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

  rr for set,

  rseq for Real_Sequence;

theorem Th108:
  1 <= r & r <= sqrt 2 implies PI/4 <= arccosec2 r & arccosec2 r <= PI/2
proof
  assume 1 <= r & r <= sqrt 2;
  then
A1: r in [.1,sqrt 2.];
  then r in dom (arccosec2 | [.1,sqrt 2.]) by Th48,RELAT_1:62;
  then (arccosec2 | [.1,sqrt 2.]).r in rng(arccosec2 | [.1,sqrt 2.]) by
FUNCT_1:def 3;
  then arccosec2 r in rng(arccosec2 | [.1,sqrt 2.]) by A1,FUNCT_1:49;
  hence thesis by Th100,XXREAL_1:1;
end;
