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

theorem Diesel3:
  for A being non empty closed_interval Subset of REAL,
      Z being open Subset of REAL,
      n being non zero Nat st
    Z = right_open_halfline 0 & A = [.n, n+1.] holds
      integral ((id Z)^,A) < 1 / n
  proof
    let A be non empty closed_interval Subset of REAL,
        Z be open Subset of REAL,
        n be non zero Nat;
    assume
aa: Z = right_open_halfline 0 & A = [.n, n+1.];
N1: not 0 in Z by aa,XXREAL_1:4;
A1: A c= Z
    proof
      let x be object;
      assume BB: x in A; then
      reconsider xx = x as Real;
      n <= xx & xx <= n + 1 by BB,aa,XXREAL_1:1;
      hence thesis by aa,XXREAL_1:235;
    end;
    set f = id Z;
a3: dom (f^) = dom f \ f"{0} by RFUNCT_1:def 2
            .= Z \ {} by Counter0,N1
            .= Z;
B1: lower_bound A = n by aa,XREAL_1:31,XXREAL_2:25;
B2: upper_bound A = n + 1 by aa,XREAL_1:31,XXREAL_2:29;
    (id Z)^ | A is continuous by ContCut,A1,N1; then
    integral ((id Z)^,A) = ln.(upper_bound A)-ln.(lower_bound A)
      by A1,aa,TAYLOR_1:18,a3,INTEGRA9:61
        .= ln.((n + 1) / n) by B1,B2,DIFF_4:4;
    hence thesis by Diesel2;
  end;
