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

  rr for set,

  rseq for Real_Sequence;

theorem Th13:
  sec|].0,PI/2.[ is increasing
proof
  for x st x in ].0,PI/2.[ holds diff(sec,x) > 0
  proof
    let x;
    assume
A1: x in ].0,PI/2.[;
    ].0,PI/2.[ c= ].-PI/2,PI/2.[ by XXREAL_1:46;
    then
A2: cos.x > 0 by A1,COMPTRIG:11;
    PI/2 < PI/1 by XREAL_1:76;
    then ].0,PI/2.[ 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,Th5;
  end;
  hence thesis by Lm9,Th1,Th5,ROLLE:9,XBOOLE_1:1;
end;
