Proof!

Let's call the grandparents of the Match and Bio-parent M, X1, X2, and N, with M & X1 being on the paternal side, and X2 & and N being on the maternal side. Further, that X1 and X2 are full sibling. Really, X1 represents the DNA received by the father from the common grandparents, while X2 represents the DNA received from them by the mother. With the father and mother being 1st cousins, the chance that X1 matches X2 is 1 in 4.

At any given location on a pair of chromosomes, Adoptee gets DNA from just one of these four (M, X1, X2, or N). At that same location, Match receives DNA from both parents, with four equally likely possibilities: (X1,X2), (X1, N), (M, X2), and (M, N).

So between Adoptee and Match, there are 16 equally likely possibilities at each location. What happens for twelve of these 16 possibilities is straightforward:

* For the 4 cases where Adoptee has DNA from M, two cases - where Match has (M, X2) or (M, N) - result in a match, but the other two cases result in no match.

* The 4 cases where Adoptee has DNA from N works the same way, with the same result, except with (X1, N) and (M, N) providing the two matches.

* The 2 cases where Adoptee has DNA from X1 or X2, and Match has (M, N) will both result in no match.

* The 2 cases where Adoptee has DNA from X1 or X2, and Match has (X1, X2) will both give a match.

In the 12 cases considered so far (out of the 16 possible), 6 result in matches; the other 6 do not. The remaining 4 cases are where Adoptee has X1 or X2, and Match has X1 or X2, but not both. Two of these cases are also straightforward:

* Adoptee with X1 and Match with (X1, N) gives a match.

* Adoptee with X2 and Match with (M, X2) gives a match.

We're up to 14 case considered (out of the 16), with 8 resulting in matches, while 6 don't. The remaining cases are:

* Adoptee with X1 and Match with (M, X2)

* Adoptee with X2 and Match with (X1, N)

Since X1 matches X2 1/4 of the time, both of these case count as a 1/4 match.

Then the total result is (8+0.25+0.25)/(14+1+1) = 8.5/16 = 17/32. Adoptee will match Match at 17/32 of the locations on their chromosome pairs. A 100% match gives about 3460cM, so a 17/32 match should be about 1838cM.