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

theorem
  Rseq is nonnegative-yielding & Partial_Sums Rseq is P-convergent implies
    P-lim(Partial_Sums Rseq) = Sum(lim_in_cod1(Partial_Sums_in_cod1 Rseq))
  & P-lim(Partial_Sums Rseq) = Sum(lim_in_cod2(Partial_Sums_in_cod2 Rseq))
proof
   assume that
A1: Rseq is nonnegative-yielding and
A2: Partial_Sums Rseq is P-convergent;
A4:Partial_Sums Rseq is convergent_in_cod1
 & Partial_Sums Rseq is convergent_in_cod2 by A1,A2,th1006a;
A5:Sum(lim_in_cod1(Partial_Sums_in_cod1 Rseq))
     = lim Partial_Sums(lim_in_cod1(Partial_Sums_in_cod1 Rseq))
         by SERIES_1:def 3;
   for e be Real st 0<e ex M be Nat st
     for m be Nat st m>=M holds
      |.(lim_in_cod1(Partial_Sums Rseq)).m
          - cod1_major_iterated_lim(Partial_Sums Rseq).| < e
             by A2,A4,DBLSEQ_1:def 7; then
   cod1_major_iterated_lim(Partial_Sums Rseq)
    = lim(lim_in_cod1(Partial_Sums Rseq)) by A2,A4,SEQ_2:def 7
   .= Sum(lim_in_cod1(Partial_Sums_in_cod1 Rseq)) by A5,A1,A2,th1006a,th03a;
   hence P-lim(Partial_Sums Rseq)
          = Sum(lim_in_cod1(Partial_Sums_in_cod1 Rseq))
             by A1,A2,th1006a,DBLSEQ_1:4;
A6:Sum(lim_in_cod2(Partial_Sums_in_cod2 Rseq))
     = lim Partial_Sums(lim_in_cod2(Partial_Sums_in_cod2 Rseq))
         by SERIES_1:def 3;
   for e be Real st 0<e ex M be Nat st
     for m be Nat st m>=M holds
      |.(lim_in_cod2(Partial_Sums Rseq)).m
          - cod2_major_iterated_lim(Partial_Sums Rseq).| < e
             by A2,A4,DBLSEQ_1:def 8; then
   cod2_major_iterated_lim(Partial_Sums Rseq)
    = lim(lim_in_cod2(Partial_Sums Rseq)) by A2,A4,SEQ_2:def 7
   .= Sum(lim_in_cod2(Partial_Sums_in_cod2 Rseq)) by A6,A1,A2,th1006a,th03b;
   hence P-lim(Partial_Sums Rseq)
          = Sum(lim_in_cod2(Partial_Sums_in_cod2 Rseq))
            by A1,A2,th1006a,DBLSEQ_1:3;
end;
