reserve a,b,c,d,e,f for Real,
        k,m for Nat,
        D for non empty set,
        V for non trivial RealLinearSpace,
        u,v,w for Element of V,
        p,q,r for Element of ProjectiveSpace(V);
reserve o,p,q,r,s,t for Point of TOP-REAL 3,
        M for Matrix of 3,F_Real;

theorem
  p <X> q = 0.TOP-REAL 3 & r = |[1,1,1]| implies p,q,r are_LinDep
  proof
    assume that
A1: p <X> q = 0.TOP-REAL 3 and
A2: r = |[1,1,1]|;
    |[ (p`2 * q`3) - (p`3 * q`2) , (p`3 * q`1) - (p`1 * q`3) ,
    (p`1 * q`2) - (p`2 * q`1) ]| = |[0,0,0]| by A1,EUCLID_5:def 4,EUCLID_5:4;
    then (p`2 * q`3) - (p`3 * q`2) = |[0,0,0]|`1 &
      (p`3 * q`1) - (p`1 * q`3) = |[0,0,0]|`2 &
      (p`1 * q`2) - (p`2 * q`1) = |[0,0,0]|`3 by EUCLID_5:14; then
A3: (p`2 * q`3) - (p`3 * q`2) = 0 & (p`3 * q`1) - (p`1 * q`3) = 0 &
      (p`1 * q`2) - (p`2 * q`1) = 0 by EUCLID_5:14;
    reconsider r = |[1,1,1]| as Element of TOP-REAL 3;
    |{p,q,r}| = 0
    proof
      r`1 = 1 & r`2 = 1 & r`3 = 1 by EUCLID_5:2;
      then |{ p,q,r }| = p`1 * q`2 * 1 - p`3* q`2 * 1 -
                            p`1*q`3* 1 + p`2*q`3* 1 -
                            p`2*q`1* 1 + p`3*q`1* 1 by Th23
                      .= 0 by A3;
      hence thesis;
    end;
    hence thesis by A2,Th37;
  end;
