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

theorem
  Partial_Sums Rseq1 is P-convergent & Partial_Sums Rseq2 is P-convergent
    implies Partial_Sums(Rseq1+Rseq2) is P-convergent
proof
   assume Partial_Sums(Rseq1) is P-convergent &
    Partial_Sums(Rseq2) is P-convergent; then
   Partial_Sums(Rseq1) + Partial_Sums(Rseq2) is P-convergent by DBLSEQ_1:11;
   hence thesis by lm74;
end;
