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,r}| * |{p,s,t}| - |{p,q,s}| * |{p,r,t}| + |{p,q,t}| * |{p,r,s}| = 0
  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;
A2: |{p,s,t}| = p`1 * s`2 * t`3 - p`3 * s`2 * t`1 - p`1 * s`3 * t`2 +
      p`2 * s`3 * t`1 - p`2 * s`1 * t`3 + p`3 * s`1 * t`2 by Th23;
A3: |{p,q,s}| = p`1 * q`2 * s`3 - p`3 * q`2 * s`1 - p`1 * q`3 * s`2 +
      p`2 * q`3 * s`1 - p`2 * q`1 * s`3 + p`3 * q`1 * s`2 by Th23;
A4: |{p,r,t}| = p`1 * r`2 * t`3 - p`3 * r`2 * t`1 - p`1 * r`3 * t`2 +
      p`2 * r`3 * t`1 - p`2 * r`1 * t`3 + p`3 * r`1 * t`2 by Th23;
A5: |{p,q,t}| = p`1 * q`2 * t`3 - p`3 * q`2 * t`1 - p`1 * q`3 * t`2 +
      p`2 * q`3 * t`1 - p`2 * q`1 * t`3 + p`3 * q`1 * t`2 by Th23;
    |{p,r,s}| = p`1 * r`2 * s`3 - p`3 * r`2 * s`1 - p`1 * r`3 * s`2 +
      p`2 * r`3 * s`1 - p`2 * r`1 * s`3 + p`3 * r`1 * s`2 by Th23;
    hence thesis by A1,A2,A3,A4,A5;
  end;
