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

theorem
A c= Z & Z c= dom (ln*cosec) & Z = dom cot & cot|A is continuous
implies integral(cot,A)=(-ln*cosec).(upper_bound A)-(-ln*cosec).(lower_bound A)
proof
  assume
A1:A c= Z & Z c= dom (ln*cosec) & Z = dom cot & cot|A is continuous;
   then
A2:cot is_integrable_on A & cot|A is bounded by INTEGRA5:10,11;
A3:-ln*cosec is_differentiable_on Z by A1,Th5;
A4:for x being Element of REAL
    st x in dom ((-ln*cosec)`|Z) holds ((-ln*cosec)`|Z).x = cot.x
   proof
     let x be Element of REAL;
     assume x in dom ((-ln*cosec)`|Z);then
  x in Z by A3,FDIFF_1:def 7;
     then ((-ln*cosec)`|Z).x=cot.x by A1,Th5;
     hence thesis;
   end;
  dom ((-ln*cosec)`|Z)=dom cot by A1,A3,FDIFF_1:def 7;
  then ((-ln*cosec)`|Z)= cot by A4,PARTFUN1:5;
  hence thesis by A1,A2,A3,INTEGRA5:13;
end;
