My usual gripe: strictly speaking, the answer is "it depends" and anyone who says otherwise is making assumptions that are unwarranted given how little we know about Mr A (Dr A?) and his situation. The type of industry matters, of course (some have a more established history of employing mathematicians than others), as well as the country that Mr A lives in. So as much as I can appreciate the encouraging words that we can give to Mr A, I want to insist that any answer must by definition have a localized domain of validity that may or may not overlap Mr A's situation.
So I don't know if the topic of Mr A's dissertation will matter, but it seems to me that it shouldn't matter to Mr A: according to the question, Mr A has essentially wrapped up the dissertation and so, to paraphrase Rumsfeld, will have to look for a job with the topic he has rather than the topic he wishes he had.
So if I can offer Mr A some advice, it would be rather to do his homework about the type of industry he wants to join. The fact that the dissertation topic was relevant to the industry may or may not matter, but what will matter for sure is your ability to make the case that you (not your dissertation) have something to offer to your prospective employer. I think scaaahu made a good point in pointing out that some employers/colleagues may have misconceptions about what mathematicians do. But more generally, you should research what the industry does and in what way you can be of help.
(For the record: I know a few mathematicians who have successful careers in various industries. As far as I can tell, all of them were very diligent in preparing for the industry in question, and they targeted one specific type of activity.)