reserve x,a,b,c for Real,
  n for Nat,
  Z for open Subset of REAL,
  f, f1,f2 for PartFunc of REAL,REAL;

theorem
  Z c= dom (sec*cos) implies sec*cos is_differentiable_on Z & for x st x
  in Z holds ((sec*cos)`|Z).x = -sin.x*sin.(cos.x)/(cos.(cos.x))^2
proof
  assume
A1: Z c= dom (sec*cos);
A2: for x st x in Z holds cos.(cos.x)<>0
  proof
    let x;
    assume x in Z;
    then cos.x in dom sec by A1,FUNCT_1:11;
    hence thesis by RFUNCT_1:3;
  end;
A3: for x st x in Z holds sec*cos is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.(cos.x)<>0 by A2;
    then cos is_differentiable_in x & sec is_differentiable_in cos.x by Th1,
SIN_COS:63;
    hence thesis by FDIFF_2:13;
  end;
  then
A4: sec*cos is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((sec*cos)`|Z).x = -sin.x*sin.(cos.x)/(cos.(cos.x ))^2
  proof
    let x;
    assume
A5: x in Z;
    then
A6: cos.(cos.x)<>0 by A2;
    then cos is_differentiable_in x & sec is_differentiable_in cos.x by Th1,
SIN_COS:63;
    then diff(sec*cos,x) = diff(sec, cos.x)*diff(cos,x) by FDIFF_2:13
      .=(sin.(cos.x)/(cos.(cos.x))^2) * diff(cos,x) by A6,Th1
      .=(-sin.x)*(sin.(cos.x)/(cos.(cos.x))^2) by SIN_COS:63;
    hence thesis by A4,A5,FDIFF_1:def 7;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
