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, q, p <X> q }| = |. p <X> q .|^2
  proof
    |{ p, q, p <X> q }| = |(q,q)| * |(p,p)| - |(q,p)| * |(p,q)| by Th45
                       .= |(p <X> q,p <X> q)| by Th46;
    hence thesis by EUCLID_2:4;
  end;
