testing an extraneous attribute in a relation using functional dependency

I am studying the book "Database System Concepts" by sudarshan . In chapter 8.4.3

in a dependency alpha->beta , If A ∈ aplha to check if A is extraneous,
let (gamma = alpha − { A}), and check if (gamma → beta) can
be inferred from F . To do so, compute gamma+(the closure of gamma ) under F
if gamma+ includes all attributes in beta, then A is extraneous in alpha

Now my problem is , will I have to include alpha-> beta too while making the closure under F. It doesn't say to compute the closure under F-{aplha->beta} rather under F. It seems counter intuitive.

But during an extraneous attribute in A ∈ beta , we indeed calculate closure of alpha+ under (F-{alpha-beta} ∪ aplha->{beta-A})