I think I get the logic of it but I don't have a proof.

A = G+H+I F = B+E+I J = B+C+G

That means they are each at least 6 and none of B, C, E, G, H, or I can be larger than 6, which means they make up 1-6 in some order.

Then, it must be impossible to arrange them such that any of A, F, or G are equal to seven without one of them being greater than 10.

That leaves 7 for D.