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

  rr for set,

  rseq for Real_Sequence;

theorem Th106:
  -sqrt 2 <= r & r <= -1 implies 3/4*PI <= arcsec2 r & arcsec2 r <= PI
proof
  assume -sqrt 2 <= r & r <= -1;
  then
A1: r in [.-sqrt 2,-1.];
  then r in dom (arcsec2 | [.-sqrt 2,-1.]) by Th46,RELAT_1:62;
  then (arcsec2 | [.-sqrt 2,-1.]).r in rng(arcsec2 | [.-sqrt 2,-1.]) by
FUNCT_1:def 3;
  then arcsec2 r in rng(arcsec2 | [.-sqrt 2,-1.]) by A1,FUNCT_1:49;
  hence thesis by Th98,XXREAL_1:1;
end;
