theorem Th39:
  for a being Real st n>=1 holds (REAL n) \ {q: |.q.| < a} <> {}
proof
  let a be Real;
A1: {q:(|.q.|)>a} c= (REAL n)\{q2:(|.q2.|)<a}
  proof
    let x be object;
    assume x in {q:(|.q.|)>a};
    then consider q such that
A2: q=x and
A3: (|.q.|)>a;
A4: now
      assume x in {q2:(|.q2.|)<a};
      then ex q2 st q2=x & (|.q2.|)<a;
      hence contradiction by A2,A3;
    end;
    q in the carrier of TOP-REAL n;
    then q in REAL n by EUCLID:22;
    hence thesis by A2,A4,XBOOLE_0:def 5;
  end;
  assume n>=1;
  hence thesis by A1,Th37,XBOOLE_1:3;
end;
