The uniqueness of sudokus after removing clues

Question

I am creating a sudoku puzzle generator from a filled sudoku and have the following doubt. Suppose I remove one element(let it be a) from a partially filled sudoku (S) and I get multiple solutions, so I don’t remove it. Then I remove some more elements from sudoku while the sudoku solution remains unique. If again now removing one number (let it be b) creates multiple solutions so I again don’t remove it. Can I deduce that from this remaining sudoku (S’) that if I again try to remove that element a, that sudoku will give multiple solutions?

Answer 1

Yes. Any puzzle that was a solution after removing $a$ the first time will be a solution after removing $a$ at a later point. That's because the order in which you removed the numbers doesn't matter.

Answer 2

Yes it will.

Let $S$ be a sudoku and $S'$ be any sudoku such that $S'\subseteq S$ (ie $S'$ has less elements than $S$). All the solution of $S$ are also solution of $S'$.