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

  rr for set,

  rseq for Real_Sequence;

theorem Th112:
  1 < r & r < sqrt 2 implies PI/4 < arccosec2 r & arccosec2 r < PI/2
proof
  assume
A1: 1 < r & r < sqrt 2;
  then PI/4 <= arccosec2 r & arccosec2 r <= PI/2 by Th108;
  then PI/4 < arccosec2 r & arccosec2 r < PI/2 or PI/4 = arccosec2 r or
  arccosec2 r = PI/2 by XXREAL_0:1;
  hence thesis by A1,Th32,Th92;
end;
