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

  rr for set,

  rseq for Real_Sequence;

theorem Th88:
  for x be set st x in [.1,sqrt 2.] holds arccosec2.x in [.PI/4,PI /2.]
proof
  let x be set;
  assume x in [.1,sqrt 2.];
  then x in ].1,sqrt 2.[ \/ {1,sqrt 2} by SQUARE_1:19,XXREAL_1:128;
  then
A1: x in ].1,sqrt 2.[ or x in {1,sqrt 2} by XBOOLE_0:def 3;
  per cases by A1,TARSKI:def 2;
  suppose
A2: x in ].1,sqrt 2.[;
    then
A3: ].1,sqrt 2.[ c= [.1,sqrt 2.] &
  ex s be Real st s=x & 1 < s & s < sqrt
    2 by XXREAL_1:25;
A4: [.1,sqrt 2.] /\ dom arccosec2 = [.1,sqrt 2.] by Th48,XBOOLE_1:28;
    then sqrt 2 in [.1,sqrt 2.] /\ dom arccosec2 by SQUARE_1:19;
    then
A5: arccosec2.x > PI/4 by A2,A4,A3,Th76,Th84,RFUNCT_2:21;
    1 in [.1,sqrt 2.] by SQUARE_1:19;
    then PI/2 > arccosec2.x by A2,A4,A3,Th76,Th84,RFUNCT_2:21;
    hence thesis by A5;
  end;
  suppose
A6: x = 1;
    PI/4 <= PI/2 by Lm8,XXREAL_1:2;
    hence thesis by A6,Th76;
  end;
  suppose
A7: x = sqrt 2;
    PI/4 <= PI/2 by Lm8,XXREAL_1:2;
    hence thesis by A7,Th76;
  end;
end;
