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

  rr for set,

  rseq for Real_Sequence;

theorem Th89:
  1 <= r & r <= sqrt 2 implies sec.(arcsec1 r) = r
proof
  assume 1 <= r & r <= sqrt 2;
  then
A1: r in [.1,sqrt 2.];
  then
A2: r in dom (arcsec1|[.1,sqrt 2.]) by Th45,RELAT_1:62;
  sec.(arcsec1 r) = ((sec|[.0,PI/4.]) qua Function).(arcsec1.r) by A1,Th85,
FUNCT_1:49
    .= ((sec|[.0,PI/4.]) qua Function).((arcsec1|[.1,sqrt 2.]).r) by A1,
FUNCT_1:49
    .= ((sec|[.0,PI/4.]) qua Function * (arcsec1|[.1,sqrt 2.])).r by A2,
FUNCT_1:13
    .= (id [.1,sqrt 2.]).r by Th41,Th49,FUNCT_1:39
    .= r by A1,FUNCT_1:18;
  hence thesis;
end;
