have : s ≤ stabilizer α t := by aesop
case inr.a.mpr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t✝ : Finset α
a : α
s : Subgroup α
t : Finset α
hst : ↑s ⊆ ↑t.mulStab
ht : t.Nonempty
hti : (image QuotientGroup.mk t).Nonempty
x : α ⧸ s
hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t
⊢ ∃ a ∈ t.mulStab, ↑a = x
case inr.a.mpr
ι : Type u_1
α : Type u_2
inst✝¹ : CommGroup α
inst✝ : DecidableEq α
s✝ t✝ : Finset α
a : α
s : Subgroup α
t : Finset α
hst : ↑s ⊆ ↑t.mulStab
ht : t.Nonempty
hti : (image QuotientGroup.mk t).Nonempty
x : α ⧸ s
hx : x • image QuotientGroup.mk t = image QuotientGroup.mk t
this : s ≤ stabilizer α t
⊢ ∃ a ∈ t.mulStab, ↑a = x