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

theorem
  F is_even_on A & F is_odd_on A implies for x st x in A holds F.x=0
proof
  assume that
A1: F is_even_on A and
A2: F is_odd_on A;
A3: F|A is even by A1;
A4: F|A is odd by A2;
  let x;
  assume
A5: x in A;
A6: A c= dom F by A1;
  then
A7: x in dom(F|A) by A5,RELAT_1:62;
  -x in A by A5,Def1;
  then
A8: -x in dom(F|A) by A6,RELAT_1:62;
      reconsider x as Element of REAL by XREAL_0:def 1;
  F.x=F/.x by A6,A5,PARTFUN1:def 6
    .=F|A/.x by A6,A5,PARTFUN2:17
    .=F|A.x by A7,PARTFUN1:def 6
    .=F|A.(-x) by A3,A7,A8,Def3
    .=-F|A.x by A4,A7,A8,Def6
    .=-F|A/.x by A7,PARTFUN1:def 6
    .=-F/.x by A6,A5,PARTFUN2:17
    .=-F.x by A6,A5,PARTFUN1:def 6;
  hence thesis;
end;
