
theorem
  for z being R_eal holds z in REAL iff - z in REAL
proof
  let z be R_eal;
A1: for z being R_eal holds z in REAL implies - z in REAL
  proof
    let z be R_eal;
A2: - z in REAL or - z in {-infty,+infty} by XBOOLE_0:def 3;
    assume z in REAL;
    hence thesis by A2,TARSKI:def 2;
  end;
  - z in REAL implies z in REAL
  proof
    assume -z in REAL;
    then - -z in REAL by A1;
    hence thesis;
  end;
  hence thesis by A1;
end;
