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 & for x st x in A holds F.x<>0) implies (A c= dom F &
  for x st x in A holds F.x / F.(-x)=-1)
proof
  assume that
A1: F is_odd_on A and
A2: for x st x in A holds F.x<>0;
A3: A c= dom F by A1;
A4: F|A is odd by A1;
  for x st x in A holds F.x / F.(-x)=-1
  proof
    let x;
    assume
A5: x in A;
    then
A6: x in dom(F|A) by A3,RELAT_1:62;
      reconsider x as Element of REAL by XREAL_0:def 1;
A7: F.x=F/.x by A3,A5,PARTFUN1:def 6
      .=F|A/.x by A3,A5,PARTFUN2:17
      .=F|A.x by A6,PARTFUN1:def 6;
A8: -x in A by A5,Def1;
    then
A9: -x in dom(F|A) by A3,RELAT_1:62;
    F.x / F.(-x)=F/.x / F.(-x) by A3,A5,PARTFUN1:def 6
      .=F/.x / F/.(-x) by A3,A8,PARTFUN1:def 6
      .=F|A/.x / F/.(-x) by A3,A5,PARTFUN2:17
      .=F|A/.x / F|A/.(-x) by A3,A8,PARTFUN2:17
      .=F|A.x / F|A/.(-x) by A6,PARTFUN1:def 6
      .=F|A.x / F|A.(-x) by A9,PARTFUN1:def 6
      .=F|A.x / (-F|A.x) by A4,A6,A9,Def6
      .=-(F|A.x / F|A.x) by XCMPLX_1:188
      .=-1 by A2,A5,A7,XCMPLX_1:60;
    hence thesis;
  end;
  hence thesis by A1;
end;
