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

theorem th1006a:
  Rseq is nonnegative-yielding & Partial_Sums Rseq is P-convergent implies
    Partial_Sums Rseq is convergent_in_cod1
  & Partial_Sums Rseq is convergent_in_cod2
proof
   assume that
A1: Rseq is nonnegative-yielding and
A2: Partial_Sums Rseq is P-convergent;
   Partial_Sums_in_cod1 Rseq is convergent_in_cod1
 & Partial_Sums_in_cod2 Rseq is convergent_in_cod2 by A1,A2,th1006;
   hence thesis by th01a,th01b;
end;
