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

  rr for set,

  rseq for Real_Sequence;

theorem Th97:
  rng(arcsec1 | [.1,sqrt 2.]) = [.0,PI/4.]
proof
  now
    let y be object;
    thus y in [.0,PI/4.] implies
ex x be object st x in dom (arcsec1 | [.1,sqrt 2
    .]) & y = (arcsec1 | [.1,sqrt 2.]).x
    proof
      assume
A1:   y in [.0,PI/4.];
      then reconsider y1=y as Real;
      y1 in [.arcsec1.1,arcsec1.(sqrt 2).] \/ [.arcsec1.(sqrt 2),arcsec1.1
      .] by A1,Th73,XBOOLE_0:def 3;
      then consider x be Real such that
A2:   x in [.1,sqrt 2.] & y1 = arcsec1.x by Th45,Th93,FCONT_2:15,SQUARE_1:19;
      take x;
      thus thesis by A2,Th45,FUNCT_1:49,RELAT_1:62;
    end;
    thus (ex x be object
st x in dom (arcsec1 | [.1,sqrt 2.]) & y = (arcsec1 | [.
    1,sqrt 2.]).x) implies y in [.0,PI/4.]
    proof
      given x be object such that
A3:   x in dom (arcsec1 | [.1,sqrt 2.]) and
A4:   y = (arcsec1 | [.1,sqrt 2.]).x;
A5:   dom (arcsec1 | [.1,sqrt 2.]) = [.1,sqrt 2.] by Th45,RELAT_1:62;
      then y = arcsec1.x by A3,A4,FUNCT_1:49;
      hence thesis by A3,A5,Th85;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
