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

  rr for set,

  rseq for Real_Sequence;

theorem Th111:
  -sqrt 2 < r & r < -1 implies -PI/2 < arccosec1 r & arccosec1 r < -PI/4
proof
  assume
A1: -sqrt 2 < r & r < -1;
  then -PI/2 <= arccosec1 r & arccosec1 r <= -PI/4 by Th107;
  then -PI/2 < arccosec1 r & arccosec1 r < -PI/4 or -PI/2 = arccosec1 r or
  arccosec1 r = -PI/4 by XXREAL_0:1;
  hence thesis by A1,Th32,Th91;
end;
