OK, I admit I'm old and grumpy and all, but just what in the hell is that sentence supposed to mean? Be kind now. I'm really not a bad guy. Honest.
The proposer of a Sudoku puzzle is implicitly asserting that there is a way to fill in the blank boxes with certain properties (existence) and that there is only one way to do so (uniqueness). Solving the puzzle proves the implicit assertion by providing a construction for the unique completion.
In mathematics there are both non-constructive existence proofs (often using the Axiom of Choice), and constructive existence proofs (which actually construct an example of what is asserted to exist). Usually when a constructive existence proof is given of a thing that is unique, there is very little extra work needed to show its uniqueness.