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 Th45:
  |{ p, q, p <X> q }| = |(q,q)| * |(p,p)| - |(q,p)| * |(p,q)|
  proof
    |{ p, q, p <X> q }| = |( p , q <X> (p <X> q) )| by EUCLID_5:def 5
                       .= |( p, |(q,q)| * p - |(q,p)| * q )| by EUCLID_5:32
                       .= |(p, |(q,q)| * p)| - |( p, |(q,p)| * q)|
                          by EUCLID_2:27
                       .= |(q,q)| * |(p,p)| - |( p, |(q,p)| * q)|
                          by EUCLID_2:20
                       .= |(q,q)| * |(p,p)| - |(q,p)| * |(p,q)| by EUCLID_2:20;
    hence thesis;
  end;
