
theorem Th63:
  for f be nonnegative Function of [:NAT,NAT:],ExtREAL, m be Element of NAT
    holds ProjMap1(Partial_Sums_in_cod2 f,m) is non-decreasing
proof
   let f be nonnegative Function of [:NAT,NAT:],ExtREAL,
       m be Element of NAT;
   set PS = ProjMap1(Partial_Sums_in_cod2 f,m);
   for n,j be Nat st j <= n holds PS.j <= PS.n
   proof
    let n,j be Nat;
    defpred Q[Nat] means PS.j <= PS.$1;
A6: for k be Nat holds PS.k <= PS.(k+1)
    proof
     let k be Nat;
     reconsider k1=k as Element of NAT by ORDINAL1:def 12;
     PS.(k+1) = (Partial_Sums_in_cod2 f).(m,k+1) by MESFUNC9:def 6
       .= (Partial_Sums_in_cod2 f).(m,k1) + f.(m,k+1) by DefCSM
       .= PS.k + f.(m,k+1) by MESFUNC9:def 6;
     hence thesis by SUPINF_2:51,XXREAL_3:39;
    end;
A8: for k be Nat st k >= j & (for l be Nat st l >= j & l < k holds Q[l])
    holds Q[k]
    proof
     let k be Nat;
     assume that
A9:  k >= j and
A10: for l be Nat st l >= j & l < k holds Q[l];
     now assume k > j; then
A11:  k >= j + 1 by NAT_1:13;
      per cases by A11,XXREAL_0:1;
      suppose k = j + 1;
       hence thesis by A6;
      end;
      suppose
A12:   k > j + 1;
       then reconsider l = k-1 as Element of NAT by NAT_1:20;
       k < k+1 by NAT_1:13; then
A13:   k > l by XREAL_1:19;
       k = l+1; then
A14:   PS.l <= PS.k by A6;
       PS.j <= PS.l by A10,A13,A12,XREAL_1:19;
       hence thesis by A14,XXREAL_0:2;
      end;
     end;
     hence thesis by A9,XXREAL_0:1;
    end;
A15: for k be Nat st k >= j holds Q[k] from NAT_1:sch 9(A8);
    assume j <= n;
    hence thesis by A15;
  end;
  hence thesis by RINFSUP2:7;
end;
