∀a:sequence C.∀l,u:C.∀H:∀i:nat.a i ∈ [l,u].
∀x:C.∀p:a order_converges x.
∀j.l ≤ (fst p) j.
-intros; cases p; clear p; simplify; cases H1; clear H1; cases H2; clear H2;
-cases (H3 j); clear H3; cases H2; cases H7; clear H2 H7;
-intro H2; cases (H8 ? H2);
-cases (H (w1+j)); apply (H12 H7);
+intros; cases p (xi yi Ux Dy Hxy); clear p; simplify;
+cases Ux (Ixi Sxi); clear Ux; cases Dy (Dyi Iyi); clear Dy;
+cases (Hxy j) (Ia Sa); clear Hxy; cases Ia (Da SSa); cases Sa (Inca SIa); clear Ia Sa;
+intro H2; cases (SSa ? H2) (w Hw); simplify in Hw;
+cases (H (w+j)) (Hal Hau); apply (Hau Hw);
qed.
lemma order_converges_smaller_upsegment: