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 ((1/a)(#)(sec*f)-id Z) & for x st x in Z holds f.x=a*x & a<>
0) implies (1/a)(#)(sec*f)-id Z is_differentiable_on Z & for x st x in Z holds
  (((1/a)(#)(sec*f)-id Z)`|Z).x= (sin.(a*x)-(cos.(a*x))^2)/(cos.(a*x))^2
proof
  assume that
A1: Z c= dom ((1/a)(#)(sec*f)-id Z) and
A2: for x st x in Z holds f.x=a*x & a<>0;
A3: Z c= dom ((1/a)(#)(sec*f)) /\ dom (id Z) by A1,VALUED_1:12;
  then
A4: Z c= dom ((1/a)(#)(sec*f)) by XBOOLE_1:18;
  then
A5: Z c= dom (sec*f) by VALUED_1:def 5;
A6: for x st x in Z holds f.x=a*x+0 by A2;
  then
A7: sec*f is_differentiable_on Z by A5,Th6;
  then
A8: (1/a)(#)(sec*f) is_differentiable_on Z by A4,FDIFF_1:20;
  set g = (1/a)(#)(sec*f);
A9: for x st x in Z holds (id Z).x = 1*x+0 by FUNCT_1:18;
A10: Z c= dom (id Z) by A3,XBOOLE_1:18;
  then
A11: id Z is_differentiable_on Z by A9,FDIFF_1:23;
A12: for x st x in Z holds cos.(f.x)<>0
  proof
    let x;
    assume x in Z;
    then f.x in dom sec by A5,FUNCT_1:11;
    hence thesis by RFUNCT_1:3;
  end;
  for x st x in Z holds (((1/a)(#)(sec*f)-id Z)`|Z).x= (sin.(a*x)-(cos.(a
  *x))^2)/(cos.(a*x))^2
  proof
    let x;
    assume
A13: x in Z;
    then
A14: f.x=a*x+0 by A2;
    cos.(f.x)<>0 by A12,A13;
    then
A15: (cos.(a*x))^2>0 by A14,SQUARE_1:12;
    ((g-id Z)`|Z).x=diff(g,x)-diff(id Z,x) by A1,A8,A11,A13,FDIFF_1:19
      .=(g`|Z).x-diff(id Z,x) by A8,A13,FDIFF_1:def 7
      .=(1/a)*diff(sec*f,x)-diff(id Z,x) by A4,A7,A13,FDIFF_1:20
      .=(1/a)*((sec*f)`|Z).x-diff(id Z,x) by A7,A13,FDIFF_1:def 7
      .=(1/a)*((sec*f)`|Z).x-((id Z)`|Z).x by A11,A13,FDIFF_1:def 7
      .=(1/a)*(a*sin.(a*x)/(cos.(a*x))^2)-((id Z)`|Z).x by A5,A6,A13,A14,Th6
      .=(1/a)*((a*sin.(a*x))/(cos.(a*x))^2) -1 by A10,A9,A13,FDIFF_1:23
      .=(1*(a*sin.(a*x)))/(a*(cos.(a*x))^2) -1 by XCMPLX_1:76
      .=sin.(a*x)/(cos.(a*x))^2 -1 by A2,A13,XCMPLX_1:91
      .=sin.(a*x)/(cos.(a*x))^2-(cos.(a*x))^2/(cos.(a*x))^2 by A15,XCMPLX_1:60
      .=(sin.(a*x)-(cos.(a*x))^2)/(cos.(a*x))^2;
    hence thesis;
  end;
  hence thesis by A1,A8,A11,FDIFF_1:19;
end;
