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

theorem
  (A c= dom F & for x st x in A holds F.x / F.(-x)=1) implies F is_even_on A
proof
  assume that
A1: A c= dom F and
A2: for x st x in A holds F.x / F.(-x)=1;
A3: dom(F|A) = A by A1,RELAT_1:62;
A4: for x st x in A holds F.(-x)=F.x
  proof
    let x;
    assume x in A;
    then F.x / F.(-x)=1 by A2;
    hence thesis by XCMPLX_1:58;
  end;
  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
A5: x in dom(F|A) and
A6: -x in dom(F|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    F|A.(-x)=F|A/.(-x) by A6,PARTFUN1:def 6
      .=F/.(-x) by A1,A3,A6,PARTFUN2:17
      .=F.(-x) by A1,A6,PARTFUN1:def 6
      .=F.x by A4,A5
      .=F/.x by A1,A5,PARTFUN1:def 6
      .=F|A/.x by A1,A3,A5,PARTFUN2:17
      .=F|A.x by A5,PARTFUN1:def 6;
    hence thesis;
  end;
  then F|A is with_symmetrical_domain quasi_even by A3;
  hence thesis by A1;
end;
