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

theorem
  Rseq is nonnegative-yielding implies
  (for i1,i2,j1,j2 being Nat st i1 <= i2 & j1 <= j2 holds
      (Partial_Sums Rseq).(i1,j1) <= (Partial_Sums Rseq).(i2,j2))
proof
   assume A1: Rseq is nonnegative-yielding;
   hereby let i1,i2,j1,j2 be Nat;
    assume i1<=i2 & j1<=j2; then
    (Partial_Sums Rseq).(i1,j1) <= (Partial_Sums Rseq).(i1,j2)
  & (Partial_Sums Rseq).(i1,j2) <= (Partial_Sums Rseq).(i2,j2)
       by A1,th105;
    hence (Partial_Sums Rseq).(i1,j1) <= (Partial_Sums Rseq).(i2,j2)
      by XXREAL_0:2;
   end;
end;
