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 implies r+F is_even_on A
proof
  assume
A1: F is_even_on A;
  then
A2: A c= dom F;
  then
A3: A c= dom (r+F) by VALUED_1:def 2;
  then
A4: dom((r+F)|A) = A by RELAT_1:62;
A5: F|A is even by A1;
  for x st x in dom((r+F)|A) & -x in dom((r+F)|A) holds (r+F)|A.(-x)=(r+F) |A.x
  proof
    let x;
    assume that
A6: x in dom((r+F)|A) and
A7: -x in dom((r+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;
    (r+F)|A.(-x)=(r+F)|A/.(-x) by A7,PARTFUN1:def 6
      .=(r+F)/.(-x) by A3,A4,A7,PARTFUN2:17
      .=(r+F).(-x) by A3,A7,PARTFUN1:def 6
      .=r + F.(-x) by A3,A7,VALUED_1:def 2
      .=r + F/.(-x) by A2,A7,PARTFUN1:def 6
      .=r + F|A/.(-x) by A2,A4,A7,PARTFUN2:17
      .=r + F|A.(-x) by A9,PARTFUN1:def 6
      .=r + F|A.x by A5,A8,A9,Def3
      .=r + F|A/.x by A8,PARTFUN1:def 6
      .=r + F/.x by A2,A4,A6,PARTFUN2:17
      .=r + F.x by A2,A6,PARTFUN1:def 6
      .=(r+F).x by A3,A6,VALUED_1:def 2
      .=(r+F)/.x by A3,A6,PARTFUN1:def 6
      .=(r+F)|A/.x by A3,A4,A6,PARTFUN2:17
      .=(r+F)|A.x by A6,PARTFUN1:def 6;
    hence thesis;
  end;
  then (r+F)|A is with_symmetrical_domain quasi_even by A4;
  hence thesis by A3;
end;
