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

  rr for set,

  rseq for Real_Sequence;

theorem Th85:
  for x be set st x in [.1,sqrt 2.] holds arcsec1.x in [.0,PI/4.]
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 arcsec1 = [.1,sqrt 2.] by Th45,XBOOLE_1:28;
    then sqrt 2 in [.1,sqrt 2.] /\ dom arcsec1 by SQUARE_1:19;
    then
A5: arcsec1.x < PI/4 by A2,A4,A3,Th73,Th81,RFUNCT_2:20;
    1 in [.1,sqrt 2.] by SQUARE_1:19;
    then 0 < arcsec1.x by A2,A4,A3,Th73,Th81,RFUNCT_2:20;
    hence thesis by A5;
  end;
  suppose
    x = 1;
    hence thesis by Th73;
  end;
  suppose
    x = sqrt 2;
    hence thesis by Th73;
  end;
end;
