 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=sin/cos/(id Z)+ln/cos^2
 & Z c= dom (ln(#)tan) & Z = dom f & f|A is continuous implies
 integral(f,A)=(ln(#)tan).(upper_bound A)-(ln(#)tan).(lower_bound A)
proof
  assume
A1:A c= Z & f=sin/cos/(id Z)+ln/cos^2
   & Z c= dom (ln(#)tan) & Z = dom f & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:(ln(#)tan) is_differentiable_on Z by A1,FDIFF_8:32;
   Z = dom (sin/cos/(id Z)) /\ dom (ln/cos^2) by A1,VALUED_1:def 1; then
A4:Z c= dom (sin/cos/(id Z)) & Z c= dom (ln/cos^2) by XBOOLE_1:18;
   dom (sin/cos/(id Z)) c= dom (sin/cos) /\ (dom (id Z) \ (id Z)"{0})
   by RFUNCT_1:def 1;then
   dom (sin/cos/(id Z)) c= dom (sin/cos) by XBOOLE_1:18;then
A5:Z c= dom (sin/cos) by A4;
A6:for x st x in Z holds f.x=sin.x/cos.x/x+ln.x/(cos.x)^2
   proof
   let x;
   assume
A7:x in Z;then
   (sin/cos/(id Z)+ln/cos^2).x=(sin/cos/(id Z)).x+(ln/cos^2).x
    by A1,VALUED_1:def 1
   .=(sin/cos).x/(id Z).x+(ln/cos^2).x by A7,A4,RFUNCT_1:def 1
   .=sin.x/cos.x/(id Z).x+(ln/cos^2).x by A5,A7,RFUNCT_1:def 1
   .=sin.x/cos.x/x+(ln/cos^2).x by A7,FUNCT_1:18
   .=sin.x/cos.x/x+ln.x/(cos^2).x by A7,A4,RFUNCT_1:def 1
   .=sin.x/cos.x/x+ln.x/(cos.x)^2 by VALUED_1:11;
    hence thesis by A1;
    end;
A8:for x being Element of REAL
     st x in dom ((ln(#)tan)`|Z) holds ((ln(#)tan)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((ln(#)tan)`|Z);then
A9:x in Z by A3,FDIFF_1:def 7;then
  ((ln(#)tan)`|Z).x=sin.x/cos.x/x+ln.x/(cos.x)^2 by A1,FDIFF_8:32
  .=f.x by A9,A6;
  hence thesis;
  end;
  dom ((ln(#)tan)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
  then ((ln(#)tan)`|Z)= f by A8,PARTFUN1:5;
  hence thesis by A1,A2,FDIFF_8:32,INTEGRA5:13;
end;
