reserve x,x0, r, s, h for Real,

  n for Element of NAT,
  rr, y for set,
  Z for open Subset of REAL,

  f, f1, f2 for PartFunc of REAL,REAL;

theorem Th56:
  rng(arccot | [.-1,1.]) = [.PI/4,3/4*PI.]
proof
  now
    let y be object;
    thus y in [.PI/4,3/4*PI.] implies
ex x be object st x in dom (arccot | [.-1,1
    .]) & y = (arccot | [.-1,1.]).x
    proof
      assume
A1:   y in [.PI/4,3/4*PI.];
      then reconsider y1=y as Real;
      y1 in [.arccot.1,arccot.(-1).] \/ [.arccot.(-1),arccot.1.] by A1,Th38
,Th40,XBOOLE_0:def 3;
      then consider x be Real such that
A2:   x in [.-1,1.] and
A3:   y1 = arccot.x by Th24,Th54,FCONT_2:15;
      take x;
      thus thesis by A2,A3,Th24,FUNCT_1:49,RELAT_1:62;
    end;
    thus (ex x be object
st x in dom (arccot | [.-1,1.]) & y = (arccot | [.-1,1.]
    ).x) implies y in [.PI/4,3/4*PI.]
    proof
      given x be object such that
A4:   x in dom (arccot | [.-1,1.]) and
A5:   y = (arccot | [.-1,1.]).x;
A6:   dom (arccot | [.-1,1.]) = [.-1,1.] by Th24,RELAT_1:62;
      then y = arccot.x by A4,A5,FUNCT_1:49;
      hence thesis by A4,A6,Th50;
    end;
  end;
  hence thesis by FUNCT_1:def 3;
end;
