Update analyticAt.lean

Nevanlinna/analyticAt.lean

@@ -165,26 +165,38 @@ theorem AnalyticAt.order_comp_CLE
     obtain ⟨g, h₁g, h₂g, h₃g⟩ := hn
     have A := eventually_nhds_comp_composition h₃g ℓ.continuous
 
+    have t₁ : AnalyticAt ℂ (fun z => ℓ z - ℓ z₀) z₀ := by
+      apply AnalyticAt.sub
+      exact ContinuousLinearEquiv.analyticAt ℓ z₀
+      exact analyticAt_const
+    have t₀ : AnalyticAt ℂ (fun z => (ℓ z - ℓ z₀) ^ n) z₀ := by
+      exact pow t₁ n
     have : AnalyticAt ℂ (fun z ↦ (ℓ z - ℓ z₀) ^ n • g (ℓ z) : ℂ → ℂ) z₀ := by
-      sorry
+      apply AnalyticAt.mul
+      exact t₀
+      apply AnalyticAt.comp h₁g
+      exact ContinuousLinearEquiv.analyticAt ℓ z₀
     rw [AnalyticAt.order_congr (hf.comp (ℓ.analyticAt z₀)) this A]
     simp
-    have t₀ : AnalyticAt ℂ (fun z => (ℓ z - ℓ z₀) ^ n) z₀ := by
-      sorry
 
     rw [AnalyticAt.order_mul t₀ ((h₁g.comp (ℓ.analyticAt z₀)))]
 
-    have t₁ : AnalyticAt ℂ (fun z => ℓ z - ℓ z₀) z₀ := by
-      sorry
     have : t₁.order = (1 : ℕ) := by
       rw [AnalyticAt.order_eq_nat_iff]
-      use (fun z ↦ ℓ 1)
+      use (fun _ ↦ ℓ 1)
       simp
       constructor
       · exact analyticAt_const
       · apply Filter.Eventually.of_forall
         intro x
-        sorry
+        calc ℓ x - ℓ z₀
+        _ = ℓ (x - z₀) := by
+          exact Eq.symm (ContinuousLinearEquiv.map_sub ℓ x z₀)
+        _ = ℓ ((x - z₀) * 1) := by
+          simp
+        _ = (x - z₀) * ℓ 1 := by
+          rw [← smul_eq_mul, ← smul_eq_mul]
+          exact ContinuousLinearEquiv.map_smul ℓ (x - z₀) 1
 
     have : t₀.order = n := by
       rw [AnalyticAt.order_pow t₁, this]