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

  rr for set,

  rseq for Real_Sequence;

theorem Th99:
  rng(arccosec1 | [.-sqrt 2,-1.]) = [.-PI/2,-PI/4.]
proof
  now
    let y be object;
    thus y in [.-PI/2,-PI/4.] implies
ex x be object st x in dom (arccosec1 | [.-
    sqrt 2,-1.]) & y = (arccosec1 | [.-sqrt 2,-1.]).x
    proof
      assume
A1:   y in [.-PI/2,-PI/4.];
      then reconsider y1=y as Real;
      -sqrt 2 < -1 & y1 in [.arccosec1.(-1),arccosec1.(-sqrt 2).] \/ [.
arccosec1.( -sqrt 2),arccosec1.(-1).] by A1,Th75,SQUARE_1:19,XBOOLE_0:def 3
,XREAL_1:24;
      then consider x be Real such that
A2:   x in [.-sqrt 2,-1.] & y1 = arccosec1.x by Th47,Th95,FCONT_2:15;
      take x;
      thus thesis by A2,Th47,FUNCT_1:49,RELAT_1:62;
    end;
    thus (ex x be object st x in dom (arccosec1 | [.-sqrt 2,-1.]) & y = (
    arccosec1 | [.-sqrt 2,-1.]).x) implies y in [.-PI/2,-PI/4.]
    proof
      given x be object such that
A3:   x in dom (arccosec1 | [.-sqrt 2,-1.]) and
A4:   y = (arccosec1 | [.-sqrt 2,-1.]).x;
A5:   dom (arccosec1 | [.-sqrt 2,-1.]) = [.-sqrt 2,-1.] by Th47,RELAT_1:62;
      then y = arccosec1.x by A3,A4,FUNCT_1:49;
      hence thesis by A3,A5,Th87;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
