 reserve n,i,k,m for Nat;
 reserve p for Prime;

theorem Impor3:
  for f1, f2 being Real_Sequence,
      n being non trivial Nat st
    (for k being non trivial Nat st k <= n holds f1.k < f2.k) holds
      Sum (f1, n, 1) < Sum (f2, n, 1)
  proof
    let f1, f2 be Real_Sequence,
        n be non trivial Nat;
    assume
A1: for k being non trivial Nat st k <= n holds f1.k < f2.k;
    defpred X[Nat] means
      (for k being non trivial Nat st k <= $1 holds
        f1.k < f2.k) implies
      Partial_Sums(f1).$1 - Partial_Sums(f1).1 <
        Partial_Sums(f2).$1 - Partial_Sums(f2).1;
     Partial_Sums(f1).2 = Partial_Sums(f1).1 + f1.(1+1) &
      Partial_Sums(f2).2 = Partial_Sums(f2).1 + f2.(1+1) by SERIES_1:def 1;
        then
a1: X[2];
A2: for n being non trivial Nat st X[n] holds X[n+1]
    proof
      let n be non trivial Nat such that
A3:   X[n];
      assume
B1:   for k being non trivial Nat st k <= n+1 holds f1.k < f2.k;
A4:   f1.(n+1) < f2.(n+1) by B1;
ZZ:   Partial_Sums(f1).(n+1) = Partial_Sums(f1).n + f1.(n+1) &
        Partial_Sums(f2).(n+1) = Partial_Sums(f2).n + f2.(n+1)
          by SERIES_1:def 1;
G1:   n <= n + 1 by NAT_1:11;
      for k being non trivial Nat st k <= n holds f1.k < f2.k
      proof
        let k be non trivial Nat;
        assume k <= n; then
        k <= n + 1 by G1,XXREAL_0:2;
        hence thesis by B1;
      end; then
      f1.(n+1) + (Partial_Sums(f1).n - Partial_Sums(f1).1) <
        f2.(n+1) + (Partial_Sums(f2).n - Partial_Sums(f2).1)
          by A3,A4,XREAL_1:8;
      hence thesis by ZZ;
    end;
    for n being non trivial Nat holds X[n] from NAT_2:sch 2(a1,A2);
    hence thesis by A1;
  end;
