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

theorem
for n,m being Nat holds
 Rseq.(n+1,m+1) = (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(n+1,m+1)
                - (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(n+1,m)
                - (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(n,m+1)
                + (Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq)).(n,m)
proof
   let n,m be Nat;
   set CPS = Partial_Sums_in_cod1(Partial_Sums_in_cod2 Rseq);
   set CS = Partial_Sums_in_cod2 Rseq;
A1:CPS.(n+1,m+1) = CS.(n+1,m+1) + CPS.(n,m+1) by DefRS
    .= Rseq.(n+1,m+1) + CS.(n+1,m) + CPS.(n,m+1) by DefCS;

   CPS.(n+1,m) = CS.(n+1,m) + CPS.(n,m) by DefRS;
   hence thesis by A1;
end;
