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

  rr for set,

  rseq for Real_Sequence;

theorem Th46:
  [.-sqrt 2,-1.] c= dom arcsec2
proof
A1: [.3/4*PI,PI.] c= ].PI/2,PI.] by Lm6,XXREAL_2:def 12;
  rng(sec | [.3/4*PI,PI.]) c= rng(sec | ].PI/2,PI.])
  proof
    let y be object;
    assume y in rng(sec | [.3/4*PI,PI.]);
    then y in sec.:[.3/4*PI,PI.] by RELAT_1:115;
    then ex x be object st x in dom sec & x in [.3/4*PI,PI.] & y = sec.x by
FUNCT_1:def 6;
    then y in sec.:].PI/2,PI.] by A1,FUNCT_1:def 6;
    hence thesis by RELAT_1:115;
  end;
  hence thesis by Th42,FUNCT_1:33;
end;
