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

theorem
  A c= ].-PI/2,PI/2.[ implies tan is_odd_on A
proof
  assume
A1: A c= ].-PI/2,PI/2.[;
  then
A2: A c= dom (tan) by SIN_COS9:1;
A3: dom(tan|A) = A by A1,RELAT_1:62,SIN_COS9:1,XBOOLE_1:1;
A4: for x st x in A holds tan.(-x) = -tan.x
  proof
    let x;
    assume
A5: x in A;
    then -x in A by Def1;
    then tan.(-x)=tan (-x) by A1,SIN_COS9:13
      .= -tan x by SIN_COS4:1
      .= -tan.x by A1,A5,SIN_COS9:13;
    hence thesis;
  end;
  for x st x in dom(tan|A) & -x in dom(tan|A) holds tan|A.(-x)=-tan|A.x
  proof
    let x;
    assume that
A6: x in dom(tan|A) and
A7: -x in dom(tan|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    tan|A.(-x)=tan|A/.(-x) by A7,PARTFUN1:def 6
      .=tan/.(-x) by A2,A3,A7,PARTFUN2:17
      .=tan.(-x) by A2,A7,PARTFUN1:def 6
      .=-tan.x by A4,A6
      .=-tan/.x by A2,A6,PARTFUN1:def 6
      .=-tan|A/.x by A2,A3,A6,PARTFUN2:17
      .=-tan|A.x by A6,PARTFUN1:def 6;
    hence thesis;
  end;
  then tan|A is with_symmetrical_domain quasi_odd by A3;
  hence thesis by A2;
end;
