reserve f,f1,f2,g for PartFunc of REAL,REAL;
reserve A for non empty closed_interval Subset of REAL;
reserve p,r,x,x0 for Real;
reserve n for Element of NAT;
reserve Z for open Subset of REAL;

theorem
  dom tanh = dom f2 & (for x st x in REAL holds f2.x = 1/(cosh.x)^2) &
  f2|A is continuous implies integral(f2,A) = tanh.(upper_bound A)-tanh.(
  lower_bound A)
proof
  assume that
A1: dom tanh = dom f2 and
A2: for x st x in REAL holds f2.x = 1/(cosh.x)^2 and
A3: f2|A is continuous;
A4: for x being Element of REAL st x in dom (tanh`|REAL)
     holds (tanh`|REAL).x = f2.x
  proof
    let x be Element of REAL;
    assume x in dom (tanh`|REAL);
    (tanh`|REAL).x =diff(tanh,x) by FDIFF_1:def 7,SIN_COS2:36
      .= 1/(cosh.x)^2 by SIN_COS2:33
      .= f2.x by A2;
    hence thesis;
  end;
  dom ((tanh)`|REAL) = dom f2 by A1,FDIFF_1:def 7,SIN_COS2:30,36;
  then
A5: ((tanh)`|REAL) = f2 by A4,PARTFUN1:5;
A6: dom tanh = REAL by FUNCT_2:def 1;
  then f2 is_integrable_on A by A1,A3,INTEGRA5:11;
  hence thesis by A1,A3,A6,A5,INTEGRA5:10,13,SIN_COS2:36;
end;
