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

theorem th1006:
  Rseq is nonnegative-yielding &
  Partial_Sums Rseq is P-convergent implies
   Partial_Sums_in_cod1 Rseq is convergent_in_cod1
 & Partial_Sums_in_cod2 Rseq is convergent_in_cod2
proof
   assume that
A1: Rseq is nonnegative-yielding and
A2: Partial_Sums Rseq is P-convergent;
   now let k be Element of NAT;
B1: ProjMap2(Partial_Sums_in_cod1 Rseq,k) is non-decreasing by A1,th1005;
    now let n be Nat;
     reconsider n1=n as Element of NAT by ORDINAL1:def 12;
B2:  (Partial_Sums Rseq).(n,k)
      = (Partial_Sums(ProjMap1(Partial_Sums_in_cod1 Rseq,n1))).k by th100;
B3:  (ProjMap2(Partial_Sums_in_cod1 Rseq,k)).n1
      = (Partial_Sums_in_cod1 Rseq).(n1,k) by MESFUNC9:def 7
     .= (ProjMap1(Partial_Sums_in_cod1 Rseq,n1)).k by MESFUNC9:def 6;
     now let d be Nat;
      ProjMap1(Partial_Sums_in_cod1 Rseq,n1) is nonnegative by A1,th1005;
      hence (ProjMap1(Partial_Sums_in_cod1 Rseq,n1)).d >= 0;
     end; then
     ProjMap1(Partial_Sums_in_cod1 Rseq,n1) is nonnegative-yielding; then
B4:  (ProjMap2(Partial_Sums_in_cod1 Rseq,k)).n1
      <= (Partial_Sums Rseq).(n,k) by B2,B3,th101;
     consider N be Nat such that
B6:   for n,m being Nat st n>=N & m>=N holds
       |. (Partial_Sums Rseq).(n,m) - P-lim(Partial_Sums Rseq) .| < 1
         by A2,DBLSEQ_1:def 2;
     reconsider N1 = max(N,max(k,n)) as Nat by TARSKI:1;
B7:  N1>=N & N1>=max(k,n) & max(k,n)>=k & max(k,n)>=n by XXREAL_0:25; then
     |.(Partial_Sums Rseq).(N1,N1)- P-lim(Partial_Sums Rseq).| < 1 by B6; then
     (Partial_Sums Rseq).(N1,N1) - P-lim(Partial_Sums Rseq) <= 1
       by ABSVALUE:5; then
     (Partial_Sums Rseq).(N1,N1) - P-lim(Partial_Sums Rseq) < 2
       by XXREAL_0:2; then
B8:  (Partial_Sums Rseq).(N1,N1) < P-lim(Partial_Sums Rseq) + 2 by XREAL_1:19;
B9:  N1>=k & N1>=n by B7,XXREAL_0:2;
     Partial_Sums Rseq is non-decreasing by A1,th80a; then
     (Partial_Sums Rseq).(n,k) <= (Partial_Sums Rseq).(N1,N1) by B9; then
     (ProjMap2(Partial_Sums_in_cod1 Rseq,k)).n
       <= (Partial_Sums Rseq).(N1,N1) by B4,XXREAL_0:2;
     hence (ProjMap2(Partial_Sums_in_cod1 Rseq,k)).n
             < P-lim(Partial_Sums Rseq) + 2 by B8,XXREAL_0:2;
    end; then
    ProjMap2(Partial_Sums_in_cod1 Rseq,k) is bounded_above by SEQ_2:def 3;
    hence ProjMap2(Partial_Sums_in_cod1 Rseq,k) is convergent by B1;
   end;
   hence Partial_Sums_in_cod1 Rseq is convergent_in_cod1;
   now let k be Element of NAT;
C1: ProjMap1(Partial_Sums_in_cod2 Rseq,k) is non-decreasing by A1,th1005;
    now let n be Nat;
     reconsider n1=n as Element of NAT by ORDINAL1:def 12;
B2:  (Partial_Sums Rseq).(k,n)
      = (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(k,n) by th103
     .= (Partial_Sums(ProjMap2(Partial_Sums_in_cod2 Rseq,n1))).k by th100;
B3:  (ProjMap1(Partial_Sums_in_cod2 Rseq,k)).n1
      = (Partial_Sums_in_cod2 Rseq).(k,n1) by MESFUNC9:def 6
     .= (ProjMap2(Partial_Sums_in_cod2 Rseq,n1)).k by MESFUNC9:def 7;
     now let d be Nat;
      ProjMap2(Partial_Sums_in_cod2 Rseq,n1) is nonnegative by A1,th1005;
      hence (ProjMap2(Partial_Sums_in_cod2 Rseq,n1)).d >= 0;
     end; then
     ProjMap2(Partial_Sums_in_cod2 Rseq,n1) is nonnegative-yielding; then
B4:  (ProjMap1(Partial_Sums_in_cod2 Rseq,k)).n1
      <= (Partial_Sums Rseq).(k,n) by B2,B3,th101;
     consider N be Nat such that
B6:   for n,m being Nat st n>=N & m>=N holds
       |. (Partial_Sums Rseq).(n,m) - P-lim(Partial_Sums Rseq) .| < 1
         by A2,DBLSEQ_1:def 2;
     reconsider N1 = max(N,max(k,n)) as Nat by TARSKI:1;
B7:  N1>=N & N1>=max(k,n) & max(k,n)>=k & max(k,n)>=n by XXREAL_0:25; then
     |.(Partial_Sums Rseq).(N1,N1)- P-lim(Partial_Sums Rseq).| < 1 by B6; then
     (Partial_Sums Rseq).(N1,N1) - P-lim(Partial_Sums Rseq) <= 1
       by ABSVALUE:5; then
     (Partial_Sums Rseq).(N1,N1) - P-lim(Partial_Sums Rseq) < 2
       by XXREAL_0:2; then
B8:  (Partial_Sums Rseq).(N1,N1) < P-lim(Partial_Sums Rseq) + 2 by XREAL_1:19;
B9:  N1>=k & N1>=n by B7,XXREAL_0:2;
     Partial_Sums Rseq is non-decreasing by A1,th80a; then
     (Partial_Sums Rseq).(k,n) <= (Partial_Sums Rseq).(N1,N1) by B9; then
     (ProjMap1(Partial_Sums_in_cod2 Rseq,k)).n
       <= (Partial_Sums Rseq).(N1,N1) by B4,XXREAL_0:2;
     hence (ProjMap1(Partial_Sums_in_cod2 Rseq,k)).n
             < P-lim(Partial_Sums Rseq) + 2 by B8,XXREAL_0:2;
    end; then
    ProjMap1(Partial_Sums_in_cod2 Rseq,k) is bounded_above by SEQ_2:def 3;
    hence ProjMap1(Partial_Sums_in_cod2 Rseq,k) is convergent by C1;
   end;
   hence Partial_Sums_in_cod2 Rseq is convergent_in_cod2;
end;
