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The current owner does not gain any extra rights in the property, after completion of the Sale, other than those rights granted to them under the AST. Your Solicitor will make sure that the Sale Contract will include a clause stipulating that the parties will enter into an AST upon completion of the Sale. Hence, the only rights the Sellers will have on completion will be a s per the AST agreement.
On a side point- I find it slightly strange that your Mortgage Lender has agreed to the above happening, on the basis that you are having a residential mortgage. However, your Solicitor will need to write to the Lender, if he has not done so already, to confirm the position and no doubt he will confirm to you once the Lender has acknowledged his letter and given you the green light.
I hope this assists and answers your question.
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