I have some question regarding Huzita–Hatori axioms.
Axiom 4 states that given a line and a point, we can make a fold passing through the point perpendicular to the line.
Axiom 5 states that given two points $p_1$ and $p_2$ and a line $l_1$, we can make a fold that places $p_1$ onto $l_1$ and passes through $p_2$ (as shown below).
Now I will show that axiom 4 can be constructed with axiom 5. Let $l$ be a line and $p$ be a point. Pick any point $q$ on the line $l$. Now by axiom 5, we can make a fold that places $q$ to $q'$ (where $q'$ lies on $l$) such that fold passes through $p$ (see below).
This will construct a perpendicular from $p$ to the line $l$ making axiom 4 redundant.
Is my argument correct or is there some flaw in it?