reserve r,s,t,x9,y9,z9,p,q for Element of RAT+;
reserve x,y,z for Element of REAL+;

theorem
  not ex y being object st [{},y] in REAL+
proof
  given y being object such that
A1: [{},y] in REAL+;
  per cases by A1,Lm3,XBOOLE_0:def 3;
  suppose
    [{},y] in RAT+;
    hence contradiction by ARYTM_3:61;
  end;
  suppose
    [{},y] in IR;
    hence contradiction by Lm1;
  end;
end;
