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

theorem
  F is odd & G is odd & dom F /\ dom G is symmetrical implies F (#) G is even
proof
  assume that
A1: F is odd and
A2: G is odd and
A3: dom F /\ dom G is symmetrical;
A4: dom F /\ dom G=dom (F (#) G) by VALUED_1:def 4;
  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 that
A7: x in dom(F (#) G) and
A8: -x in dom(F (#) G);
    (F (#) G).(-x)=F.(-x) * G.(-x) by A8,VALUED_1:def 4
      .=(-F.x) * G.(-x) by A1,A6,A7,A8,Def6
      .=(-F.x) * (-G.x) by A2,A5,A7,A8,Def6
      .=(F.x * G.x)
      .=(F (#) G).x by A7,VALUED_1:def 4;
    hence thesis;
  end;
  then (F (#) G) is with_symmetrical_domain quasi_even by A3,A4;
  hence thesis;
end;
