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 cosh is_even_on A
proof
  let A be symmetrical Subset of REAL;
A1: dom cosh = REAL by FUNCT_2:def 1;
  then
A2: dom(cosh|A) = A by RELAT_1:62;
  for x st x in dom(cosh|A) & -x in dom(cosh|A) holds cosh|A.(-x)=cosh|A.x
  proof
    let x;
    assume that
A3: x in dom(cosh|A) and
A4: -x in dom(cosh|A);
      reconsider x as Element of REAL by XREAL_0:def 1;
    cosh|A.(-x)=cosh|A/.(-x) by A4,PARTFUN1:def 6
      .=cosh/.(-x) by A1,A4,PARTFUN2:17
      .=cosh/.x by SIN_COS2:19
      .=cosh|A/.x by A1,A3,PARTFUN2:17
      .=cosh|A.x by A3,PARTFUN1:def 6;
    hence thesis;
  end;
  then cosh|A is with_symmetrical_domain quasi_even by A2;
  hence thesis by A1;
end;
