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

  rr for set,

  rseq for Real_Sequence;

theorem
  sec.:].PI/2,PI.[ is open
proof
  for x0 st x0 in ].PI/2,PI.[ holds diff(sec,x0) > 0
  proof
    let x0;
    assume
A1: x0 in ].PI/2,PI.[;
    ].PI/2,PI.[ c= ].PI/2,3/2*PI.[ by COMPTRIG:5,XXREAL_1:46;
    then
A2: cos.x0 < 0 by A1,COMPTRIG:13;
    ].PI/2,PI.[ c= ].0,PI.[ by XXREAL_1:46;
    then sin.x0 > 0 by A1,COMPTRIG:7;
    then sin.x0/(cos.x0)^2 > 0/(cos.x0)^2 by A2;
    hence thesis by A1,Th6;
  end;
  then rng(sec|].PI/2,PI.[) is open by Lm12,Th6,FDIFF_2:41;
  hence thesis by RELAT_1:115;
end;
