rw [hx, hy]
case intro.intro.intro.intro.intro.intro.intro.intro
M : Type ?u.8793
N : Type ?u.8796
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
hp : Prime p
x y : R
hxy : x * y ∈ (Submonoid.powers p).toSubsemigroup
m : ℕ
hm : 1 * p ^ m = x * y
i j : ℕ
w✝¹ w✝ : R
left✝ : i + j = m
hbc : 1 = w✝¹ * w✝
hx : x = w✝¹ * p ^ i
hy : y = w✝ * p ^ j
⊢ (∃ z ∈ (Submonoid.powers p).toSubsemigroup, Associated x z) ∧
    ∃ z ∈ (Submonoid.powers p).toSubsemigroup, Associated y z
case intro.intro.intro.intro.intro.intro.intro.intro
M : Type ?u.8793
N : Type ?u.8796
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
hp : Prime p
x y : R
hxy : x * y ∈ (Submonoid.powers p).toSubsemigroup
m : ℕ
hm : 1 * p ^ m = x * y
i j : ℕ
w✝¹ w✝ : R
left✝ : i + j = m
hbc : 1 = w✝¹ * w✝
hx : x = w✝¹ * p ^ i
hy : y = w✝ * p ^ j
⊢ (∃ z ∈ (Submonoid.powers p).toSubsemigroup, Associated (w✝¹ * p ^ i) z) ∧
    ∃ z ∈ (Submonoid.powers p).toSubsemigroup, Associated (w✝ * p ^ j) z