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

theorem
  F is_odd_on A & G is_odd_on A implies F - G is_odd_on A
proof
  assume that
A1: F is_odd_on A and
A2: G is_odd_on A;
A3: A c= dom G by A2;
A4: G|A is odd by A2;
A5: A c= dom F by A1;
  then
A6: A c= dom F /\ dom G by A3,XBOOLE_1:19;
A7: dom F /\ dom G=dom (F - G) by VALUED_1:12;
  then
A8: dom((F - G)|A) = A by A5,A3,RELAT_1:62,XBOOLE_1:19;
A9: F|A is odd by A1;
  for x st x in dom((F - G)|A) & -x in dom((F - G)|A) holds (F - G)|A.(-x
  )=-(F - G)|A.x
  proof
    let x;
    assume that
A10: x in dom((F - G)|A) and
A11: -x in dom((F - G)|A);
A12: x in dom(F|A) by A5,A8,A10,RELAT_1:62;
A13: x in dom(G|A) by A3,A8,A10,RELAT_1:62;
A14: -x in dom(F|A) by A5,A8,A11,RELAT_1:62;
A15: -x in dom(G|A) by A3,A8,A11,RELAT_1:62;
      reconsider x as Element of REAL by XREAL_0:def 1;
    (F - G)|A.(-x)=(F - G)|A/.(-x) by A11,PARTFUN1:def 6
      .=(F - G)/.(-x) by A6,A7,A8,A11,PARTFUN2:17
      .=(F - G).(-x) by A6,A7,A11,PARTFUN1:def 6
      .=F.(-x) - G.(-x) by A6,A7,A11,VALUED_1:13
      .=F/.(-x) - G.(-x) by A5,A11,PARTFUN1:def 6
      .=F/.(-x) - G/.(-x) by A3,A11,PARTFUN1:def 6
      .=F|A/.(-x) - G/.(-x) by A5,A8,A11,PARTFUN2:17
      .=F|A/.(-x) - G|A/.(-x) by A3,A8,A11,PARTFUN2:17
      .=F|A.(-x) - G|A/.(-x) by A14,PARTFUN1:def 6
      .=F|A.(-x) - G|A.(-x) by A15,PARTFUN1:def 6
      .=(-F|A.x) - G|A.(-x) by A9,A12,A14,Def6
      .=(-F|A.x) - (-G|A.x) by A4,A13,A15,Def6
      .=-(F|A.x - G|A.x)
      .=-(F|A/.x - G|A.x) by A12,PARTFUN1:def 6
      .=-(F|A/.x - G|A/.x) by A13,PARTFUN1:def 6
      .=-(F/.x - G|A/.x) by A5,A8,A10,PARTFUN2:17
      .=-(F/.x - G/.x) by A3,A8,A10,PARTFUN2:17
      .=-(F.x - G/.x) by A5,A10,PARTFUN1:def 6
      .=-(F.x - G.x) by A3,A10,PARTFUN1:def 6
      .=-(F - G).x by A6,A7,A10,VALUED_1:13
      .=-(F - G)/.x by A6,A7,A10,PARTFUN1:def 6
      .=-(F - G)|A/.x by A6,A7,A8,A10,PARTFUN2:17
      .=-(F - G)|A.x by A10,PARTFUN1:def 6;
    hence thesis;
  end;
  then (F - G)|A is with_symmetrical_domain quasi_odd by A8;
  hence thesis by A6,A7;
end;
