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

theorem Th65:
  for A being symmetrical Subset of REAL holds sin is_odd_on A
proof
  let A be symmetrical Subset of REAL;
A1: dom(sin|A) = A by RELAT_1:62,SIN_COS:24;
  for x st x in dom(sin|A) & -x in dom(sin|A) holds sin|A.(-x)=-sin|A.x
  proof
    let x;
    assume that
A2: x in dom(sin|A) and
A3: -x in dom(sin|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    sin|A.(-x)=sin|A/.(-x) by A3,PARTFUN1:def 6
      .=sin/.(-x) by A3,PARTFUN2:17,SIN_COS:24
      .=-sin/.x by SIN_COS:30
      .=-sin|A/.x by A2,PARTFUN2:17,SIN_COS:24
      .=-sin|A.x by A2,PARTFUN1:def 6;
    hence thesis;
  end;
  then sin|A is with_symmetrical_domain quasi_odd by A1;
  hence thesis by SIN_COS:24;
end;
