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

  rr for set,

  rseq for Real_Sequence;

theorem Th91:
  -sqrt 2 <= r & r <= -1 implies cosec.(arccosec1 r) = r
proof
  assume -sqrt 2 <= r & r <= -1;
  then
A1: r in [.-sqrt 2,-1.];
  then
A2: r in dom (arccosec1|[.-sqrt 2,-1.]) by Th47,RELAT_1:62;
  cosec.(arccosec1 r) = ((cosec|[.-PI/2,-PI/4.]) qua Function).(arccosec1.
  r) by A1,Th87,FUNCT_1:49
    .= ((cosec|[.-PI/2,-PI/4.]) qua Function).((arccosec1|[.-sqrt 2,-1.]).r)
  by A1,FUNCT_1:49
    .= ((cosec|[.-PI/2,-PI/4.]) qua Function * (arccosec1|[.-sqrt 2,-1.])).r
  by A2,FUNCT_1:13
    .= (id [.-sqrt 2,-1.]).r by Th43,Th51,FUNCT_1:39
    .= r by A1,FUNCT_1:18;
  hence thesis;
end;
