For example: HSQLDB does not use indexes to improve sorting of query results.
But indexes have a crucial role in improving query speed.
If no index is used in a query on a single table, such as a DELETE query, then all the rows of the table must be examined.
With an index on one of the columns that is in the WHERE clause, it is often possible to start directly from the first candidate row and reduce the number of rows that are examined.
If the query also has a condition on t1, e.g., then an index on t1.c3 would eliminate the need for checking all the rows of t1 one by one, and will reduce query time to less than a millisecond per returned row.