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

  rr for set,

  rseq for Real_Sequence;

theorem Th107:
  -sqrt 2 <= r & r <= -1 implies -PI/2 <= arccosec1 r & arccosec1 r <= -PI/4
proof
  assume -sqrt 2 <= r & r <= -1;
  then
A1: r in [.-sqrt 2,-1.];
  then r in dom (arccosec1 | [.-sqrt 2,-1.]) by Th47,RELAT_1:62;
  then (arccosec1 | [.-sqrt 2,-1.]).r in rng(arccosec1 | [.-sqrt 2,-1.]) by
FUNCT_1:def 3;
  then arccosec1 r in rng(arccosec1 | [.-sqrt 2,-1.]) by A1,FUNCT_1:49;
  hence thesis by Th99,XXREAL_1:1;
end;
