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

theorem th03a:
  Partial_Sums Rseq is convergent_in_cod1 implies
    lim_in_cod1(Partial_Sums Rseq)
      = Partial_Sums(lim_in_cod1(Partial_Sums_in_cod1 Rseq))
proof
   assume AS: Partial_Sums Rseq is convergent_in_cod1;
   now let m be Nat;
    reconsider m1=m as Element of NAT by ORDINAL1:def 12;
    defpred P[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: P[0];
A2: for k being Nat st P[k] holds P[k+1]
    proof
     let k be Nat;
     assume A3: P[k];
     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 thesis by AS,th02a;
    end;
    for k being Nat holds P[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;
   end;
   hence thesis;
end;
