In order to demonstrate the non-trivial Lookup, we need to remember q is the product of small primes of the order of root c.
The product of q and c give c'.
At [e',1] we know there are many values of n' for c' because we have forced c' to have many factors.
We also know that the product of a and some values of q will appear before N'c' in [e',1] with one of the n' values.
The non-trivial Lookup makes this process simple.