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

  rr for set,

  rseq for Real_Sequence;

theorem Th100:
  rng(arccosec2 | [.1,sqrt 2.]) = [.PI/4,PI/2.]
proof
  now
    let y be object;
    thus y in [.PI/4,PI/2.] implies
ex x be object st x in dom (arccosec2 | [.1,
    sqrt 2.]) & y = (arccosec2 | [.1,sqrt 2.]).x
    proof
      assume
A1:   y in [.PI/4,PI/2.];
      then reconsider y1=y as Real;
      y1 in [.arccosec2.(sqrt 2),arccosec2.1.] \/ [.arccosec2.1,arccosec2.
      (sqrt 2).] by A1,Th76,XBOOLE_0:def 3;
      then consider x be Real such that
A2:   x in [.1,sqrt 2.] & y1 = arccosec2.x by Th48,Th96,FCONT_2:15,SQUARE_1:19;
      take x;
      thus thesis by A2,Th48,FUNCT_1:49,RELAT_1:62;
    end;
    thus (ex x be object
st x in dom (arccosec2 | [.1,sqrt 2.]) & y = (arccosec2
    | [.1,sqrt 2.]).x) implies y in [.PI/4,PI/2.]
    proof
      given x be object such that
A3:   x in dom (arccosec2 | [.1,sqrt 2.]) and
A4:   y = (arccosec2 | [.1,sqrt 2.]).x;
A5:   dom (arccosec2 | [.1,sqrt 2.]) = [.1,sqrt 2.] by Th48,RELAT_1:62;
      then y = arccosec2.x by A3,A4,FUNCT_1:49;
      hence thesis by A3,A5,Th88;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
