Is Stable class DL

lemma isStable_classDL [SecondCountableTopology ι] [PseudoMetrizableSpace ι] :    IsStable 𝓕 (ClassDL (E := E) · 𝓕 P) := by  refine fun X ⟨hX_prog, hUI_X⟩ τ hτ ↦ ⟨isStable_isStronglyProgressive X hX_prog τ hτ, fun t ↦ ?_⟩  let embed : {T | IsStoppingTime 𝓕 T ∧ ∀ ω, T ω ≤ t} →              {T | IsStoppingTime 𝓕 T ∧ ∀ ω, T ω ≠ ⊤} :=    fun σ ↦ ⟨σ.1, σ.2.1, fun ω ↦ ne_top_of_le_ne_top WithTop.coe_ne_top (σ.2.2 ω)⟩  refine uniformIntegrable_of_dominated (hUI_X t) (fun σ ↦ ?_) (fun σ ↦ ?_)  · exact (ProgMeasurable.stronglyMeasurable_stoppedValue_stoppedProcess hX_prog hτ      (embed σ)).aestronglyMeasurable  · rcases stoppedValue_stoppedProcess_dominated_le X hτ (embed σ) with ⟨rho, h_le_sigma, h_dom⟩    let rho_bounded : {T | IsStoppingTime 𝓕 T ∧ ∀ ω, T ω ≤ ↑t} :=      ⟨rho.1, rho.2.1, fun ω ↦ le_trans (h_le_sigma ω) (σ.2.2 ω)⟩    exact ⟨rho_bounded, h_dom⟩