induction p using Formula.rec' <;> simp
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p : Formula α
hp : p ∈ maximalConsistentTheory consisT
⊢ { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p
case hfalsum
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
hp : ⊥ ∈ maximalConsistentTheory consisT
⊢ False

case hatom
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
a✝ : α
hp : Formula.atom a✝ ∈ maximalConsistentTheory consisT
⊢ Formula.atom a✝ ∈ maximalConsistentTheory consisT

case hnatom
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
a✝ : α
hp : Formula.natom a✝ ∈ maximalConsistentTheory consisT
⊢ Formula.atom a✝ ∉ maximalConsistentTheory consisT

case hand
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p✝ q✝ : Formula α
a✝¹ : p✝ ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p✝
a✝ : q✝ ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ q✝
hp : p✝ ⋏ q✝ ∈ maximalConsistentTheory consisT
⊢ { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p✝ ∧
    { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ q✝

case hor
α : Type u_1
Δ : Sequent α
T : Theory α
consisT : Consistent T
p✝ q✝ : Formula α
a✝¹ : p✝ ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p✝
a✝ : q✝ ∈ maximalConsistentTheory consisT → { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ q✝
hp : p✝ ⋎ q✝ ∈ maximalConsistentTheory consisT
⊢ { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ p✝ ∨
    { val := fun x => Formula.atom x ∈ maximalConsistentTheory consisT } ⊧ q✝