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 Th46:
  |(p <X> q, p <X> q)| = |(q,q)| * |(p,p)| - |(q,p)| * |(p,q)|
  proof
    set r1 = (p <X> q)`1, r2 = (p <X> q)`2, r3 = (p <X> q)`3;
    p <X> q = |[ (p`2 * q`3) - (p`3 * q`2) , (p`3 * q`1) - (p`1 * q`3) ,
      (p`1 * q`2) - (p`2 * q`1) ]| by EUCLID_5:def 4; then
A1: r1 = p`2 * q`3 - p`3 * q`2 & r2 = p`3 * q`1 - p`1 * q`3 &
      r3 = p`1 * q`2 - p`2 * q`1 by EUCLID_5:2;
A2: |(p <X> q,p <X> q)| = r1 * r1 + r2 * r2 + r3 * r3 by EUCLID_5:29;
    |(q,q)| * |(p,p)| - |(q,p)| * |(p,q)| =
     (q`1 * q`1 + q`2 * q`2 + q`3 * q`3) * |(p,p)| - |(q,p)| * |(p,q)|
     by EUCLID_5:29
                        .= (q`1 * q`1 + q`2 * q`2 + q`3 * q`3) *
       (p`1 * p`1 + p`2 * p`2 + p`3 * p`3) - |(q,p)| * |(p,q)|
       by EUCLID_5:29
                        .= (q`1 * q`1 + q`2 * q`2 + q`3 * q`3) *
       (p`1 * p`1 + p`2 * p`2 + p`3 * p`3) - (q`1*p`1+q`2*p`2+q`3*p`3)
       * |(p,q)| by EUCLID_5:29
                        .= (q`1 * q`1 + q`2 * q`2 + q`3 * q`3) *
       (p`1 * p`1 + p`2 * p`2 + p`3 * p`3) - (q`1*p`1+q`2*p`2+q`3*p`3) *
       (p`1*q`1+p`2*q`2+p`3*q`3) by EUCLID_5:29;
    hence thesis by A1,A2;
  end;
