
theorem
  for u,v,w being non zero Element of TOP-REAL 3 st |{u,v,w}| = 0
  holds ex p being non zero Element of TOP-REAL 3 st
  |(p,u)| = 0 & |(p,v)| = 0 & |(p,w)| = 0
  proof
    let u,v,w be non zero Element of TOP-REAL 3;
    assume
A1: |{u,v,w}| = 0;
    reconsider p = |[u`1,v`1,w`1]|,
               q = |[u`2,v`2,w`2]|,
               r = |[u`3,v`3,w`3]| as Element of TOP-REAL 3;
A3: |{ p,q,r }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 -
      p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27;
    |{u,v,w}| = u`1 * v`2 * w`3 - u`3*v`2*w`1 - u`1*v`3*w`2 + u`2*v`3*w`1 -
      u`2*v`1*w`3 + u`3*v`1*w`2 by ANPROJ_8:27;
    then consider a,b,c be Real such that
A4: a * p + b * q + c * r = 0.TOP-REAL 3 and
A5: a <> 0 or b <> 0 or c <> 0 by A1,A3,ANPROJ_8:42;
A6: |[0,0,0]|
        = |[a * p`1,a * p`2,a * p`3]| + b * q + c * r by A4,EUCLID_5:4,7
       .= |[a * p`1,a * p`2,a * p`3]| + |[b * q`1,b * q`2,b * q`3]| + c * r
         by EUCLID_5:7
       .= |[a * p`1,a * p`2,a * p`3]| + |[b * q`1,b * q`2,b * q`3]|
         + |[c * r`1,c * r`2,c * r`3]| by EUCLID_5:7
       .= |[a * p`1+b*q`1,a * p`2+b*q`2,a * p`3+b*q`3]|
         + |[c * r`1,c * r`2,c * r`3]| by EUCLID_5:6
       .= |[a * p`1+b*q`1+c*r`1,a * p`2+b*q`2+c*r`2,a * p`3+b*q`3+c*r`3]|
         by EUCLID_5:6;
    reconsider p = |[a,b,c]| as non zero Element of TOP-REAL 3 by A5;
    take p;
    thus |(p,u)| = p`1 * u`1 + p`2 * u`2 + p`3 * u`3 by EUCLID_5:29
                .= a * u`1+p`2*u`2+p`3*u`3
                .= a * u`1+b*u`2+p`3*u`3
                .= a * u`1+b*u`2+c*u`3
                .= 0 by A6,FINSEQ_1:78;
    thus |(p,v)| = p`1 * v`1 + p`2 * v`2 + p`3 * v`3 by EUCLID_5:29
                .= a * v`1+p`2*v`2+p`3*v`3
                .= a * v`1+b*v`2+p`3*v`3
                .= a * v`1+b*v`2+c*v`3
                .= 0 by A6,FINSEQ_1:78;
    thus |(p,w)| = p`1 * w`1 + p`2 * w`2 + p`3 * w`3 by EUCLID_5:29
                .= a * w`1+p`2*w`2+p`3*w`3
                .= a * w`1+b*w`2+p`3*w`3
                .= a * w`1+b*w`2+c*w`3
                .= 0 by A6,FINSEQ_1:78;
  end;
