Let $P(X_1,X_2)$ be a discrete bivariate distribution that has the form shown in the figure below, i.e. its support can be split into blocks that do not overlap on either dimensions.

Let's build $P'(B_1,B_2)$ obtained from $P(X_1,X_2)$ by integrating (summing) the values within each block. I would like to show that the following inequality holds $$ H(X_1) + H(X_2) - H(X_1,X_2) \ge H'(B_1) + H'(B_2) - H'(B_1,B_2) $$ where $H$ denotes entropy values computed with respect to $P(X_1,X_2)$ and $H'$ entropy values computed using $P'(X_1,X_2)$. Proving the inequality will allow me to prove this theorem.

**Question 1**: Any suggestion on how to prove this?

**Question 2**: How would you call a matrix like the one above? According to wikipedia the name "*block diagonal matrix*" applies only if the matrix and the blocks are squares.