T (t0; i; ) = X X  T 2(B)T Pr ST = sT; b T =  T ; b T =  T X sT 2(S)T  T 2(B)T jt0 i = G; TG = t0; KG = i; bt0 i =   X T t=0 R(sT [t℄; A( T [t℄); ( T [t℄))  X X X  T 2(B)T Pr ST = sT; b T =  T ; b T =  T sT 2(S)T  T 2(B)T jt0 i = null; TG = t0; KG = i; bt0 i =   X T t=0 R(sT [t℄; A( T [t℄); ( T [t℄)) (17)

We an rewrite these summations more simply using our various shorthand notations:

= X X  T 2(B)T Pr((h0; T i ; s;  T )jt0 i = G) sT 2(S)T X T t=0 R(sT [t℄; A(( T [t℄; G )); G ( T [t℄))  X X  T 2(B)T Pr((h0; T i ; s;  T )jt0 i = null) sT 2(S)T X T t=0 R(sT [t℄; A(( T [t℄;  null));  null( T [t℄)) (18)

The remaining derivation exploits our Markovian assumptions to rearrange the summations and an el like terms to produ e the theorem's result. 2 Theorem 7 states, informally, that we prefer sending G whenever the the 
ost of exeution after a hieving G outweighs the ost of ommuni ation of the fa t that G has been a hieved. More pre isely, the outer summations on the left-hand side of the inequality iterate over all possible past histories of world and belief states, produ ing a probability distribution over the possible states the team an be in at time t0. For ea
h su
h state, the expression inside the parentheses omputes the dieren e in domain-level reward, over all possible future sequen es of world and belief states, between sending and not sending G.

By our theorem's assumption that no ommuni ation other than G is possible after G has been a hieved, we an ignore any ommuni ation osts in the future. However, if we relax this assumption, we an extend the left-hand side in a straightforward manner into a longer beliefs (both pre- and postommuni ation):
