
theorem Th89:
for f be Function of [:NAT,NAT:],ExtREAL, i,j,k be Nat st
 (for m be Element of NAT holds ProjMap2(f,m) is non-decreasing) & i<=j
holds (Partial_Sums_in_cod2 f).(i,k) <= (Partial_Sums_in_cod2 f).(j,k)
proof
   let f be Function of [:NAT,NAT:],ExtREAL, i,j,k be Nat;
   assume that
A1: for m be Element of NAT holds ProjMap2(f,m) is non-decreasing and
A2: i <= j;
   reconsider i1=i, j1=j as Element of NAT by ORDINAL1:def 12;
   defpred P[Nat] means
    (Partial_Sums_in_cod2 f).(i,$1) <= (Partial_Sums_in_cod2 f).(j,$1);
   ProjMap2(f,0).i1 = f.(i,0) & ProjMap2(f,0).j1 = f.(j,0)
     by MESFUNC9:def 7; then
   ProjMap2(f,0).i1 = (Partial_Sums_in_cod2 f).(i,0) &
   ProjMap2(f,0).j1 = (Partial_Sums_in_cod2 f).(j,0) by DefCSM; then
A4:P[0] by A1,A2,RINFSUP2:7;
A5:for n be Nat st P[n] holds P[n+1]
   proof
    let n be Nat;
    assume A6: P[n];
A7: (Partial_Sums_in_cod2 f).(i,n+1)
     = (Partial_Sums_in_cod2 f).(i,n) + f.(i,n+1)
  & (Partial_Sums_in_cod2 f).(j,n+1)
     = (Partial_Sums_in_cod2 f).(j,n) + f.(j,n+1) by DefCSM;
A8: ProjMap2(f,n+1).i <= ProjMap2(f,n+1).j by A1,A2,RINFSUP2:7;
    ProjMap2(f,n+1).i1 = f.(i,n+1) & ProjMap2(f,n+1).j1 = f.(j,n+1)
      by MESFUNC9:def 7;
    hence P[n+1] by A6,A7,A8,XXREAL_3:36;
   end;
   for n be Nat holds P[n] from NAT_1:sch 2(A4,A5);
   hence thesis;
end;
