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

  rr for set,

  rseq for Real_Sequence;

theorem Th14:
  sec|].PI/2,PI.[ is increasing
proof
  for x st x in ].PI/2,PI.[ holds diff(sec,x) > 0
  proof
    let x;
    assume
A1: x in ].PI/2,PI.[;
    PI <= 3/2*PI by XREAL_1:151;
    then ].PI/2,PI.[ c= ].PI/2,3/2*PI.[ by XXREAL_1:46;
    then
A2: cos.x < 0 by A1,COMPTRIG:13;
    ].PI/2,PI.[ c= ].0,PI.[ by XXREAL_1:46;
    then sin.x > 0 by A1,COMPTRIG:7;
    then sin.x/(cos.x)^2 > 0/(cos.x)^2 by A2;
    hence thesis by A1,Th6;
  end;
  hence thesis by Lm11,Th2,Th6,ROLLE:9,XBOOLE_1:1;
end;
