 reserve Rseq, Rseq1, Rseq2 for Function of [:NAT,NAT:],REAL;

theorem
  Rseq is nonnegative-yielding &
  Partial_Sums Rseq is P-convergent implies
   lim_in_cod1(Partial_Sums_in_cod1 Rseq) is summable
 & lim_in_cod2(Partial_Sums_in_cod2 Rseq) is summable
proof
   assume that
A1: Rseq is nonnegative-yielding and
A2: Partial_Sums Rseq is P-convergent;
   Partial_Sums Rseq is convergent_in_cod1
 & Partial_Sums Rseq is convergent_in_cod2 by A1,A2,th1006a; then
A3:lim_in_cod1(Partial_Sums Rseq) is convergent
 & lim_in_cod2(Partial_Sums Rseq) is convergent by A2; then
   Partial_Sums(lim_in_cod1(Partial_Sums_in_cod1 Rseq)) is convergent
     by A1,A2,th1006a,th03a;
   hence lim_in_cod1(Partial_Sums_in_cod1 Rseq) is summable by SERIES_1:def 2;
   Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 Rseq)) is convergent
     by A3,A1,A2,th1006a,th03b;
   hence lim_in_cod2(Partial_Sums_in_cod2 Rseq) is summable by SERIES_1:def 2;
end;
