cases v
j : ℕ
z : String
v : Var
⊢ v.freeVarSet ⊆ (Var.open j (free_ z) v).freeVarSet
case free_
j : ℕ
z a✝ : String
⊢ (free_ a✝).freeVarSet ⊆ (Var.open j (free_ z) (free_ a✝)).freeVarSet

case bound_
j : ℕ
z : String
a✝ : ℕ
⊢ (bound_ a✝).freeVarSet ⊆ (Var.open j (free_ z) (bound_ a✝)).freeVarSet