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

theorem Th18:
  F is_odd_on A implies F" is_odd_on A
proof
  assume
A1: F is_odd_on A;
  then
A2: A c= dom F;
  then
A3: A c= dom (F") by VALUED_1:def 7;
  then
A4: dom((F")|A) = A by RELAT_1:62;
A5: F|A is odd by A1;
  for x st x in dom((F")|A) & -x in dom((F")|A) holds (F")|A.(-x)=-(F")|A. x
  proof
    let x;
    assume that
A6: x in dom((F")|A) and
A7: -x in dom((F")|A);
A8: x in dom(F|A) by A2,A4,A6,RELAT_1:62;
A9: -x in dom(F|A) by A2,A4,A7,RELAT_1:62;
      reconsider x as Element of REAL by XREAL_0:def 1;
    (F")|A.(-x)=(F")|A/.(-x) by A7,PARTFUN1:def 6
      .=(F")/.(-x) by A3,A4,A7,PARTFUN2:17
      .=(F").(-x) by A3,A7,PARTFUN1:def 6
      .=(F.(-x))" by A3,A7,VALUED_1:def 7
      .=(F/.(-x))" by A2,A7,PARTFUN1:def 6
      .=(F|A/.(-x))" by A2,A4,A7,PARTFUN2:17
      .=(F|A.(-x))" by A9,PARTFUN1:def 6
      .=(-F|A.x)" by A5,A8,A9,Def6
      .=(-F|A/.x)" by A8,PARTFUN1:def 6
      .=(-F/.x)" by A2,A4,A6,PARTFUN2:17
      .=(-F.x)" by A2,A6,PARTFUN1:def 6
      .=-(F.x)" by XCMPLX_1:222
      .=-(F").x by A3,A6,VALUED_1:def 7
      .=-(F")/.x by A3,A6,PARTFUN1:def 6
      .=-(F")|A/.x by A3,A4,A6,PARTFUN2:17
      .=-(F")|A.x by A6,PARTFUN1:def 6;
    hence thesis;
  end;
  then (F")|A is with_symmetrical_domain quasi_odd by A4;
  hence thesis by A3;
end;
