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 Th49:
  for x be set st x in [.-1,1.] holds arctan.x in [.-PI/4,PI/4.]
proof
  let x be set;
  assume x in [.-1,1.];
  then x in ].-1,1.[ \/ {-1,1} by XXREAL_1:128;
  then
A1: x in ].-1,1.[ or x in {-1,1} by XBOOLE_0:def 3;
  per cases by A1,TARSKI:def 2;
  suppose
A2: x in ].-1,1.[;
    then x in { s where s is Real: -1 < s & s < 1 } by RCOMP_1:def 2;
    then
A3: ex s be Real st s=x & -1 < s & s < 1;
A4: ].-1,1.[ c= [.-1,1.] by XXREAL_1:25;
A5: [.-1,1.] /\ dom arctan = [.-1,1.] by Th23,XBOOLE_1:28;
    then 1 in [.-1,1.] /\ dom arctan by XXREAL_1:1;
    then
A6: arctan.x < PI/4 by A2,A5,A4,A3,Th39,Th47,RFUNCT_2:20;
    -1 in [.-1,1.] by XXREAL_1:1;
    then -PI/4 < arctan.x by A2,A5,A4,A3,Th37,Th47,RFUNCT_2:20;
    hence thesis by A6,XXREAL_1:1;
  end;
  suppose
    x = -1;
    hence thesis by Th37,XXREAL_1:1;
  end;
  suppose
    x = 1;
    hence thesis by Th39,XXREAL_1:1;
  end;
end;
