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

theorem
  A c= Z & dom sec = Z & (for x st x in Z holds f.x =sin.x/(cos.x)^2) &
Z = dom f & f|A is continuous implies integral(f,A) = sec.(upper_bound A)-sec.(
  lower_bound A)
proof
  assume that
A1: A c= Z and
A2: dom sec = Z and
A3: for x st x in Z holds f.x =sin.x/(cos.x)^2 and
A4: Z = dom f and
A5: f|A is continuous;
A6: sec is_differentiable_on Z by A2,FDIFF_9:4;
A7: for x being Element of REAL st x in dom ((sec)`|Z) holds ((sec)`|Z).x = f.x
  proof
    let x be Element of REAL;
    assume x in dom ((sec)`|Z);
    then
A8: x in Z by A6,FDIFF_1:def 7;
    then ((sec)`|Z).x = sin.x/(cos.x)^2 by A2,FDIFF_9:4
      .= f.x by A3,A8;
    hence thesis;
  end;
  dom ((sec)`|Z) = dom f by A4,A6,FDIFF_1:def 7;
  then
A9: ((sec)`|Z) = f by A7,PARTFUN1:5;
  f is_integrable_on A & f|A is bounded by A1,A4,A5,INTEGRA5:10,11;
  hence thesis by A1,A2,A9,FDIFF_9:4,INTEGRA5:13;
end;
