Thompson's is basically wax. It's a parafin disolved into mineral spirits, when the spirits evaporate, it leaves the wax. As Steve said, it doesn't have much durability, in most cases, it has to be re-applied once or twice each year to get any consistant protection out of it.
Does your patio get any sun? Or does the deck keep it completly shaded?
I'd try Steves suggestion with the vinegar first, following with a strong pressure washer (3500 psi or better). If you've never used a machine like this, repost and we'll give you some tips.
If that doesn't work, I'd try a liquid form of a Sodium Hydroxide deck stripper. Follow it with a rinsing with the same power washer, and then spray white vinegar to neuturalize the alkaline of the stripper, and rinse again.
When this is finished, there will be no sealer left and something will have to be put on it, or your patio will be land fill in several years. Any type of sealer like Thompson's will be slippery when first applied, but that will go away with some rains and a few months wear. Keep in mind that Thompson's and the like do not hold up. they will need to be re-applied once each year in the shade, twice in the sun.
There used to be a liquified sillicon sealant made by a company called "LifeGuard". This stuff was slippery as heck for several months after it was first applied, but it's lifespan was up to 15 years. I used it quite a bit when I lived in MI and always had good results. Somewhat tricky to use in that you have to be carefull not to get it on vinyl siding or any surface that you do not want to be somewhat shiney for a while. Another thing to be aware of is that when a sillicon sealant is applied, nothing else can be applied over it until the sillicon is worn away completly.