One thing to notice is that support is not cut if the supporting unit is attacked from the province he's supporting into, and is not dislodged. This is pretty difficult to explain, lots of stuff in that sentence. But take the following situation:
There is a French army in Paris, and German armies in Gascony and Burgundy. If Burgundy attacks Paris and Gascony supports, and Paris has no support hold from anywhere else, the attack will go through and Paris is lost, even if Paris goes to Gascony. He can't cut support aimed at himself.
If there's an army in Brest, and Germany does the same attack, but France attacks Paris-Gascony supported by Brest, Paris will succeed and force Gascony to retreat, but since he left Paris, Burgundy can walk into Paris.
However! If there is another French army in Picardy who moves to Paris... I'll repeat the moves first.
Germany has armies in Burgundy and Gascony. Burgundy attacks Paris, Gascony supports.
France has armies in Paris and Brest. Paris attacks Gascony, Brest supports.
Picardy moves to Paris.
Now Gascony's support is cut, despite the fact that it was Paris cutting support into Paris. Dislodging trumps anything. Burgundy will bounce with Picardy and Paris will not be lost, because with Gascony retreating, Burgundy now has only 1 strength.
A lot to take in and hard to explain, but I hope that makes it clear. Is that a Swedish name, by the way?