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

theorem th01a:
  Partial_Sums Rseq is convergent_in_cod1 iff
    Partial_Sums_in_cod1 Rseq is convergent_in_cod1
proof
   hereby assume A1: Partial_Sums Rseq is convergent_in_cod1;
    now let m be Element of NAT;
     defpred P[Nat] means
      for k be Element of NAT st k = $1 holds
       ProjMap2(Partial_Sums_in_cod1 Rseq,k) is convergent;
     now let k be Element of NAT;
      assume k = 0; then
      ProjMap2(Partial_Sums Rseq,k) = ProjMap2(Partial_Sums_in_cod1 Rseq,k)
        by th00;
      hence ProjMap2(Partial_Sums_in_cod1 Rseq,k) is convergent by A1;
     end; then
A3:  P[0];
A4:  for m1 being Nat st P[m1] holds P[m1+1]
     proof
      let m1 be Nat;
      reconsider m=m1 as Element of NAT by ORDINAL1:def 12;
      assume P[m1];
      hereby let k be Element of NAT;
       assume B2: k = m1+1; then
       reconsider k1 = k-1 as Element of NAT by NAT_1:11,21;
B4:    ProjMap2(Partial_Sums Rseq,m) is convergent
     & ProjMap2(Partial_Sums Rseq,k) is convergent by A1;
       now let e be Real;
        assume B6: 0<e; then
        consider N1 be Nat such that
B7:      for n being Nat st n>=N1 holds
          |.ProjMap2(Partial_Sums Rseq,m).n
             - lim(ProjMap2(Partial_Sums Rseq,m)).| < e/2 by B4,SEQ_2:def 7;
        consider N2 be Nat such that
B8:      for n being Nat st n>=N2 holds
          |.ProjMap2(Partial_Sums Rseq,k).n
             - lim(ProjMap2(Partial_Sums Rseq,k)).| < e/2
                by B4,B6,SEQ_2:def 7;
        reconsider N=max(N1,N2) as Nat by TARSKI:1;
        take N;
        hereby let n be Nat;
         assume B9: n>=N;
         N >= N1 & N >= N2 by XXREAL_0:25; then
         n >= N1 & n >= N2 by B9,XXREAL_0:2; then
         |.ProjMap2(Partial_Sums Rseq,m).n
             - lim(ProjMap2(Partial_Sums Rseq,m)).| < e/2
      & |.ProjMap2(Partial_Sums Rseq,k).n
             - lim(ProjMap2(Partial_Sums Rseq,k)).| < e/2 by B7,B8; then
B12:     |.ProjMap2(Partial_Sums Rseq,m).n
             - lim(ProjMap2(Partial_Sums Rseq,m)).|
          + |.ProjMap2(Partial_Sums Rseq,k).n
               - lim(ProjMap2(Partial_Sums Rseq,k)).| < e/2 + e/2
               by XREAL_1:8;
         reconsider n1=n as Element of NAT by ORDINAL1:def 12;
         ProjMap2(Partial_Sums Rseq,k).n
          = (Partial_Sums Rseq).(n1,k) by MESFUNC9:def 7
         .= (Partial_Sums Rseq).(n1,m)
              + (Partial_Sums_in_cod1 Rseq).(n1,k) by B2,DefCS
         .= ProjMap2(Partial_Sums Rseq,m).n
              + (Partial_Sums_in_cod1 Rseq).(n1,k) by MESFUNC9:def 7
         .= ProjMap2(Partial_Sums Rseq,m).n
              + ProjMap2(Partial_Sums_in_cod1 Rseq,k).n by MESFUNC9:def 7; then
         |.ProjMap2(Partial_Sums_in_cod1 Rseq,k).n
             - ( lim(ProjMap2(Partial_Sums Rseq,k))
                - lim(ProjMap2(Partial_Sums Rseq,m)) ).|
           = |.( ProjMap2(Partial_Sums Rseq,k).n
                   - lim(ProjMap2(Partial_Sums Rseq,k)) )
              -( ProjMap2(Partial_Sums Rseq,m).n
                   - lim(ProjMap2(Partial_Sums Rseq,m)) ).|; then
         |.ProjMap2(Partial_Sums_in_cod1 Rseq,k).n
             - ( lim(ProjMap2(Partial_Sums Rseq,k))
                - lim(ProjMap2(Partial_Sums Rseq,m)) ).|
           <= |.ProjMap2(Partial_Sums Rseq,k).n
                   - lim(ProjMap2(Partial_Sums Rseq,k)).|
              + |.ProjMap2(Partial_Sums Rseq,m).n
                   - lim(ProjMap2(Partial_Sums Rseq,m)).| by COMPLEX1:57;
         hence |.ProjMap2(Partial_Sums_in_cod1 Rseq,k).n
             - ( lim(ProjMap2(Partial_Sums Rseq,k))
                - lim(ProjMap2(Partial_Sums Rseq,m)) ).| < e by B12,XXREAL_0:2;
        end;
       end;
       hence ProjMap2(Partial_Sums_in_cod1 Rseq,k) is convergent
         by SEQ_2:def 6;
      end;
     end;
     for m1 being Nat holds P[m1] from NAT_1:sch 2(A3,A4);
     hence ProjMap2(Partial_Sums_in_cod1 Rseq,m) is convergent;
    end;
    hence Partial_Sums_in_cod1 Rseq is convergent_in_cod1;
   end;
   assume C0: Partial_Sums_in_cod1 Rseq is convergent_in_cod1;
   now let m be Element of NAT;
    defpred P[Nat] means
     for k being Element of NAT st k = $1 holds
     ProjMap2(Partial_Sums Rseq,k) is convergent;
    ProjMap2(Partial_Sums Rseq,0) = ProjMap2(Partial_Sums_in_cod1 Rseq,0)
      by th00; then
C1: P[0] by C0;
C2: for m being Nat st P[m] holds P[m+1]
    proof
     let m be Nat;
     assume C3: P[m];
     reconsider m1=m as Element of NAT by ORDINAL1:def 12;
     hereby let k be Element of NAT;
      assume C4: k=m+1; then
      reconsider k1=k-1 as Element of NAT by NAT_1:11,21;
      for n being Element of NAT holds
       ProjMap2(Partial_Sums Rseq,k).n
       = (ProjMap2(Partial_Sums Rseq,m1)
         + ProjMap2(Partial_Sums_in_cod1 Rseq,m1+1)).n
      proof
       let n be Element of NAT;
       ProjMap2(Partial_Sums Rseq,k).n
        = (Partial_Sums Rseq).(n,m1+1) by C4,MESFUNC9:def 7
       .= (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(n,m1+1) by th103a
       .= (Partial_Sums_in_cod1 Rseq).(n,m1+1)
            + (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(n,m1) by ThRS
       .= (Partial_Sums_in_cod1 Rseq).(n,m1+1)
            + (Partial_Sums Rseq).(n,m1) by th103a
       .= ProjMap2(Partial_Sums_in_cod1 Rseq,m1+1).n
            + (Partial_Sums Rseq).(n,m1) by MESFUNC9:def 7
       .= ProjMap2(Partial_Sums_in_cod1 Rseq,m1+1).n
            + ProjMap2(Partial_Sums Rseq,m1).n by MESFUNC9:def 7;
       hence thesis by VALUED_1:1;
      end; then
C5:   ProjMap2(Partial_Sums Rseq,k)
       = ProjMap2(Partial_Sums Rseq,m1)
         + ProjMap2(Partial_Sums_in_cod1 Rseq,m1+1);
      ProjMap2(Partial_Sums Rseq,m1) is convergent
    & ProjMap2(Partial_Sums_in_cod1 Rseq,m1+1) is convergent by C3,C0;
      hence ProjMap2(Partial_Sums Rseq,k) is convergent by C5,SEQ_2:5;
     end;
    end;
    for m being Nat holds P[m] from NAT_1:sch 2(C1,C2);
    hence ProjMap2(Partial_Sums Rseq,m) is convergent;
   end;
   hence Partial_Sums Rseq is convergent_in_cod1;
end;
