 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;

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