Update analyticAt.lean

Nevanlinna/analyticAt.lean

@@ -135,9 +135,22 @@ theorem AnalyticAt.order_congr
   (hf : f₁ =ᶠ[nhds z₀] f₂) :
   hf₁.order = (hf₁.congr hf).order := by
 
-
-
-  sorry
+  by_cases h₁f₁ : hf₁.order = ⊤
+  rw [h₁f₁, eq_comm, AnalyticAt.order_eq_top_iff]
+  rw [AnalyticAt.order_eq_top_iff] at h₁f₁
+  exact Filter.EventuallyEq.rw h₁f₁ (fun x => Eq (f₂ x)) (id (Filter.EventuallyEq.symm hf))
+  --
+  let n := hf₁.order.toNat
+  have hn : hf₁.order = n := Eq.symm (ENat.coe_toNat h₁f₁)
+  rw [hn, eq_comm, AnalyticAt.order_eq_nat_iff]
+  rw [AnalyticAt.order_eq_nat_iff] at hn
+  obtain ⟨g, h₁g, h₂g, h₃g⟩ := hn
+  use g
+  constructor
+  · assumption
+  · constructor
+    · assumption
+    · exact Filter.EventuallyEq.rw h₃g (fun x => Eq (f₂ x)) (id (Filter.EventuallyEq.symm hf))
 
 
 theorem AnalyticAt.order_comp_CLE
@@ -154,7 +167,8 @@ theorem AnalyticAt.order_comp_CLE
     simp at A
     rw [AnalyticAt.order_congr (hf.comp (ℓ.analyticAt z₀)) A]
 
-    have : AnalyticAt ℂ (0 : ℂ → ℂ) z₀ := by apply analyticAt_const
+    have : AnalyticAt ℂ (0 : ℂ → ℂ) z₀ := by
+      apply analyticAt_const
     have : this.order = ⊤ := by
       rw [AnalyticAt.order_eq_top_iff]
       simp