If I have $(P,Q,R,X,Y,Z,...)\barwedge(P,Q,R,X',Y',Z',...)$ and all points lie on line $l$,
How do I go about proving $(X,Y,Z,...)=(X',Y',Z',...)$?
By a theorem, there exists a unique projectivity taking P,Q,R to itself.
If I define it by $f$, such that, $f(a)=a$,
Can I say, since there is no other projectivity apart from $f$ that takes P,Q,R to itself, in any tuple taking these points to themselves, the only possible projectivity is f. hence $f(X,Y,Z) = X', Y', Z'$
I can understand that it has to be true, but what is the correct way to prove this?