reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;

theorem
  a * u + (-a) * u = 0.TOP-REAL 3
  proof
    a* u = |[ a * u`1,a * u`2 , a * u`3 ]| by EUCLID_5:7;
    then a * u +(-a) * u = |[ a * u`1,a * u`2 , a * u`3 ]| +
      |[ (-a) * u`1,(-a) * u`2 , (-a) * u`3 ]| by EUCLID_5:7
                        .= |[ a * u`1 + (-a) * u`1 ,a * u`2 + (-a) * u`2,
      a * u`3 + (-a) *u`3 ]| by EUCLID_5:6
                        .= |[0,0,0]|;
    hence thesis by EUCLID_5:4;
  end;
