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 Th23:
  |{ p,q,r }| = p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 -
    p`2*q`1*r`3 + p`3*q`1*r`2
  proof
A1: p = |[ p`1,p`2,p`3 ]| by EUCLID_5:3;
A2: q <X> r = |[ (q`2 * r`3) - (q`3 * r`2) , (q`3 * r`1) - (q`1 * r`3) ,
      (q`1 * r`2) - (q`2 * r`1) ]| by EUCLID_5:def 4;
    |( p, q <X> r )| = p`1*((q`2 * r`3) - (q`3 * r`2))+p`2*((q`3 * r`1) -
      (q`1 * r`3))+ p`3*((q`1 * r`2) - (q`2 * r`1)) by A1,A2,EUCLID_5:30;
    hence thesis by EUCLID_5:def 5;
  end;
