cod1 (Partial_Sums Rseq)) . $1;
(Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq))) . 0 = (lim_in_cod1 (Partial_Sums_in_cod1 Rseq)) . 0 by SERIES_1:def 1
.= lim (ProjMap2 ((Partial_Sums_in_cod1 Rseq),0)) by DBLSEQ_1:def 5
.= lim (ProjMap2 ((Partial_Sums Rseq),0)) by th00
.= (lim_in_cod1 (Partial_Sums Rseq)) . 0 by DBLSEQ_1:def 5 ;
then A1: S₁[ 0 ] ;
A2: for k being Nat st S₁[k] holds
S₁[k + 1]

proof

let k be Nat; :: thesis: ( S₁[k] implies S₁[k + 1] )
assume A3: S₁[k] ; :: thesis: S₁[k + 1]
reconsider k1 = k as Element of NAT by ORDINAL1:def 12;
(Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq))) . (k + 1) = ((lim_in_cod1 (Partial_Sums Rseq)) . k) + ((lim_in_cod1 (Partial_Sums_in_cod1 Rseq)) . (k + 1)) by A3, SERIES_1:def 1;
hence S₁[k + 1] by AS, th02a; :: thesis: verum

end;

for k being Nat holds S₁[k] from NAT_1:sch 2(A1, A2);
hence (lim_in_cod1 (Partial_Sums Rseq)) . m = (Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq))) . m ; :: thesis: verum

end;

hence lim_in_cod1 (Partial_Sums Rseq) = Partial_Sums (lim_in_cod1 (Partial_Sums_in_cod1 Rseq)) ; :: thesis: verum