A computational proof of the theorem of Pappus in projective geometry

Projective points A1, B1,C1 lie on one line

A1 := {1, 0, 0};
B1 :=  {0, 1, 0};
C1 :=  A1  +  p*B1

Points A2, B2, C2 lie on a different line

A2 := {0, 0, 1};
B2 :=  {1, 1, 1};
C2 :=  A2  +  q*B2

Now find where line A1B2 intersects B1A2.

L = (A1\[Cross]B2)\[Cross](B1\[Cross]A2)

Find the other two points of intersection.

M = (A1\[Cross]C2)\[Cross](C1\[Cross]
              A2);
NN = (B1\[Cross]C2)\[Cross](C1\[Cross]B2);

Test whether L, M, NN are collinear.

(L\[Cross]M) . NN // Simplify

And the theorem is proved.