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

theorem
  F is even implies F-r is even
proof
A1: dom F=dom(F-r ) by VALUED_1:3;
  assume
A2: F is even;
  for x st x in dom(F-r) & -x in dom(F-r) holds (F-r).(-x)=(F-r).x
  proof
    let x;
    assume that
A3: x in dom(F-r) and
A4: -x in dom(F-r);
A5: x in dom F by A3,VALUED_1:3;
    -x in dom F by A4,VALUED_1:3;
    then (F-r).(-x)=F.(-x)-r by VALUED_1:3
      .=F.x -r by A2,A1,A3,A4,Def3
      .=(F-r).x by A5,VALUED_1:3;
    hence thesis;
  end;
  then F-r is with_symmetrical_domain quasi_even by A2,A1;
  hence thesis;
end;
