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

  rr for set,

  rseq for Real_Sequence;

theorem Th98:
  rng(arcsec2 | [.-sqrt 2,-1.]) = [.3/4*PI,PI.]
proof
  now
    let y be object;
    thus y in [.3/4*PI,PI.] implies
ex x be object st x in dom (arcsec2 | [.-sqrt
    2,-1.]) & y = (arcsec2 | [.-sqrt 2,-1.]).x
    proof
      assume
A1:   y in [.3/4*PI,PI.];
      then reconsider y1=y as Real;
      -sqrt 2 < -1 & y1 in [.arcsec2.(-sqrt 2),arcsec2.(-1).] \/ [.arcsec2
.(-1), arcsec2.(-sqrt 2).] by A1,Th74,SQUARE_1:19,XBOOLE_0:def 3,XREAL_1:24;
      then consider x be Real such that
A2:   x in [.-sqrt 2,-1.] & y1 = arcsec2.x by Th46,Th94,FCONT_2:15;
      take x;
      thus thesis by A2,Th46,FUNCT_1:49,RELAT_1:62;
    end;
    thus (ex x be object
 st x in dom (arcsec2 | [.-sqrt 2,-1.]) & y = (arcsec2 |
    [.-sqrt 2,-1.]).x) implies y in [.3/4*PI,PI.]
    proof
      given x be object such that
A3:   x in dom (arcsec2 | [.-sqrt 2,-1.]) and
A4:   y = (arcsec2 | [.-sqrt 2,-1.]).x;
A5:   dom (arcsec2 | [.-sqrt 2,-1.]) = [.-sqrt 2,-1.] by Th46,RELAT_1:62;
      then y = arcsec2.x by A3,A4,FUNCT_1:49;
      hence thesis by A3,A5,Th86;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
