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

  rr for set,

  rseq for Real_Sequence;

theorem Th109:
  1 < r & r < sqrt 2 implies 0 < arcsec1 r & arcsec1 r < PI/4
proof
  assume
A1: 1 < r & r < sqrt 2;
  then arcsec1 r <= PI/4 by Th105;
  then
  0 < arcsec1 r & arcsec1 r < PI/4 or 0 = arcsec1 r or arcsec1 r = PI/4 by A1
,Th105,XXREAL_0:1;
  hence thesis by A1,Th31,Th89;
end;
