Oh no! It's a mare...

If, when you dug over the plot, you took out all the weed roots you could find I wouldn't bother trying to dig the mares tails roots out, apparently been on the earth longer than man and can go down about 30ft.
Just keep pulling them out as they appear and you will weaken the root but you'll never get rid completely. Took about 2 years on ours but the odd bit still comes up after 4 years.
Think the odd saying goes "never let them see Sunday"
I was told they were brought to this country from China and used to bind the soil along railway tracks.

I agree with your comments Terry, we are on clay soil and my neighbours have the dredded weed.  When it rears its uggly head in my soil from under the fence I pull it out or treat it with roundup.  If you persist with it this weakens it and it dies off.  My neighbour gave me a lovely ornamental fir tree so I have put it in a large tub and have been treating the mares tail each year (for 2 years) as it rears its ugly head.  The trick is to get it before it matures, I am quite hopefull that it should be clear this year and safe to plant into the garden without the risk of contaminating my garden.