. intros h y z x red_x_y red_x_z
  have h1 : refl_trans_clos R x y :=
    by
      apply (trans_clos_idempotent _ _ _).1; trivial
  have h2 : refl_trans_clos R x z :=
    by
      apply (trans_clos_idempotent _ _ _).1; trivial
  cases' (h _ _ _ h1 h2) with w h
  cases' h with h1 h2
  exists w; constructor <;> apply (trans_clos_idempotent _ _ _).2 <;> trivial
case mpr
A : Type
R : A → A → Prop
inhabited_A : Nonempty A
S : A → A → Prop
⊢ (∀ (y z x : A),
      refl_trans_clos R x y → refl_trans_clos R x z → ∃ z_1, refl_trans_clos R y z_1 ∧ refl_trans_clos R z z_1) →
    ∀ (y z x : A),
      refl_trans_clos (refl_trans_clos R) x y →
        refl_trans_clos (refl_trans_clos R) x z →
          ∃ z_1, refl_trans_clos (refl_trans_clos R) y z_1 ∧ refl_trans_clos (refl_trans_clos R) z z_1
no goals