have : ¬Satisfiable (u' : Theory (languageFinset u)) := by
  intro h
  have : Satisfiable (u : Theory L) := by
    rw[←image_u']
    simpa using (satisfiable_lMap L.ofSubLanguage (fun k ↦ Subtype.val_injective) (fun _ ↦ Subtype.val_injective) h)
  contradiction
case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝¹ : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence (languageFinset u)) := u.imageOfFinset fun x hσ => toSubLanguageFinsetSelf hσ
image_u' : Finset.image (⇑(lMap L.ofSubLanguage)) u' = u
⊢ T ⊢! σ
case intro
L : Language
T : Theory L
σ : Sentence L
h : T ⊨ σ
this✝² : (α : Type u) → DecidableEq α := Classical.typeDecidableEq
u : Finset (Sentence L)
ssu : ↑u ⊆ insert (~σ) T
hu : ¬Satisfiable ↑u
this✝¹ : (k : ℕ) → Encodable ((languageFinset u).Func k)
this✝ : (k : ℕ) → Encodable ((languageFinset u).Rel k)
u' : Finset (Sentence (languageFinset u)) := u.imageOfFinset fun x hσ => toSubLanguageFinsetSelf hσ
image_u' : Finset.image (⇑(lMap L.ofSubLanguage)) u' = u
this : ¬Satisfiable ↑u'
⊢ T ⊢! σ