To establish the last claim, let ......
Their remarkable achievement seemed to validate John's claim. However, it soon turned out that ......
[see also: assert]
Lax claimed to obtain a formula for ......, but the argument turned out to be incomplete.
Note that for semilattices one could not claim that $A$ is algebraically closed in $B$.
We claim that, by setting $w$ to zero on this interval, the value of $F(w)$ is reduced.
So $F$ must be constant on $H$, thereby showing that $A=B$, as claimed $\langle$desired/required$\rangle$.
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