Hi Tom: I think your question is about Rule 37, (re-opening the betting) if not, please LMK.
UTG makes it $10 total. That was a raise over something I presume, so let's say the SB / BB are 2-4. So BB is 4 and UTG makes it a total of 10 (a raise of 6 over the BB).
The key is that when the bet comes back to UTG, in order for him to re-raise the total bet to him will have to be at least a full legal raise above whatever bet he last made. In this illustration a full legal raise over his last total bet would be 6 more for a total of 16.
In your illustration, UTG+2 raises it to 20 (a raise of 10 over the UTG total bet). Right there the betting is re-opened for the UTG b/c he will be facing a full legal raise. Doesn't really matter what the 3rd guy does.
Ditto if you treat the UTG total bet of 10 as an initial bet... Then a full legal raise is 10 and the UTG+2 raises it by 10 to total of 20, so again re-raising is opened for the UTG, doesn't matter what the all-in does.
In these illustrations, the following wouldn't re-open the betting and the UTG could only call:
UTG raises the 4 BB by 6 for total of 10... The UTG+2 calls and the 3rd guy goes all-in for a total of 15. 15 is an all-in wager that's not a full raise over the UTG. (we need 6, he only raises by 5).
Second case UTG opens for 10, UTG+2 goes all in for 18, the 3rd guy (not all in) calls the 18. The UTG needs a total bet of at least 20 (a full legal raise of 10 over his opener of 10) in order to re-raise. The total is only 18 so he can just call the 18.