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.|) 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.|;
A3: dom(F|A) = A by A1,RELAT_1:62;
A4: for x st x in A holds -x in A by Def1;
A5: for x st x in A holds F.(-x) = F.(x)
  proof
    let x;
    assume
A6: x in A;
    per cases;
    suppose
      x < 0;
      then F.(-x)=F. |.x.| by ABSVALUE:def 1
        .=F.x by A2,A6;
      hence thesis;
    end;
    suppose
      0 < x;
      then |.-x.|=-(-x) by ABSVALUE:def 1
        .=x;
      hence thesis by A2,A4,A6;
    end;
    suppose
      x = 0;
      hence thesis;
    end;
  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
A7: x in dom(F|A) and
A8: -x in dom(F|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    F|A.(-x)=F|A/.(-x) by A8,PARTFUN1:def 6
      .=F/.(-x) by A1,A3,A8,PARTFUN2:17
      .=F.(-x) by A1,A8,PARTFUN1:def 6
      .=F.x by A5,A7
      .=F/.x by A1,A7,PARTFUN1:def 6
      .=F|A/.x by A1,A3,A7,PARTFUN2:17
      .=F|A.x by A7,PARTFUN1:def 6;
    hence thesis;
  end;
  then F|A is with_symmetrical_domain quasi_even by A3;
  hence thesis by A1;
end;
