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

  rr for set,

  rseq for Real_Sequence;

theorem Th5:
  sec is_differentiable_on ].0,PI/2.[ & for x st x in ].0,PI/2.[
  holds diff(sec,x) = sin.x/(cos.x)^2
proof
  set Z = ].0,PI/2.[;
  [.0,PI/2.[ = Z \/ {0} by XXREAL_1:131;
  then Z c= [.0,PI/2.[ by XBOOLE_1:7;
  then
A1: Z c= dom sec by Th1;
  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;
