Union of Finitely Generated Congruences on Ground Term Algebra

We show that for any ground term equation systems $E$ and $F$, (1) the union of the generated congruences by $E$ and $F$ is a congruence on the ground term algebra if and only if there exists a ground term equation system $H$ such that the congruence generated by $H$ is equal to the union of the congruences generated by $E$ and $F$ if and only if the congruence generated by the union of $E$ and $F$ is equal to the union of the congruences generated by $E$ and $F$, and (2) it is decidable in square time whether the congruence generated by the union of $E$ and $F$ is equal to the union of the congruences generated by $E$ and $F$, where the size of the input is the number of occurrences of symbols in $E$ plus the number of occurrences of symbols in $F$.