rw [orthSpace_eq, orthSpace_eq, LinearEquiv.finrank_map_eq', LinearEquiv.dualAnnihilator_map_eq,
  LinearEquiv.finrank_map_eq', ← Subspace.finrank_dualCoannihilator_eq,
  Subspace.dualAnnihilator_dualCoannihilator_eq]
case refine_2
α : Type u_1
R : Type u_2
η✝ : Type u_3
ι : Type u_4
η : Type u_5
inst✝² : CommRing R
x : η → R
U✝¹ V✝ : Submodule R (η → R)
K : Type u_6
inst✝¹ : Field K
inst✝ : Fintype η
U✝ V U : Subspace K (η → K)
⊢ FiniteDimensional.finrank K ↥(orthSpace U).orthSpace = FiniteDimensional.finrank K ↥U
no goals