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 odd
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_1:12;
  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_1:13
      .=(-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_1:13;
    hence thesis;
  end;
  then (F - G) is with_symmetrical_domain quasi_odd by A3,A4;
  hence thesis;
end;
