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(#)tan) implies (exp_R(#)tan) is_differentiable_on Z &
for x st x in Z holds ((exp_R(#)tan)`|Z).x = exp_R.x*sin.x/cos.x+exp_R.x/(cos.x
  )^2
proof
A1: for x st x in Z holds exp_R is_differentiable_in x by SIN_COS:65;
  assume
A2: Z c= dom (exp_R(#)tan);
  then
A3: Z c= dom (exp_R) /\ dom tan by VALUED_1:def 4;
  then
A4: Z c= dom tan by XBOOLE_1:18;
  Z c= dom (exp_R) by A3,XBOOLE_1:18;
  then
A5: exp_R is_differentiable_on Z by A1,FDIFF_1:9;
A6: for x st x in Z holds tan is_differentiable_in x & diff(tan, x)=1/(cos.x
  )^2
  proof
    let x;
    assume x in Z;
    then cos.x<>0 by A4,Th1;
    hence thesis by FDIFF_7:46;
  end;
  then for x st x in Z holds tan is_differentiable_in x;
  then
A7: tan is_differentiable_on Z by A4,FDIFF_1:9;
  for x st x in Z holds ((exp_R(#)tan)`|Z).x = exp_R.x*sin.x/cos.x+exp_R.
  x/(cos.x)^2
  proof
    let x;
    assume
A8: x in Z;
    then
    ((exp_R(#)tan)`|Z).x = (tan.x)*diff(exp_R,x)+(exp_R.x)*diff(tan,x) by A2,A5
,A7,FDIFF_1:21
      .=(tan.x)*exp_R.x+(exp_R.x)*diff(tan,x) by SIN_COS:65
      .=(tan.x)*exp_R.x+(exp_R.x)*(1/(cos.x)^2) by A6,A8
      .=(sin.x/cos.x)*(exp_R.x/1)+(exp_R.x)/(cos.x)^2 by A4,A8,RFUNCT_1:def 1
      .=exp_R.x*sin.x/cos.x+exp_R.x/(cos.x)^2;
    hence thesis;
  end;
  hence thesis by A2,A5,A7,FDIFF_1:21;
end;
