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Commit 3a17b60d authored by François Clément's avatar François Clément
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Use new generic results from Set_system_any.v.

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...@@ -46,18 +46,16 @@ unfold PB_A; intros B HB; apply Open_Gen; easy. ...@@ -46,18 +46,16 @@ unfold PB_A; intros B HB; apply Open_Gen; easy.
intros B HB; apply Open_Gen; auto. intros B HB; apply Open_Gen; auto.
Qed. Qed.
Lemma Basisp_covers_all : unionp_any PB = fullset. Lemma Basisp_fullset : unionp_any PB = fullset.
Proof. Proof.
rewrite (proj2 HPB fullset). rewrite <- full_equiv; apply is_Basisp_full with T; try easy.
apply set_ext_equiv; split; intros x [B HB]; exists B; easy. apply is_Open_equiv; easy.
rewrite <- HT; apply Open_wFull.
Qed. Qed.
Lemma Basisp_Union_any_inter : Union_any_inter PB. Lemma Basisp_Union_any_inter : Union_any_inter PB.
Proof. Proof.
destruct HPB as [HPB1 HPB2]; intros B1 B2 HB1 HB2. apply is_Basisp_inter with T; try easy.
exists (fun B => PB B /\ incl B (inter B1 B2)); split; try easy. apply is_Open_equiv; easy.
apply HPB2; rewrite <- HT; apply Open_Inter; apply Open_Gen; auto.
Qed. Qed.
End Basis_Facts3. End Basis_Facts3.
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