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(#)arctan) & Z c= ].-1,1.[ implies (sec(#)arctan)
is_differentiable_on Z & for x st x in Z holds ((sec(#)arctan)`|Z).x = (sin.x*
  arctan.x)/(cos.x)^2+1/(cos.x*(1+x^2))
proof
  assume that
A1: Z c= dom (sec(#)arctan) and
A2: Z c= ].-1,1.[;
A3: arctan is_differentiable_on Z by A2,SIN_COS9:81;
  Z c= dom sec /\ dom arctan by A1,VALUED_1:def 4;
  then
A4: Z c= dom sec by XBOOLE_1:18;
  for x st x in Z holds sec is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.x <> 0 by A4,RFUNCT_1:3;
    hence thesis by FDIFF_9:1;
  end;
  then
A5: sec is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((sec(#)arctan)`|Z).x = (sin.x*arctan.x)/(cos.x)
  ^2+1/(cos.x*(1+x^2))
  proof
    let x;
    assume
A6: x in Z;
    then
A7: cos.x <> 0 by A4,RFUNCT_1:3;
    ((sec(#)arctan)`|Z).x = (arctan.x)*diff(sec,x)+(sec.x)*diff(arctan,x)
    by A1,A5,A3,A6,FDIFF_1:21
      .= (arctan.x)*(sin.x/(cos.x)^2)+(sec.x)*diff(arctan,x) by A7,FDIFF_9:1
      .= (sin.x*arctan.x)/(cos.x)^2+(sec.x)*((arctan)`|Z).x by A3,A6,
FDIFF_1:def 7
      .= (sin.x*arctan.x)/(cos.x)^2+(sec.x)*(1/(1+x^2)) by A2,A6,SIN_COS9:81
      .= (sin.x*arctan.x)/(cos.x)^2+(1/cos.x)*(1/(1+x^2)) by A4,A6,
RFUNCT_1:def 2
      .= (sin.x*arctan.x)/(cos.x)^2+1/(cos.x*(1+x^2)) by XCMPLX_1:102;
    hence thesis;
  end;
  hence thesis by A1,A5,A3,FDIFF_1:21;
end;
