Update laplace2.lean

Nevanlinna/laplace2.lean

@@ -12,6 +12,28 @@ variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDim
 variable {F : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
 
 
+lemma vectorPresentation'
+  [Fintype ι]
+  (b : OrthonormalBasis ι ℝ E)
+  --(hb : Orthonormal ℝ b)
+  (v : E) :
+  v = ∑ i, ⟪b i, v⟫_ℝ • (b i) := by
+
+  let A :=  b.sum_repr v
+  let i : ι := by sorry
+  let B := b.repr v i
+
+
+  nth_rw 1 [← (b.sum_repr v)]
+  apply Fintype.sum_congr
+  intro i
+  --let A := b.repr v
+  --have : (b.repr v) = ((OrthonormalBasis.toBasis b).repr v) := by tauto
+  rw [← Orthonormal.inner_right_finsupp hb (b.repr v) i]
+  simp
+
+
+
 lemma vectorPresentation
   [Fintype ι]
   (b : Basis ι ℝ E)
@@ -232,4 +254,3 @@ theorem Complex.Laplace'
   rw [LaplaceIndep''' Complex.orthonormalBasisOneI.toBasis Complex.orthonormalBasisOneI.orthonormal f]
   unfold Laplace_wrt_basis
   simp
-