Superposition calculus: Elimination of redundant atoms
This one seems to be a case of a not very careful choice of notation in this paper.

So we have a clause $C = \Gamma, u\approx v, u\approx v \to \Delta$ in $\mathcal{N}$.

We could use the simplification rule as follows. Let $\Gamma_1 = \Gamma, u\approx v$. Then the following two conditions hold:


$C = \Gamma_1, u\approx v \to \Delta$ in $\mathcal{N}$
$\Gamma_1 \to u\approx v, \Delta$ is a tautology therefore it can be added to $\mathcal{N}$ without loss of soundness.


Now applying the simplification rule we can replace $C$ with $\Gamma_1\to \Delta$. From the definition of $\Gamma_1$ we have replaced $C$ with the clause $\Gamma, u\approx v\to\Delta$. Effectively the second occurrence of the atom $u\approx v$ has been removed.

They also claim that the same procedure works for removing of multiple occurrences in the succedent.

Also note that this is not specific to equational theorem proving and applies to any atoms.