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 (tan-cos^) & for x st x in Z holds (1-sin.x)<>0 & (1+sin.x)
<>0) implies tan-cos^ is_differentiable_on Z & for x st x in Z holds ((tan-cos^
  )`|Z).x = 1/(1+sin.x)
proof
  assume that
A1: Z c= dom (tan-cos^) and
A2: for x st x in Z holds (1-sin.x)<>0 & (1+sin.x)<>0;
  Z c= dom tan /\ dom (cos^) by A1,VALUED_1:12;
  then
A3: Z c= dom tan by XBOOLE_1:18;
  then
A4: for x st x in Z holds cos.x<>0 by Th1;
  then
A5: cos^ is_differentiable_on Z by FDIFF_4:39;
  for x st x in Z holds tan is_differentiable_in x
  proof
    let x;
    assume x in Z;
    then cos.x<>0 by A3,Th1;
    hence thesis by FDIFF_7:46;
  end;
  then
A6: tan is_differentiable_on Z by A3,FDIFF_1:9;
  for x st x in Z holds ((tan-cos^)`|Z).x = 1/(1+sin.x)
  proof
    let x;
    assume
A7: x in Z;
    then
A8: (1-sin.x)<>0 by A2;
A9: cos.x<>0 by A3,A7,Th1;
    ((tan-cos^)`|Z).x = diff(tan,x) - diff(cos^,x) by A1,A5,A6,A7,FDIFF_1:19
      .=1/(cos.x)^2 - diff(cos^,x) by A9,FDIFF_7:46
      .=1/(cos.x)^2 - ((cos^)`|Z).x by A5,A7,FDIFF_1:def 7
      .=1/(cos.x)^2 - sin.x/(cos.x)^2 by A4,A7,FDIFF_4:39
      .=(1-sin.x)/((cos.x)^2+(sin.x)^2-(sin.x)^2)
      .=(1-sin.x)/(1-(sin.x)^2) by SIN_COS:28
      .=(1-sin.x)/((1-sin.x)*(1+sin.x))
      .=(1-sin.x)/(1-sin.x)/(1+sin.x) by XCMPLX_1:78
      .=1/(1+sin.x) by A8,XCMPLX_1:60;
    hence thesis;
  end;
  hence thesis by A1,A5,A6,FDIFF_1:19;
end;
