have sup₀ : support [(a, x₀)] = [x₀] := by
  rfl
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
a : R
non_zero : a ≠ 0
x₀ x₁ : X
hyp : coords [(a, x₀)] = coords [(a, x₁)]
⊢ x₀ = x₁
R : Type u_1
inst✝² : Ring R
inst✝¹ : DecidableEq R
X : Type u_2
inst✝ : DecidableEq X
a : R
non_zero : a ≠ 0
x₀ x₁ : X
hyp : coords [(a, x₀)] = coords [(a, x₁)]
sup₀ : support [(a, x₀)] = [x₀]
⊢ x₀ = x₁