End Free_Variables.
Section Logic.

Lemma existSys :
forall (T : System) (f g : Formula) (v : nat),
~ In_freeVarSys LNN v T ->
~ In v (freeVarFormula LNN g) ->
SysPrf (Ensembles.Add _ T f) g -> SysPrf (Ensembles.Add _ T (existH v f)) g.