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

  rr for set,

  rseq for Real_Sequence;

theorem Th44:
  rng(cosec | [.PI/4,PI/2.]) = [.1,sqrt 2.]
proof
  now
    let y be object;
    thus y in [.1,sqrt 2.] implies
ex x be object st x in dom (cosec | [.PI/4,PI/
    2.]) & y = (cosec | [.PI/4,PI/2.]).x
    proof
      [.PI/4,PI/2.] c= ].0,PI/2.] by Lm8,XXREAL_2:def 12;
      then
A1:   cosec|[.PI/4,PI/2.] is continuous by Th40,FCONT_1:16;
      assume
A2:   y in [.1,sqrt 2.];
      then reconsider y1=y as Real;
A3:   PI/4 <= PI/2 by Lm8,XXREAL_1:2;
      y1 in [.cosec.(PI/2),cosec.(PI/4).] \/ [.cosec.(PI/4),cosec.(PI/2).]
      by A2,Th32,XBOOLE_0:def 3;
      then consider x be Real such that
A4:   x in [.PI/4,PI/2.] & y1 = cosec.x by A3,A1,Lm20,Th4,FCONT_2:15,XBOOLE_1:1
;
      take x;
      thus thesis by A4,Lm32,FUNCT_1:49;
    end;
    thus (ex x be object
   st x in dom (cosec | [.PI/4,PI/2.]) & y = (cosec | [.PI/
    4,PI/2.]).x) implies y in [.1,sqrt 2.]
    proof
      given x be object such that
A5:   x in dom (cosec | [.PI/4,PI/2.]) and
A6:   y = (cosec | [.PI/4,PI/2.]).x;
      reconsider x1=x as Real by A5;
      y = cosec.x1 by A5,A6,Lm32,FUNCT_1:49;
      hence thesis by A5,Lm32,Th36;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
