1
To establish the last claim, let......
Their remarkable achievement seemed to validate John's claim. However, it soon turned out that......
2
[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.
So F must be constant on H, thereby showing that A=B, as claimed <desired/required>.