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_1:8;
  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
A3: x in dom(-F) & -x in dom(-F);
    (-F).(-x)=-F.(-x) by VALUED_1:8
      .=-(-F.x) by A2,A1,A3,Def6
      .=-(-F).x by VALUED_1:8;
    hence thesis;
  end;
  then -F is with_symmetrical_domain quasi_odd by A2,A1;
  hence thesis;
end;
