theorem Th13:
  <e1> <> 0.TOP-REAL 3 &
  <e2> <> 0.TOP-REAL 3 &
  <e3> <> 0.TOP-REAL 3
  proof
    |[1,0,0]| <> |[0,0,0]|
    proof
      assume |[1,0,0]| = |[0,0,0]|;
      then 1 = |[0,0,0]|`1 by EUCLID_5:2;
      hence thesis by EUCLID_5:2;
    end;
    hence <e1> <> 0.TOP-REAL 3 by EUCLID_5:4,EUCLID_8:def 1;
    |[0,1,0]| <> |[0,0,0]|
    proof
      assume |[0,1,0]| = |[0,0,0]|;
      then 1 = |[0,0,0]|`2 by EUCLID_5:2;
      hence thesis by EUCLID_5:2;
    end;
    hence <e2> <> 0.TOP-REAL 3 by EUCLID_5:4,EUCLID_8:def 2;
    |[0,0,1]| <> |[0,0,0]|
    proof
      assume |[0,0,1]| = |[0,0,0]|;
      then 1 = |[0,0,0]|`3 by EUCLID_5:2;
      hence thesis by EUCLID_5:2;
    end;
    hence <e3> <> 0.TOP-REAL 3 by EUCLID_5:4,EUCLID_8:def 3;
  end;
