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

  rr for set,

  rseq for Real_Sequence;

theorem Th90:
  -sqrt 2 <= r & r <= -1 implies sec.(arcsec2 r ) = r
proof
  assume -sqrt 2 <= r & r <= -1;
  then
A1: r in [.-sqrt 2,-1.];
  then
A2: r in dom (arcsec2|[.-sqrt 2,-1.]) by Th46,RELAT_1:62;
  sec.(arcsec2 r) = ((sec|[.3/4*PI,PI.]) qua Function).(arcsec2.r) by A1,Th86,
FUNCT_1:49
    .= ((sec|[.3/4*PI,PI.]) qua Function).((arcsec2|[.-sqrt 2,-1.]).r) by A1,
FUNCT_1:49
    .= ((sec|[.3/4*PI,PI.]) qua Function * (arcsec2|[.-sqrt 2,-1.])).r by A2,
FUNCT_1:13
    .= (id [.-sqrt 2,-1.]).r by Th42,Th50,FUNCT_1:39
    .= r by A1,FUNCT_1:18;
  hence thesis;
end;
