equal
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replaced
466 have "finite {i \<in> I. inv_into (carrier (G i)) (f i) (x i) \<noteq> \<one>\<^bsub>G i\<^esub>}" |
466 have "finite {i \<in> I. inv_into (carrier (G i)) (f i) (x i) \<noteq> \<one>\<^bsub>G i\<^esub>}" |
467 apply (rule finite_subset [OF _ fin]) |
467 apply (rule finite_subset [OF _ fin]) |
468 using G H group.subgroup_self hom_one homf injf inv_into_f_eq subgroup.one_closed by fastforce |
468 using G H group.subgroup_self hom_one homf injf inv_into_f_eq subgroup.one_closed by fastforce |
469 with x show "?g \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" |
469 with x show "?g \<in> {x \<in> \<Pi>\<^sub>E i\<in>I. carrier (G i). finite {i \<in> I. x i \<noteq> \<one>\<^bsub>G i\<^esub>}}" |
470 apply (auto simp: PiE_iff inv_into_into conj_commute cong: conj_cong) |
470 apply (auto simp: PiE_iff inv_into_into conj_commute cong: conj_cong) |
471 by (metis (no_types, hide_lams) iso_iff f inv_into_into) |
471 by (metis (no_types, opaque_lifting) iso_iff f inv_into_into) |
472 qed |
472 qed |
473 qed |
473 qed |
474 qed |
474 qed |
475 qed |
475 qed |
476 then show ?thesis |
476 then show ?thesis |