subst def_l
case intro.intro.intro.intro.intro
x : ℕ
F : List Formula
δ : List Program
α β : Program
x_notin : x ∉ vocabOfProgram α ∧ x ∉ vocabOfProgram β
y : Program
y_in : y ∈ δ
l : List (List Formula × List Program)
Fδ_in_l : (F, δ) ∈ l
F' : List Formula
δ' : List Program
Fδ'_in : (F', δ') ∈ H α
def_l : (if δ' = [] then (List.map (fun x => [(F' ∪ x.1, x.2)]) (H β)).join else [(F', δ' ++ [β])]) = l
⊢ x ∉ vocabOfProgram y
case intro.intro.intro.intro.intro
x : ℕ
F : List Formula
δ : List Program
α β : Program
x_notin : x ∉ vocabOfProgram α ∧ x ∉ vocabOfProgram β
y : Program
y_in : y ∈ δ
F' : List Formula
δ' : List Program
Fδ'_in : (F', δ') ∈ H α
Fδ_in_l : (F, δ) ∈ if δ' = [] then (List.map (fun x => [(F' ∪ x.1, x.2)]) (H β)).join else [(F', δ' ++ [β])]
⊢ x ∉ vocabOfProgram y