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 Th40:
  |{ o, p, (o <X> p) <X> (q <X> r) }| = 0 &
  |{ q, r, (o <X> p) <X> (q <X> r) }| = 0
  proof
    set s = (o <X> p) <X> (q <X> r);
    thus |{o,p,s}| = |( o <X> p , s)| by EUCLID_5:35
                  .= 0 by Th38;
    thus |{q,r,s}| = |( q <X> r, (o <X> p) <X> (q <X> r) )| by EUCLID_5:35
                  .= 0 by Th39;
  end;
