+(* Part of proposition 9.9 *)
+lemma f_minus_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻ p ≤ R⎻ q.
+ intros;
+ apply (. (or_prop2 : ?));
+ apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop2 : ?)^ -1); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_minus_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R⎻* p ≤ R⎻* q.
+ intros;
+ apply (. (or_prop2 : ?)^ -1);
+ apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop2 : ?)); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R p ≤ R q.
+ intros;
+ apply (. (or_prop1 : ?));
+ apply oa_leq_trans; [2: apply f; | skip | apply (. (or_prop1 : ?)^ -1); apply oa_leq_refl;]
+qed.
+
+(* Part of proposition 9.9 *)
+lemma f_star_image_monotone: ∀S,T.∀R:arrows2 OA S T.∀p,q. p ≤ q → R* p ≤ R* q.
+ intros;
+ apply (. (or_prop1 : ?)^ -1);
+ apply oa_leq_trans; [3: apply f; | skip | apply (. (or_prop1 : ?)); apply oa_leq_refl;]
+qed.
+
+lemma lemma_10_2_a: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R⎻* (R⎻ p).
+ intros;
+ apply (. (or_prop2 : ?)^-1);
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_b: ∀S,T.∀R:arrows2 OA S T.∀p. R⎻ (R⎻* p) ≤ p.
+ intros;
+ apply (. (or_prop2 : ?));
+ apply oa_leq_refl.
+qed.
+
+lemma lemma_10_2_c: ∀S,T.∀R:arrows2 OA S T.∀p. p ≤ R* (R p).
+ intros;
+ apply (. (or_prop1 : ?)^-1);
+ apply oa_leq_refl.