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That's an easy one if only 6'. I would attach the deck to the house and not free standing. Really, you think 6' is enough? That's just enough for two people to stand there and have a conversation. I will try and get you a pic of my almost finish deck my 13 and 15 year old built tomorrow. Its dark out now.
You doing this yourself?
Underpinning that corner isn't difficult, but it is hard work. Dig a hole depending on how far it dips, normally, we going about 3' from both directions on the corner and under house. excavate under the foundation and go down about 4'. Epoxy rebars to existing footing. Pour that puppy and you should be done. If you like, the building can be jacked up an inch with minimal interior damage. You are basically reinforcing that corner with a giant concrete pad. I have done many hill side home like it to keep it stable. Again, I have not seen the house but that's the general idea.
There's also Ram Jack. I am not sure about your building dept's approval with them. LA is a no go last time I check. If they did, I would have made millions with all the hill side home we have that are falling down the hill. Its much cheaper and so much less work. There's a guy out of the central cost that do that type of work. I wanted him to jack up an apt unit I was interested in buying at Taft but the deal fell through.
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