7 Fixed typo
Lemma 1 and $r$ is such that $K \subset L_r$, where $K$ is as in Lemma 1, then for
6 Added more reasons to despair.

Things are seeming rather hopeless at this point. In fact, things are worsethan they seem! But, before we can revel in this despair, we must introduce two definitions:

Definition (Simply Connected at Infinity):Let $Z$ be a topological manifold. $Z$ is simply connected at infinity if forany compact subset $C$ of $Z$ there exists a compact subset $C'$ of $Z$ that contains $C$ and is such that the inclusion $Z - C' \rightarrow Z - C$ inducesthe trivial map $\pi_1(Z - C') \rightarrow \pi_1(Z - C)$.

Definition (End Sum):Let $Z_1$ and $Z_2$ be non-compact oriented smooth four-manifolds that aresimply connected at infinity. Choose two proper smooth embeddings $\gamma_i : [0, \infty) \rightarrow Z_i$. Remove a tubular neighborhood of $\gamma_i((0, \infty))$ fromeach $Z_i$ and glue the resulting $\mathbb{R}^3$ boundaries together respectingorientations. The result is the end sum $Z_1 \natural Z_2$ of $Z_1$ and $Z_2$.

Remark:The requirement that $Z_i$ is simply connected at infinity guaranteesthat $\gamma_i$ is unique up to ambient isotopy and thus $Z_1 \natural Z_2$is unique up to diffeomorphism (Gompf and Stipsicz Definition 9.4.6).

Remark:If $R_1$ and $R_2$ are exotic $\mathbb{R}^4$, then they are non-compact oriented smooth four-manifolds that are simply connected at infinity and $R_1 \natural R_2$ is a smooth manifold homeomorphic to $\mathbb{R}^4$.

Remark:$X$ of Lemma 1 is simply connected at infinity.

Theorem 2:If $\{L_t | 0 < t \le \infty \}$ is a radial family for an $L$ as appears in Lemma 1 and $r$ is such that $K \subset L_r$, where $K$ as in Lemma 1, then for$R$ an exotic $\mathbb{R}^4$ and $t$ such that $r \le t \le \infty$ there existsno flip of $R \natural L_t$.

Proof:The proof is basically a slight variation on the above theme. Assume one could flipthe end of $R \natural L_t$ for $r \le t \le \infty$. Thus, one could use this flipto glue $R \natural L_t$ to the "end" of $X$ less the image of $L - L_t$ end summedwith $R$, in other words with the flip glue $R \natural L_t$ to $R \natural (X - (L - L_t))$, and obtain a simply connected closed smooth four-manifold with negative definite intersection form not isomorphic to $n\langle-1\rangle$. Again, according to Donaldson's Theorem (Gompf and Stipsicz Theorem 1.2.30) there exists no such manifold. Thus, thereexists no such flip.QED

Now we can revel in this despair!

5 Sorry for the noise; fixed incorrect theorem number.
4 A final oblation of back-ticks to the MathJax gods that rule the realm of OS X-Chrome and iPad-Safari seem to have won rendering favor
3 Removed all \langle and \rangle uses as they render incorrectly on the iPad
2 Removed unneeded int()
1