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

  rr for set,

  rseq for Real_Sequence;

theorem Th45:
  [.1,sqrt 2.] c= dom arcsec1
proof
A1: [.0,PI/4.] c= [.0,PI/2.[ by Lm5,XXREAL_2:def 12;
  rng(sec | [.0,PI/4.]) c= rng(sec | [.0,PI/2.[)
  proof
    let y be object;
    assume y in rng(sec | [.0,PI/4.]);
    then y in sec.:[.0,PI/4.] by RELAT_1:115;
    then ex x be object st x in dom sec & x in [.0,PI/4.] & y = sec .x by
FUNCT_1:def 6;
    then y in sec.:[.0,PI/2.[ by A1,FUNCT_1:def 6;
    hence thesis by RELAT_1:115;
  end;
  hence thesis by Th41,FUNCT_1:33;
end;
