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

theorem
  F is even & G is odd & dom F /\ dom G is symmetrical implies F /" G is odd
proof
  assume that
A1: F is even and
A2: G is odd and
A3: dom F /\ dom G is symmetrical;
A4: dom F /\ dom G=dom (F /" G) by VALUED_1:16;
  then
A5: dom (F /" G) c= dom G by XBOOLE_1:17;
A6: dom (F /" G) c= dom F by A4,XBOOLE_1:17;
  for x st x in dom(F /" G) & -x in dom(F /" G) holds (F /" G).(-x)=-(F /" G).x
  proof
    let x;
    assume
A7: x in dom(F /" G) & -x in dom(F /" G);
    (F /" G).(-x)=F.(-x) / G.(-x) by VALUED_1:17
      .=F.x / G.(-x) by A1,A6,A7,Def3
      .=F.x / (-G.x) by A2,A5,A7,Def6
      .=-(F.x / G.x) by XCMPLX_1:188
      .=-(F /" G).x by VALUED_1:17;
    hence thesis;
  end;
  then (F /" G) is with_symmetrical_domain quasi_odd by A3,A4;
  hence thesis;
end;
