\subset A -> C \subset B -> C \subset A \cap B. Proof. move=> A B C HC_subset_A HC_subset_B x. split. - by apply HC_subset_A. - by apply HC_subset_B. Qed. (* (2.4)' *) Theorem cap_diag: forall A, A \cap A = A. Proof. move=> A. apply eq_subset' => x; rewrite -cap_and. - by case. - by split => //. Qed. 11