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

theorem
  for Z being open Subset of REAL,
      A being non empty closed_interval Subset of REAL
    st Z = right_open_halfline 0 & A = [.1,n+1.] holds
  integral ((id Z)^,A) = ln.(n + 1)
  proof
    let Z be open Subset of REAL,
        A be non empty closed_interval Subset of REAL;
    assume
Z1: Z = right_open_halfline 0 & A = [.1,n+1.]; then
N1: not 0 in Z by XXREAL_1:4;
A1: A c= Z
    proof
      let x be object;
      assume aa: x in A; then
      reconsider xx = x as Real;
      1 <= xx & xx <= n + 1 by aa,Z1,XXREAL_1:1;
      hence thesis by Z1,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 = 1 by Z1,XREAL_1:31,XXREAL_2:25;
B2: upper_bound A = n + 1 by Z1,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,Z1,TAYLOR_1:18,a3,INTEGRA9:61
        .= ln.(n + 1) - (1 - 1) by ENTROPY1:2,B1,B2
        .= ln.(n + 1);
    hence thesis;
  end;
