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

theorem
  A c= dom tan implies tan is_odd_on A
proof
  assume that
A1: A c= dom tan;
A2: dom(tan|A) = A by A1,RELAT_1:62;
A3: for x st x in A holds tan.(-x) = -tan.x
  proof
    let x;
    assume
A4: x in A; then
A5: cos.x <> 0 by A1,FDIFF_8:1;
    -x in A by Def1,A4; then
    tan.(-x) = tan (-x) by A1,FDIFF_8:1,SIN_COS9:15
      .=-tan x by SIN_COS4:1
      .=-tan.x by A5,SIN_COS9:15;
    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 A1,A2,A7,PARTFUN2:17
      .=tan.(-x) by A1,A7,PARTFUN1:def 6
      .=-tan.x by A3,A6
      .=-tan/.x by A1,A6,PARTFUN1:def 6
      .=-tan|A/.x by A1,A2,A6,PARTFUN2:17
      .=-tan|A.x by A6,PARTFUN1:def 6;
    hence thesis;
  end;
  then tan|A is with_symmetrical_domain quasi_odd by A2;
  hence thesis by A1;
end;
