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 (exp_R*sec) implies exp_R*sec is_differentiable_on Z & for x
  st x in Z holds ((exp_R*sec)`|Z).x = exp_R.(sec.x)*sin.x/(cos.x)^2
proof
  assume
A1: Z c= dom (exp_R*sec);
A2: for x st x in Z holds cos.x<>0
  proof
    let x;
    assume x in Z;
    then x in dom sec by A1,FUNCT_1:11;
    hence thesis by RFUNCT_1:3;
  end;
A3: for x st x in Z holds exp_R*sec is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.x<>0 by A2;
    then
A4: sec is_differentiable_in x by Th1;
    exp_R is_differentiable_in sec.x by SIN_COS:65;
    hence thesis by A4,FDIFF_2:13;
  end;
  then
A5: exp_R*sec is_differentiable_on Z by A1,FDIFF_1:9;
  for x st x in Z holds ((exp_R*sec)`|Z).x = exp_R.(sec.x)*sin.x/(cos.x) ^2
  proof
    let x;
A6: exp_R is_differentiable_in sec.x by SIN_COS:65;
    assume
A7: x in Z;
    then
A8: cos.x<>0 by A2;
    then sec is_differentiable_in x by Th1;
    then diff(exp_R*sec,x) = diff(exp_R,sec.x)*diff(sec,x) by A6,FDIFF_2:13
      .=diff(exp_R,sec.x)*(sin.x/(cos.x)^2) by A8,Th1
      .=exp_R.(sec.x)*(sin.x/(cos.x)^2) by SIN_COS:65;
    hence thesis by A5,A7,FDIFF_1:def 7;
  end;
  hence thesis by A1,A3,FDIFF_1:9;
end;
