Well, next is the process whereby you get a survey done to ensure that there are no problems with the house, and also tell your mortgage lender that you are going ahead, because they will get a valuation of the property done: don't be concerned by that, it makes sense - they need to make absolutely sure that if they are lending you money which is what a mortgage is, that the property is suitably valuable to be worth the money that they are lending you!
Now when you get the survey done, there will be lots of information that comes back. Some of this may be problems with the property that you didn't know about. If that happens, then don't panic - that is why it is absolutely essential to get a survey done to avoid nasty surprises. Typically what will happen next is that you ask the vendor to fix anything that needs to be sorted out and make that a condition of completion, or alternatively you renegotiate the sale price so that you can cover the costs.
There are all sorts of things that can show up on surveys, and it may be up to you to make a judgement call on some of them. For instance if there is a lean-to attached to the building made of wood then the survey may say that it could fall over one day, that it is structurally insecure and is also a fire risk: it is up to you what you do with that information. If you get told there is asbestos up in the attic, then the chances are you are going to want to make the removal of that a pre-condition of a sale and so forth. Clearly if there are signs of subsidence then that may well mean that you want to pull out of buying the property, as a minimum you'll want it thoroughly investigated and sorted beyond all reasonable doubt before you can think about continuing! The point is simply that you may need to exercise your judgement as to what you yourself consider to be a problem significant enough that you want to renegotiate on it, make it a condition of exchange or simply walk away from the purchase on the back off.
Once your offer is accepted, of course, you'll also need a solicitor to manage the legal side: this may be one that you find yourself or perhaps one recommended by the estate agent that you are buying the property through. There are several types of checks and searches that need to be done, and this is what the solicitor does, as well as working out things like the amount of stamp duty you have to pay: though of course as a first-time buyer, depending on the value of the property, you may not have to pay this. Remember the vendor also has an estate agent so the two will talk to each other during the process.
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