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

theorem Th66:
  for A being symmetrical Subset of REAL holds cos is_even_on A
proof
  let A be symmetrical Subset of REAL;
A1: dom(cos|A) = A by RELAT_1:62,SIN_COS:24;
  for x st x in dom(cos|A) & -x in dom(cos|A) holds cos|A.(-x)=cos|A.x
  proof
    let x;
    assume that
A2: x in dom(cos|A) and
A3: -x in dom(cos|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    cos|A.(-x)=cos|A/.(-x) by A3,PARTFUN1:def 6
      .=cos/.(-x) by A3,PARTFUN2:17,SIN_COS:24
      .=cos/.x by SIN_COS:30
      .=cos|A/.x by A2,PARTFUN2:17,SIN_COS:24
      .=cos|A.x by A2,PARTFUN1:def 6;
    hence thesis;
  end;
  then cos|A is with_symmetrical_domain quasi_even by A1;
  hence thesis by SIN_COS:24;
end;
