
theorem
for f be Function of [:NAT,NAT:],ExtREAL, i,j,k be Nat st
 (for n be Element of NAT holds ProjMap1(f,n) is non-decreasing) & i<=j
holds (Partial_Sums_in_cod1 f).(k,i) <= (Partial_Sums_in_cod1 f).(k,j)
proof
   let f be Function of [:NAT,NAT:],ExtREAL, i,j,k be Nat;
   assume that
A1: for n be Element of NAT holds ProjMap1(f,n) is non-decreasing and
A2: i <= j;
   for n be Element of NAT holds ProjMap2(~f,n) is non-decreasing
   proof
    let n be Element of NAT;
    ProjMap1(f,n) = ProjMap2(~f,n) by Th32;
    hence thesis by A1;
   end; then
   (Partial_Sums_in_cod2 ~f).(i,k) <= (Partial_Sums_in_cod2 ~f).(j,k)
     by A2,Th89; then
   (Partial_Sums_in_cod1 f).(k,i) <= (Partial_Sums_in_cod2 ~f).(j,k)
     by Th39;
   hence thesis by Th39;
end;
