rintro x y hxy
M : Type ?u.8793
N : Type ?u.8796
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDomain R
p : R
hp : Prime p
⊢ ProdProperty (Submonoid.powers p).toSubsemigroup
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
⊢ (∃ z ∈ (Submonoid.powers p).toSubsemigroup, Associated x z) ∧
    ∃ z ∈ (Submonoid.powers p).toSubsemigroup, Associated y z