 reserve a,b,x,r for Real;
 reserve y for set;
 reserve n for Element of NAT;
 reserve A for non empty closed_interval Subset of REAL;
 reserve f,g,f1,f2,g1,g2 for PartFunc of REAL,REAL;
 reserve Z for open Subset of REAL;

theorem
 A c= Z & (for x st x in Z holds f.x=(sin.x)^2/(cos.x)^2)
 & Z c= dom (tan-id Z) & Z = dom f & f|A is continuous implies
 integral(f,A)=(tan-id Z).(upper_bound A)-(tan-id Z).(lower_bound A)
proof
  assume
A1:A c= Z & (for x st x in Z holds f.x=(sin.x)^2/(cos.x)^2)
   & Z c= dom (tan-id Z) & Z = dom f & f|A is continuous;then
A2:f is_integrable_on A & f|A is bounded by INTEGRA5:10,11;
A3:tan-id Z is_differentiable_on Z by A1,FDIFF_8:24;
A4:for x being Element of REAL st x in dom ((tan-id Z)`|Z)
holds ((tan-id Z)`|Z).x=f.x
   proof
   let x be Element of REAL;
   assume x in dom ((tan-id Z)`|Z);then
A5:x in Z by A3,FDIFF_1:def 7;then
   ((tan-id Z)`|Z).x=(sin.x)^2/(cos.x)^2 by A1,FDIFF_8:24
   .=f.x by A1,A5;
   hence thesis;
   end;
   dom ((tan-id Z)`|Z)=dom f by A1,A3,FDIFF_1:def 7;
   then ((tan-id Z)`|Z)= f by A4,PARTFUN1:5;
    hence thesis by A1,A2,FDIFF_8:24,INTEGRA5:13;
end;
