apply Finset.mem_inter_of_mem
φ : Formula
L R : Finset Formula
valid_proj : ~(□φ) ∈ L
ℓ : Char
ℓ_in_proj : ℓ ∈ voc (diamondProjectTNode (Sum.inl φ) (L, R)).1 ∩ voc (diamondProjectTNode (Sum.inl φ) (L, R)).2
⊢ ℓ ∈ voc (L, R).1 ∩ voc (L, R).2
case a
φ : Formula
L R : Finset Formula
valid_proj : ~(□φ) ∈ L
ℓ : Char
ℓ_in_proj : ℓ ∈ voc (diamondProjectTNode (Sum.inl φ) (L, R)).1 ∩ voc (diamondProjectTNode (Sum.inl φ) (L, R)).2
⊢ ℓ ∈ voc (L, R).1

case a
φ : Formula
L R : Finset Formula
valid_proj : ~(□φ) ∈ L
ℓ : Char
ℓ_in_proj : ℓ ∈ voc (diamondProjectTNode (Sum.inl φ) (L, R)).1 ∩ voc (diamondProjectTNode (Sum.inl φ) (L, R)).2
⊢ ℓ ∈ voc (L, R).2