I started reading Thomassen's paper A Theorem on Paths in Planar Graphs, where he proves one of Plummer's conjectures: Every $4$-connected planar graph is Hamiltonian-connected.
Context. Recall that a graph $G$ is called Hamiltonian-connected if for every two vertices $x$ and $y$ in $G$, we can find a Hamiltonian path starting from $x$ and ending at $y$. This is a rather strong condition, and in particular implies that $G$ must have a Hamiltonian circuit (indeed, we can take two adjacent vertices $x$ and $y$, find a Hamiltonian path from $y$ to $x$, and then add the edge $xy$ back to get a Hamiltonian circuit). In particular, Plummer's conjecture already implies Tutte's celebrated theorem that every $4$-connected planar graph is Hamiltonian (i.e. contains a Hamiltonian circuit).
For my question below to make sense, it also helps to be familiar with the definition of $H$-component where $H$ is a subgraph of $G$. Here is the definition, taken from Thomassen's paper:
Finally, an outer cycle just means the cycle bounding the outside (infinite) face of a planar graph in its plane drawing. With all the definitions out of the way, Thomassen's main theorem is the following:
Thomassen claims that the theorem above immediately implies
Plummer's conjecture: Every $4$-connected planar graph is Hamiltonian-connected.
My question is: Can somebody explain how to see this implication? I presume that we need to remove two points $x$ and $y$, which would result in a $2$-connected graph. After that, do we apply the Theorem above? But it is not clear to me what the vertex $u$ or the edge $e$ should be. I would very much appreciate if someone could elaborate and explain the details. Thanks!