Bayes' Theorem

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thanks for the help in advance, really appreciate it.

A borrower has a credit rating f if lenders assign the borrower a probability f to being type G and 1 - f to being type B.
Denote the credit rating (conditional probability that a borrower is of type G) given an upgrade by fu, and rating given a downgrade by fd.
assume that fu > f > fd: the probability of being of type G is greater given an upgrade than given a downgrade.
All borrowers receive either an upgrade or a downgrade. For simplicity, assume that all type B borrowers receive a downgrade, implying that fu = 1 because only type G's ever receive an upgrade.
"All of my results hold when there is a positive probability of an upgrade of a type B, as long as the probability differs from that of a type G, but this extra parameter is not needed".
Let e denote the probability that a type G borrower receives an "erroneous" downgrade (1 - e is the probability of an upgrade for a type G).
Because all type B's are downgraded, Bayer's Law implies that e, the probability that a type G borrower receives a downgrade is e = [fd(l - f)]/f(1 - fd), where fd is the credit rating given a downgrade and f is the initial (date 0) credit rating of the borrower.

my question is how to demonstrate e = [fd(l - f)]/f(1 - fd)?

the text is from "Debt Maturity Structure and Liquidity Risk, Douglas W. Diamond 1991"