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 Th36:
  |{p,q,r}| = 0 implies ex a,b,c st a*p + b*q + c*r = 0.(TOP-REAL 3) &
  (a<>0 or b<>0 or c <>0)
  proof
    assume
A1: |{p,q,r}| = 0;
    reconsider M = <*<*p`1,p`2,p`3*>,<*q`1,q`2,q`3*>,<*r`1,r`2,r`3*>*>
      as Matrix of 3,F_Real by Th16;
    Det M = 0 by A1,Th29
         .= 0.F_Real by STRUCT_0:def 6;
    then the_rank_of M < 3 by Th30;
    hence thesis by Th34;
  end;
