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

  rr for set,

  rseq for Real_Sequence;

theorem
  (-sqrt 2 <= r & r <= -1 & arccosec1 r = -PI/2 implies r = -1) & (-sqrt
  2 <= r & r <= -1 & arccosec1 r = -PI/4 implies r = -sqrt 2) by Th32,Th91;
