Dodgon's determinant-evaluation rule proved by TWO-TIMING MEN and WOMEN (Q1378547)

I give a bijective proof of the Reverend Charles Lutwidge Dodgson's rule: \[ \text{det}\Biggl[(a_{i,j})_{_{\substack{ 1\leq i\leq n\\ 1\leq j\leq n}}}\Biggr]\cdot \text{det}\Biggl[(a_{i,j})_{_{\substack{ 2\leq i\leq n-1\\ 2\leq j\leq n-1}}}\Biggr]= \] \[ \text{det}\Biggl[(a_{i,j})_{_{\substack{ 1\leq i\leq n-1\\ 1\leq j\leq n-1}}}\Biggr]\cdot\text{det} \Biggl[(a_{i,j})_{_{\substack{ 2\leq i\leq n\\ 2\leq j\leq n}}}\Biggr]- \text{det}\Biggl[(a_{i,j})_{_{\substack{ 1\leq i\leq n-1\\ 2\leq j\leq n}}}\Biggr]\cdot \text{det}\Biggr[(a_{i, j})_{_{\substack{ 2\leq i\leq n\\ 1\leq j\leq n-1}}}\Biggr] (\text{Alice}). \]