How can I prove that my database schema actually is in 3rd normal form?
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Your database is in 3NF if you meet the rules of 1NF, 2NF and 3NF.
As a rough guide.
Have no duplicate data in a single row. (columns where the same data is displayed in different way)
Eliminate data subsets that are recurring multiple times (I think of this as ‘reference’ data such as lookups. Things like CustomerRelationshipType)
Remove any data that is not directly related to the key of the table (This is a check to ensure that you’re not carrying anything unnecessarily or possibly storing something in the wrong table).