have that : ↥(stabilizer α t) = ↥t.mulStab := by
  rw [← SetLike.coe_sort_coe, ← coe_mulStab ht, Finset.coe_sort_coe]
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t✝ : Finset α
a : α
s t : Finset α
ht : t.Nonempty
this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s)
⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card
case inr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t✝ : Finset α
a : α
s t : Finset α
ht : t.Nonempty
this : ↑(QuotientGroup.mk ⁻¹' (QuotientGroup.mk '' ↑s)) ≃ ↥(stabilizer α t) × ↑(QuotientGroup.mk '' ↑s)
that : ↥(stabilizer α t) = { x // x ∈ t.mulStab }
⊢ t.mulStab.card * (image QuotientGroup.mk s).card = (s * t.mulStab).card