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