- [ exists [apply a1]
- split;
- assumption
- | cases H2 in j H H1; clear H2 a1; intros;
- cases H; clear H;
- cases (H4 i); clear H4; cases H; clear H;
- apply (H2 w); clear H2;
- assumption]
-qed.
\ No newline at end of file
+ [ exists [apply x] assumption
+ | cases H2 in j H H1; clear H2 a1; intros;
+ cases (H1 i); clear H1; apply (H3 a1); assumption]
+qed.
+
+definition leq ≝ λA:axiom_set.λa,b:A. a ◃ {y|b=y}.
+
+interpretation "covered by one" 'leq a b = (leq ? a b).
+
+theorem leq_refl: ∀A:axiom_set.∀a:A. a ≤ a.
+ intros;
+ apply refl;
+ normalize;
+ reflexivity.
+qed.
+
+theorem leq_trans: ∀A:axiom_set.∀a,b,c:A. a ≤ b → b ≤ c → a ≤ c.
+ intros;
+ unfold in H H1 ⊢ %;
+ apply (transitivity ???? H);
+ constructor 1;
+ intros;
+ normalize in H2;
+ rewrite < H2;
+ assumption.
+qed.
+
+definition uparrow ≝ λA:axiom_set.λa:A.mk_powerset ? (λb:A. a ≤ b).
+
+interpretation "uparrow" 'uparrow a = (uparrow ? a).
+
+definition downarrow ≝ λA:axiom_set.λU:Ω \sup A.mk_powerset ? (λa:A. (↑a) ≬ U).
+
+interpretation "downarrow" 'downarrow a = (downarrow ? a).
+
+definition fintersects ≝ λA:axiom_set.λU,V:Ω \sup A.↓U ∩ ↓V.
+
+interpretation "fintersects" 'fintersects U V = (fintersects ? U V).
+
+record convergent_generated_topology : Type ≝
+ { AA:> axiom_set;
+ convergence: ∀a:AA.∀U,V:Ω \sup AA. a ◃ U → a ◃ V → a ◃ (U ↓ V)
+ }.