The definition of the SELECT and PROJECT operators used below may be found in Chapter 6 of "Relational Database Design and Implementation", 4th Edition, Harrington, Jan L.

The SQL equivalent of the PROJECT (resp. RESTRICT) is SELECT (resp. WHERE with a predicate to reduce the number of elements in a relation). Let us use b_pred for the application of the predicate "pred" to a relation to reduce its elements. Then a(b_pred(relation))=b_pred(a(relation)) iff b_pred does not eliminate the column(s) supporting the predicate "pred". Furthermore if b_pred uses a column which is removed by a, the RHS expression is incorrect.

Is the result always correct when the RESTRICT operation is performed first?

What is a formal proof of that?

Why would we be interested in the opposite order of the operations?

Maybe performance is the only reason.