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