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 F-r is_even_on A
proof
  assume
A1: F is_even_on A;
  then
A2: A c= dom F;
  then
A3: A c= dom (F-r) by VALUED_1:3;
  then
A4: dom((F-r)|A) = A by RELAT_1:62;
A5: F|A is even by A1;
  for x st x in dom((F-r)|A) & -x in dom((F-r)|A) holds (F-r)|A.(-x)=(F-r) |A.x
  proof
    let x;
    assume that
A6: x in dom((F-r)|A) and
A7: -x in dom((F-r)|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;
    (F-r)|A.(-x)=(F-r)|A/.(-x) by A7,PARTFUN1:def 6
      .=(F-r)/.(-x) by A3,A4,A7,PARTFUN2:17
      .=(F-r).(-x) by A3,A7,PARTFUN1:def 6
      .=F.(-x)-r by A2,A7,VALUED_1:3
      .=F/.(-x)-r by A2,A7,PARTFUN1:def 6
      .=F|A/.(-x)-r by A2,A4,A7,PARTFUN2:17
      .=F|A.(-x)-r by A9,PARTFUN1:def 6
      .=F|A.x-r by A5,A8,A9,Def3
      .=F|A/.x-r by A8,PARTFUN1:def 6
      .=F/.x-r by A2,A4,A6,PARTFUN2:17
      .=F.x-r by A2,A6,PARTFUN1:def 6
      .=(F-r).x by A2,A6,VALUED_1:3
      .=(F-r)/.x by A3,A6,PARTFUN1:def 6
      .=(F-r)|A/.x by A3,A4,A6,PARTFUN2:17
      .=(F-r)|A.x by A6,PARTFUN1:def 6;
    hence thesis;
  end;
  then (F-r)|A is with_symmetrical_domain quasi_even by A4;
  hence thesis by A3;
end;
