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

theorem thRS2:
  for n,m being Nat holds
 Rseq.(n+1,m+1) = (Partial_Sums Rseq).(n+1,m+1)
                - (Partial_Sums Rseq).(n,m+1)
                - (Partial_Sums Rseq).(n+1,m)
                + (Partial_Sums Rseq).(n,m)
proof
   let n,m be Nat;
   set RPS = Partial_Sums_in_cod2(Partial_Sums_in_cod1 Rseq);
A1:RPS.(n+1,m+1)
    = (Partial_Sums_in_cod1(Rseq)).(n+1,m+1) + RPS.(n+1,m) by DefCS
   .= Rseq.(n+1,m+1) + (Partial_Sums_in_cod1(Rseq)).(n,m+1)
     + RPS.(n+1,m) by DefRS;
   RPS.(n,m+1) = (Partial_Sums_in_cod1(Rseq)).(n,m+1) + RPS.(n,m) by DefCS;
   hence thesis by A1;
end;
