Puzzle #196: Books

Georgina has 1000 books. They consist of 500 science fiction books, 400 horror, and 300 romances. There is obviously some overlap, and every book is in at least one of the three categories.

Twice as many books are horror-romances as are sf-romances. (Warning: An sf-horror-romance is in both groups.) She has four times as many romance-only books than she has sf-romance-only. She has the same number of sf-horror-only as she has sf-romance.

How many books of each combination of the book categories does she have?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, March 8, 2017 at noon Pacific Time. I will post the answer shortly after.