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
  M = <*<*p`1,q`1,r`1*>,<*p`2,q`2,r`2*>,<*p`3,q`3,r`3*>*> implies
  |{ p,q,r }| = Det M
  proof
    assume M = <*<*p`1,q`1,r`1*>,<*p`2,q`2,r`2*>,<*p`3,q`3,r`3*>*>; then
A1: M@ = <*<*p`1,p`2,p`3*>,<*q`1,q`2,q`3*>,<*r`1,r`2,r`3*>*> by Th19;
A2: p = |[ p`1,p`2,p`3 ]| by EUCLID_5:3;
A3: 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;
A4: |( 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 A2,A3,EUCLID_5:30;
    reconsider p1 = p`1,p2 = p`2,p3 = p`3,
               q1 = q`1,q2 = q`2,q3 = q`3,
               r1 = r`1,r2 = r`2,r3 = r`3 as Element of F_Real
      by XREAL_0:def 1;
    Det M = Det M@ by MATRIX_7:37
         .= p1*q2*r3 - p3*q2*r1 - p1*q3*r2
              + p2*q3*r1 - p2*q1*r3 + p3*q1*r2
      by A1,MATRIX_9:46;
    hence thesis by A4,EUCLID_5:def 5;
  end;
