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 cosec.(arcsec1 r) = r/sqrt(r^2-1)
proof
  set x = arcsec1 r;
  ].0,PI/2.] = ].0,PI/2.[ \/ {PI/2} by XXREAL_1:132;
  then
A1: ].0,PI/2.[ c= ].0,PI/2.] by XBOOLE_1:7;
  PI/4 < PI/2 by XREAL_1:76;
  then ].0,PI/4.[ c= ].0,PI/2.[ by XXREAL_1:46;
  then ].0,PI/4.[ c= ].0,PI/2.] by A1;
  then
A2: ].0,PI/4.[ c= dom cosec by Th4;
  assume
A3: 1 < r & r < sqrt 2;
  then 0 < arcsec1 r & arcsec1 r < PI/4 by Th109;
  then x in ].0,PI/4.[;
  then cosec.x = 1/sin.x by A2,RFUNCT_1:def 2
    .= 1/(sqrt(r^2-1)/r) by A3,Th113
    .= r/sqrt(r^2-1) by XCMPLX_1:57;
  hence thesis;
end;
