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

  rr for set,

  rseq for Real_Sequence;

theorem Th34:
  for x be set st x in [.3/4*PI,PI.] holds sec.x in [.-sqrt 2,-1.]
proof
  let x be set;
A1: PI/4 < PI/2 by XREAL_1:76;
  then
A2: PI/4+PI/2 < PI/2+PI/2 by XREAL_1:8;
  assume x in [.3/4*PI,PI.];
  then x in ].3/4*PI,PI.[ \/ {3/4*PI,PI} by A2,XXREAL_1:128;
  then
A3: x in ].3/4*PI,PI.[ or x in {3/4*PI,PI} by XBOOLE_0:def 3;
  per cases by A3,TARSKI:def 2;
  suppose
A4: x in ].3/4*PI,PI.[;
    PI/4+PI/4 < PI/2+PI/4 by A1,XREAL_1:8;
    then
A5: 3/4*PI in ].PI/2,PI.] by A2;
    PI in ].PI/2,PI.] by COMPTRIG:5;
    then
A6: [.3/4*PI,PI.] c= ].PI/2,PI.] by A5,XXREAL_2:def 12;
    then
A7: sec|[.3/4*PI,PI.] is increasing by Th18,RFUNCT_2:28;
A8: ex s be Real st s=x & 3/4*PI < s & s < PI by A4;
A9: ex s be Real st s=x & 3/4*PI < s & s < PI by A4;
A10: ].3/4*PI,PI.[ c= [.3/4*PI,PI.] by XXREAL_1:25;
A11: [.3/4*PI,PI.] /\ dom sec = [.3/4*PI,PI.] by A6,Th2,XBOOLE_1:1,28;
    then PI in [.3/4*PI,PI.] /\ dom sec by A2;
    then
A12: sec.x < -1 by A4,A7,A11,A10,A9,Th31,RFUNCT_2:20;
    3/4*PI in [.3/4*PI,PI.] by A2;
    then -sqrt 2 < sec.x by A4,A7,A11,A10,A8,Th31,RFUNCT_2:20;
    hence thesis by A12;
  end;
  suppose
A13: x = 3/4*PI;
    -sqrt 2 < -1 by SQUARE_1:19,XREAL_1:24;
    hence thesis by A13,Th31;
  end;
  suppose
A14: x = PI;
    -sqrt 2 < -1 by SQUARE_1:19,XREAL_1:24;
    hence thesis by A14,Th31;
  end;
end;
