rintro h x y hxy
M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
s : Submonoid M
t : Submonoid N
⊢ Subsemigroup.ProdProperty (prod s t).toSubsemigroup → Subsemigroup.ProdProperty t.toSubsemigroup
M : Type u_1
N : Type u_2
inst✝¹ : Monoid M
inst✝ : Monoid N
s : Submonoid M
t : Submonoid N
h : Subsemigroup.ProdProperty (prod s t).toSubsemigroup
x y : N
hxy : x * y ∈ t.toSubsemigroup
⊢ (∃ z ∈ t.toSubsemigroup, Associated x z) ∧ ∃ z ∈ t.toSubsemigroup, Associated y z