 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 Th3:
  Z c= dom (cosec*ln) implies -cosec*ln is_differentiable_on Z &
  for x st x in Z holds ((-cosec*ln)`|Z).x = cos.(ln.x)/(x*(sin.(ln.x))^2)
proof
  assume
A1:Z c= dom (cosec*ln);
then A2:Z c= dom (-cosec*ln) by VALUED_1:8;
A3:cosec*ln is_differentiable_on Z by A1,FDIFF_9:15;
then A4:(-1)(#)(cosec*ln) is_differentiable_on Z by A2,FDIFF_1:20;
  for x st x in Z holds ((-cosec*ln)`|Z).x = cos.(ln.x)/(x*(sin.(ln.x))^2)
  proof
    let x;
    assume
A5: x in Z;
 ((-cosec*ln)`|Z).x=((-1)(#)((cosec*ln)`|Z)).x by A3,FDIFF_2:19
                 .=(-1)*(((cosec*ln)`|Z).x) by VALUED_1:6
                 .=(-1)*(-cos.(ln.x)/(x*(sin.(ln.x))^2))
   by A1,A5,FDIFF_9:15
                  .=cos.(ln.x)/(x*(sin.(ln.x))^2);
    hence thesis;
  end;
  hence thesis by A4;
end;
