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

  rr for set,

  rseq for Real_Sequence;

theorem
  1 < r & r < sqrt 2 implies sec.(arccosec2 r) = r/sqrt(r^2-1)
proof
  set x = arccosec2 r;
  [.0,PI/2.[ = ].0,PI/2.[ \/ {0} by XXREAL_1:131;
  then ].PI/4,PI/2.[ c= ].0,PI/2.[ & ].0,PI/2.[ c= [.0,PI/2.[ by XBOOLE_1:7
,XXREAL_1:46;
  then ].PI/4,PI/2.[ c= [.0,PI/2.[;
  then
A1: ].PI/4,PI/2.[ c= dom sec by Th1;
  assume
A2: 1 < r & r < sqrt 2;
  then PI/4 < arccosec2 r & arccosec2 r < PI/2 by Th112;
  then x in ].PI/4,PI/2.[;
  then sec.x = 1/cos.x by A1,RFUNCT_1:def 2
    .= 1/(sqrt(r^2-1)/r) by A2,Th116
    .= r/sqrt(r^2-1) by XCMPLX_1:57;
  hence thesis;
end;
