have := BProof.mp (BProof.adj prf₅ this) BTheorem.distRight
case cons.nil
t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
w₂ : Form
l₅ : w₁¦w₂ ∈ Δ
prf₅ :
  BProof (lindenbaumSequence t Δ (i, j + 0))
    ((Denumerable.ofNat (Form × Form) (j + 0)).fst¦(Denumerable.ofNat (Form × Form) (j + 0)).snd)
head : Form
tail : List Form
l₆ : { x | x ∈ head :: tail } ⊆ lindenbaumSequence t Δ (i, j + 0)
l₇ : { x | x ∈ [] } ⊆ lindenbaumSequence t Δ (i, j + 0)
thm₃ :
  BTheorem
    (Form.conjoinList (Denumerable.ofNat (Form × Form) (j + 0)).fst
          []¦Form.conjoinList (Denumerable.ofNat (Form × Form) (j + 0)).snd (head :: tail)⊃w₁¦w₂)
this : BProof (lindenbaumSequence t Δ (i, j + 0)) (Form.conjoinList head tail)
⊢ False
case cons.nil
t : Th
Δ : Ctx
h₁ : ↑t ∩ Δ = ∅
h₂ : isDisjunctionClosed Δ
i j : ℕ
w₁ : Form
l₃ : ▲lindenbaumSequence t Δ (i, j) ∩ Δ = ∅
w₂ : Form
l₅ : w₁¦w₂ ∈ Δ
prf₅ :
  BProof (lindenbaumSequence t Δ (i, j + 0))
    ((Denumerable.ofNat (Form × Form) (j + 0)).fst¦(Denumerable.ofNat (Form × Form) (j + 0)).snd)
head : Form
tail : List Form
l₆ : { x | x ∈ head :: tail } ⊆ lindenbaumSequence t Δ (i, j + 0)
l₇ : { x | x ∈ [] } ⊆ lindenbaumSequence t Δ (i, j + 0)
thm₃ :
  BTheorem
    (Form.conjoinList (Denumerable.ofNat (Form × Form) (j + 0)).fst
          []¦Form.conjoinList (Denumerable.ofNat (Form × Form) (j + 0)).snd (head :: tail)⊃w₁¦w₂)
this✝ : BProof (lindenbaumSequence t Δ (i, j + 0)) (Form.conjoinList head tail)
this :
  BProof (lindenbaumSequence t Δ (i, j + 0))
    (((Denumerable.ofNat (Form × Form)
            (j +
              0)).fst&Form.conjoinList head
          tail)¦(Denumerable.ofNat (Form × Form) (j + 0)).snd&Form.conjoinList head tail)
⊢ False