
theorem Th8:
  for a,u,v being non zero Element of TOP-REAL 3 st not are_Prop u,v &
  |(a,u)| = 0 & |(a,v)| = 0 holds are_Prop a,u <X> v
  proof
    let a,u,v be non zero Element of TOP-REAL 3;
    assume that
A1: not are_Prop u,v and
A2: |(a,u)| = 0 and
A3: |(a,v)| = 0;
    u <X> v is non zero by A1,ANPROJ_8:51;
    then reconsider uv = u <X> v as non zero Element of TOP-REAL 3;
A4: a`1 * u`1 + a`2 * u`2 + a`3 * u`3 = 0 &
      a`1 * v`1 + a`2 * v`2 + a`3 * v`3 = 0 by A2,A3,EUCLID_5:29;
    per cases by EUCLID_5:3,4;
    suppose
A5:   a`1 <> 0;
      then
A6:   u`1 = -a`2/a`1 * u`2 - a`3/a`1 * u`3 &
        v`1 = -a`2/a`1 * v`2 - a`3/a`1 * v`3 by A4,ANPROJ_8:13;
      set p1 = u,p2 = v;
      now
        reconsider r = a`1 as Real;
        thus
A7:     u <X> v = |[ 1 *( p1`2 * p2`3 - p1`3 * p2`2),
                     a`2/a`1 *( p1`2 * p2`3 - p1`3 * p2`2),
                     (a`3/a`1) * (- p1`3*p2`2 + p1`2*p2`3) ]| by A6
         .= ( p1`2 * p2`3 - p1`3 * p2`2) * |[ 1 ,a`2/a`1, a`3/a`1 ]|
           by EUCLID_5:8
         .= ( p1`2 * p2`3 - p1`3 * p2`2) * |[ a`1 / r, a`2 / r, a`3 / r ]|
           by A5,XCMPLX_1:60
         .= ( p1`2 * p2`3 - p1`3 * p2`2) * ((1/a`1) * a) by EUCLID_5:7
         .= (( p1`2 * p2`3 - p1`3 * p2`2) * (1/a`1)) * a by RVSUM_1:49;
        p1`2 * p2`3 - p1`3 * p2`2 <> 0
        proof
          assume p1`2 * p2`3 - p1`3 * p2`2 = 0;
          then u <X> v = |[0 * a`1,0 * a`2,0 * a`3]| by A7,EUCLID_5:7
                      .= 0.TOP-REAL 3 by EUCLID_5:4;
          hence thesis by A1,ANPROJ_8:51;
        end;
        hence (p1`2 * p2`3 - p1`3 * p2`2) * (1/a`1) <> 0 by A5;
      end;
      hence thesis by ANPROJ_1:1;
    end;
    suppose
A8:   a`2 <> 0;
      then
A9:   u`2 = -a`1/a`2 * u`1 - a`3/a`2 * u`3 &
        v`2 = -a`1/a`2 * v`1 - a`3/a`2 * v`3 by A4,ANPROJ_8:13;
      set p1 = u, p2 = v;
      now
        reconsider r = a`2 as Real;
        thus
A10:     u <X> v = |[ (a`1/a`2) *( p1`3 * p2`1 - p1`1 * p2`3),
                      1 *( p1`3 * p2`1 - p1`1 * p2`3),
                      (a`3/a`2) * ( p1`3*p2`1 - p1`1*p2`3) ]| by A9
               .= (p1`3*p2`1-p1`1*p2`3) * |[a`1/a`2,1,a`3/a`2]| by EUCLID_5:8
               .= (p1`3*p2`1-p1`1*p2`3) * |[a`1/r,r/r,a`3/r]| by A8,XCMPLX_1:60
               .= (p1`3*p2`1-p1`1*p2`3) * ((1/a`2) * a) by EUCLID_5:7
               .= ((p1`3*p2`1-p1`1*p2`3) * (1/a`2)) * a by RVSUM_1:49;
        p1`3*p2`1-p1`1*p2`3 <> 0
        proof
          assume p1`3*p2`1-p1`1*p2`3 = 0;
          then u <X> v = |[0 * a`1,0 * a`2,0 * a`3]| by A10,EUCLID_5:7
                      .= 0.TOP-REAL 3 by EUCLID_5:4;
          hence thesis by A1,ANPROJ_8:51;
        end;
        hence (p1`3*p2`1-p1`1*p2`3) * (1/a`2) <> 0 by A8;
      end;
      hence thesis by ANPROJ_1:1;
    end;
    suppose
A11:  a`3 <> 0;
      a`3 * u`3 + a`1 * u`1 + a`2 * u`2 = 0 &
        a`3 * v`3 + a`1 * v`1 + a`2 * v`2 = 0 by A4;
      then
A12:   u`3 = -a`1/a`3 * u`1 - a`2/a`3 * u`2 &
        v`3 = -a`1/a`3 * v`1 - a`2/a`3 * v`2 by A11,ANPROJ_8:13;
      set p1 = u, p2 = v;
      now
        reconsider r = a`3 as Real;
        thus
A13:    u <X> v = |[ (a`1/a`3) * (p1`1 * p2`2 - p1`2 * p2`1),
                      a`2/a`3 * (p1`1 * p2`2 - p1`2 * p2`1),
                      1 * (p1`1 * p2`2 - p1`2 * p2`1) ]| by A12
               .= (p1`1*p2`2-p1`2*p2`1) * |[a`1/a`3,a`2/a`3,1]| by EUCLID_5:8
               .= (p1`1*p2`2-p1`2*p2`1) * |[a`1/r,a`2/r,r/r]|
                 by A11,XCMPLX_1:60
               .= (p1`1*p2`2-p1`2*p2`1) * ((1/a`3) * a) by EUCLID_5:7
               .= ((p1`1*p2`2-p1`2*p2`1) * (1/a`3)) * a by RVSUM_1:49;
        p1`1*p2`2-p1`2*p2`1 <> 0
        proof
          assume p1`1*p2`2-p1`2*p2`1 = 0;
          then u <X> v = |[0 * a`1,0 * a`2,0 * a`3]| by A13,EUCLID_5:7
                      .= 0.TOP-REAL 3 by EUCLID_5:4;
          hence thesis by A1,ANPROJ_8:51;
        end;
        hence (p1`1*p2`2-p1`2*p2`1) * (1/a`3) <> 0 by A11;
      end;
      hence thesis by ANPROJ_1:1;
    end;
  end;
