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  Th24:
  |{ p,q,r }| = - |{ p,r,q }|
  proof
A1: |{ 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 by Th23;
    |{ p,r,q }| = p`1 * r`2 * q`3 - p`3*r`2*q`1 - p`1*r`3*q`2 +
      p`2*r`3*q`1 - p`2*r`1*q`3 + p`3*r`1*q`2 by Th23;
    hence thesis by A1;
  end;
