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

theorem
  F is odd implies F" is odd
proof
A1: dom F=dom(F") by VALUED_1:def 7;
  assume
A2: F is odd;
  for x st x in dom(F") & -x in dom(F") holds (F").(-x)=-(F").x
  proof
    let x;
    assume that
A3: x in dom(F") and
A4: -x in dom(F");
    (F").(-x)=(F.(-x))" by A4,VALUED_1:def 7
      .=(-F.x)" by A2,A1,A3,A4,Def6
      .=-(F.x)" by XCMPLX_1:222
      .=-(F").x by A3,VALUED_1:def 7;
    hence thesis;
  end;
  then F" is with_symmetrical_domain quasi_odd by A2,A1;
  hence thesis;
end;
