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