Well, here's how the removal was accomplished.
Since the drain in the shower is just like the one in most kitchen sinks, the top flange is bedded in a strip of caulk.
Then, under the shower pan, there is a rubber gasket and a large metal flange which is pressed tightly against the bottom of the pan by a large PVC nut, which is threaded on the outside of the drain tube.
So, in order to remove the drain assembly (of which the corroded top piece is a part) you must use a "basket wrench" to undo that PVC nut.
Only trouble is that after being in there for 16 years, the nut, the rubber gasket and the metal flange were quite well bonded to one another.
The greatest amount of effort I could put into the basket wrench just caused it to slip off the (VERY thin) nut.
In the end, I used my Dremel tool with a heavy duty cut off wheel and split the PVC nut.
The rest came out quite easily.
Next time, I'm going to START with the Dremel!
Bob