reserve x for Real,

  n for Element of NAT,
   y for set,
  Z for open Subset of REAL,

     g for PartFunc of REAL,REAL;

theorem
  Z c= dom (cosec*arccot) & Z c= ].-1,1.[ implies cosec*arccot
  is_differentiable_on Z & for x st x in Z holds ((cosec*arccot)`|Z).x = cos.(
  arccot.x)/((sin.(arccot.x))^2*(1+x^2))
proof
  assume that
A1: Z c= dom (cosec*arccot) and
A2: Z c= ].-1,1.[;
A3: for x st x in Z holds sin.(arccot.x) <> 0
  proof
    let x;
    assume x in Z;
    then arccot.x in dom cosec by A1,FUNCT_1:11;
    hence thesis by RFUNCT_1:3;
  end;
A4: for x st x in Z holds cosec*arccot is_differentiable_in x
  proof
    let x;
    assume
A5: x in Z;
    then sin.(arccot.x) <> 0 by A3;
    then
A6: cosec is_differentiable_in arccot.x by FDIFF_9:2;
    arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then arccot is_differentiable_in x by A5,FDIFF_1:9;
    hence thesis by A6,FDIFF_2:13;
  end;
  then
A7: cosec*arccot is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((cosec*arccot)`|Z).x = cos.(arccot.x)/((sin.(
  arccot.x))^2*(1+x^2))
  proof
    let x;
    assume
A8: x in Z;
    then
A9: sin.(arccot.x) <> 0 by A3;
    sin.(arccot.x) <> 0 by A3,A8;
    then
A10: cosec is_differentiable_in arccot.x by FDIFF_9:2;
A11: arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then
A12: arccot is_differentiable_in x by A8,FDIFF_1:9;
    ((cosec*arccot)`|Z).x = diff(cosec*arccot,x) by A7,A8,FDIFF_1:def 7
      .= diff(cosec,arccot.x)*diff(arccot,x) by A12,A10,FDIFF_2:13
      .= (-cos.(arccot.x)/(sin.(arccot.x))^2)*diff(arccot,x) by A9,FDIFF_9:2
      .= -(cos.(arccot.x)/(sin.(arccot.x))^2)*diff(arccot,x)
      .= -(cos.(arccot.x)/(sin.(arccot.x))^2)*((arccot)`|Z).x by A8,A11,
FDIFF_1:def 7
      .= -(cos.(arccot.x)/(sin.(arccot.x))^2)*(-1/(1+x^2)) by A2,A8,SIN_COS9:82
      .= (cos.(arccot.x)/(sin.(arccot.x))^2)*(1/(1+x^2))
      .= (cos.(arccot.x)*1)/((sin.(arccot.x))^2*(1+x^2)) by XCMPLX_1:76
      .= cos.(arccot.x)/((sin.(arccot.x))^2*(1+x^2));
    hence thesis;
  end;
  hence thesis by A1,A4,FDIFF_1:9;
end;
