
theorem Th5:
  for u being Element of TOP-REAL 3 holds u is zero iff |( u, u )| = 0
  proof
    let u be Element of TOP-REAL 3;
    reconsider un = u as Element of REAL 3 by EUCLID:22;
    hereby
      assume u is zero;
      then 0.REAL 3 = u by EUCLID:66;
      then |( un,un )| = 0 by EUCLID_4:17;
      hence |( u, u )| = 0;
    end;
    assume |( u, u )| = 0;
    then un = 0.REAL 3 by EUCLID_4:17;
    hence thesis by EUCLID:66;
  end;
