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

theorem th02b:
  Partial_Sums Rseq is convergent_in_cod2 implies
    for k being Nat holds
  (lim_in_cod2(Partial_Sums Rseq)).(k+1)
    = (lim_in_cod2(Partial_Sums Rseq)).k
    + (lim_in_cod2(Partial_Sums_in_cod2 Rseq)).(k+1)
proof
   assume A1: Partial_Sums Rseq is convergent_in_cod2;
   let k be Nat;
   reconsider k1=k as Element of NAT by ORDINAL1:def 12;
   Partial_Sums_in_cod2 Rseq is convergent_in_cod2 by A1,th01b; then
A2:ProjMap1(Partial_Sums Rseq,k1) is convergent
 & ProjMap1(Partial_Sums_in_cod2 Rseq,k1+1) is convergent by A1;
A3:(lim_in_cod2(Partial_Sums Rseq)).(k+1)
    = lim ProjMap1(Partial_Sums Rseq,k1+1) by DBLSEQ_1:def 6;
A4:(lim_in_cod2(Partial_Sums Rseq)).k
    = lim ProjMap1(Partial_Sums Rseq,k1)
& (lim_in_cod2(Partial_Sums_in_cod2 Rseq)).(k+1)
    = lim ProjMap1(Partial_Sums_in_cod2 Rseq,k1+1) by DBLSEQ_1:def 6;
   now let j be Element of NAT;
B1: ProjMap1(Partial_Sums Rseq,k1).j
     = (Partial_Sums Rseq).(k1,j) by MESFUNC9:def 6
    .= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(k1,j) by th103a;
    ProjMap1(Partial_Sums_in_cod2 Rseq,k1+1).j
     = (Partial_Sums_in_cod2 Rseq).(k1+1,j) by MESFUNC9:def 6; then
    ProjMap1(Partial_Sums Rseq,k1).j
      + ProjMap1(Partial_Sums_in_cod2 Rseq,k1+1).j
     = (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(k1+1,j)
         by B1,DefRS
    .= (Partial_Sums Rseq).(k1+1,j) by th103a
    .= ProjMap1(Partial_Sums Rseq,k1+1).j by MESFUNC9:def 6;
    hence ProjMap1(Partial_Sums Rseq,k1+1).j
      = (ProjMap1(Partial_Sums Rseq,k1)
         + ProjMap1(Partial_Sums_in_cod2 Rseq,k1+1)).j by VALUED_1:1;
   end; then
   ProjMap1(Partial_Sums Rseq,k1+1)
   = ProjMap1(Partial_Sums Rseq,k1)
     + ProjMap1(Partial_Sums_in_cod2 Rseq,k1+1);
   hence thesis by A3,A4,A2,SEQ_2:6;
end;
