Hello Tom,
Take a look here:-http://www.gunproof.com/Proofing/proofing.html
This is the Website of the Birmingham (England) Proof House where your gun was almost certainly proofed originally. Some of their 'Proof Marks' are illustrated.
The term 'Out of proof' can mean several things including wear which makes the tollerances allowed in a particular chamber or bore to be outside the permitted limits for that calibre or bore.
From what I can see from your pictures your shotgun is Nitro Proofed, probably in the 1930-40 period but the details don't show up all that well. You could send the gun to be re-proofed, but it would cost quite a bit when you include shipping and insurance charges from the USA and back.
On the other hand, if it failed re-proof you could console yourself by thinking of the heavy hospital bills you may have had to pay if it blew up in use, providing of course you survived.
Harry
Always trying to bring a little happiness to someone else. lol.
