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

  rr for set,

  rseq for Real_Sequence;

theorem Th105:
  1 <= r & r <= sqrt 2 implies 0 <= arcsec1 r & arcsec1 r <= PI/4
proof
  assume 1 <= r & r <= sqrt 2;
  then
A1: r in [.1,sqrt 2.];
  then r in dom (arcsec1 | [.1,sqrt 2.]) by Th45,RELAT_1:62;
  then (arcsec1 | [.1,sqrt 2.]).r in rng(arcsec1 | [.1,sqrt 2.]) by
FUNCT_1:def 3;
  then arcsec1 r in rng(arcsec1 | [.1,sqrt 2.]) by A1,FUNCT_1:49;
  hence thesis by Th97,XXREAL_1:1;
end;
