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

theorem
 A c= Z & f=exp_R*tan/cos^2 & Z = dom f & f|A is continuous
 implies integral(f,A)=(exp_R*tan).(upper_bound A)-(exp_R*tan).(lower_bound A)
proof
  assume
A1:A c= Z & f=exp_R*tan/cos^2 & Z = dom f & f|A is continuous; then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
   Z = dom (exp_R*tan) /\ (dom (cos^2) \ (cos^2)"{0}) by A1,RFUNCT_1:def 1;then
A3:Z c= dom (exp_R*tan) by XBOOLE_1:18;
then A4:exp_R*tan is_differentiable_on Z by FDIFF_8:16;
A5:for x st x in Z holds f.x=exp_R.(tan.x)/(cos.x)^2
    proof
    let x;
    assume
A6:x in Z;
   then (exp_R*tan/cos^2).x=(exp_R*tan).x/(cos^2).x by A1,RFUNCT_1:def 1
  .=exp_R.(tan.x)/(cos^2).x by A3,A6,FUNCT_1:12
  .=exp_R.(tan.x)/(cos.x)^2 by VALUED_1:11;
   hence thesis by A1;
   end;
A7:for x being Element of REAL
     st x in dom ((exp_R*tan)`|Z) holds ((exp_R*tan)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((exp_R*tan)`|Z);then
A8:x in Z by A4,FDIFF_1:def 7;then
  ((exp_R*tan)`|Z).x=exp_R.(tan.x)/(cos.x)^2 by A3,FDIFF_8:16
  .=f.x by A5,A8;
  hence thesis;
  end;
  dom ((exp_R*tan)`|Z)=dom f by A1,A4,FDIFF_1:def 7;
  then ((exp_R*tan)`|Z)= f by A7,PARTFUN1:5;
  hence thesis by A1,A2,A3,FDIFF_8:16,INTEGRA5:13;
end;
