have ⟨Ω, hΩ⟩ := MaximalParametricConsistentTheory.lindenbaum (L := L) (T := ((□''⁻¹^[t.m]Ω₂.theory) ∪ (□''⁻¹^[t.n]Ω₃.theory))) $ by
  apply intro_union_ParametricConsistent;
  intro Γ Δ hΓ hΔ hC;
  replace hΓ : ∀ p ∈ Γ, □^[t.m]p ∈ Ω₂.theory := by simpa using hΓ;
  have hΓconj : □^[t.m](Γ.conj') ∈ Ω₂.theory := iff_mem_multibox_conj'.mpr hΓ;
  replace hΔ : ∀ p ∈ Δ, □^[t.n]p ∈ Ω₃.theory := by simpa using hΔ;
  have : □^[t.n](Δ.conj') ∈ Ω₃.theory := iff_mem_multibox_conj'.mpr hΔ;
  have : □^[t.j](◇^[t.n](Γ.conj')) ∈ Ω₁.theory := iff_mem_imp.mp
    (membership_iff.mpr $ Context.of! ⟨Deduction.maxm (by apply hAx; simp_all)⟩)
    (multiframe_def_multidia.mp r₁₂ hΓconj)
  have : ◇^[t.n]Γ.conj' ∈ Ω₃.theory := multiframe_def_multibox.mp r₁₃ this;
  have : L ⊢! □^[t.n](Δ.conj') ⋏ ◇^[t.n](Γ.conj') ⟶ ⊥ := by
    apply andImplyIffImplyImply'!.mpr;
    exact imp_trans!
      (show L ⊢! □^[t.n](Δ.conj') ⟶ □^[t.n](~Γ.conj') by exact imply_multibox_distribute'! $ contra₁'! $ andImplyIffImplyImply'!.mp hC)
      (show L ⊢! □^[t.n](~Γ.conj') ⟶ ~(◇^[t.n]Γ.conj') by exact contra₁'! $ conj₁'! $ multidiaDuality!);
  have : L ⊬! □^[t.n](Δ.conj') ⋏ ◇^[t.n](Γ.conj') ⟶ ⊥ := by simpa using Ω₃.consistent (Γ := [□^[t.n](Δ.conj'), ◇^[t.n](Γ.conj')]) (by simp_all)
  contradiction;
α : Type u_1
Λ : AxiomSet α
inst✝³ : Inhabited α
inst✝² : DecidableEq α
t : Axioms.Geach.Taple
L : DeductionParameter α
inst✝¹ : L.HasNecOnly
includeK : L.IncludeK
inst✝ : Inhabited (MCT L)
hAx : 𝗴𝗲(t) ⊆ Ax(L)
this : L.Normal
Ω₁ Ω₂ Ω₃ : (CanonicalFrame L).World
r₁₂ : Ω₁ ≺^[t.i] Ω₂
r₁₃ : Ω₁ ≺^[t.j] Ω₃
⊢ ∃ u, Ω₂ ≺^[t.m] u ∧ Ω₃ ≺^[t.n] u
α : Type u_1
Λ : AxiomSet α
inst✝³ : Inhabited α
inst✝² : DecidableEq α
t : Axioms.Geach.Taple
L : DeductionParameter α
inst✝¹ : L.HasNecOnly
includeK : L.IncludeK
inst✝ : Inhabited (MCT L)
hAx : 𝗴𝗲(t) ⊆ Ax(L)
this : L.Normal
Ω₁ Ω₂ Ω₃ : (CanonicalFrame L).World
r₁₂ : Ω₁ ≺^[t.i] Ω₂
r₁₃ : Ω₁ ≺^[t.j] Ω₃
Ω : MCT L
hΩ : □''⁻¹^[t.m]Ω₂.theory ∪ □''⁻¹^[t.n]Ω₃.theory ⊆ Ω.theory
⊢ ∃ u, Ω₂ ≺^[t.m] u ∧ Ω₃ ≺^[t.n] u