Module InjectExtra.

Lemma inject_step U W this s1 s2 s1' s2' :
valid (U \+ W) ->
s1 \In Coh U -> s2 \In Coh U ->
network_step (U \+ W) this (s1 \+ s1') (s2 \+ s2') ->
network_step U this s1 s2 /\ s1' = s2' \/
network_step W this s1' s2' /\ s1 = s2.