# Wallis' axiom for parallel lines

I want to prove, using the typical tools from a Hilbert plane, that the Wallis' axiom implies ($P_{\leq 1}$), where

Wallis' axiom: Given a triangle $\Delta ABC$ and given a line segment $DE$, there exists a similar triangle $\Delta A'B'C'$, having side $A'B' \geq DE$.

$P_{\leq 1}$: For each line $l$ and for each point $P\notin l$, there is at most one line containing $P$ that is parallel to $l$

I have already proved Proclo's axiom is equivalent to $P_{\leq 1}$, but I got no idea how to solve this problem...

Any help would be appreciate.

• Hilbert space? Don't you mean Euclidean plane? – Han de Bruijn May 20 '18 at 19:53
• That's a mistake. It's Hilbert plane, sorry... – user326159 May 20 '18 at 20:04