TPTP Problem File: SLH0018^1.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain : Archive of Formal Proofs
% Problem :
% Version : Especial.
% English :
% Refs : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source : [Des23]
% Names : Finite_Fields/0025_Finite_Fields_Isomorphic/prob_00084_003162__18456474_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1623 ( 371 unt; 344 typ; 0 def)
% Number of atoms : 4121 (1144 equ; 0 cnn)
% Maximal formula atoms : 11 ( 3 avg)
% Number of connectives : 27555 ( 299 ~; 38 |; 145 &;24937 @)
% ( 0 <=>;2136 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 8 avg)
% Number of types : 40 ( 39 usr)
% Number of type conns : 669 ( 669 >; 0 *; 0 +; 0 <<)
% Number of symbols : 306 ( 305 usr; 17 con; 0-4 aty)
% Number of variables : 2707 ( 52 ^;2584 !; 71 ?;2707 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:24:02.803
%------------------------------------------------------------------------------
% Could-be-implicit typings (39)
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia7052216462576429980t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia7913147511651046390t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia7144473111789202860t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia3352279114339913222t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia6652711048488803900t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia3601206958761670294t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2956882679547061052t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J_J,type,
partia7496981018696276118t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_It__Int__Oint_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J_J,type,
partia4934656038542163276t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__List__Olist_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
partia2670972154091845814t_unit: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_It__Set__Oset_Itf__a_J_Mt__Group__Omonoid__Omonoid____ext_It__Set__Oset_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J_J_J,type,
partia6043505979758434576t_unit: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
produc2061375659605276261et_int: $tType ).
thf(ty_n_t__Congruence__Opartial____object__Opartial____object____ext_Itf__a_Mt__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J_J,type,
partia2175431115845679010xt_a_b: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_I_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J_J,type,
set_se6837996996060283045et_int: $tType ).
thf(ty_n_t__Set__Oset_I_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
set_li8746675981475697647et_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
produc6642921209907155909et_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
set_li7397999359881343176et_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
set_se8558717844084009108et_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
list_l8867605966578153106et_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
set_li2838760105183549752et_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
set_set_list_set_int: $tType ).
thf(ty_n_t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
produc9164743771328383783list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
set_set_set_set_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
list_list_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
set_list_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
set_set_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__List__Olist_Itf__a_J_J_J,type,
set_list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
set_set_set_a: $tType ).
thf(ty_n_t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
list_set_int: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
set_set_int: $tType ).
thf(ty_n_t__List__Olist_It__List__Olist_Itf__a_J_J,type,
list_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_Itf__a_J_J,type,
set_list_a: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
set_set_a: $tType ).
thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
set_int: $tType ).
thf(ty_n_t__List__Olist_Itf__a_J,type,
list_a: $tType ).
thf(ty_n_t__Set__Oset_Itf__a_J,type,
set_a: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_t__Int__Oint,type,
int: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (305)
thf(sy_c_AbelCoset_OA__RCOSETS_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_RCOS5818729163757401641t_unit: partia6652711048488803900t_unit > set_list_set_int > set_set_list_set_int ).
thf(sy_c_AbelCoset_OA__RCOSETS_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_RCOS5559887075240879033t_unit: partia4934656038542163276t_unit > set_set_int > set_set_set_int ).
thf(sy_c_AbelCoset_OA__SET__INV_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_SET_4289452989722768907t_unit: partia6652711048488803900t_unit > set_list_set_int > set_list_set_int ).
thf(sy_c_AbelCoset_Oa__l__coset_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_l_co143929110759192528t_unit: partia6652711048488803900t_unit > list_set_int > set_list_set_int > set_list_set_int ).
thf(sy_c_AbelCoset_Oa__l__coset_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_l_co3504123944629134560t_unit: partia4934656038542163276t_unit > set_int > set_set_int > set_set_int ).
thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
a_r_co2531912773843275078t_unit: partia7144473111789202860t_unit > set_li2838760105183549752et_int > list_list_set_int > set_li2838760105183549752et_int ).
thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_r_co912825291596486358t_unit: partia6652711048488803900t_unit > set_list_set_int > list_set_int > set_list_set_int ).
thf(sy_c_AbelCoset_Oa__r__coset_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_r_co1577572119920843660t_unit: partia2670972154091845814t_unit > set_list_a > list_a > set_list_a ).
thf(sy_c_AbelCoset_Oa__r__coset_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_r_co692709266861932262t_unit: partia4934656038542163276t_unit > set_set_int > set_int > set_set_int ).
thf(sy_c_AbelCoset_Oa__r__coset_001tf__a_001tf__b,type,
a_r_coset_a_b: partia2175431115845679010xt_a_b > set_a > a > set_a ).
thf(sy_c_AbelCoset_Oabelian__subgroup_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
abelia4826137221796835534t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_AbelCoset_Oadditive__subgroup_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
additi4441901441605882380t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_AbelCoset_Oadditive__subgroup_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
additi7073586575563672860t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_AbelCoset_Oset__add_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
set_ad2159827153730365080t_unit: partia6652711048488803900t_unit > set_list_set_int > set_list_set_int > set_list_set_int ).
thf(sy_c_AbelCoset_Oset__add_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
set_ad273131178244904872t_unit: partia4934656038542163276t_unit > set_set_int > set_set_int > set_set_int ).
thf(sy_c_AbelCoset_Oset__add_001tf__a_001tf__b,type,
set_add_a_b: partia2175431115845679010xt_a_b > set_a > set_a > set_a ).
thf(sy_c_Card__Irreducible__Polynomials__Aux_Ogauss__poly_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
card_I2978652876270886323t_unit: partia4934656038542163276t_unit > nat > list_set_int ).
thf(sy_c_Card__Irreducible__Polynomials__Aux_Ogauss__poly_001tf__a_001tf__b,type,
card_I2373409586816755191ly_a_b: partia2175431115845679010xt_a_b > nat > list_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
comple4195941263769839116et_int: set_li8746675981475697647et_int > list_set_int > set_list_set_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_I_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
comple3138861590707219906et_int: set_se6837996996060283045et_int > set_li8746675981475697647et_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Int__Oint_J,type,
comple3221217463730067765et_int: set_set_int > set_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
comple3081917594375107323et_int: set_set_list_set_int > set_list_set_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
comple7281953568134767595et_int: set_set_set_int > set_set_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
comple148233303328197937et_int: set_se8558717844084009108et_int > set_set_list_set_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
comple1756060948637632417et_int: set_set_set_set_int > set_set_set_int ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
comple3958522678809307947_set_a: set_set_set_a > set_set_a ).
thf(sy_c_Complete__Lattices_OSup__class_OSup_001t__Set__Oset_Itf__a_J,type,
comple2307003609928055243_set_a: set_set_a > set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_Mt__Product____Type__Ounit_J_J,type,
partia4593313445827799451t_unit: partia7052216462576429980t_unit > set_li7397999359881343176et_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J_J,type,
partia3237421854401958059t_unit: partia7144473111789202860t_unit > set_li2838760105183549752et_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__List__Olist_Itf__a_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_Itf__a_J_J_Mt__Product____Type__Ounit_J_J,type,
partia2464479390973590831t_unit: partia2956882679547061052t_unit > set_list_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J_J,type,
partia5243588073521899067t_unit: partia6652711048488803900t_unit > set_list_set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__List__Olist_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__List__Olist_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia5361259788508890537t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__Int__Oint_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Int__Oint_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J_J,type,
partia966996272515721803t_unit: partia4934656038542163276t_unit > set_set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J_J,type,
partia7954853462395303941t_unit: partia3352279114339913222t_unit > set_set_list_set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J_J,type,
partia1981568256555877781t_unit: partia3601206958761670294t_unit > set_set_set_int ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001t__Set__Oset_Itf__a_J_001t__Group__Omonoid__Omonoid____ext_It__Set__Oset_Itf__a_J_Mt__Ring__Oring__Oring____ext_It__Set__Oset_Itf__a_J_Mt__Product____Type__Ounit_J_J,type,
partia5907974310037520643t_unit: partia6043505979758434576t_unit > set_set_a ).
thf(sy_c_Congruence_Opartial__object_Ocarrier_001tf__a_001t__Group__Omonoid__Omonoid____ext_Itf__a_Mt__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J_J,type,
partia707051561876973205xt_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Divisibility_Ofactor_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J,type,
factor668791974002123612t_unit: partia7144473111789202860t_unit > list_list_set_int > list_list_set_int > $o ).
thf(sy_c_Divisibility_Ofactor_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
factor7280686822248936060t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int > $o ).
thf(sy_c_Divisibility_Ofactor_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
factor1757716651909850160t_unit: partia2670972154091845814t_unit > list_a > list_a > $o ).
thf(sy_c_Divisibility_Ofactor_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
factor5186451337065598620t_unit: partia4934656038542163276t_unit > set_int > set_int > $o ).
thf(sy_c_Divisibility_Ofactor_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
factor8216151070175719842xt_a_b: partia2175431115845679010xt_a_b > a > a > $o ).
thf(sy_c_Divisibility_Oisgcd_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
isgcd_1196993390438183731t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int > list_set_int > $o ).
thf(sy_c_Divisibility_Oisgcd_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
isgcd_4636411027072998995t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int > $o ).
thf(sy_c_Divisibility_Oisgcd_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
isgcd_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
monoid1136863209656779155t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Divisibility_Omonoid__cancel_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
monoid497721730651901107t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Divisibility_Oprime_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
prime_7199642643890090290t_unit: partia6652711048488803900t_unit > list_set_int > $o ).
thf(sy_c_Divisibility_Oprime_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
prime_2746729220581880658t_unit: partia4934656038542163276t_unit > set_int > $o ).
thf(sy_c_Divisibility_Oprime_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
prime_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Divisibility_Osomegcd_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
somegc4977764851446611037t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int > list_set_int ).
thf(sy_c_Divisibility_Osomegcd_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
somegc3856857245158089085t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Divisibility_Osomegcd_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
somegc1600592057159103747xt_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Embedded__Algebras_Oring_Ofinite__dimension_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
embedd4385354970001408291t_unit: partia6652711048488803900t_unit > set_list_set_int > set_list_set_int > $o ).
thf(sy_c_Embedded__Algebras_Oring_Oline__extension_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
embedd942231993492570031t_unit: partia6652711048488803900t_unit > set_list_set_int > list_set_int > set_list_set_int > set_list_set_int ).
thf(sy_c_Embedded__Algebras_Osubalgebra_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
embedd3025314616789055568t_unit: set_list_set_int > set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
finite7825103155802847241et_int: set_list_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Int__Oint_J,type,
finite6197958912794628473et_int: set_set_int > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Formal__Polynomial__Derivatives_Opderiv_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
formal2278089127137682990t_unit: partia6652711048488803900t_unit > list_list_set_int > list_list_set_int ).
thf(sy_c_Group_OUnits_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J,type,
units_2543522835650936210t_unit: partia7144473111789202860t_unit > set_li2838760105183549752et_int ).
thf(sy_c_Group_OUnits_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
units_4332284670423708338t_unit: partia6652711048488803900t_unit > set_list_set_int ).
thf(sy_c_Group_OUnits_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
units_2932844235741507942t_unit: partia2670972154091845814t_unit > set_list_a ).
thf(sy_c_Group_OUnits_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
units_4038138251425117394t_unit: partia4934656038542163276t_unit > set_set_int ).
thf(sy_c_Group_OUnits_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
units_a_ring_ext_a_b: partia2175431115845679010xt_a_b > set_a ).
thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J,type,
mult_l2322401601159563005t_unit: partia7144473111789202860t_unit > list_list_set_int > list_list_set_int > list_list_set_int ).
thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
mult_l3261022468980662045t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int > list_set_int ).
thf(sy_c_Group_Omonoid_Omult_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
mult_l7073676228092353617t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Group_Omonoid_Omult_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
mult_s3864001451298473021t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Group_Omonoid_Omult_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
mult_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Group_Omonoid_Oone_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
one_li3514222801933830793t_unit: partia6652711048488803900t_unit > list_set_int ).
thf(sy_c_Group_Omonoid_Oone_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
one_a_ring_ext_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_I_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
minus_6937832704872953928et_int: set_li8746675981475697647et_int > set_li8746675981475697647et_int > set_li8746675981475697647et_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
minus_4948633356380776833et_int: set_list_set_int > set_list_set_int > set_list_set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
minus_8897228262479074673et_int: set_set_int > set_set_int > set_set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
minus_8524384805321567159et_int: set_set_list_set_int > set_set_list_set_int > set_set_list_set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
minus_6857623457997529383et_int: set_set_set_int > set_set_set_int > set_set_set_int ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
minus_5736297505244876581_set_a: set_set_a > set_set_a > set_set_a ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
minus_minus_set_a: set_a > set_a > set_a ).
thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_Mt__Product____Type__Ounit_J,type,
cgenid7864102925118505423t_unit: partia7144473111789202860t_unit > list_list_set_int > set_li2838760105183549752et_int ).
thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Ring__Oring__Oring____ext_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Product____Type__Ounit_J,type,
cgenid1447163913751362543t_unit: partia6652711048488803900t_unit > list_set_int > set_list_set_int ).
thf(sy_c_Ideal_Ocgenideal_001t__List__Olist_Itf__a_J_001t__Ring__Oring__Oring____ext_It__List__Olist_Itf__a_J_Mt__Product____Type__Ounit_J,type,
cgenid9131348535277946915t_unit: partia2670972154091845814t_unit > list_a > set_list_a ).
thf(sy_c_Ideal_Ocgenideal_001t__Set__Oset_It__Int__Oint_J_001t__Ring__Oring__Oring____ext_It__Set__Oset_It__Int__Oint_J_Mt__Product____Type__Ounit_J,type,
cgenid8502489213727343375t_unit: partia4934656038542163276t_unit > set_int > set_set_int ).
thf(sy_c_Ideal_Ocgenideal_001tf__a_001t__Ring__Oring__Oring____ext_Itf__a_Mtf__b_J,type,
cgenid547466209912283029xt_a_b: partia2175431115845679010xt_a_b > a > set_a ).
thf(sy_c_Ideal_Ogenideal_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
genide4509635305121348427t_unit: partia6652711048488803900t_unit > set_list_set_int > set_list_set_int ).
thf(sy_c_Ideal_Ogenideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
genide1545711809618862555t_unit: partia4934656038542163276t_unit > set_set_int > set_set_int ).
thf(sy_c_Ideal_Ogenideal_001tf__a_001tf__b,type,
genideal_a_b: partia2175431115845679010xt_a_b > set_a > set_a ).
thf(sy_c_Ideal_Oideal_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ideal_3307802554209726317t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Ideal_Oideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ideal_7262958097527394045t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Ideal_Oideal_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
ideal_7892184242768911799t_unit: set_set_list_set_int > partia3352279114339913222t_unit > $o ).
thf(sy_c_Ideal_Oideal_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ideal_1992573625310827463t_unit: set_set_set_int > partia3601206958761670294t_unit > $o ).
thf(sy_c_Ideal_Oideal_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
ideal_4463284918206690523t_unit: set_set_a > partia6043505979758434576t_unit > $o ).
thf(sy_c_Ideal_Oideal_001tf__a_001tf__b,type,
ideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Omaximalideal_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
maxima7372504373286735342t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
maxima6262477034536100350t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Ideal_Omaximalideal_001tf__a_001tf__b,type,
maximalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprimeideal_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
primei7054412349171268410t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Ideal_Oprimeideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
primei350866878734230858t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Ideal_Oprimeideal_001tf__a_001tf__b,type,
primeideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ideal_Oprincipalideal_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
princi2884690910141361289t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
princi8860937869964495385t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Ideal_Oprincipalideal_001tf__a_001tf__b,type,
principalideal_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_IntRing_OZFact,type,
zFact: int > partia4934656038542163276t_unit ).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
cons_list_set_int: list_set_int > list_list_set_int > list_list_set_int ).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
nil_list_set_int: list_list_set_int ).
thf(sy_c_List_Olist_ONil_001t__Set__Oset_It__Int__Oint_J,type,
nil_set_int: list_set_int ).
thf(sy_c_List_Olist_ONil_001tf__a,type,
nil_a: list_a ).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
map_li4936211705635448506et_int: ( list_set_int > list_list_set_int ) > list_list_set_int > list_l8867605966578153106et_int ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Int__Oint_J_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
map_se7207430148037755802et_int: ( set_int > list_set_int ) > list_set_int > list_list_set_int ).
thf(sy_c_List_Olist_Omap_001t__Set__Oset_It__Int__Oint_J_001tf__a,type,
map_set_int_a: ( set_int > a ) > list_set_int > list_a ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__List__Olist_Itf__a_J,type,
map_a_list_a: ( a > list_a ) > list_a > list_list_a ).
thf(sy_c_Monic__Polynomial__Factorization_Omonic__poly_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
monic_3404647755478805438t_unit: partia4934656038542163276t_unit > list_set_int > $o ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint,type,
semiri1314217659103216013at_int: nat > int ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_M_Eo_J,type,
bot_bo1744859523148385737_int_o: list_set_int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_It__Set__Oset_It__Int__Oint_J_M_Eo_J,type,
bot_bot_set_int_o: set_int > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001_062_Itf__a_M_Eo_J,type,
bot_bot_a_o: a > $o ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_I_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
bot_bo7587639734496279683et_int: set_li8746675981475697647et_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
bot_bot_set_int: set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
bot_bo7344713902834881468et_int: set_list_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
bot_bot_set_set_int: set_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
bot_bo4503415735588907506et_int: set_set_list_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
bot_bo2384636101374064866et_int: set_set_set_int ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
bot_bot_set_set_a: set_set_a ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_Itf__a_J,type,
bot_bot_set_a: set_a ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
ord_le4060840825881956505et_int: ( list_set_int > set_list_set_int ) > ( list_set_int > set_list_set_int ) > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_I_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
ord_le7834723768769537103et_int: set_li8746675981475697647et_int > set_li8746675981475697647et_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Int__Oint_J,type,
ord_less_eq_set_int: set_int > set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le4527193859438875528et_int: set_list_set_int > set_list_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
ord_le4403425263959731960et_int: set_set_int > set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
ord_le8104031629355235774et_int: set_set_list_set_int > set_set_list_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
ord_le4317611570275147438et_int: set_set_set_int > set_set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
ord_le5799834111343130868et_int: set_se8558717844084009108et_int > set_se8558717844084009108et_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
ord_le3130544380672780132et_int: set_set_set_set_int > set_set_set_set_int > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_It__Set__Oset_Itf__a_J_J_J,type,
ord_le5722252365846178494_set_a: set_set_set_a > set_set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
ord_le3724670747650509150_set_a: set_set_a > set_set_a > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_Itf__a_J,type,
ord_less_eq_set_a: set_a > set_a > $o ).
thf(sy_c_Polynomial__Divisibility_Opdivides_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
polyno5399072443184722328t_unit: partia6652711048488803900t_unit > list_list_set_int > list_list_set_int > $o ).
thf(sy_c_Polynomial__Divisibility_Opdivides_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
polyno1573582608993444520t_unit: partia4934656038542163276t_unit > list_set_int > list_set_int > $o ).
thf(sy_c_Polynomial__Divisibility_Opdivides_001tf__a_001tf__b,type,
polyno5814909790663948098es_a_b: partia2175431115845679010xt_a_b > list_a > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
polyno5593083006719613942t_unit: partia6652711048488803900t_unit > list_list_set_int > list_set_int > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
polyno6587735422648734726t_unit: partia4934656038542163276t_unit > list_set_int > set_int > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Ois__root_001tf__a_001tf__b,type,
polyno4133073214067823460ot_a_b: partia2175431115845679010xt_a_b > list_a > a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Olong__divides_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
polyno7057773693803059378t_unit: partia6652711048488803900t_unit > list_list_set_int > list_list_set_int > produc2061375659605276261et_int > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Olong__divides_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
polyno4953732871684447938t_unit: partia4934656038542163276t_unit > list_set_int > list_set_int > produc6642921209907155909et_int > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Olong__divides_001tf__a_001tf__b,type,
polyno2806191415236617128es_a_b: partia2175431115845679010xt_a_b > list_a > list_a > produc9164743771328383783list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Opdiv_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
polyno5348356287797398593t_unit: partia6652711048488803900t_unit > list_list_set_int > list_list_set_int > list_list_set_int ).
thf(sy_c_Polynomial__Divisibility_Oring_Opdiv_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
polyno2948870938714006993t_unit: partia4934656038542163276t_unit > list_set_int > list_set_int > list_set_int ).
thf(sy_c_Polynomial__Divisibility_Oring_Opdiv_001tf__a_001tf__b,type,
polynomial_pdiv_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Opmod_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
polyno6644449323419600656t_unit: partia6652711048488803900t_unit > list_list_set_int > list_list_set_int > list_list_set_int ).
thf(sy_c_Polynomial__Divisibility_Oring_Opmod_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
polyno6711862844499874848t_unit: partia4934656038542163276t_unit > list_set_int > list_set_int > list_set_int ).
thf(sy_c_Polynomial__Divisibility_Oring_Opmod_001tf__a_001tf__b,type,
polynomial_pmod_a_b: partia2175431115845679010xt_a_b > list_a > list_a > list_a ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
polyno2166899109720230297t_unit: partia4934656038542163276t_unit > list_set_int > $o ).
thf(sy_c_Polynomial__Divisibility_Oring_Osplitted_001tf__a_001tf__b,type,
polyno8329700637149614481ed_a_b: partia2175431115845679010xt_a_b > list_a > $o ).
thf(sy_c_Polynomial__Divisibility_Orupture_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
polyno4866556214859693103t_unit: partia6652711048488803900t_unit > set_list_set_int > list_list_set_int > partia7913147511651046390t_unit ).
thf(sy_c_Polynomial__Divisibility_Orupture_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
polyno6134647158531630271t_unit: partia4934656038542163276t_unit > set_set_int > list_set_int > partia3352279114339913222t_unit ).
thf(sy_c_Polynomial__Divisibility_Orupture_001tf__a_001tf__b,type,
polyno5459750281392823787re_a_b: partia2175431115845679010xt_a_b > set_a > list_a > partia7496981018696276118t_unit ).
thf(sy_c_Polynomials_Oring_Oconst__term_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
const_2217831012996398173t_unit: partia6652711048488803900t_unit > list_list_set_int > list_set_int ).
thf(sy_c_Polynomials_Oring_Oeval_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
eval_l2744756199357571242t_unit: partia6652711048488803900t_unit > list_list_set_int > list_set_int > list_set_int ).
thf(sy_c_Polynomials_Oring_Oeval_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
eval_s3798525405962683066t_unit: partia4934656038542163276t_unit > list_set_int > set_int > set_int ).
thf(sy_c_Polynomials_Oring_Oeval_001tf__a_001tf__b,type,
eval_a_b: partia2175431115845679010xt_a_b > list_a > a > a ).
thf(sy_c_Polynomials_Oring_Onormalize_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
normal3791444541944621327t_unit: partia7144473111789202860t_unit > list_l8867605966578153106et_int > list_l8867605966578153106et_int ).
thf(sy_c_Polynomials_Oring_Onormalize_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
normal6870180608241117983t_unit: partia6652711048488803900t_unit > list_list_set_int > list_list_set_int ).
thf(sy_c_Polynomials_Oring_Onormalize_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
normal637505603836502915t_unit: partia2670972154091845814t_unit > list_list_a > list_list_a ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
poly_o1782926711940423372t_unit: partia6652711048488803900t_unit > list_set_int > list_list_set_int ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
poly_o1995067004181805532t_unit: partia4934656038542163276t_unit > set_int > list_set_int ).
thf(sy_c_Polynomials_Oring_Opoly__of__const_001tf__a_001tf__b,type,
poly_of_const_a_b: partia2175431115845679010xt_a_b > a > list_a ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
univ_p4950887682782688900t_unit: partia7144473111789202860t_unit > set_li2838760105183549752et_int > partia7052216462576429980t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
univ_p7181306125638219540t_unit: partia6652711048488803900t_unit > set_list_set_int > partia7144473111789202860t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
univ_p7953238456130426574t_unit: partia2670972154091845814t_unit > set_list_a > partia2956882679547061052t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
univ_p8160232691235018276t_unit: partia4934656038542163276t_unit > set_set_int > partia6652711048488803900t_unit ).
thf(sy_c_Polynomials_Ouniv__poly_001tf__a_001tf__b,type,
univ_poly_a_b: partia2175431115845679010xt_a_b > set_a > partia2670972154091845814t_unit ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
produc7276067142252306077et_int: list_list_set_int > list_list_set_int > produc2061375659605276261et_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
produc1446815805028224253et_int: list_set_int > list_set_int > produc6642921209907155909et_int ).
thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__a_J_001t__List__Olist_Itf__a_J,type,
produc6837034575241423639list_a: list_a > list_a > produc9164743771328383783list_a ).
thf(sy_c_QuotRing_OFactRing_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
factRi8995599497114439928t_unit: partia6652711048488803900t_unit > set_list_set_int > partia3352279114339913222t_unit ).
thf(sy_c_QuotRing_OFactRing_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
factRi3149420076008518152t_unit: partia4934656038542163276t_unit > set_set_int > partia3601206958761670294t_unit ).
thf(sy_c_QuotRing_OFactRing_001tf__a_001tf__b,type,
factRing_a_b: partia2175431115845679010xt_a_b > set_a > partia6043505979758434576t_unit ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
is_rin8255480188913837456t_unit: partia6652711048488803900t_unit > partia3352279114339913222t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
is_rin4964436081320486288t_unit: partia4934656038542163276t_unit > partia3601206958761670294t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
is_rin7318007858330770044t_unit: partia3352279114339913222t_unit > partia6652711048488803900t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
is_rin1729965262454186108t_unit: partia3601206958761670294t_unit > partia4934656038542163276t_unit > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit_001tf__a_001tf__b,type,
is_rin6001486760346555702it_a_b: partia6043505979758434576t_unit > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_QuotRing_Ois__ring__iso_001tf__a_001tf__b_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
is_rin9099215527551818550t_unit: partia2175431115845679010xt_a_b > partia6043505979758434576t_unit > $o ).
thf(sy_c_QuotRing_Orcoset__mult_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
rcoset3375355112047469280t_unit: partia6652711048488803900t_unit > set_list_set_int > set_list_set_int > set_list_set_int > set_list_set_int ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
a_inv_9003328026414236349t_unit: partia7144473111789202860t_unit > list_list_set_int > list_list_set_int ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_inv_7608866662416757709t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int ).
thf(sy_c_Ring_Oa__inv_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_inv_8944721093294617173t_unit: partia2670972154091845814t_unit > list_a > list_a ).
thf(sy_c_Ring_Oa__inv_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_inv_5951419416477254493t_unit: partia4934656038542163276t_unit > set_int > set_int ).
thf(sy_c_Ring_Oa__inv_001tf__a_001tf__b,type,
a_inv_a_b: partia2175431115845679010xt_a_b > a > a ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
a_minu1053138715659470590t_unit: partia7144473111789202860t_unit > list_list_set_int > list_list_set_int > list_list_set_int ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
a_minu3289003089969826958t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int > list_set_int ).
thf(sy_c_Ring_Oa__minus_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
a_minu3984020753470702548t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oa__minus_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
a_minu5974516859897376926t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Ring_Odomain_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
domain1893962698156629705t_unit: partia7144473111789202860t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
domain1631227219952682457t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
domain6553523120543210313t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
domain6183376680155302761t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
domain6977320662053078051t_unit: partia3352279114339913222t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
domain2593739924244766771t_unit: partia3601206958761670294t_unit > $o ).
thf(sy_c_Ring_Odomain_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
domain4236798911309298543t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring_Odomain_001tf__a_001tf__b,type,
domain_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Ofield_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
field_4857232450668044075t_unit: partia7144473111789202860t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
field_7411436548976123707t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
field_6388047844668329575t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
field_5943785737635511755t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_001t__Product____Type__Ounit,type,
field_7531927466519412469t_unit: partia7913147511651046390t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
field_6001781496977528197t_unit: partia3352279114339913222t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
field_26233345952514695t_unit: partia7496981018696276118t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
field_4969826516285655701t_unit: partia3601206958761670294t_unit > $o ).
thf(sy_c_Ring_Ofield_001t__Set__Oset_Itf__a_J_001t__Product____Type__Ounit,type,
field_6045675692312731021t_unit: partia6043505979758434576t_unit > $o ).
thf(sy_c_Ring_Ofield_001tf__a_001tf__b,type,
field_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
ring_l6429277945453189341t_unit: partia7144473111789202860t_unit > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ring_l5834914389771393517t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring_Oring_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_l6212528067271185461t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring_Oring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_s5316885176909347197t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring_Oring_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
ring_s3147478420139132983t_unit: partia3352279114339913222t_unit > $o ).
thf(sy_c_Ring_Oring_001tf__a_001tf__b,type,
ring_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
add_li3203230194967740868t_unit: partia7144473111789202860t_unit > list_list_set_int > list_list_set_int > list_list_set_int ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
add_li6184372983183467092t_unit: partia6652711048488803900t_unit > list_set_int > list_set_int > list_set_int ).
thf(sy_c_Ring_Oring_Oadd_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
add_li7652885771158616974t_unit: partia2670972154091845814t_unit > list_a > list_a > list_a ).
thf(sy_c_Ring_Oring_Oadd_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
add_se5859248395121729892t_unit: partia4934656038542163276t_unit > set_int > set_int > set_int ).
thf(sy_c_Ring_Oring_Oadd_001tf__a_001tf__b,type,
add_a_b: partia2175431115845679010xt_a_b > a > a > a ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
zero_l4263514192383796165t_unit: partia7144473111789202860t_unit > list_list_set_int ).
thf(sy_c_Ring_Oring_Ozero_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
zero_l5318887016005761237t_unit: partia6652711048488803900t_unit > list_set_int ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
zero_s6269048424454532197t_unit: partia4934656038542163276t_unit > set_int ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_001t__Product____Type__Ounit,type,
zero_s7811746193054374799t_unit: partia7913147511651046390t_unit > set_li2838760105183549752et_int ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
zero_s8971087499367319327t_unit: partia3352279114339913222t_unit > set_list_set_int ).
thf(sy_c_Ring_Oring_Ozero_001t__Set__Oset_It__List__Olist_Itf__a_J_J_001t__Product____Type__Ounit,type,
zero_s2910681146719230829t_unit: partia7496981018696276118t_unit > set_list_a ).
thf(sy_c_Ring_Oring_Ozero_001tf__a_001tf__b,type,
zero_a_b: partia2175431115845679010xt_a_b > a ).
thf(sy_c_Ring_Oring__hom_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
ring_h7954335661761407236t_unit: partia6652711048488803900t_unit > partia3352279114339913222t_unit > set_li8746675981475697647et_int ).
thf(sy_c_Ring_Osemiring_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
semiri2117368926114562431t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ochar_001tf__a_001tf__b,type,
ring_char_a_b: partia2175431115845679010xt_a_b > nat ).
thf(sy_c_Ring__Characteristic_Ochar__iso_001tf__a_001tf__b,type,
ring_char_iso_a_b: partia2175431115845679010xt_a_b > set_int > a ).
thf(sy_c_Ring__Characteristic_Ofinite__field_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_f302724563095964181t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Characteristic_Ofinite__field_001tf__a_001tf__b,type,
ring_f9002592259703987221ld_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ring_f3415241477199771139t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_f5979754897444624531t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Divisibility_Ofactorial__domain_001tf__a_001tf__b,type,
ring_f5272581269873410839in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ring_n3580647943193786865t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_n4948453042392689537t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__domain_001tf__a_001tf__b,type,
ring_n4045954140777738665in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ring_n7202549095058397701t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_n4918880612521764245t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Divisibility_Onoetherian__ring_001tf__a_001tf__b,type,
ring_n3639167112692572309ng_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
ring_p1105522777314784042t_unit: partia7144473111789202860t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ring_p1500425057661402810t_unit: partia6652711048488803900t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_p8098905331641078952t_unit: partia2670972154091845814t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_p2413783099608688842t_unit: partia4934656038542163276t_unit > $o ).
thf(sy_c_Ring__Divisibility_Oprincipal__domain_001tf__a_001tf__b,type,
ring_p8803135361686045600in_a_b: partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
ring_r5263416876369801476t_unit: partia7144473111789202860t_unit > list_list_set_int > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ring_r7480558297017414036t_unit: partia6652711048488803900t_unit > list_set_int > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r932985474545269838t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_r5206707500676108452t_unit: partia4934656038542163276t_unit > set_int > $o ).
thf(sy_c_Ring__Divisibility_Oring__irreducible_001tf__a_001tf__b,type,
ring_r999134135267193926le_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_001t__Product____Type__Ounit,type,
ring_r7068619596278734765t_unit: partia7144473111789202860t_unit > list_list_set_int > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
ring_r7101923692879464637t_unit: partia6652711048488803900t_unit > list_set_int > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__List__Olist_Itf__a_J_001t__Product____Type__Ounit,type,
ring_r6430282645014804837t_unit: partia2670972154091845814t_unit > list_a > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
ring_r2768735206093983821t_unit: partia4934656038542163276t_unit > set_int > $o ).
thf(sy_c_Ring__Divisibility_Oring__prime_001tf__a_001tf__b,type,
ring_ring_prime_a_b: partia2175431115845679010xt_a_b > a > $o ).
thf(sy_c_Set_OCollect_001_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
collec223697692861505422et_int: ( ( list_set_int > set_list_set_int ) > $o ) > set_li8746675981475697647et_int ).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
collect_list_set_int: ( list_set_int > $o ) > set_list_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Int__Oint_J,type,
collect_set_int: ( set_int > $o ) > set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
collec8150298066296752253et_int: ( set_list_set_int > $o ) > set_set_list_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
collect_set_set_int: ( set_set_int > $o ) > set_set_set_int ).
thf(sy_c_Set_OCollect_001t__Set__Oset_Itf__a_J,type,
collect_set_a: ( set_a > $o ) > set_set_a ).
thf(sy_c_Set_OCollect_001tf__a,type,
collect_a: ( a > $o ) > set_a ).
thf(sy_c_Set_OPow_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
pow_list_set_int: set_list_set_int > set_set_list_set_int ).
thf(sy_c_Set_OPow_001t__Set__Oset_It__Int__Oint_J,type,
pow_set_int: set_set_int > set_set_set_int ).
thf(sy_c_Set_Oinsert_001_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
insert1512894373432338025et_int: ( list_set_int > set_list_set_int ) > set_li8746675981475697647et_int > set_li8746675981475697647et_int ).
thf(sy_c_Set_Oinsert_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
insert_list_set_int: list_set_int > set_list_set_int > set_list_set_int ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Int__Oint_J,type,
insert_set_int: set_int > set_set_int > set_set_int ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
insert4179525362955487576et_int: set_list_set_int > set_set_list_set_int > set_set_list_set_int ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
insert_set_set_int: set_set_int > set_set_set_int > set_set_set_int ).
thf(sy_c_Set_Oinsert_001t__Set__Oset_Itf__a_J,type,
insert_set_a: set_a > set_set_a > set_set_a ).
thf(sy_c_Set_Oinsert_001tf__a,type,
insert_a: a > set_a > set_a ).
thf(sy_c_Subrings_Osubcring_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
subcri4185096033226064695t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Subrings_Osubdomain_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
subdom6469697055745072292t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
subfie3135231139456689584t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subfie3888952257595785920t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubfield_001tf__a_001tf__b,type,
subfield_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_Subrings_Osubring_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J_001t__Product____Type__Ounit,type,
subrin291743070294386232t_unit: set_list_set_int > partia6652711048488803900t_unit > $o ).
thf(sy_c_Subrings_Osubring_001t__Set__Oset_It__Int__Oint_J_001t__Product____Type__Ounit,type,
subrin7689096310803670856t_unit: set_set_int > partia4934656038542163276t_unit > $o ).
thf(sy_c_Subrings_Osubring_001tf__a_001tf__b,type,
subring_a_b: set_a > partia2175431115845679010xt_a_b > $o ).
thf(sy_c_member_001_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
member5554658343060950864et_int: ( list_set_int > set_list_set_int ) > set_li8746675981475697647et_int > $o ).
thf(sy_c_member_001t__Int__Oint,type,
member_int: int > set_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
member6183427284096218281et_int: list_l8867605966578153106et_int > set_li7397999359881343176et_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
member7440485371949894425et_int: list_list_set_int > set_li2838760105183549752et_int > $o ).
thf(sy_c_member_001t__List__Olist_It__List__Olist_Itf__a_J_J,type,
member_list_list_a: list_list_a > set_list_list_a > $o ).
thf(sy_c_member_001t__List__Olist_It__Set__Oset_It__Int__Oint_J_J,type,
member_list_set_int: list_set_int > set_list_set_int > $o ).
thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
member_list_a: list_a > set_list_a > $o ).
thf(sy_c_member_001t__Set__Oset_I_062_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_Mt__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J_J,type,
member1341228145649960710et_int: set_li8746675981475697647et_int > set_se6837996996060283045et_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
member_set_int: set_int > set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J,type,
member4487585027056014399et_int: set_list_set_int > set_set_list_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Int__Oint_J_J,type,
member_set_set_int: set_set_int > set_set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__List__Olist_It__Set__Oset_It__Int__Oint_J_J_J_J,type,
member8308387497365429877et_int: set_set_list_set_int > set_se8558717844084009108et_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_It__Set__Oset_It__Int__Oint_J_J_J,type,
member7356822600254261989et_int: set_set_set_int > set_set_set_set_int > $o ).
thf(sy_c_member_001t__Set__Oset_It__Set__Oset_Itf__a_J_J,type,
member_set_set_a: set_set_a > set_set_set_a > $o ).
thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
member_set_a: set_a > set_set_a > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_R,type,
r: partia2175431115845679010xt_a_b ).
thf(sy_v_f,type,
f: list_set_int ).
thf(sy_v_p,type,
p: nat ).
thf(sy_v_x,type,
x: a ).
thf(sy_v_y_H____,type,
y: list_set_int ).
thf(sy_v_y____,type,
y2: list_set_int ).
% Relevant facts (1278)
thf(fact_0_finite__field__axioms,axiom,
ring_f9002592259703987221ld_a_b @ r ).
% finite_field_axioms
thf(fact_1_factorial__domain__axioms,axiom,
ring_f5272581269873410839in_a_b @ r ).
% factorial_domain_axioms
thf(fact_2_p__def,axiom,
( p
= ( ring_char_a_b @ r ) ) ).
% p_def
thf(fact_3_assms_I5_J,axiom,
member_a @ x @ ( partia707051561876973205xt_a_b @ r ) ).
% assms(5)
thf(fact_4_noetherian__domain__axioms,axiom,
ring_n4045954140777738665in_a_b @ r ).
% noetherian_domain_axioms
thf(fact_5_y_H__def_I2_J,axiom,
( y2
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ y @ f ) ) ).
% y'_def(2)
thf(fact_6_zf_Ofactorial__domain__axioms,axiom,
ring_f5979754897444624531t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.factorial_domain_axioms
thf(fact_7_pzf_Ofactorial__domain__axioms,axiom,
ring_f3415241477199771139t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.factorial_domain_axioms
thf(fact_8_zf_Onoetherian__domain__axioms,axiom,
ring_n4948453042392689537t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.noetherian_domain_axioms
thf(fact_9_assms_I2_J,axiom,
member_list_set_int @ f @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% assms(2)
thf(fact_10_y_H__def_I1_J,axiom,
member_list_set_int @ y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% y'_def(1)
thf(fact_11_assms_I3_J,axiom,
ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ).
% assms(3)
thf(fact_12_assms_I6_J,axiom,
( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ f ) @ x )
= ( zero_a_b @ r ) ) ).
% assms(6)
thf(fact_13_pzf_Onoetherian__domain__axioms,axiom,
ring_n3580647943193786865t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.noetherian_domain_axioms
thf(fact_14_calculation,axiom,
( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ y2 ) @ x )
= ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ y @ f ) ) @ x ) ) ).
% calculation
thf(fact_15_zf_Oprincipal__domain__axioms,axiom,
ring_p2413783099608688842t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.principal_domain_axioms
thf(fact_16_principal__domain__axioms,axiom,
ring_p8803135361686045600in_a_b @ r ).
% principal_domain_axioms
thf(fact_17_pzf_Onoetherian__ring__axioms,axiom,
ring_n7202549095058397701t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.noetherian_ring_axioms
thf(fact_18_pzf_Oprincipal__domain__axioms,axiom,
ring_p1500425057661402810t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.principal_domain_axioms
thf(fact_19_zf_Onoetherian__ring__axioms,axiom,
ring_n4918880612521764245t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.noetherian_ring_axioms
thf(fact_20_zf_Oonepideal,axiom,
princi8860937869964495385t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.onepideal
thf(fact_21_noetherian__ring__axioms,axiom,
ring_n3639167112692572309ng_a_b @ r ).
% noetherian_ring_axioms
thf(fact_22_pzf_Osemiring__axioms,axiom,
semiri2117368926114562431t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.semiring_axioms
thf(fact_23_zf_Ofinite__field__axioms,axiom,
ring_f302724563095964181t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.finite_field_axioms
thf(fact_24_zf_Omonic__poly__mult,axiom,
! [F: list_set_int,G: list_set_int] :
( ( monic_3404647755478805438t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ F )
=> ( ( monic_3404647755478805438t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ G )
=> ( monic_3404647755478805438t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F @ G ) ) ) ) ).
% zf.monic_poly_mult
thf(fact_25_pzf_Om__assoc,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ Z )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ Z ) ) ) ) ) ) ).
% pzf.m_assoc
thf(fact_26_pzf_Om__comm,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X ) ) ) ) ).
% pzf.m_comm
thf(fact_27_m__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) )
= ( mult_a_ring_ext_a_b @ r @ Y @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% m_lcomm
thf(fact_28_m__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ Y @ X ) ) ) ) ).
% m_comm
thf(fact_29_m__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ Z )
= ( mult_a_ring_ext_a_b @ r @ X @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% m_assoc
thf(fact_30_m__rcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ B @ A )
= ( mult_a_ring_ext_a_b @ r @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% m_rcancel
thf(fact_31_m__lcancel,axiom,
! [A: a,B: a,C: a] :
( ( A
!= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( mult_a_ring_ext_a_b @ r @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% m_lcancel
thf(fact_32_integral__iff,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
= ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% integral_iff
thf(fact_33_local_Ointegral,axiom,
! [A: a,B: a] :
( ( ( mult_a_ring_ext_a_b @ r @ A @ B )
= ( zero_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( zero_a_b @ r ) )
| ( B
= ( zero_a_b @ r ) ) ) ) ) ) ).
% local.integral
thf(fact_34_f__comm__group__1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
!= ( zero_a_b @ r ) )
=> ( ( Y
!= ( zero_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
!= ( zero_a_b @ r ) ) ) ) ) ) ).
% f_comm_group_1
thf(fact_35_pzf_Om__lcomm,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ Z ) )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Z ) ) ) ) ) ) ).
% pzf.m_lcomm
thf(fact_36_zf_Omonic__poly__carr,axiom,
! [F: list_set_int] :
( ( monic_3404647755478805438t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ F )
=> ( member_list_set_int @ F @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.monic_poly_carr
thf(fact_37__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062y_H_O_A_092_060lbrakk_062y_H_A_092_060in_062_Acarrier_A_Ipoly__ring_A_IZFact_A_Iint_Ap_J_J_J_059_Ay_A_061_Ay_H_A_092_060otimes_062_092_060_094bsub_062poly__ring_A_IZFact_A_Iint_Ap_J_J_092_060_094esub_062_Af_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
~ ! [Y2: list_set_int] :
( ( member_list_set_int @ Y2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( y2
!= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y2 @ f ) ) ) ).
% \<open>\<And>thesis. (\<And>y'. \<lbrakk>y' \<in> carrier (poly_ring (ZFact (int p))); y = y' \<otimes>\<^bsub>poly_ring (ZFact (int p))\<^esub> f\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_38_h_Ohom__closed,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% h.hom_closed
thf(fact_39_zero__closed,axiom,
member_a @ ( zero_a_b @ r ) @ ( partia707051561876973205xt_a_b @ r ) ).
% zero_closed
thf(fact_40_m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% m_closed
thf(fact_41_r__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ) ).
% r_null
thf(fact_42_l__null,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_null
thf(fact_43_pzf_Om__closed,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.m_closed
thf(fact_44_h_Ohom__mult,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ Y ) @ x ) ) ) ) ) ).
% h.hom_mult
thf(fact_45_mem__Collect__eq,axiom,
! [A: set_set_int,P: set_set_int > $o] :
( ( member_set_set_int @ A @ ( collect_set_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_46_mem__Collect__eq,axiom,
! [A: set_int,P: set_int > $o] :
( ( member_set_int @ A @ ( collect_set_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_47_mem__Collect__eq,axiom,
! [A: set_a,P: set_a > $o] :
( ( member_set_a @ A @ ( collect_set_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_48_mem__Collect__eq,axiom,
! [A: list_set_int > set_list_set_int,P: ( list_set_int > set_list_set_int ) > $o] :
( ( member5554658343060950864et_int @ A @ ( collec223697692861505422et_int @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_49_mem__Collect__eq,axiom,
! [A: a,P: a > $o] :
( ( member_a @ A @ ( collect_a @ P ) )
= ( P @ A ) ) ).
% mem_Collect_eq
thf(fact_50_Collect__mem__eq,axiom,
! [A2: set_set_set_int] :
( ( collect_set_set_int
@ ^ [X2: set_set_int] : ( member_set_set_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_51_Collect__mem__eq,axiom,
! [A2: set_set_int] :
( ( collect_set_int
@ ^ [X2: set_int] : ( member_set_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_52_Collect__mem__eq,axiom,
! [A2: set_set_a] :
( ( collect_set_a
@ ^ [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_53_Collect__mem__eq,axiom,
! [A2: set_li8746675981475697647et_int] :
( ( collec223697692861505422et_int
@ ^ [X2: list_set_int > set_list_set_int] : ( member5554658343060950864et_int @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_54_Collect__mem__eq,axiom,
! [A2: set_a] :
( ( collect_a
@ ^ [X2: a] : ( member_a @ X2 @ A2 ) )
= A2 ) ).
% Collect_mem_eq
thf(fact_55__092_060open_062y_A_092_060in_062_APIdl_092_060_094bsub_062poly__ring_A_IZFact_A_Iint_Ap_J_J_092_060_094esub_062_Af_092_060close_062,axiom,
member_list_set_int @ y2 @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ).
% \<open>y \<in> PIdl\<^bsub>poly_ring (ZFact (int p))\<^esub> f\<close>
thf(fact_56_pzf_Oonepideal,axiom,
princi2884690910141361289t_unit @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.onepideal
thf(fact_57_zf_Ogauss__poly__carr,axiom,
! [N: nat] : ( member_list_set_int @ ( card_I2978652876270886323t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ N ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.gauss_poly_carr
thf(fact_58_onepideal,axiom,
principalideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% onepideal
thf(fact_59_pzf_Omonoid__cancelI,axiom,
( ! [A3: list_set_int,B2: list_set_int,C2: list_set_int] :
( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C2 @ A3 )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C2 @ B2 ) )
=> ( ( member_list_set_int @ A3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: list_set_int,B2: list_set_int,C2: list_set_int] :
( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A3 @ C2 )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 @ C2 ) )
=> ( ( member_list_set_int @ A3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid1136863209656779155t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.monoid_cancelI
thf(fact_60_pzf_Oprimeness__condition,axiom,
! [P2: list_set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
= ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).
% pzf.primeness_condition
thf(fact_61_zf_Ois__root__poly__mult__imp__is__root,axiom,
! [P2: list_set_int,Q: list_set_int,X: set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( polyno6587735422648734726t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) @ X )
=> ( ( polyno6587735422648734726t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ X )
| ( polyno6587735422648734726t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Q @ X ) ) ) ) ) ).
% zf.is_root_poly_mult_imp_is_root
thf(fact_62_h_Ozero__closed,axiom,
member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ).
% h.zero_closed
thf(fact_63_ring__primeE_I1_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( P2
!= ( zero_a_b @ r ) ) ) ) ).
% ring_primeE(1)
thf(fact_64_h_Oa__inv__closed,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_a @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ x ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% h.a_inv_closed
thf(fact_65_ring__irreducibleE_I1_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( R
!= ( zero_a_b @ r ) ) ) ) ).
% ring_irreducibleE(1)
thf(fact_66_primeness__condition,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeness_condition
thf(fact_67_pzf_Ocgenideal__self,axiom,
! [I: list_set_int] :
( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ I @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I ) ) ) ).
% pzf.cgenideal_self
thf(fact_68_pzf_Ointegral,axiom,
! [A: list_set_int,B: list_set_int] :
( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( A
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
| ( B
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.integral
thf(fact_69_pzf_Ointegral__iff,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= ( ( A
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
| ( B
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.integral_iff
thf(fact_70_pzf_Om__lcancel,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( A
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% pzf.m_lcancel
thf(fact_71_pzf_Om__rcancel,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( A
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ A )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% pzf.m_rcancel
thf(fact_72_pzf_Ol__minus,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ Y )
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) ) ) ) ) ).
% pzf.l_minus
thf(fact_73_pzf_Or__minus,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) )
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) ) ) ) ) ).
% pzf.r_minus
thf(fact_74_pzf_Oring__irreducibleE_I1_J,axiom,
! [R: list_set_int] :
( ( member_list_set_int @ R @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ R )
=> ( R
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.ring_irreducibleE(1)
thf(fact_75_pzf_Oring__primeE_I1_J,axiom,
! [P2: list_set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
=> ( P2
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.ring_primeE(1)
thf(fact_76_i_OIcarr,axiom,
! [I: list_set_int] :
( ( member_list_set_int @ I @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% i.Icarr
thf(fact_77_i_Oa__Hcarr,axiom,
! [H: list_set_int] :
( ( member_list_set_int @ H @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% i.a_Hcarr
thf(fact_78_i_Ozero__closed,axiom,
member_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ).
% i.zero_closed
thf(fact_79_pzf_Ocgenideal__is__principalideal,axiom,
! [I: list_set_int] :
( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( princi2884690910141361289t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.cgenideal_is_principalideal
thf(fact_80_i_OI__l__closed,axiom,
! [A: list_set_int,X: list_set_int] :
( ( member_list_set_int @ A @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ A ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.I_l_closed
thf(fact_81_i_OI__r__closed,axiom,
! [A: list_set_int,X: list_set_int] :
( ( member_list_set_int @ A @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ X ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.I_r_closed
thf(fact_82_i_Ohelper__I__closed,axiom,
! [A: list_set_int,X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ X ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ) ) ).
% i.helper_I_closed
thf(fact_83_pzf_Ozero__closed,axiom,
member_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% pzf.zero_closed
thf(fact_84_pzf_Oadd_Oinv__closed,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.add.inv_closed
thf(fact_85_pzf_Ominus__minus,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) )
= X ) ) ).
% pzf.minus_minus
thf(fact_86_pzf_Ominus__zero,axiom,
( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.minus_zero
thf(fact_87_pzf_Ol__null,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.l_null
thf(fact_88_pzf_Or__null,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.r_null
thf(fact_89_pzf_Oadd_Oinv__eq__1__iff,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= ( X
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.add.inv_eq_1_iff
thf(fact_90_i_Oa__inv__closed,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ).
% i.a_inv_closed
thf(fact_91_h_Ohom__zero,axiom,
( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ x )
= ( zero_a_b @ r ) ) ).
% h.hom_zero
thf(fact_92_pzf_Oirreducible__imp__maximalideal,axiom,
! [P2: list_set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
=> ( maxima7372504373286735342t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.irreducible_imp_maximalideal
thf(fact_93_pzf_Oring__primeI,axiom,
! [P2: list_set_int] :
( ( P2
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( prime_7199642643890090290t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
=> ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).
% pzf.ring_primeI
thf(fact_94_pzf_Oring__primeE_I3_J,axiom,
! [P2: list_set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
=> ( prime_7199642643890090290t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).
% pzf.ring_primeE(3)
thf(fact_95_pzf_Ozero__is__prime_I1_J,axiom,
prime_7199642643890090290t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% pzf.zero_is_prime(1)
thf(fact_96_i_Ois__additive__subgroup,axiom,
additi4441901441605882380t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% i.is_additive_subgroup
thf(fact_97_ring__primeI,axiom,
! [P2: a] :
( ( P2
!= ( zero_a_b @ r ) )
=> ( ( prime_a_ring_ext_a_b @ r @ P2 )
=> ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% ring_primeI
thf(fact_98_ring__primeE_I3_J,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_ring_prime_a_b @ r @ P2 )
=> ( prime_a_ring_ext_a_b @ r @ P2 ) ) ) ).
% ring_primeE(3)
thf(fact_99_i_Oa__subgroup__in__rcosets,axiom,
member4487585027056014399et_int @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ).
% i.a_subgroup_in_rcosets
thf(fact_100_i_Oa__inv__op__closed1,axiom,
! [X: list_set_int,H: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ H @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ H ) @ X ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.a_inv_op_closed1
thf(fact_101_i_Oa__inv__op__closed2,axiom,
! [X: list_set_int,H: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ H @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ H ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.a_inv_op_closed2
thf(fact_102_i_Oa__subset,axiom,
ord_le4527193859438875528et_int @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% i.a_subset
thf(fact_103_zero__is__prime_I1_J,axiom,
prime_a_ring_ext_a_b @ r @ ( zero_a_b @ r ) ).
% zero_is_prime(1)
thf(fact_104_pzf_Oadd_Or__cancel,axiom,
! [A: list_set_int,C: list_set_int,B: list_set_int] :
( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ C ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% pzf.add.r_cancel
thf(fact_105_pzf_Oadd_Om__lcomm,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ Z ) )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Z ) ) ) ) ) ) ).
% pzf.add.m_lcomm
thf(fact_106_pzf_Oadd_Om__comm,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X ) ) ) ) ).
% pzf.add.m_comm
thf(fact_107_pzf_Oadd_Om__assoc,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ Z )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ Z ) ) ) ) ) ) ).
% pzf.add.m_assoc
thf(fact_108_pzf_Oadd_Ol__cancel,axiom,
! [C: list_set_int,A: list_set_int,B: list_set_int] :
( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ A )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ B ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( A = B ) ) ) ) ) ).
% pzf.add.l_cancel
thf(fact_109_pzf_Ominus__unique,axiom,
! [Y: list_set_int,X: list_set_int,Y3: list_set_int] :
( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y3 )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% pzf.minus_unique
thf(fact_110_pzf_Oadd_Or__inv__ex,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ? [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ X3 )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.add.r_inv_ex
thf(fact_111_pzf_Oadd_Oone__unique,axiom,
! [U: list_set_int] :
( ( member_list_set_int @ U @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.add.one_unique
thf(fact_112_pzf_Oadd_Ol__inv__ex,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ? [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.add.l_inv_ex
thf(fact_113_pzf_Oadd_Oinv__comm,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.add.inv_comm
thf(fact_114_pzf_Or__distr,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Z @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Z @ X ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Z @ Y ) ) ) ) ) ) ).
% pzf.r_distr
thf(fact_115_pzf_Ol__distr,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ Z )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Z ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ Z ) ) ) ) ) ) ).
% pzf.l_distr
thf(fact_116_pzf_Or__neg2,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% pzf.r_neg2
thf(fact_117_pzf_Or__neg1,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) )
= Y ) ) ) ).
% pzf.r_neg1
thf(fact_118_pzf_Ominus__add,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) ) ) ) ).
% pzf.minus_add
thf(fact_119_pzf_Oadd_Oinv__solve__right_H,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C ) )
= A )
= ( B
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) ) ) ) ) ) ).
% pzf.add.inv_solve_right'
thf(fact_120_pzf_Oadd_Oinv__solve__right,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( A
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C ) ) )
= ( B
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) ) ) ) ) ) ).
% pzf.add.inv_solve_right
thf(fact_121_pzf_Oadd_Oinv__solve__left_H,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B ) @ C )
= A )
= ( C
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ A ) ) ) ) ) ) ).
% pzf.add.inv_solve_left'
thf(fact_122_pzf_Oadd_Oinv__solve__left,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( A
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B ) @ C ) )
= ( C
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ A ) ) ) ) ) ) ).
% pzf.add.inv_solve_left
thf(fact_123_pzf_Oadd_Oinv__mult__group,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) ) ) ) ).
% pzf.add.inv_mult_group
thf(fact_124_pzf_Oa__transpose__inv,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= Z )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% pzf.a_transpose_inv
thf(fact_125_pzf_Osum__zero__eq__neg,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( X
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) ) ) ) ).
% pzf.sum_zero_eq_neg
thf(fact_126_pzf_Or__neg,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.r_neg
thf(fact_127_pzf_Ominus__equality,axiom,
! [Y: list_set_int,X: list_set_int] :
( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X )
= Y ) ) ) ) ).
% pzf.minus_equality
thf(fact_128_pzf_Ol__neg,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.l_neg
thf(fact_129_i_Oa__rcosets__carrier,axiom,
! [X4: set_list_set_int] :
( ( member4487585027056014399et_int @ X4 @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ord_le4527193859438875528et_int @ X4 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% i.a_rcosets_carrier
thf(fact_130_pzf_Oadd_Oright__cancel,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% pzf.add.right_cancel
thf(fact_131_pzf_Oadd_Om__closed,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.add.m_closed
thf(fact_132_pzf_Or__zero,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= X ) ) ).
% pzf.r_zero
thf(fact_133_pzf_Ol__zero,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ X )
= X ) ) ).
% pzf.l_zero
thf(fact_134_pzf_Oadd_Or__cancel__one_H,axiom,
! [X: list_set_int,A: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( X
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ X ) )
= ( A
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.add.r_cancel_one'
thf(fact_135_pzf_Oadd_Or__cancel__one,axiom,
! [X: list_set_int,A: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ X )
= X )
= ( A
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.add.r_cancel_one
thf(fact_136_pzf_Oadd_Ol__cancel__one_H,axiom,
! [X: list_set_int,A: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( X
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ A ) )
= ( A
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.add.l_cancel_one'
thf(fact_137_pzf_Oadd_Ol__cancel__one,axiom,
! [X: list_set_int,A: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ A )
= X )
= ( A
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.add.l_cancel_one
thf(fact_138_i_Oa__closed,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( ( member_list_set_int @ Y @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.a_closed
thf(fact_139_i_Oa__setinv__closed,axiom,
! [K: set_list_set_int] :
( ( member4487585027056014399et_int @ K @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( member4487585027056014399et_int @ ( a_SET_4289452989722768907t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.a_setinv_closed
thf(fact_140_pzf_Oadd_Oone__in__subset,axiom,
! [H2: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( H2 != bot_bo7344713902834881468et_int )
=> ( ! [X3: list_set_int] :
( ( member_list_set_int @ X3 @ H2 )
=> ( member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 ) @ H2 ) )
=> ( ! [X3: list_set_int] :
( ( member_list_set_int @ X3 @ H2 )
=> ! [Xa: list_set_int] :
( ( member_list_set_int @ Xa @ H2 )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 @ Xa ) @ H2 ) ) )
=> ( member_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ H2 ) ) ) ) ) ).
% pzf.add.one_in_subset
thf(fact_141_pzf_Oa__lcos__m__assoc,axiom,
! [M: set_list_set_int,G: list_set_int,H: list_set_int] :
( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ G @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G @ ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ M ) )
= ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G @ H ) @ M ) ) ) ) ) ).
% pzf.a_lcos_m_assoc
thf(fact_142_pzf_Oa__lcos__mult__one,axiom,
! [M: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ M )
= M ) ) ).
% pzf.a_lcos_mult_one
thf(fact_143_i_Oa__cosets__finite,axiom,
! [C: set_list_set_int] :
( ( member4487585027056014399et_int @ C @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ( ord_le4527193859438875528et_int @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( finite7825103155802847241et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( finite7825103155802847241et_int @ C ) ) ) ) ).
% i.a_cosets_finite
thf(fact_144_pzf_Omaximalideal__prime,axiom,
! [I2: set_list_set_int] :
( ( maxima7372504373286735342t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( primei7054412349171268410t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.maximalideal_prime
thf(fact_145_pzf_Osubalgebra__in__carrier,axiom,
! [K: set_list_set_int,V: set_list_set_int] :
( ( embedd3025314616789055568t_unit @ K @ V @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ V @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.subalgebra_in_carrier
thf(fact_146_pzf_Ocarrier__is__subalgebra,axiom,
! [K: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( embedd3025314616789055568t_unit @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.carrier_is_subalgebra
thf(fact_147_pzf_Oa__l__coset__subset__G,axiom,
! [H2: set_list_set_int,X: list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ H2 ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.a_l_coset_subset_G
thf(fact_148_pzf_Osubset__Idl__subset,axiom,
! [I2: set_list_set_int,H2: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ H2 @ I2 )
=> ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 ) @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) ) ) ).
% pzf.subset_Idl_subset
thf(fact_149_pzf_Ogenideal__self,axiom,
! [S: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ S @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ S @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) ) ) ).
% pzf.genideal_self
thf(fact_150_pzf_Ocarrier__not__empty,axiom,
( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
!= bot_bo7344713902834881468et_int ) ).
% pzf.carrier_not_empty
thf(fact_151_pzf_Ominus__eq,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) ) ).
% pzf.minus_eq
thf(fact_152_i_Oa__rcosets__inv__mult__group__eq,axiom,
! [M: set_list_set_int] :
( ( member4487585027056014399et_int @ M @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_SET_4289452989722768907t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M ) @ M )
= ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ).
% i.a_rcosets_inv_mult_group_eq
thf(fact_153_i_Ois__abelian__subgroup,axiom,
abelia4826137221796835534t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% i.is_abelian_subgroup
thf(fact_154_i_Oa__rcosets__part__G,axiom,
( ( comple3081917594375107323et_int @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
= ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% i.a_rcosets_part_G
thf(fact_155_pzf_Ocgenideal__eq__genideal,axiom,
! [I: list_set_int] :
( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I )
= ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ I @ bot_bo7344713902834881468et_int ) ) ) ) ).
% pzf.cgenideal_eq_genideal
thf(fact_156_pzf_OIdl__subset__ideal_H,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int ) ) )
= ( member_list_set_int @ A @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int ) ) ) ) ) ) ).
% pzf.Idl_subset_ideal'
thf(fact_157_pzf_Oline__extension__mem__iff,axiom,
! [U: list_set_int,K: set_list_set_int,A: list_set_int,E: set_list_set_int] :
( ( member_list_set_int @ U @ ( embedd942231993492570031t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ A @ E ) )
= ( ? [X2: list_set_int] :
( ( member_list_set_int @ X2 @ K )
& ? [Y4: list_set_int] :
( ( member_list_set_int @ Y4 @ E )
& ( U
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X2 @ A ) @ Y4 ) ) ) ) ) ) ).
% pzf.line_extension_mem_iff
thf(fact_158_i_Oa__rcos__inv,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_SET_4289452989722768907t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) ) ) ).
% i.a_rcos_inv
thf(fact_159_pzf_Oline__extension__in__carrier,axiom,
! [K: set_list_set_int,A: list_set_int,E: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ E @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ ( embedd942231993492570031t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ A @ E ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.line_extension_in_carrier
thf(fact_160_pzf_Oadd__additive__subgroups,axiom,
! [H2: set_list_set_int,K: set_list_set_int] :
( ( additi4441901441605882380t_unit @ H2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( additi4441901441605882380t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( additi4441901441605882380t_unit @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ K ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.add_additive_subgroups
thf(fact_161_pzf_Oa__r__coset__subset__G,axiom,
! [H2: set_list_set_int,X: list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ X ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.a_r_coset_subset_G
thf(fact_162_pzf_Osetadd__subset__G,axiom,
! [H2: set_list_set_int,K: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ K ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.setadd_subset_G
thf(fact_163_pzf_Oset__add__comm,axiom,
! [I2: set_list_set_int,J: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ I2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ J @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ J )
= ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ J @ I2 ) ) ) ) ).
% pzf.set_add_comm
thf(fact_164_pzf_Oset__add__closed,axiom,
! [A2: set_list_set_int,B3: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ B3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A2 @ B3 ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.set_add_closed
thf(fact_165_i_Oa__rcos__const,axiom,
! [H: list_set_int] :
( ( member_list_set_int @ H @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ H )
= ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ).
% i.a_rcos_const
thf(fact_166_pzf_Oa__rcosI,axiom,
! [H: list_set_int,H2: set_list_set_int,X: list_set_int] :
( ( member_list_set_int @ H @ H2 )
=> ( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H @ X ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ X ) ) ) ) ) ).
% pzf.a_rcosI
thf(fact_167_pzf_Oa__coset__add__assoc,axiom,
! [M: set_list_set_int,G: list_set_int,H: list_set_int] :
( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ G @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ H @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ G ) @ H )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G @ H ) ) ) ) ) ) ).
% pzf.a_coset_add_assoc
thf(fact_168_pzf_Oa__setmult__rcos__assoc,axiom,
! [H2: set_list_set_int,K: set_list_set_int,X: list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ X ) )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ K ) @ X ) ) ) ) ) ).
% pzf.a_setmult_rcos_assoc
thf(fact_169_pzf_Oa__rcosetsI,axiom,
! [H2: set_list_set_int,X: list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member4487585027056014399et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ X ) @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 ) ) ) ) ).
% pzf.a_rcosetsI
thf(fact_170_pzf_Ogenideal__self_H,axiom,
! [I: list_set_int] :
( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ I @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ I @ bot_bo7344713902834881468et_int ) ) ) ) ).
% pzf.genideal_self'
thf(fact_171_pzf_Ogenideal__zero,axiom,
( ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
= ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) ).
% pzf.genideal_zero
thf(fact_172_pzf_Ozeroprimeideal,axiom,
primei7054412349171268410t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.zeroprimeideal
thf(fact_173_i_Orcos__const__imp__mem,axiom,
! [I: list_set_int] :
( ( member_list_set_int @ I @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ I )
= ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ I @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.rcos_const_imp_mem
thf(fact_174_i_Oa__repr__independenceD,axiom,
! [Y: list_set_int,X: list_set_int] :
( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ Y ) )
=> ( member_list_set_int @ Y @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) ) ) ) ).
% i.a_repr_independenceD
thf(fact_175_i_Oa__repr__independence_H,axiom,
! [Y: list_set_int,X: list_set_int] :
( ( member_list_set_int @ Y @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ Y ) ) ) ) ).
% i.a_repr_independence'
thf(fact_176_i_Oa__rcos__self,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ X @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) ) ) ).
% i.a_rcos_self
thf(fact_177_i_Oa__elemrcos__carrier,axiom,
! [A: list_set_int,A4: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A4 @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ A ) )
=> ( member_list_set_int @ A4 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% i.a_elemrcos_carrier
thf(fact_178_pzf_Ozeropideal,axiom,
princi2884690910141361289t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.zeropideal
thf(fact_179_pzf_Oa__coset__add__inv2,axiom,
! [M: set_list_set_int,X: list_set_int,Y: list_set_int] :
( ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ X )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ Y ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) )
= M ) ) ) ) ) ).
% pzf.a_coset_add_inv2
thf(fact_180_pzf_Oa__coset__add__inv1,axiom,
! [M: set_list_set_int,X: list_set_int,Y: list_set_int] :
( ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y ) ) )
= M )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ X )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ Y ) ) ) ) ) ) ).
% pzf.a_coset_add_inv1
thf(fact_181_i_Orcosets__add__eq,axiom,
! [M: set_list_set_int] :
( ( member4487585027056014399et_int @ M @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ M )
= M ) ) ).
% i.rcosets_add_eq
thf(fact_182_i_Oa__setmult__closed,axiom,
! [K1: set_list_set_int,K2: set_list_set_int] :
( ( member4487585027056014399et_int @ K1 @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ( member4487585027056014399et_int @ K2 @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( member4487585027056014399et_int @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K1 @ K2 ) @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ) ).
% i.a_setmult_closed
thf(fact_183_i_Oa__rcosets__assoc,axiom,
! [M1: set_list_set_int,M2: set_list_set_int,M3: set_list_set_int] :
( ( member4487585027056014399et_int @ M1 @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ( member4487585027056014399et_int @ M2 @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ( member4487585027056014399et_int @ M3 @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M1 @ M2 ) @ M3 )
= ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M1 @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M2 @ M3 ) ) ) ) ) ) ).
% i.a_rcosets_assoc
thf(fact_184_pzf_Oa__rcos__assoc__lcos,axiom,
! [H2: set_list_set_int,K: set_list_set_int,X: list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ X ) @ K )
= ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 @ ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ K ) ) ) ) ) ) ).
% pzf.a_rcos_assoc_lcos
thf(fact_185_i_Oa__coset__eq,axiom,
! [X5: list_set_int] :
( ( member_list_set_int @ X5 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X5 )
= ( a_l_co143929110759192528t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X5 @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.a_coset_eq
thf(fact_186_pzf_Oset__add__zero,axiom,
! [A2: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ A2 )
= A2 ) ) ).
% pzf.set_add_zero
thf(fact_187_i_Oa__rcos__module__rev,axiom,
! [X: list_set_int,X6: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X6 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X6 @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( member_list_set_int @ X6 @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) ) ) ) ) ).
% i.a_rcos_module_rev
thf(fact_188_i_Oa__rcos__module__imp,axiom,
! [X: list_set_int,X6: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X6 @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X6 @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ).
% i.a_rcos_module_imp
thf(fact_189_i_Oa__rcos__module,axiom,
! [X: list_set_int,X6: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X6 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X6 @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) )
= ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X6 @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ) ).
% i.a_rcos_module
thf(fact_190_i_Oa__rcos__sum,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ A ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ B ) )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) ) ) ) ) ).
% i.a_rcos_sum
thf(fact_191_pzf_Ominus__closed,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.minus_closed
thf(fact_192_i_Orcoset__mult__add,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( rcoset3375355112047469280t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ Y ) )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) ) ) ) ) ).
% i.rcoset_mult_add
thf(fact_193_pzf_Or__right__minus__eq,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= ( A = B ) ) ) ) ).
% pzf.r_right_minus_eq
thf(fact_194_pzf_Oa__coset__add__zero,axiom,
! [M: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ M @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ M @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= M ) ) ).
% pzf.a_coset_add_zero
thf(fact_195_pzf_Oideal__sum__iff__gcd,axiom,
! [A: list_set_int,B: list_set_int,D: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ D @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ D )
= ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B ) ) )
= ( isgcd_1196993390438183731t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ D @ A @ B ) ) ) ) ) ).
% pzf.ideal_sum_iff_gcd
thf(fact_196_pzf_Obezout__identity,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B ) )
= ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( somegc4977764851446611037t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) ) ) ) ) ).
% pzf.bezout_identity
thf(fact_197_pzf_Oprimeideal__iff__prime,axiom,
! [P2: list_set_int] :
( ( member_list_set_int @ P2 @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( ( primei7054412349171268410t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ).
% pzf.primeideal_iff_prime
thf(fact_198_i_Oa__rcos__module__minus,axiom,
! [X: list_set_int,X6: list_set_int] :
( ( ring_l5834914389771393517t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X6 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X6 @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ X ) )
= ( member_list_set_int @ ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X6 @ X ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ) ) ) ).
% i.a_rcos_module_minus
thf(fact_199_cSup__singleton,axiom,
! [X: set_int] :
( ( comple3221217463730067765et_int @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= X ) ).
% cSup_singleton
thf(fact_200_cSup__singleton,axiom,
! [X: set_list_set_int] :
( ( comple3081917594375107323et_int @ ( insert4179525362955487576et_int @ X @ bot_bo4503415735588907506et_int ) )
= X ) ).
% cSup_singleton
thf(fact_201_cSup__singleton,axiom,
! [X: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% cSup_singleton
thf(fact_202_cSup__singleton,axiom,
! [X: set_set_int] :
( ( comple7281953568134767595et_int @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) )
= X ) ).
% cSup_singleton
thf(fact_203_ccpo__Sup__singleton,axiom,
! [X: set_int] :
( ( comple3221217463730067765et_int @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_204_ccpo__Sup__singleton,axiom,
! [X: set_list_set_int] :
( ( comple3081917594375107323et_int @ ( insert4179525362955487576et_int @ X @ bot_bo4503415735588907506et_int ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_205_ccpo__Sup__singleton,axiom,
! [X: set_a] :
( ( comple2307003609928055243_set_a @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_206_ccpo__Sup__singleton,axiom,
! [X: set_set_int] :
( ( comple7281953568134767595et_int @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) )
= X ) ).
% ccpo_Sup_singleton
thf(fact_207_Sup__empty,axiom,
( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% Sup_empty
thf(fact_208_Sup__empty,axiom,
( ( comple3081917594375107323et_int @ bot_bo4503415735588907506et_int )
= bot_bo7344713902834881468et_int ) ).
% Sup_empty
thf(fact_209_Sup__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Sup_empty
thf(fact_210_Sup__empty,axiom,
( ( comple7281953568134767595et_int @ bot_bo2384636101374064866et_int )
= bot_bot_set_set_int ) ).
% Sup_empty
thf(fact_211_singleton__insert__inj__eq_H,axiom,
! [A: list_set_int,A2: set_list_set_int,B: list_set_int] :
( ( ( insert_list_set_int @ A @ A2 )
= ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int ) )
= ( ( A = B )
& ( ord_le4527193859438875528et_int @ A2 @ ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_212_singleton__insert__inj__eq_H,axiom,
! [A: set_list_set_int,A2: set_set_list_set_int,B: set_list_set_int] :
( ( ( insert4179525362955487576et_int @ A @ A2 )
= ( insert4179525362955487576et_int @ B @ bot_bo4503415735588907506et_int ) )
= ( ( A = B )
& ( ord_le8104031629355235774et_int @ A2 @ ( insert4179525362955487576et_int @ B @ bot_bo4503415735588907506et_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_213_singleton__insert__inj__eq_H,axiom,
! [A: set_set_int,A2: set_set_set_int,B: set_set_int] :
( ( ( insert_set_set_int @ A @ A2 )
= ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) )
= ( ( A = B )
& ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_214_singleton__insert__inj__eq_H,axiom,
! [A: set_int,A2: set_set_int,B: set_int] :
( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_215_singleton__insert__inj__eq_H,axiom,
! [A: set_a,A2: set_set_a,B: set_a] :
( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ bot_bot_set_set_a ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_216_singleton__insert__inj__eq_H,axiom,
! [A: a,A2: set_a,B: a] :
( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ bot_bot_set_a ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_217_singleton__insert__inj__eq,axiom,
! [B: list_set_int,A: list_set_int,A2: set_list_set_int] :
( ( ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int )
= ( insert_list_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4527193859438875528et_int @ A2 @ ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_218_singleton__insert__inj__eq,axiom,
! [B: set_list_set_int,A: set_list_set_int,A2: set_set_list_set_int] :
( ( ( insert4179525362955487576et_int @ B @ bot_bo4503415735588907506et_int )
= ( insert4179525362955487576et_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le8104031629355235774et_int @ A2 @ ( insert4179525362955487576et_int @ B @ bot_bo4503415735588907506et_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_219_singleton__insert__inj__eq,axiom,
! [B: set_set_int,A: set_set_int,A2: set_set_set_int] :
( ( ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int )
= ( insert_set_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ bot_bo2384636101374064866et_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_220_singleton__insert__inj__eq,axiom,
! [B: set_int,A: set_int,A2: set_set_int] :
( ( ( insert_set_int @ B @ bot_bot_set_set_int )
= ( insert_set_int @ A @ A2 ) )
= ( ( A = B )
& ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_221_singleton__insert__inj__eq,axiom,
! [B: set_a,A: set_a,A2: set_set_a] :
( ( ( insert_set_a @ B @ bot_bot_set_set_a )
= ( insert_set_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ bot_bot_set_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_222_singleton__insert__inj__eq,axiom,
! [B: a,A: a,A2: set_a] :
( ( ( insert_a @ B @ bot_bot_set_a )
= ( insert_a @ A @ A2 ) )
= ( ( A = B )
& ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_223_carrier__not__empty,axiom,
( ( partia707051561876973205xt_a_b @ r )
!= bot_bot_set_a ) ).
% carrier_not_empty
thf(fact_224_zf_Ocarrier__not__empty,axiom,
( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
!= bot_bot_set_set_int ) ).
% zf.carrier_not_empty
thf(fact_225_zeropideal,axiom,
principalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeropideal
thf(fact_226_subset__antisym,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ( ( ord_le8104031629355235774et_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_227_subset__antisym,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( ord_le4317611570275147438et_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_228_subset__antisym,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_229_subset__antisym,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_230_subset__antisym,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ A2 )
=> ( A2 = B3 ) ) ) ).
% subset_antisym
thf(fact_231_subsetI,axiom,
! [A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int] :
( ! [X3: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ X3 @ A2 )
=> ( member5554658343060950864et_int @ X3 @ B3 ) )
=> ( ord_le7834723768769537103et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_232_subsetI,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ! [X3: set_list_set_int] :
( ( member4487585027056014399et_int @ X3 @ A2 )
=> ( member4487585027056014399et_int @ X3 @ B3 ) )
=> ( ord_le8104031629355235774et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_233_subsetI,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ! [X3: set_set_int] :
( ( member_set_set_int @ X3 @ A2 )
=> ( member_set_set_int @ X3 @ B3 ) )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_234_subsetI,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
=> ( member_set_int @ X3 @ B3 ) )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ).
% subsetI
thf(fact_235_subsetI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( member_set_a @ X3 @ B3 ) )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_236_subsetI,axiom,
! [A2: set_a,B3: set_a] :
( ! [X3: a] :
( ( member_a @ X3 @ A2 )
=> ( member_a @ X3 @ B3 ) )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% subsetI
thf(fact_237_empty__Collect__eq,axiom,
! [P: list_set_int > $o] :
( ( bot_bo7344713902834881468et_int
= ( collect_list_set_int @ P ) )
= ( ! [X2: list_set_int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_238_empty__Collect__eq,axiom,
! [P: a > $o] :
( ( bot_bot_set_a
= ( collect_a @ P ) )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_239_empty__Collect__eq,axiom,
! [P: set_int > $o] :
( ( bot_bot_set_set_int
= ( collect_set_int @ P ) )
= ( ! [X2: set_int] :
~ ( P @ X2 ) ) ) ).
% empty_Collect_eq
thf(fact_240_Collect__empty__eq,axiom,
! [P: list_set_int > $o] :
( ( ( collect_list_set_int @ P )
= bot_bo7344713902834881468et_int )
= ( ! [X2: list_set_int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_241_Collect__empty__eq,axiom,
! [P: a > $o] :
( ( ( collect_a @ P )
= bot_bot_set_a )
= ( ! [X2: a] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_242_Collect__empty__eq,axiom,
! [P: set_int > $o] :
( ( ( collect_set_int @ P )
= bot_bot_set_set_int )
= ( ! [X2: set_int] :
~ ( P @ X2 ) ) ) ).
% Collect_empty_eq
thf(fact_243_all__not__in__conv,axiom,
! [A2: set_set_set_int] :
( ( ! [X2: set_set_int] :
~ ( member_set_set_int @ X2 @ A2 ) )
= ( A2 = bot_bo2384636101374064866et_int ) ) ).
% all_not_in_conv
thf(fact_244_all__not__in__conv,axiom,
! [A2: set_set_a] :
( ( ! [X2: set_a] :
~ ( member_set_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_a ) ) ).
% all_not_in_conv
thf(fact_245_all__not__in__conv,axiom,
! [A2: set_li8746675981475697647et_int] :
( ( ! [X2: list_set_int > set_list_set_int] :
~ ( member5554658343060950864et_int @ X2 @ A2 ) )
= ( A2 = bot_bo7587639734496279683et_int ) ) ).
% all_not_in_conv
thf(fact_246_all__not__in__conv,axiom,
! [A2: set_list_set_int] :
( ( ! [X2: list_set_int] :
~ ( member_list_set_int @ X2 @ A2 ) )
= ( A2 = bot_bo7344713902834881468et_int ) ) ).
% all_not_in_conv
thf(fact_247_all__not__in__conv,axiom,
! [A2: set_a] :
( ( ! [X2: a] :
~ ( member_a @ X2 @ A2 ) )
= ( A2 = bot_bot_set_a ) ) ).
% all_not_in_conv
thf(fact_248_all__not__in__conv,axiom,
! [A2: set_set_int] :
( ( ! [X2: set_int] :
~ ( member_set_int @ X2 @ A2 ) )
= ( A2 = bot_bot_set_set_int ) ) ).
% all_not_in_conv
thf(fact_249_empty__iff,axiom,
! [C: set_set_int] :
~ ( member_set_set_int @ C @ bot_bo2384636101374064866et_int ) ).
% empty_iff
thf(fact_250_empty__iff,axiom,
! [C: set_a] :
~ ( member_set_a @ C @ bot_bot_set_set_a ) ).
% empty_iff
thf(fact_251_empty__iff,axiom,
! [C: list_set_int > set_list_set_int] :
~ ( member5554658343060950864et_int @ C @ bot_bo7587639734496279683et_int ) ).
% empty_iff
thf(fact_252_empty__iff,axiom,
! [C: list_set_int] :
~ ( member_list_set_int @ C @ bot_bo7344713902834881468et_int ) ).
% empty_iff
thf(fact_253_empty__iff,axiom,
! [C: a] :
~ ( member_a @ C @ bot_bot_set_a ) ).
% empty_iff
thf(fact_254_empty__iff,axiom,
! [C: set_int] :
~ ( member_set_int @ C @ bot_bot_set_set_int ) ).
% empty_iff
thf(fact_255_insert__absorb2,axiom,
! [X: list_set_int,A2: set_list_set_int] :
( ( insert_list_set_int @ X @ ( insert_list_set_int @ X @ A2 ) )
= ( insert_list_set_int @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_256_insert__absorb2,axiom,
! [X: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ X @ A2 ) )
= ( insert_a @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_257_insert__absorb2,axiom,
! [X: set_int,A2: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ X @ A2 ) )
= ( insert_set_int @ X @ A2 ) ) ).
% insert_absorb2
thf(fact_258_insert__iff,axiom,
! [A: list_set_int,B: list_set_int,A2: set_list_set_int] :
( ( member_list_set_int @ A @ ( insert_list_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_list_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_259_insert__iff,axiom,
! [A: set_set_int,B: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ ( insert_set_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_set_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_260_insert__iff,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
= ( ( A = B )
| ( member_set_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_261_insert__iff,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
= ( ( A = B )
| ( member_set_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_262_insert__iff,axiom,
! [A: list_set_int > set_list_set_int,B: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ A @ ( insert1512894373432338025et_int @ B @ A2 ) )
= ( ( A = B )
| ( member5554658343060950864et_int @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_263_insert__iff,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
= ( ( A = B )
| ( member_a @ A @ A2 ) ) ) ).
% insert_iff
thf(fact_264_insertCI,axiom,
! [A: list_set_int,B3: set_list_set_int,B: list_set_int] :
( ( ~ ( member_list_set_int @ A @ B3 )
=> ( A = B ) )
=> ( member_list_set_int @ A @ ( insert_list_set_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_265_insertCI,axiom,
! [A: set_set_int,B3: set_set_set_int,B: set_set_int] :
( ( ~ ( member_set_set_int @ A @ B3 )
=> ( A = B ) )
=> ( member_set_set_int @ A @ ( insert_set_set_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_266_insertCI,axiom,
! [A: set_int,B3: set_set_int,B: set_int] :
( ( ~ ( member_set_int @ A @ B3 )
=> ( A = B ) )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_267_insertCI,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( ~ ( member_set_a @ A @ B3 )
=> ( A = B ) )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_268_insertCI,axiom,
! [A: list_set_int > set_list_set_int,B3: set_li8746675981475697647et_int,B: list_set_int > set_list_set_int] :
( ( ~ ( member5554658343060950864et_int @ A @ B3 )
=> ( A = B ) )
=> ( member5554658343060950864et_int @ A @ ( insert1512894373432338025et_int @ B @ B3 ) ) ) ).
% insertCI
thf(fact_269_insertCI,axiom,
! [A: a,B3: set_a,B: a] :
( ( ~ ( member_a @ A @ B3 )
=> ( A = B ) )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertCI
thf(fact_270_Union__iff,axiom,
! [A2: set_set_int,C3: set_set_set_set_int] :
( ( member_set_set_int @ A2 @ ( comple1756060948637632417et_int @ C3 ) )
= ( ? [X2: set_set_set_int] :
( ( member7356822600254261989et_int @ X2 @ C3 )
& ( member_set_set_int @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_271_Union__iff,axiom,
! [A2: set_a,C3: set_set_set_a] :
( ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C3 ) )
= ( ? [X2: set_set_a] :
( ( member_set_set_a @ X2 @ C3 )
& ( member_set_a @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_272_Union__iff,axiom,
! [A2: list_set_int > set_list_set_int,C3: set_se6837996996060283045et_int] :
( ( member5554658343060950864et_int @ A2 @ ( comple3138861590707219906et_int @ C3 ) )
= ( ? [X2: set_li8746675981475697647et_int] :
( ( member1341228145649960710et_int @ X2 @ C3 )
& ( member5554658343060950864et_int @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_273_Union__iff,axiom,
! [A2: list_set_int,C3: set_set_list_set_int] :
( ( member_list_set_int @ A2 @ ( comple3081917594375107323et_int @ C3 ) )
= ( ? [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ C3 )
& ( member_list_set_int @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_274_Union__iff,axiom,
! [A2: a,C3: set_set_a] :
( ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C3 ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ C3 )
& ( member_a @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_275_Union__iff,axiom,
! [A2: set_int,C3: set_set_set_int] :
( ( member_set_int @ A2 @ ( comple7281953568134767595et_int @ C3 ) )
= ( ? [X2: set_set_int] :
( ( member_set_set_int @ X2 @ C3 )
& ( member_set_int @ A2 @ X2 ) ) ) ) ).
% Union_iff
thf(fact_276_UnionI,axiom,
! [X4: set_set_set_int,C3: set_set_set_set_int,A2: set_set_int] :
( ( member7356822600254261989et_int @ X4 @ C3 )
=> ( ( member_set_set_int @ A2 @ X4 )
=> ( member_set_set_int @ A2 @ ( comple1756060948637632417et_int @ C3 ) ) ) ) ).
% UnionI
thf(fact_277_UnionI,axiom,
! [X4: set_set_a,C3: set_set_set_a,A2: set_a] :
( ( member_set_set_a @ X4 @ C3 )
=> ( ( member_set_a @ A2 @ X4 )
=> ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_278_UnionI,axiom,
! [X4: set_li8746675981475697647et_int,C3: set_se6837996996060283045et_int,A2: list_set_int > set_list_set_int] :
( ( member1341228145649960710et_int @ X4 @ C3 )
=> ( ( member5554658343060950864et_int @ A2 @ X4 )
=> ( member5554658343060950864et_int @ A2 @ ( comple3138861590707219906et_int @ C3 ) ) ) ) ).
% UnionI
thf(fact_279_UnionI,axiom,
! [X4: set_int,C3: set_set_int,A2: int] :
( ( member_set_int @ X4 @ C3 )
=> ( ( member_int @ A2 @ X4 )
=> ( member_int @ A2 @ ( comple3221217463730067765et_int @ C3 ) ) ) ) ).
% UnionI
thf(fact_280_UnionI,axiom,
! [X4: set_list_set_int,C3: set_set_list_set_int,A2: list_set_int] :
( ( member4487585027056014399et_int @ X4 @ C3 )
=> ( ( member_list_set_int @ A2 @ X4 )
=> ( member_list_set_int @ A2 @ ( comple3081917594375107323et_int @ C3 ) ) ) ) ).
% UnionI
thf(fact_281_UnionI,axiom,
! [X4: set_a,C3: set_set_a,A2: a] :
( ( member_set_a @ X4 @ C3 )
=> ( ( member_a @ A2 @ X4 )
=> ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C3 ) ) ) ) ).
% UnionI
thf(fact_282_UnionI,axiom,
! [X4: set_set_int,C3: set_set_set_int,A2: set_int] :
( ( member_set_set_int @ X4 @ C3 )
=> ( ( member_set_int @ A2 @ X4 )
=> ( member_set_int @ A2 @ ( comple7281953568134767595et_int @ C3 ) ) ) ) ).
% UnionI
thf(fact_283_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_list_set_int,P: list_set_int > $o] :
( ( ! [X2: list_set_int] :
( ( member_list_set_int @ X2 @ ( comple3081917594375107323et_int @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A2 )
=> ! [Y4: list_set_int] :
( ( member_list_set_int @ Y4 @ X2 )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_284_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_a,P: a > $o] :
( ( ! [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ! [Y4: a] :
( ( member_a @ Y4 @ X2 )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_285_UN__ball__bex__simps_I1_J,axiom,
! [A2: set_set_set_int,P: set_int > $o] :
( ( ! [X2: set_int] :
( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ A2 ) )
=> ( P @ X2 ) ) )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ! [Y4: set_int] :
( ( member_set_int @ Y4 @ X2 )
=> ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(1)
thf(fact_286_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_list_set_int,P: list_set_int > $o] :
( ( ? [X2: list_set_int] :
( ( member_list_set_int @ X2 @ ( comple3081917594375107323et_int @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A2 )
& ? [Y4: list_set_int] :
( ( member_list_set_int @ Y4 @ X2 )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_287_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_a,P: a > $o] :
( ( ? [X2: a] :
( ( member_a @ X2 @ ( comple2307003609928055243_set_a @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
& ? [Y4: a] :
( ( member_a @ Y4 @ X2 )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_288_UN__ball__bex__simps_I3_J,axiom,
! [A2: set_set_set_int,P: set_int > $o] :
( ( ? [X2: set_int] :
( ( member_set_int @ X2 @ ( comple7281953568134767595et_int @ A2 ) )
& ( P @ X2 ) ) )
= ( ? [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
& ? [Y4: set_int] :
( ( member_set_int @ Y4 @ X2 )
& ( P @ Y4 ) ) ) ) ) ).
% UN_ball_bex_simps(3)
thf(fact_289_pzf_Oring__axioms,axiom,
ring_l5834914389771393517t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.ring_axioms
thf(fact_290_empty__subsetI,axiom,
! [A2: set_list_set_int] : ( ord_le4527193859438875528et_int @ bot_bo7344713902834881468et_int @ A2 ) ).
% empty_subsetI
thf(fact_291_empty__subsetI,axiom,
! [A2: set_set_list_set_int] : ( ord_le8104031629355235774et_int @ bot_bo4503415735588907506et_int @ A2 ) ).
% empty_subsetI
thf(fact_292_empty__subsetI,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ bot_bo2384636101374064866et_int @ A2 ) ).
% empty_subsetI
thf(fact_293_empty__subsetI,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ bot_bot_set_set_int @ A2 ) ).
% empty_subsetI
thf(fact_294_empty__subsetI,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ bot_bot_set_set_a @ A2 ) ).
% empty_subsetI
thf(fact_295_empty__subsetI,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ bot_bot_set_a @ A2 ) ).
% empty_subsetI
thf(fact_296_subset__empty,axiom,
! [A2: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ bot_bo7344713902834881468et_int )
= ( A2 = bot_bo7344713902834881468et_int ) ) ).
% subset_empty
thf(fact_297_subset__empty,axiom,
! [A2: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ bot_bo4503415735588907506et_int )
= ( A2 = bot_bo4503415735588907506et_int ) ) ).
% subset_empty
thf(fact_298_subset__empty,axiom,
! [A2: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ bot_bo2384636101374064866et_int )
= ( A2 = bot_bo2384636101374064866et_int ) ) ).
% subset_empty
thf(fact_299_subset__empty,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ bot_bot_set_set_int )
= ( A2 = bot_bot_set_set_int ) ) ).
% subset_empty
thf(fact_300_subset__empty,axiom,
! [A2: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ bot_bot_set_set_a )
= ( A2 = bot_bot_set_set_a ) ) ).
% subset_empty
thf(fact_301_subset__empty,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ bot_bot_set_a )
= ( A2 = bot_bot_set_a ) ) ).
% subset_empty
thf(fact_302_insert__subset,axiom,
! [X: list_set_int,A2: set_list_set_int,B3: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ ( insert_list_set_int @ X @ A2 ) @ B3 )
= ( ( member_list_set_int @ X @ B3 )
& ( ord_le4527193859438875528et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_303_insert__subset,axiom,
! [X: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int] :
( ( ord_le7834723768769537103et_int @ ( insert1512894373432338025et_int @ X @ A2 ) @ B3 )
= ( ( member5554658343060950864et_int @ X @ B3 )
& ( ord_le7834723768769537103et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_304_insert__subset,axiom,
! [X: set_list_set_int,A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ ( insert4179525362955487576et_int @ X @ A2 ) @ B3 )
= ( ( member4487585027056014399et_int @ X @ B3 )
& ( ord_le8104031629355235774et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_305_insert__subset,axiom,
! [X: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( insert_set_set_int @ X @ A2 ) @ B3 )
= ( ( member_set_set_int @ X @ B3 )
& ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_306_insert__subset,axiom,
! [X: set_int,A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( insert_set_int @ X @ A2 ) @ B3 )
= ( ( member_set_int @ X @ B3 )
& ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_307_insert__subset,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( ( member_set_a @ X @ B3 )
& ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_308_insert__subset,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( ( member_a @ X @ B3 )
& ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% insert_subset
thf(fact_309_singletonI,axiom,
! [A: set_set_int] : ( member_set_set_int @ A @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) ).
% singletonI
thf(fact_310_singletonI,axiom,
! [A: set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) ).
% singletonI
thf(fact_311_singletonI,axiom,
! [A: list_set_int > set_list_set_int] : ( member5554658343060950864et_int @ A @ ( insert1512894373432338025et_int @ A @ bot_bo7587639734496279683et_int ) ) ).
% singletonI
thf(fact_312_singletonI,axiom,
! [A: list_set_int] : ( member_list_set_int @ A @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) ).
% singletonI
thf(fact_313_singletonI,axiom,
! [A: a] : ( member_a @ A @ ( insert_a @ A @ bot_bot_set_a ) ) ).
% singletonI
thf(fact_314_singletonI,axiom,
! [A: set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ).
% singletonI
thf(fact_315_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_list_set_int] :
( ( ( comple3081917594375107323et_int @ A2 )
= bot_bo7344713902834881468et_int )
= ( ! [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A2 )
=> ( X2 = bot_bo7344713902834881468et_int ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_316_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_317_Sup__bot__conv_I1_J,axiom,
! [A2: set_set_set_int] :
( ( ( comple7281953568134767595et_int @ A2 )
= bot_bot_set_set_int )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_int ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_318_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_list_set_int] :
( ( bot_bo7344713902834881468et_int
= ( comple3081917594375107323et_int @ A2 ) )
= ( ! [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A2 )
=> ( X2 = bot_bo7344713902834881468et_int ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_319_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_320_Sup__bot__conv_I2_J,axiom,
! [A2: set_set_set_int] :
( ( bot_bot_set_set_int
= ( comple7281953568134767595et_int @ A2 ) )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_int ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_321_Diff__cancel,axiom,
! [A2: set_list_set_int] :
( ( minus_4948633356380776833et_int @ A2 @ A2 )
= bot_bo7344713902834881468et_int ) ).
% Diff_cancel
thf(fact_322_Diff__cancel,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ A2 )
= bot_bot_set_a ) ).
% Diff_cancel
thf(fact_323_Diff__cancel,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ A2 )
= bot_bot_set_set_int ) ).
% Diff_cancel
thf(fact_324_empty__Diff,axiom,
! [A2: set_list_set_int] :
( ( minus_4948633356380776833et_int @ bot_bo7344713902834881468et_int @ A2 )
= bot_bo7344713902834881468et_int ) ).
% empty_Diff
thf(fact_325_empty__Diff,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ bot_bot_set_a @ A2 )
= bot_bot_set_a ) ).
% empty_Diff
thf(fact_326_empty__Diff,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ bot_bot_set_set_int @ A2 )
= bot_bot_set_set_int ) ).
% empty_Diff
thf(fact_327_Diff__empty,axiom,
! [A2: set_list_set_int] :
( ( minus_4948633356380776833et_int @ A2 @ bot_bo7344713902834881468et_int )
= A2 ) ).
% Diff_empty
thf(fact_328_Diff__empty,axiom,
! [A2: set_a] :
( ( minus_minus_set_a @ A2 @ bot_bot_set_a )
= A2 ) ).
% Diff_empty
thf(fact_329_Diff__empty,axiom,
! [A2: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ bot_bot_set_set_int )
= A2 ) ).
% Diff_empty
thf(fact_330_insert__Diff1,axiom,
! [X: set_set_int,B3: set_set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ X @ B3 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X @ A2 ) @ B3 )
= ( minus_6857623457997529383et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_331_insert__Diff1,axiom,
! [X: set_a,B3: set_set_a,A2: set_set_a] :
( ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_332_insert__Diff1,axiom,
! [X: list_set_int > set_list_set_int,B3: set_li8746675981475697647et_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ X @ B3 )
=> ( ( minus_6937832704872953928et_int @ ( insert1512894373432338025et_int @ X @ A2 ) @ B3 )
= ( minus_6937832704872953928et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_333_insert__Diff1,axiom,
! [X: list_set_int,B3: set_list_set_int,A2: set_list_set_int] :
( ( member_list_set_int @ X @ B3 )
=> ( ( minus_4948633356380776833et_int @ ( insert_list_set_int @ X @ A2 ) @ B3 )
= ( minus_4948633356380776833et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_334_insert__Diff1,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_335_insert__Diff1,axiom,
! [X: set_int,B3: set_set_int,A2: set_set_int] :
( ( member_set_int @ X @ B3 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ B3 )
= ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ) ).
% insert_Diff1
thf(fact_336_Diff__insert0,axiom,
! [X: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ X @ A2 )
=> ( ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ X @ B3 ) )
= ( minus_6857623457997529383et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_337_Diff__insert0,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_338_Diff__insert0,axiom,
! [X: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int] :
( ~ ( member5554658343060950864et_int @ X @ A2 )
=> ( ( minus_6937832704872953928et_int @ A2 @ ( insert1512894373432338025et_int @ X @ B3 ) )
= ( minus_6937832704872953928et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_339_Diff__insert0,axiom,
! [X: list_set_int,A2: set_list_set_int,B3: set_list_set_int] :
( ~ ( member_list_set_int @ X @ A2 )
=> ( ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ X @ B3 ) )
= ( minus_4948633356380776833et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_340_Diff__insert0,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( minus_minus_set_a @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_341_Diff__insert0,axiom,
! [X: set_int,A2: set_set_int,B3: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X @ B3 ) )
= ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ) ).
% Diff_insert0
thf(fact_342_Diff__eq__empty__iff,axiom,
! [A2: set_list_set_int,B3: set_list_set_int] :
( ( ( minus_4948633356380776833et_int @ A2 @ B3 )
= bot_bo7344713902834881468et_int )
= ( ord_le4527193859438875528et_int @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_343_Diff__eq__empty__iff,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( ( minus_8524384805321567159et_int @ A2 @ B3 )
= bot_bo4503415735588907506et_int )
= ( ord_le8104031629355235774et_int @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_344_Diff__eq__empty__iff,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ( minus_6857623457997529383et_int @ A2 @ B3 )
= bot_bo2384636101374064866et_int )
= ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_345_Diff__eq__empty__iff,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ( minus_8897228262479074673et_int @ A2 @ B3 )
= bot_bot_set_set_int )
= ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_346_Diff__eq__empty__iff,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ( minus_5736297505244876581_set_a @ A2 @ B3 )
= bot_bot_set_set_a )
= ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_347_Diff__eq__empty__iff,axiom,
! [A2: set_a,B3: set_a] :
( ( ( minus_minus_set_a @ A2 @ B3 )
= bot_bot_set_a )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% Diff_eq_empty_iff
thf(fact_348_insert__Diff__single,axiom,
! [A: list_set_int,A2: set_list_set_int] :
( ( insert_list_set_int @ A @ ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) )
= ( insert_list_set_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_349_insert__Diff__single,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= ( insert_a @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_350_insert__Diff__single,axiom,
! [A: set_int,A2: set_set_int] :
( ( insert_set_int @ A @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
= ( insert_set_int @ A @ A2 ) ) ).
% insert_Diff_single
thf(fact_351_insert__Diff__if,axiom,
! [X: set_set_int,B3: set_set_set_int,A2: set_set_set_int] :
( ( ( member_set_set_int @ X @ B3 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X @ A2 ) @ B3 )
= ( minus_6857623457997529383et_int @ A2 @ B3 ) ) )
& ( ~ ( member_set_set_int @ X @ B3 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X @ A2 ) @ B3 )
= ( insert_set_set_int @ X @ ( minus_6857623457997529383et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_352_insert__Diff__if,axiom,
! [X: set_a,B3: set_set_a,A2: set_set_a] :
( ( ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_set_a @ X @ B3 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ B3 )
= ( insert_set_a @ X @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_353_insert__Diff__if,axiom,
! [X: list_set_int > set_list_set_int,B3: set_li8746675981475697647et_int,A2: set_li8746675981475697647et_int] :
( ( ( member5554658343060950864et_int @ X @ B3 )
=> ( ( minus_6937832704872953928et_int @ ( insert1512894373432338025et_int @ X @ A2 ) @ B3 )
= ( minus_6937832704872953928et_int @ A2 @ B3 ) ) )
& ( ~ ( member5554658343060950864et_int @ X @ B3 )
=> ( ( minus_6937832704872953928et_int @ ( insert1512894373432338025et_int @ X @ A2 ) @ B3 )
= ( insert1512894373432338025et_int @ X @ ( minus_6937832704872953928et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_354_insert__Diff__if,axiom,
! [X: list_set_int,B3: set_list_set_int,A2: set_list_set_int] :
( ( ( member_list_set_int @ X @ B3 )
=> ( ( minus_4948633356380776833et_int @ ( insert_list_set_int @ X @ A2 ) @ B3 )
= ( minus_4948633356380776833et_int @ A2 @ B3 ) ) )
& ( ~ ( member_list_set_int @ X @ B3 )
=> ( ( minus_4948633356380776833et_int @ ( insert_list_set_int @ X @ A2 ) @ B3 )
= ( insert_list_set_int @ X @ ( minus_4948633356380776833et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_355_insert__Diff__if,axiom,
! [X: a,B3: set_a,A2: set_a] :
( ( ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( minus_minus_set_a @ A2 @ B3 ) ) )
& ( ~ ( member_a @ X @ B3 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ B3 )
= ( insert_a @ X @ ( minus_minus_set_a @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_356_insert__Diff__if,axiom,
! [X: set_int,B3: set_set_int,A2: set_set_int] :
( ( ( member_set_int @ X @ B3 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ B3 )
= ( minus_8897228262479074673et_int @ A2 @ B3 ) ) )
& ( ~ ( member_set_int @ X @ B3 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ B3 )
= ( insert_set_int @ X @ ( minus_8897228262479074673et_int @ A2 @ B3 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_357_Diff__mono,axiom,
! [A2: set_list_set_int,C3: set_list_set_int,D2: set_list_set_int,B3: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ C3 )
=> ( ( ord_le4527193859438875528et_int @ D2 @ B3 )
=> ( ord_le4527193859438875528et_int @ ( minus_4948633356380776833et_int @ A2 @ B3 ) @ ( minus_4948633356380776833et_int @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_358_Diff__mono,axiom,
! [A2: set_set_list_set_int,C3: set_set_list_set_int,D2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ C3 )
=> ( ( ord_le8104031629355235774et_int @ D2 @ B3 )
=> ( ord_le8104031629355235774et_int @ ( minus_8524384805321567159et_int @ A2 @ B3 ) @ ( minus_8524384805321567159et_int @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_359_Diff__mono,axiom,
! [A2: set_set_set_int,C3: set_set_set_int,D2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ C3 )
=> ( ( ord_le4317611570275147438et_int @ D2 @ B3 )
=> ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ B3 ) @ ( minus_6857623457997529383et_int @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_360_Diff__mono,axiom,
! [A2: set_set_int,C3: set_set_int,D2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ C3 )
=> ( ( ord_le4403425263959731960et_int @ D2 @ B3 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B3 ) @ ( minus_8897228262479074673et_int @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_361_Diff__mono,axiom,
! [A2: set_set_a,C3: set_set_a,D2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ C3 )
=> ( ( ord_le3724670747650509150_set_a @ D2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) @ ( minus_5736297505244876581_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_362_Diff__mono,axiom,
! [A2: set_a,C3: set_a,D2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ C3 )
=> ( ( ord_less_eq_set_a @ D2 @ B3 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( minus_minus_set_a @ C3 @ D2 ) ) ) ) ).
% Diff_mono
thf(fact_363_Diff__subset,axiom,
! [A2: set_list_set_int,B3: set_list_set_int] : ( ord_le4527193859438875528et_int @ ( minus_4948633356380776833et_int @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_364_Diff__subset,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] : ( ord_le8104031629355235774et_int @ ( minus_8524384805321567159et_int @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_365_Diff__subset,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] : ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_366_Diff__subset,axiom,
! [A2: set_set_int,B3: set_set_int] : ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_367_Diff__subset,axiom,
! [A2: set_set_a,B3: set_set_a] : ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_368_Diff__subset,axiom,
! [A2: set_a,B3: set_a] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ A2 ) ).
% Diff_subset
thf(fact_369_double__diff,axiom,
! [A2: set_list_set_int,B3: set_list_set_int,C3: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ B3 )
=> ( ( ord_le4527193859438875528et_int @ B3 @ C3 )
=> ( ( minus_4948633356380776833et_int @ B3 @ ( minus_4948633356380776833et_int @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_370_double__diff,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int,C3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ( ( ord_le8104031629355235774et_int @ B3 @ C3 )
=> ( ( minus_8524384805321567159et_int @ B3 @ ( minus_8524384805321567159et_int @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_371_double__diff,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( ord_le4317611570275147438et_int @ B3 @ C3 )
=> ( ( minus_6857623457997529383et_int @ B3 @ ( minus_6857623457997529383et_int @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_372_double__diff,axiom,
! [A2: set_set_int,B3: set_set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ C3 )
=> ( ( minus_8897228262479074673et_int @ B3 @ ( minus_8897228262479074673et_int @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_373_double__diff,axiom,
! [A2: set_set_a,B3: set_set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C3 )
=> ( ( minus_5736297505244876581_set_a @ B3 @ ( minus_5736297505244876581_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_374_double__diff,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ( minus_minus_set_a @ B3 @ ( minus_minus_set_a @ C3 @ A2 ) )
= A2 ) ) ) ).
% double_diff
thf(fact_375_bot__set__def,axiom,
( bot_bo7344713902834881468et_int
= ( collect_list_set_int @ bot_bo1744859523148385737_int_o ) ) ).
% bot_set_def
thf(fact_376_bot__set__def,axiom,
( bot_bot_set_a
= ( collect_a @ bot_bot_a_o ) ) ).
% bot_set_def
thf(fact_377_bot__set__def,axiom,
( bot_bot_set_set_int
= ( collect_set_int @ bot_bot_set_int_o ) ) ).
% bot_set_def
thf(fact_378_subset__Diff__insert,axiom,
! [A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int,X: list_set_int > set_list_set_int,C3: set_li8746675981475697647et_int] :
( ( ord_le7834723768769537103et_int @ A2 @ ( minus_6937832704872953928et_int @ B3 @ ( insert1512894373432338025et_int @ X @ C3 ) ) )
= ( ( ord_le7834723768769537103et_int @ A2 @ ( minus_6937832704872953928et_int @ B3 @ C3 ) )
& ~ ( member5554658343060950864et_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_379_subset__Diff__insert,axiom,
! [A2: set_list_set_int,B3: set_list_set_int,X: list_set_int,C3: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ ( minus_4948633356380776833et_int @ B3 @ ( insert_list_set_int @ X @ C3 ) ) )
= ( ( ord_le4527193859438875528et_int @ A2 @ ( minus_4948633356380776833et_int @ B3 @ C3 ) )
& ~ ( member_list_set_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_380_subset__Diff__insert,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int,X: set_list_set_int,C3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ ( minus_8524384805321567159et_int @ B3 @ ( insert4179525362955487576et_int @ X @ C3 ) ) )
= ( ( ord_le8104031629355235774et_int @ A2 @ ( minus_8524384805321567159et_int @ B3 @ C3 ) )
& ~ ( member4487585027056014399et_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_381_subset__Diff__insert,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,X: set_set_int,C3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ ( minus_6857623457997529383et_int @ B3 @ ( insert_set_set_int @ X @ C3 ) ) )
= ( ( ord_le4317611570275147438et_int @ A2 @ ( minus_6857623457997529383et_int @ B3 @ C3 ) )
& ~ ( member_set_set_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_382_subset__Diff__insert,axiom,
! [A2: set_set_int,B3: set_set_int,X: set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B3 @ ( insert_set_int @ X @ C3 ) ) )
= ( ( ord_le4403425263959731960et_int @ A2 @ ( minus_8897228262479074673et_int @ B3 @ C3 ) )
& ~ ( member_set_int @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_383_subset__Diff__insert,axiom,
! [A2: set_set_a,B3: set_set_a,X: set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B3 @ ( insert_set_a @ X @ C3 ) ) )
= ( ( ord_le3724670747650509150_set_a @ A2 @ ( minus_5736297505244876581_set_a @ B3 @ C3 ) )
& ~ ( member_set_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_384_subset__Diff__insert,axiom,
! [A2: set_a,B3: set_a,X: a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ ( insert_a @ X @ C3 ) ) )
= ( ( ord_less_eq_set_a @ A2 @ ( minus_minus_set_a @ B3 @ C3 ) )
& ~ ( member_a @ X @ A2 ) ) ) ).
% subset_Diff_insert
thf(fact_385_Diff__insert,axiom,
! [A2: set_list_set_int,A: list_set_int,B3: set_list_set_int] :
( ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ A @ B3 ) )
= ( minus_4948633356380776833et_int @ ( minus_4948633356380776833et_int @ A2 @ B3 ) @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) ) ).
% Diff_insert
thf(fact_386_Diff__insert,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ B3 ) @ ( insert_a @ A @ bot_bot_set_a ) ) ) ).
% Diff_insert
thf(fact_387_Diff__insert,axiom,
! [A2: set_set_int,A: set_int,B3: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B3 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ B3 ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) ) ).
% Diff_insert
thf(fact_388_insert__Diff,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ( ( insert_set_set_int @ A @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_389_insert__Diff,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ A @ bot_bot_set_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_390_insert__Diff,axiom,
! [A: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ A @ A2 )
=> ( ( insert1512894373432338025et_int @ A @ ( minus_6937832704872953928et_int @ A2 @ ( insert1512894373432338025et_int @ A @ bot_bo7587639734496279683et_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_391_insert__Diff,axiom,
! [A: list_set_int,A2: set_list_set_int] :
( ( member_list_set_int @ A @ A2 )
=> ( ( insert_list_set_int @ A @ ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_392_insert__Diff,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_393_insert__Diff,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( insert_set_int @ A @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) )
= A2 ) ) ).
% insert_Diff
thf(fact_394_Diff__insert2,axiom,
! [A2: set_list_set_int,A: list_set_int,B3: set_list_set_int] :
( ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ A @ B3 ) )
= ( minus_4948633356380776833et_int @ ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_395_Diff__insert2,axiom,
! [A2: set_a,A: a,B3: set_a] :
( ( minus_minus_set_a @ A2 @ ( insert_a @ A @ B3 ) )
= ( minus_minus_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ A @ bot_bot_set_a ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_396_Diff__insert2,axiom,
! [A2: set_set_int,A: set_int,B3: set_set_int] :
( ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ B3 ) )
= ( minus_8897228262479074673et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ B3 ) ) ).
% Diff_insert2
thf(fact_397_Diff__insert__absorb,axiom,
! [X: set_set_int,A2: set_set_set_int] :
( ~ ( member_set_set_int @ X @ A2 )
=> ( ( minus_6857623457997529383et_int @ ( insert_set_set_int @ X @ A2 ) @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_398_Diff__insert__absorb,axiom,
! [X: set_a,A2: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( minus_5736297505244876581_set_a @ ( insert_set_a @ X @ A2 ) @ ( insert_set_a @ X @ bot_bot_set_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_399_Diff__insert__absorb,axiom,
! [X: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ~ ( member5554658343060950864et_int @ X @ A2 )
=> ( ( minus_6937832704872953928et_int @ ( insert1512894373432338025et_int @ X @ A2 ) @ ( insert1512894373432338025et_int @ X @ bot_bo7587639734496279683et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_400_Diff__insert__absorb,axiom,
! [X: list_set_int,A2: set_list_set_int] :
( ~ ( member_list_set_int @ X @ A2 )
=> ( ( minus_4948633356380776833et_int @ ( insert_list_set_int @ X @ A2 ) @ ( insert_list_set_int @ X @ bot_bo7344713902834881468et_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_401_Diff__insert__absorb,axiom,
! [X: a,A2: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( minus_minus_set_a @ ( insert_a @ X @ A2 ) @ ( insert_a @ X @ bot_bot_set_a ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_402_Diff__insert__absorb,axiom,
! [X: set_int,A2: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ( minus_8897228262479074673et_int @ ( insert_set_int @ X @ A2 ) @ ( insert_set_int @ X @ bot_bot_set_set_int ) )
= A2 ) ) ).
% Diff_insert_absorb
thf(fact_403_subset__insert__iff,axiom,
! [A2: set_li8746675981475697647et_int,X: list_set_int > set_list_set_int,B3: set_li8746675981475697647et_int] :
( ( ord_le7834723768769537103et_int @ A2 @ ( insert1512894373432338025et_int @ X @ B3 ) )
= ( ( ( member5554658343060950864et_int @ X @ A2 )
=> ( ord_le7834723768769537103et_int @ ( minus_6937832704872953928et_int @ A2 @ ( insert1512894373432338025et_int @ X @ bot_bo7587639734496279683et_int ) ) @ B3 ) )
& ( ~ ( member5554658343060950864et_int @ X @ A2 )
=> ( ord_le7834723768769537103et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_404_subset__insert__iff,axiom,
! [A2: set_list_set_int,X: list_set_int,B3: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ ( insert_list_set_int @ X @ B3 ) )
= ( ( ( member_list_set_int @ X @ A2 )
=> ( ord_le4527193859438875528et_int @ ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ X @ bot_bo7344713902834881468et_int ) ) @ B3 ) )
& ( ~ ( member_list_set_int @ X @ A2 )
=> ( ord_le4527193859438875528et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_405_subset__insert__iff,axiom,
! [A2: set_set_list_set_int,X: set_list_set_int,B3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ ( insert4179525362955487576et_int @ X @ B3 ) )
= ( ( ( member4487585027056014399et_int @ X @ A2 )
=> ( ord_le8104031629355235774et_int @ ( minus_8524384805321567159et_int @ A2 @ ( insert4179525362955487576et_int @ X @ bot_bo4503415735588907506et_int ) ) @ B3 ) )
& ( ~ ( member4487585027056014399et_int @ X @ A2 )
=> ( ord_le8104031629355235774et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_406_subset__insert__iff,axiom,
! [A2: set_set_set_int,X: set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X @ B3 ) )
= ( ( ( member_set_set_int @ X @ A2 )
=> ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) ) @ B3 ) )
& ( ~ ( member_set_set_int @ X @ A2 )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_407_subset__insert__iff,axiom,
! [A2: set_set_int,X: set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ B3 ) )
= ( ( ( member_set_int @ X @ A2 )
=> ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X @ bot_bot_set_set_int ) ) @ B3 ) )
& ( ~ ( member_set_int @ X @ A2 )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_408_subset__insert__iff,axiom,
! [A2: set_set_a,X: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( ( ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 ) )
& ( ~ ( member_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_409_subset__insert__iff,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ( ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 ) )
& ( ~ ( member_a @ X @ A2 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ) ).
% subset_insert_iff
thf(fact_410_Diff__single__insert,axiom,
! [A2: set_list_set_int,X: list_set_int,B3: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ ( minus_4948633356380776833et_int @ A2 @ ( insert_list_set_int @ X @ bot_bo7344713902834881468et_int ) ) @ B3 )
=> ( ord_le4527193859438875528et_int @ A2 @ ( insert_list_set_int @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_411_Diff__single__insert,axiom,
! [A2: set_set_list_set_int,X: set_list_set_int,B3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ ( minus_8524384805321567159et_int @ A2 @ ( insert4179525362955487576et_int @ X @ bot_bo4503415735588907506et_int ) ) @ B3 )
=> ( ord_le8104031629355235774et_int @ A2 @ ( insert4179525362955487576et_int @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_412_Diff__single__insert,axiom,
! [A2: set_set_set_int,X: set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( minus_6857623457997529383et_int @ A2 @ ( insert_set_set_int @ X @ bot_bo2384636101374064866et_int ) ) @ B3 )
=> ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_413_Diff__single__insert,axiom,
! [A2: set_set_int,X: set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( minus_8897228262479074673et_int @ A2 @ ( insert_set_int @ X @ bot_bot_set_set_int ) ) @ B3 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_414_Diff__single__insert,axiom,
! [A2: set_set_a,X: set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( minus_5736297505244876581_set_a @ A2 @ ( insert_set_a @ X @ bot_bot_set_set_a ) ) @ B3 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_415_Diff__single__insert,axiom,
! [A2: set_a,X: a,B3: set_a] :
( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A2 @ ( insert_a @ X @ bot_bot_set_a ) ) @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) ) ) ).
% Diff_single_insert
thf(fact_416_Collect__mono__iff,axiom,
! [P: set_list_set_int > $o,Q2: set_list_set_int > $o] :
( ( ord_le8104031629355235774et_int @ ( collec8150298066296752253et_int @ P ) @ ( collec8150298066296752253et_int @ Q2 ) )
= ( ! [X2: set_list_set_int] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_417_Collect__mono__iff,axiom,
! [P: set_set_int > $o,Q2: set_set_int > $o] :
( ( ord_le4317611570275147438et_int @ ( collect_set_set_int @ P ) @ ( collect_set_set_int @ Q2 ) )
= ( ! [X2: set_set_int] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_418_Collect__mono__iff,axiom,
! [P: set_int > $o,Q2: set_int > $o] :
( ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q2 ) )
= ( ! [X2: set_int] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_419_Collect__mono__iff,axiom,
! [P: set_a > $o,Q2: set_a > $o] :
( ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q2 ) )
= ( ! [X2: set_a] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_420_Collect__mono__iff,axiom,
! [P: a > $o,Q2: a > $o] :
( ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q2 ) )
= ( ! [X2: a] :
( ( P @ X2 )
=> ( Q2 @ X2 ) ) ) ) ).
% Collect_mono_iff
thf(fact_421_set__eq__subset,axiom,
( ( ^ [Y5: set_set_list_set_int,Z2: set_set_list_set_int] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_list_set_int,B4: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A5 @ B4 )
& ( ord_le8104031629355235774et_int @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_422_set__eq__subset,axiom,
( ( ^ [Y5: set_set_set_int,Z2: set_set_set_int] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_set_int,B4: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A5 @ B4 )
& ( ord_le4317611570275147438et_int @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_423_set__eq__subset,axiom,
( ( ^ [Y5: set_set_int,Z2: set_set_int] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_int,B4: set_set_int] :
( ( ord_le4403425263959731960et_int @ A5 @ B4 )
& ( ord_le4403425263959731960et_int @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_424_set__eq__subset,axiom,
( ( ^ [Y5: set_set_a,Z2: set_set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_set_a,B4: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A5 @ B4 )
& ( ord_le3724670747650509150_set_a @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_425_set__eq__subset,axiom,
( ( ^ [Y5: set_a,Z2: set_a] : ( Y5 = Z2 ) )
= ( ^ [A5: set_a,B4: set_a] :
( ( ord_less_eq_set_a @ A5 @ B4 )
& ( ord_less_eq_set_a @ B4 @ A5 ) ) ) ) ).
% set_eq_subset
thf(fact_426_subset__trans,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int,C3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ( ( ord_le8104031629355235774et_int @ B3 @ C3 )
=> ( ord_le8104031629355235774et_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_427_subset__trans,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( ord_le4317611570275147438et_int @ B3 @ C3 )
=> ( ord_le4317611570275147438et_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_428_subset__trans,axiom,
! [A2: set_set_int,B3: set_set_int,C3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( ord_le4403425263959731960et_int @ B3 @ C3 )
=> ( ord_le4403425263959731960et_int @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_429_subset__trans,axiom,
! [A2: set_set_a,B3: set_set_a,C3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( ord_le3724670747650509150_set_a @ B3 @ C3 )
=> ( ord_le3724670747650509150_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_430_subset__trans,axiom,
! [A2: set_a,B3: set_a,C3: set_a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( ord_less_eq_set_a @ B3 @ C3 )
=> ( ord_less_eq_set_a @ A2 @ C3 ) ) ) ).
% subset_trans
thf(fact_431_Collect__mono,axiom,
! [P: set_list_set_int > $o,Q2: set_list_set_int > $o] :
( ! [X3: set_list_set_int] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le8104031629355235774et_int @ ( collec8150298066296752253et_int @ P ) @ ( collec8150298066296752253et_int @ Q2 ) ) ) ).
% Collect_mono
thf(fact_432_Collect__mono,axiom,
! [P: set_set_int > $o,Q2: set_set_int > $o] :
( ! [X3: set_set_int] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le4317611570275147438et_int @ ( collect_set_set_int @ P ) @ ( collect_set_set_int @ Q2 ) ) ) ).
% Collect_mono
thf(fact_433_Collect__mono,axiom,
! [P: set_int > $o,Q2: set_int > $o] :
( ! [X3: set_int] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le4403425263959731960et_int @ ( collect_set_int @ P ) @ ( collect_set_int @ Q2 ) ) ) ).
% Collect_mono
thf(fact_434_Collect__mono,axiom,
! [P: set_a > $o,Q2: set_a > $o] :
( ! [X3: set_a] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_le3724670747650509150_set_a @ ( collect_set_a @ P ) @ ( collect_set_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_435_Collect__mono,axiom,
! [P: a > $o,Q2: a > $o] :
( ! [X3: a] :
( ( P @ X3 )
=> ( Q2 @ X3 ) )
=> ( ord_less_eq_set_a @ ( collect_a @ P ) @ ( collect_a @ Q2 ) ) ) ).
% Collect_mono
thf(fact_436_subset__refl,axiom,
! [A2: set_set_list_set_int] : ( ord_le8104031629355235774et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_437_subset__refl,axiom,
! [A2: set_set_set_int] : ( ord_le4317611570275147438et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_438_subset__refl,axiom,
! [A2: set_set_int] : ( ord_le4403425263959731960et_int @ A2 @ A2 ) ).
% subset_refl
thf(fact_439_subset__refl,axiom,
! [A2: set_set_a] : ( ord_le3724670747650509150_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_440_subset__refl,axiom,
! [A2: set_a] : ( ord_less_eq_set_a @ A2 @ A2 ) ).
% subset_refl
thf(fact_441_subset__iff,axiom,
( ord_le7834723768769537103et_int
= ( ^ [A5: set_li8746675981475697647et_int,B4: set_li8746675981475697647et_int] :
! [T: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ T @ A5 )
=> ( member5554658343060950864et_int @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_442_subset__iff,axiom,
( ord_le8104031629355235774et_int
= ( ^ [A5: set_set_list_set_int,B4: set_set_list_set_int] :
! [T: set_list_set_int] :
( ( member4487585027056014399et_int @ T @ A5 )
=> ( member4487585027056014399et_int @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_443_subset__iff,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A5: set_set_set_int,B4: set_set_set_int] :
! [T: set_set_int] :
( ( member_set_set_int @ T @ A5 )
=> ( member_set_set_int @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_444_subset__iff,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A5: set_set_int,B4: set_set_int] :
! [T: set_int] :
( ( member_set_int @ T @ A5 )
=> ( member_set_int @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_445_subset__iff,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
! [T: set_a] :
( ( member_set_a @ T @ A5 )
=> ( member_set_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_446_subset__iff,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [T: a] :
( ( member_a @ T @ A5 )
=> ( member_a @ T @ B4 ) ) ) ) ).
% subset_iff
thf(fact_447_equalityD2,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( A2 = B3 )
=> ( ord_le8104031629355235774et_int @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_448_equalityD2,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( A2 = B3 )
=> ( ord_le4317611570275147438et_int @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_449_equalityD2,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( A2 = B3 )
=> ( ord_le4403425263959731960et_int @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_450_equalityD2,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( A2 = B3 )
=> ( ord_le3724670747650509150_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_451_equalityD2,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ B3 @ A2 ) ) ).
% equalityD2
thf(fact_452_equalityD1,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( A2 = B3 )
=> ( ord_le8104031629355235774et_int @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_453_equalityD1,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( A2 = B3 )
=> ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_454_equalityD1,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( A2 = B3 )
=> ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_455_equalityD1,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( A2 = B3 )
=> ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_456_equalityD1,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ( ord_less_eq_set_a @ A2 @ B3 ) ) ).
% equalityD1
thf(fact_457_subset__eq,axiom,
( ord_le7834723768769537103et_int
= ( ^ [A5: set_li8746675981475697647et_int,B4: set_li8746675981475697647et_int] :
! [X2: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ X2 @ A5 )
=> ( member5554658343060950864et_int @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_458_subset__eq,axiom,
( ord_le8104031629355235774et_int
= ( ^ [A5: set_set_list_set_int,B4: set_set_list_set_int] :
! [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A5 )
=> ( member4487585027056014399et_int @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_459_subset__eq,axiom,
( ord_le4317611570275147438et_int
= ( ^ [A5: set_set_set_int,B4: set_set_set_int] :
! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A5 )
=> ( member_set_set_int @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_460_subset__eq,axiom,
( ord_le4403425263959731960et_int
= ( ^ [A5: set_set_int,B4: set_set_int] :
! [X2: set_int] :
( ( member_set_int @ X2 @ A5 )
=> ( member_set_int @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_461_subset__eq,axiom,
( ord_le3724670747650509150_set_a
= ( ^ [A5: set_set_a,B4: set_set_a] :
! [X2: set_a] :
( ( member_set_a @ X2 @ A5 )
=> ( member_set_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_462_subset__eq,axiom,
( ord_less_eq_set_a
= ( ^ [A5: set_a,B4: set_a] :
! [X2: a] :
( ( member_a @ X2 @ A5 )
=> ( member_a @ X2 @ B4 ) ) ) ) ).
% subset_eq
thf(fact_463_equalityE,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( A2 = B3 )
=> ~ ( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ~ ( ord_le8104031629355235774et_int @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_464_equalityE,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( A2 = B3 )
=> ~ ( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ~ ( ord_le4317611570275147438et_int @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_465_equalityE,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( A2 = B3 )
=> ~ ( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ~ ( ord_le4403425263959731960et_int @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_466_equalityE,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ~ ( ord_le3724670747650509150_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_467_equalityE,axiom,
! [A2: set_a,B3: set_a] :
( ( A2 = B3 )
=> ~ ( ( ord_less_eq_set_a @ A2 @ B3 )
=> ~ ( ord_less_eq_set_a @ B3 @ A2 ) ) ) ).
% equalityE
thf(fact_468_subsetD,axiom,
! [A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int,C: list_set_int > set_list_set_int] :
( ( ord_le7834723768769537103et_int @ A2 @ B3 )
=> ( ( member5554658343060950864et_int @ C @ A2 )
=> ( member5554658343060950864et_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_469_subsetD,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int,C: set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ( ( member4487585027056014399et_int @ C @ A2 )
=> ( member4487585027056014399et_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_470_subsetD,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,C: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( member_set_set_int @ C @ A2 )
=> ( member_set_set_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_471_subsetD,axiom,
! [A2: set_set_int,B3: set_set_int,C: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( member_set_int @ C @ A2 )
=> ( member_set_int @ C @ B3 ) ) ) ).
% subsetD
thf(fact_472_subsetD,axiom,
! [A2: set_set_a,B3: set_set_a,C: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ C @ A2 )
=> ( member_set_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_473_subsetD,axiom,
! [A2: set_a,B3: set_a,C: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ C @ A2 )
=> ( member_a @ C @ B3 ) ) ) ).
% subsetD
thf(fact_474_in__mono,axiom,
! [A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int,X: list_set_int > set_list_set_int] :
( ( ord_le7834723768769537103et_int @ A2 @ B3 )
=> ( ( member5554658343060950864et_int @ X @ A2 )
=> ( member5554658343060950864et_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_475_in__mono,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int,X: set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ( ( member4487585027056014399et_int @ X @ A2 )
=> ( member4487585027056014399et_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_476_in__mono,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,X: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ( member_set_set_int @ X @ A2 )
=> ( member_set_set_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_477_in__mono,axiom,
! [A2: set_set_int,B3: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ( member_set_int @ X @ A2 )
=> ( member_set_int @ X @ B3 ) ) ) ).
% in_mono
thf(fact_478_in__mono,axiom,
! [A2: set_set_a,B3: set_set_a,X: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ( member_set_a @ X @ A2 )
=> ( member_set_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_479_in__mono,axiom,
! [A2: set_a,B3: set_a,X: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ( member_a @ X @ A2 )
=> ( member_a @ X @ B3 ) ) ) ).
% in_mono
thf(fact_480_ex__in__conv,axiom,
! [A2: set_set_set_int] :
( ( ? [X2: set_set_int] : ( member_set_set_int @ X2 @ A2 ) )
= ( A2 != bot_bo2384636101374064866et_int ) ) ).
% ex_in_conv
thf(fact_481_ex__in__conv,axiom,
! [A2: set_set_a] :
( ( ? [X2: set_a] : ( member_set_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_a ) ) ).
% ex_in_conv
thf(fact_482_ex__in__conv,axiom,
! [A2: set_li8746675981475697647et_int] :
( ( ? [X2: list_set_int > set_list_set_int] : ( member5554658343060950864et_int @ X2 @ A2 ) )
= ( A2 != bot_bo7587639734496279683et_int ) ) ).
% ex_in_conv
thf(fact_483_ex__in__conv,axiom,
! [A2: set_list_set_int] :
( ( ? [X2: list_set_int] : ( member_list_set_int @ X2 @ A2 ) )
= ( A2 != bot_bo7344713902834881468et_int ) ) ).
% ex_in_conv
thf(fact_484_ex__in__conv,axiom,
! [A2: set_a] :
( ( ? [X2: a] : ( member_a @ X2 @ A2 ) )
= ( A2 != bot_bot_set_a ) ) ).
% ex_in_conv
thf(fact_485_ex__in__conv,axiom,
! [A2: set_set_int] :
( ( ? [X2: set_int] : ( member_set_int @ X2 @ A2 ) )
= ( A2 != bot_bot_set_set_int ) ) ).
% ex_in_conv
thf(fact_486_equals0I,axiom,
! [A2: set_set_set_int] :
( ! [Y6: set_set_int] :
~ ( member_set_set_int @ Y6 @ A2 )
=> ( A2 = bot_bo2384636101374064866et_int ) ) ).
% equals0I
thf(fact_487_equals0I,axiom,
! [A2: set_set_a] :
( ! [Y6: set_a] :
~ ( member_set_a @ Y6 @ A2 )
=> ( A2 = bot_bot_set_set_a ) ) ).
% equals0I
thf(fact_488_equals0I,axiom,
! [A2: set_li8746675981475697647et_int] :
( ! [Y6: list_set_int > set_list_set_int] :
~ ( member5554658343060950864et_int @ Y6 @ A2 )
=> ( A2 = bot_bo7587639734496279683et_int ) ) ).
% equals0I
thf(fact_489_equals0I,axiom,
! [A2: set_list_set_int] :
( ! [Y6: list_set_int] :
~ ( member_list_set_int @ Y6 @ A2 )
=> ( A2 = bot_bo7344713902834881468et_int ) ) ).
% equals0I
thf(fact_490_equals0I,axiom,
! [A2: set_a] :
( ! [Y6: a] :
~ ( member_a @ Y6 @ A2 )
=> ( A2 = bot_bot_set_a ) ) ).
% equals0I
thf(fact_491_equals0I,axiom,
! [A2: set_set_int] :
( ! [Y6: set_int] :
~ ( member_set_int @ Y6 @ A2 )
=> ( A2 = bot_bot_set_set_int ) ) ).
% equals0I
thf(fact_492_equals0D,axiom,
! [A2: set_set_set_int,A: set_set_int] :
( ( A2 = bot_bo2384636101374064866et_int )
=> ~ ( member_set_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_493_equals0D,axiom,
! [A2: set_set_a,A: set_a] :
( ( A2 = bot_bot_set_set_a )
=> ~ ( member_set_a @ A @ A2 ) ) ).
% equals0D
thf(fact_494_equals0D,axiom,
! [A2: set_li8746675981475697647et_int,A: list_set_int > set_list_set_int] :
( ( A2 = bot_bo7587639734496279683et_int )
=> ~ ( member5554658343060950864et_int @ A @ A2 ) ) ).
% equals0D
thf(fact_495_equals0D,axiom,
! [A2: set_list_set_int,A: list_set_int] :
( ( A2 = bot_bo7344713902834881468et_int )
=> ~ ( member_list_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_496_equals0D,axiom,
! [A2: set_a,A: a] :
( ( A2 = bot_bot_set_a )
=> ~ ( member_a @ A @ A2 ) ) ).
% equals0D
thf(fact_497_equals0D,axiom,
! [A2: set_set_int,A: set_int] :
( ( A2 = bot_bot_set_set_int )
=> ~ ( member_set_int @ A @ A2 ) ) ).
% equals0D
thf(fact_498_emptyE,axiom,
! [A: set_set_int] :
~ ( member_set_set_int @ A @ bot_bo2384636101374064866et_int ) ).
% emptyE
thf(fact_499_emptyE,axiom,
! [A: set_a] :
~ ( member_set_a @ A @ bot_bot_set_set_a ) ).
% emptyE
thf(fact_500_emptyE,axiom,
! [A: list_set_int > set_list_set_int] :
~ ( member5554658343060950864et_int @ A @ bot_bo7587639734496279683et_int ) ).
% emptyE
thf(fact_501_emptyE,axiom,
! [A: list_set_int] :
~ ( member_list_set_int @ A @ bot_bo7344713902834881468et_int ) ).
% emptyE
thf(fact_502_emptyE,axiom,
! [A: a] :
~ ( member_a @ A @ bot_bot_set_a ) ).
% emptyE
thf(fact_503_emptyE,axiom,
! [A: set_int] :
~ ( member_set_int @ A @ bot_bot_set_set_int ) ).
% emptyE
thf(fact_504_mk__disjoint__insert,axiom,
! [A: list_set_int,A2: set_list_set_int] :
( ( member_list_set_int @ A @ A2 )
=> ? [B5: set_list_set_int] :
( ( A2
= ( insert_list_set_int @ A @ B5 ) )
& ~ ( member_list_set_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_505_mk__disjoint__insert,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ? [B5: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ A @ B5 ) )
& ~ ( member_set_set_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_506_mk__disjoint__insert,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ? [B5: set_set_int] :
( ( A2
= ( insert_set_int @ A @ B5 ) )
& ~ ( member_set_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_507_mk__disjoint__insert,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ? [B5: set_set_a] :
( ( A2
= ( insert_set_a @ A @ B5 ) )
& ~ ( member_set_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_508_mk__disjoint__insert,axiom,
! [A: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ A @ A2 )
=> ? [B5: set_li8746675981475697647et_int] :
( ( A2
= ( insert1512894373432338025et_int @ A @ B5 ) )
& ~ ( member5554658343060950864et_int @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_509_mk__disjoint__insert,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ? [B5: set_a] :
( ( A2
= ( insert_a @ A @ B5 ) )
& ~ ( member_a @ A @ B5 ) ) ) ).
% mk_disjoint_insert
thf(fact_510_insert__commute,axiom,
! [X: list_set_int,Y: list_set_int,A2: set_list_set_int] :
( ( insert_list_set_int @ X @ ( insert_list_set_int @ Y @ A2 ) )
= ( insert_list_set_int @ Y @ ( insert_list_set_int @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_511_insert__commute,axiom,
! [X: a,Y: a,A2: set_a] :
( ( insert_a @ X @ ( insert_a @ Y @ A2 ) )
= ( insert_a @ Y @ ( insert_a @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_512_insert__commute,axiom,
! [X: set_int,Y: set_int,A2: set_set_int] :
( ( insert_set_int @ X @ ( insert_set_int @ Y @ A2 ) )
= ( insert_set_int @ Y @ ( insert_set_int @ X @ A2 ) ) ) ).
% insert_commute
thf(fact_513_insert__eq__iff,axiom,
! [A: list_set_int,A2: set_list_set_int,B: list_set_int,B3: set_list_set_int] :
( ~ ( member_list_set_int @ A @ A2 )
=> ( ~ ( member_list_set_int @ B @ B3 )
=> ( ( ( insert_list_set_int @ A @ A2 )
= ( insert_list_set_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_list_set_int] :
( ( A2
= ( insert_list_set_int @ B @ C4 ) )
& ~ ( member_list_set_int @ B @ C4 )
& ( B3
= ( insert_list_set_int @ A @ C4 ) )
& ~ ( member_list_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_514_insert__eq__iff,axiom,
! [A: set_set_int,A2: set_set_set_int,B: set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ A @ A2 )
=> ( ~ ( member_set_set_int @ B @ B3 )
=> ( ( ( insert_set_set_int @ A @ A2 )
= ( insert_set_set_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ B @ C4 ) )
& ~ ( member_set_set_int @ B @ C4 )
& ( B3
= ( insert_set_set_int @ A @ C4 ) )
& ~ ( member_set_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_515_insert__eq__iff,axiom,
! [A: set_int,A2: set_set_int,B: set_int,B3: set_set_int] :
( ~ ( member_set_int @ A @ A2 )
=> ( ~ ( member_set_int @ B @ B3 )
=> ( ( ( insert_set_int @ A @ A2 )
= ( insert_set_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_set_int] :
( ( A2
= ( insert_set_int @ B @ C4 ) )
& ~ ( member_set_int @ B @ C4 )
& ( B3
= ( insert_set_int @ A @ C4 ) )
& ~ ( member_set_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_516_insert__eq__iff,axiom,
! [A: set_a,A2: set_set_a,B: set_a,B3: set_set_a] :
( ~ ( member_set_a @ A @ A2 )
=> ( ~ ( member_set_a @ B @ B3 )
=> ( ( ( insert_set_a @ A @ A2 )
= ( insert_set_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_set_a] :
( ( A2
= ( insert_set_a @ B @ C4 ) )
& ~ ( member_set_a @ B @ C4 )
& ( B3
= ( insert_set_a @ A @ C4 ) )
& ~ ( member_set_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_517_insert__eq__iff,axiom,
! [A: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int,B: list_set_int > set_list_set_int,B3: set_li8746675981475697647et_int] :
( ~ ( member5554658343060950864et_int @ A @ A2 )
=> ( ~ ( member5554658343060950864et_int @ B @ B3 )
=> ( ( ( insert1512894373432338025et_int @ A @ A2 )
= ( insert1512894373432338025et_int @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_li8746675981475697647et_int] :
( ( A2
= ( insert1512894373432338025et_int @ B @ C4 ) )
& ~ ( member5554658343060950864et_int @ B @ C4 )
& ( B3
= ( insert1512894373432338025et_int @ A @ C4 ) )
& ~ ( member5554658343060950864et_int @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_518_insert__eq__iff,axiom,
! [A: a,A2: set_a,B: a,B3: set_a] :
( ~ ( member_a @ A @ A2 )
=> ( ~ ( member_a @ B @ B3 )
=> ( ( ( insert_a @ A @ A2 )
= ( insert_a @ B @ B3 ) )
= ( ( ( A = B )
=> ( A2 = B3 ) )
& ( ( A != B )
=> ? [C4: set_a] :
( ( A2
= ( insert_a @ B @ C4 ) )
& ~ ( member_a @ B @ C4 )
& ( B3
= ( insert_a @ A @ C4 ) )
& ~ ( member_a @ A @ C4 ) ) ) ) ) ) ) ).
% insert_eq_iff
thf(fact_519_insert__absorb,axiom,
! [A: list_set_int,A2: set_list_set_int] :
( ( member_list_set_int @ A @ A2 )
=> ( ( insert_list_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_520_insert__absorb,axiom,
! [A: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ A2 )
=> ( ( insert_set_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_521_insert__absorb,axiom,
! [A: set_int,A2: set_set_int] :
( ( member_set_int @ A @ A2 )
=> ( ( insert_set_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_522_insert__absorb,axiom,
! [A: set_a,A2: set_set_a] :
( ( member_set_a @ A @ A2 )
=> ( ( insert_set_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_523_insert__absorb,axiom,
! [A: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ A @ A2 )
=> ( ( insert1512894373432338025et_int @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_524_insert__absorb,axiom,
! [A: a,A2: set_a] :
( ( member_a @ A @ A2 )
=> ( ( insert_a @ A @ A2 )
= A2 ) ) ).
% insert_absorb
thf(fact_525_insert__ident,axiom,
! [X: list_set_int,A2: set_list_set_int,B3: set_list_set_int] :
( ~ ( member_list_set_int @ X @ A2 )
=> ( ~ ( member_list_set_int @ X @ B3 )
=> ( ( ( insert_list_set_int @ X @ A2 )
= ( insert_list_set_int @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_526_insert__ident,axiom,
! [X: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ X @ A2 )
=> ( ~ ( member_set_set_int @ X @ B3 )
=> ( ( ( insert_set_set_int @ X @ A2 )
= ( insert_set_set_int @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_527_insert__ident,axiom,
! [X: set_int,A2: set_set_int,B3: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ~ ( member_set_int @ X @ B3 )
=> ( ( ( insert_set_int @ X @ A2 )
= ( insert_set_int @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_528_insert__ident,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ~ ( member_set_a @ X @ B3 )
=> ( ( ( insert_set_a @ X @ A2 )
= ( insert_set_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_529_insert__ident,axiom,
! [X: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int] :
( ~ ( member5554658343060950864et_int @ X @ A2 )
=> ( ~ ( member5554658343060950864et_int @ X @ B3 )
=> ( ( ( insert1512894373432338025et_int @ X @ A2 )
= ( insert1512894373432338025et_int @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_530_insert__ident,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ~ ( member_a @ X @ B3 )
=> ( ( ( insert_a @ X @ A2 )
= ( insert_a @ X @ B3 ) )
= ( A2 = B3 ) ) ) ) ).
% insert_ident
thf(fact_531_Set_Oset__insert,axiom,
! [X: list_set_int,A2: set_list_set_int] :
( ( member_list_set_int @ X @ A2 )
=> ~ ! [B5: set_list_set_int] :
( ( A2
= ( insert_list_set_int @ X @ B5 ) )
=> ( member_list_set_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_532_Set_Oset__insert,axiom,
! [X: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ X @ A2 )
=> ~ ! [B5: set_set_set_int] :
( ( A2
= ( insert_set_set_int @ X @ B5 ) )
=> ( member_set_set_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_533_Set_Oset__insert,axiom,
! [X: set_int,A2: set_set_int] :
( ( member_set_int @ X @ A2 )
=> ~ ! [B5: set_set_int] :
( ( A2
= ( insert_set_int @ X @ B5 ) )
=> ( member_set_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_534_Set_Oset__insert,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ~ ! [B5: set_set_a] :
( ( A2
= ( insert_set_a @ X @ B5 ) )
=> ( member_set_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_535_Set_Oset__insert,axiom,
! [X: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ X @ A2 )
=> ~ ! [B5: set_li8746675981475697647et_int] :
( ( A2
= ( insert1512894373432338025et_int @ X @ B5 ) )
=> ( member5554658343060950864et_int @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_536_Set_Oset__insert,axiom,
! [X: a,A2: set_a] :
( ( member_a @ X @ A2 )
=> ~ ! [B5: set_a] :
( ( A2
= ( insert_a @ X @ B5 ) )
=> ( member_a @ X @ B5 ) ) ) ).
% Set.set_insert
thf(fact_537_insertI2,axiom,
! [A: list_set_int,B3: set_list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ B3 )
=> ( member_list_set_int @ A @ ( insert_list_set_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_538_insertI2,axiom,
! [A: set_set_int,B3: set_set_set_int,B: set_set_int] :
( ( member_set_set_int @ A @ B3 )
=> ( member_set_set_int @ A @ ( insert_set_set_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_539_insertI2,axiom,
! [A: set_int,B3: set_set_int,B: set_int] :
( ( member_set_int @ A @ B3 )
=> ( member_set_int @ A @ ( insert_set_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_540_insertI2,axiom,
! [A: set_a,B3: set_set_a,B: set_a] :
( ( member_set_a @ A @ B3 )
=> ( member_set_a @ A @ ( insert_set_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_541_insertI2,axiom,
! [A: list_set_int > set_list_set_int,B3: set_li8746675981475697647et_int,B: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ A @ B3 )
=> ( member5554658343060950864et_int @ A @ ( insert1512894373432338025et_int @ B @ B3 ) ) ) ).
% insertI2
thf(fact_542_insertI2,axiom,
! [A: a,B3: set_a,B: a] :
( ( member_a @ A @ B3 )
=> ( member_a @ A @ ( insert_a @ B @ B3 ) ) ) ).
% insertI2
thf(fact_543_insertI1,axiom,
! [A: list_set_int,B3: set_list_set_int] : ( member_list_set_int @ A @ ( insert_list_set_int @ A @ B3 ) ) ).
% insertI1
thf(fact_544_insertI1,axiom,
! [A: set_set_int,B3: set_set_set_int] : ( member_set_set_int @ A @ ( insert_set_set_int @ A @ B3 ) ) ).
% insertI1
thf(fact_545_insertI1,axiom,
! [A: set_int,B3: set_set_int] : ( member_set_int @ A @ ( insert_set_int @ A @ B3 ) ) ).
% insertI1
thf(fact_546_insertI1,axiom,
! [A: set_a,B3: set_set_a] : ( member_set_a @ A @ ( insert_set_a @ A @ B3 ) ) ).
% insertI1
thf(fact_547_insertI1,axiom,
! [A: list_set_int > set_list_set_int,B3: set_li8746675981475697647et_int] : ( member5554658343060950864et_int @ A @ ( insert1512894373432338025et_int @ A @ B3 ) ) ).
% insertI1
thf(fact_548_insertI1,axiom,
! [A: a,B3: set_a] : ( member_a @ A @ ( insert_a @ A @ B3 ) ) ).
% insertI1
thf(fact_549_insertE,axiom,
! [A: list_set_int,B: list_set_int,A2: set_list_set_int] :
( ( member_list_set_int @ A @ ( insert_list_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_list_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_550_insertE,axiom,
! [A: set_set_int,B: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ A @ ( insert_set_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_551_insertE,axiom,
! [A: set_int,B: set_int,A2: set_set_int] :
( ( member_set_int @ A @ ( insert_set_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_552_insertE,axiom,
! [A: set_a,B: set_a,A2: set_set_a] :
( ( member_set_a @ A @ ( insert_set_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_set_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_553_insertE,axiom,
! [A: list_set_int > set_list_set_int,B: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ A @ ( insert1512894373432338025et_int @ B @ A2 ) )
=> ( ( A != B )
=> ( member5554658343060950864et_int @ A @ A2 ) ) ) ).
% insertE
thf(fact_554_insertE,axiom,
! [A: a,B: a,A2: set_a] :
( ( member_a @ A @ ( insert_a @ B @ A2 ) )
=> ( ( A != B )
=> ( member_a @ A @ A2 ) ) ) ).
% insertE
thf(fact_555_UnionE,axiom,
! [A2: int,C3: set_set_int] :
( ( member_int @ A2 @ ( comple3221217463730067765et_int @ C3 ) )
=> ~ ! [X7: set_int] :
( ( member_int @ A2 @ X7 )
=> ~ ( member_set_int @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_556_UnionE,axiom,
! [A2: set_set_int,C3: set_set_set_set_int] :
( ( member_set_set_int @ A2 @ ( comple1756060948637632417et_int @ C3 ) )
=> ~ ! [X7: set_set_set_int] :
( ( member_set_set_int @ A2 @ X7 )
=> ~ ( member7356822600254261989et_int @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_557_UnionE,axiom,
! [A2: set_a,C3: set_set_set_a] :
( ( member_set_a @ A2 @ ( comple3958522678809307947_set_a @ C3 ) )
=> ~ ! [X7: set_set_a] :
( ( member_set_a @ A2 @ X7 )
=> ~ ( member_set_set_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_558_UnionE,axiom,
! [A2: list_set_int > set_list_set_int,C3: set_se6837996996060283045et_int] :
( ( member5554658343060950864et_int @ A2 @ ( comple3138861590707219906et_int @ C3 ) )
=> ~ ! [X7: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ A2 @ X7 )
=> ~ ( member1341228145649960710et_int @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_559_UnionE,axiom,
! [A2: list_set_int,C3: set_set_list_set_int] :
( ( member_list_set_int @ A2 @ ( comple3081917594375107323et_int @ C3 ) )
=> ~ ! [X7: set_list_set_int] :
( ( member_list_set_int @ A2 @ X7 )
=> ~ ( member4487585027056014399et_int @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_560_UnionE,axiom,
! [A2: a,C3: set_set_a] :
( ( member_a @ A2 @ ( comple2307003609928055243_set_a @ C3 ) )
=> ~ ! [X7: set_a] :
( ( member_a @ A2 @ X7 )
=> ~ ( member_set_a @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_561_UnionE,axiom,
! [A2: set_int,C3: set_set_set_int] :
( ( member_set_int @ A2 @ ( comple7281953568134767595et_int @ C3 ) )
=> ~ ! [X7: set_set_int] :
( ( member_set_int @ A2 @ X7 )
=> ~ ( member_set_set_int @ X7 @ C3 ) ) ) ).
% UnionE
thf(fact_562_insert__mono,axiom,
! [C3: set_list_set_int,D2: set_list_set_int,A: list_set_int] :
( ( ord_le4527193859438875528et_int @ C3 @ D2 )
=> ( ord_le4527193859438875528et_int @ ( insert_list_set_int @ A @ C3 ) @ ( insert_list_set_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_563_insert__mono,axiom,
! [C3: set_set_list_set_int,D2: set_set_list_set_int,A: set_list_set_int] :
( ( ord_le8104031629355235774et_int @ C3 @ D2 )
=> ( ord_le8104031629355235774et_int @ ( insert4179525362955487576et_int @ A @ C3 ) @ ( insert4179525362955487576et_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_564_insert__mono,axiom,
! [C3: set_set_set_int,D2: set_set_set_int,A: set_set_int] :
( ( ord_le4317611570275147438et_int @ C3 @ D2 )
=> ( ord_le4317611570275147438et_int @ ( insert_set_set_int @ A @ C3 ) @ ( insert_set_set_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_565_insert__mono,axiom,
! [C3: set_set_int,D2: set_set_int,A: set_int] :
( ( ord_le4403425263959731960et_int @ C3 @ D2 )
=> ( ord_le4403425263959731960et_int @ ( insert_set_int @ A @ C3 ) @ ( insert_set_int @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_566_insert__mono,axiom,
! [C3: set_set_a,D2: set_set_a,A: set_a] :
( ( ord_le3724670747650509150_set_a @ C3 @ D2 )
=> ( ord_le3724670747650509150_set_a @ ( insert_set_a @ A @ C3 ) @ ( insert_set_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_567_insert__mono,axiom,
! [C3: set_a,D2: set_a,A: a] :
( ( ord_less_eq_set_a @ C3 @ D2 )
=> ( ord_less_eq_set_a @ ( insert_a @ A @ C3 ) @ ( insert_a @ A @ D2 ) ) ) ).
% insert_mono
thf(fact_568_subset__insert,axiom,
! [X: list_set_int,A2: set_list_set_int,B3: set_list_set_int] :
( ~ ( member_list_set_int @ X @ A2 )
=> ( ( ord_le4527193859438875528et_int @ A2 @ ( insert_list_set_int @ X @ B3 ) )
= ( ord_le4527193859438875528et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_569_subset__insert,axiom,
! [X: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int] :
( ~ ( member5554658343060950864et_int @ X @ A2 )
=> ( ( ord_le7834723768769537103et_int @ A2 @ ( insert1512894373432338025et_int @ X @ B3 ) )
= ( ord_le7834723768769537103et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_570_subset__insert,axiom,
! [X: set_list_set_int,A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ~ ( member4487585027056014399et_int @ X @ A2 )
=> ( ( ord_le8104031629355235774et_int @ A2 @ ( insert4179525362955487576et_int @ X @ B3 ) )
= ( ord_le8104031629355235774et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_571_subset__insert,axiom,
! [X: set_set_int,A2: set_set_set_int,B3: set_set_set_int] :
( ~ ( member_set_set_int @ X @ A2 )
=> ( ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ X @ B3 ) )
= ( ord_le4317611570275147438et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_572_subset__insert,axiom,
! [X: set_int,A2: set_set_int,B3: set_set_int] :
( ~ ( member_set_int @ X @ A2 )
=> ( ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ X @ B3 ) )
= ( ord_le4403425263959731960et_int @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_573_subset__insert,axiom,
! [X: set_a,A2: set_set_a,B3: set_set_a] :
( ~ ( member_set_a @ X @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ X @ B3 ) )
= ( ord_le3724670747650509150_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_574_subset__insert,axiom,
! [X: a,A2: set_a,B3: set_a] :
( ~ ( member_a @ X @ A2 )
=> ( ( ord_less_eq_set_a @ A2 @ ( insert_a @ X @ B3 ) )
= ( ord_less_eq_set_a @ A2 @ B3 ) ) ) ).
% subset_insert
thf(fact_575_subset__insertI,axiom,
! [B3: set_list_set_int,A: list_set_int] : ( ord_le4527193859438875528et_int @ B3 @ ( insert_list_set_int @ A @ B3 ) ) ).
% subset_insertI
thf(fact_576_subset__insertI,axiom,
! [B3: set_set_list_set_int,A: set_list_set_int] : ( ord_le8104031629355235774et_int @ B3 @ ( insert4179525362955487576et_int @ A @ B3 ) ) ).
% subset_insertI
thf(fact_577_subset__insertI,axiom,
! [B3: set_set_set_int,A: set_set_int] : ( ord_le4317611570275147438et_int @ B3 @ ( insert_set_set_int @ A @ B3 ) ) ).
% subset_insertI
thf(fact_578_subset__insertI,axiom,
! [B3: set_set_int,A: set_int] : ( ord_le4403425263959731960et_int @ B3 @ ( insert_set_int @ A @ B3 ) ) ).
% subset_insertI
thf(fact_579_subset__insertI,axiom,
! [B3: set_set_a,A: set_a] : ( ord_le3724670747650509150_set_a @ B3 @ ( insert_set_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_580_subset__insertI,axiom,
! [B3: set_a,A: a] : ( ord_less_eq_set_a @ B3 @ ( insert_a @ A @ B3 ) ) ).
% subset_insertI
thf(fact_581_subset__insertI2,axiom,
! [A2: set_list_set_int,B3: set_list_set_int,B: list_set_int] :
( ( ord_le4527193859438875528et_int @ A2 @ B3 )
=> ( ord_le4527193859438875528et_int @ A2 @ ( insert_list_set_int @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_582_subset__insertI2,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int,B: set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ( ord_le8104031629355235774et_int @ A2 @ ( insert4179525362955487576et_int @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_583_subset__insertI2,axiom,
! [A2: set_set_set_int,B3: set_set_set_int,B: set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ord_le4317611570275147438et_int @ A2 @ ( insert_set_set_int @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_584_subset__insertI2,axiom,
! [A2: set_set_int,B3: set_set_int,B: set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ord_le4403425263959731960et_int @ A2 @ ( insert_set_int @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_585_subset__insertI2,axiom,
! [A2: set_set_a,B3: set_set_a,B: set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ A2 @ ( insert_set_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_586_subset__insertI2,axiom,
! [A2: set_a,B3: set_a,B: a] :
( ( ord_less_eq_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ A2 @ ( insert_a @ B @ B3 ) ) ) ).
% subset_insertI2
thf(fact_587_Sup__eqI,axiom,
! [A2: set_set_int,X: set_int] :
( ! [Y6: set_int] :
( ( member_set_int @ Y6 @ A2 )
=> ( ord_less_eq_set_int @ Y6 @ X ) )
=> ( ! [Y6: set_int] :
( ! [Z3: set_int] :
( ( member_set_int @ Z3 @ A2 )
=> ( ord_less_eq_set_int @ Z3 @ Y6 ) )
=> ( ord_less_eq_set_int @ X @ Y6 ) )
=> ( ( comple3221217463730067765et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_588_Sup__eqI,axiom,
! [A2: set_li8746675981475697647et_int,X: list_set_int > set_list_set_int] :
( ! [Y6: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ Y6 @ A2 )
=> ( ord_le4060840825881956505et_int @ Y6 @ X ) )
=> ( ! [Y6: list_set_int > set_list_set_int] :
( ! [Z3: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ Z3 @ A2 )
=> ( ord_le4060840825881956505et_int @ Z3 @ Y6 ) )
=> ( ord_le4060840825881956505et_int @ X @ Y6 ) )
=> ( ( comple4195941263769839116et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_589_Sup__eqI,axiom,
! [A2: set_se8558717844084009108et_int,X: set_set_list_set_int] :
( ! [Y6: set_set_list_set_int] :
( ( member8308387497365429877et_int @ Y6 @ A2 )
=> ( ord_le8104031629355235774et_int @ Y6 @ X ) )
=> ( ! [Y6: set_set_list_set_int] :
( ! [Z3: set_set_list_set_int] :
( ( member8308387497365429877et_int @ Z3 @ A2 )
=> ( ord_le8104031629355235774et_int @ Z3 @ Y6 ) )
=> ( ord_le8104031629355235774et_int @ X @ Y6 ) )
=> ( ( comple148233303328197937et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_590_Sup__eqI,axiom,
! [A2: set_set_set_set_int,X: set_set_set_int] :
( ! [Y6: set_set_set_int] :
( ( member7356822600254261989et_int @ Y6 @ A2 )
=> ( ord_le4317611570275147438et_int @ Y6 @ X ) )
=> ( ! [Y6: set_set_set_int] :
( ! [Z3: set_set_set_int] :
( ( member7356822600254261989et_int @ Z3 @ A2 )
=> ( ord_le4317611570275147438et_int @ Z3 @ Y6 ) )
=> ( ord_le4317611570275147438et_int @ X @ Y6 ) )
=> ( ( comple1756060948637632417et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_591_Sup__eqI,axiom,
! [A2: set_set_set_a,X: set_set_a] :
( ! [Y6: set_set_a] :
( ( member_set_set_a @ Y6 @ A2 )
=> ( ord_le3724670747650509150_set_a @ Y6 @ X ) )
=> ( ! [Y6: set_set_a] :
( ! [Z3: set_set_a] :
( ( member_set_set_a @ Z3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ Z3 @ Y6 ) )
=> ( ord_le3724670747650509150_set_a @ X @ Y6 ) )
=> ( ( comple3958522678809307947_set_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_592_Sup__eqI,axiom,
! [A2: set_set_list_set_int,X: set_list_set_int] :
( ! [Y6: set_list_set_int] :
( ( member4487585027056014399et_int @ Y6 @ A2 )
=> ( ord_le4527193859438875528et_int @ Y6 @ X ) )
=> ( ! [Y6: set_list_set_int] :
( ! [Z3: set_list_set_int] :
( ( member4487585027056014399et_int @ Z3 @ A2 )
=> ( ord_le4527193859438875528et_int @ Z3 @ Y6 ) )
=> ( ord_le4527193859438875528et_int @ X @ Y6 ) )
=> ( ( comple3081917594375107323et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_593_Sup__eqI,axiom,
! [A2: set_set_a,X: set_a] :
( ! [Y6: set_a] :
( ( member_set_a @ Y6 @ A2 )
=> ( ord_less_eq_set_a @ Y6 @ X ) )
=> ( ! [Y6: set_a] :
( ! [Z3: set_a] :
( ( member_set_a @ Z3 @ A2 )
=> ( ord_less_eq_set_a @ Z3 @ Y6 ) )
=> ( ord_less_eq_set_a @ X @ Y6 ) )
=> ( ( comple2307003609928055243_set_a @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_594_Sup__eqI,axiom,
! [A2: set_set_set_int,X: set_set_int] :
( ! [Y6: set_set_int] :
( ( member_set_set_int @ Y6 @ A2 )
=> ( ord_le4403425263959731960et_int @ Y6 @ X ) )
=> ( ! [Y6: set_set_int] :
( ! [Z3: set_set_int] :
( ( member_set_set_int @ Z3 @ A2 )
=> ( ord_le4403425263959731960et_int @ Z3 @ Y6 ) )
=> ( ord_le4403425263959731960et_int @ X @ Y6 ) )
=> ( ( comple7281953568134767595et_int @ A2 )
= X ) ) ) ).
% Sup_eqI
thf(fact_595_Sup__mono,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ! [A3: set_int] :
( ( member_set_int @ A3 @ A2 )
=> ? [X5: set_int] :
( ( member_set_int @ X5 @ B3 )
& ( ord_less_eq_set_int @ A3 @ X5 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_596_Sup__mono,axiom,
! [A2: set_li8746675981475697647et_int,B3: set_li8746675981475697647et_int] :
( ! [A3: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ A3 @ A2 )
=> ? [X5: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ X5 @ B3 )
& ( ord_le4060840825881956505et_int @ A3 @ X5 ) ) )
=> ( ord_le4060840825881956505et_int @ ( comple4195941263769839116et_int @ A2 ) @ ( comple4195941263769839116et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_597_Sup__mono,axiom,
! [A2: set_se8558717844084009108et_int,B3: set_se8558717844084009108et_int] :
( ! [A3: set_set_list_set_int] :
( ( member8308387497365429877et_int @ A3 @ A2 )
=> ? [X5: set_set_list_set_int] :
( ( member8308387497365429877et_int @ X5 @ B3 )
& ( ord_le8104031629355235774et_int @ A3 @ X5 ) ) )
=> ( ord_le8104031629355235774et_int @ ( comple148233303328197937et_int @ A2 ) @ ( comple148233303328197937et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_598_Sup__mono,axiom,
! [A2: set_set_set_set_int,B3: set_set_set_set_int] :
( ! [A3: set_set_set_int] :
( ( member7356822600254261989et_int @ A3 @ A2 )
=> ? [X5: set_set_set_int] :
( ( member7356822600254261989et_int @ X5 @ B3 )
& ( ord_le4317611570275147438et_int @ A3 @ X5 ) ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ ( comple1756060948637632417et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_599_Sup__mono,axiom,
! [A2: set_set_set_a,B3: set_set_set_a] :
( ! [A3: set_set_a] :
( ( member_set_set_a @ A3 @ A2 )
=> ? [X5: set_set_a] :
( ( member_set_set_a @ X5 @ B3 )
& ( ord_le3724670747650509150_set_a @ A3 @ X5 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ ( comple3958522678809307947_set_a @ B3 ) ) ) ).
% Sup_mono
thf(fact_600_Sup__mono,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ! [A3: set_list_set_int] :
( ( member4487585027056014399et_int @ A3 @ A2 )
=> ? [X5: set_list_set_int] :
( ( member4487585027056014399et_int @ X5 @ B3 )
& ( ord_le4527193859438875528et_int @ A3 @ X5 ) ) )
=> ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ A2 ) @ ( comple3081917594375107323et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_601_Sup__mono,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [A3: set_a] :
( ( member_set_a @ A3 @ A2 )
=> ? [X5: set_a] :
( ( member_set_a @ X5 @ B3 )
& ( ord_less_eq_set_a @ A3 @ X5 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Sup_mono
thf(fact_602_Sup__mono,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ! [A3: set_set_int] :
( ( member_set_set_int @ A3 @ A2 )
=> ? [X5: set_set_int] :
( ( member_set_set_int @ X5 @ B3 )
& ( ord_le4403425263959731960et_int @ A3 @ X5 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Sup_mono
thf(fact_603_Sup__least,axiom,
! [A2: set_set_int,Z: set_int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
=> ( ord_less_eq_set_int @ X3 @ Z ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_604_Sup__least,axiom,
! [A2: set_li8746675981475697647et_int,Z: list_set_int > set_list_set_int] :
( ! [X3: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ X3 @ A2 )
=> ( ord_le4060840825881956505et_int @ X3 @ Z ) )
=> ( ord_le4060840825881956505et_int @ ( comple4195941263769839116et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_605_Sup__least,axiom,
! [A2: set_se8558717844084009108et_int,Z: set_set_list_set_int] :
( ! [X3: set_set_list_set_int] :
( ( member8308387497365429877et_int @ X3 @ A2 )
=> ( ord_le8104031629355235774et_int @ X3 @ Z ) )
=> ( ord_le8104031629355235774et_int @ ( comple148233303328197937et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_606_Sup__least,axiom,
! [A2: set_set_set_set_int,Z: set_set_set_int] :
( ! [X3: set_set_set_int] :
( ( member7356822600254261989et_int @ X3 @ A2 )
=> ( ord_le4317611570275147438et_int @ X3 @ Z ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_607_Sup__least,axiom,
! [A2: set_set_set_a,Z: set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_608_Sup__least,axiom,
! [A2: set_set_list_set_int,Z: set_list_set_int] :
( ! [X3: set_list_set_int] :
( ( member4487585027056014399et_int @ X3 @ A2 )
=> ( ord_le4527193859438875528et_int @ X3 @ Z ) )
=> ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_609_Sup__least,axiom,
! [A2: set_set_a,Z: set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ( ord_less_eq_set_a @ X3 @ Z ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_610_Sup__least,axiom,
! [A2: set_set_set_int,Z: set_set_int] :
( ! [X3: set_set_int] :
( ( member_set_set_int @ X3 @ A2 )
=> ( ord_le4403425263959731960et_int @ X3 @ Z ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ Z ) ) ).
% Sup_least
thf(fact_611_Sup__upper,axiom,
! [X: set_int,A2: set_set_int] :
( ( member_set_int @ X @ A2 )
=> ( ord_less_eq_set_int @ X @ ( comple3221217463730067765et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_612_Sup__upper,axiom,
! [X: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ X @ A2 )
=> ( ord_le4060840825881956505et_int @ X @ ( comple4195941263769839116et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_613_Sup__upper,axiom,
! [X: set_set_list_set_int,A2: set_se8558717844084009108et_int] :
( ( member8308387497365429877et_int @ X @ A2 )
=> ( ord_le8104031629355235774et_int @ X @ ( comple148233303328197937et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_614_Sup__upper,axiom,
! [X: set_set_set_int,A2: set_set_set_set_int] :
( ( member7356822600254261989et_int @ X @ A2 )
=> ( ord_le4317611570275147438et_int @ X @ ( comple1756060948637632417et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_615_Sup__upper,axiom,
! [X: set_set_a,A2: set_set_set_a] :
( ( member_set_set_a @ X @ A2 )
=> ( ord_le3724670747650509150_set_a @ X @ ( comple3958522678809307947_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_616_Sup__upper,axiom,
! [X: set_list_set_int,A2: set_set_list_set_int] :
( ( member4487585027056014399et_int @ X @ A2 )
=> ( ord_le4527193859438875528et_int @ X @ ( comple3081917594375107323et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_617_Sup__upper,axiom,
! [X: set_a,A2: set_set_a] :
( ( member_set_a @ X @ A2 )
=> ( ord_less_eq_set_a @ X @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Sup_upper
thf(fact_618_Sup__upper,axiom,
! [X: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ X @ A2 )
=> ( ord_le4403425263959731960et_int @ X @ ( comple7281953568134767595et_int @ A2 ) ) ) ).
% Sup_upper
thf(fact_619_Sup__le__iff,axiom,
! [A2: set_se8558717844084009108et_int,B: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ ( comple148233303328197937et_int @ A2 ) @ B )
= ( ! [X2: set_set_list_set_int] :
( ( member8308387497365429877et_int @ X2 @ A2 )
=> ( ord_le8104031629355235774et_int @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_620_Sup__le__iff,axiom,
! [A2: set_set_set_set_int,B: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ B )
= ( ! [X2: set_set_set_int] :
( ( member7356822600254261989et_int @ X2 @ A2 )
=> ( ord_le4317611570275147438et_int @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_621_Sup__le__iff,axiom,
! [A2: set_set_set_a,B: set_set_a] :
( ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ B )
= ( ! [X2: set_set_a] :
( ( member_set_set_a @ X2 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_622_Sup__le__iff,axiom,
! [A2: set_set_list_set_int,B: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ A2 ) @ B )
= ( ! [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A2 )
=> ( ord_le4527193859438875528et_int @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_623_Sup__le__iff,axiom,
! [A2: set_set_a,B: set_a] :
( ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ B )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( ord_less_eq_set_a @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_624_Sup__le__iff,axiom,
! [A2: set_set_set_int,B: set_set_int] :
( ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ B )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( ord_le4403425263959731960et_int @ X2 @ B ) ) ) ) ).
% Sup_le_iff
thf(fact_625_Sup__upper2,axiom,
! [U: set_int,A2: set_set_int,V2: set_int] :
( ( member_set_int @ U @ A2 )
=> ( ( ord_less_eq_set_int @ V2 @ U )
=> ( ord_less_eq_set_int @ V2 @ ( comple3221217463730067765et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_626_Sup__upper2,axiom,
! [U: list_set_int > set_list_set_int,A2: set_li8746675981475697647et_int,V2: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ U @ A2 )
=> ( ( ord_le4060840825881956505et_int @ V2 @ U )
=> ( ord_le4060840825881956505et_int @ V2 @ ( comple4195941263769839116et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_627_Sup__upper2,axiom,
! [U: set_set_list_set_int,A2: set_se8558717844084009108et_int,V2: set_set_list_set_int] :
( ( member8308387497365429877et_int @ U @ A2 )
=> ( ( ord_le8104031629355235774et_int @ V2 @ U )
=> ( ord_le8104031629355235774et_int @ V2 @ ( comple148233303328197937et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_628_Sup__upper2,axiom,
! [U: set_set_set_int,A2: set_set_set_set_int,V2: set_set_set_int] :
( ( member7356822600254261989et_int @ U @ A2 )
=> ( ( ord_le4317611570275147438et_int @ V2 @ U )
=> ( ord_le4317611570275147438et_int @ V2 @ ( comple1756060948637632417et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_629_Sup__upper2,axiom,
! [U: set_set_a,A2: set_set_set_a,V2: set_set_a] :
( ( member_set_set_a @ U @ A2 )
=> ( ( ord_le3724670747650509150_set_a @ V2 @ U )
=> ( ord_le3724670747650509150_set_a @ V2 @ ( comple3958522678809307947_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_630_Sup__upper2,axiom,
! [U: set_list_set_int,A2: set_set_list_set_int,V2: set_list_set_int] :
( ( member4487585027056014399et_int @ U @ A2 )
=> ( ( ord_le4527193859438875528et_int @ V2 @ U )
=> ( ord_le4527193859438875528et_int @ V2 @ ( comple3081917594375107323et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_631_Sup__upper2,axiom,
! [U: set_a,A2: set_set_a,V2: set_a] :
( ( member_set_a @ U @ A2 )
=> ( ( ord_less_eq_set_a @ V2 @ U )
=> ( ord_less_eq_set_a @ V2 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_632_Sup__upper2,axiom,
! [U: set_set_int,A2: set_set_set_int,V2: set_set_int] :
( ( member_set_set_int @ U @ A2 )
=> ( ( ord_le4403425263959731960et_int @ V2 @ U )
=> ( ord_le4403425263959731960et_int @ V2 @ ( comple7281953568134767595et_int @ A2 ) ) ) ) ).
% Sup_upper2
thf(fact_633_cSup__eq__maximum,axiom,
! [Z: set_int,X4: set_set_int] :
( ( member_set_int @ Z @ X4 )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ X4 )
=> ( ord_less_eq_set_int @ X3 @ Z ) )
=> ( ( comple3221217463730067765et_int @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_634_cSup__eq__maximum,axiom,
! [Z: list_set_int > set_list_set_int,X4: set_li8746675981475697647et_int] :
( ( member5554658343060950864et_int @ Z @ X4 )
=> ( ! [X3: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ X3 @ X4 )
=> ( ord_le4060840825881956505et_int @ X3 @ Z ) )
=> ( ( comple4195941263769839116et_int @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_635_cSup__eq__maximum,axiom,
! [Z: set_set_list_set_int,X4: set_se8558717844084009108et_int] :
( ( member8308387497365429877et_int @ Z @ X4 )
=> ( ! [X3: set_set_list_set_int] :
( ( member8308387497365429877et_int @ X3 @ X4 )
=> ( ord_le8104031629355235774et_int @ X3 @ Z ) )
=> ( ( comple148233303328197937et_int @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_636_cSup__eq__maximum,axiom,
! [Z: set_set_set_int,X4: set_set_set_set_int] :
( ( member7356822600254261989et_int @ Z @ X4 )
=> ( ! [X3: set_set_set_int] :
( ( member7356822600254261989et_int @ X3 @ X4 )
=> ( ord_le4317611570275147438et_int @ X3 @ Z ) )
=> ( ( comple1756060948637632417et_int @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_637_cSup__eq__maximum,axiom,
! [Z: set_set_a,X4: set_set_set_a] :
( ( member_set_set_a @ Z @ X4 )
=> ( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ X4 )
=> ( ord_le3724670747650509150_set_a @ X3 @ Z ) )
=> ( ( comple3958522678809307947_set_a @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_638_cSup__eq__maximum,axiom,
! [Z: set_list_set_int,X4: set_set_list_set_int] :
( ( member4487585027056014399et_int @ Z @ X4 )
=> ( ! [X3: set_list_set_int] :
( ( member4487585027056014399et_int @ X3 @ X4 )
=> ( ord_le4527193859438875528et_int @ X3 @ Z ) )
=> ( ( comple3081917594375107323et_int @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_639_cSup__eq__maximum,axiom,
! [Z: set_a,X4: set_set_a] :
( ( member_set_a @ Z @ X4 )
=> ( ! [X3: set_a] :
( ( member_set_a @ X3 @ X4 )
=> ( ord_less_eq_set_a @ X3 @ Z ) )
=> ( ( comple2307003609928055243_set_a @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_640_cSup__eq__maximum,axiom,
! [Z: set_set_int,X4: set_set_set_int] :
( ( member_set_set_int @ Z @ X4 )
=> ( ! [X3: set_set_int] :
( ( member_set_set_int @ X3 @ X4 )
=> ( ord_le4403425263959731960et_int @ X3 @ Z ) )
=> ( ( comple7281953568134767595et_int @ X4 )
= Z ) ) ) ).
% cSup_eq_maximum
thf(fact_641_singletonD,axiom,
! [B: set_set_int,A: set_set_int] :
( ( member_set_set_int @ B @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_642_singletonD,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_643_singletonD,axiom,
! [B: list_set_int > set_list_set_int,A: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ B @ ( insert1512894373432338025et_int @ A @ bot_bo7587639734496279683et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_644_singletonD,axiom,
! [B: list_set_int,A: list_set_int] :
( ( member_list_set_int @ B @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_645_singletonD,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_646_singletonD,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
=> ( B = A ) ) ).
% singletonD
thf(fact_647_singleton__iff,axiom,
! [B: set_set_int,A: set_set_int] :
( ( member_set_set_int @ B @ ( insert_set_set_int @ A @ bot_bo2384636101374064866et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_648_singleton__iff,axiom,
! [B: set_a,A: set_a] :
( ( member_set_a @ B @ ( insert_set_a @ A @ bot_bot_set_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_649_singleton__iff,axiom,
! [B: list_set_int > set_list_set_int,A: list_set_int > set_list_set_int] :
( ( member5554658343060950864et_int @ B @ ( insert1512894373432338025et_int @ A @ bot_bo7587639734496279683et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_650_singleton__iff,axiom,
! [B: list_set_int,A: list_set_int] :
( ( member_list_set_int @ B @ ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_651_singleton__iff,axiom,
! [B: a,A: a] :
( ( member_a @ B @ ( insert_a @ A @ bot_bot_set_a ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_652_singleton__iff,axiom,
! [B: set_int,A: set_int] :
( ( member_set_int @ B @ ( insert_set_int @ A @ bot_bot_set_set_int ) )
= ( B = A ) ) ).
% singleton_iff
thf(fact_653_doubleton__eq__iff,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int,D: list_set_int] :
( ( ( insert_list_set_int @ A @ ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int ) )
= ( insert_list_set_int @ C @ ( insert_list_set_int @ D @ bot_bo7344713902834881468et_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_654_doubleton__eq__iff,axiom,
! [A: a,B: a,C: a,D: a] :
( ( ( insert_a @ A @ ( insert_a @ B @ bot_bot_set_a ) )
= ( insert_a @ C @ ( insert_a @ D @ bot_bot_set_a ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_655_doubleton__eq__iff,axiom,
! [A: set_int,B: set_int,C: set_int,D: set_int] :
( ( ( insert_set_int @ A @ ( insert_set_int @ B @ bot_bot_set_set_int ) )
= ( insert_set_int @ C @ ( insert_set_int @ D @ bot_bot_set_set_int ) ) )
= ( ( ( A = C )
& ( B = D ) )
| ( ( A = D )
& ( B = C ) ) ) ) ).
% doubleton_eq_iff
thf(fact_656_insert__not__empty,axiom,
! [A: list_set_int,A2: set_list_set_int] :
( ( insert_list_set_int @ A @ A2 )
!= bot_bo7344713902834881468et_int ) ).
% insert_not_empty
thf(fact_657_insert__not__empty,axiom,
! [A: a,A2: set_a] :
( ( insert_a @ A @ A2 )
!= bot_bot_set_a ) ).
% insert_not_empty
thf(fact_658_insert__not__empty,axiom,
! [A: set_int,A2: set_set_int] :
( ( insert_set_int @ A @ A2 )
!= bot_bot_set_set_int ) ).
% insert_not_empty
thf(fact_659_singleton__inject,axiom,
! [A: list_set_int,B: list_set_int] :
( ( ( insert_list_set_int @ A @ bot_bo7344713902834881468et_int )
= ( insert_list_set_int @ B @ bot_bo7344713902834881468et_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_660_singleton__inject,axiom,
! [A: a,B: a] :
( ( ( insert_a @ A @ bot_bot_set_a )
= ( insert_a @ B @ bot_bot_set_a ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_661_singleton__inject,axiom,
! [A: set_int,B: set_int] :
( ( ( insert_set_int @ A @ bot_bot_set_set_int )
= ( insert_set_int @ B @ bot_bot_set_set_int ) )
=> ( A = B ) ) ).
% singleton_inject
thf(fact_662_Union__mono,axiom,
! [A2: set_se8558717844084009108et_int,B3: set_se8558717844084009108et_int] :
( ( ord_le5799834111343130868et_int @ A2 @ B3 )
=> ( ord_le8104031629355235774et_int @ ( comple148233303328197937et_int @ A2 ) @ ( comple148233303328197937et_int @ B3 ) ) ) ).
% Union_mono
thf(fact_663_Union__mono,axiom,
! [A2: set_set_set_set_int,B3: set_set_set_set_int] :
( ( ord_le3130544380672780132et_int @ A2 @ B3 )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ ( comple1756060948637632417et_int @ B3 ) ) ) ).
% Union_mono
thf(fact_664_Union__mono,axiom,
! [A2: set_set_set_a,B3: set_set_set_a] :
( ( ord_le5722252365846178494_set_a @ A2 @ B3 )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ ( comple3958522678809307947_set_a @ B3 ) ) ) ).
% Union_mono
thf(fact_665_Union__mono,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ B3 )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).
% Union_mono
thf(fact_666_Union__mono,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ( ord_le8104031629355235774et_int @ A2 @ B3 )
=> ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ A2 ) @ ( comple3081917594375107323et_int @ B3 ) ) ) ).
% Union_mono
thf(fact_667_Union__mono,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ( ord_le3724670747650509150_set_a @ A2 @ B3 )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Union_mono
thf(fact_668_Union__mono,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Union_mono
thf(fact_669_Union__least,axiom,
! [A2: set_set_int,C3: set_int] :
( ! [X7: set_int] :
( ( member_set_int @ X7 @ A2 )
=> ( ord_less_eq_set_int @ X7 @ C3 ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_670_Union__least,axiom,
! [A2: set_se8558717844084009108et_int,C3: set_set_list_set_int] :
( ! [X7: set_set_list_set_int] :
( ( member8308387497365429877et_int @ X7 @ A2 )
=> ( ord_le8104031629355235774et_int @ X7 @ C3 ) )
=> ( ord_le8104031629355235774et_int @ ( comple148233303328197937et_int @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_671_Union__least,axiom,
! [A2: set_set_set_set_int,C3: set_set_set_int] :
( ! [X7: set_set_set_int] :
( ( member7356822600254261989et_int @ X7 @ A2 )
=> ( ord_le4317611570275147438et_int @ X7 @ C3 ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_672_Union__least,axiom,
! [A2: set_set_set_a,C3: set_set_a] :
( ! [X7: set_set_a] :
( ( member_set_set_a @ X7 @ A2 )
=> ( ord_le3724670747650509150_set_a @ X7 @ C3 ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_673_Union__least,axiom,
! [A2: set_set_list_set_int,C3: set_list_set_int] :
( ! [X7: set_list_set_int] :
( ( member4487585027056014399et_int @ X7 @ A2 )
=> ( ord_le4527193859438875528et_int @ X7 @ C3 ) )
=> ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_674_Union__least,axiom,
! [A2: set_set_a,C3: set_a] :
( ! [X7: set_a] :
( ( member_set_a @ X7 @ A2 )
=> ( ord_less_eq_set_a @ X7 @ C3 ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_675_Union__least,axiom,
! [A2: set_set_set_int,C3: set_set_int] :
( ! [X7: set_set_int] :
( ( member_set_set_int @ X7 @ A2 )
=> ( ord_le4403425263959731960et_int @ X7 @ C3 ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ C3 ) ) ).
% Union_least
thf(fact_676_Union__upper,axiom,
! [B3: set_int,A2: set_set_int] :
( ( member_set_int @ B3 @ A2 )
=> ( ord_less_eq_set_int @ B3 @ ( comple3221217463730067765et_int @ A2 ) ) ) ).
% Union_upper
thf(fact_677_Union__upper,axiom,
! [B3: set_set_list_set_int,A2: set_se8558717844084009108et_int] :
( ( member8308387497365429877et_int @ B3 @ A2 )
=> ( ord_le8104031629355235774et_int @ B3 @ ( comple148233303328197937et_int @ A2 ) ) ) ).
% Union_upper
thf(fact_678_Union__upper,axiom,
! [B3: set_set_set_int,A2: set_set_set_set_int] :
( ( member7356822600254261989et_int @ B3 @ A2 )
=> ( ord_le4317611570275147438et_int @ B3 @ ( comple1756060948637632417et_int @ A2 ) ) ) ).
% Union_upper
thf(fact_679_Union__upper,axiom,
! [B3: set_set_a,A2: set_set_set_a] :
( ( member_set_set_a @ B3 @ A2 )
=> ( ord_le3724670747650509150_set_a @ B3 @ ( comple3958522678809307947_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_680_Union__upper,axiom,
! [B3: set_list_set_int,A2: set_set_list_set_int] :
( ( member4487585027056014399et_int @ B3 @ A2 )
=> ( ord_le4527193859438875528et_int @ B3 @ ( comple3081917594375107323et_int @ A2 ) ) ) ).
% Union_upper
thf(fact_681_Union__upper,axiom,
! [B3: set_a,A2: set_set_a] :
( ( member_set_a @ B3 @ A2 )
=> ( ord_less_eq_set_a @ B3 @ ( comple2307003609928055243_set_a @ A2 ) ) ) ).
% Union_upper
thf(fact_682_Union__upper,axiom,
! [B3: set_set_int,A2: set_set_set_int] :
( ( member_set_set_int @ B3 @ A2 )
=> ( ord_le4403425263959731960et_int @ B3 @ ( comple7281953568134767595et_int @ A2 ) ) ) ).
% Union_upper
thf(fact_683_Union__subsetI,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ! [X3: set_int] :
( ( member_set_int @ X3 @ A2 )
=> ? [Y7: set_int] :
( ( member_set_int @ Y7 @ B3 )
& ( ord_less_eq_set_int @ X3 @ Y7 ) ) )
=> ( ord_less_eq_set_int @ ( comple3221217463730067765et_int @ A2 ) @ ( comple3221217463730067765et_int @ B3 ) ) ) ).
% Union_subsetI
thf(fact_684_Union__subsetI,axiom,
! [A2: set_se8558717844084009108et_int,B3: set_se8558717844084009108et_int] :
( ! [X3: set_set_list_set_int] :
( ( member8308387497365429877et_int @ X3 @ A2 )
=> ? [Y7: set_set_list_set_int] :
( ( member8308387497365429877et_int @ Y7 @ B3 )
& ( ord_le8104031629355235774et_int @ X3 @ Y7 ) ) )
=> ( ord_le8104031629355235774et_int @ ( comple148233303328197937et_int @ A2 ) @ ( comple148233303328197937et_int @ B3 ) ) ) ).
% Union_subsetI
thf(fact_685_Union__subsetI,axiom,
! [A2: set_set_set_set_int,B3: set_set_set_set_int] :
( ! [X3: set_set_set_int] :
( ( member7356822600254261989et_int @ X3 @ A2 )
=> ? [Y7: set_set_set_int] :
( ( member7356822600254261989et_int @ Y7 @ B3 )
& ( ord_le4317611570275147438et_int @ X3 @ Y7 ) ) )
=> ( ord_le4317611570275147438et_int @ ( comple1756060948637632417et_int @ A2 ) @ ( comple1756060948637632417et_int @ B3 ) ) ) ).
% Union_subsetI
thf(fact_686_Union__subsetI,axiom,
! [A2: set_set_set_a,B3: set_set_set_a] :
( ! [X3: set_set_a] :
( ( member_set_set_a @ X3 @ A2 )
=> ? [Y7: set_set_a] :
( ( member_set_set_a @ Y7 @ B3 )
& ( ord_le3724670747650509150_set_a @ X3 @ Y7 ) ) )
=> ( ord_le3724670747650509150_set_a @ ( comple3958522678809307947_set_a @ A2 ) @ ( comple3958522678809307947_set_a @ B3 ) ) ) ).
% Union_subsetI
thf(fact_687_Union__subsetI,axiom,
! [A2: set_set_list_set_int,B3: set_set_list_set_int] :
( ! [X3: set_list_set_int] :
( ( member4487585027056014399et_int @ X3 @ A2 )
=> ? [Y7: set_list_set_int] :
( ( member4487585027056014399et_int @ Y7 @ B3 )
& ( ord_le4527193859438875528et_int @ X3 @ Y7 ) ) )
=> ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ A2 ) @ ( comple3081917594375107323et_int @ B3 ) ) ) ).
% Union_subsetI
thf(fact_688_Union__subsetI,axiom,
! [A2: set_set_a,B3: set_set_a] :
( ! [X3: set_a] :
( ( member_set_a @ X3 @ A2 )
=> ? [Y7: set_a] :
( ( member_set_a @ Y7 @ B3 )
& ( ord_less_eq_set_a @ X3 @ Y7 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ A2 ) @ ( comple2307003609928055243_set_a @ B3 ) ) ) ).
% Union_subsetI
thf(fact_689_Union__subsetI,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ! [X3: set_set_int] :
( ( member_set_set_int @ X3 @ A2 )
=> ? [Y7: set_set_int] :
( ( member_set_set_int @ Y7 @ B3 )
& ( ord_le4403425263959731960et_int @ X3 @ Y7 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Union_subsetI
thf(fact_690_Union__empty,axiom,
( ( comple3221217463730067765et_int @ bot_bot_set_set_int )
= bot_bot_set_int ) ).
% Union_empty
thf(fact_691_Union__empty,axiom,
( ( comple3081917594375107323et_int @ bot_bo4503415735588907506et_int )
= bot_bo7344713902834881468et_int ) ).
% Union_empty
thf(fact_692_Union__empty,axiom,
( ( comple2307003609928055243_set_a @ bot_bot_set_set_a )
= bot_bot_set_a ) ).
% Union_empty
thf(fact_693_Union__empty,axiom,
( ( comple7281953568134767595et_int @ bot_bo2384636101374064866et_int )
= bot_bot_set_set_int ) ).
% Union_empty
thf(fact_694_Union__empty__conv,axiom,
! [A2: set_set_list_set_int] :
( ( ( comple3081917594375107323et_int @ A2 )
= bot_bo7344713902834881468et_int )
= ( ! [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A2 )
=> ( X2 = bot_bo7344713902834881468et_int ) ) ) ) ).
% Union_empty_conv
thf(fact_695_Union__empty__conv,axiom,
! [A2: set_set_a] :
( ( ( comple2307003609928055243_set_a @ A2 )
= bot_bot_set_a )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% Union_empty_conv
thf(fact_696_Union__empty__conv,axiom,
! [A2: set_set_set_int] :
( ( ( comple7281953568134767595et_int @ A2 )
= bot_bot_set_set_int )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_int ) ) ) ) ).
% Union_empty_conv
thf(fact_697_empty__Union__conv,axiom,
! [A2: set_set_list_set_int] :
( ( bot_bo7344713902834881468et_int
= ( comple3081917594375107323et_int @ A2 ) )
= ( ! [X2: set_list_set_int] :
( ( member4487585027056014399et_int @ X2 @ A2 )
=> ( X2 = bot_bo7344713902834881468et_int ) ) ) ) ).
% empty_Union_conv
thf(fact_698_empty__Union__conv,axiom,
! [A2: set_set_a] :
( ( bot_bot_set_a
= ( comple2307003609928055243_set_a @ A2 ) )
= ( ! [X2: set_a] :
( ( member_set_a @ X2 @ A2 )
=> ( X2 = bot_bot_set_a ) ) ) ) ).
% empty_Union_conv
thf(fact_699_empty__Union__conv,axiom,
! [A2: set_set_set_int] :
( ( bot_bot_set_set_int
= ( comple7281953568134767595et_int @ A2 ) )
= ( ! [X2: set_set_int] :
( ( member_set_set_int @ X2 @ A2 )
=> ( X2 = bot_bot_set_set_int ) ) ) ) ).
% empty_Union_conv
thf(fact_700_Sup__subset__mono,axiom,
! [A2: set_set_set_int,B3: set_set_set_int] :
( ( ord_le4317611570275147438et_int @ A2 @ B3 )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ A2 ) @ ( comple7281953568134767595et_int @ B3 ) ) ) ).
% Sup_subset_mono
thf(fact_701_pzf_Oring__irreducibleI,axiom,
! [R: list_set_int] :
( ( member_list_set_int @ R @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( ~ ( member_list_set_int @ R @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [A3: list_set_int,B2: list_set_int] :
( ( member_list_set_int @ A3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( R
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A3 @ B2 ) )
=> ( ( member_list_set_int @ A3 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
| ( member_list_set_int @ B2 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) )
=> ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ R ) ) ) ) ).
% pzf.ring_irreducibleI
thf(fact_702_pzf_Osubfield__m__inv__simprule,axiom,
! [K: set_list_set_int,K3: list_set_int,A: list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ K3 @ ( minus_4948633356380776833et_int @ K @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K3 @ A ) @ K )
=> ( member_list_set_int @ A @ K ) ) ) ) ) ).
% pzf.subfield_m_inv_simprule
thf(fact_703_pzf_Ofinite__domain__units,axiom,
( ( finite7825103155802847241et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) ) ) ).
% pzf.finite_domain_units
thf(fact_704_pzf_Omult__divides,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ A ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ B ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) ) ) ) ) ).
% pzf.mult_divides
thf(fact_705_pzf_Orcosets__subset__PowG,axiom,
! [H2: set_list_set_int] :
( ( additi4441901441605882380t_unit @ H2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ord_le8104031629355235774et_int @ ( a_RCOS5818729163757401641t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 ) @ ( pow_list_set_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.rcosets_subset_PowG
thf(fact_706_pzf_Onoetherian__ringI,axiom,
( ! [I3: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I3 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ? [A6: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A6 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( finite7825103155802847241et_int @ A6 )
& ( I3
= ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A6 ) ) ) )
=> ( ring_n7202549095058397701t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.noetherian_ringI
thf(fact_707_local_Oring__axioms,axiom,
ring_a_b @ r ).
% local.ring_axioms
thf(fact_708_setadd__subset__G,axiom,
! [H2: set_a,K: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ H2 @ K ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% setadd_subset_G
thf(fact_709_set__add__comm,axiom,
! [I2: set_a,J: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ J @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ I2 @ J )
= ( set_add_a_b @ r @ J @ I2 ) ) ) ) ).
% set_add_comm
thf(fact_710_set__add__closed,axiom,
! [A2: set_a,B3: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ B3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( set_add_a_b @ r @ A2 @ B3 ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% set_add_closed
thf(fact_711_finite__carrier,axiom,
finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) ).
% finite_carrier
thf(fact_712_zf_Oring__axioms,axiom,
ring_s5316885176909347197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.ring_axioms
thf(fact_713_zf_Osetadd__subset__G,axiom,
! [H2: set_set_int,K: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ K ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.setadd_subset_G
thf(fact_714_zf_Oset__add__comm,axiom,
! [I2: set_set_int,J: set_set_int] :
( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ J @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ J )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ J @ I2 ) ) ) ) ).
% zf.set_add_comm
thf(fact_715_zf_Oset__add__closed,axiom,
! [A2: set_set_int,B3: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ B3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A2 @ B3 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.set_add_closed
thf(fact_716_zf_Ofinite__carrier,axiom,
finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ).
% zf.finite_carrier
thf(fact_717_zf_Oset__add__zero,axiom,
! [A2: set_set_int] :
( ( ord_le4403425263959731960et_int @ A2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ A2 )
= A2 ) ) ).
% zf.set_add_zero
thf(fact_718_set__add__zero,axiom,
! [A2: set_a] :
( ( ord_less_eq_set_a @ A2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ A2 )
= A2 ) ) ).
% set_add_zero
thf(fact_719_zf_Ozeropideal,axiom,
princi8860937869964495385t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.zeropideal
thf(fact_720_pzf_Osubring__props_I2_J,axiom,
! [K: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( member_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ K ) ) ).
% pzf.subring_props(2)
thf(fact_721_pzf_Osubring__props_I7_J,axiom,
! [K: set_list_set_int,H1: list_set_int,H22: list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ H1 @ K )
=> ( ( member_list_set_int @ H22 @ K )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% pzf.subring_props(7)
thf(fact_722_pzf_OUnits__closed,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.Units_closed
thf(fact_723_pzf_Osubring__props_I6_J,axiom,
! [K: set_list_set_int,H1: list_set_int,H22: list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ H1 @ K )
=> ( ( member_list_set_int @ H22 @ K )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% pzf.subring_props(6)
thf(fact_724_pzf_Odivides__trans,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ C )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) ) ) ) ).
% pzf.divides_trans
thf(fact_725_pzf_Ozero__divides,axiom,
! [A: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ A )
= ( A
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.zero_divides
thf(fact_726_pzf_Ooneideal,axiom,
ideal_3307802554209726317t_unit @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.oneideal
thf(fact_727_pzf_Osubring__props_I4_J,axiom,
! [K: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( K != bot_bo7344713902834881468et_int ) ) ).
% pzf.subring_props(4)
thf(fact_728_pzf_Osubring__props_I5_J,axiom,
! [K: set_list_set_int,H: list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ H @ K )
=> ( member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H ) @ K ) ) ) ).
% pzf.subring_props(5)
thf(fact_729_pzf_Oa__rcos__zero,axiom,
! [I2: set_list_set_int,I: list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ I @ I2 )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ I )
= I2 ) ) ) ).
% pzf.a_rcos_zero
thf(fact_730_pzf_Oadd__ideals,axiom,
! [I2: set_list_set_int,J: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ideal_3307802554209726317t_unit @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ J ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.add_ideals
thf(fact_731_pzf_Osubring__props_I1_J,axiom,
! [K: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.subring_props(1)
thf(fact_732_pzf_Odivides__zero,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.divides_zero
thf(fact_733_pzf_Odiv__sum__iff,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ C ) )
= ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) ) ) ) ) ) ).
% pzf.div_sum_iff
thf(fact_734_pzf_Odiv__sum,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ C ) ) ) ) ) ) ) ).
% pzf.div_sum
thf(fact_735_pzf_Ounit__factor,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.unit_factor
thf(fact_736_pzf_Oprod__unit__r,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.prod_unit_r
thf(fact_737_pzf_Oprod__unit__l,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ B @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.prod_unit_l
thf(fact_738_pzf_Odivides__prod__r,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ C ) ) ) ) ) ).
% pzf.divides_prod_r
thf(fact_739_pzf_Odivides__prod__l,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ B ) ) ) ) ) ) ).
% pzf.divides_prod_l
thf(fact_740_pzf_Odivides__mult,axiom,
! [A: list_set_int,C: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ A ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ B ) ) ) ) ) ).
% pzf.divides_mult
thf(fact_741_pzf_Oideal__eq__carrier__iff,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) )
= ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.ideal_eq_carrier_iff
thf(fact_742_pzf_Odiv__neg,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B ) ) ) ) ) ).
% pzf.div_neg
thf(fact_743_pzf_Oexists__gen,axiom,
! [I2: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ? [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( I2
= ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 ) ) ) ) ).
% pzf.exists_gen
thf(fact_744_pzf_Ocgenideal__ideal,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ideal_3307802554209726317t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.cgenideal_ideal
thf(fact_745_pzf_Ounit__divides,axiom,
! [U: list_set_int,A: list_set_int] :
( ( member_list_set_int @ U @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ U @ A ) ) ) ).
% pzf.unit_divides
thf(fact_746_pzf_Odivides__unit,axiom,
! [A: list_set_int,U: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ U )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ U @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.divides_unit
thf(fact_747_pzf_Ocgenideal__minimal,axiom,
! [J: set_list_set_int,A: list_set_int] :
( ( ideal_3307802554209726317t_unit @ J @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ A @ J )
=> ( ord_le4527193859438875528et_int @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) @ J ) ) ) ).
% pzf.cgenideal_minimal
thf(fact_748_pzf_Oring__irreducibleE_I4_J,axiom,
! [R: list_set_int] :
( ( member_list_set_int @ R @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ R )
=> ~ ( member_list_set_int @ R @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.ring_irreducibleE(4)
thf(fact_749_pzf_Ogenideal__minimal,axiom,
! [I2: set_list_set_int,S: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ S @ I2 )
=> ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) @ I2 ) ) ) ).
% pzf.genideal_minimal
thf(fact_750_i_Ois__ideal,axiom,
ideal_3307802554209726317t_unit @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% i.is_ideal
thf(fact_751_pzf_Oisgcd__divides__r,axiom,
! [B: list_set_int,A: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ A )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( isgcd_1196993390438183731t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ A @ B ) ) ) ) ).
% pzf.isgcd_divides_r
thf(fact_752_pzf_Oisgcd__divides__l,axiom,
! [A: list_set_int,B: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( isgcd_1196993390438183731t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ A @ B ) ) ) ) ).
% pzf.isgcd_divides_l
thf(fact_753_pzf_Oto__contain__is__to__divide,axiom,
! [A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) )
= ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) ) ) ) ).
% pzf.to_contain_is_to_divide
thf(fact_754_pzf_Ozeroideal,axiom,
ideal_3307802554209726317t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.zeroideal
thf(fact_755_pzf_Oring__irreducibleE_I5_J,axiom,
! [R: list_set_int,A: list_set_int,B: list_set_int] :
( ( member_list_set_int @ R @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ R )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( R
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) )
=> ( ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
| ( member_list_set_int @ B @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ) ) ).
% pzf.ring_irreducibleE(5)
thf(fact_756_pzf_Ogenideal__ideal,axiom,
! [S: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ S @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ideal_3307802554209726317t_unit @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ S ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.genideal_ideal
thf(fact_757_pzf_OIdl__subset__ideal,axiom,
! [I2: set_list_set_int,H2: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H2 ) @ I2 )
= ( ord_le4527193859438875528et_int @ H2 @ I2 ) ) ) ) ).
% pzf.Idl_subset_ideal
thf(fact_758_pzf_Oideal__is__subalgebra,axiom,
! [K: set_list_set_int,I2: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ K @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( embedd3025314616789055568t_unit @ K @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.ideal_is_subalgebra
thf(fact_759_pzf_Oquotient__eq__iff__same__a__r__cos,axiom,
! [I2: set_list_set_int,A: list_set_int,B: list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B ) @ I2 )
= ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ A )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ B ) ) ) ) ) ) ).
% pzf.quotient_eq_iff_same_a_r_cos
thf(fact_760_pzf_Oline__extension__smult__closed,axiom,
! [K: set_list_set_int,E: set_list_set_int,A: list_set_int,K3: list_set_int,U: list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ! [K4: list_set_int,V3: list_set_int] :
( ( member_list_set_int @ K4 @ K )
=> ( ( member_list_set_int @ V3 @ E )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K4 @ V3 ) @ E ) ) )
=> ( ( ord_le4527193859438875528et_int @ E @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ K3 @ K )
=> ( ( member_list_set_int @ U @ ( embedd942231993492570031t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ A @ E ) )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K3 @ U ) @ ( embedd942231993492570031t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ A @ E ) ) ) ) ) ) ) ) ).
% pzf.line_extension_smult_closed
thf(fact_761_pzf_Ofinetely__gen,axiom,
! [I2: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ? [A7: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ A7 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( finite7825103155802847241et_int @ A7 )
& ( I2
= ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A7 ) ) ) ) ).
% pzf.finetely_gen
thf(fact_762_pzf_Oexists__irreducible__divisor,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( ~ ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ~ ! [B2: list_set_int] :
( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 )
=> ~ ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B2 @ A ) ) ) ) ) ).
% pzf.exists_irreducible_divisor
thf(fact_763_zf_Ozero__closed,axiom,
member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ).
% zf.zero_closed
thf(fact_764_pzf_Odivides__refl,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ A ) ) ).
% pzf.divides_refl
thf(fact_765_pzf_OUnits__m__closed,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.Units_m_closed
thf(fact_766_pzf_OUnits__l__cancel,axiom,
! [X: list_set_int,Y: list_set_int,Z: list_set_int] :
( ( member_list_set_int @ X @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Z @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% pzf.Units_l_cancel
thf(fact_767_pzf_Odivides__mult__rI,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ C ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ C ) ) ) ) ) ) ).
% pzf.divides_mult_rI
thf(fact_768_pzf_Odivides__mult__lI,axiom,
! [A: list_set_int,B: list_set_int,C: list_set_int] :
( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ B )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ A ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ C @ B ) ) ) ) ) ).
% pzf.divides_mult_lI
thf(fact_769_pzf_Ofinite__ring__finite__units,axiom,
( ( finite7825103155802847241et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( finite7825103155802847241et_int @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.finite_ring_finite_units
thf(fact_770_zf_Oring__primeE_I1_J,axiom,
! [P2: set_int] :
( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
=> ( P2
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.ring_primeE(1)
thf(fact_771_zeromaximalideal,axiom,
maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeromaximalideal
thf(fact_772_zf_Ozeromaximalideal,axiom,
maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.zeromaximalideal
thf(fact_773_zf_Oring__irreducibleE_I1_J,axiom,
! [R: set_int] :
( ( member_set_int @ R @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ R )
=> ( R
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.ring_irreducibleE(1)
thf(fact_774_rupt__carr,axiom,
! [Y: set_list_set_int] :
( ( member4487585027056014399et_int @ Y @ ( partia7954853462395303941t_unit @ ( polyno6134647158531630271t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ f ) ) )
=> ( ord_le4527193859438875528et_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% rupt_carr
thf(fact_775_divides__unit,axiom,
! [A: a,U: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ U )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% divides_unit
thf(fact_776_unit__divides,axiom,
! [U: a,A: a] :
( ( member_a @ U @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ U @ A ) ) ) ).
% unit_divides
thf(fact_777_Units__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% Units_closed
thf(fact_778_divides__trans,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ B @ C )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ).
% divides_trans
thf(fact_779_zero__divides,axiom,
! [A: a] :
( ( factor8216151070175719842xt_a_b @ r @ ( zero_a_b @ r ) @ A )
= ( A
= ( zero_a_b @ r ) ) ) ).
% zero_divides
thf(fact_780_divides__zero,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( zero_a_b @ r ) ) ) ).
% divides_zero
thf(fact_781_prod__unit__l,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_l
thf(fact_782_prod__unit__r,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ).
% prod_unit_r
thf(fact_783_unit__factor,axiom,
! [A: a,B: a] :
( ( member_a @ ( mult_a_ring_ext_a_b @ r @ A @ B ) @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ).
% unit_factor
thf(fact_784_local_Odivides__mult,axiom,
! [A: a,C: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% local.divides_mult
thf(fact_785_divides__prod__l,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ) ).
% divides_prod_l
thf(fact_786_divides__prod__r,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ).
% divides_prod_r
thf(fact_787_ring__irreducibleE_I4_J,axiom,
! [R: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ring_irreducibleE(4)
thf(fact_788_zf_Odivides__unit,axiom,
! [A: set_int,U: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ U )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.divides_unit
thf(fact_789_zf_Ounit__divides,axiom,
! [U: set_int,A: set_int] :
( ( member_set_int @ U @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ U @ A ) ) ) ).
% zf.unit_divides
thf(fact_790_zf_OUnits__closed,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.Units_closed
thf(fact_791_zf_Odivides__trans,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ C )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) ) ) ) ).
% zf.divides_trans
thf(fact_792_zf_Ozero__divides,axiom,
! [A: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ A )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.zero_divides
thf(fact_793_ring__irreducibleE_I5_J,axiom,
! [R: a,A: a,B: a] :
( ( member_a @ R @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ R )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ r @ A @ B ) )
=> ( ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) ) ) ) ).
% ring_irreducibleE(5)
thf(fact_794_zf_Odivides__zero,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.divides_zero
thf(fact_795_local_Ofield__Units,axiom,
( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ).
% local.field_Units
thf(fact_796_zf_Oring__irreducibleE_I4_J,axiom,
! [R: set_int] :
( ( member_set_int @ R @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ R )
=> ~ ( member_set_int @ R @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.ring_irreducibleE(4)
thf(fact_797_mult__divides,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ) ).
% mult_divides
thf(fact_798_finite__domain__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ) ) ).
% finite_domain_units
thf(fact_799_exists__irreducible__divisor,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) )
=> ~ ! [B2: a] :
( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ B2 )
=> ~ ( factor8216151070175719842xt_a_b @ r @ B2 @ A ) ) ) ) ) ).
% exists_irreducible_divisor
thf(fact_800_zf_Oprimeness__condition,axiom,
! [P2: set_int] :
( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
= ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).
% zf.primeness_condition
thf(fact_801_rupt__is__ring,axiom,
ring_s3147478420139132983t_unit @ ( polyno6134647158531630271t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ f ) ).
% rupt_is_ring
thf(fact_802_zf_Ofield__Units,axiom,
( ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
= ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) ) ).
% zf.field_Units
thf(fact_803_ring__irreducibleI,axiom,
! [R: a] :
( ( member_a @ R @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ~ ( member_a @ R @ ( units_a_ring_ext_a_b @ r ) )
=> ( ! [A3: a,B2: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( R
= ( mult_a_ring_ext_a_b @ r @ A3 @ B2 ) )
=> ( ( member_a @ A3 @ ( units_a_ring_ext_a_b @ r ) )
| ( member_a @ B2 @ ( units_a_ring_ext_a_b @ r ) ) ) ) ) )
=> ( ring_r999134135267193926le_a_b @ r @ R ) ) ) ) ).
% ring_irreducibleI
thf(fact_804_zf_Ofinite__domain__units,axiom,
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
= ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) ) ) ).
% zf.finite_domain_units
thf(fact_805_zf_Oexists__irreducible__divisor,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ~ ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ~ ! [B2: set_int] :
( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 )
=> ~ ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 @ A ) ) ) ) ) ).
% zf.exists_irreducible_divisor
thf(fact_806_divides__refl,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ A @ A ) ) ).
% divides_refl
thf(fact_807_Units__m__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( units_a_ring_ext_a_b @ r ) )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% Units_m_closed
thf(fact_808_Units__l__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( units_a_ring_ext_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( mult_a_ring_ext_a_b @ r @ X @ Y )
= ( mult_a_ring_ext_a_b @ r @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% Units_l_cancel
thf(fact_809_divides__mult__lI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ C @ A ) @ ( mult_a_ring_ext_a_b @ r @ C @ B ) ) ) ) ) ).
% divides_mult_lI
thf(fact_810_divides__mult__rI,axiom,
! [A: a,B: a,C: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( factor8216151070175719842xt_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ A @ C ) @ ( mult_a_ring_ext_a_b @ r @ B @ C ) ) ) ) ) ) ).
% divides_mult_rI
thf(fact_811_finite__ring__finite__units,axiom,
( ( finite_finite_a @ ( partia707051561876973205xt_a_b @ r ) )
=> ( finite_finite_a @ ( units_a_ring_ext_a_b @ r ) ) ) ).
% finite_ring_finite_units
thf(fact_812_zf_Odivides__refl,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ A ) ) ).
% zf.divides_refl
thf(fact_813_zf_Ofinite__ring__finite__units,axiom,
( ( finite6197958912794628473et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( finite6197958912794628473et_int @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.finite_ring_finite_units
thf(fact_814_zf_Oisgcd__divides__l,axiom,
! [A: set_int,B: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ A @ B ) ) ) ) ).
% zf.isgcd_divides_l
thf(fact_815_zf_Oisgcd__divides__r,axiom,
! [B: set_int,A: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ A )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ A @ B ) ) ) ) ).
% zf.isgcd_divides_r
thf(fact_816_zf_Oring__primeI,axiom,
! [P2: set_int] :
( ( P2
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( prime_2746729220581880658t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
=> ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).
% zf.ring_primeI
thf(fact_817_zf_Oring__primeE_I3_J,axiom,
! [P2: set_int] :
( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
=> ( prime_2746729220581880658t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).
% zf.ring_primeE(3)
thf(fact_818_pzf_Ocanonical__proj__vimage__mem__iff,axiom,
! [I2: set_list_set_int,J: set_set_list_set_int,A: list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ord_le8104031629355235774et_int @ J @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ A @ ( comple3081917594375107323et_int @ J ) )
= ( member4487585027056014399et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 @ A ) @ J ) ) ) ) ) ).
% pzf.canonical_proj_vimage_mem_iff
thf(fact_819_exists__gen,axiom,
! [I2: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( I2
= ( cgenid547466209912283029xt_a_b @ r @ X3 ) ) ) ) ).
% exists_gen
thf(fact_820_cgenideal__ideal,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ r ) ) ).
% cgenideal_ideal
thf(fact_821_oneideal,axiom,
ideal_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% oneideal
thf(fact_822_cgenideal__self,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( cgenid547466209912283029xt_a_b @ r @ I ) ) ) ).
% cgenideal_self
thf(fact_823_cgenideal__minimal,axiom,
! [J: set_a,A: a] :
( ( ideal_a_b @ J @ r )
=> ( ( member_a @ A @ J )
=> ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ J ) ) ) ).
% cgenideal_minimal
thf(fact_824_genideal__minimal,axiom,
! [I2: set_a,S: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_less_eq_set_a @ S @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ S ) @ I2 ) ) ) ).
% genideal_minimal
thf(fact_825_add__ideals,axiom,
! [I2: set_a,J: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_a_b @ J @ r )
=> ( ideal_a_b @ ( set_add_a_b @ r @ I2 @ J ) @ r ) ) ) ).
% add_ideals
thf(fact_826_Idl__subset__ideal,axiom,
! [I2: set_a,H2: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ I2 )
= ( ord_less_eq_set_a @ H2 @ I2 ) ) ) ) ).
% Idl_subset_ideal
thf(fact_827_genideal__ideal,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ideal_a_b @ ( genideal_a_b @ r @ S ) @ r ) ) ).
% genideal_ideal
thf(fact_828_subset__Idl__subset,axiom,
! [I2: set_a,H2: set_a] :
( ( ord_less_eq_set_a @ I2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ H2 @ I2 )
=> ( ord_less_eq_set_a @ ( genideal_a_b @ r @ H2 ) @ ( genideal_a_b @ r @ I2 ) ) ) ) ).
% subset_Idl_subset
thf(fact_829_genideal__self,axiom,
! [S: set_a] :
( ( ord_less_eq_set_a @ S @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ S @ ( genideal_a_b @ r @ S ) ) ) ).
% genideal_self
thf(fact_830_ideal__eq__carrier__iff,axiom,
! [A: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( partia707051561876973205xt_a_b @ r )
= ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( member_a @ A @ ( units_a_ring_ext_a_b @ r ) ) ) ) ).
% ideal_eq_carrier_iff
thf(fact_831_isgcd__divides__r,axiom,
! [B: a,A: a] :
( ( factor8216151070175719842xt_a_b @ r @ B @ A )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ B @ A @ B ) ) ) ) ).
% isgcd_divides_r
thf(fact_832_isgcd__divides__l,axiom,
! [A: a,B: a] :
( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( isgcd_a_ring_ext_a_b @ r @ A @ A @ B ) ) ) ) ).
% isgcd_divides_l
thf(fact_833_bezout__identity,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B ) )
= ( cgenid547466209912283029xt_a_b @ r @ ( somegc1600592057159103747xt_a_b @ r @ A @ B ) ) ) ) ) ).
% bezout_identity
thf(fact_834_ideal__sum__iff__gcd,axiom,
! [A: a,B: a,D: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ D @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( cgenid547466209912283029xt_a_b @ r @ D )
= ( set_add_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) @ ( cgenid547466209912283029xt_a_b @ r @ B ) ) )
= ( isgcd_a_ring_ext_a_b @ r @ D @ A @ B ) ) ) ) ) ).
% ideal_sum_iff_gcd
thf(fact_835_zf_Oexists__gen,axiom,
! [I2: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
& ( I2
= ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 ) ) ) ) ).
% zf.exists_gen
thf(fact_836_zf_Ocgenideal__ideal,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.cgenideal_ideal
thf(fact_837_zf_Ooneideal,axiom,
ideal_7262958097527394045t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.oneideal
thf(fact_838_zf_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ Z ) )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Z ) ) ) ) ) ) ).
% zf.m_lcomm
thf(fact_839_zf_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X ) ) ) ) ).
% zf.m_comm
thf(fact_840_zf_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ Z )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ Z ) ) ) ) ) ) ).
% zf.m_assoc
thf(fact_841_zf_Ocgenideal__self,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ I @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I ) ) ) ).
% zf.cgenideal_self
thf(fact_842_zf_Ocgenideal__minimal,axiom,
! [J: set_set_int,A: set_int] :
( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_int @ A @ J )
=> ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ J ) ) ) ).
% zf.cgenideal_minimal
thf(fact_843_zf_Ogenideal__minimal,axiom,
! [I2: set_set_int,S: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( ord_le4403425263959731960et_int @ S @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) @ I2 ) ) ) ).
% zf.genideal_minimal
thf(fact_844_cgenideal__is__principalideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( principalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ I ) @ r ) ) ).
% cgenideal_is_principalideal
thf(fact_845_cgenideal__eq__genideal,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( cgenid547466209912283029xt_a_b @ r @ I )
= ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% cgenideal_eq_genideal
thf(fact_846_genideal__self_H,axiom,
! [I: a] :
( ( member_a @ I @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ I @ ( genideal_a_b @ r @ ( insert_a @ I @ bot_bot_set_a ) ) ) ) ).
% genideal_self'
thf(fact_847_zf_Oadd__ideals,axiom,
! [I2: set_set_int,J: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( ideal_7262958097527394045t_unit @ J @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ideal_7262958097527394045t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ J ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.add_ideals
thf(fact_848_zeroideal,axiom,
ideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroideal
thf(fact_849_genideal__zero,axiom,
( ( genideal_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) )
= ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% genideal_zero
thf(fact_850_to__contain__is__to__divide,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( cgenid547466209912283029xt_a_b @ r @ B ) @ ( cgenid547466209912283029xt_a_b @ r @ A ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ B ) ) ) ) ).
% to_contain_is_to_divide
thf(fact_851_finetely__gen,axiom,
! [I2: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ? [A7: set_a] :
( ( ord_less_eq_set_a @ A7 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A7 )
& ( I2
= ( genideal_a_b @ r @ A7 ) ) ) ) ).
% finetely_gen
thf(fact_852_zf_Om__rcancel,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( A
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ A )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ A ) )
= ( B = C ) ) ) ) ) ) ).
% zf.m_rcancel
thf(fact_853_zf_Om__lcancel,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( A
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) )
= ( B = C ) ) ) ) ) ) ).
% zf.m_lcancel
thf(fact_854_zf_Ointegral__iff,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
= ( ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
| ( B
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% zf.integral_iff
thf(fact_855_zf_Ointegral,axiom,
! [A: set_int,B: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
| ( B
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% zf.integral
thf(fact_856_zf_Of__comm__group__1,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( X
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( Y
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% zf.f_comm_group_1
thf(fact_857_zf_Ogenideal__ideal,axiom,
! [S: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ideal_7262958097527394045t_unit @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.genideal_ideal
thf(fact_858_zf_OIdl__subset__ideal,axiom,
! [I2: set_set_int,H2: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 ) @ I2 )
= ( ord_le4403425263959731960et_int @ H2 @ I2 ) ) ) ) ).
% zf.Idl_subset_ideal
thf(fact_859_zf_Osubset__Idl__subset,axiom,
! [I2: set_set_int,H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ I2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ H2 @ I2 )
=> ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) ) ) ).
% zf.subset_Idl_subset
thf(fact_860_zf_Ogenideal__self,axiom,
! [S: set_set_int] :
( ( ord_le4403425263959731960et_int @ S @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ord_le4403425263959731960et_int @ S @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ S ) ) ) ).
% zf.genideal_self
thf(fact_861_zf_Ounit__factor,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.unit_factor
thf(fact_862_zf_Oprod__unit__r,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% zf.prod_unit_r
thf(fact_863_zf_Oprod__unit__l,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ B @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% zf.prod_unit_l
thf(fact_864_zf_Oideal__eq__carrier__iff,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
= ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) )
= ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.ideal_eq_carrier_iff
thf(fact_865_zf_Odivides__prod__r,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ C ) ) ) ) ) ).
% zf.divides_prod_r
thf(fact_866_zf_Odivides__prod__l,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ B ) ) ) ) ) ) ).
% zf.divides_prod_l
thf(fact_867_zf_Odivides__mult,axiom,
! [A: set_int,C: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ A ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ B ) ) ) ) ) ).
% zf.divides_mult
thf(fact_868_irreducible__imp__maximalideal,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ring_r999134135267193926le_a_b @ r @ P2 )
=> ( maximalideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r ) ) ) ).
% irreducible_imp_maximalideal
thf(fact_869_zf_Obezout__identity,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B ) )
= ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( somegc3856857245158089085t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) ) ) ) ) ).
% zf.bezout_identity
thf(fact_870_Idl__subset__ideal_H,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ ( genideal_a_b @ r @ ( insert_a @ A @ bot_bot_set_a ) ) @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) )
= ( member_a @ A @ ( genideal_a_b @ r @ ( insert_a @ B @ bot_bot_set_a ) ) ) ) ) ) ).
% Idl_subset_ideal'
thf(fact_871_zf_Ozero__is__prime_I1_J,axiom,
prime_2746729220581880658t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ).
% zf.zero_is_prime(1)
thf(fact_872_zf_Ocgenideal__is__principalideal,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( princi8860937869964495385t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.cgenideal_is_principalideal
thf(fact_873_zf_Ocgenideal__eq__genideal,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I )
= ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).
% zf.cgenideal_eq_genideal
thf(fact_874_zf_Ogenideal__self_H,axiom,
! [I: set_int] :
( ( member_set_int @ I @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ I @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ I @ bot_bot_set_set_int ) ) ) ) ).
% zf.genideal_self'
thf(fact_875_zf_Oto__contain__is__to__divide,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) )
= ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) ) ) ) ).
% zf.to_contain_is_to_divide
thf(fact_876_zf_Ofinetely__gen,axiom,
! [I2: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ? [A7: set_set_int] :
( ( ord_le4403425263959731960et_int @ A7 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
& ( finite6197958912794628473et_int @ A7 )
& ( I2
= ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A7 ) ) ) ) ).
% zf.finetely_gen
thf(fact_877_zf_Ozeroideal,axiom,
ideal_7262958097527394045t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.zeroideal
thf(fact_878_zf_Ogenideal__zero,axiom,
( ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) )
= ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) ).
% zf.genideal_zero
thf(fact_879_noetherian__ringI,axiom,
( ! [I3: set_a] :
( ( ideal_a_b @ I3 @ r )
=> ? [A6: set_a] :
( ( ord_less_eq_set_a @ A6 @ ( partia707051561876973205xt_a_b @ r ) )
& ( finite_finite_a @ A6 )
& ( I3
= ( genideal_a_b @ r @ A6 ) ) ) )
=> ( ring_n3639167112692572309ng_a_b @ r ) ) ).
% noetherian_ringI
thf(fact_880_zf_Oring__irreducibleE_I5_J,axiom,
! [R: set_int,A: set_int,B: set_int] :
( ( member_set_int @ R @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ R )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( R
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) )
=> ( ( member_set_int @ A @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
| ( member_set_int @ B @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% zf.ring_irreducibleE(5)
thf(fact_881_zf_Oirreducible__imp__maximalideal,axiom,
! [P2: set_int] :
( ( member_set_int @ P2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 )
=> ( maxima6262477034536100350t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.irreducible_imp_maximalideal
thf(fact_882_zf_OIdl__subset__ideal_H,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ A @ bot_bot_set_set_int ) ) @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) )
= ( member_set_int @ A @ ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ B @ bot_bot_set_set_int ) ) ) ) ) ) ).
% zf.Idl_subset_ideal'
thf(fact_883_zf_Oideal__sum__iff__gcd,axiom,
! [A: set_int,B: set_int,D: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ D @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ D )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B ) ) )
= ( isgcd_4636411027072998995t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ D @ A @ B ) ) ) ) ) ).
% zf.ideal_sum_iff_gcd
thf(fact_884_zf_Onoetherian__ringI,axiom,
( ! [I3: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I3 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ? [A6: set_set_int] :
( ( ord_le4403425263959731960et_int @ A6 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
& ( finite6197958912794628473et_int @ A6 )
& ( I3
= ( genide1545711809618862555t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A6 ) ) ) )
=> ( ring_n4918880612521764245t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.noetherian_ringI
thf(fact_885_pzf_Oquot__ideal__imp__ring__ideal,axiom,
! [I2: set_list_set_int,J: set_set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ideal_7892184242768911799t_unit @ J @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) )
=> ( ideal_3307802554209726317t_unit @ ( comple3081917594375107323et_int @ J ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.quot_ideal_imp_ring_ideal
thf(fact_886_zf_Omult__divides,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ A ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ B ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) ) ) ) ) ).
% zf.mult_divides
thf(fact_887_pzf_Oquot__carr,axiom,
! [I2: set_list_set_int,Y: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member4487585027056014399et_int @ Y @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) )
=> ( ord_le4527193859438875528et_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.quot_carr
thf(fact_888_zf_Oring__irreducibleI,axiom,
! [R: set_int] :
( ( member_set_int @ R @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ~ ( member_set_int @ R @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ! [A3: set_int,B2: set_int] :
( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( R
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A3 @ B2 ) )
=> ( ( member_set_int @ A3 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
| ( member_set_int @ B2 @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
=> ( ring_r5206707500676108452t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ R ) ) ) ) ).
% zf.ring_irreducibleI
thf(fact_889_i_Oquotient__is__ring,axiom,
ring_s3147478420139132983t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ).
% i.quotient_is_ring
thf(fact_890_pzf_Ocanonical__proj__vimage__in__carrier,axiom,
! [I2: set_list_set_int,J: set_set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ I2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ord_le8104031629355235774et_int @ J @ ( partia7954853462395303941t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ I2 ) ) )
=> ( ord_le4527193859438875528et_int @ ( comple3081917594375107323et_int @ J ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.canonical_proj_vimage_in_carrier
thf(fact_891_zf_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.m_closed
thf(fact_892_zf_OUnits__m__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.Units_m_closed
thf(fact_893_zf_Or__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.r_null
thf(fact_894_zf_Ol__null,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.l_null
thf(fact_895_zf_OUnits__l__cancel,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Z ) )
= ( Y = Z ) ) ) ) ) ).
% zf.Units_l_cancel
thf(fact_896_zf_Odivides__mult__rI,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ C ) ) ) ) ) ) ).
% zf.divides_mult_rI
thf(fact_897_zf_Odivides__mult__lI,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ A ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ B ) ) ) ) ) ).
% zf.divides_mult_lI
thf(fact_898_pzf_OFactRing__zeroideal_I2_J,axiom,
is_rin8255480188913837456t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) ).
% pzf.FactRing_zeroideal(2)
thf(fact_899_pzf_OFactRing__zeroideal_I1_J,axiom,
is_rin7318007858330770044t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.FactRing_zeroideal(1)
thf(fact_900_zf_Oprimeideal__iff__prime,axiom,
! [P2: set_int] :
( ( member_set_int @ P2 @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( primei350866878734230858t_unit @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
= ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ).
% zf.primeideal_iff_prime
thf(fact_901_primeideal__iff__prime,axiom,
! [P2: a] :
( ( member_a @ P2 @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( primeideal_a_b @ ( cgenid547466209912283029xt_a_b @ r @ P2 ) @ r )
= ( ring_ring_prime_a_b @ r @ P2 ) ) ) ).
% primeideal_iff_prime
thf(fact_902_pzf_Odomain__iff__prime,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( ( domain6977320662053078051t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) )
= ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) ) ).
% pzf.domain_iff_prime
thf(fact_903_i_Orcos__ring__hom,axiom,
member5554658343060950864et_int @ ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) @ ( ring_h7954335661761407236t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) ) ) ).
% i.rcos_ring_hom
thf(fact_904_quot__ideal__imp__ring__ideal,axiom,
! [I2: set_a,J: set_set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ideal_4463284918206690523t_unit @ J @ ( factRing_a_b @ r @ I2 ) )
=> ( ideal_a_b @ ( comple2307003609928055243_set_a @ J ) @ r ) ) ) ).
% quot_ideal_imp_ring_ideal
thf(fact_905_zf_Oquot__ideal__imp__ring__ideal,axiom,
! [I2: set_set_int,J: set_set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( ideal_1992573625310827463t_unit @ J @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) )
=> ( ideal_7262958097527394045t_unit @ ( comple7281953568134767595et_int @ J ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.quot_ideal_imp_ring_ideal
thf(fact_906_maximalideal__prime,axiom,
! [I2: set_a] :
( ( maximalideal_a_b @ I2 @ r )
=> ( primeideal_a_b @ I2 @ r ) ) ).
% maximalideal_prime
thf(fact_907_canonical__proj__vimage__in__carrier,axiom,
! [I2: set_a,J: set_set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
=> ( ord_less_eq_set_a @ ( comple2307003609928055243_set_a @ J ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% canonical_proj_vimage_in_carrier
thf(fact_908_quot__carr,axiom,
! [I2: set_a,Y: set_a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( member_set_a @ Y @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
=> ( ord_less_eq_set_a @ Y @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% quot_carr
thf(fact_909_zeroprimeideal,axiom,
primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ).
% zeroprimeideal
thf(fact_910_zf_Ocanonical__proj__vimage__in__carrier,axiom,
! [I2: set_set_int,J: set_set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) )
=> ( ord_le4403425263959731960et_int @ ( comple7281953568134767595et_int @ J ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.canonical_proj_vimage_in_carrier
thf(fact_911_zf_Oquot__carr,axiom,
! [I2: set_set_int,Y: set_set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_set_int @ Y @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) )
=> ( ord_le4403425263959731960et_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.quot_carr
thf(fact_912_zf_Omaximalideal__prime,axiom,
! [I2: set_set_int] :
( ( maxima6262477034536100350t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( primei350866878734230858t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.maximalideal_prime
thf(fact_913_zf_Ozeroprimeideal,axiom,
primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.zeroprimeideal
thf(fact_914_pzf_Oquot__domain__iff__primeideal,axiom,
! [P: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ P @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( domain6977320662053078051t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P ) )
= ( primei7054412349171268410t_unit @ P @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.quot_domain_iff_primeideal
thf(fact_915_pzf_Oquot__domain__imp__primeideal,axiom,
! [P: set_list_set_int] :
( ( ideal_3307802554209726317t_unit @ P @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( domain6977320662053078051t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P ) )
=> ( primei7054412349171268410t_unit @ P @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.quot_domain_imp_primeideal
thf(fact_916_FactRing__zeroideal_I2_J,axiom,
is_rin9099215527551818550t_unit @ r @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) ).
% FactRing_zeroideal(2)
thf(fact_917_FactRing__zeroideal_I1_J,axiom,
is_rin6001486760346555702it_a_b @ ( factRing_a_b @ r @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) @ r ).
% FactRing_zeroideal(1)
thf(fact_918_zf_OFactRing__zeroideal_I2_J,axiom,
is_rin4964436081320486288t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) ).
% zf.FactRing_zeroideal(2)
thf(fact_919_zf_OFactRing__zeroideal_I1_J,axiom,
is_rin1729965262454186108t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.FactRing_zeroideal(1)
thf(fact_920_zf_Omonoid__cancelI,axiom,
( ! [A3: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C2 @ A3 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C2 @ B2 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( ! [A3: set_int,B2: set_int,C2: set_int] :
( ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A3 @ C2 )
= ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B2 @ C2 ) )
=> ( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( A3 = B2 ) ) ) ) )
=> ( monoid497721730651901107t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.monoid_cancelI
thf(fact_921_domain__axioms,axiom,
domain_a_b @ r ).
% domain_axioms
thf(fact_922_r__neg2,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y ) )
= Y ) ) ) ).
% r_neg2
thf(fact_923_r__neg1,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( add_a_b @ r @ X @ Y ) )
= Y ) ) ) ).
% r_neg1
thf(fact_924_local_Ominus__add,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% local.minus_add
thf(fact_925_a__transpose__inv,axiom,
! [X: a,Y: a,Z: a] :
( ( ( add_a_b @ r @ X @ Y )
= Z )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Z )
= Y ) ) ) ) ) ).
% a_transpose_inv
thf(fact_926_add_Oinv__solve__right_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) )
= A )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right'
thf(fact_927_add_Oinv__solve__right,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ B @ ( a_inv_a_b @ r @ C ) ) )
= ( B
= ( add_a_b @ r @ A @ C ) ) ) ) ) ) ).
% add.inv_solve_right
thf(fact_928_add_Oinv__solve__left_H,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C )
= A )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left'
thf(fact_929_add_Oinv__solve__left,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( A
= ( add_a_b @ r @ ( a_inv_a_b @ r @ B ) @ C ) )
= ( C
= ( add_a_b @ r @ B @ A ) ) ) ) ) ) ).
% add.inv_solve_left
thf(fact_930_add_Oinv__mult__group,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( a_inv_a_b @ r @ Y ) @ ( a_inv_a_b @ r @ X ) ) ) ) ) ).
% add.inv_mult_group
thf(fact_931_a__lcomm,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) )
= ( add_a_b @ r @ Y @ ( add_a_b @ r @ X @ Z ) ) ) ) ) ) ).
% a_lcomm
thf(fact_932_a__comm,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ Y )
= ( add_a_b @ r @ Y @ X ) ) ) ) ).
% a_comm
thf(fact_933_a__assoc,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ X @ ( add_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% a_assoc
thf(fact_934_add_Or__cancel,axiom,
! [A: a,C: a,B: a] :
( ( ( add_a_b @ r @ A @ C )
= ( add_a_b @ r @ B @ C ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.r_cancel
thf(fact_935_add_Ol__cancel,axiom,
! [C: a,A: a,B: a] :
( ( ( add_a_b @ r @ C @ A )
= ( add_a_b @ r @ C @ B ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( A = B ) ) ) ) ) ).
% add.l_cancel
thf(fact_936_a__rcos__zero,axiom,
! [I2: set_a,I: a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( member_a @ I @ I2 )
=> ( ( a_r_coset_a_b @ r @ I2 @ I )
= I2 ) ) ) ).
% a_rcos_zero
thf(fact_937_zf_Odomain__axioms,axiom,
domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.domain_axioms
thf(fact_938_l__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( a_inv_a_b @ r @ X ) @ X )
= ( zero_a_b @ r ) ) ) ).
% l_neg
thf(fact_939_minus__equality,axiom,
! [Y: a,X: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ X )
= Y ) ) ) ) ).
% minus_equality
thf(fact_940_r__neg,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ X ) )
= ( zero_a_b @ r ) ) ) ).
% r_neg
thf(fact_941_sum__zero__eq__neg,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( X
= ( a_inv_a_b @ r @ Y ) ) ) ) ) ).
% sum_zero_eq_neg
thf(fact_942_local_Ominus__unique,axiom,
! [Y: a,X: a,Y3: a] :
( ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ Y3 )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% local.minus_unique
thf(fact_943_add_Or__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X @ X3 )
= ( zero_a_b @ r ) ) ) ) ).
% add.r_inv_ex
thf(fact_944_add_Oone__unique,axiom,
! [U: a] :
( ( member_a @ U @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_a_b @ r ) ) ) ) ).
% add.one_unique
thf(fact_945_add_Ol__inv__ex,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ? [X3: a] :
( ( member_a @ X3 @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( add_a_b @ r @ X3 @ X )
= ( zero_a_b @ r ) ) ) ) ).
% add.l_inv_ex
thf(fact_946_add_Oinv__comm,axiom,
! [X: a,Y: a] :
( ( ( add_a_b @ r @ X @ Y )
= ( zero_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ Y @ X )
= ( zero_a_b @ r ) ) ) ) ) ).
% add.inv_comm
thf(fact_947_a__coset__add__inv2,axiom,
! [M: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M @ X )
= ( a_r_coset_a_b @ r @ M @ Y ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M ) ) ) ) ) ).
% a_coset_add_inv2
thf(fact_948_a__coset__add__inv1,axiom,
! [M: set_a,X: a,Y: a] :
( ( ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) ) )
= M )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M @ X )
= ( a_r_coset_a_b @ r @ M @ Y ) ) ) ) ) ) ).
% a_coset_add_inv1
thf(fact_949_a__coset__add__assoc,axiom,
! [M: set_a,G: a,H: a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ G @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ H @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ ( a_r_coset_a_b @ r @ M @ G ) @ H )
= ( a_r_coset_a_b @ r @ M @ ( add_a_b @ r @ G @ H ) ) ) ) ) ) ).
% a_coset_add_assoc
thf(fact_950_a__rcosI,axiom,
! [H: a,H2: set_a,X: a] :
( ( member_a @ H @ H2 )
=> ( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ H @ X ) @ ( a_r_coset_a_b @ r @ H2 @ X ) ) ) ) ) ).
% a_rcosI
thf(fact_951_a__r__coset__subset__G,axiom,
! [H2: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ord_less_eq_set_a @ ( a_r_coset_a_b @ r @ H2 @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_r_coset_subset_G
thf(fact_952_r__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ X @ ( a_inv_a_b @ r @ Y ) )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% r_minus
thf(fact_953_l__minus,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( a_inv_a_b @ r @ X ) @ Y )
= ( a_inv_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Y ) ) ) ) ) ).
% l_minus
thf(fact_954_r__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ Z @ ( add_a_b @ r @ X @ Y ) )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ Z @ X ) @ ( mult_a_ring_ext_a_b @ r @ Z @ Y ) ) ) ) ) ) ).
% r_distr
thf(fact_955_l__distr,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( mult_a_ring_ext_a_b @ r @ ( add_a_b @ r @ X @ Y ) @ Z )
= ( add_a_b @ r @ ( mult_a_ring_ext_a_b @ r @ X @ Z ) @ ( mult_a_ring_ext_a_b @ r @ Y @ Z ) ) ) ) ) ) ).
% l_distr
thf(fact_956_div__neg,axiom,
! [A: a,B: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( a_inv_a_b @ r @ B ) ) ) ) ) ).
% div_neg
thf(fact_957_div__sum__iff,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) )
= ( factor8216151070175719842xt_a_b @ r @ A @ C ) ) ) ) ) ) ).
% div_sum_iff
thf(fact_958_div__sum,axiom,
! [A: a,B: a,C: a] :
( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ B @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ C @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ B )
=> ( ( factor8216151070175719842xt_a_b @ r @ A @ C )
=> ( factor8216151070175719842xt_a_b @ r @ A @ ( add_a_b @ r @ B @ C ) ) ) ) ) ) ) ).
% div_sum
thf(fact_959_zf_Or__neg2,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ Y ) )
= Y ) ) ) ).
% zf.r_neg2
thf(fact_960_zf_Or__neg1,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) )
= Y ) ) ) ).
% zf.r_neg1
thf(fact_961_zf_Ominus__add,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) ) ) ) ).
% zf.minus_add
thf(fact_962_zf_Oadd_Oinv__solve__right_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C ) )
= A )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) ) ) ) ) ) ).
% zf.add.inv_solve_right'
thf(fact_963_zf_Oadd_Oinv__solve__right,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C ) ) )
= ( B
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) ) ) ) ) ) ).
% zf.add.inv_solve_right
thf(fact_964_zf_Oadd_Oinv__solve__left_H,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B ) @ C )
= A )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ A ) ) ) ) ) ) ).
% zf.add.inv_solve_left'
thf(fact_965_zf_Oadd_Oinv__solve__left,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( A
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B ) @ C ) )
= ( C
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ A ) ) ) ) ) ) ).
% zf.add.inv_solve_left
thf(fact_966_zf_Oadd_Oinv__mult__group,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) ) ) ) ) ).
% zf.add.inv_mult_group
thf(fact_967_zf_Oa__transpose__inv,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
= Z )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ Z )
= Y ) ) ) ) ) ).
% zf.a_transpose_inv
thf(fact_968_zf_Oadd_Or__cancel,axiom,
! [A: set_int,C: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ C ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( A = B ) ) ) ) ) ).
% zf.add.r_cancel
thf(fact_969_zf_Oadd_Om__lcomm,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ Z ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Z ) ) ) ) ) ) ).
% zf.add.m_lcomm
thf(fact_970_zf_Oadd_Om__comm,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X ) ) ) ) ).
% zf.add.m_comm
thf(fact_971_zf_Oadd_Om__assoc,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ Z ) ) ) ) ) ) ).
% zf.add.m_assoc
thf(fact_972_zf_Oadd_Ol__cancel,axiom,
! [C: set_int,A: set_int,B: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ A )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ C @ B ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( A = B ) ) ) ) ) ).
% zf.add.l_cancel
thf(fact_973_zf_Oa__rcos__zero,axiom,
! [I2: set_set_int,I: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_int @ I @ I2 )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ I )
= I2 ) ) ) ).
% zf.a_rcos_zero
thf(fact_974_a__setmult__rcos__assoc,axiom,
! [H2: set_a,K: set_a,X: a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( set_add_a_b @ r @ H2 @ ( a_r_coset_a_b @ r @ K @ X ) )
= ( a_r_coset_a_b @ r @ ( set_add_a_b @ r @ H2 @ K ) @ X ) ) ) ) ) ).
% a_setmult_rcos_assoc
thf(fact_975_pzf_Odomain__axioms,axiom,
domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.domain_axioms
thf(fact_976_zf_Osum__zero__eq__neg,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( X
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) ) ) ) ).
% zf.sum_zero_eq_neg
thf(fact_977_zf_Or__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.r_neg
thf(fact_978_zf_Ominus__equality,axiom,
! [Y: set_int,X: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X )
= Y ) ) ) ) ).
% zf.minus_equality
thf(fact_979_zf_Ol__neg,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.l_neg
thf(fact_980_zf_Ominus__unique,axiom,
! [Y: set_int,X: set_int,Y3: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y3 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% zf.minus_unique
thf(fact_981_zf_Oadd_Or__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ X3 )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.add.r_inv_ex
thf(fact_982_zf_Oadd_Oone__unique,axiom,
! [U: set_int] :
( ( member_set_int @ U @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.add.one_unique
thf(fact_983_zf_Oadd_Ol__inv__ex,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ? [X3: set_int] :
( ( member_set_int @ X3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
& ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.add.l_inv_ex
thf(fact_984_zf_Oadd_Oinv__comm,axiom,
! [X: set_int,Y: set_int] :
( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.add.inv_comm
thf(fact_985_zf_Oa__coset__add__inv2,axiom,
! [M: set_set_int,X: set_int,Y: set_int] :
( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ X )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ Y ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) )
= M ) ) ) ) ) ).
% zf.a_coset_add_inv2
thf(fact_986_zf_Oa__coset__add__inv1,axiom,
! [M: set_set_int,X: set_int,Y: set_int] :
( ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) )
= M )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ X )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ Y ) ) ) ) ) ) ).
% zf.a_coset_add_inv1
thf(fact_987_zf_Oa__rcosI,axiom,
! [H: set_int,H2: set_set_int,X: set_int] :
( ( member_set_int @ H @ H2 )
=> ( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ X ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ X ) ) ) ) ) ).
% zf.a_rcosI
thf(fact_988_zf_Oa__coset__add__assoc,axiom,
! [M: set_set_int,G: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ G ) @ H )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ G @ H ) ) ) ) ) ) ).
% zf.a_coset_add_assoc
thf(fact_989_zf_Oa__r__coset__subset__G,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.a_r_coset_subset_G
thf(fact_990_zf_Or__minus,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) ) ) ) ) ).
% zf.r_minus
thf(fact_991_zf_Ol__minus,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ Y )
= ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) ) ) ) ) ).
% zf.l_minus
thf(fact_992_zf_Or__distr,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Z @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Z @ X ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Z @ Y ) ) ) ) ) ) ).
% zf.r_distr
thf(fact_993_zf_Ol__distr,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ Z )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Z ) @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ Z ) ) ) ) ) ) ).
% zf.l_distr
thf(fact_994_zf_Odiv__neg,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B ) ) ) ) ) ).
% zf.div_neg
thf(fact_995_zf_Odiv__sum__iff,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ C ) )
= ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C ) ) ) ) ) ) ).
% zf.div_sum_iff
thf(fact_996_zf_Odiv__sum,axiom,
! [A: set_int,B: set_int,C: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ C @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
=> ( ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ C )
=> ( factor5186451337065598620t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ C ) ) ) ) ) ) ) ).
% zf.div_sum
thf(fact_997_quot__domain__imp__primeideal,axiom,
! [P: set_a] :
( ( ideal_a_b @ P @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P ) )
=> ( primeideal_a_b @ P @ r ) ) ) ).
% quot_domain_imp_primeideal
thf(fact_998_quot__domain__iff__primeideal,axiom,
! [P: set_a] :
( ( ideal_a_b @ P @ r )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ P ) )
= ( primeideal_a_b @ P @ r ) ) ) ).
% quot_domain_iff_primeideal
thf(fact_999_add_Oone__in__subset,axiom,
! [H2: set_a] :
( ( ord_less_eq_set_a @ H2 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( H2 != bot_bot_set_a )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H2 )
=> ( member_a @ ( a_inv_a_b @ r @ X3 ) @ H2 ) )
=> ( ! [X3: a] :
( ( member_a @ X3 @ H2 )
=> ! [Xa: a] :
( ( member_a @ Xa @ H2 )
=> ( member_a @ ( add_a_b @ r @ X3 @ Xa ) @ H2 ) ) )
=> ( member_a @ ( zero_a_b @ r ) @ H2 ) ) ) ) ) ).
% add.one_in_subset
thf(fact_1000_domain__eq__zeroprimeideal,axiom,
( ( domain_a_b @ r )
= ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r ) ) ).
% domain_eq_zeroprimeideal
thf(fact_1001_zeroprimeideal__domainI,axiom,
( ( primeideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( domain_a_b @ r ) ) ).
% zeroprimeideal_domainI
thf(fact_1002_pzf_Opprime__iff__pirreducible,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ring_r7068619596278734765t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
= ( ring_r5263416876369801476t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) ) ) ) ).
% pzf.pprime_iff_pirreducible
thf(fact_1003_pzf_OpprimeE_I2_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ring_r7068619596278734765t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
=> ~ ( member7440485371949894425et_int @ P2 @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ) ).
% pzf.pprimeE(2)
thf(fact_1004_pzf_Ouniv__poly__is__principal,axiom,
! [K: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ring_p1105522777314784042t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).
% pzf.univ_poly_is_principal
thf(fact_1005_zf_Oa__setmult__rcos__assoc,axiom,
! [H2: set_set_int,K: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K @ X ) )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ K ) @ X ) ) ) ) ) ).
% zf.a_setmult_rcos_assoc
thf(fact_1006_zf_Oadd_Oone__in__subset,axiom,
! [H2: set_set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( H2 != bot_bot_set_set_int )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ H2 )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 ) @ H2 ) )
=> ( ! [X3: set_int] :
( ( member_set_int @ X3 @ H2 )
=> ! [Xa: set_int] :
( ( member_set_int @ Xa @ H2 )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X3 @ Xa ) @ H2 ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ H2 ) ) ) ) ) ).
% zf.add.one_in_subset
thf(fact_1007_zf_Oquot__domain__iff__primeideal,axiom,
! [P: set_set_int] :
( ( ideal_7262958097527394045t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P ) )
= ( primei350866878734230858t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.quot_domain_iff_primeideal
thf(fact_1008_zf_Oquot__domain__imp__primeideal,axiom,
! [P: set_set_int] :
( ( ideal_7262958097527394045t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P ) )
=> ( primei350866878734230858t_unit @ P @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.quot_domain_imp_primeideal
thf(fact_1009_zf_Ozeroprimeideal__domainI,axiom,
( ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.zeroprimeideal_domainI
thf(fact_1010_zf_Odomain__eq__zeroprimeideal,axiom,
( ( domain6183376680155302761t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
= ( primei350866878734230858t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.domain_eq_zeroprimeideal
thf(fact_1011_canonical__proj__vimage__mem__iff,axiom,
! [I2: set_a,J: set_set_a,A: a] :
( ( ideal_a_b @ I2 @ r )
=> ( ( ord_le3724670747650509150_set_a @ J @ ( partia5907974310037520643t_unit @ ( factRing_a_b @ r @ I2 ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( comple2307003609928055243_set_a @ J ) )
= ( member_set_a @ ( a_r_coset_a_b @ r @ I2 @ A ) @ J ) ) ) ) ) ).
% canonical_proj_vimage_mem_iff
thf(fact_1012_pzf_Ozeroprimeideal__domainI,axiom,
( ( primei7054412349171268410t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.zeroprimeideal_domainI
thf(fact_1013_pzf_Odomain__eq__zeroprimeideal,axiom,
( ( domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( primei7054412349171268410t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.domain_eq_zeroprimeideal
thf(fact_1014_local_Ominus__minus,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_inv_a_b @ r @ ( a_inv_a_b @ r @ X ) )
= X ) ) ).
% local.minus_minus
thf(fact_1015_a__inv__closed,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( a_inv_a_b @ r @ X ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% a_inv_closed
thf(fact_1016_a__closed,axiom,
! [X: a,Y: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_a @ ( add_a_b @ r @ X @ Y ) @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% a_closed
thf(fact_1017_local_Oadd_Oright__cancel,axiom,
! [X: a,Y: a,Z: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Y @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ Z @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ Y @ X )
= ( add_a_b @ r @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% local.add.right_cancel
thf(fact_1018_local_Ominus__zero,axiom,
( ( a_inv_a_b @ r @ ( zero_a_b @ r ) )
= ( zero_a_b @ r ) ) ).
% local.minus_zero
thf(fact_1019_domain__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( domain4236798911309298543t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% domain_iff_prime
thf(fact_1020_zf_Ocanonical__proj__vimage__mem__iff,axiom,
! [I2: set_set_int,J: set_set_set_int,A: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( ord_le4317611570275147438et_int @ J @ ( partia1981568256555877781t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( comple7281953568134767595et_int @ J ) )
= ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ A ) @ J ) ) ) ) ) ).
% zf.canonical_proj_vimage_mem_iff
thf(fact_1021_zf_Odomain__iff__prime,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( domain2593739924244766771t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) )
= ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) ) ).
% zf.domain_iff_prime
thf(fact_1022_add_Oinv__eq__1__iff,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( a_inv_a_b @ r @ X )
= ( zero_a_b @ r ) )
= ( X
= ( zero_a_b @ r ) ) ) ) ).
% add.inv_eq_1_iff
thf(fact_1023_r__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ X @ ( zero_a_b @ r ) )
= X ) ) ).
% r_zero
thf(fact_1024_l__zero,axiom,
! [X: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( add_a_b @ r @ ( zero_a_b @ r ) @ X )
= X ) ) ).
% l_zero
thf(fact_1025_add_Or__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ A @ X ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one'
thf(fact_1026_add_Or__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ A @ X )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.r_cancel_one
thf(fact_1027_add_Ol__cancel__one_H,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( X
= ( add_a_b @ r @ X @ A ) )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one'
thf(fact_1028_add_Ol__cancel__one,axiom,
! [X: a,A: a] :
( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( add_a_b @ r @ X @ A )
= X )
= ( A
= ( zero_a_b @ r ) ) ) ) ) ).
% add.l_cancel_one
thf(fact_1029_zf_Ominus__minus,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) )
= X ) ) ).
% zf.minus_minus
thf(fact_1030_zf_Oadd_Oinv__closed,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.add.inv_closed
thf(fact_1031_zf_Oadd_Oright__cancel,axiom,
! [X: set_int,Y: set_int,Z: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Z @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y @ X )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Z @ X ) )
= ( Y = Z ) ) ) ) ) ).
% zf.add.right_cancel
thf(fact_1032_zf_Oadd_Om__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.add.m_closed
thf(fact_1033_zf_Ominus__zero,axiom,
( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.minus_zero
thf(fact_1034_a__coset__add__zero,axiom,
! [M: set_a] :
( ( ord_less_eq_set_a @ M @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( a_r_coset_a_b @ r @ M @ ( zero_a_b @ r ) )
= M ) ) ).
% a_coset_add_zero
thf(fact_1035_zf_Oadd_Oinv__eq__1__iff,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
= ( X
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.add.inv_eq_1_iff
thf(fact_1036_zf_Or__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
= X ) ) ).
% zf.r_zero
thf(fact_1037_zf_Ol__zero,axiom,
! [X: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ X )
= X ) ) ).
% zf.l_zero
thf(fact_1038_zf_Oadd_Or__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ X ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.add.r_cancel_one'
thf(fact_1039_zf_Oadd_Or__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ X )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.add.r_cancel_one
thf(fact_1040_zf_Oadd_Ol__cancel__one_H,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( X
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ A ) )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.add.l_cancel_one'
thf(fact_1041_zf_Oadd_Ol__cancel__one,axiom,
! [X: set_int,A: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ A )
= X )
= ( A
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% zf.add.l_cancel_one
thf(fact_1042_zf_Oa__coset__add__zero,axiom,
! [M: set_set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ M @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
= M ) ) ).
% zf.a_coset_add_zero
thf(fact_1043_h_Ohom__add,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y ) ) @ x )
= ( add_a_b @ r @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ Y ) @ x ) ) ) ) ) ).
% h.hom_add
thf(fact_1044_h_Ohom__a__inv,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) ) @ x )
= ( a_inv_a_b @ r @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ X ) @ x ) ) ) ) ).
% h.hom_a_inv
thf(fact_1045_pzf_Olong__division__a__inv_I1_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) @ Q )
= ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) ) ) ) ) ).
% pzf.long_division_a_inv(1)
thf(fact_1046_pzf_Olong__division__add_I1_J,axiom,
! [K: set_list_set_int,A: list_list_set_int,B: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ A @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ B @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ A @ B ) @ Q )
= ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ Q ) @ ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ Q ) ) ) ) ) ) ) ).
% pzf.long_division_add(1)
thf(fact_1047_pzf_Olong__division__closed_I1_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( member7440485371949894425et_int @ ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ) ).
% pzf.long_division_closed(1)
thf(fact_1048_pzf_Ois__root__poly__mult__imp__is__root,axiom,
! [P2: list_list_set_int,Q: list_list_set_int,X: list_set_int] :
( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
=> ( ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ P2 @ Q ) @ X )
=> ( ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ X )
| ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q @ X ) ) ) ) ) ).
% pzf.is_root_poly_mult_imp_is_root
thf(fact_1049_zf_Oquotient__eq__iff__same__a__r__cos,axiom,
! [I2: set_set_int,A: set_int,B: set_int] :
( ( ideal_7262958097527394045t_unit @ I2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B ) @ I2 )
= ( ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ A )
= ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ I2 @ B ) ) ) ) ) ) ).
% zf.quotient_eq_iff_same_a_r_cos
thf(fact_1050_zf_Oa__rcosetsI,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_set_int @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ X ) @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 ) ) ) ) ).
% zf.a_rcosetsI
thf(fact_1051_zf_Ominus__eq,axiom,
! [X: set_int,Y: set_int] :
( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y )
= ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Y ) ) ) ).
% zf.minus_eq
thf(fact_1052_zf_Ominus__closed,axiom,
! [X: set_int,Y: set_int] :
( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ Y @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_set_int @ ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ Y ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.minus_closed
thf(fact_1053_zf_Or__right__minus__eq,axiom,
! [A: set_int,B: set_int] :
( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ B @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( a_minu5974516859897376926t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ B )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
= ( A = B ) ) ) ) ).
% zf.r_right_minus_eq
thf(fact_1054_zf_Orcosets__subset__PowG,axiom,
! [H2: set_set_int] :
( ( additi7073586575563672860t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ord_le4317611570275147438et_int @ ( a_RCOS5559887075240879033t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 ) @ ( pow_set_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.rcosets_subset_PowG
thf(fact_1055_pzf_Opdiv__pmod,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( P2
= ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ Q @ ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) ) ) ) ) ).
% pzf.pdiv_pmod
thf(fact_1056_zf_Oa__rcos__assoc__lcos,axiom,
! [H2: set_set_int,K: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_r_co692709266861932262t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ X ) @ K )
= ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ K ) ) ) ) ) ) ).
% zf.a_rcos_assoc_lcos
thf(fact_1057_zf_Oadd__additive__subgroups,axiom,
! [H2: set_set_int,K: set_set_int] :
( ( additi7073586575563672860t_unit @ H2 @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( additi7073586575563672860t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( additi7073586575563672860t_unit @ ( set_ad273131178244904872t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H2 @ K ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.add_additive_subgroups
thf(fact_1058_zf_Oa__l__coset__subset__G,axiom,
! [H2: set_set_int,X: set_int] :
( ( ord_le4403425263959731960et_int @ H2 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ord_le4403425263959731960et_int @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ X @ H2 ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% zf.a_l_coset_subset_G
thf(fact_1059_zf_Oa__lcos__mult__one,axiom,
! [M: set_set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ M )
= M ) ) ).
% zf.a_lcos_mult_one
thf(fact_1060_zf_Oa__lcos__m__assoc,axiom,
! [M: set_set_int,G: set_int,H: set_int] :
( ( ord_le4403425263959731960et_int @ M @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ G @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ H @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ G @ ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H @ M ) )
= ( a_l_co3504123944629134560t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ G @ H ) @ M ) ) ) ) ) ).
% zf.a_lcos_m_assoc
thf(fact_1061_pzf_Olong__division__closed_I2_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( member7440485371949894425et_int @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ) ).
% pzf.long_division_closed(2)
thf(fact_1062_pzf_Orupture__surj__composed__with__pmod,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( a_r_co2531912773843275078t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( cgenid7864102925118505423t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) @ Q )
= ( a_r_co2531912773843275078t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( cgenid7864102925118505423t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q @ P2 ) ) ) ) ) ) ).
% pzf.rupture_surj_composed_with_pmod
thf(fact_1063_pzf_Olong__division__add_I2_J,axiom,
! [K: set_list_set_int,A: list_list_set_int,B: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ A @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ B @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ A @ B ) @ Q )
= ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ Q ) @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ Q ) ) ) ) ) ) ) ).
% pzf.long_division_add(2)
thf(fact_1064_pzf_Olong__division__add__iff,axiom,
! [K: set_list_set_int,A: list_list_set_int,B: list_list_set_int,C: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ A @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ B @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ C @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ Q )
= ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ Q ) )
= ( ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ A @ C ) @ Q )
= ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% pzf.long_division_add_iff
thf(fact_1065_pzf_Olong__division__a__inv_I2_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) @ Q )
= ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) ) ) ) ) ).
% pzf.long_division_a_inv(2)
thf(fact_1066_pzf_OpprimeE_I1_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ring_r7068619596278734765t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
=> ( P2 != nil_list_set_int ) ) ) ) ).
% pzf.pprimeE(1)
thf(fact_1067_pzf_Olong__division__zero_I1_J,axiom,
! [K: set_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ nil_list_set_int @ Q )
= nil_list_set_int ) ) ) ).
% pzf.long_division_zero(1)
thf(fact_1068_pzf_Omap__in__poly__ring__carrier,axiom,
! [P2: list_list_set_int,F: list_set_int > list_list_set_int] :
( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
=> ( ! [A3: list_set_int] :
( ( member_list_set_int @ A3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member7440485371949894425et_int @ ( F @ A3 ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) )
=> ( ! [A3: list_set_int] :
( ( A3
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( F @ A3 )
!= nil_list_set_int ) )
=> ( member6183427284096218281et_int @ ( map_li4936211705635448506et_int @ F @ P2 ) @ ( partia4593313445827799451t_unit @ ( univ_p4950887682782688900t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% pzf.map_in_poly_ring_carrier
thf(fact_1069_pzf_Olong__division__zero_I2_J,axiom,
! [K: set_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ nil_list_set_int @ Q )
= nil_list_set_int ) ) ) ).
% pzf.long_division_zero(2)
thf(fact_1070_pzf_Oexists__unique__long__division,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( Q != nil_list_set_int )
=> ? [X3: produc2061375659605276261et_int] :
( ( polyno7057773693803059378t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q @ X3 )
& ! [Y7: produc2061375659605276261et_int] :
( ( polyno7057773693803059378t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q @ Y7 )
=> ( Y7 = X3 ) ) ) ) ) ) ) ).
% pzf.exists_unique_long_division
thf(fact_1071_pzf_OpprimeI,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( P2 != nil_list_set_int )
=> ( ~ ( member7440485371949894425et_int @ P2 @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ! [Q3: list_list_set_int,R2: list_list_set_int] :
( ( member7440485371949894425et_int @ Q3 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ R2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ Q3 @ R2 ) )
=> ( ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q3 )
| ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ R2 ) ) ) ) )
=> ( ring_r7068619596278734765t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) ) ) ) ) ) ).
% pzf.pprimeI
thf(fact_1072_pzf_Ozero__pdivides,axiom,
! [P2: list_list_set_int] :
( ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ nil_list_set_int @ P2 )
= ( P2 = nil_list_set_int ) ) ).
% pzf.zero_pdivides
thf(fact_1073_pzf_Ozero__pdivides__zero,axiom,
polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ nil_list_set_int @ nil_list_set_int ).
% pzf.zero_pdivides_zero
thf(fact_1074_pzf_Opdivides__iff__shell,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q )
= ( factor668791974002123612t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 @ Q ) ) ) ) ) ).
% pzf.pdivides_iff_shell
thf(fact_1075_pzf_Opmod__zero__iff__pdivides,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q )
= nil_list_set_int )
= ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q @ P2 ) ) ) ) ) ).
% pzf.pmod_zero_iff_pdivides
thf(fact_1076_pzf_OpprimeE_I3_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int,R: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ring_r7068619596278734765t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ R @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ Q @ R ) )
=> ( ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q )
| ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ R ) ) ) ) ) ) ) ) ).
% pzf.pprimeE(3)
thf(fact_1077_pzf_Osame__pmod__iff__pdivides,axiom,
! [K: set_list_set_int,A: list_list_set_int,B: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ A @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ B @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ Q )
= ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ B @ Q ) )
= ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q @ ( a_minu1053138715659470590t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ A @ B ) ) ) ) ) ) ) ).
% pzf.same_pmod_iff_pdivides
thf(fact_1078_pzf_Orupture__eq__0__iff,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ( a_r_co2531912773843275078t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( cgenid7864102925118505423t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) @ Q )
= ( zero_s7811746193054374799t_unit @ ( polyno4866556214859693103t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ P2 ) ) )
= ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) ) ) ) ).
% pzf.rupture_eq_0_iff
thf(fact_1079_rupture__eq__0__iff,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
= ( zero_s2910681146719230829t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) ) )
= ( polyno5814909790663948098es_a_b @ r @ P2 @ Q ) ) ) ) ) ).
% rupture_eq_0_iff
thf(fact_1080_pprimeE_I3_J,axiom,
! [K: set_a,P2: list_a,Q: list_a,R: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R ) ) ) ) ) ) ) ) ).
% pprimeE(3)
thf(fact_1081_pdivides__iff__shell,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
= ( factor1757716651909850160t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q ) ) ) ) ) ).
% pdivides_iff_shell
thf(fact_1082_pprimeE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pprimeE(2)
thf(fact_1083_pprime__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% pprime_iff_pirreducible
thf(fact_1084_univ__poly__is__principal,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ring_p8098905331641078952t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_principal
thf(fact_1085_carrier__is__subfield,axiom,
subfield_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subfield
thf(fact_1086_subring__props_I2_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( zero_a_b @ r ) @ K ) ) ).
% subring_props(2)
thf(fact_1087_subring__props_I7_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( add_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(7)
thf(fact_1088_subring__props_I6_J,axiom,
! [K: set_a,H1: a,H22: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H1 @ K )
=> ( ( member_a @ H22 @ K )
=> ( member_a @ ( mult_a_ring_ext_a_b @ r @ H1 @ H22 ) @ K ) ) ) ) ).
% subring_props(6)
thf(fact_1089_subring__props_I4_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( K != bot_bot_set_a ) ) ).
% subring_props(4)
thf(fact_1090_subring__props_I5_J,axiom,
! [K: set_a,H: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ H @ K )
=> ( member_a @ ( a_inv_a_b @ r @ H ) @ K ) ) ) ).
% subring_props(5)
thf(fact_1091_subring__props_I1_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( ord_less_eq_set_a @ K @ ( partia707051561876973205xt_a_b @ r ) ) ) ).
% subring_props(1)
thf(fact_1092_zf_Ocarrier__is__subfield,axiom,
subfie3888952257595785920t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.carrier_is_subfield
thf(fact_1093_zf_Osubring__props_I2_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( member_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ K ) ) ).
% zf.subring_props(2)
thf(fact_1094_zf_Osubring__props_I7_J,axiom,
! [K: set_set_int,H1: set_int,H22: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H22 @ K )
=> ( member_set_int @ ( add_se5859248395121729892t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% zf.subring_props(7)
thf(fact_1095_zf_Osubring__props_I6_J,axiom,
! [K: set_set_int,H1: set_int,H22: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_int @ H1 @ K )
=> ( ( member_set_int @ H22 @ K )
=> ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H1 @ H22 ) @ K ) ) ) ) ).
% zf.subring_props(6)
thf(fact_1096_zf_Osubring__props_I4_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( K != bot_bot_set_set_int ) ) ).
% zf.subring_props(4)
thf(fact_1097_zf_Osubring__props_I5_J,axiom,
! [K: set_set_int,H: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_int @ H @ K )
=> ( member_set_int @ ( a_inv_5951419416477254493t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ H ) @ K ) ) ) ).
% zf.subring_props(5)
thf(fact_1098_pdivides__imp__root__sharing,axiom,
! [P2: list_a,Q: list_a,A: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( ( eval_a_b @ r @ P2 @ A )
= ( zero_a_b @ r ) )
=> ( ( eval_a_b @ r @ Q @ A )
= ( zero_a_b @ r ) ) ) ) ) ) ).
% pdivides_imp_root_sharing
thf(fact_1099_zf_Osubring__props_I1_J,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ord_le4403425263959731960et_int @ K @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.subring_props(1)
thf(fact_1100_zf_Opdivides__iff__shell,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q )
= ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 @ Q ) ) ) ) ) ).
% zf.pdivides_iff_shell
thf(fact_1101_zf_Ouniv__poly__is__principal,axiom,
! [K: set_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ring_p1500425057661402810t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).
% zf.univ_poly_is_principal
thf(fact_1102_zf_OpprimeE_I3_J,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int,R: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ R @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ Q @ R ) )
=> ( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q )
| ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ R ) ) ) ) ) ) ) ) ).
% zf.pprimeE(3)
thf(fact_1103_zf_OpprimeE_I2_J,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
=> ~ ( member_list_set_int @ P2 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ) ) ).
% zf.pprimeE(2)
thf(fact_1104_subfield__m__inv__simprule,axiom,
! [K: set_a,K3: a,A: a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_a @ K3 @ ( minus_minus_set_a @ K @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( member_a @ A @ ( partia707051561876973205xt_a_b @ r ) )
=> ( ( member_a @ ( mult_a_ring_ext_a_b @ r @ K3 @ A ) @ K )
=> ( member_a @ A @ K ) ) ) ) ) ).
% subfield_m_inv_simprule
thf(fact_1105_zf_Opprime__iff__pirreducible,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
= ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) ) ) ) ).
% zf.pprime_iff_pirreducible
thf(fact_1106_zf_Orupture__eq__0__iff,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) @ Q )
= ( zero_s8971087499367319327t_unit @ ( polyno6134647158531630271t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K @ P2 ) ) )
= ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) ) ) ) ) ).
% zf.rupture_eq_0_iff
thf(fact_1107_zf_Osubfield__m__inv__simprule,axiom,
! [K: set_set_int,K3: set_int,A: set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_set_int @ K3 @ ( minus_8897228262479074673et_int @ K @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( member_set_int @ ( mult_s3864001451298473021t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K3 @ A ) @ K )
=> ( member_set_int @ A @ K ) ) ) ) ) ).
% zf.subfield_m_inv_simprule
thf(fact_1108_pzf_Olong__divisionI,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int,B: list_list_set_int,R: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( Q != nil_list_set_int )
=> ( ( polyno7057773693803059378t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q @ ( produc7276067142252306077et_int @ B @ R ) )
=> ( ( produc7276067142252306077et_int @ B @ R )
= ( produc7276067142252306077et_int @ ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) ) ) ) ) ) ) ).
% pzf.long_divisionI
thf(fact_1109_pzf_Olong__divisionE,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( Q != nil_list_set_int )
=> ( polyno7057773693803059378t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q @ ( produc7276067142252306077et_int @ ( polyno5348356287797398593t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) @ ( polyno6644449323419600656t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) ) ) ) ) ) ).
% pzf.long_divisionE
thf(fact_1110_pzf_Opoly__add_Ocases,axiom,
! [X: produc2061375659605276261et_int] :
~ ! [P1: list_list_set_int,P22: list_list_set_int] :
( X
!= ( produc7276067142252306077et_int @ P1 @ P22 ) ) ).
% pzf.poly_add.cases
thf(fact_1111_pzf_Oexists__long__division,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( Q != nil_list_set_int )
=> ~ ! [B2: list_list_set_int] :
( ( member7440485371949894425et_int @ B2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ! [R2: list_list_set_int] :
( ( member7440485371949894425et_int @ R2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ~ ( polyno7057773693803059378t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q @ ( produc7276067142252306077et_int @ B2 @ R2 ) ) ) ) ) ) ) ) ).
% pzf.exists_long_division
thf(fact_1112_zf_Olong__division__a__inv_I1_J,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) @ Q )
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) ) ) ) ) ) ).
% zf.long_division_a_inv(1)
thf(fact_1113_zf_Olong__division__add_I1_J,axiom,
! [K: set_set_int,A: list_set_int,B: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ A @ B ) @ Q )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ Q ) @ ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ Q ) ) ) ) ) ) ) ).
% zf.long_division_add(1)
thf(fact_1114_zf_Olong__division__closed_I1_J,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( member_list_set_int @ ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ) ) ).
% zf.long_division_closed(1)
thf(fact_1115_pzf_Onorm__map__in__poly__ring__carrier,axiom,
! [P2: list_list_set_int,F: list_set_int > list_list_set_int] :
( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
=> ( ! [A3: list_set_int] :
( ( member_list_set_int @ A3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( member7440485371949894425et_int @ ( F @ A3 ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) )
=> ( member6183427284096218281et_int @ ( normal3791444541944621327t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ ( map_li4936211705635448506et_int @ F @ P2 ) ) @ ( partia4593313445827799451t_unit @ ( univ_p4950887682782688900t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ) ) ).
% pzf.norm_map_in_poly_ring_carrier
thf(fact_1116_gauss__poly__carr,axiom,
! [N: nat] : ( member_list_a @ ( card_I2373409586816755191ly_a_b @ r @ N ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ).
% gauss_poly_carr
thf(fact_1117_zero__pdivides,axiom,
! [P2: list_a] :
( ( polyno5814909790663948098es_a_b @ r @ nil_a @ P2 )
= ( P2 = nil_a ) ) ).
% zero_pdivides
thf(fact_1118_zero__pdivides__zero,axiom,
polyno5814909790663948098es_a_b @ r @ nil_a @ nil_a ).
% zero_pdivides_zero
thf(fact_1119_eval_Osimps_I1_J,axiom,
( ( eval_a_b @ r @ nil_a )
= ( ^ [Uu: a] : ( zero_a_b @ r ) ) ) ).
% eval.simps(1)
thf(fact_1120_zf_Ozero__pdivides__zero,axiom,
polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ nil_set_int @ nil_set_int ).
% zf.zero_pdivides_zero
thf(fact_1121_zf_Ozero__pdivides,axiom,
! [P2: list_set_int] :
( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ nil_set_int @ P2 )
= ( P2 = nil_set_int ) ) ).
% zf.zero_pdivides
thf(fact_1122_exists__long__division,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ~ ! [B2: list_a] :
( ( member_list_a @ B2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ! [R2: list_a] :
( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ~ ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ B2 @ R2 ) ) ) ) ) ) ) ) ).
% exists_long_division
thf(fact_1123_long__divisionE,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ) ).
% long_divisionE
thf(fact_1124_long__divisionI,axiom,
! [K: set_a,P2: list_a,Q: list_a,B: list_a,R: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ ( produc6837034575241423639list_a @ B @ R ) )
=> ( ( produc6837034575241423639list_a @ B @ R )
= ( produc6837034575241423639list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ) ) ).
% long_divisionI
thf(fact_1125_long__division__zero_I2_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(2)
thf(fact_1126_long__division__zero_I1_J,axiom,
! [K: set_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ nil_a @ Q )
= nil_a ) ) ) ).
% long_division_zero(1)
thf(fact_1127_exists__unique__long__division,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( Q != nil_a )
=> ? [X3: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ X3 )
& ! [Y7: produc9164743771328383783list_a] :
( ( polyno2806191415236617128es_a_b @ r @ P2 @ Q @ Y7 )
=> ( Y7 = X3 ) ) ) ) ) ) ) ).
% exists_unique_long_division
thf(fact_1128_long__division__add__iff,axiom,
! [K: set_a,A: list_a,B: list_a,C: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ C @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ C ) @ Q )
= ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% long_division_add_iff
thf(fact_1129_long__division__add_I2_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ A @ Q ) @ ( polynomial_pmod_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(2)
thf(fact_1130_long__division__closed_I2_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(2)
thf(fact_1131_long__division__a__inv_I2_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pmod_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).
% long_division_a_inv(2)
thf(fact_1132_long__division__a__inv_I1_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).
% long_division_a_inv(1)
thf(fact_1133_long__division__closed_I1_J,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% long_division_closed(1)
thf(fact_1134_long__division__add_I1_J,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polynomial_pdiv_a_b @ r @ ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) @ Q )
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( polynomial_pdiv_a_b @ r @ A @ Q ) @ ( polynomial_pdiv_a_b @ r @ B @ Q ) ) ) ) ) ) ) ).
% long_division_add(1)
thf(fact_1135_map__in__poly__ring__carrier,axiom,
! [P2: list_a,F: a > list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( ! [A3: a] :
( ( A3
!= ( zero_a_b @ r ) )
=> ( ( F @ A3 )
!= nil_a ) )
=> ( member_list_list_a @ ( map_a_list_a @ F @ P2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ) ).
% map_in_poly_ring_carrier
thf(fact_1136_pmod__zero__iff__pdivides,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ P2 @ Q )
= nil_a )
= ( polyno5814909790663948098es_a_b @ r @ Q @ P2 ) ) ) ) ) ).
% pmod_zero_iff_pdivides
thf(fact_1137_pdiv__pmod,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( P2
= ( add_li7652885771158616974t_unit @ ( univ_poly_a_b @ r @ K ) @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ ( polynomial_pdiv_a_b @ r @ P2 @ Q ) ) @ ( polynomial_pmod_a_b @ r @ P2 @ Q ) ) ) ) ) ) ).
% pdiv_pmod
thf(fact_1138_pprimeE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pprimeE(1)
thf(fact_1139_zf_Oexists__long__division,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( Q != nil_set_int )
=> ~ ! [B2: list_set_int] :
( ( member_list_set_int @ B2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ! [R2: list_set_int] :
( ( member_list_set_int @ R2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ~ ( polyno4953732871684447938t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q @ ( produc1446815805028224253et_int @ B2 @ R2 ) ) ) ) ) ) ) ) ).
% zf.exists_long_division
thf(fact_1140_zf_Olong__division__zero_I2_J,axiom,
! [K: set_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ nil_set_int @ Q )
= nil_set_int ) ) ) ).
% zf.long_division_zero(2)
thf(fact_1141_zf_Oexists__unique__long__division,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( Q != nil_set_int )
=> ? [X3: produc6642921209907155909et_int] :
( ( polyno4953732871684447938t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q @ X3 )
& ! [Y7: produc6642921209907155909et_int] :
( ( polyno4953732871684447938t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q @ Y7 )
=> ( Y7 = X3 ) ) ) ) ) ) ) ).
% zf.exists_unique_long_division
thf(fact_1142_zf_Olong__division__closed_I2_J,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( member_list_set_int @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ) ) ).
% zf.long_division_closed(2)
thf(fact_1143_rupture__surj__composed__with__pmod,axiom,
! [K: set_a,P2: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ Q )
= ( a_r_co1577572119920843660t_unit @ ( univ_poly_a_b @ r @ K ) @ ( cgenid9131348535277946915t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) @ ( polynomial_pmod_a_b @ r @ Q @ P2 ) ) ) ) ) ) ).
% rupture_surj_composed_with_pmod
thf(fact_1144_zf_Olong__division__add__iff,axiom,
! [K: set_set_int,A: list_set_int,B: list_set_int,C: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ C @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ Q )
= ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ Q ) )
= ( ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ A @ C ) @ Q )
= ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ B @ C ) @ Q ) ) ) ) ) ) ) ) ).
% zf.long_division_add_iff
thf(fact_1145_zf_Olong__division__add_I2_J,axiom,
! [K: set_set_int,A: list_set_int,B: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ A @ B ) @ Q )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ Q ) @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ Q ) ) ) ) ) ) ) ).
% zf.long_division_add(2)
thf(fact_1146_zf_Olong__division__a__inv_I2_J,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) @ Q )
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) ) ) ) ) ) ).
% zf.long_division_a_inv(2)
thf(fact_1147_zf_Opmod__zero__iff__pdivides,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q )
= nil_set_int )
= ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Q @ P2 ) ) ) ) ) ).
% zf.pmod_zero_iff_pdivides
thf(fact_1148_zf_OpprimeE_I1_J,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
=> ( P2 != nil_set_int ) ) ) ) ).
% zf.pprimeE(1)
thf(fact_1149_zf_Olong__divisionI,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int,B: list_set_int,R: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( Q != nil_set_int )
=> ( ( polyno4953732871684447938t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q @ ( produc1446815805028224253et_int @ B @ R ) )
=> ( ( produc1446815805028224253et_int @ B @ R )
= ( produc1446815805028224253et_int @ ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) ) ) ) ) ) ) ) ).
% zf.long_divisionI
thf(fact_1150_zf_Olong__divisionE,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( Q != nil_set_int )
=> ( polyno4953732871684447938t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q @ ( produc1446815805028224253et_int @ ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) ) ) ) ) ) ) ).
% zf.long_divisionE
thf(fact_1151_zf_Olong__division__zero_I1_J,axiom,
! [K: set_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ nil_set_int @ Q )
= nil_set_int ) ) ) ).
% zf.long_division_zero(1)
thf(fact_1152_zf_Orupture__surj__composed__with__pmod,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) @ Q )
= ( a_r_co912825291596486358t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Q @ P2 ) ) ) ) ) ) ).
% zf.rupture_surj_composed_with_pmod
thf(fact_1153_zf_Osame__pmod__iff__pdivides,axiom,
! [K: set_set_int,A: list_set_int,B: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ B @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A @ Q )
= ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ B @ Q ) )
= ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Q @ ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ A @ B ) ) ) ) ) ) ) ).
% zf.same_pmod_iff_pdivides
thf(fact_1154_zf_Omap__in__poly__ring__carrier,axiom,
! [P2: list_set_int,F: set_int > list_set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [A3: set_int] :
( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_list_set_int @ ( F @ A3 ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) )
=> ( ! [A3: set_int] :
( ( A3
!= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( F @ A3 )
!= nil_set_int ) )
=> ( member7440485371949894425et_int @ ( map_se7207430148037755802et_int @ F @ P2 ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ) ).
% zf.map_in_poly_ring_carrier
thf(fact_1155_zf_Opdiv__pmod,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( P2
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ Q @ ( polyno2948870938714006993t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) ) @ ( polyno6711862844499874848t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q ) ) ) ) ) ) ).
% zf.pdiv_pmod
thf(fact_1156_pprimeI,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R2: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R2 ) )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q3 )
| ( polyno5814909790663948098es_a_b @ r @ P2 @ R2 ) ) ) ) )
=> ( ring_r6430282645014804837t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pprimeI
thf(fact_1157_zf_OpprimeI,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( P2 != nil_set_int )
=> ( ~ ( member_list_set_int @ P2 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ! [Q3: list_set_int,R2: list_set_int] :
( ( member_list_set_int @ Q3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ R2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ Q3 @ R2 ) )
=> ( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q3 )
| ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ R2 ) ) ) ) )
=> ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) ) ) ) ) ) ).
% zf.pprimeI
thf(fact_1158_zf_Opdivides__imp__splitted,axiom,
! [P2: list_set_int,Q: list_set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( Q != nil_set_int )
=> ( ( polyno2166899109720230297t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Q )
=> ( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q )
=> ( polyno2166899109720230297t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 ) ) ) ) ) ) ).
% zf.pdivides_imp_splitted
thf(fact_1159_pdivides__imp__splitted,axiom,
! [P2: list_a,Q: list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( Q != nil_a )
=> ( ( polyno8329700637149614481ed_a_b @ r @ Q )
=> ( ( polyno5814909790663948098es_a_b @ r @ P2 @ Q )
=> ( polyno8329700637149614481ed_a_b @ r @ P2 ) ) ) ) ) ) ).
% pdivides_imp_splitted
thf(fact_1160_zf_Opoly__add_Ocases,axiom,
! [X: produc6642921209907155909et_int] :
~ ! [P1: list_set_int,P22: list_set_int] :
( X
!= ( produc1446815805028224253et_int @ P1 @ P22 ) ) ).
% zf.poly_add.cases
thf(fact_1161_poly__add_Ocases,axiom,
! [X: produc9164743771328383783list_a] :
~ ! [P1: list_a,P22: list_a] :
( X
!= ( produc6837034575241423639list_a @ P1 @ P22 ) ) ).
% poly_add.cases
thf(fact_1162_zf_Oeval_Osimps_I1_J,axiom,
( ( eval_s3798525405962683066t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ nil_set_int )
= ( ^ [Uu: set_int] : ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.eval.simps(1)
thf(fact_1163_is__root__def,axiom,
! [P2: list_a,X: a] :
( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
= ( ( member_a @ X @ ( partia707051561876973205xt_a_b @ r ) )
& ( ( eval_a_b @ r @ P2 @ X )
= ( zero_a_b @ r ) )
& ( P2 != nil_a ) ) ) ).
% is_root_def
thf(fact_1164_is__root__poly__mult__imp__is__root,axiom,
! [P2: list_a,Q: list_a,X: a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ( polyno4133073214067823460ot_a_b @ r @ ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) @ X )
=> ( ( polyno4133073214067823460ot_a_b @ r @ P2 @ X )
| ( polyno4133073214067823460ot_a_b @ r @ Q @ X ) ) ) ) ) ).
% is_root_poly_mult_imp_is_root
thf(fact_1165_pzf_Onormalize_Osimps_I1_J,axiom,
( ( normal6870180608241117983t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ nil_list_set_int )
= nil_list_set_int ) ).
% pzf.normalize.simps(1)
thf(fact_1166_same__pmod__iff__pdivides,axiom,
! [K: set_a,A: list_a,B: list_a,Q: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ A @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ B @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ( polynomial_pmod_a_b @ r @ A @ Q )
= ( polynomial_pmod_a_b @ r @ B @ Q ) )
= ( polyno5814909790663948098es_a_b @ r @ Q @ ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ A @ B ) ) ) ) ) ) ) ).
% same_pmod_iff_pdivides
thf(fact_1167_pzf_Oeval_Osimps_I1_J,axiom,
( ( eval_l2744756199357571242t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ nil_list_set_int )
= ( ^ [Uu: list_set_int] : ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.eval.simps(1)
thf(fact_1168_zf_Ois__root__def,axiom,
! [P2: list_set_int,X: set_int] :
( ( polyno6587735422648734726t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ X )
= ( ( member_set_int @ X @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
& ( ( eval_s3798525405962683066t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ X )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
& ( P2 != nil_set_int ) ) ) ).
% zf.is_root_def
thf(fact_1169_zf_Opdivides__imp__root__sharing,axiom,
! [P2: list_set_int,Q: list_set_int,A: set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ Q )
=> ( ( member_set_int @ A @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( ( eval_s3798525405962683066t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ A )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( ( eval_s3798525405962683066t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ Q @ A )
= ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% zf.pdivides_imp_root_sharing
thf(fact_1170_pzf_Ois__root__def,axiom,
! [P2: list_list_set_int,X: list_set_int] :
( ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ X )
= ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( ( eval_l2744756199357571242t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ X )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( P2 != nil_list_set_int ) ) ) ).
% pzf.is_root_def
thf(fact_1171_zf_Onorm__map__in__poly__ring__carrier,axiom,
! [P2: list_set_int,F: set_int > list_set_int] :
( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [A3: set_int] :
( ( member_set_int @ A3 @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) )
=> ( member_list_set_int @ ( F @ A3 ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) )
=> ( member7440485371949894425et_int @ ( normal6870180608241117983t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( map_se7207430148037755802et_int @ F @ P2 ) ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% zf.norm_map_in_poly_ring_carrier
thf(fact_1172_norm__map__in__poly__ring__carrier,axiom,
! [P2: list_a,F: a > list_a] :
( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) )
=> ( ! [A3: a] :
( ( member_a @ A3 @ ( partia707051561876973205xt_a_b @ r ) )
=> ( member_list_a @ ( F @ A3 ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) )
=> ( member_list_list_a @ ( normal637505603836502915t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( map_a_list_a @ F @ P2 ) ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ) ) ).
% norm_map_in_poly_ring_carrier
thf(fact_1173_pzf_Opdivides__imp__root__sharing,axiom,
! [P2: list_list_set_int,Q: list_list_set_int,A: list_set_int] :
( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
=> ( ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q )
=> ( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( eval_l2744756199357571242t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ A )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( eval_l2744756199357571242t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q @ A )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.pdivides_imp_root_sharing
thf(fact_1174_pzf_OpirreducibleI,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( P2 != nil_list_set_int )
=> ( ~ ( member7440485371949894425et_int @ P2 @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ! [Q3: list_list_set_int,R2: list_list_set_int] :
( ( member7440485371949894425et_int @ Q3 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ R2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( P2
= ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ Q3 @ R2 ) )
=> ( ( member7440485371949894425et_int @ Q3 @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
| ( member7440485371949894425et_int @ R2 @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ) )
=> ( ring_r5263416876369801476t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) ) ) ) ) ) ).
% pzf.pirreducibleI
thf(fact_1175_pzf_Ofield__iff__prime,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( ( field_6001781496977528197t_unit @ ( factRi8995599497114439928t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) )
= ( ring_r7101923692879464637t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A ) ) ) ).
% pzf.field_iff_prime
thf(fact_1176_pzf_Ocarrier__is__subring,axiom,
subrin291743070294386232t_unit @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% pzf.carrier_is_subring
thf(fact_1177_pzf_Ouniv__poly__is__domain,axiom,
! [K: set_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( domain1893962698156629705t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).
% pzf.univ_poly_is_domain
thf(fact_1178_pzf_Ouniv__poly__is__ring,axiom,
! [K: set_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ring_l6429277945453189341t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).
% pzf.univ_poly_is_ring
thf(fact_1179_pzf_Opdivides__zero,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ nil_list_set_int ) ) ) ).
% pzf.pdivides_zero
thf(fact_1180_pzf_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
= ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ P2 ) ) ) ) ).
% pzf.univ_poly_a_inv_consistent
thf(fact_1181_pzf_OpirreducibleE_I1_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ring_r5263416876369801476t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
=> ( P2 != nil_list_set_int ) ) ) ) ).
% pzf.pirreducibleE(1)
thf(fact_1182_zf_Orupture__is__field__iff__pirreducible,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subfie3888952257595785920t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( field_6001781496977528197t_unit @ ( polyno6134647158531630271t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K @ P2 ) )
= ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) ) ) ) ).
% zf.rupture_is_field_iff_pirreducible
thf(fact_1183_pzf_OpirreducibleE_I2_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ring_r5263416876369801476t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
=> ~ ( member7440485371949894425et_int @ P2 @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ) ).
% pzf.pirreducibleE(2)
thf(fact_1184_pzf_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_list_set_int,Q: list_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( a_minu1053138715659470590t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 @ Q )
= ( a_minu1053138715659470590t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ P2 @ Q ) ) ) ) ).
% pzf.univ_poly_a_minus_consistent
thf(fact_1185_pzf_OpirreducibleE_I3_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int,R: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( ring_r5263416876369801476t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ R @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( P2
= ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ Q @ R ) )
=> ( ( member7440485371949894425et_int @ Q @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
| ( member7440485371949894425et_int @ R @ ( units_2543522835650936210t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ) ) ) ) ) ).
% pzf.pirreducibleE(3)
thf(fact_1186_pzf_Ocarrier__polynomial__shell,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.carrier_polynomial_shell
thf(fact_1187_pzf_Omap__norm__in__poly__ring__carrier,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( member6183427284096218281et_int @ ( map_li4936211705635448506et_int @ ( poly_o1782926711940423372t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ P2 ) @ ( partia4593313445827799451t_unit @ ( univ_p4950887682782688900t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ) ) ).
% pzf.map_norm_in_poly_ring_carrier
thf(fact_1188_pzf_Opderiv__mult,axiom,
! [K: set_list_set_int,F: list_list_set_int,G: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ G @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F @ G ) )
= ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) @ G ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G ) ) ) ) ) ) ) ).
% pzf.pderiv_mult
thf(fact_1189_local_Ofield__axioms,axiom,
field_a_b @ r ).
% local.field_axioms
thf(fact_1190_carrier__is__subring,axiom,
subring_a_b @ ( partia707051561876973205xt_a_b @ r ) @ r ).
% carrier_is_subring
thf(fact_1191_univ__poly__not__field,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ~ ( field_6388047844668329575t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_not_field
thf(fact_1192_univ__poly__is__ring,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( ring_l6212528067271185461t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_ring
thf(fact_1193_univ__poly__is__domain,axiom,
! [K: set_a] :
( ( subring_a_b @ K @ r )
=> ( domain6553523120543210313t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ).
% univ_poly_is_domain
thf(fact_1194_zf_Ofield__axioms,axiom,
field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.field_axioms
thf(fact_1195_zf_Ocarrier__is__subring,axiom,
subrin7689096310803670856t_unit @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ).
% zf.carrier_is_subring
thf(fact_1196_zf_Ouniv__poly__not__field,axiom,
! [K: set_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ~ ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).
% zf.univ_poly_not_field
thf(fact_1197_zf_Ouniv__poly__is__domain,axiom,
! [K: set_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( domain1631227219952682457t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).
% zf.univ_poly_is_domain
thf(fact_1198_pdivides__zero,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( polyno5814909790663948098es_a_b @ r @ P2 @ nil_a ) ) ) ).
% pdivides_zero
thf(fact_1199_zf_Ouniv__poly__is__ring,axiom,
! [K: set_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ring_l5834914389771393517t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ).
% zf.univ_poly_is_ring
thf(fact_1200_univ__poly__a__inv__consistent,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
= ( a_inv_8944721093294617173t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 ) ) ) ) ).
% univ_poly_a_inv_consistent
thf(fact_1201_pirreducibleE_I1_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( P2 != nil_a ) ) ) ) ).
% pirreducibleE(1)
thf(fact_1202_univ__poly__a__minus__consistent,axiom,
! [K: set_a,Q: list_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 @ Q )
= ( a_minu3984020753470702548t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) @ P2 @ Q ) ) ) ) ).
% univ_poly_a_minus_consistent
thf(fact_1203_pirreducibleE_I2_J,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ).
% pirreducibleE(2)
thf(fact_1204_zeromaximalideal__fieldI,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
=> ( field_a_b @ r ) ) ).
% zeromaximalideal_fieldI
thf(fact_1205_zeromaximalideal__eq__field,axiom,
( ( maximalideal_a_b @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) @ r )
= ( field_a_b @ r ) ) ).
% zeromaximalideal_eq_field
thf(fact_1206_zf_Ouniv__poly__a__inv__consistent,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ) ).
% zf.univ_poly_a_inv_consistent
thf(fact_1207_rupture__is__field__iff__pirreducible,axiom,
! [K: set_a,P2: list_a] :
( ( subfield_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( field_26233345952514695t_unit @ ( polyno5459750281392823787re_a_b @ r @ K @ P2 ) )
= ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ).
% rupture_is_field_iff_pirreducible
thf(fact_1208_pzf_Ouniv__poly__not__field,axiom,
! [K: set_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ~ ( field_4857232450668044075t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).
% pzf.univ_poly_not_field
thf(fact_1209_pirreducibleE_I3_J,axiom,
! [K: set_a,P2: list_a,Q: list_a,R: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 )
=> ( ( member_list_a @ Q @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q @ R ) )
=> ( ( member_list_a @ Q @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ) ) ) ).
% pirreducibleE(3)
thf(fact_1210_zf_Opdivides__zero,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( polyno1573582608993444520t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ P2 @ nil_set_int ) ) ) ).
% zf.pdivides_zero
thf(fact_1211_zf_Ouniv__poly__a__minus__consistent,axiom,
! [K: set_set_int,Q: list_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 @ Q )
= ( a_minu3289003089969826958t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ Q ) ) ) ) ).
% zf.univ_poly_a_minus_consistent
thf(fact_1212_zf_OpirreducibleE_I1_J,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
=> ( P2 != nil_set_int ) ) ) ) ).
% zf.pirreducibleE(1)
thf(fact_1213_pzf_Opderiv__zero,axiom,
! [K: set_list_set_int] :
( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( zero_l4263514192383796165t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
= ( zero_l4263514192383796165t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ).
% pzf.pderiv_zero
thf(fact_1214_zf_OpirreducibleE_I2_J,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
=> ~ ( member_list_set_int @ P2 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ) ) ).
% zf.pirreducibleE(2)
thf(fact_1215_cring__fieldI,axiom,
( ( ( units_a_ring_ext_a_b @ r )
= ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( field_a_b @ r ) ) ).
% cring_fieldI
thf(fact_1216_zf_Ozeromaximalideal__eq__field,axiom,
( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
= ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.zeromaximalideal_eq_field
thf(fact_1217_zf_Ozeromaximalideal__fieldI,axiom,
( ( maxima6262477034536100350t_unit @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.zeromaximalideal_fieldI
thf(fact_1218_pirreducibleI,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2 != nil_a )
=> ( ~ ( member_list_a @ P2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ! [Q3: list_a,R2: list_a] :
( ( member_list_a @ Q3 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( member_list_a @ R2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( ( P2
= ( mult_l7073676228092353617t_unit @ ( univ_poly_a_b @ r @ K ) @ Q3 @ R2 ) )
=> ( ( member_list_a @ Q3 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) )
| ( member_list_a @ R2 @ ( units_2932844235741507942t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) )
=> ( ring_r932985474545269838t_unit @ ( univ_poly_a_b @ r @ K ) @ P2 ) ) ) ) ) ) ).
% pirreducibleI
thf(fact_1219_zf_OpirreducibleE_I3_J,axiom,
! [K: set_set_int,P2: list_set_int,Q: list_set_int,R: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 )
=> ( ( member_list_set_int @ Q @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ R @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( P2
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ Q @ R ) )
=> ( ( member_list_set_int @ Q @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
| ( member_list_set_int @ R @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ) ) ) ) ) ) ).
% zf.pirreducibleE(3)
thf(fact_1220_zf_Ocring__fieldI,axiom,
( ( ( units_4038138251425117394t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
= ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( field_5943785737635511755t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ).
% zf.cring_fieldI
thf(fact_1221_zf_OpirreducibleI,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( P2 != nil_set_int )
=> ( ~ ( member_list_set_int @ P2 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ! [Q3: list_set_int,R2: list_set_int] :
( ( member_list_set_int @ Q3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( member_list_set_int @ R2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( ( P2
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ Q3 @ R2 ) )
=> ( ( member_list_set_int @ Q3 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
| ( member_list_set_int @ R2 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ) ) )
=> ( ring_r7480558297017414036t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ P2 ) ) ) ) ) ) ).
% zf.pirreducibleI
thf(fact_1222_pzf_Opderiv__carr,axiom,
! [K: set_list_set_int,F: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( member7440485371949894425et_int @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) ) ) ) ).
% pzf.pderiv_carr
thf(fact_1223_pzf_Ozeromaximalideal__eq__field,axiom,
( ( maxima7372504373286735342t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.zeromaximalideal_eq_field
thf(fact_1224_pzf_Ozeromaximalideal__fieldI,axiom,
( ( maxima7372504373286735342t_unit @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.zeromaximalideal_fieldI
thf(fact_1225_field__iff__prime,axiom,
! [A: a] :
( ( member_a @ A @ ( minus_minus_set_a @ ( partia707051561876973205xt_a_b @ r ) @ ( insert_a @ ( zero_a_b @ r ) @ bot_bot_set_a ) ) )
=> ( ( field_6045675692312731021t_unit @ ( factRing_a_b @ r @ ( cgenid547466209912283029xt_a_b @ r @ A ) ) )
= ( ring_ring_prime_a_b @ r @ A ) ) ) ).
% field_iff_prime
thf(fact_1226_pzf_Opderiv__add,axiom,
! [K: set_list_set_int,F: list_list_set_int,G: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ G @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F @ G ) )
= ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ G ) ) ) ) ) ) ).
% pzf.pderiv_add
thf(fact_1227_pzf_Orupture__is__field__iff__pirreducible,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( field_7531927466519412469t_unit @ ( polyno4866556214859693103t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ P2 ) )
= ( ring_r5263416876369801476t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) ) ) ) ).
% pzf.rupture_is_field_iff_pirreducible
thf(fact_1228_pzf_Opderiv__inv,axiom,
! [K: set_list_set_int,F: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ F @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ F ) )
= ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ ( formal2278089127137682990t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F ) ) ) ) ) ).
% pzf.pderiv_inv
thf(fact_1229_pzf_Ocring__fieldI,axiom,
( ( ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.cring_fieldI
thf(fact_1230_zf_Ofield__iff__prime,axiom,
! [A: set_int] :
( ( member_set_int @ A @ ( minus_8897228262479074673et_int @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ ( insert_set_int @ ( zero_s6269048424454532197t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ bot_bot_set_set_int ) ) )
=> ( ( field_4969826516285655701t_unit @ ( factRi3149420076008518152t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( cgenid8502489213727343375t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) )
= ( ring_r2768735206093983821t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ A ) ) ) ).
% zf.field_iff_prime
thf(fact_1231_carrier__polynomial__shell,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ ( partia707051561876973205xt_a_b @ r ) ) ) ) ) ) ).
% carrier_polynomial_shell
thf(fact_1232_zf_Ocarrier__polynomial__shell,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% zf.carrier_polynomial_shell
thf(fact_1233_pzf_OsubdomainI_H,axiom,
! [H2: set_list_set_int] :
( ( subrin291743070294386232t_unit @ H2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( subdom6469697055745072292t_unit @ H2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.subdomainI'
thf(fact_1234_pzf_Oconst__term__simprules__shell_I4_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( a_inv_9003328026414236349t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 ) )
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) ) ) ) ) ).
% pzf.const_term_simprules_shell(4)
thf(fact_1235_pzf_Oconst__term__not__zero,axiom,
! [P2: list_list_set_int] :
( ( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( P2 != nil_list_set_int ) ) ).
% pzf.const_term_not_zero
thf(fact_1236_pzf_Oconst__term__def,axiom,
! [P2: list_list_set_int] :
( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 )
= ( eval_l2744756199357571242t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.const_term_def
thf(fact_1237_pzf_Oconst__term__simprules__shell_I1_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( member_list_set_int @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ K ) ) ) ).
% pzf.const_term_simprules_shell(1)
thf(fact_1238_zf_Omap__norm__in__poly__ring__carrier,axiom,
! [K: set_set_int,P2: list_set_int] :
( ( subrin7689096310803670856t_unit @ K @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) )
=> ( ( member_list_set_int @ P2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) )
=> ( member7440485371949894425et_int @ ( map_se7207430148037755802et_int @ ( poly_o1995067004181805532t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) @ P2 ) @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ K ) ) ) ) ) ) ) ).
% zf.map_norm_in_poly_ring_carrier
thf(fact_1239_map__norm__in__poly__ring__carrier,axiom,
! [K: set_a,P2: list_a] :
( ( subring_a_b @ K @ r )
=> ( ( member_list_a @ P2 @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) )
=> ( member_list_list_a @ ( map_a_list_a @ ( poly_of_const_a_b @ r ) @ P2 ) @ ( partia2464479390973590831t_unit @ ( univ_p7953238456130426574t_unit @ ( univ_poly_a_b @ r @ K ) @ ( partia5361259788508890537t_unit @ ( univ_poly_a_b @ r @ K ) ) ) ) ) ) ) ).
% map_norm_in_poly_ring_carrier
thf(fact_1240_pzf_Oconst__term__simprules__shell_I2_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( mult_l2322401601159563005t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 @ Q ) )
= ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q ) ) ) ) ) ) ).
% pzf.const_term_simprules_shell(2)
thf(fact_1241_pzf_Oconst__term__simprules__shell_I3_J,axiom,
! [K: set_list_set_int,P2: list_list_set_int,Q: list_list_set_int] :
( ( subrin291743070294386232t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( member7440485371949894425et_int @ Q @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) ) )
=> ( ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( add_li3203230194967740868t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K ) @ P2 @ Q ) )
= ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 ) @ ( const_2217831012996398173t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Q ) ) ) ) ) ) ).
% pzf.const_term_simprules_shell(3)
thf(fact_1242_pzf_Ofield__intro2,axiom,
( ( ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
!= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( minus_4948633356380776833et_int @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) )
=> ( member_list_set_int @ X3 @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) )
=> ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.field_intro2
thf(fact_1243_pzf_Osubalbegra__incl__imp__finite__dimension,axiom,
! [K: set_list_set_int,E: set_list_set_int,V: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ E )
=> ( ( embedd3025314616789055568t_unit @ K @ V @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ord_le4527193859438875528et_int @ V @ E )
=> ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ V ) ) ) ) ) ).
% pzf.subalbegra_incl_imp_finite_dimension
thf(fact_1244_pzf_Ozero__not__one,axiom,
( ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
!= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.zero_not_one
thf(fact_1245_pzf_Osubring__props_I3_J,axiom,
! [K: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( member_list_set_int @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ K ) ) ).
% pzf.subring_props(3)
thf(fact_1246_zf_Omonic__poly__one,axiom,
monic_3404647755478805438t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% zf.monic_poly_one
thf(fact_1247_pzf_Otelescopic__base__dim_I1_J,axiom,
! [K: set_list_set_int,F2: set_list_set_int,E: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( subfie3135231139456689584t_unit @ F2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ F2 )
=> ( ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ F2 @ E )
=> ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ E ) ) ) ) ) ).
% pzf.telescopic_base_dim(1)
thf(fact_1248_pzf_Oinv__unique,axiom,
! [Y: list_set_int,X: list_set_int,Y3: list_set_int] :
( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y3 )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( Y = Y3 ) ) ) ) ) ) ).
% pzf.inv_unique
thf(fact_1249_pzf_Oone__unique,axiom,
! [U: list_set_int] :
( ( member_list_set_int @ U @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ U @ X3 )
= X3 ) )
=> ( U
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.one_unique
thf(fact_1250_pzf_Oone__divides,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ A ) ) ).
% pzf.one_divides
thf(fact_1251_pzf_OUnits__inv__comm,axiom,
! [X: list_set_int,Y: list_set_int] :
( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ Y )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ X @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ Y @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ Y @ X )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ).
% pzf.Units_inv_comm
thf(fact_1252_pzf_Osum__space__dim_I1_J,axiom,
! [K: set_list_set_int,E: set_list_set_int,F2: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ E )
=> ( ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ F2 )
=> ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ ( set_ad2159827153730365080t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ E @ F2 ) ) ) ) ) ).
% pzf.sum_space_dim(1)
thf(fact_1253_pzf_Ofinite__dimension__imp__subalgebra,axiom,
! [K: set_list_set_int,E: set_list_set_int] :
( ( subfie3135231139456689584t_unit @ K @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( embedd4385354970001408291t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ K @ E )
=> ( embedd3025314616789055568t_unit @ K @ E @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.finite_dimension_imp_subalgebra
thf(fact_1254_pzf_Osquare__eq__one,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ X )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( X
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
| ( X
= ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.square_eq_one
thf(fact_1255_pzf_OUnits__l__inv__ex,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ? [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X3 @ X )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.Units_l_inv_ex
thf(fact_1256_pzf_OUnits__r__inv__ex,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ? [X3: list_set_int] :
( ( member_list_set_int @ X3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ X3 )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.Units_r_inv_ex
thf(fact_1257_pzf_OUnit__eq__dividesone,axiom,
! [U: list_set_int] :
( ( member_list_set_int @ U @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ U @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ U @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.Unit_eq_dividesone
thf(fact_1258_pzf_Odivides__one,axiom,
! [A: list_set_int] :
( ( member_list_set_int @ A @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( factor7280686822248936060t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= ( member_list_set_int @ A @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.divides_one
thf(fact_1259_i_Oone__imp__carrier,axiom,
( ( member_list_set_int @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f ) )
=> ( ( cgenid1447163913751362543t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ f )
= ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% i.one_imp_carrier
thf(fact_1260_pzf_Ocarrier__one__not__zero,axiom,
( ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
!= ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
= ( ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.carrier_one_not_zero
thf(fact_1261_pzf_Ocarrier__one__zero,axiom,
( ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
= ( ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.carrier_one_zero
thf(fact_1262_pzf_Oone__zeroD,axiom,
( ( ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) ) ) ).
% pzf.one_zeroD
thf(fact_1263_pzf_Oone__zeroI,axiom,
( ( ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( insert_list_set_int @ ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
=> ( ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.one_zeroI
thf(fact_1264_pzf_Ocring__fieldI2,axiom,
( ( ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
!= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [A3: list_set_int] :
( ( member_list_set_int @ A3 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( A3
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ? [X5: list_set_int] :
( ( member_list_set_int @ X5 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
& ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ A3 @ X5 )
= ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) )
=> ( field_7411436548976123707t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ).
% pzf.cring_fieldI2
thf(fact_1265_pzf_Ogenideal__one,axiom,
( ( genide4509635305121348427t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( insert_list_set_int @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ bot_bo7344713902834881468et_int ) )
= ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ).
% pzf.genideal_one
thf(fact_1266_pzf_OsubringI,axiom,
! [H2: set_list_set_int] :
( ( ord_le4527193859438875528et_int @ H2 @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( member_list_set_int @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ H2 )
=> ( ! [H3: list_set_int] :
( ( member_list_set_int @ H3 @ H2 )
=> ( member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H3 ) @ H2 ) )
=> ( ! [H12: list_set_int,H23: list_set_int] :
( ( member_list_set_int @ H12 @ H2 )
=> ( ( member_list_set_int @ H23 @ H2 )
=> ( member_list_set_int @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H12 @ H23 ) @ H2 ) ) )
=> ( ! [H12: list_set_int,H23: list_set_int] :
( ( member_list_set_int @ H12 @ H2 )
=> ( ( member_list_set_int @ H23 @ H2 )
=> ( member_list_set_int @ ( add_li6184372983183467092t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H12 @ H23 ) @ H2 ) ) )
=> ( subrin291743070294386232t_unit @ H2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) ) ).
% pzf.subringI
thf(fact_1267_pzf_Oone__closed,axiom,
member_list_set_int @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% pzf.one_closed
thf(fact_1268_pzf_OUnits__one__closed,axiom,
member_list_set_int @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% pzf.Units_one_closed
thf(fact_1269_pzf_Ol__one,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ X )
= X ) ) ).
% pzf.l_one
thf(fact_1270_pzf_Or__one,axiom,
! [X: list_set_int] :
( ( member_list_set_int @ X @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
= X ) ) ).
% pzf.r_one
thf(fact_1271_pzf_OUnits__minus__one__closed,axiom,
member_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) @ ( units_4332284670423708338t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ).
% pzf.Units_minus_one_closed
thf(fact_1272_pzf_Ois__root__imp__pdivides,axiom,
! [P2: list_list_set_int,X: list_set_int] :
( ( member7440485371949894425et_int @ P2 @ ( partia3237421854401958059t_unit @ ( univ_p7181306125638219540t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( partia5243588073521899067t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) )
=> ( ( polyno5593083006719613942t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ P2 @ X )
=> ( polyno5399072443184722328t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ ( cons_list_set_int @ ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) @ ( cons_list_set_int @ ( a_inv_7608866662416757709t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ X ) @ nil_list_set_int ) ) @ P2 ) ) ) ).
% pzf.is_root_imp_pdivides
thf(fact_1273_pzf_OsubdomainI,axiom,
! [H2: set_list_set_int] :
( ( subcri4185096033226064695t_unit @ H2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
=> ( ( ( one_li3514222801933830793t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) )
!= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ! [H12: list_set_int,H23: list_set_int] :
( ( member_list_set_int @ H12 @ H2 )
=> ( ( member_list_set_int @ H23 @ H2 )
=> ( ( ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ H12 @ H23 )
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
=> ( ( H12
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) )
| ( H23
= ( zero_l5318887016005761237t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ) )
=> ( subdom6469697055745072292t_unit @ H2 @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) ) ) ) ) ).
% pzf.subdomainI
thf(fact_1274_pzf_Onormalize_Ocases,axiom,
! [X: list_list_set_int] :
( ( X != nil_list_set_int )
=> ~ ! [V3: list_set_int,Va: list_list_set_int] :
( X
!= ( cons_list_set_int @ V3 @ Va ) ) ) ).
% pzf.normalize.cases
thf(fact_1275_zero__not__one,axiom,
( ( zero_a_b @ r )
!= ( one_a_ring_ext_a_b @ r ) ) ).
% zero_not_one
thf(fact_1276_subring__props_I3_J,axiom,
! [K: set_a] :
( ( subfield_a_b @ K @ r )
=> ( member_a @ ( one_a_ring_ext_a_b @ r ) @ K ) ) ).
% subring_props(3)
thf(fact_1277_pzf_Opoly__mult_Ocases,axiom,
! [X: produc2061375659605276261et_int] :
( ! [P22: list_list_set_int] :
( X
!= ( produc7276067142252306077et_int @ nil_list_set_int @ P22 ) )
=> ~ ! [V3: list_set_int,Va: list_list_set_int,P22: list_list_set_int] :
( X
!= ( produc7276067142252306077et_int @ ( cons_list_set_int @ V3 @ Va ) @ P22 ) ) ) ).
% pzf.poly_mult.cases
% Conjectures (1)
thf(conj_0,conjecture,
( ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ ( mult_l3261022468980662045t_unit @ ( univ_p8160232691235018276t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) @ ( partia966996272515721803t_unit @ ( zFact @ ( semiri1314217659103216013at_int @ p ) ) ) ) @ y @ f ) ) @ x )
= ( mult_a_ring_ext_a_b @ r @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ y ) @ x ) @ ( eval_a_b @ r @ ( map_set_int_a @ ( ring_char_iso_a_b @ r ) @ f ) @ x ) ) ) ).
%------------------------------------------------------------------------------