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

theorem
  for A being symmetrical Subset of REAL holds absreal is_even_on A
proof
  let A be symmetrical Subset of REAL;
A1: dom absreal = REAL by FUNCT_2:def 1;
  then
A2: dom(absreal|A) = A by RELAT_1:62;
  for x st x in dom(absreal|A) & -x in dom(absreal|A) holds absreal|A.(-x)
  =absreal|A.x
  proof
    let x;
    assume that
A3: x in dom(absreal|A) and
A4: -x in dom(absreal|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    absreal|A.(-x)=absreal|A/.(-x) by A4,PARTFUN1:def 6
      .=absreal/.(-x) by A1,A4,PARTFUN2:17
      .=absreal/.x by Th63
      .=absreal|A/.x by A1,A3,PARTFUN2:17
      .=absreal|A.x by A3,PARTFUN1:def 6;
    hence thesis;
  end;
  then absreal|A is with_symmetrical_domain quasi_even by A2;
  hence thesis by A1;
end;
