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_even_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_1:16;
  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 VALUED_1:17
      .=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 by XCMPLX_1:191
      .=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 VALUED_1:17
      .=(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_even by A8;
  hence thesis by A6,A7;
end;
