reserve x1, x2, x3, x4, x5, x6, x7 for set;

theorem Th16:
  for x being Real holds x is irrational iff x in IRRAT
proof
  let x be Real;
A1: x in REAL by XREAL_0:def 1;
  hereby
    assume x is irrational;
    then not x in RAT;
    hence x in IRRAT by A1,XBOOLE_0:def 5;
  end;
  assume x in IRRAT;
  then not x in RAT by XBOOLE_0:def 5;
  hence thesis by RAT_1:def 2;
end;
