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 Th37:
  p,q,r are_LinDep iff |{p,q,r}| = 0
  proof
    hereby assume p,q,r are_LinDep;
      then ex a,b,c st a*p + b*q + c*r = 0.TOP-REAL 3 &
      (a<>0 or b<>0 or c <>0) by ANPROJ_1:def 2;
      hence |{p,q,r}| = 0 by Th35;
    end;
    assume |{p,q,r}| = 0;
    then ex a,b,c st a*p + b*q + c*r = 0.TOP-REAL 3 &
    (a<>0 or b<>0 or c <>0) by Th36;
    hence p,q,r are_LinDep by ANPROJ_1:def 2;
  end;
