reserve x, r for Real;
reserve A for symmetrical Subset of COMPLEX;
reserve F,G for PartFunc of REAL, REAL;

theorem
  A c= [.-1,1.] implies arctan is_odd_on A
proof
  assume
A1: A c= [.-1,1.];
  then
A2: A c= dom arctan by SIN_COS9:23;
A3: dom(arctan|A) = A by A1,RELAT_1:62,SIN_COS9:23,XBOOLE_1:1;
A4: for x st x in A holds arctan.(-x) = -arctan.x
  proof
    let x;
    assume x in A;
    then -1 <= x & x <= 1 by A1,XXREAL_1:1;
    then arctan x = -arctan (-x) by SIN_COS9:67;
    hence thesis;
  end;
  for x st x in dom(arctan|A) & -x in dom(arctan|A) holds arctan|A.(-x)=-
  arctan|A.x
  proof
    let x;
    assume that
A5: x in dom(arctan|A) and
A6: -x in dom(arctan|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    arctan|A.(-x)=arctan|A/.(-x) by A6,PARTFUN1:def 6
      .=arctan/.(-x) by A2,A3,A6,PARTFUN2:17
      .=arctan.(-x) by A2,A6,PARTFUN1:def 6
      .=-arctan.x by A4,A5
      .=-arctan/.x by A2,A5,PARTFUN1:def 6
      .=-arctan|A/.x by A2,A3,A5,PARTFUN2:17
      .=-arctan|A.x by A5,PARTFUN1:def 6;
    hence thesis;
  end;
  then arctan|A is with_symmetrical_domain quasi_odd by A3;
  hence thesis by A2;
end;
