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 (cos*arccot) & Z c= ].-1,1.[ implies cos*arccot
  is_differentiable_on Z & for x st x in Z holds ((cos*arccot)`|Z).x = sin.(
  arccot.x)/(1+x^2)
proof
  assume that
A1: Z c= dom (cos*arccot) and
A2: Z c= ].-1,1.[;
A3: for x st x in Z holds cos*arccot is_differentiable_in x
  proof
    let x;
    assume
A4: x in Z;
A5: cos is_differentiable_in arccot.x by SIN_COS:63;
    arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then arccot is_differentiable_in x by A4,FDIFF_1:9;
    hence thesis by A5,FDIFF_2:13;
  end;
  then
A6: cos*arccot is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((cos*arccot)`|Z).x = sin.(arccot.x)/(1+x^2)
  proof
    let x;
    assume
A7: x in Z;
A8: arccot is_differentiable_on Z by A2,SIN_COS9:82;
    then
A9: arccot is_differentiable_in x by A7,FDIFF_1:9;
A10: cos is_differentiable_in arccot.x by SIN_COS:63;
    ((cos*arccot)`|Z).x = diff(cos*arccot,x) by A6,A7,FDIFF_1:def 7
      .= diff(cos,arccot.x)*diff(arccot,x) by A9,A10,FDIFF_2:13
      .= (-sin.(arccot.x))*diff(arccot,x) by SIN_COS:63
      .= -sin.(arccot.x)*diff(arccot,x)
      .= -sin.(arccot.x)*((arccot)`|Z).x by A7,A8,FDIFF_1:def 7
      .= -sin.(arccot.x)*(-1/(1+x^2)) by A2,A7,SIN_COS9:82
      .= sin.(arccot.x)/(1+x^2);
    hence thesis;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
