reserve x,x0, r,r1,r2 for Real,
      th for Real,

  rr for set,

  rseq for Real_Sequence;

theorem Th6:
  sec is_differentiable_on ].PI/2,PI.[ & for x st x in ].PI/2,PI.[
  holds diff(sec,x) = sin.x/(cos.x)^2
proof
  set Z = ].PI/2,PI.[;
  PI/2 < PI/1 by XREAL_1:76;
  then ].PI/2,PI.] = Z \/ {PI} by XXREAL_1:132;
  then Z c= ].PI/2,PI.] by XBOOLE_1:7;
  then
A1: Z c= dom sec by Th2;
  then
A2: sec is_differentiable_on Z by FDIFF_9:4;
  for x st x in Z holds diff(sec,x) = sin.x/(cos.x)^2
  proof
    let x;
    assume
A3: x in Z;
    then diff(sec,x) = ((sec)`|Z).x by A2,FDIFF_1:def 7
      .= sin.x/(cos.x)^2 by A1,A3,FDIFF_9:4;
    hence thesis;
  end;
  hence thesis by A1,FDIFF_9:4;
end;
