TPTP Problem File: SLH0731^1.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 : Eval_FO/0005_Ailamazyan/prob_00964_034623__15654590_1 [Des23]
% Status : Theorem
% Rating : ? v8.2.0
% Syntax : Number of formulae : 1560 ( 687 unt; 282 typ; 0 def)
% Number of atoms : 2995 (1219 equ; 0 cnn)
% Maximal formula atoms : 12 ( 2 avg)
% Number of connectives : 9252 ( 165 ~; 25 |; 201 &;7745 @)
% ( 0 <=>;1116 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 6 avg)
% Number of types : 50 ( 49 usr)
% Number of type conns : 839 ( 839 >; 0 *; 0 +; 0 <<)
% Number of symbols : 236 ( 233 usr; 38 con; 0-3 aty)
% Number of variables : 3157 ( 131 ^;2913 !; 113 ?;3157 :)
% SPC : TH0_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2023-01-19 13:06:09.832
%------------------------------------------------------------------------------
% Could-be-implicit typings (49)
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
set_Pr5094982260447487303t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
set_Su4110612849109743515t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__Product____Type__Ounit_J_J,type,
set_Pr8892099060450471616t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__String__Oliteral_J_J,type,
set_Pr2282710299206288910iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Product____Type__Ounit_J_J,type,
set_Su7870427913035841300t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__String__Oliteral_J_J,type,
set_Su1261039151791658594iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
set_Pr5326375054644909319iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
set_Pr1763845938948868674it_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
set_Pr4334478416066269672t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
set_Su681948970226885467iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
set_Su4968945780807083758it_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
set_Su7539578257924484756t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
set_Pr2368984730104727433al_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
set_Pr4449086587592242223iteral: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
sum_su8719719018421925477t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
set_Su2755845247344502325al_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
set_Su4835947104832017115iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Pr1261947904930325089at_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__String__Oliteral_Mt__Product____Type__Ounit_J,type,
sum_su4378694014359939550t_unit: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__String__Oliteral_J,type,
sum_su7983299865237571628iteral: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
set_Sum_sum_nat_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__String__Oliteral_Mt__String__Oliteral_J,type,
sum_su3757740610105557285iteral: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
sum_su7788004701277271566it_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
sum_su7713564395780796596t_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Product____Type__Ounit_J_J,type,
set_op3165557761946182707t_unit: $tType ).
thf(ty_n_t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
list_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
set_Sum_sum_a_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J,type,
sum_sum_literal_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J,type,
sum_sum_nat_literal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
set_set_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__String__Oliteral_J_J,type,
set_option_literal: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__String__Oliteral_J_J,type,
set_set_literal: $tType ).
thf(ty_n_t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
sum_sum_nat_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
set_option_nat: $tType ).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
set_list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
set_set_nat: $tType ).
thf(ty_n_t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
sum_sum_a_nat: $tType ).
thf(ty_n_t__List__Olist_It__Product____Type__Ounit_J,type,
list_Product_unit: $tType ).
thf(ty_n_t__Set__Oset_It__Product____Type__Ounit_J,type,
set_Product_unit: $tType ).
thf(ty_n_t__List__Olist_It__String__Oliteral_J,type,
list_literal: $tType ).
thf(ty_n_t__Set__Oset_It__String__Oliteral_J,type,
set_literal: $tType ).
thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
list_nat: $tType ).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
set_nat: $tType ).
thf(ty_n_t__Product____Type__Ounit,type,
product_unit: $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__String__Oliteral,type,
literal: $tType ).
thf(ty_n_t__Nat__Onat,type,
nat: $tType ).
thf(ty_n_tf__a,type,
a: $tType ).
% Explicit typings (233)
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001t__Nat__Onat,type,
ad_agr_list_a_nat: set_a > list_Sum_sum_a_nat > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Ofo__nmlzd_001tf__a,type,
fo_nmlzd_a: set_a > list_Sum_sum_a_nat > $o ).
thf(sy_c_Ailamazyan_Orremdups_001t__Nat__Onat,type,
rremdups_nat: list_nat > list_nat ).
thf(sy_c_Ailamazyan_Orremdups_001t__Product____Type__Ounit,type,
rremdu6278002517723239495t_unit: list_Product_unit > list_Product_unit ).
thf(sy_c_Ailamazyan_Orremdups_001t__String__Oliteral,type,
rremdups_literal: list_literal > list_literal ).
thf(sy_c_Ailamazyan_Orremdups_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
rremdu8304153113908149561_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Finite__Set_Ocard_001t__Nat__Onat,type,
finite_card_nat: set_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Product____Type__Ounit,type,
finite410649719033368117t_unit: set_Product_unit > nat ).
thf(sy_c_Finite__Set_Ocard_001t__String__Oliteral,type,
finite_card_literal: set_literal > nat ).
thf(sy_c_Finite__Set_Ocard_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
finite6080979521523705895_a_nat: set_Sum_sum_a_nat > nat ).
thf(sy_c_Finite__Set_Ocard_001tf__a,type,
finite_card_a: set_a > nat ).
thf(sy_c_Finite__Set_Ofinite_001t__Nat__Onat,type,
finite_finite_nat: set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Nat__Onat_J,type,
finite5523153139673422903on_nat: set_option_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__Product____Type__Ounit_J,type,
finite1445617369574913404t_unit: set_op3165557761946182707t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Option__Ooption_It__String__Oliteral_J,type,
finite5071707688241699267iteral: set_option_literal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6177210948735845034at_nat: set_Pr1261947904930325089at_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
finite5113082511001691337t_unit: set_Pr4334478416066269672t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Nat__Onat_Mt__String__Oliteral_J,type,
finite211349803975347344iteral: set_Pr4449086587592242223iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
finite5187522816498166307it_nat: set_Pr1763845938948868674it_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
finite6816719414181127824t_unit: set_Pr5094982260447487303t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__Product____Type__Ounit_Mt__String__Oliteral_J,type,
finite885979093102766295iteral: set_Pr2282710299206288910iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__String__Oliteral_Mt__Nat__Onat_J,type,
finite4899330813501287658al_nat: set_Pr2368984730104727433al_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__String__Oliteral_Mt__Product____Type__Ounit_J,type,
finite6504745279079910025t_unit: set_Pr8892099060450471616t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Oprod_It__String__Oliteral_Mt__String__Oliteral_J,type,
finite1012402614658241104iteral: set_Pr5326375054644909319iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Product____Type__Ounit,type,
finite4290736615968046902t_unit: set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Nat__Onat_J,type,
finite1152437895449049373et_nat: set_set_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__Product____Type__Ounit_J,type,
finite1772178364199683094t_unit: set_set_Product_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Set__Oset_It__String__Oliteral_J,type,
finite2869373537460367197iteral: set_set_literal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__String__Oliteral,type,
finite5847741373460823677iteral: set_literal > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
finite6187706683773761046at_nat: set_Sum_sum_nat_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
finite4327512606132785245t_unit: set_Su7539578257924484756t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J,type,
finite7336130560110450212iteral: set_Su4835947104832017115iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
finite4401952911629260215it_nat: set_Su4968945780807083758it_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
finite3146551501593861116t_unit: set_Su4110612849109743515t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__String__Oliteral_J,type,
finite2327689956804853827iteral: set_Su1261039151791658594iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J,type,
finite2800739532781614718al_nat: set_Su2755845247344502325al_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__String__Oliteral_Mt__Product____Type__Ounit_J,type,
finite7946456142781997557t_unit: set_Su7870427913035841300t_unit > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_It__String__Oliteral_Mt__String__Oliteral_J,type,
finite8615155063774727100iteral: set_Su681948970226885467iteral > $o ).
thf(sy_c_Finite__Set_Ofinite_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
finite502105017643426984_a_nat: set_Sum_sum_a_nat > $o ).
thf(sy_c_Finite__Set_Ofinite_001tf__a,type,
finite_finite_a: set_a > $o ).
thf(sy_c_Fun_Oinj__on_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
inj_on3049792774292151987st_nat: ( list_nat > list_nat ) > set_list_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Nat__Onat,type,
inj_on_nat_nat: ( nat > nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Product____Type__Ounit,type,
inj_on7061601236592826506t_unit: ( nat > product_unit ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__String__Oliteral,type,
inj_on_nat_literal: ( nat > literal ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
inj_on4348161877322679292_a_nat: ( nat > sum_sum_a_nat ) > set_nat > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Nat__Onat,type,
inj_on8430439091780834860it_nat: ( product_unit > nat ) > set_Product_unit > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
inj_on8151373323710067377t_unit: ( product_unit > product_unit ) > set_Product_unit > $o ).
thf(sy_c_Fun_Oinj__on_001t__Product____Type__Ounit_001t__String__Oliteral,type,
inj_on3356833730873875064iteral: ( product_unit > literal ) > set_Product_unit > $o ).
thf(sy_c_Fun_Oinj__on_001t__String__Oliteral_001t__Nat__Onat,type,
inj_on_literal_nat: ( literal > nat ) > set_literal > $o ).
thf(sy_c_Fun_Oinj__on_001t__String__Oliteral_001t__Product____Type__Ounit,type,
inj_on4267703138521210538t_unit: ( literal > product_unit ) > set_literal > $o ).
thf(sy_c_Fun_Oinj__on_001t__String__Oliteral_001t__String__Oliteral,type,
inj_on602069361295035377iteral: ( literal > literal ) > set_literal > $o ).
thf(sy_c_Fun_Oinj__on_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Ounit,type,
inj_on4315315934791609919t_unit: ( sum_sum_a_nat > product_unit ) > set_Sum_sum_a_nat > $o ).
thf(sy_c_Fun_Oinj__on_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
inj_on3092108040974901106_a_nat: ( a > sum_sum_a_nat ) > set_a > $o ).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
minus_minus_nat: nat > nat > nat ).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
one_one_nat: nat ).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
plus_plus_nat: nat > nat > nat ).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat,type,
times_times_nat: nat > nat > nat ).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
zero_zero_nat: nat ).
thf(sy_c_If_001t__Nat__Onat,type,
if_nat: $o > nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Nat__Onat,type,
inf_inf_nat: nat > nat > nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Nat__Onat_J,type,
inf_inf_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Product____Type__Ounit_J,type,
inf_in4660618365625256667t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__String__Oliteral_J,type,
inf_inf_set_literal: set_literal > set_literal > set_literal ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
inf_in7084830621192376909_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Lattices_Oinf__class_Oinf_001t__Set__Oset_Itf__a_J,type,
inf_inf_set_a: set_a > set_a > set_a ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
sup_sup_nat: nat > nat > nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
sup_sup_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Ounit_J,type,
sup_su793286257634532545t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__String__Oliteral_J,type,
sup_sup_set_literal: set_literal > set_literal > set_literal ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
sup_su3567568935942035937at_nat: set_Sum_sum_nat_nat > set_Sum_sum_nat_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
sup_su4295710226244385384t_unit: set_Su7539578257924484756t_unit > set_Su7539578257924484756t_unit > set_Su7539578257924484756t_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
sup_su5939970013262617775iteral: set_Su4835947104832017115iteral > set_Su4835947104832017115iteral > set_Su4835947104832017115iteral ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
sup_su1725077749126984386it_nat: set_Su4968945780807083758it_nat > set_Su4968945780807083758it_nat > set_Su4968945780807083758it_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
sup_su9179690136711093447t_unit: set_Su4110612849109743515t_unit > set_Su4110612849109743515t_unit > set_Su4110612849109743515t_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__String__Oliteral_J_J,type,
sup_su6064231614352391566iteral: set_Su1261039151791658594iteral > set_Su1261039151791658594iteral > set_Su1261039151791658594iteral ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
sup_su3859868155775102985al_nat: set_Su2755845247344502325al_nat > set_Su2755845247344502325al_nat > set_Su2755845247344502325al_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Product____Type__Ounit_J_J,type,
sup_su3450248338741798464t_unit: set_Su7870427913035841300t_unit > set_Su7870427913035841300t_unit > set_Su7870427913035841300t_unit ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
sup_su7822573100775623815iteral: set_Su681948970226885467iteral > set_Su681948970226885467iteral > set_Su681948970226885467iteral ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
sup_su6804446743777130803_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
sup_sup_set_a: set_a > set_a > set_a ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
map_nat_nat: ( nat > nat ) > list_nat > list_nat ).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_na823391071729141993_a_nat: ( nat > sum_sum_a_nat ) > list_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_Su2790769393171190532_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Omap_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
map_a_Sum_sum_a_nat: ( a > sum_sum_a_nat ) > list_a > list_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
set_nat2: list_nat > set_nat ).
thf(sy_c_List_Olist_Oset_001t__Product____Type__Ounit,type,
set_Product_unit2: list_Product_unit > set_Product_unit ).
thf(sy_c_List_Olist_Oset_001t__String__Oliteral,type,
set_literal2: list_literal > set_literal ).
thf(sy_c_List_Olist_Oset_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
set_Sum_sum_a_nat2: list_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_List_Olist_Oset_001tf__a,type,
set_a2: list_a > set_a ).
thf(sy_c_List_Ounion_001t__Nat__Onat,type,
union_nat: list_nat > list_nat > list_nat ).
thf(sy_c_List_Ounion_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
union_Sum_sum_a_nat: list_Sum_sum_a_nat > list_Sum_sum_a_nat > list_Sum_sum_a_nat ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
size_size_list_nat: list_nat > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Ounit_J,type,
size_s245203480648594047t_unit: list_Product_unit > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__String__Oliteral_J,type,
size_s2501651207091587910iteral: list_literal > nat ).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
size_s5283204784079214577_a_nat: list_Sum_sum_a_nat > nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
bot_bot_nat: nat ).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
bot_bot_set_nat: set_nat ).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
ord_less_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
ord_less_eq_nat: nat > nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Nat__Onat_J,type,
ord_less_eq_set_nat: set_nat > set_nat > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_le3507040750410214029t_unit: set_Product_unit > set_Product_unit > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__String__Oliteral_J,type,
ord_le7307670543136651348iteral: set_literal > set_literal > $o ).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
ord_le1325389633284124927_a_nat: set_Sum_sum_a_nat > set_Sum_sum_a_nat > $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_Orderings_Oord__class_Oless__eq_001t__String__Oliteral,type,
ord_less_eq_literal: literal > literal > $o ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
ord_min_nat: nat > nat > nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Nat__Onat_J,type,
ord_min_set_nat: set_nat > set_nat > set_nat ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__Product____Type__Ounit_J,type,
ord_mi7693666415311043668t_unit: set_Product_unit > set_Product_unit > set_Product_unit ).
thf(sy_c_Orderings_Oord__class_Omin_001t__Set__Oset_It__String__Oliteral_J,type,
ord_min_set_literal: set_literal > set_literal > set_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Nat__Onat_M_Eo_J,type,
top_top_nat_o: nat > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__Product____Type__Ounit_M_Eo_J,type,
top_to2465898995584390880unit_o: product_unit > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001_062_It__String__Oliteral_M_Eo_J,type,
top_top_literal_o: literal > $o ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
top_top_set_list_nat: set_list_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
top_top_set_nat: set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Nat__Onat_J_J,type,
top_to8920198386146353926on_nat: set_option_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__Product____Type__Ounit_J_J,type,
top_to2690860209552263555t_unit: set_op3165557761946182707t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Option__Ooption_It__String__Oliteral_J_J,type,
top_to8248435444729185354iteral: set_option_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to4669805908274784177at_nat: set_Pr1261947904930325089at_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to8544742955230171288t_unit: set_Pr4334478416066269672t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
top_to6658620532179778271iteral: set_Pr4449086587592242223iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
top_to5974110478112770290it_nat: set_Pr1763845938948868674it_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
top_to1835807148980544151t_unit: set_Pr5094982260447487303t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Ounit_Mt__String__Oliteral_J_J,type,
top_to1987565607987313502iteral: set_Pr2282710299206288910iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
top_to4578518674692263481al_nat: set_Pr2368984730104727433al_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__Product____Type__Ounit_J_J,type,
top_to8596954369231496208t_unit: set_Pr8892099060450471616t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Oprod_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
top_to594236051165446743iteral: set_Pr5326375054644909319iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
top_to1996260823553986621t_unit: set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
top_top_set_set_nat: set_set_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__Product____Type__Ounit_J_J,type,
top_to1767297665138865437t_unit: set_set_Product_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Set__Oset_It__String__Oliteral_J_J,type,
top_to5694933271948605156iteral: set_set_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Oliteral_J,type,
top_top_set_literal: set_literal ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
top_to6661820994512907621at_nat: set_Sum_sum_nat_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J_J,type,
top_to5465250082899874788t_unit: set_Su7539578257924484756t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J_J,type,
top_to148093990134820907iteral: set_Su4835947104832017115iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J_J,type,
top_to2894617605782473790it_nat: set_Su4968945780807083758it_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J_J,type,
top_to2771918933716375115t_unit: set_Su4110612849109743515t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Ounit_Mt__String__Oliteral_J_J,type,
top_to8707194323715505426iteral: set_Su1261039151791658594iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J_J,type,
top_to7291364169502081925al_nat: set_Su2755845247344502325al_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__Product____Type__Ounit_J_J,type,
top_to6093211048104912324t_unit: set_Su7870427913035841300t_unit ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_It__String__Oliteral_Mt__String__Oliteral_J_J,type,
top_to8459027286199865867iteral: set_Su681948970226885467iteral ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
top_to795618464972521135_a_nat: set_Sum_sum_a_nat ).
thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_Itf__a_J,type,
top_top_set_a: set_a ).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
power_power_nat: nat > nat > nat ).
thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
collect_nat: ( nat > $o ) > set_nat ).
thf(sy_c_Set_OCollect_001t__Product____Type__Ounit,type,
collect_Product_unit: ( product_unit > $o ) > set_Product_unit ).
thf(sy_c_Set_OCollect_001t__String__Oliteral,type,
collect_literal: ( literal > $o ) > set_literal ).
thf(sy_c_Set_OCollect_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
collec7073057861543223018_a_nat: ( sum_sum_a_nat > $o ) > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Nat__Onat,type,
image_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
image_8730104196221521654t_unit: ( nat > product_unit ) > set_nat > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__String__Oliteral,type,
image_nat_literal: ( nat > literal ) > set_nat > set_literal ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
image_678696785212003926at_nat: ( nat > sum_sum_nat_nat ) > set_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
image_5493879627330158621t_unit: ( nat > sum_su7713564395780796596t_unit ) > set_nat > set_Su7539578257924484756t_unit ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J,type,
image_948809277473826276iteral: ( nat > sum_sum_nat_literal ) > set_nat > set_Su4835947104832017115iteral ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
image_5568319932826633591it_nat: ( nat > sum_su7788004701277271566it_nat ) > set_nat > set_Su4968945780807083758it_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J,type,
image_5636790286999766590al_nat: ( nat > sum_sum_literal_nat ) > set_nat > set_Su2755845247344502325al_nat ).
thf(sy_c_Set_Oimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7293268710728258664_a_nat: ( nat > sum_sum_a_nat ) > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
image_875570014554754200it_nat: ( product_unit > nat ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
image_405062704495631173t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__String__Oliteral,type,
image_5876984745897992460iteral: ( product_unit > literal ) > set_Product_unit > set_literal ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Product____Type__Ounit_J,type,
image_5632832758920211948t_unit: ( product_unit > sum_su7713564395780796596t_unit ) > set_Product_unit > set_Su7539578257924484756t_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Nat__Onat_J,type,
image_5707273064416686918it_nat: ( product_unit > sum_su7788004701277271566it_nat ) > set_Product_unit > set_Su4968945780807083758it_nat ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__Product____Type__Ounit_J,type,
image_2037928093122670381t_unit: ( product_unit > sum_su8719719018421925477t_unit ) > set_Product_unit > set_Su4110612849109743515t_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__String__Oliteral_J,type,
image_1445046369150594292iteral: ( product_unit > sum_su7983299865237571628iteral ) > set_Product_unit > set_Su1261039151791658594iteral ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Sum____Type__Osum_It__String__Oliteral_Mt__Product____Type__Ounit_J,type,
image_7063812555127738022t_unit: ( product_unit > sum_su4378694014359939550t_unit ) > set_Product_unit > set_Su7870427913035841300t_unit ).
thf(sy_c_Set_Oimage_001t__Product____Type__Ounit_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_8388312959667285303_a_nat: ( product_unit > sum_sum_a_nat ) > set_Product_unit > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Nat__Onat,type,
image_literal_nat: ( literal > nat ) > set_literal > set_nat ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Product____Type__Ounit,type,
image_6787854153545327934t_unit: ( literal > product_unit ) > set_literal > set_Product_unit ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__String__Oliteral,type,
image_8195128725298311301iteral: ( literal > literal ) > set_literal > set_literal ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Sum____Type__Osum_It__Nat__Onat_Mt__String__Oliteral_J,type,
image_505656761235946540iteral: ( literal > sum_sum_nat_literal ) > set_literal > set_Su4835947104832017115iteral ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Sum____Type__Osum_It__Product____Type__Ounit_Mt__String__Oliteral_J,type,
image_731553875274544187iteral: ( literal > sum_su7983299865237571628iteral ) > set_literal > set_Su1261039151791658594iteral ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Sum____Type__Osum_It__String__Oliteral_Mt__Nat__Onat_J,type,
image_5193637770761886854al_nat: ( literal > sum_sum_literal_nat ) > set_literal > set_Su2755845247344502325al_nat ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Sum____Type__Osum_It__String__Oliteral_Mt__Product____Type__Ounit_J,type,
image_6350320061251687917t_unit: ( literal > sum_su4378694014359939550t_unit ) > set_literal > set_Su7870427913035841300t_unit ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Sum____Type__Osum_It__String__Oliteral_Mt__String__Oliteral_J,type,
image_8650502459363062708iteral: ( literal > sum_su3757740610105557285iteral ) > set_literal > set_Su681948970226885467iteral ).
thf(sy_c_Set_Oimage_001t__String__Oliteral_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_605486080324953776_a_nat: ( literal > sum_sum_a_nat ) > set_literal > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
image_2473878607534554506at_nat: ( sum_sum_a_nat > nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Oimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7142520692256960453_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Oimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
image_7873763678140191238_a_nat: ( a > sum_sum_a_nat ) > set_a > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Nat__Onat,type,
vimage_nat_nat: ( nat > nat ) > set_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Product____Type__Ounit,type,
vimage4884490618288580032t_unit: ( nat > product_unit ) > set_Product_unit > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__String__Oliteral,type,
vimage_nat_literal: ( nat > literal ) > set_literal > set_nat ).
thf(sy_c_Set_Ovimage_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage3040984495076556338_a_nat: ( nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_nat ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__Nat__Onat,type,
vimage6253328473476588386it_nat: ( product_unit > nat ) > set_nat > set_Product_unit ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
vimage7995052115951654139t_unit: ( product_unit > product_unit ) > set_Product_unit > set_Product_unit ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__String__Oliteral,type,
vimage5137003825767189442iteral: ( product_unit > literal ) > set_literal > set_Product_unit ).
thf(sy_c_Set_Ovimage_001t__Product____Type__Ounit_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage5806748310446461933_a_nat: ( product_unit > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Product_unit ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__Nat__Onat,type,
vimage_literal_nat: ( literal > nat ) > set_nat > set_literal ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__Product____Type__Ounit,type,
vimage6047873233414524916t_unit: ( literal > product_unit ) > set_Product_unit > set_literal ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__String__Oliteral,type,
vimage8238609917233974331iteral: ( literal > literal ) > set_literal > set_literal ).
thf(sy_c_Set_Ovimage_001t__String__Oliteral_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage1582777469784555110_a_nat: ( literal > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_literal ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Nat__Onat,type,
vimage7444966428737627988at_nat: ( sum_sum_a_nat > nat ) > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Product____Type__Ounit,type,
vimage2477388680732255113t_unit: ( sum_sum_a_nat > product_unit ) > set_Product_unit > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage6545432446589551483_a_nat: ( sum_sum_a_nat > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_Sum_sum_a_nat ).
thf(sy_c_Set_Ovimage_001tf__a_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
vimage4607795094749595324_a_nat: ( a > sum_sum_a_nat ) > set_Sum_sum_a_nat > set_a ).
thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Nat__Onat,type,
set_ord_lessThan_nat: nat > set_nat ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inl_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__Product____Type__Ounit,type,
sum_In8759356770566287949t_unit: nat > sum_su7713564395780796596t_unit ).
thf(sy_c_Sum__Type_OInl_001t__Nat__Onat_001t__String__Oliteral,type,
sum_Inl_nat_literal: nat > sum_sum_nat_literal ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Ounit_001t__Nat__Onat,type,
sum_In904822588899520495it_nat: product_unit > sum_su7788004701277271566it_nat ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sum_In8043524862774375086t_unit: product_unit > sum_su8719719018421925477t_unit ).
thf(sy_c_Sum__Type_OInl_001t__Product____Type__Ounit_001t__String__Oliteral,type,
sum_In8061226877302236661iteral: product_unit > sum_su7983299865237571628iteral ).
thf(sy_c_Sum__Type_OInl_001t__String__Oliteral_001t__Nat__Onat,type,
sum_Inl_literal_nat: literal > sum_sum_literal_nat ).
thf(sy_c_Sum__Type_OInl_001t__String__Oliteral_001t__Product____Type__Ounit,type,
sum_In8972096284949572135t_unit: literal > sum_su4378694014359939550t_unit ).
thf(sy_c_Sum__Type_OInl_001t__String__Oliteral_001t__String__Oliteral,type,
sum_In8673369501829128942iteral: literal > sum_su3757740610105557285iteral ).
thf(sy_c_Sum__Type_OInl_001tf__a_001t__Nat__Onat,type,
sum_Inl_a_nat: a > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Inr_nat_nat: nat > sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__Product____Type__Ounit,type,
sum_In4922714968575391175t_unit: nat > sum_su7788004701277271566it_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001t__String__Oliteral,type,
sum_Inr_nat_literal: nat > sum_sum_literal_nat ).
thf(sy_c_Sum__Type_OInr_001t__Nat__Onat_001tf__a,type,
sum_Inr_nat_a: nat > sum_sum_a_nat ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Ounit_001t__Nat__Onat,type,
sum_In6291552823763399529it_nat: product_unit > sum_su7713564395780796596t_unit ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sum_In1458446571380145076t_unit: product_unit > sum_su8719719018421925477t_unit ).
thf(sy_c_Sum__Type_OInr_001t__Product____Type__Ounit_001t__String__Oliteral,type,
sum_In4586030566214162427iteral: product_unit > sum_su4378694014359939550t_unit ).
thf(sy_c_Sum__Type_OInr_001t__String__Oliteral_001t__Nat__Onat,type,
sum_Inr_literal_nat: literal > sum_sum_nat_literal ).
thf(sy_c_Sum__Type_OInr_001t__String__Oliteral_001t__Product____Type__Ounit,type,
sum_In5496899973861497901t_unit: literal > sum_su7983299865237571628iteral ).
thf(sy_c_Sum__Type_OInr_001t__String__Oliteral_001t__String__Oliteral,type,
sum_In6960570357259124212iteral: literal > sum_su3757740610105557285iteral ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__Nat__Onat,type,
sum_Plus_nat_nat: set_nat > set_nat > set_Sum_sum_nat_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__Product____Type__Ounit,type,
sum_Pl8139761314266719112t_unit: set_nat > set_Product_unit > set_Su7539578257924484756t_unit ).
thf(sy_c_Sum__Type_OPlus_001t__Nat__Onat_001t__String__Oliteral,type,
sum_Plus_nat_literal: set_nat > set_literal > set_Su4835947104832017115iteral ).
thf(sy_c_Sum__Type_OPlus_001t__Product____Type__Ounit_001t__Nat__Onat,type,
sum_Pl285227132599951658it_nat: set_Product_unit > set_nat > set_Su4968945780807083758it_nat ).
thf(sy_c_Sum__Type_OPlus_001t__Product____Type__Ounit_001t__Product____Type__Ounit,type,
sum_Pl144888763950457139t_unit: set_Product_unit > set_Product_unit > set_Su4110612849109743515t_unit ).
thf(sy_c_Sum__Type_OPlus_001t__Product____Type__Ounit_001t__String__Oliteral,type,
sum_Pl603064936740334586iteral: set_Product_unit > set_literal > set_Su1261039151791658594iteral ).
thf(sy_c_Sum__Type_OPlus_001t__String__Oliteral_001t__Nat__Onat,type,
sum_Plus_literal_nat: set_literal > set_nat > set_Su2755845247344502325al_nat ).
thf(sy_c_Sum__Type_OPlus_001t__String__Oliteral_001t__Product____Type__Ounit,type,
sum_Pl1513934344387670060t_unit: set_literal > set_Product_unit > set_Su7870427913035841300t_unit ).
thf(sy_c_Sum__Type_OPlus_001t__String__Oliteral_001t__String__Oliteral,type,
sum_Pl2008226088555200627iteral: set_literal > set_literal > set_Su681948970226885467iteral ).
thf(sy_c_Sum__Type_OPlus_001tf__a_001t__Nat__Onat,type,
sum_Plus_a_nat: set_a > set_nat > set_Sum_sum_a_nat ).
thf(sy_c_member_001t__Nat__Onat,type,
member_nat: nat > set_nat > $o ).
thf(sy_c_member_001t__Product____Type__Ounit,type,
member_Product_unit: product_unit > set_Product_unit > $o ).
thf(sy_c_member_001t__String__Oliteral,type,
member_literal: literal > set_literal > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_It__Nat__Onat_Mt__Nat__Onat_J,type,
member8583185029347631382at_nat: sum_sum_nat_nat > set_Sum_sum_nat_nat > $o ).
thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
member_Sum_sum_a_nat: sum_sum_a_nat > set_Sum_sum_a_nat > $o ).
thf(sy_c_member_001tf__a,type,
member_a: a > set_a > $o ).
thf(sy_v_AD,type,
ad: set_a ).
thf(sy_v_xs,type,
xs: list_Sum_sum_a_nat ).
% Relevant facts (1274)
thf(fact_0_assms,axiom,
fo_nmlzd_a @ ad @ xs ).
% assms
thf(fact_1_card__lessThan,axiom,
! [U: nat] :
( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_2_vimage__Un,axiom,
! [F: sum_sum_a_nat > nat,A: set_nat,B: set_nat] :
( ( vimage7444966428737627988at_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( vimage7444966428737627988at_nat @ F @ A ) @ ( vimage7444966428737627988at_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_3_vimage__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_4_vimage__Un,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_5_vimage__Un,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage6545432446589551483_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( vimage6545432446589551483_a_nat @ F @ A ) @ ( vimage6545432446589551483_a_nat @ F @ B ) ) ) ).
% vimage_Un
thf(fact_6_vimage__Int,axiom,
! [F: sum_sum_a_nat > nat,A: set_nat,B: set_nat] :
( ( vimage7444966428737627988at_nat @ F @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_in7084830621192376909_a_nat @ ( vimage7444966428737627988at_nat @ F @ A ) @ ( vimage7444966428737627988at_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_7_vimage__Int,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( vimage_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ ( vimage_nat_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_8_vimage__Int,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_9_vimage__Int,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( vimage6545432446589551483_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( inf_in7084830621192376909_a_nat @ ( vimage6545432446589551483_a_nat @ F @ A ) @ ( vimage6545432446589551483_a_nat @ F @ B ) ) ) ).
% vimage_Int
thf(fact_10_Inl__Inr__False,axiom,
! [X: a,Y: nat] :
( ( sum_Inl_a_nat @ X )
!= ( sum_Inr_nat_a @ Y ) ) ).
% Inl_Inr_False
thf(fact_11_Inr__Inl__False,axiom,
! [X: nat,Y: a] :
( ( sum_Inr_nat_a @ X )
!= ( sum_Inl_a_nat @ Y ) ) ).
% Inr_Inl_False
thf(fact_12_Int__Un__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( sup_sup_set_nat @ T @ ( inf_inf_set_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_13_Int__Un__eq_I4_J,axiom,
! [T: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ T @ ( inf_in7084830621192376909_a_nat @ S @ T ) )
= T ) ).
% Int_Un_eq(4)
thf(fact_14_Int__Un__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ S @ ( inf_inf_set_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_15_Int__Un__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ S @ ( inf_in7084830621192376909_a_nat @ S @ T ) )
= S ) ).
% Int_Un_eq(3)
thf(fact_16_Int__Un__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_17_Int__Un__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T ) @ T )
= T ) ).
% Int_Un_eq(2)
thf(fact_18_Int__Un__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_19_Int__Un__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ S @ T ) @ S )
= S ) ).
% Int_Un_eq(1)
thf(fact_20_Un__Int__eq_I4_J,axiom,
! [T: set_Sum_sum_a_nat,S: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ T @ ( sup_su6804446743777130803_a_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_21_Un__Int__eq_I4_J,axiom,
! [T: set_nat,S: set_nat] :
( ( inf_inf_set_nat @ T @ ( sup_sup_set_nat @ S @ T ) )
= T ) ).
% Un_Int_eq(4)
thf(fact_22_Un__Int__eq_I3_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ S @ ( sup_su6804446743777130803_a_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_23_Un__Int__eq_I3_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ S @ ( sup_sup_set_nat @ S @ T ) )
= S ) ).
% Un_Int_eq(3)
thf(fact_24_Un__Int__eq_I2_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_25_Un__Int__eq_I2_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ T )
= T ) ).
% Un_Int_eq(2)
thf(fact_26_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X: a,A: set_a] :
( ( B2
= ( F @ X ) )
=> ( ( member_a @ X @ A )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_27_image__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_28_image__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat,A: set_nat] :
( ( B2
= ( F @ X ) )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% image_eqI
thf(fact_29_Int__iff,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A )
& ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_30_Int__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
& ( member_nat @ C @ B ) ) ) ).
% Int_iff
thf(fact_31_IntI,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A )
=> ( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_32_IntI,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% IntI
thf(fact_33_Un__iff,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( ( member_Sum_sum_a_nat @ C @ A )
| ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_34_Un__iff,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
= ( ( member_nat @ C @ A )
| ( member_nat @ C @ B ) ) ) ).
% Un_iff
thf(fact_35_UnCI,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ~ ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ A ) )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_36_UnCI,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( ~ ( member_nat @ C @ B )
=> ( member_nat @ C @ A ) )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnCI
thf(fact_37_lessThan__eq__iff,axiom,
! [X: nat,Y: nat] :
( ( ( set_ord_lessThan_nat @ X )
= ( set_ord_lessThan_nat @ Y ) )
= ( X = Y ) ) ).
% lessThan_eq_iff
thf(fact_38_vimage__eq,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
= ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_39_vimage__eq,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_40_vimage__eq,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ B ) )
= ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimage_eq
thf(fact_41_vimageI,axiom,
! [F: nat > nat,A2: nat,B2: nat,B: set_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_42_vimageI,axiom,
! [F: nat > sum_sum_a_nat,A2: nat,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_Sum_sum_a_nat @ B2 @ B )
=> ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_43_vimageI,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: sum_sum_a_nat,B2: sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( F @ A2 )
= B2 )
=> ( ( member_Sum_sum_a_nat @ B2 @ B )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ B ) ) ) ) ).
% vimageI
thf(fact_44_Un__Int__eq_I1_J,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_45_Un__Int__eq_I1_J,axiom,
! [S: set_nat,T: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ S @ T ) @ S )
= S ) ).
% Un_Int_eq(1)
thf(fact_46_rev__image__eqI,axiom,
! [X: a,A: set_a,B2: sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( member_a @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_47_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_48_rev__image__eqI,axiom,
! [X: nat,A: set_nat,B2: nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ A ) ) ) ) ).
% rev_image_eqI
thf(fact_49_ball__imageD,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X3: a] :
( ( member_a @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_50_ball__imageD,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_51_ball__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( image_nat_nat @ F @ A ) )
=> ( P @ X2 ) )
=> ! [X3: nat] :
( ( member_nat @ X3 @ A )
=> ( P @ ( F @ X3 ) ) ) ) ).
% ball_imageD
thf(fact_52_image__cong,axiom,
! [M: set_a,N: set_a,F: a > sum_sum_a_nat,G: a > sum_sum_a_nat] :
( ( M = N )
=> ( ! [X2: a] :
( ( member_a @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7873763678140191238_a_nat @ F @ M )
= ( image_7873763678140191238_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_53_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_7293268710728258664_a_nat @ F @ M )
= ( image_7293268710728258664_a_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_54_image__cong,axiom,
! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
( ( M = N )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ N )
=> ( ( F @ X2 )
= ( G @ X2 ) ) )
=> ( ( image_nat_nat @ F @ M )
= ( image_nat_nat @ G @ N ) ) ) ) ).
% image_cong
thf(fact_55_bex__imageD,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: sum_sum_a_nat > $o] :
( ? [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( image_7873763678140191238_a_nat @ F @ A ) )
& ( P @ X3 ) )
=> ? [X2: a] :
( ( member_a @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_56_bex__imageD,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: sum_sum_a_nat > $o] :
( ? [X3: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X3 @ ( image_7293268710728258664_a_nat @ F @ A ) )
& ( P @ X3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_57_bex__imageD,axiom,
! [F: nat > nat,A: set_nat,P: nat > $o] :
( ? [X3: nat] :
( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
& ( P @ X3 ) )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( P @ ( F @ X2 ) ) ) ) ).
% bex_imageD
thf(fact_58_image__iff,axiom,
! [Z: sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( member_Sum_sum_a_nat @ Z @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( ? [X4: a] :
( ( member_a @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_59_image__iff,axiom,
! [Z: sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( member_Sum_sum_a_nat @ Z @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_60_image__iff,axiom,
! [Z: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ Z @ ( image_nat_nat @ F @ A ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( Z
= ( F @ X4 ) ) ) ) ) ).
% image_iff
thf(fact_61_imageI,axiom,
! [X: a,A: set_a,F: a > sum_sum_a_nat] :
( ( member_a @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_62_imageI,axiom,
! [X: nat,A: set_nat,F: nat > sum_sum_a_nat] :
( ( member_nat @ X @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X ) @ ( image_7293268710728258664_a_nat @ F @ A ) ) ) ).
% imageI
thf(fact_63_imageI,axiom,
! [X: nat,A: set_nat,F: nat > nat] :
( ( member_nat @ X @ A )
=> ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A ) ) ) ).
% imageI
thf(fact_64_Int__left__commute,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ B @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_65_Int__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ B @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_left_commute
thf(fact_66_Int__left__absorb,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( inf_in7084830621192376909_a_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_67_Int__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ A @ B ) )
= ( inf_inf_set_nat @ A @ B ) ) ).
% Int_left_absorb
thf(fact_68_Int__commute,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_69_Int__commute,axiom,
( inf_inf_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( inf_inf_set_nat @ B3 @ A3 ) ) ) ).
% Int_commute
thf(fact_70_Int__absorb,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_71_Int__absorb,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ A )
= A ) ).
% Int_absorb
thf(fact_72_Int__assoc,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ C2 )
= ( inf_in7084830621192376909_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_73_Int__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) ) ) ).
% Int_assoc
thf(fact_74_IntD2,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ).
% IntD2
thf(fact_75_IntD2,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ B ) ) ).
% IntD2
thf(fact_76_IntD1,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
=> ( member_Sum_sum_a_nat @ C @ A ) ) ).
% IntD1
thf(fact_77_IntD1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ( member_nat @ C @ A ) ) ).
% IntD1
thf(fact_78_IntE,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
=> ~ ( ( member_Sum_sum_a_nat @ C @ A )
=> ~ ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% IntE
thf(fact_79_IntE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( inf_inf_set_nat @ A @ B ) )
=> ~ ( ( member_nat @ C @ A )
=> ~ ( member_nat @ C @ B ) ) ) ).
% IntE
thf(fact_80_Un__left__commute,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_81_Un__left__commute,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ B @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_left_commute
thf(fact_82_Un__left__absorb,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_83_Un__left__absorb,axiom,
! [A: set_nat,B: set_nat] :
( ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ A @ B ) ) ).
% Un_left_absorb
thf(fact_84_Un__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_85_Un__commute,axiom,
( sup_sup_set_nat
= ( ^ [A3: set_nat,B3: set_nat] : ( sup_sup_set_nat @ B3 @ A3 ) ) ) ).
% Un_commute
thf(fact_86_Un__absorb,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_87_Un__absorb,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ A )
= A ) ).
% Un_absorb
thf(fact_88_Un__assoc,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 )
= ( sup_su6804446743777130803_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_89_Un__assoc,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( sup_sup_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) ) ).
% Un_assoc
thf(fact_90_ball__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_91_ball__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ! [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) )
=> ( P @ X4 ) ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( P @ X4 ) )
& ! [X4: nat] :
( ( member_nat @ X4 @ B )
=> ( P @ X4 ) ) ) ) ).
% ball_Un
thf(fact_92_bex__Un,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o] :
( ( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_93_bex__Un,axiom,
! [A: set_nat,B: set_nat,P: nat > $o] :
( ( ? [X4: nat] :
( ( member_nat @ X4 @ ( sup_sup_set_nat @ A @ B ) )
& ( P @ X4 ) ) )
= ( ? [X4: nat] :
( ( member_nat @ X4 @ A )
& ( P @ X4 ) )
| ? [X4: nat] :
( ( member_nat @ X4 @ B )
& ( P @ X4 ) ) ) ) ).
% bex_Un
thf(fact_94_UnI2,axiom,
! [C: sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ B )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_95_UnI2,axiom,
! [C: nat,B: set_nat,A: set_nat] :
( ( member_nat @ C @ B )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI2
thf(fact_96_UnI1,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_97_UnI1,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ A )
=> ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) ) ) ).
% UnI1
thf(fact_98_mem__Collect__eq,axiom,
! [A2: nat,P: nat > $o] :
( ( member_nat @ A2 @ ( collect_nat @ P ) )
= ( P @ A2 ) ) ).
% mem_Collect_eq
thf(fact_99_Collect__mem__eq,axiom,
! [A: set_nat] :
( ( collect_nat
@ ^ [X4: nat] : ( member_nat @ X4 @ A ) )
= A ) ).
% Collect_mem_eq
thf(fact_100_UnE,axiom,
! [C: sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ( ~ ( member_Sum_sum_a_nat @ C @ A )
=> ( member_Sum_sum_a_nat @ C @ B ) ) ) ).
% UnE
thf(fact_101_UnE,axiom,
! [C: nat,A: set_nat,B: set_nat] :
( ( member_nat @ C @ ( sup_sup_set_nat @ A @ B ) )
=> ( ~ ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% UnE
thf(fact_102_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: nat > sum_sum_a_nat,Q: nat > $o] :
( ! [X2: nat] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage3040984495076556338_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collect_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_103_vimage__Collect,axiom,
! [P: sum_sum_a_nat > $o,F: sum_sum_a_nat > sum_sum_a_nat,Q: sum_sum_a_nat > $o] :
( ! [X2: sum_sum_a_nat] :
( ( P @ ( F @ X2 ) )
= ( Q @ X2 ) )
=> ( ( vimage6545432446589551483_a_nat @ F @ ( collec7073057861543223018_a_nat @ P ) )
= ( collec7073057861543223018_a_nat @ Q ) ) ) ).
% vimage_Collect
thf(fact_104_vimageI2,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( member_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_105_vimageI2,axiom,
! [F: nat > sum_sum_a_nat,A2: nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_106_vimageI2,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A2: sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A )
=> ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ A ) ) ) ).
% vimageI2
thf(fact_107_vimageE,axiom,
! [A2: nat,F: nat > nat,B: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ B ) )
=> ( member_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_108_vimageE,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ B ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_109_vimageE,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ B ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ B ) ) ).
% vimageE
thf(fact_110_vimageD,axiom,
! [A2: nat,F: nat > nat,A: set_nat] :
( ( member_nat @ A2 @ ( vimage_nat_nat @ F @ A ) )
=> ( member_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_111_vimageD,axiom,
! [A2: nat,F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_nat @ A2 @ ( vimage3040984495076556338_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_112_vimageD,axiom,
! [A2: sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ A2 @ ( vimage6545432446589551483_a_nat @ F @ A ) )
=> ( member_Sum_sum_a_nat @ ( F @ A2 ) @ A ) ) ).
% vimageD
thf(fact_113_image__Un,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_a] :
( ( image_7873763678140191238_a_nat @ F @ ( sup_sup_set_a @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_114_image__Un,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_115_image__Un,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( image_2473878607534554506at_nat @ F @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_2473878607534554506at_nat @ F @ A ) @ ( image_2473878607534554506at_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_116_image__Un,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_su6804446743777130803_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_117_image__Un,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A @ B ) )
= ( sup_sup_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Un
thf(fact_118_Un__Int__distrib2,axiom,
! [B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) @ A )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B @ A ) @ ( sup_su6804446743777130803_a_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_119_Un__Int__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ C2 ) @ A )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ A ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_distrib2
thf(fact_120_Int__Un__distrib2,axiom,
! [B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) @ A )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ B @ A ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_121_Int__Un__distrib2,axiom,
! [B: set_nat,C2: set_nat,A: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ B @ C2 ) @ A )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ B @ A ) @ ( inf_inf_set_nat @ C2 @ A ) ) ) ).
% Int_Un_distrib2
thf(fact_122_Un__Int__distrib,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_123_Un__Int__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ A @ ( inf_inf_set_nat @ B @ C2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ A @ C2 ) ) ) ).
% Un_Int_distrib
thf(fact_124_Int__Un__distrib,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_125_Int__Un__distrib,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ A @ C2 ) ) ) ).
% Int_Un_distrib
thf(fact_126_Un__Int__crazy,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ B @ C2 ) ) @ ( inf_in7084830621192376909_a_nat @ C2 @ A ) )
= ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) @ ( sup_su6804446743777130803_a_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_127_Un__Int__crazy,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ B @ C2 ) ) @ ( inf_inf_set_nat @ C2 @ A ) )
= ( inf_inf_set_nat @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ B @ C2 ) ) @ ( sup_sup_set_nat @ C2 @ A ) ) ) ).
% Un_Int_crazy
thf(fact_128_vimage__inter__cong,axiom,
! [S: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,G: sum_sum_a_nat > sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [W: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ ( vimage6545432446589551483_a_nat @ F @ Y ) @ S )
= ( inf_in7084830621192376909_a_nat @ ( vimage6545432446589551483_a_nat @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_129_vimage__inter__cong,axiom,
! [S: set_nat,F: nat > sum_sum_a_nat,G: nat > sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [W: nat] :
( ( member_nat @ W @ S )
=> ( ( F @ W )
= ( G @ W ) ) )
=> ( ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ F @ Y ) @ S )
= ( inf_inf_set_nat @ ( vimage3040984495076556338_a_nat @ G @ Y ) @ S ) ) ) ).
% vimage_inter_cong
thf(fact_130_greaterThan__Int__greaterThan,axiom,
! [A2: nat,B2: nat] :
( ( inf_inf_set_nat @ ( set_ord_lessThan_nat @ A2 ) @ ( set_ord_lessThan_nat @ B2 ) )
= ( set_ord_lessThan_nat @ ( ord_min_nat @ A2 @ B2 ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_131_sup__inf__absorb,axiom,
! [X: nat,Y: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_132_sup__inf__absorb,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_133_sup__inf__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= X ) ).
% sup_inf_absorb
thf(fact_134_inf__sup__absorb,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_135_inf__sup__absorb,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_136_inf__sup__absorb,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= X ) ).
% inf_sup_absorb
thf(fact_137_min_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 )
= ( ord_min_nat @ A2 @ B2 ) ) ).
% min.right_idem
thf(fact_138_min_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( ord_min_nat @ A2 @ ( ord_min_nat @ A2 @ B2 ) )
= ( ord_min_nat @ A2 @ B2 ) ) ).
% min.left_idem
thf(fact_139_min_Oidem,axiom,
! [A2: nat] :
( ( ord_min_nat @ A2 @ A2 )
= A2 ) ).
% min.idem
thf(fact_140_old_Osum_Oinject_I2_J,axiom,
! [B2: nat,B4: nat] :
( ( ( sum_Inr_nat_a @ B2 )
= ( sum_Inr_nat_a @ B4 ) )
= ( B2 = B4 ) ) ).
% old.sum.inject(2)
thf(fact_141_sum_Oinject_I2_J,axiom,
! [X22: nat,Y2: nat] :
( ( ( sum_Inr_nat_a @ X22 )
= ( sum_Inr_nat_a @ Y2 ) )
= ( X22 = Y2 ) ) ).
% sum.inject(2)
thf(fact_142_old_Osum_Oinject_I1_J,axiom,
! [A2: a,A4: a] :
( ( ( sum_Inl_a_nat @ A2 )
= ( sum_Inl_a_nat @ A4 ) )
= ( A2 = A4 ) ) ).
% old.sum.inject(1)
thf(fact_143_sum_Oinject_I1_J,axiom,
! [X1: a,Y1: a] :
( ( ( sum_Inl_a_nat @ X1 )
= ( sum_Inl_a_nat @ Y1 ) )
= ( X1 = Y1 ) ) ).
% sum.inject(1)
thf(fact_144_inf__right__idem,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ Y )
= ( inf_in7084830621192376909_a_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_145_inf__right__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Y )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_146_inf__right__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_right_idem
thf(fact_147_inf_Oright__idem,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ B2 )
= ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_148_inf_Oright__idem,axiom,
! [A2: nat,B2: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_149_inf_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.right_idem
thf(fact_150_sup_Oidem,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_151_sup_Oidem,axiom,
! [A2: set_nat] :
( ( sup_sup_set_nat @ A2 @ A2 )
= A2 ) ).
% sup.idem
thf(fact_152_sup__idem,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_153_sup__idem,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ X )
= X ) ).
% sup_idem
thf(fact_154_sup_Oleft__idem,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_155_sup_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.left_idem
thf(fact_156_sup__left__idem,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) )
= ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_157_sup__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_left_idem
thf(fact_158_sup_Oright__idem,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ B2 )
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_159_sup_Oright__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ B2 )
= ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.right_idem
thf(fact_160_inf_Oidem,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_161_inf_Oidem,axiom,
! [A2: nat] :
( ( inf_inf_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_162_inf_Oidem,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ A2 )
= A2 ) ).
% inf.idem
thf(fact_163_inf__idem,axiom,
! [X: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_164_inf__idem,axiom,
! [X: nat] :
( ( inf_inf_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_165_inf__idem,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ X )
= X ) ).
% inf_idem
thf(fact_166_inf_Oleft__idem,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) )
= ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_167_inf_Oleft__idem,axiom,
! [A2: nat,B2: nat] :
( ( inf_inf_nat @ A2 @ ( inf_inf_nat @ A2 @ B2 ) )
= ( inf_inf_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_168_inf_Oleft__idem,axiom,
! [A2: set_nat,B2: set_nat] :
( ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ A2 @ B2 ) )
= ( inf_inf_set_nat @ A2 @ B2 ) ) ).
% inf.left_idem
thf(fact_169_inf__left__idem,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ X @ Y ) )
= ( inf_in7084830621192376909_a_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_170_inf__left__idem,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_171_inf__left__idem,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_left_idem
thf(fact_172_inf__sup__aci_I8_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) )
= ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_173_inf__sup__aci_I8_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) )
= ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_aci(8)
thf(fact_174_inf__sup__aci_I7_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_175_inf__sup__aci_I7_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(7)
thf(fact_176_inf__sup__aci_I6_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ Z )
= ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_177_inf__sup__aci_I6_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(6)
thf(fact_178_inf__sup__aci_I5_J,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X4: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_179_inf__sup__aci_I5_J,axiom,
( sup_sup_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(5)
thf(fact_180_sup_Oassoc,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ C )
= ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_181_sup_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ C )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.assoc
thf(fact_182_sup__assoc,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ Z )
= ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_183_sup__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% sup_assoc
thf(fact_184_sup_Ocommute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_185_sup_Ocommute,axiom,
( sup_sup_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).
% sup.commute
thf(fact_186_sup__commute,axiom,
( sup_su6804446743777130803_a_nat
= ( ^ [X4: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( sup_su6804446743777130803_a_nat @ Y3 @ X4 ) ) ) ).
% sup_commute
thf(fact_187_sup__commute,axiom,
( sup_sup_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] : ( sup_sup_set_nat @ Y3 @ X4 ) ) ) ).
% sup_commute
thf(fact_188_sup_Oleft__commute,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A2 @ C ) )
= ( sup_su6804446743777130803_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_189_sup_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( sup_sup_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ C ) )
= ( sup_sup_set_nat @ A2 @ ( sup_sup_set_nat @ B2 @ C ) ) ) ).
% sup.left_commute
thf(fact_190_sup__left__commute,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_191_sup__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ Y @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_left_commute
thf(fact_192_inf__sup__aci_I4_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ X @ Y ) )
= ( inf_in7084830621192376909_a_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_193_inf__sup__aci_I4_J,axiom,
! [X: nat,Y: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ X @ Y ) )
= ( inf_inf_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_194_inf__sup__aci_I4_J,axiom,
! [X: set_nat,Y: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ X @ Y ) )
= ( inf_inf_set_nat @ X @ Y ) ) ).
% inf_sup_aci(4)
thf(fact_195_inf__sup__aci_I3_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ Y @ ( inf_in7084830621192376909_a_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_196_inf__sup__aci_I3_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_197_inf__sup__aci_I3_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_aci(3)
thf(fact_198_inf__sup__aci_I2_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ Z )
= ( inf_in7084830621192376909_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_199_inf__sup__aci_I2_J,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_200_inf__sup__aci_I2_J,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_sup_aci(2)
thf(fact_201_inf__sup__aci_I1_J,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [X4: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_202_inf__sup__aci_I1_J,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_203_inf__sup__aci_I1_J,axiom,
( inf_inf_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X4 ) ) ) ).
% inf_sup_aci(1)
thf(fact_204_inf_Oassoc,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ C )
= ( inf_in7084830621192376909_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_205_inf_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C )
= ( inf_inf_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_206_inf_Oassoc,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.assoc
thf(fact_207_inf__assoc,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ Z )
= ( inf_in7084830621192376909_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_208_inf__assoc,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ ( inf_inf_nat @ X @ Y ) @ Z )
= ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_209_inf__assoc,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Z )
= ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ).
% inf_assoc
thf(fact_210_inf_Ocommute,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_211_inf_Ocommute,axiom,
( inf_inf_nat
= ( ^ [A5: nat,B5: nat] : ( inf_inf_nat @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_212_inf_Ocommute,axiom,
( inf_inf_set_nat
= ( ^ [A5: set_nat,B5: set_nat] : ( inf_inf_set_nat @ B5 @ A5 ) ) ) ).
% inf.commute
thf(fact_213_inf__commute,axiom,
( inf_in7084830621192376909_a_nat
= ( ^ [X4: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] : ( inf_in7084830621192376909_a_nat @ Y3 @ X4 ) ) ) ).
% inf_commute
thf(fact_214_inf__commute,axiom,
( inf_inf_nat
= ( ^ [X4: nat,Y3: nat] : ( inf_inf_nat @ Y3 @ X4 ) ) ) ).
% inf_commute
thf(fact_215_inf__commute,axiom,
( inf_inf_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] : ( inf_inf_set_nat @ Y3 @ X4 ) ) ) ).
% inf_commute
thf(fact_216_inf_Oleft__commute,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ B2 @ ( inf_in7084830621192376909_a_nat @ A2 @ C ) )
= ( inf_in7084830621192376909_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_217_inf_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( inf_inf_nat @ B2 @ ( inf_inf_nat @ A2 @ C ) )
= ( inf_inf_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_218_inf_Oleft__commute,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( inf_inf_set_nat @ B2 @ ( inf_inf_set_nat @ A2 @ C ) )
= ( inf_inf_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ).
% inf.left_commute
thf(fact_219_inf__left__commute,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ Y @ ( inf_in7084830621192376909_a_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_220_inf__left__commute,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ Y @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_221_inf__left__commute,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ Y @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_left_commute
thf(fact_222_Inl__inject,axiom,
! [X: a,Y: a] :
( ( ( sum_Inl_a_nat @ X )
= ( sum_Inl_a_nat @ Y ) )
=> ( X = Y ) ) ).
% Inl_inject
thf(fact_223_Inr__inject,axiom,
! [X: nat,Y: nat] :
( ( ( sum_Inr_nat_a @ X )
= ( sum_Inr_nat_a @ Y ) )
=> ( X = Y ) ) ).
% Inr_inject
thf(fact_224_min_Oassoc,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_min_nat @ ( ord_min_nat @ A2 @ B2 ) @ C )
= ( ord_min_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ).
% min.assoc
thf(fact_225_min_Ocommute,axiom,
( ord_min_nat
= ( ^ [A5: nat,B5: nat] : ( ord_min_nat @ B5 @ A5 ) ) ) ).
% min.commute
thf(fact_226_min_Oleft__commute,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_min_nat @ B2 @ ( ord_min_nat @ A2 @ C ) )
= ( ord_min_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ).
% min.left_commute
thf(fact_227_distrib__imp1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y4: nat,Z2: nat] :
( ( inf_inf_nat @ X2 @ ( sup_sup_nat @ Y4 @ Z2 ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X2 @ Y4 ) @ ( inf_inf_nat @ X2 @ Z2 ) ) )
=> ( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_228_distrib__imp1,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X2 @ ( sup_su6804446743777130803_a_nat @ Y4 @ Z2 ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X2 @ Y4 ) @ ( inf_in7084830621192376909_a_nat @ X2 @ Z2 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ ( sup_su6804446743777130803_a_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_229_distrib__imp1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( inf_inf_set_nat @ X2 @ ( sup_sup_set_nat @ Y4 @ Z2 ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X2 @ Y4 ) @ ( inf_inf_set_nat @ X2 @ Z2 ) ) )
=> ( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp1
thf(fact_230_distrib__imp2,axiom,
! [X: nat,Y: nat,Z: nat] :
( ! [X2: nat,Y4: nat,Z2: nat] :
( ( sup_sup_nat @ X2 @ ( inf_inf_nat @ Y4 @ Z2 ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X2 @ Y4 ) @ ( sup_sup_nat @ X2 @ Z2 ) ) )
=> ( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_231_distrib__imp2,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X2 @ ( inf_in7084830621192376909_a_nat @ Y4 @ Z2 ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X2 @ Y4 ) @ ( sup_su6804446743777130803_a_nat @ X2 @ Z2 ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ ( inf_in7084830621192376909_a_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_232_distrib__imp2,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( sup_sup_set_nat @ X2 @ ( inf_inf_set_nat @ Y4 @ Z2 ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X2 @ Y4 ) @ ( sup_sup_set_nat @ X2 @ Z2 ) ) )
=> ( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ) ).
% distrib_imp2
thf(fact_233_inf__sup__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) )
= ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_234_inf__sup__distrib1,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ ( inf_in7084830621192376909_a_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_235_inf__sup__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% inf_sup_distrib1
thf(fact_236_inf__sup__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( inf_inf_nat @ ( sup_sup_nat @ Y @ Z ) @ X )
= ( sup_sup_nat @ ( inf_inf_nat @ Y @ X ) @ ( inf_inf_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_237_inf__sup__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) @ X )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ X ) @ ( inf_in7084830621192376909_a_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_238_inf__sup__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).
% inf_sup_distrib2
thf(fact_239_sup__inf__distrib1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_240_sup__inf__distrib1,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ ( sup_su6804446743777130803_a_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_241_sup__inf__distrib1,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% sup_inf_distrib1
thf(fact_242_sup__inf__distrib2,axiom,
! [Y: nat,Z: nat,X: nat] :
( ( sup_sup_nat @ ( inf_inf_nat @ Y @ Z ) @ X )
= ( inf_inf_nat @ ( sup_sup_nat @ Y @ X ) @ ( sup_sup_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_243_sup__inf__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) @ X )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ X ) @ ( sup_su6804446743777130803_a_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_244_sup__inf__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).
% sup_inf_distrib2
thf(fact_245_inf__min,axiom,
inf_inf_nat = ord_min_nat ).
% inf_min
thf(fact_246_sum_Odistinct_I1_J,axiom,
! [X1: a,X22: nat] :
( ( sum_Inl_a_nat @ X1 )
!= ( sum_Inr_nat_a @ X22 ) ) ).
% sum.distinct(1)
thf(fact_247_old_Osum_Odistinct_I2_J,axiom,
! [B4: nat,A2: a] :
( ( sum_Inr_nat_a @ B4 )
!= ( sum_Inl_a_nat @ A2 ) ) ).
% old.sum.distinct(2)
thf(fact_248_old_Osum_Odistinct_I1_J,axiom,
! [A2: a,B4: nat] :
( ( sum_Inl_a_nat @ A2 )
!= ( sum_Inr_nat_a @ B4 ) ) ).
% old.sum.distinct(1)
thf(fact_249_old_Osum_Oexhaust,axiom,
! [Y: sum_sum_a_nat] :
( ! [A6: a] :
( Y
!= ( sum_Inl_a_nat @ A6 ) )
=> ~ ! [B6: nat] :
( Y
!= ( sum_Inr_nat_a @ B6 ) ) ) ).
% old.sum.exhaust
thf(fact_250_sumE,axiom,
! [S2: sum_sum_a_nat] :
( ! [X2: a] :
( S2
!= ( sum_Inl_a_nat @ X2 ) )
=> ~ ! [Y4: nat] :
( S2
!= ( sum_Inr_nat_a @ Y4 ) ) ) ).
% sumE
thf(fact_251_Inr__not__Inl,axiom,
! [B2: nat,A2: a] :
( ( sum_Inr_nat_a @ B2 )
!= ( sum_Inl_a_nat @ A2 ) ) ).
% Inr_not_Inl
thf(fact_252_split__sum__ex,axiom,
( ( ^ [P2: sum_sum_a_nat > $o] :
? [X5: sum_sum_a_nat] : ( P2 @ X5 ) )
= ( ^ [P3: sum_sum_a_nat > $o] :
( ? [X4: a] : ( P3 @ ( sum_Inl_a_nat @ X4 ) )
| ? [X4: nat] : ( P3 @ ( sum_Inr_nat_a @ X4 ) ) ) ) ) ).
% split_sum_ex
thf(fact_253_split__sum__all,axiom,
( ( ^ [P2: sum_sum_a_nat > $o] :
! [X5: sum_sum_a_nat] : ( P2 @ X5 ) )
= ( ^ [P3: sum_sum_a_nat > $o] :
( ! [X4: a] : ( P3 @ ( sum_Inl_a_nat @ X4 ) )
& ! [X4: nat] : ( P3 @ ( sum_Inr_nat_a @ X4 ) ) ) ) ) ).
% split_sum_all
thf(fact_254_obj__sumE,axiom,
! [S2: sum_sum_a_nat] :
( ! [X2: a] :
( S2
!= ( sum_Inl_a_nat @ X2 ) )
=> ~ ! [X2: nat] :
( S2
!= ( sum_Inr_nat_a @ X2 ) ) ) ).
% obj_sumE
thf(fact_255_fo__nmlzd__all__AD,axiom,
! [Xs: list_Sum_sum_a_nat,AD: set_a] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ AD ) )
=> ( fo_nmlzd_a @ AD @ Xs ) ) ).
% fo_nmlzd_all_AD
thf(fact_256_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ ( inf_in7084830621192376909_a_nat @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_257_boolean__algebra_Oconj__disj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) ) ).
% boolean_algebra.conj_disj_distrib
thf(fact_258_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ ( sup_su6804446743777130803_a_nat @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_259_boolean__algebra_Odisj__conj__distrib,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% boolean_algebra.disj_conj_distrib
thf(fact_260_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) @ X )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ X ) @ ( inf_in7084830621192376909_a_nat @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_261_boolean__algebra_Oconj__disj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X )
= ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ X ) @ ( inf_inf_set_nat @ Z @ X ) ) ) ).
% boolean_algebra.conj_disj_distrib2
thf(fact_262_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) @ X )
= ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ X ) @ ( sup_su6804446743777130803_a_nat @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_263_boolean__algebra_Odisj__conj__distrib2,axiom,
! [Y: set_nat,Z: set_nat,X: set_nat] :
( ( sup_sup_set_nat @ ( inf_inf_set_nat @ Y @ Z ) @ X )
= ( inf_inf_set_nat @ ( sup_sup_set_nat @ Y @ X ) @ ( sup_sup_set_nat @ Z @ X ) ) ) ).
% boolean_algebra.disj_conj_distrib2
thf(fact_264_card__Inr__vimage__le__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_Inr_vimage_le_length
thf(fact_265_subsetI,axiom,
! [A: set_nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ X2 @ B ) )
=> ( ord_less_eq_set_nat @ A @ B ) ) ).
% subsetI
thf(fact_266_sup_Obounded__iff,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A2 )
= ( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
& ( ord_le1325389633284124927_a_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_267_sup_Obounded__iff,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_set_nat @ B2 @ A2 )
& ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_268_sup_Obounded__iff,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.bounded_iff
thf(fact_269_le__sup__iff,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ Z )
= ( ( ord_le1325389633284124927_a_nat @ X @ Z )
& ( ord_le1325389633284124927_a_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_270_le__sup__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_set_nat @ X @ Z )
& ( ord_less_eq_set_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_271_le__sup__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
& ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% le_sup_iff
thf(fact_272_inf_Obounded__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
= ( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
& ( ord_le1325389633284124927_a_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_273_inf_Obounded__iff,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
= ( ( ord_less_eq_set_nat @ A2 @ B2 )
& ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_274_inf_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.bounded_iff
thf(fact_275_le__inf__iff,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) )
= ( ( ord_le1325389633284124927_a_nat @ X @ Y )
& ( ord_le1325389633284124927_a_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_276_le__inf__iff,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) )
= ( ( ord_less_eq_set_nat @ X @ Y )
& ( ord_less_eq_set_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_277_le__inf__iff,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) )
= ( ( ord_less_eq_nat @ X @ Y )
& ( ord_less_eq_nat @ X @ Z ) ) ) ).
% le_inf_iff
thf(fact_278_Int__subset__iff,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A @ B ) )
= ( ( ord_le1325389633284124927_a_nat @ C2 @ A )
& ( ord_le1325389633284124927_a_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_279_Int__subset__iff,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) )
= ( ( ord_less_eq_set_nat @ C2 @ A )
& ( ord_less_eq_set_nat @ C2 @ B ) ) ) ).
% Int_subset_iff
thf(fact_280_Un__subset__iff,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 )
= ( ( ord_le1325389633284124927_a_nat @ A @ C2 )
& ( ord_le1325389633284124927_a_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_281_Un__subset__iff,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 )
= ( ( ord_less_eq_set_nat @ A @ C2 )
& ( ord_less_eq_set_nat @ B @ C2 ) ) ) ).
% Un_subset_iff
thf(fact_282_min_Obounded__iff,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
= ( ( ord_less_eq_nat @ A2 @ B2 )
& ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.bounded_iff
thf(fact_283_min_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= B2 ) ) ).
% min.absorb2
thf(fact_284_min_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_min_nat @ A2 @ B2 )
= A2 ) ) ).
% min.absorb1
thf(fact_285_lessThan__subset__iff,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X ) @ ( set_ord_lessThan_nat @ Y ) )
= ( ord_less_eq_nat @ X @ Y ) ) ).
% lessThan_subset_iff
thf(fact_286_bounded__Max__nat,axiom,
! [P: nat > $o,X: nat,M: nat] :
( ( P @ X )
=> ( ! [X2: nat] :
( ( P @ X2 )
=> ( ord_less_eq_nat @ X2 @ M ) )
=> ~ ! [M2: nat] :
( ( P @ M2 )
=> ~ ! [X3: nat] :
( ( P @ X3 )
=> ( ord_less_eq_nat @ X3 @ M2 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_287_subset__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [T2: nat] :
( ( member_nat @ T2 @ A3 )
=> ( member_nat @ T2 @ B3 ) ) ) ) ).
% subset_iff
thf(fact_288_subset__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
! [X4: nat] :
( ( member_nat @ X4 @ A3 )
=> ( member_nat @ X4 @ B3 ) ) ) ) ).
% subset_eq
thf(fact_289_subsetD,axiom,
! [A: set_nat,B: set_nat,C: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ C @ A )
=> ( member_nat @ C @ B ) ) ) ).
% subsetD
thf(fact_290_in__mono,axiom,
! [A: set_nat,B: set_nat,X: nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ X @ A )
=> ( member_nat @ X @ B ) ) ) ).
% in_mono
thf(fact_291_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( ? [AA: set_a] :
( ( ord_less_eq_set_a @ AA @ A )
& ( B
= ( image_7873763678140191238_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_292_subset__image__iff,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_7293268710728258664_a_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_293_subset__image__iff,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
= ( ? [AA: set_nat] :
( ( ord_less_eq_set_nat @ AA @ A )
& ( B
= ( image_nat_nat @ F @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_294_image__subset__iff,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ B )
= ( ! [X4: a] :
( ( member_a @ X4 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_295_image__subset__iff,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_296_image__subset__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% image_subset_iff
thf(fact_297_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ~ ! [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
=> ( B
!= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_298_subset__imageE,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_299_subset__imageE,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ~ ! [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
=> ( B
!= ( image_nat_nat @ F @ C3 ) ) ) ) ).
% subset_imageE
thf(fact_300_image__subsetI,axiom,
! [A: set_a,F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_301_image__subsetI,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X2 ) @ B ) )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_302_image__subsetI,axiom,
! [A: set_nat,F: nat > nat,B: set_nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( member_nat @ ( F @ X2 ) @ B ) )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B ) ) ).
% image_subsetI
thf(fact_303_image__mono,axiom,
! [A: set_a,B: set_a,F: a > sum_sum_a_nat] :
( ( ord_less_eq_set_a @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_304_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_305_image__mono,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_mono
thf(fact_306_sup_OcoboundedI2,axiom,
! [C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ B2 )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_307_sup_OcoboundedI2,axiom,
! [C: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ C @ B2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_308_sup_OcoboundedI2,axiom,
! [C: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI2
thf(fact_309_sup_OcoboundedI1,axiom,
! [C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A2 )
=> ( ord_le1325389633284124927_a_nat @ C @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_310_sup_OcoboundedI1,axiom,
! [C: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_311_sup_OcoboundedI1,axiom,
! [C: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.coboundedI1
thf(fact_312_sup_Oabsorb__iff2,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_313_sup_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_314_sup_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= B5 ) ) ) ).
% sup.absorb_iff2
thf(fact_315_sup_Oabsorb__iff1,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B5: set_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_316_sup_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( sup_sup_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_317_sup_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( sup_sup_nat @ A5 @ B5 )
= A5 ) ) ) ).
% sup.absorb_iff1
thf(fact_318_sup_Ocobounded2,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_319_sup_Ocobounded2,axiom,
! [B2: set_nat,A2: set_nat] : ( ord_less_eq_set_nat @ B2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_320_sup_Ocobounded2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq_nat @ B2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded2
thf(fact_321_sup_Ocobounded1,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A2 @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_322_sup_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ A2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_323_sup_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ A2 @ ( sup_sup_nat @ A2 @ B2 ) ) ).
% sup.cobounded1
thf(fact_324_sup_Oorder__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B5: set_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
( A5
= ( sup_su6804446743777130803_a_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_325_sup_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( A5
= ( sup_sup_set_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_326_sup_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( A5
= ( sup_sup_nat @ A5 @ B5 ) ) ) ) ).
% sup.order_iff
thf(fact_327_sup_OboundedI,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ C @ A2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_328_sup_OboundedI,axiom,
! [B2: set_nat,A2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_329_sup_OboundedI,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ A2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 ) ) ) ).
% sup.boundedI
thf(fact_330_sup_OboundedE,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ~ ( ord_le1325389633284124927_a_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_331_sup_OboundedE,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_set_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_332_sup_OboundedE,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
=> ~ ( ( ord_less_eq_nat @ B2 @ A2 )
=> ~ ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% sup.boundedE
thf(fact_333_sup__absorb2,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ Y )
=> ( ( sup_su6804446743777130803_a_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_334_sup__absorb2,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( sup_sup_set_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_335_sup__absorb2,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( sup_sup_nat @ X @ Y )
= Y ) ) ).
% sup_absorb2
thf(fact_336_sup__absorb1,axiom,
! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X )
=> ( ( sup_su6804446743777130803_a_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_337_sup__absorb1,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( sup_sup_set_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_338_sup__absorb1,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( sup_sup_nat @ X @ Y )
= X ) ) ).
% sup_absorb1
thf(fact_339_sup_Oabsorb2,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_340_sup_Oabsorb2,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_341_sup_Oabsorb2,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= B2 ) ) ).
% sup.absorb2
thf(fact_342_sup_Oabsorb1,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_343_sup_Oabsorb1,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( sup_sup_set_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_344_sup_Oabsorb1,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( sup_sup_nat @ A2 @ B2 )
= A2 ) ) ).
% sup.absorb1
thf(fact_345_sup__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y4 @ X2 )
=> ( ( ord_le1325389633284124927_a_nat @ Z2 @ X2 )
=> ( ord_le1325389633284124927_a_nat @ ( F @ Y4 @ Z2 ) @ X2 ) ) )
=> ( ( sup_su6804446743777130803_a_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_346_sup__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_set_nat @ Z2 @ X2 )
=> ( ord_less_eq_set_nat @ ( F @ Y4 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_347_sup__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ X2 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ ( F @ X2 @ Y4 ) )
=> ( ! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ Y4 @ X2 )
=> ( ( ord_less_eq_nat @ Z2 @ X2 )
=> ( ord_less_eq_nat @ ( F @ Y4 @ Z2 ) @ X2 ) ) )
=> ( ( sup_sup_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% sup_unique
thf(fact_348_sup_OorderI,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A2
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_349_sup_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_350_sup_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( sup_sup_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ B2 @ A2 ) ) ).
% sup.orderI
thf(fact_351_sup_OorderE,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( A2
= ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_352_sup_OorderE,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_353_sup_OorderE,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2
= ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% sup.orderE
thf(fact_354_le__iff__sup,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X4: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_355_le__iff__sup,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( sup_sup_set_nat @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_356_le__iff__sup,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( sup_sup_nat @ X4 @ Y3 )
= Y3 ) ) ) ).
% le_iff_sup
thf(fact_357_sup__least,axiom,
! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X )
=> ( ( ord_le1325389633284124927_a_nat @ Z @ X )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_358_sup__least,axiom,
! [Y: set_nat,X: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( ord_less_eq_set_nat @ Z @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_359_sup__least,axiom,
! [Y: nat,X: nat,Z: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ Z @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ Y @ Z ) @ X ) ) ) ).
% sup_least
thf(fact_360_sup__mono,axiom,
! [A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ ( sup_su6804446743777130803_a_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_361_sup__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ ( sup_sup_set_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_362_sup__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ ( sup_sup_nat @ C @ D ) ) ) ) ).
% sup_mono
thf(fact_363_sup_Omono,axiom,
! [C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ D @ B2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ C @ D ) @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_364_sup_Omono,axiom,
! [C: set_nat,A2: set_nat,D: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ C @ A2 )
=> ( ( ord_less_eq_set_nat @ D @ B2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ C @ D ) @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_365_sup_Omono,axiom,
! [C: nat,A2: nat,D: nat,B2: nat] :
( ( ord_less_eq_nat @ C @ A2 )
=> ( ( ord_less_eq_nat @ D @ B2 )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ C @ D ) @ ( sup_sup_nat @ A2 @ B2 ) ) ) ) ).
% sup.mono
thf(fact_366_le__supI2,axiom,
! [X: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_367_le__supI2,axiom,
! [X: set_nat,B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_368_le__supI2,axiom,
! [X: nat,B2: nat,A2: nat] :
( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI2
thf(fact_369_le__supI1,axiom,
! [X: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ A2 )
=> ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_370_le__supI1,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_371_le__supI1,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ord_less_eq_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).
% le_supI1
thf(fact_372_sup__ge2,axiom,
! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_373_sup__ge2,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_374_sup__ge2,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge2
thf(fact_375_sup__ge1,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_376_sup__ge1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_377_sup__ge1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% sup_ge1
thf(fact_378_le__supI,axiom,
! [A2: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ X )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ X )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_379_le__supI,axiom,
! [A2: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_380_le__supI,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X ) ) ) ).
% le_supI
thf(fact_381_le__supE,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A2 @ X )
=> ~ ( ord_le1325389633284124927_a_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_382_le__supE,axiom,
! [A2: set_nat,B2: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ X )
=> ~ ( ord_less_eq_set_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_383_le__supE,axiom,
! [A2: nat,B2: nat,X: nat] :
( ( ord_less_eq_nat @ ( sup_sup_nat @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq_nat @ A2 @ X )
=> ~ ( ord_less_eq_nat @ B2 @ X ) ) ) ).
% le_supE
thf(fact_384_inf__sup__ord_I3_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_385_inf__sup__ord_I3_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ X @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_386_inf__sup__ord_I3_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ X @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(3)
thf(fact_387_inf__sup__ord_I4_J,axiom,
! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ Y @ ( sup_su6804446743777130803_a_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_388_inf__sup__ord_I4_J,axiom,
! [Y: set_nat,X: set_nat] : ( ord_less_eq_set_nat @ Y @ ( sup_sup_set_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_389_inf__sup__ord_I4_J,axiom,
! [Y: nat,X: nat] : ( ord_less_eq_nat @ Y @ ( sup_sup_nat @ X @ Y ) ) ).
% inf_sup_ord(4)
thf(fact_390_inf_OcoboundedI2,axiom,
! [B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ C )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_391_inf_OcoboundedI2,axiom,
! [B2: set_nat,C: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_392_inf_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI2
thf(fact_393_inf_OcoboundedI1,axiom,
! [A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_394_inf_OcoboundedI1,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_395_inf_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ C ) ) ).
% inf.coboundedI1
thf(fact_396_inf_Oabsorb__iff2,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [B5: set_Sum_sum_a_nat,A5: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_397_inf_Oabsorb__iff2,axiom,
( ord_less_eq_set_nat
= ( ^ [B5: set_nat,A5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_398_inf_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= B5 ) ) ) ).
% inf.absorb_iff2
thf(fact_399_inf_Oabsorb__iff1,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_400_inf_Oabsorb__iff1,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( ( inf_inf_set_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_401_inf_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( inf_inf_nat @ A5 @ B5 )
= A5 ) ) ) ).
% inf.absorb_iff1
thf(fact_402_inf_Ocobounded2,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_403_inf_Ocobounded2,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_404_inf_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ B2 ) ).
% inf.cobounded2
thf(fact_405_inf_Ocobounded1,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_406_inf_Ocobounded1,axiom,
! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_407_inf_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ A2 ) ).
% inf.cobounded1
thf(fact_408_inf_Oorder__iff,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A5: set_Sum_sum_a_nat,B5: set_Sum_sum_a_nat] :
( A5
= ( inf_in7084830621192376909_a_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_409_inf_Oorder__iff,axiom,
( ord_less_eq_set_nat
= ( ^ [A5: set_nat,B5: set_nat] :
( A5
= ( inf_inf_set_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_410_inf_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( inf_inf_nat @ A5 @ B5 ) ) ) ) ).
% inf.order_iff
thf(fact_411_inf__greatest,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ Y )
=> ( ( ord_le1325389633284124927_a_nat @ X @ Z )
=> ( ord_le1325389633284124927_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_412_inf__greatest,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( ord_less_eq_set_nat @ X @ Z )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_413_inf__greatest,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ X @ Z )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) ) ) ).
% inf_greatest
thf(fact_414_inf_OboundedI,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( ord_le1325389633284124927_a_nat @ A2 @ C )
=> ( ord_le1325389633284124927_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_415_inf_OboundedI,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_416_inf_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) ) ) ) ).
% inf.boundedI
thf(fact_417_inf_OboundedE,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ ( inf_in7084830621192376909_a_nat @ B2 @ C ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ~ ( ord_le1325389633284124927_a_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_418_inf_OboundedE,axiom,
! [A2: set_nat,B2: set_nat,C: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ ( inf_inf_set_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_set_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_419_inf_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( inf_inf_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% inf.boundedE
thf(fact_420_inf__absorb2,axiom,
! [Y: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ Y @ X )
=> ( ( inf_in7084830621192376909_a_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_421_inf__absorb2,axiom,
! [Y: set_nat,X: set_nat] :
( ( ord_less_eq_set_nat @ Y @ X )
=> ( ( inf_inf_set_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_422_inf__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( inf_inf_nat @ X @ Y )
= Y ) ) ).
% inf_absorb2
thf(fact_423_inf__absorb1,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ Y )
=> ( ( inf_in7084830621192376909_a_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_424_inf__absorb1,axiom,
! [X: set_nat,Y: set_nat] :
( ( ord_less_eq_set_nat @ X @ Y )
=> ( ( inf_inf_set_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_425_inf__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( inf_inf_nat @ X @ Y )
= X ) ) ).
% inf_absorb1
thf(fact_426_inf_Oabsorb2,axiom,
! [B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ A2 )
=> ( ( inf_in7084830621192376909_a_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_427_inf_Oabsorb2,axiom,
! [B2: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ A2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_428_inf_Oabsorb2,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= B2 ) ) ).
% inf.absorb2
thf(fact_429_inf_Oabsorb1,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( ( inf_in7084830621192376909_a_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_430_inf_Oabsorb1,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( ( inf_inf_set_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_431_inf_Oabsorb1,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( inf_inf_nat @ A2 @ B2 )
= A2 ) ) ).
% inf.absorb1
thf(fact_432_le__iff__inf,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [X4: set_Sum_sum_a_nat,Y3: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_433_le__iff__inf,axiom,
( ord_less_eq_set_nat
= ( ^ [X4: set_nat,Y3: set_nat] :
( ( inf_inf_set_nat @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_434_le__iff__inf,axiom,
( ord_less_eq_nat
= ( ^ [X4: nat,Y3: nat] :
( ( inf_inf_nat @ X4 @ Y3 )
= X4 ) ) ) ).
% le_iff_inf
thf(fact_435_inf__unique,axiom,
! [F: set_Sum_sum_a_nat > set_Sum_sum_a_nat > set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_Sum_sum_a_nat,Y4: set_Sum_sum_a_nat,Z2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X2 @ Y4 )
=> ( ( ord_le1325389633284124927_a_nat @ X2 @ Z2 )
=> ( ord_le1325389633284124927_a_nat @ X2 @ ( F @ Y4 @ Z2 ) ) ) )
=> ( ( inf_in7084830621192376909_a_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_436_inf__unique,axiom,
! [F: set_nat > set_nat > set_nat,X: set_nat,Y: set_nat] :
( ! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: set_nat,Y4: set_nat] : ( ord_less_eq_set_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: set_nat,Y4: set_nat,Z2: set_nat] :
( ( ord_less_eq_set_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_set_nat @ X2 @ Z2 )
=> ( ord_less_eq_set_nat @ X2 @ ( F @ Y4 @ Z2 ) ) ) )
=> ( ( inf_inf_set_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_437_inf__unique,axiom,
! [F: nat > nat > nat,X: nat,Y: nat] :
( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ X2 )
=> ( ! [X2: nat,Y4: nat] : ( ord_less_eq_nat @ ( F @ X2 @ Y4 ) @ Y4 )
=> ( ! [X2: nat,Y4: nat,Z2: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ( ord_less_eq_nat @ X2 @ Z2 )
=> ( ord_less_eq_nat @ X2 @ ( F @ Y4 @ Z2 ) ) ) )
=> ( ( inf_inf_nat @ X @ Y )
= ( F @ X @ Y ) ) ) ) ) ).
% inf_unique
thf(fact_438_inf_OorderI,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A2
= ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) )
=> ( ord_le1325389633284124927_a_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_439_inf_OorderI,axiom,
! [A2: set_nat,B2: set_nat] :
( ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) )
=> ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_440_inf_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( inf_inf_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% inf.orderI
thf(fact_441_inf_OorderE,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ B2 )
=> ( A2
= ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_442_inf_OorderE,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_set_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_443_inf_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( inf_inf_nat @ A2 @ B2 ) ) ) ).
% inf.orderE
thf(fact_444_le__infI2,axiom,
! [B2: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B2 @ X )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_445_le__infI2,axiom,
! [B2: set_nat,X: set_nat,A2: set_nat] :
( ( ord_less_eq_set_nat @ B2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_446_le__infI2,axiom,
! [B2: nat,X: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI2
thf(fact_447_le__infI1,axiom,
! [A2: set_Sum_sum_a_nat,X: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ X )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_448_le__infI1,axiom,
! [A2: set_nat,X: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ X )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_449_le__infI1,axiom,
! [A2: nat,X: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ X )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ X ) ) ).
% le_infI1
thf(fact_450_inf__mono,axiom,
! [A2: set_Sum_sum_a_nat,C: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,D: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A2 @ C )
=> ( ( ord_le1325389633284124927_a_nat @ B2 @ D )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) @ ( inf_in7084830621192376909_a_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_451_inf__mono,axiom,
! [A2: set_nat,C: set_nat,B2: set_nat,D: set_nat] :
( ( ord_less_eq_set_nat @ A2 @ C )
=> ( ( ord_less_eq_set_nat @ B2 @ D )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A2 @ B2 ) @ ( inf_inf_set_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_452_inf__mono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( inf_inf_nat @ A2 @ B2 ) @ ( inf_inf_nat @ C @ D ) ) ) ) ).
% inf_mono
thf(fact_453_le__infI,axiom,
! [X: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ A2 )
=> ( ( ord_le1325389633284124927_a_nat @ X @ B2 )
=> ( ord_le1325389633284124927_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_454_le__infI,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ A2 )
=> ( ( ord_less_eq_set_nat @ X @ B2 )
=> ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_455_le__infI,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ A2 )
=> ( ( ord_less_eq_nat @ X @ B2 )
=> ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% le_infI
thf(fact_456_le__infE,axiom,
! [X: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) )
=> ~ ( ( ord_le1325389633284124927_a_nat @ X @ A2 )
=> ~ ( ord_le1325389633284124927_a_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_457_le__infE,axiom,
! [X: set_nat,A2: set_nat,B2: set_nat] :
( ( ord_less_eq_set_nat @ X @ ( inf_inf_set_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_set_nat @ X @ A2 )
=> ~ ( ord_less_eq_set_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_458_le__infE,axiom,
! [X: nat,A2: nat,B2: nat] :
( ( ord_less_eq_nat @ X @ ( inf_inf_nat @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq_nat @ X @ A2 )
=> ~ ( ord_less_eq_nat @ X @ B2 ) ) ) ).
% le_infE
thf(fact_459_inf__le2,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_460_inf__le2,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_461_inf__le2,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_le2
thf(fact_462_inf__le1,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_463_inf__le1,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_464_inf__le1,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_le1
thf(fact_465_inf__sup__ord_I1_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_466_inf__sup__ord_I1_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_467_inf__sup__ord_I1_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ X ) ).
% inf_sup_ord(1)
thf(fact_468_inf__sup__ord_I2_J,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_469_inf__sup__ord_I2_J,axiom,
! [X: set_nat,Y: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_470_inf__sup__ord_I2_J,axiom,
! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X @ Y ) @ Y ) ).
% inf_sup_ord(2)
thf(fact_471_Int__Collect__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,P: sum_sum_a_nat > $o,Q: sum_sum_a_nat > $o] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ ( collec7073057861543223018_a_nat @ P ) ) @ ( inf_in7084830621192376909_a_nat @ B @ ( collec7073057861543223018_a_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_472_Int__Collect__mono,axiom,
! [A: set_nat,B: set_nat,P: nat > $o,Q: nat > $o] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( ( P @ X2 )
=> ( Q @ X2 ) ) )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ ( collect_nat @ P ) ) @ ( inf_inf_set_nat @ B @ ( collect_nat @ Q ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_473_Int__greatest,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ A )
=> ( ( ord_le1325389633284124927_a_nat @ C2 @ B )
=> ( ord_le1325389633284124927_a_nat @ C2 @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_474_Int__greatest,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ A )
=> ( ( ord_less_eq_set_nat @ C2 @ B )
=> ( ord_less_eq_set_nat @ C2 @ ( inf_inf_set_nat @ A @ B ) ) ) ) ).
% Int_greatest
thf(fact_475_Int__absorb2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_476_Int__absorb2,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( inf_inf_set_nat @ A @ B )
= A ) ) ).
% Int_absorb2
thf(fact_477_Int__absorb1,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_478_Int__absorb1,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( inf_inf_set_nat @ A @ B )
= B ) ) ).
% Int_absorb1
thf(fact_479_Int__lower2,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_480_Int__lower2,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ B ) ).
% Int_lower2
thf(fact_481_Int__lower1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_482_Int__lower1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ A ) ).
% Int_lower1
thf(fact_483_Int__mono,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ ( inf_in7084830621192376909_a_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_484_Int__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( inf_inf_set_nat @ A @ B ) @ ( inf_inf_set_nat @ C2 @ D2 ) ) ) ) ).
% Int_mono
thf(fact_485_subset__Un__eq,axiom,
( ord_le1325389633284124927_a_nat
= ( ^ [A3: set_Sum_sum_a_nat,B3: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_486_subset__Un__eq,axiom,
( ord_less_eq_set_nat
= ( ^ [A3: set_nat,B3: set_nat] :
( ( sup_sup_set_nat @ A3 @ B3 )
= B3 ) ) ) ).
% subset_Un_eq
thf(fact_487_subset__UnE,axiom,
! [C2: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ C2 @ ( sup_su6804446743777130803_a_nat @ A @ B ) )
=> ~ ! [A7: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A7 @ A )
=> ! [B7: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B7 @ B )
=> ( C2
!= ( sup_su6804446743777130803_a_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_488_subset__UnE,axiom,
! [C2: set_nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ C2 @ ( sup_sup_set_nat @ A @ B ) )
=> ~ ! [A7: set_nat] :
( ( ord_less_eq_set_nat @ A7 @ A )
=> ! [B7: set_nat] :
( ( ord_less_eq_set_nat @ B7 @ B )
=> ( C2
!= ( sup_sup_set_nat @ A7 @ B7 ) ) ) ) ) ).
% subset_UnE
thf(fact_489_Un__absorb2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B @ A )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_490_Un__absorb2,axiom,
! [B: set_nat,A: set_nat] :
( ( ord_less_eq_set_nat @ B @ A )
=> ( ( sup_sup_set_nat @ A @ B )
= A ) ) ).
% Un_absorb2
thf(fact_491_Un__absorb1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_492_Un__absorb1,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( sup_sup_set_nat @ A @ B )
= B ) ) ).
% Un_absorb1
thf(fact_493_Un__upper2,axiom,
! [B: set_Sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ B @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_494_Un__upper2,axiom,
! [B: set_nat,A: set_nat] : ( ord_less_eq_set_nat @ B @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper2
thf(fact_495_Un__upper1,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_496_Un__upper1,axiom,
! [A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ A @ ( sup_sup_set_nat @ A @ B ) ) ).
% Un_upper1
thf(fact_497_Un__least,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ C2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_498_Un__least,axiom,
! [A: set_nat,C2: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ C2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ C2 ) ) ) ).
% Un_least
thf(fact_499_Un__mono,axiom,
! [A: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,D2: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ C2 )
=> ( ( ord_le1325389633284124927_a_nat @ B @ D2 )
=> ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ A @ B ) @ ( sup_su6804446743777130803_a_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_500_Un__mono,axiom,
! [A: set_nat,C2: set_nat,B: set_nat,D2: set_nat] :
( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( ord_less_eq_set_nat @ B @ D2 )
=> ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ A @ B ) @ ( sup_sup_set_nat @ C2 @ D2 ) ) ) ) ).
% Un_mono
thf(fact_501_min__le__iff__disj,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ ( ord_min_nat @ X @ Y ) @ Z )
= ( ( ord_less_eq_nat @ X @ Z )
| ( ord_less_eq_nat @ Y @ Z ) ) ) ).
% min_le_iff_disj
thf(fact_502_min_OcoboundedI2,axiom,
! [B2: nat,C: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).
% min.coboundedI2
thf(fact_503_min_OcoboundedI1,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ C ) ) ).
% min.coboundedI1
thf(fact_504_min_Oabsorb__iff2,axiom,
( ord_less_eq_nat
= ( ^ [B5: nat,A5: nat] :
( ( ord_min_nat @ A5 @ B5 )
= B5 ) ) ) ).
% min.absorb_iff2
thf(fact_505_min_Oabsorb__iff1,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( ( ord_min_nat @ A5 @ B5 )
= A5 ) ) ) ).
% min.absorb_iff1
thf(fact_506_min_Ocobounded2,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ B2 ) ).
% min.cobounded2
thf(fact_507_min_Ocobounded1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ A2 ) ).
% min.cobounded1
thf(fact_508_min_Oorder__iff,axiom,
( ord_less_eq_nat
= ( ^ [A5: nat,B5: nat] :
( A5
= ( ord_min_nat @ A5 @ B5 ) ) ) ) ).
% min.order_iff
thf(fact_509_min_OboundedI,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ A2 @ C )
=> ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) ) ) ) ).
% min.boundedI
thf(fact_510_min_OboundedE,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( ord_min_nat @ B2 @ C ) )
=> ~ ( ( ord_less_eq_nat @ A2 @ B2 )
=> ~ ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% min.boundedE
thf(fact_511_min_OorderI,axiom,
! [A2: nat,B2: nat] :
( ( A2
= ( ord_min_nat @ A2 @ B2 ) )
=> ( ord_less_eq_nat @ A2 @ B2 ) ) ).
% min.orderI
thf(fact_512_min_OorderE,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2
= ( ord_min_nat @ A2 @ B2 ) ) ) ).
% min.orderE
thf(fact_513_min_Omono,axiom,
! [A2: nat,C: nat,B2: nat,D: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ D )
=> ( ord_less_eq_nat @ ( ord_min_nat @ A2 @ B2 ) @ ( ord_min_nat @ C @ D ) ) ) ) ).
% min.mono
thf(fact_514_subset__vimage__iff,axiom,
! [A: set_nat,F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_less_eq_set_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B ) )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_515_subset__vimage__iff,axiom,
! [A: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ ( vimage6545432446589551483_a_nat @ F @ B ) )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ A )
=> ( member_Sum_sum_a_nat @ ( F @ X4 ) @ B ) ) ) ) ).
% subset_vimage_iff
thf(fact_516_vimage__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_less_eq_set_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_517_vimage__mono,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ A @ B )
=> ( ord_le1325389633284124927_a_nat @ ( vimage6545432446589551483_a_nat @ F @ A ) @ ( vimage6545432446589551483_a_nat @ F @ B ) ) ) ).
% vimage_mono
thf(fact_518_distrib__sup__le,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ ( inf_in7084830621192376909_a_nat @ Y @ Z ) ) @ ( inf_in7084830621192376909_a_nat @ ( sup_su6804446743777130803_a_nat @ X @ Y ) @ ( sup_su6804446743777130803_a_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_519_distrib__sup__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ X @ ( inf_inf_set_nat @ Y @ Z ) ) @ ( inf_inf_set_nat @ ( sup_sup_set_nat @ X @ Y ) @ ( sup_sup_set_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_520_distrib__sup__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ X @ ( inf_inf_nat @ Y @ Z ) ) @ ( inf_inf_nat @ ( sup_sup_nat @ X @ Y ) @ ( sup_sup_nat @ X @ Z ) ) ) ).
% distrib_sup_le
thf(fact_521_distrib__inf__le,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat,Z: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ X @ Y ) @ ( inf_in7084830621192376909_a_nat @ X @ Z ) ) @ ( inf_in7084830621192376909_a_nat @ X @ ( sup_su6804446743777130803_a_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_522_distrib__inf__le,axiom,
! [X: set_nat,Y: set_nat,Z: set_nat] : ( ord_less_eq_set_nat @ ( sup_sup_set_nat @ ( inf_inf_set_nat @ X @ Y ) @ ( inf_inf_set_nat @ X @ Z ) ) @ ( inf_inf_set_nat @ X @ ( sup_sup_set_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_523_distrib__inf__le,axiom,
! [X: nat,Y: nat,Z: nat] : ( ord_less_eq_nat @ ( sup_sup_nat @ ( inf_inf_nat @ X @ Y ) @ ( inf_inf_nat @ X @ Z ) ) @ ( inf_inf_nat @ X @ ( sup_sup_nat @ Y @ Z ) ) ) ).
% distrib_inf_le
thf(fact_524_image__Int__subset,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_a] : ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( inf_inf_set_a @ A @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ ( image_7873763678140191238_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_525_image__Int__subset,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ ( image_7142520692256960453_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_526_image__Int__subset,axiom,
! [F: sum_sum_a_nat > nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] : ( ord_less_eq_set_nat @ ( image_2473878607534554506at_nat @ F @ ( inf_in7084830621192376909_a_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_2473878607534554506at_nat @ F @ A ) @ ( image_2473878607534554506at_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_527_image__Int__subset,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_in7084830621192376909_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ ( image_7293268710728258664_a_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_528_image__Int__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) @ ( inf_inf_set_nat @ ( image_nat_nat @ F @ A ) @ ( image_nat_nat @ F @ B ) ) ) ).
% image_Int_subset
thf(fact_529_Un__Int__assoc__eq,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat,C2: set_Sum_sum_a_nat] :
( ( ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ A @ B ) @ C2 )
= ( inf_in7084830621192376909_a_nat @ A @ ( sup_su6804446743777130803_a_nat @ B @ C2 ) ) )
= ( ord_le1325389633284124927_a_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_530_Un__Int__assoc__eq,axiom,
! [A: set_nat,B: set_nat,C2: set_nat] :
( ( ( sup_sup_set_nat @ ( inf_inf_set_nat @ A @ B ) @ C2 )
= ( inf_inf_set_nat @ A @ ( sup_sup_set_nat @ B @ C2 ) ) )
= ( ord_less_eq_set_nat @ C2 @ A ) ) ).
% Un_Int_assoc_eq
thf(fact_531_image__subset__iff__subset__vimage,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) @ B )
= ( ord_less_eq_set_a @ A @ ( vimage4607795094749595324_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_532_image__subset__iff__subset__vimage,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage_nat_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_533_image__subset__iff__subset__vimage,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) @ B )
= ( ord_less_eq_set_nat @ A @ ( vimage3040984495076556338_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_534_image__subset__iff__subset__vimage,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ A ) @ B )
= ( ord_le1325389633284124927_a_nat @ A @ ( vimage6545432446589551483_a_nat @ F @ B ) ) ) ).
% image_subset_iff_subset_vimage
thf(fact_535_image__vimage__subset,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ ( vimage4607795094749595324_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_536_image__vimage__subset,axiom,
! [F: nat > nat,A: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_537_image__vimage__subset,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ ( vimage3040984495076556338_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_538_image__vimage__subset,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] : ( ord_le1325389633284124927_a_nat @ ( image_7142520692256960453_a_nat @ F @ ( vimage6545432446589551483_a_nat @ F @ A ) ) @ A ) ).
% image_vimage_subset
thf(fact_539_Sup_OSUP__cong,axiom,
! [A: set_a,B: set_a,C2: a > sum_sum_a_nat,D2: a > sum_sum_a_nat,Sup: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_7873763678140191238_a_nat @ C2 @ A ) )
= ( Sup @ ( image_7873763678140191238_a_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_540_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > sum_sum_a_nat,D2: nat > sum_sum_a_nat,Sup: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_7293268710728258664_a_nat @ C2 @ A ) )
= ( Sup @ ( image_7293268710728258664_a_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_541_Sup_OSUP__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D2: nat > nat,Sup: set_nat > nat] :
( ( A = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Sup @ ( image_nat_nat @ C2 @ A ) )
= ( Sup @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Sup.SUP_cong
thf(fact_542_Inf_OINF__cong,axiom,
! [A: set_a,B: set_a,C2: a > sum_sum_a_nat,D2: a > sum_sum_a_nat,Inf: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A = B )
=> ( ! [X2: a] :
( ( member_a @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_7873763678140191238_a_nat @ C2 @ A ) )
= ( Inf @ ( image_7873763678140191238_a_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_543_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > sum_sum_a_nat,D2: nat > sum_sum_a_nat,Inf: set_Sum_sum_a_nat > sum_sum_a_nat] :
( ( A = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_7293268710728258664_a_nat @ C2 @ A ) )
= ( Inf @ ( image_7293268710728258664_a_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_544_Inf_OINF__cong,axiom,
! [A: set_nat,B: set_nat,C2: nat > nat,D2: nat > nat,Inf: set_nat > nat] :
( ( A = B )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ B )
=> ( ( C2 @ X2 )
= ( D2 @ X2 ) ) )
=> ( ( Inf @ ( image_nat_nat @ C2 @ A ) )
= ( Inf @ ( image_nat_nat @ D2 @ B ) ) ) ) ) ).
% Inf.INF_cong
thf(fact_545_boolean__algebra__cancel_Osup2,axiom,
! [B: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( B
= ( sup_su6804446743777130803_a_nat @ K @ B2 ) )
=> ( ( sup_su6804446743777130803_a_nat @ A2 @ B )
= ( sup_su6804446743777130803_a_nat @ K @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_546_boolean__algebra__cancel_Osup2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( sup_sup_set_nat @ K @ B2 ) )
=> ( ( sup_sup_set_nat @ A2 @ B )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup2
thf(fact_547_boolean__algebra__cancel_Osup1,axiom,
! [A: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A
= ( sup_su6804446743777130803_a_nat @ K @ A2 ) )
=> ( ( sup_su6804446743777130803_a_nat @ A @ B2 )
= ( sup_su6804446743777130803_a_nat @ K @ ( sup_su6804446743777130803_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_548_boolean__algebra__cancel_Osup1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( sup_sup_set_nat @ K @ A2 ) )
=> ( ( sup_sup_set_nat @ A @ B2 )
= ( sup_sup_set_nat @ K @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.sup1
thf(fact_549_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat] :
( ( B
= ( inf_in7084830621192376909_a_nat @ K @ B2 ) )
=> ( ( inf_in7084830621192376909_a_nat @ A2 @ B )
= ( inf_in7084830621192376909_a_nat @ K @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_550_boolean__algebra__cancel_Oinf2,axiom,
! [B: nat,K: nat,B2: nat,A2: nat] :
( ( B
= ( inf_inf_nat @ K @ B2 ) )
=> ( ( inf_inf_nat @ A2 @ B )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_551_boolean__algebra__cancel_Oinf2,axiom,
! [B: set_nat,K: set_nat,B2: set_nat,A2: set_nat] :
( ( B
= ( inf_inf_set_nat @ K @ B2 ) )
=> ( ( inf_inf_set_nat @ A2 @ B )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf2
thf(fact_552_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_Sum_sum_a_nat,K: set_Sum_sum_a_nat,A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( A
= ( inf_in7084830621192376909_a_nat @ K @ A2 ) )
=> ( ( inf_in7084830621192376909_a_nat @ A @ B2 )
= ( inf_in7084830621192376909_a_nat @ K @ ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_553_boolean__algebra__cancel_Oinf1,axiom,
! [A: nat,K: nat,A2: nat,B2: nat] :
( ( A
= ( inf_inf_nat @ K @ A2 ) )
=> ( ( inf_inf_nat @ A @ B2 )
= ( inf_inf_nat @ K @ ( inf_inf_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_554_boolean__algebra__cancel_Oinf1,axiom,
! [A: set_nat,K: set_nat,A2: set_nat,B2: set_nat] :
( ( A
= ( inf_inf_set_nat @ K @ A2 ) )
=> ( ( inf_inf_set_nat @ A @ B2 )
= ( inf_inf_set_nat @ K @ ( inf_inf_set_nat @ A2 @ B2 ) ) ) ) ).
% boolean_algebra_cancel.inf1
thf(fact_555_not__arg__cong__Inr,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ( sum_Inr_nat_a @ X )
!= ( sum_Inr_nat_a @ Y ) ) ) ).
% not_arg_cong_Inr
thf(fact_556_card__length,axiom,
! [Xs: list_nat] : ( ord_less_eq_nat @ ( finite_card_nat @ ( set_nat2 @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ).
% card_length
thf(fact_557_card__length,axiom,
! [Xs: list_literal] : ( ord_less_eq_nat @ ( finite_card_literal @ ( set_literal2 @ Xs ) ) @ ( size_s2501651207091587910iteral @ Xs ) ) ).
% card_length
thf(fact_558_card__length,axiom,
! [Xs: list_Product_unit] : ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( set_Product_unit2 @ Xs ) ) @ ( size_s245203480648594047t_unit @ Xs ) ) ).
% card_length
thf(fact_559_card__length,axiom,
! [Xs: list_Sum_sum_a_nat] : ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% card_length
thf(fact_560_order__refl,axiom,
! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).
% order_refl
thf(fact_561_dual__order_Orefl,axiom,
! [A2: nat] : ( ord_less_eq_nat @ A2 @ A2 ) ).
% dual_order.refl
thf(fact_562_min__absorb2,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_min_nat @ X @ Y )
= Y ) ) ).
% min_absorb2
thf(fact_563_min__absorb1,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_min_nat @ X @ Y )
= X ) ) ).
% min_absorb1
thf(fact_564_min__def,axiom,
( ord_min_nat
= ( ^ [A5: nat,B5: nat] : ( if_nat @ ( ord_less_eq_nat @ A5 @ B5 ) @ A5 @ B5 ) ) ) ).
% min_def
thf(fact_565_ad__agr__list__rev__mono,axiom,
! [Y5: set_a,X6: set_a,Ys: list_Sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( ord_less_eq_set_a @ Y5 @ X6 )
=> ( ( ad_agr_list_a_nat @ Y5 @ Ys @ Xs )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) @ Y5 )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ ( set_Sum_sum_a_nat2 @ Ys ) ) @ Y5 )
=> ( ad_agr_list_a_nat @ X6 @ Ys @ Xs ) ) ) ) ) ).
% ad_agr_list_rev_mono
thf(fact_566_subset__code_I1_J,axiom,
! [Xs: list_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B )
= ( ! [X4: nat] :
( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
=> ( member_nat @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_567_subset__code_I1_J,axiom,
! [Xs: list_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ B )
= ( ! [X4: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X4 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( member_Sum_sum_a_nat @ X4 @ B ) ) ) ) ).
% subset_code(1)
thf(fact_568_all__subset__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ord_less_eq_set_a @ B3 @ A )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_569_all__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_570_all__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ord_less_eq_set_nat @ B3 @ A )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_subset_image
thf(fact_571_ad__agr__list__length,axiom,
! [X6: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ad_agr_list_a_nat @ X6 @ Xs @ Ys )
=> ( ( size_s5283204784079214577_a_nat @ Xs )
= ( size_s5283204784079214577_a_nat @ Ys ) ) ) ).
% ad_agr_list_length
thf(fact_572_order__antisym__conv,axiom,
! [Y: nat,X: nat] :
( ( ord_less_eq_nat @ Y @ X )
=> ( ( ord_less_eq_nat @ X @ Y )
= ( X = Y ) ) ) ).
% order_antisym_conv
thf(fact_573_linorder__le__cases,axiom,
! [X: nat,Y: nat] :
( ~ ( ord_less_eq_nat @ X @ Y )
=> ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_le_cases
thf(fact_574_ord__le__eq__subst,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ( F @ B2 )
= C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% ord_le_eq_subst
thf(fact_575_ord__eq__le__subst,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( A2
= ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_576_linorder__linear,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
| ( ord_less_eq_nat @ Y @ X ) ) ).
% linorder_linear
thf(fact_577_order__eq__refl,axiom,
! [X: nat,Y: nat] :
( ( X = Y )
=> ( ord_less_eq_nat @ X @ Y ) ) ).
% order_eq_refl
thf(fact_578_order__subst2,axiom,
! [A2: nat,B2: nat,F: nat > nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ ( F @ B2 ) @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ ( F @ A2 ) @ C ) ) ) ) ).
% order_subst2
thf(fact_579_order__subst1,axiom,
! [A2: nat,F: nat > nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ ( F @ B2 ) )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ! [X2: nat,Y4: nat] :
( ( ord_less_eq_nat @ X2 @ Y4 )
=> ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y4 ) ) )
=> ( ord_less_eq_nat @ A2 @ ( F @ C ) ) ) ) ) ).
% order_subst1
thf(fact_580_Orderings_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ A5 @ B5 )
& ( ord_less_eq_nat @ B5 @ A5 ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_581_antisym,axiom,
! [A2: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ).
% antisym
thf(fact_582_dual__order_Otrans,axiom,
! [B2: nat,A2: nat,C: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ C @ B2 )
=> ( ord_less_eq_nat @ C @ A2 ) ) ) ).
% dual_order.trans
thf(fact_583_dual__order_Oantisym,axiom,
! [B2: nat,A2: nat] :
( ( ord_less_eq_nat @ B2 @ A2 )
=> ( ( ord_less_eq_nat @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ).
% dual_order.antisym
thf(fact_584_dual__order_Oeq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [A5: nat,B5: nat] :
( ( ord_less_eq_nat @ B5 @ A5 )
& ( ord_less_eq_nat @ A5 @ B5 ) ) ) ) ).
% dual_order.eq_iff
thf(fact_585_linorder__wlog,axiom,
! [P: nat > nat > $o,A2: nat,B2: nat] :
( ! [A6: nat,B6: nat] :
( ( ord_less_eq_nat @ A6 @ B6 )
=> ( P @ A6 @ B6 ) )
=> ( ! [A6: nat,B6: nat] :
( ( P @ B6 @ A6 )
=> ( P @ A6 @ B6 ) )
=> ( P @ A2 @ B2 ) ) ) ).
% linorder_wlog
thf(fact_586_order__trans,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ Z )
=> ( ord_less_eq_nat @ X @ Z ) ) ) ).
% order_trans
thf(fact_587_order_Otrans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% order.trans
thf(fact_588_order__antisym,axiom,
! [X: nat,Y: nat] :
( ( ord_less_eq_nat @ X @ Y )
=> ( ( ord_less_eq_nat @ Y @ X )
=> ( X = Y ) ) ) ).
% order_antisym
thf(fact_589_ord__le__eq__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( ord_less_eq_nat @ A2 @ B2 )
=> ( ( B2 = C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_le_eq_trans
thf(fact_590_ord__eq__le__trans,axiom,
! [A2: nat,B2: nat,C: nat] :
( ( A2 = B2 )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ord_less_eq_nat @ A2 @ C ) ) ) ).
% ord_eq_le_trans
thf(fact_591_order__class_Oorder__eq__iff,axiom,
( ( ^ [Y6: nat,Z3: nat] : ( Y6 = Z3 ) )
= ( ^ [X4: nat,Y3: nat] :
( ( ord_less_eq_nat @ X4 @ Y3 )
& ( ord_less_eq_nat @ Y3 @ X4 ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_592_le__cases3,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( ( ord_less_eq_nat @ X @ Y )
=> ~ ( ord_less_eq_nat @ Y @ Z ) )
=> ( ( ( ord_less_eq_nat @ Y @ X )
=> ~ ( ord_less_eq_nat @ X @ Z ) )
=> ( ( ( ord_less_eq_nat @ X @ Z )
=> ~ ( ord_less_eq_nat @ Z @ Y ) )
=> ( ( ( ord_less_eq_nat @ Z @ Y )
=> ~ ( ord_less_eq_nat @ Y @ X ) )
=> ( ( ( ord_less_eq_nat @ Y @ Z )
=> ~ ( ord_less_eq_nat @ Z @ X ) )
=> ~ ( ( ord_less_eq_nat @ Z @ X )
=> ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_593_nle__le,axiom,
! [A2: nat,B2: nat] :
( ( ~ ( ord_less_eq_nat @ A2 @ B2 ) )
= ( ( ord_less_eq_nat @ B2 @ A2 )
& ( B2 != A2 ) ) ) ).
% nle_le
thf(fact_594_Ex__list__of__length,axiom,
! [N2: nat] :
? [Xs2: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ Xs2 )
= N2 ) ).
% Ex_list_of_length
thf(fact_595_neq__if__length__neq,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ Xs )
!= ( size_s5283204784079214577_a_nat @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_596_ad__agr__list__set,axiom,
! [X6: set_a,Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat,Y: a] :
( ( ad_agr_list_a_nat @ X6 @ Xs @ Ys )
=> ( ( member_a @ Y @ X6 )
=> ( ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y ) @ ( set_Sum_sum_a_nat2 @ Ys ) )
=> ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ Y ) @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ) ) ).
% ad_agr_list_set
thf(fact_597_set__union,axiom,
! [Xs: list_Sum_sum_a_nat,Ys: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( union_Sum_sum_a_nat @ Xs @ Ys ) )
= ( sup_su6804446743777130803_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) @ ( set_Sum_sum_a_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_598_set__union,axiom,
! [Xs: list_nat,Ys: list_nat] :
( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
= ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).
% set_union
thf(fact_599_Plus__def,axiom,
( sum_Plus_a_nat
= ( ^ [A3: set_a,B3: set_nat] : ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ A3 ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ B3 ) ) ) ) ).
% Plus_def
thf(fact_600_length__rremdups,axiom,
! [Xs: list_nat] :
( ( size_size_list_nat @ ( rremdups_nat @ Xs ) )
= ( finite_card_nat @ ( set_nat2 @ Xs ) ) ) ).
% length_rremdups
thf(fact_601_length__rremdups,axiom,
! [Xs: list_literal] :
( ( size_s2501651207091587910iteral @ ( rremdups_literal @ Xs ) )
= ( finite_card_literal @ ( set_literal2 @ Xs ) ) ) ).
% length_rremdups
thf(fact_602_length__rremdups,axiom,
! [Xs: list_Product_unit] :
( ( size_s245203480648594047t_unit @ ( rremdu6278002517723239495t_unit @ Xs ) )
= ( finite410649719033368117t_unit @ ( set_Product_unit2 @ Xs ) ) ) ).
% length_rremdups
thf(fact_603_length__rremdups,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( rremdu8304153113908149561_a_nat @ Xs ) )
= ( finite6080979521523705895_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) ) ).
% length_rremdups
thf(fact_604_image__vimage__eq,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( image_7873763678140191238_a_nat @ F @ ( vimage4607795094749595324_a_nat @ F @ A ) )
= ( inf_in7084830621192376909_a_nat @ A @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) ) ) ).
% image_vimage_eq
thf(fact_605_image__vimage__eq,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( vimage6545432446589551483_a_nat @ F @ A ) )
= ( inf_in7084830621192376909_a_nat @ A @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) ) ) ).
% image_vimage_eq
thf(fact_606_image__vimage__eq,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( vimage3040984495076556338_a_nat @ F @ A ) )
= ( inf_in7084830621192376909_a_nat @ A @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_607_image__vimage__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% image_vimage_eq
thf(fact_608_image__vimage__eq,axiom,
! [F: literal > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( image_605486080324953776_a_nat @ F @ ( vimage1582777469784555110_a_nat @ F @ A ) )
= ( inf_in7084830621192376909_a_nat @ A @ ( image_605486080324953776_a_nat @ F @ top_top_set_literal ) ) ) ).
% image_vimage_eq
thf(fact_609_image__vimage__eq,axiom,
! [F: literal > nat,A: set_nat] :
( ( image_literal_nat @ F @ ( vimage_literal_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_literal_nat @ F @ top_top_set_literal ) ) ) ).
% image_vimage_eq
thf(fact_610_image__vimage__eq,axiom,
! [F: product_unit > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( image_8388312959667285303_a_nat @ F @ ( vimage5806748310446461933_a_nat @ F @ A ) )
= ( inf_in7084830621192376909_a_nat @ A @ ( image_8388312959667285303_a_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% image_vimage_eq
thf(fact_611_image__vimage__eq,axiom,
! [F: product_unit > nat,A: set_nat] :
( ( image_875570014554754200it_nat @ F @ ( vimage6253328473476588386it_nat @ F @ A ) )
= ( inf_inf_set_nat @ A @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% image_vimage_eq
thf(fact_612_surj__card__le,axiom,
! [A: set_Product_unit,B: set_nat,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_875570014554754200it_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_613_surj__card__le,axiom,
! [A: set_Product_unit,B: set_literal,F: product_unit > literal] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_5876984745897992460iteral @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_614_surj__card__le,axiom,
! [A: set_Product_unit,B: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( ord_le3507040750410214029t_unit @ B @ ( image_405062704495631173t_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% surj_card_le
thf(fact_615_surj__card__le,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_616_surj__card__le,axiom,
! [A: set_nat,B: set_literal,F: nat > literal] :
( ( finite_finite_nat @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_nat_literal @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_617_surj__card__le,axiom,
! [A: set_nat,B: set_Product_unit,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ( ord_le3507040750410214029t_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite_card_nat @ A ) ) ) ) ).
% surj_card_le
thf(fact_618_surj__card__le,axiom,
! [A: set_literal,B: set_nat,F: literal > nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_literal_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_literal @ A ) ) ) ) ).
% surj_card_le
thf(fact_619_surj__card__le,axiom,
! [A: set_literal,B: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_8195128725298311301iteral @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite_card_literal @ A ) ) ) ) ).
% surj_card_le
thf(fact_620_surj__card__le,axiom,
! [A: set_literal,B: set_Product_unit,F: literal > product_unit] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_le3507040750410214029t_unit @ B @ ( image_6787854153545327934t_unit @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite_card_literal @ A ) ) ) ) ).
% surj_card_le
thf(fact_621_surj__card__le,axiom,
! [A: set_a,B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( finite_finite_a @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ B ) @ ( finite_card_a @ A ) ) ) ) ).
% surj_card_le
thf(fact_622_inf__nat__def,axiom,
inf_inf_nat = ord_min_nat ).
% inf_nat_def
thf(fact_623_ad__agr__list__eq,axiom,
! [Ys: list_a,AD: set_a,Xs: list_a] :
( ( ord_less_eq_set_a @ ( set_a2 @ Ys ) @ AD )
=> ( ( ad_agr_list_a_nat @ AD @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Xs ) @ ( map_a_Sum_sum_a_nat @ sum_Inl_a_nat @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% ad_agr_list_eq
thf(fact_624_UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% UNIV_I
thf(fact_625_UNIV__I,axiom,
! [X: literal] : ( member_literal @ X @ top_top_set_literal ) ).
% UNIV_I
thf(fact_626_UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% UNIV_I
thf(fact_627_finite__lessThan,axiom,
! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).
% finite_lessThan
thf(fact_628_finite__Plus__UNIV__iff,axiom,
( ( finite6187706683773761046at_nat @ top_to6661820994512907621at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_629_finite__Plus__UNIV__iff,axiom,
( ( finite7336130560110450212iteral @ top_to148093990134820907iteral )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_630_finite__Plus__UNIV__iff,axiom,
( ( finite4327512606132785245t_unit @ top_to5465250082899874788t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_631_finite__Plus__UNIV__iff,axiom,
( ( finite2800739532781614718al_nat @ top_to7291364169502081925al_nat )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_632_finite__Plus__UNIV__iff,axiom,
( ( finite8615155063774727100iteral @ top_to8459027286199865867iteral )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_633_finite__Plus__UNIV__iff,axiom,
( ( finite7946456142781997557t_unit @ top_to6093211048104912324t_unit )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_634_finite__Plus__UNIV__iff,axiom,
( ( finite4401952911629260215it_nat @ top_to2894617605782473790it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_635_finite__Plus__UNIV__iff,axiom,
( ( finite2327689956804853827iteral @ top_to8707194323715505426iteral )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_636_finite__Plus__UNIV__iff,axiom,
( ( finite3146551501593861116t_unit @ top_to2771918933716375115t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_637_finite__imageI,axiom,
! [F2: set_a,H: a > sum_sum_a_nat] :
( ( finite_finite_a @ F2 )
=> ( finite502105017643426984_a_nat @ ( image_7873763678140191238_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_638_finite__imageI,axiom,
! [F2: set_nat,H: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ F2 )
=> ( finite502105017643426984_a_nat @ ( image_7293268710728258664_a_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_639_finite__imageI,axiom,
! [F2: set_nat,H: nat > nat] :
( ( finite_finite_nat @ F2 )
=> ( finite_finite_nat @ ( image_nat_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_640_finite__imageI,axiom,
! [F2: set_nat,H: nat > literal] :
( ( finite_finite_nat @ F2 )
=> ( finite5847741373460823677iteral @ ( image_nat_literal @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_641_finite__imageI,axiom,
! [F2: set_literal,H: literal > nat] :
( ( finite5847741373460823677iteral @ F2 )
=> ( finite_finite_nat @ ( image_literal_nat @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_642_finite__imageI,axiom,
! [F2: set_literal,H: literal > literal] :
( ( finite5847741373460823677iteral @ F2 )
=> ( finite5847741373460823677iteral @ ( image_8195128725298311301iteral @ H @ F2 ) ) ) ).
% finite_imageI
thf(fact_643_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_644_boolean__algebra_Odisj__one__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_right
thf(fact_645_boolean__algebra_Odisj__one__right,axiom,
! [X: set_literal] :
( ( sup_sup_set_literal @ X @ top_top_set_literal )
= top_top_set_literal ) ).
% boolean_algebra.disj_one_right
thf(fact_646_boolean__algebra_Odisj__one__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_right
thf(fact_647_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ X )
= top_to795618464972521135_a_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_648_boolean__algebra_Odisj__one__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% boolean_algebra.disj_one_left
thf(fact_649_boolean__algebra_Odisj__one__left,axiom,
! [X: set_literal] :
( ( sup_sup_set_literal @ top_top_set_literal @ X )
= top_top_set_literal ) ).
% boolean_algebra.disj_one_left
thf(fact_650_boolean__algebra_Odisj__one__left,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
= top_to1996260823553986621t_unit ) ).
% boolean_algebra.disj_one_left
thf(fact_651_sup__top__right,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ X @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% sup_top_right
thf(fact_652_sup__top__right,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ X @ top_top_set_nat )
= top_top_set_nat ) ).
% sup_top_right
thf(fact_653_sup__top__right,axiom,
! [X: set_literal] :
( ( sup_sup_set_literal @ X @ top_top_set_literal )
= top_top_set_literal ) ).
% sup_top_right
thf(fact_654_sup__top__right,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ X @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% sup_top_right
thf(fact_655_sup__top__left,axiom,
! [X: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ X )
= top_to795618464972521135_a_nat ) ).
% sup_top_left
thf(fact_656_sup__top__left,axiom,
! [X: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ X )
= top_top_set_nat ) ).
% sup_top_left
thf(fact_657_sup__top__left,axiom,
! [X: set_literal] :
( ( sup_sup_set_literal @ top_top_set_literal @ X )
= top_top_set_literal ) ).
% sup_top_left
thf(fact_658_sup__top__left,axiom,
! [X: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ X )
= top_to1996260823553986621t_unit ) ).
% sup_top_left
thf(fact_659_inf__top_Oright__neutral,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A2 @ top_to795618464972521135_a_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_660_inf__top_Oright__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ A2 @ top_top_set_nat )
= A2 ) ).
% inf_top.right_neutral
thf(fact_661_inf__top_Oright__neutral,axiom,
! [A2: set_literal] :
( ( inf_inf_set_literal @ A2 @ top_top_set_literal )
= A2 ) ).
% inf_top.right_neutral
thf(fact_662_inf__top_Oright__neutral,axiom,
! [A2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A2 @ top_to1996260823553986621t_unit )
= A2 ) ).
% inf_top.right_neutral
thf(fact_663_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ A2 @ B2 ) )
= ( ( A2 = top_to795618464972521135_a_nat )
& ( B2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_664_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ A2 @ B2 ) )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_665_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_literal,B2: set_literal] :
( ( top_top_set_literal
= ( inf_inf_set_literal @ A2 @ B2 ) )
= ( ( A2 = top_top_set_literal )
& ( B2 = top_top_set_literal ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_666_inf__top_Oneutr__eq__iff,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ A2 @ B2 ) )
= ( ( A2 = top_to1996260823553986621t_unit )
& ( B2 = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_667_inf__top_Oleft__neutral,axiom,
! [A2: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_668_inf__top_Oleft__neutral,axiom,
! [A2: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_669_inf__top_Oleft__neutral,axiom,
! [A2: set_literal] :
( ( inf_inf_set_literal @ top_top_set_literal @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_670_inf__top_Oleft__neutral,axiom,
! [A2: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ A2 )
= A2 ) ).
% inf_top.left_neutral
thf(fact_671_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Sum_sum_a_nat,B2: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A2 @ B2 )
= top_to795618464972521135_a_nat )
= ( ( A2 = top_to795618464972521135_a_nat )
& ( B2 = top_to795618464972521135_a_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_672_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_nat,B2: set_nat] :
( ( ( inf_inf_set_nat @ A2 @ B2 )
= top_top_set_nat )
= ( ( A2 = top_top_set_nat )
& ( B2 = top_top_set_nat ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_673_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_literal,B2: set_literal] :
( ( ( inf_inf_set_literal @ A2 @ B2 )
= top_top_set_literal )
= ( ( A2 = top_top_set_literal )
& ( B2 = top_top_set_literal ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_674_inf__top_Oeq__neutr__iff,axiom,
! [A2: set_Product_unit,B2: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A2 @ B2 )
= top_to1996260823553986621t_unit )
= ( ( A2 = top_to1996260823553986621t_unit )
& ( B2 = top_to1996260823553986621t_unit ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_675_top__eq__inf__iff,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( top_to795618464972521135_a_nat
= ( inf_in7084830621192376909_a_nat @ X @ Y ) )
= ( ( X = top_to795618464972521135_a_nat )
& ( Y = top_to795618464972521135_a_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_676_top__eq__inf__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( top_top_set_nat
= ( inf_inf_set_nat @ X @ Y ) )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% top_eq_inf_iff
thf(fact_677_top__eq__inf__iff,axiom,
! [X: set_literal,Y: set_literal] :
( ( top_top_set_literal
= ( inf_inf_set_literal @ X @ Y ) )
= ( ( X = top_top_set_literal )
& ( Y = top_top_set_literal ) ) ) ).
% top_eq_inf_iff
thf(fact_678_top__eq__inf__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( top_to1996260823553986621t_unit
= ( inf_in4660618365625256667t_unit @ X @ Y ) )
= ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% top_eq_inf_iff
thf(fact_679_inf__eq__top__iff,axiom,
! [X: set_Sum_sum_a_nat,Y: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ X @ Y )
= top_to795618464972521135_a_nat )
= ( ( X = top_to795618464972521135_a_nat )
& ( Y = top_to795618464972521135_a_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_680_inf__eq__top__iff,axiom,
! [X: set_nat,Y: set_nat] :
( ( ( inf_inf_set_nat @ X @ Y )
= top_top_set_nat )
= ( ( X = top_top_set_nat )
& ( Y = top_top_set_nat ) ) ) ).
% inf_eq_top_iff
thf(fact_681_inf__eq__top__iff,axiom,
! [X: set_literal,Y: set_literal] :
( ( ( inf_inf_set_literal @ X @ Y )
= top_top_set_literal )
= ( ( X = top_top_set_literal )
& ( Y = top_top_set_literal ) ) ) ).
% inf_eq_top_iff
thf(fact_682_inf__eq__top__iff,axiom,
! [X: set_Product_unit,Y: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ X @ Y )
= top_to1996260823553986621t_unit )
= ( ( X = top_to1996260823553986621t_unit )
& ( Y = top_to1996260823553986621t_unit ) ) ) ).
% inf_eq_top_iff
thf(fact_683_inf__top__right,axiom,
! [X: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ top_to795618464972521135_a_nat )
= X ) ).
% inf_top_right
thf(fact_684_inf__top__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% inf_top_right
thf(fact_685_inf__top__right,axiom,
! [X: set_literal] :
( ( inf_inf_set_literal @ X @ top_top_set_literal )
= X ) ).
% inf_top_right
thf(fact_686_inf__top__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
= X ) ).
% inf_top_right
thf(fact_687_inf__top__left,axiom,
! [X: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ X )
= X ) ).
% inf_top_left
thf(fact_688_inf__top__left,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ X )
= X ) ).
% inf_top_left
thf(fact_689_inf__top__left,axiom,
! [X: set_literal] :
( ( inf_inf_set_literal @ top_top_set_literal @ X )
= X ) ).
% inf_top_left
thf(fact_690_inf__top__left,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ X )
= X ) ).
% inf_top_left
thf(fact_691_List_Ofinite__set,axiom,
! [Xs: list_Sum_sum_a_nat] : ( finite502105017643426984_a_nat @ ( set_Sum_sum_a_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_692_List_Ofinite__set,axiom,
! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).
% List.finite_set
thf(fact_693_List_Ofinite__set,axiom,
! [Xs: list_literal] : ( finite5847741373460823677iteral @ ( set_literal2 @ Xs ) ) ).
% List.finite_set
thf(fact_694_Int__UNIV,axiom,
! [A: set_Sum_sum_a_nat,B: set_Sum_sum_a_nat] :
( ( ( inf_in7084830621192376909_a_nat @ A @ B )
= top_to795618464972521135_a_nat )
= ( ( A = top_to795618464972521135_a_nat )
& ( B = top_to795618464972521135_a_nat ) ) ) ).
% Int_UNIV
thf(fact_695_Int__UNIV,axiom,
! [A: set_nat,B: set_nat] :
( ( ( inf_inf_set_nat @ A @ B )
= top_top_set_nat )
= ( ( A = top_top_set_nat )
& ( B = top_top_set_nat ) ) ) ).
% Int_UNIV
thf(fact_696_Int__UNIV,axiom,
! [A: set_literal,B: set_literal] :
( ( ( inf_inf_set_literal @ A @ B )
= top_top_set_literal )
= ( ( A = top_top_set_literal )
& ( B = top_top_set_literal ) ) ) ).
% Int_UNIV
thf(fact_697_Int__UNIV,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( ( inf_in4660618365625256667t_unit @ A @ B )
= top_to1996260823553986621t_unit )
= ( ( A = top_to1996260823553986621t_unit )
& ( B = top_to1996260823553986621t_unit ) ) ) ).
% Int_UNIV
thf(fact_698_finite__Int,axiom,
! [F2: set_literal,G2: set_literal] :
( ( ( finite5847741373460823677iteral @ F2 )
| ( finite5847741373460823677iteral @ G2 ) )
=> ( finite5847741373460823677iteral @ ( inf_inf_set_literal @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_699_finite__Int,axiom,
! [F2: set_Sum_sum_a_nat,G2: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ F2 )
| ( finite502105017643426984_a_nat @ G2 ) )
=> ( finite502105017643426984_a_nat @ ( inf_in7084830621192376909_a_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_700_finite__Int,axiom,
! [F2: set_nat,G2: set_nat] :
( ( ( finite_finite_nat @ F2 )
| ( finite_finite_nat @ G2 ) )
=> ( finite_finite_nat @ ( inf_inf_set_nat @ F2 @ G2 ) ) ) ).
% finite_Int
thf(fact_701_finite__Un,axiom,
! [F2: set_literal,G2: set_literal] :
( ( finite5847741373460823677iteral @ ( sup_sup_set_literal @ F2 @ G2 ) )
= ( ( finite5847741373460823677iteral @ F2 )
& ( finite5847741373460823677iteral @ G2 ) ) ) ).
% finite_Un
thf(fact_702_finite__Un,axiom,
! [F2: set_Sum_sum_a_nat,G2: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ F2 @ G2 ) )
= ( ( finite502105017643426984_a_nat @ F2 )
& ( finite502105017643426984_a_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_703_finite__Un,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) )
= ( ( finite_finite_nat @ F2 )
& ( finite_finite_nat @ G2 ) ) ) ).
% finite_Un
thf(fact_704_length__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( size_s5283204784079214577_a_nat @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) )
= ( size_s5283204784079214577_a_nat @ Xs ) ) ).
% length_map
thf(fact_705_min__top2,axiom,
! [X: set_nat] :
( ( ord_min_set_nat @ X @ top_top_set_nat )
= X ) ).
% min_top2
thf(fact_706_min__top2,axiom,
! [X: set_literal] :
( ( ord_min_set_literal @ X @ top_top_set_literal )
= X ) ).
% min_top2
thf(fact_707_min__top2,axiom,
! [X: set_Product_unit] :
( ( ord_mi7693666415311043668t_unit @ X @ top_to1996260823553986621t_unit )
= X ) ).
% min_top2
thf(fact_708_min__top,axiom,
! [X: set_nat] :
( ( ord_min_set_nat @ top_top_set_nat @ X )
= X ) ).
% min_top
thf(fact_709_min__top,axiom,
! [X: set_literal] :
( ( ord_min_set_literal @ top_top_set_literal @ X )
= X ) ).
% min_top
thf(fact_710_min__top,axiom,
! [X: set_Product_unit] :
( ( ord_mi7693666415311043668t_unit @ top_to1996260823553986621t_unit @ X )
= X ) ).
% min_top
thf(fact_711_vimage__UNIV,axiom,
! [F: nat > nat] :
( ( vimage_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_712_vimage__UNIV,axiom,
! [F: literal > nat] :
( ( vimage_literal_nat @ F @ top_top_set_nat )
= top_top_set_literal ) ).
% vimage_UNIV
thf(fact_713_vimage__UNIV,axiom,
! [F: product_unit > nat] :
( ( vimage6253328473476588386it_nat @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_714_vimage__UNIV,axiom,
! [F: nat > literal] :
( ( vimage_nat_literal @ F @ top_top_set_literal )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_715_vimage__UNIV,axiom,
! [F: literal > literal] :
( ( vimage8238609917233974331iteral @ F @ top_top_set_literal )
= top_top_set_literal ) ).
% vimage_UNIV
thf(fact_716_vimage__UNIV,axiom,
! [F: product_unit > literal] :
( ( vimage5137003825767189442iteral @ F @ top_top_set_literal )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_717_vimage__UNIV,axiom,
! [F: nat > product_unit] :
( ( vimage4884490618288580032t_unit @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_718_vimage__UNIV,axiom,
! [F: literal > product_unit] :
( ( vimage6047873233414524916t_unit @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal ) ).
% vimage_UNIV
thf(fact_719_vimage__UNIV,axiom,
! [F: product_unit > product_unit] :
( ( vimage7995052115951654139t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% vimage_UNIV
thf(fact_720_vimage__UNIV,axiom,
! [F: nat > sum_sum_a_nat] :
( ( vimage3040984495076556338_a_nat @ F @ top_to795618464972521135_a_nat )
= top_top_set_nat ) ).
% vimage_UNIV
thf(fact_721_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_nat_nat @ top_top_set_nat @ top_top_set_nat )
= top_to6661820994512907621at_nat ) ).
% UNIV_Plus_UNIV
thf(fact_722_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_nat_literal @ top_top_set_nat @ top_top_set_literal )
= top_to148093990134820907iteral ) ).
% UNIV_Plus_UNIV
thf(fact_723_UNIV__Plus__UNIV,axiom,
( ( sum_Pl8139761314266719112t_unit @ top_top_set_nat @ top_to1996260823553986621t_unit )
= top_to5465250082899874788t_unit ) ).
% UNIV_Plus_UNIV
thf(fact_724_UNIV__Plus__UNIV,axiom,
( ( sum_Plus_literal_nat @ top_top_set_literal @ top_top_set_nat )
= top_to7291364169502081925al_nat ) ).
% UNIV_Plus_UNIV
thf(fact_725_UNIV__Plus__UNIV,axiom,
( ( sum_Pl2008226088555200627iteral @ top_top_set_literal @ top_top_set_literal )
= top_to8459027286199865867iteral ) ).
% UNIV_Plus_UNIV
thf(fact_726_UNIV__Plus__UNIV,axiom,
( ( sum_Pl1513934344387670060t_unit @ top_top_set_literal @ top_to1996260823553986621t_unit )
= top_to6093211048104912324t_unit ) ).
% UNIV_Plus_UNIV
thf(fact_727_UNIV__Plus__UNIV,axiom,
( ( sum_Pl285227132599951658it_nat @ top_to1996260823553986621t_unit @ top_top_set_nat )
= top_to2894617605782473790it_nat ) ).
% UNIV_Plus_UNIV
thf(fact_728_UNIV__Plus__UNIV,axiom,
( ( sum_Pl603064936740334586iteral @ top_to1996260823553986621t_unit @ top_top_set_literal )
= top_to8707194323715505426iteral ) ).
% UNIV_Plus_UNIV
thf(fact_729_UNIV__Plus__UNIV,axiom,
( ( sum_Pl144888763950457139t_unit @ top_to1996260823553986621t_unit @ top_to1996260823553986621t_unit )
= top_to2771918933716375115t_unit ) ).
% UNIV_Plus_UNIV
thf(fact_730_finite__Plus__iff,axiom,
! [A: set_nat,B: set_nat] :
( ( finite6187706683773761046at_nat @ ( sum_Plus_nat_nat @ A @ B ) )
= ( ( finite_finite_nat @ A )
& ( finite_finite_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_731_finite__Plus__iff,axiom,
! [A: set_nat,B: set_literal] :
( ( finite7336130560110450212iteral @ ( sum_Plus_nat_literal @ A @ B ) )
= ( ( finite_finite_nat @ A )
& ( finite5847741373460823677iteral @ B ) ) ) ).
% finite_Plus_iff
thf(fact_732_finite__Plus__iff,axiom,
! [A: set_literal,B: set_nat] :
( ( finite2800739532781614718al_nat @ ( sum_Plus_literal_nat @ A @ B ) )
= ( ( finite5847741373460823677iteral @ A )
& ( finite_finite_nat @ B ) ) ) ).
% finite_Plus_iff
thf(fact_733_finite__Plus__iff,axiom,
! [A: set_literal,B: set_literal] :
( ( finite8615155063774727100iteral @ ( sum_Pl2008226088555200627iteral @ A @ B ) )
= ( ( finite5847741373460823677iteral @ A )
& ( finite5847741373460823677iteral @ B ) ) ) ).
% finite_Plus_iff
thf(fact_734_InlI,axiom,
! [A2: a,A: set_a,B: set_nat] :
( ( member_a @ A2 @ A )
=> ( member_Sum_sum_a_nat @ ( sum_Inl_a_nat @ A2 ) @ ( sum_Plus_a_nat @ A @ B ) ) ) ).
% InlI
thf(fact_735_InrI,axiom,
! [B2: nat,B: set_nat,A: set_a] :
( ( member_nat @ B2 @ B )
=> ( member_Sum_sum_a_nat @ ( sum_Inr_nat_a @ B2 ) @ ( sum_Plus_a_nat @ A @ B ) ) ) ).
% InrI
thf(fact_736_list_Oset__map,axiom,
! [F: nat > nat,V: list_nat] :
( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
= ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_737_list_Oset__map,axiom,
! [F: a > sum_sum_a_nat,V: list_a] :
( ( set_Sum_sum_a_nat2 @ ( map_a_Sum_sum_a_nat @ F @ V ) )
= ( image_7873763678140191238_a_nat @ F @ ( set_a2 @ V ) ) ) ).
% list.set_map
thf(fact_738_list_Oset__map,axiom,
! [F: nat > sum_sum_a_nat,V: list_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ V ) )
= ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_739_list_Oset__map,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,V: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ V ) )
= ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ V ) ) ) ).
% list.set_map
thf(fact_740_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ top_top_set_nat ) )
=> ( A = top_top_set_nat ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_741_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( ( finite_card_literal @ A )
= ( finite_card_literal @ top_top_set_literal ) )
=> ( A = top_top_set_literal ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_742_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit ) )
=> ( A = top_to1996260823553986621t_unit ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_743_infinite__UNIV__char__0,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_char_0
thf(fact_744_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_745_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite211349803975347344iteral @ top_to6658620532179778271iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_746_finite__Prod__UNIV,axiom,
( ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_747_finite__Prod__UNIV,axiom,
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite4899330813501287658al_nat @ top_to4578518674692263481al_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_748_finite__Prod__UNIV,axiom,
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite1012402614658241104iteral @ top_to594236051165446743iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_749_finite__Prod__UNIV,axiom,
( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6504745279079910025t_unit @ top_to8596954369231496208t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_750_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite_finite_nat @ top_top_set_nat )
=> ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat ) ) ) ).
% finite_Prod_UNIV
thf(fact_751_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( finite885979093102766295iteral @ top_to1987565607987313502iteral ) ) ) ).
% finite_Prod_UNIV
thf(fact_752_finite__Prod__UNIV,axiom,
( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit ) ) ) ).
% finite_Prod_UNIV
thf(fact_753_ex__new__if__finite,axiom,
! [A: set_nat] :
( ~ ( finite_finite_nat @ top_top_set_nat )
=> ( ( finite_finite_nat @ A )
=> ? [A6: nat] :
~ ( member_nat @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_754_ex__new__if__finite,axiom,
! [A: set_literal] :
( ~ ( finite5847741373460823677iteral @ top_top_set_literal )
=> ( ( finite5847741373460823677iteral @ A )
=> ? [A6: literal] :
~ ( member_literal @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_755_ex__new__if__finite,axiom,
! [A: set_Product_unit] :
( ~ ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
=> ( ( finite4290736615968046902t_unit @ A )
=> ? [A6: product_unit] :
~ ( member_Product_unit @ A6 @ A ) ) ) ).
% ex_new_if_finite
thf(fact_756_finite__prod,axiom,
( ( finite6177210948735845034at_nat @ top_to4669805908274784177at_nat )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_757_finite__prod,axiom,
( ( finite211349803975347344iteral @ top_to6658620532179778271iteral )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_758_finite__prod,axiom,
( ( finite5113082511001691337t_unit @ top_to8544742955230171288t_unit )
= ( ( finite_finite_nat @ top_top_set_nat )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_759_finite__prod,axiom,
( ( finite4899330813501287658al_nat @ top_to4578518674692263481al_nat )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_760_finite__prod,axiom,
( ( finite1012402614658241104iteral @ top_to594236051165446743iteral )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_761_finite__prod,axiom,
( ( finite6504745279079910025t_unit @ top_to8596954369231496208t_unit )
= ( ( finite5847741373460823677iteral @ top_top_set_literal )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_762_finite__prod,axiom,
( ( finite5187522816498166307it_nat @ top_to5974110478112770290it_nat )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite_finite_nat @ top_top_set_nat ) ) ) ).
% finite_prod
thf(fact_763_finite__prod,axiom,
( ( finite885979093102766295iteral @ top_to1987565607987313502iteral )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite5847741373460823677iteral @ top_top_set_literal ) ) ) ).
% finite_prod
thf(fact_764_finite__prod,axiom,
( ( finite6816719414181127824t_unit @ top_to1835807148980544151t_unit )
= ( ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit )
& ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% finite_prod
thf(fact_765_finite__Plus,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite6187706683773761046at_nat @ ( sum_Plus_nat_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_766_finite__Plus,axiom,
! [A: set_nat,B: set_literal] :
( ( finite_finite_nat @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( finite7336130560110450212iteral @ ( sum_Plus_nat_literal @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_767_finite__Plus,axiom,
! [A: set_literal,B: set_nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite_finite_nat @ B )
=> ( finite2800739532781614718al_nat @ ( sum_Plus_literal_nat @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_768_finite__Plus,axiom,
! [A: set_literal,B: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( finite8615155063774727100iteral @ ( sum_Pl2008226088555200627iteral @ A @ B ) ) ) ) ).
% finite_Plus
thf(fact_769_Finite__Set_Ofinite__set,axiom,
( ( finite1152437895449049373et_nat @ top_top_set_set_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% Finite_Set.finite_set
thf(fact_770_Finite__Set_Ofinite__set,axiom,
( ( finite2869373537460367197iteral @ top_to5694933271948605156iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% Finite_Set.finite_set
thf(fact_771_Finite__Set_Ofinite__set,axiom,
( ( finite1772178364199683094t_unit @ top_to1767297665138865437t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% Finite_Set.finite_set
thf(fact_772_finite__class_Ofinite__UNIV,axiom,
finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ).
% finite_class.finite_UNIV
thf(fact_773_UNIV__witness,axiom,
? [X2: nat] : ( member_nat @ X2 @ top_top_set_nat ) ).
% UNIV_witness
thf(fact_774_UNIV__witness,axiom,
? [X2: literal] : ( member_literal @ X2 @ top_top_set_literal ) ).
% UNIV_witness
thf(fact_775_UNIV__witness,axiom,
? [X2: product_unit] : ( member_Product_unit @ X2 @ top_to1996260823553986621t_unit ) ).
% UNIV_witness
thf(fact_776_UNIV__eq__I,axiom,
! [A: set_nat] :
( ! [X2: nat] : ( member_nat @ X2 @ A )
=> ( top_top_set_nat = A ) ) ).
% UNIV_eq_I
thf(fact_777_UNIV__eq__I,axiom,
! [A: set_literal] :
( ! [X2: literal] : ( member_literal @ X2 @ A )
=> ( top_top_set_literal = A ) ) ).
% UNIV_eq_I
thf(fact_778_UNIV__eq__I,axiom,
! [A: set_Product_unit] :
( ! [X2: product_unit] : ( member_Product_unit @ X2 @ A )
=> ( top_to1996260823553986621t_unit = A ) ) ).
% UNIV_eq_I
thf(fact_779_list_Omap__ident__strong,axiom,
! [T3: list_nat,F: nat > nat] :
( ! [Z2: nat] :
( ( member_nat @ Z2 @ ( set_nat2 @ T3 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_nat_nat @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_780_list_Omap__ident__strong,axiom,
! [T3: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [Z2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ Z2 @ ( set_Sum_sum_a_nat2 @ T3 ) )
=> ( ( F @ Z2 )
= Z2 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ T3 )
= T3 ) ) ).
% list.map_ident_strong
thf(fact_781_map__idI,axiom,
! [Xs: list_nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_nat_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_782_map__idI,axiom,
! [Xs: list_Sum_sum_a_nat,F: sum_sum_a_nat > sum_sum_a_nat] :
( ! [X2: sum_sum_a_nat] :
( ( member_Sum_sum_a_nat @ X2 @ ( set_Sum_sum_a_nat2 @ Xs ) )
=> ( ( F @ X2 )
= X2 ) )
=> ( ( map_Su2790769393171190532_a_nat @ F @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_783_top_Oextremum__uniqueI,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
=> ( A2 = top_top_set_nat ) ) ).
% top.extremum_uniqueI
thf(fact_784_top_Oextremum__uniqueI,axiom,
! [A2: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A2 )
=> ( A2 = top_top_set_literal ) ) ).
% top.extremum_uniqueI
thf(fact_785_top_Oextremum__uniqueI,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
=> ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_uniqueI
thf(fact_786_top_Oextremum__unique,axiom,
! [A2: set_nat] :
( ( ord_less_eq_set_nat @ top_top_set_nat @ A2 )
= ( A2 = top_top_set_nat ) ) ).
% top.extremum_unique
thf(fact_787_top_Oextremum__unique,axiom,
! [A2: set_literal] :
( ( ord_le7307670543136651348iteral @ top_top_set_literal @ A2 )
= ( A2 = top_top_set_literal ) ) ).
% top.extremum_unique
thf(fact_788_top_Oextremum__unique,axiom,
! [A2: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ top_to1996260823553986621t_unit @ A2 )
= ( A2 = top_to1996260823553986621t_unit ) ) ).
% top.extremum_unique
thf(fact_789_top__greatest,axiom,
! [A2: set_nat] : ( ord_less_eq_set_nat @ A2 @ top_top_set_nat ) ).
% top_greatest
thf(fact_790_top__greatest,axiom,
! [A2: set_literal] : ( ord_le7307670543136651348iteral @ A2 @ top_top_set_literal ) ).
% top_greatest
thf(fact_791_top__greatest,axiom,
! [A2: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A2 @ top_to1996260823553986621t_unit ) ).
% top_greatest
thf(fact_792_finite__vimageD,axiom,
! [H: nat > nat,F2: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ H @ F2 ) )
=> ( ( ( image_nat_nat @ H @ top_top_set_nat )
= top_top_set_nat )
=> ( finite_finite_nat @ F2 ) ) ) ).
% finite_vimageD
thf(fact_793_finite__vimageD,axiom,
! [H: nat > literal,F2: set_literal] :
( ( finite_finite_nat @ ( vimage_nat_literal @ H @ F2 ) )
=> ( ( ( image_nat_literal @ H @ top_top_set_nat )
= top_top_set_literal )
=> ( finite5847741373460823677iteral @ F2 ) ) ) ).
% finite_vimageD
thf(fact_794_finite__vimageD,axiom,
! [H: nat > product_unit,F2: set_Product_unit] :
( ( finite_finite_nat @ ( vimage4884490618288580032t_unit @ H @ F2 ) )
=> ( ( ( image_8730104196221521654t_unit @ H @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F2 ) ) ) ).
% finite_vimageD
thf(fact_795_finite__vimageD,axiom,
! [H: literal > nat,F2: set_nat] :
( ( finite5847741373460823677iteral @ ( vimage_literal_nat @ H @ F2 ) )
=> ( ( ( image_literal_nat @ H @ top_top_set_literal )
= top_top_set_nat )
=> ( finite_finite_nat @ F2 ) ) ) ).
% finite_vimageD
thf(fact_796_finite__vimageD,axiom,
! [H: literal > literal,F2: set_literal] :
( ( finite5847741373460823677iteral @ ( vimage8238609917233974331iteral @ H @ F2 ) )
=> ( ( ( image_8195128725298311301iteral @ H @ top_top_set_literal )
= top_top_set_literal )
=> ( finite5847741373460823677iteral @ F2 ) ) ) ).
% finite_vimageD
thf(fact_797_finite__vimageD,axiom,
! [H: literal > product_unit,F2: set_Product_unit] :
( ( finite5847741373460823677iteral @ ( vimage6047873233414524916t_unit @ H @ F2 ) )
=> ( ( ( image_6787854153545327934t_unit @ H @ top_top_set_literal )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F2 ) ) ) ).
% finite_vimageD
thf(fact_798_finite__vimageD,axiom,
! [H: product_unit > nat,F2: set_nat] :
( ( finite4290736615968046902t_unit @ ( vimage6253328473476588386it_nat @ H @ F2 ) )
=> ( ( ( image_875570014554754200it_nat @ H @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( finite_finite_nat @ F2 ) ) ) ).
% finite_vimageD
thf(fact_799_finite__vimageD,axiom,
! [H: product_unit > literal,F2: set_literal] :
( ( finite4290736615968046902t_unit @ ( vimage5137003825767189442iteral @ H @ F2 ) )
=> ( ( ( image_5876984745897992460iteral @ H @ top_to1996260823553986621t_unit )
= top_top_set_literal )
=> ( finite5847741373460823677iteral @ F2 ) ) ) ).
% finite_vimageD
thf(fact_800_finite__vimageD,axiom,
! [H: product_unit > product_unit,F2: set_Product_unit] :
( ( finite4290736615968046902t_unit @ ( vimage7995052115951654139t_unit @ H @ F2 ) )
=> ( ( ( image_405062704495631173t_unit @ H @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( finite4290736615968046902t_unit @ F2 ) ) ) ).
% finite_vimageD
thf(fact_801_finite__vimageD,axiom,
! [H: a > sum_sum_a_nat,F2: set_Sum_sum_a_nat] :
( ( finite_finite_a @ ( vimage4607795094749595324_a_nat @ H @ F2 ) )
=> ( ( ( image_7873763678140191238_a_nat @ H @ top_top_set_a )
= top_to795618464972521135_a_nat )
=> ( finite502105017643426984_a_nat @ F2 ) ) ) ).
% finite_vimageD
thf(fact_802_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ X @ top_to795618464972521135_a_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_803_boolean__algebra_Oconj__one__right,axiom,
! [X: set_nat] :
( ( inf_inf_set_nat @ X @ top_top_set_nat )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_804_boolean__algebra_Oconj__one__right,axiom,
! [X: set_literal] :
( ( inf_inf_set_literal @ X @ top_top_set_literal )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_805_boolean__algebra_Oconj__one__right,axiom,
! [X: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ X @ top_to1996260823553986621t_unit )
= X ) ).
% boolean_algebra.conj_one_right
thf(fact_806_finite__has__minimal2,axiom,
! [A: set_literal,A2: literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( member_literal @ A2 @ A )
=> ? [X2: literal] :
( ( member_literal @ X2 @ A )
& ( ord_less_eq_literal @ X2 @ A2 )
& ! [Xa: literal] :
( ( member_literal @ Xa @ A )
=> ( ( ord_less_eq_literal @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_807_finite__has__minimal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ X2 @ A2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ Xa @ X2 )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_808_finite__has__maximal2,axiom,
! [A: set_literal,A2: literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( member_literal @ A2 @ A )
=> ? [X2: literal] :
( ( member_literal @ X2 @ A )
& ( ord_less_eq_literal @ A2 @ X2 )
& ! [Xa: literal] :
( ( member_literal @ Xa @ A )
=> ( ( ord_less_eq_literal @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_809_finite__has__maximal2,axiom,
! [A: set_nat,A2: nat] :
( ( finite_finite_nat @ A )
=> ( ( member_nat @ A2 @ A )
=> ? [X2: nat] :
( ( member_nat @ X2 @ A )
& ( ord_less_eq_nat @ A2 @ X2 )
& ! [Xa: nat] :
( ( member_nat @ Xa @ A )
=> ( ( ord_less_eq_nat @ X2 @ Xa )
=> ( X2 = Xa ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_810_rev__finite__subset,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% rev_finite_subset
thf(fact_811_rev__finite__subset,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( finite5847741373460823677iteral @ A ) ) ) ).
% rev_finite_subset
thf(fact_812_infinite__super,axiom,
! [S: set_nat,T: set_nat] :
( ( ord_less_eq_set_nat @ S @ T )
=> ( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_super
thf(fact_813_infinite__super,axiom,
! [S: set_literal,T: set_literal] :
( ( ord_le7307670543136651348iteral @ S @ T )
=> ( ~ ( finite5847741373460823677iteral @ S )
=> ~ ( finite5847741373460823677iteral @ T ) ) ) ).
% infinite_super
thf(fact_814_finite__subset,axiom,
! [A: set_nat,B: set_nat] :
( ( ord_less_eq_set_nat @ A @ B )
=> ( ( finite_finite_nat @ B )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_subset
thf(fact_815_finite__subset,axiom,
! [A: set_literal,B: set_literal] :
( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ( finite5847741373460823677iteral @ B )
=> ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_subset
thf(fact_816_finite__list,axiom,
! [A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ A )
=> ? [Xs2: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_817_finite__list,axiom,
! [A: set_nat] :
( ( finite_finite_nat @ A )
=> ? [Xs2: list_nat] :
( ( set_nat2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_818_finite__list,axiom,
! [A: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ? [Xs2: list_literal] :
( ( set_literal2 @ Xs2 )
= A ) ) ).
% finite_list
thf(fact_819_finite__nat__set__iff__bounded__le,axiom,
( finite_finite_nat
= ( ^ [N3: set_nat] :
? [M3: nat] :
! [X4: nat] :
( ( member_nat @ X4 @ N3 )
=> ( ord_less_eq_nat @ X4 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_820_infinite__Un,axiom,
! [S: set_literal,T: set_literal] :
( ( ~ ( finite5847741373460823677iteral @ ( sup_sup_set_literal @ S @ T ) ) )
= ( ~ ( finite5847741373460823677iteral @ S )
| ~ ( finite5847741373460823677iteral @ T ) ) ) ).
% infinite_Un
thf(fact_821_infinite__Un,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ( ~ ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) ) )
= ( ~ ( finite502105017643426984_a_nat @ S )
| ~ ( finite502105017643426984_a_nat @ T ) ) ) ).
% infinite_Un
thf(fact_822_infinite__Un,axiom,
! [S: set_nat,T: set_nat] :
( ( ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) )
= ( ~ ( finite_finite_nat @ S )
| ~ ( finite_finite_nat @ T ) ) ) ).
% infinite_Un
thf(fact_823_Un__infinite,axiom,
! [S: set_literal,T: set_literal] :
( ~ ( finite5847741373460823677iteral @ S )
=> ~ ( finite5847741373460823677iteral @ ( sup_sup_set_literal @ S @ T ) ) ) ).
% Un_infinite
thf(fact_824_Un__infinite,axiom,
! [S: set_Sum_sum_a_nat,T: set_Sum_sum_a_nat] :
( ~ ( finite502105017643426984_a_nat @ S )
=> ~ ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_825_Un__infinite,axiom,
! [S: set_nat,T: set_nat] :
( ~ ( finite_finite_nat @ S )
=> ~ ( finite_finite_nat @ ( sup_sup_set_nat @ S @ T ) ) ) ).
% Un_infinite
thf(fact_826_finite__UnI,axiom,
! [F2: set_literal,G2: set_literal] :
( ( finite5847741373460823677iteral @ F2 )
=> ( ( finite5847741373460823677iteral @ G2 )
=> ( finite5847741373460823677iteral @ ( sup_sup_set_literal @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_827_finite__UnI,axiom,
! [F2: set_Sum_sum_a_nat,G2: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ F2 )
=> ( ( finite502105017643426984_a_nat @ G2 )
=> ( finite502105017643426984_a_nat @ ( sup_su6804446743777130803_a_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_828_finite__UnI,axiom,
! [F2: set_nat,G2: set_nat] :
( ( finite_finite_nat @ F2 )
=> ( ( finite_finite_nat @ G2 )
=> ( finite_finite_nat @ ( sup_sup_set_nat @ F2 @ G2 ) ) ) ) ).
% finite_UnI
thf(fact_829_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: a > sum_sum_a_nat,X: a] :
( ( B2
= ( F @ X ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) ) ) ).
% range_eqI
thf(fact_830_range__eqI,axiom,
! [B2: sum_sum_a_nat,F: nat > sum_sum_a_nat,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_831_range__eqI,axiom,
! [B2: nat,F: nat > nat,X: nat] :
( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) ) ) ).
% range_eqI
thf(fact_832_range__eqI,axiom,
! [B2: nat,F: literal > nat,X: literal] :
( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_literal_nat @ F @ top_top_set_literal ) ) ) ).
% range_eqI
thf(fact_833_range__eqI,axiom,
! [B2: nat,F: product_unit > nat,X: product_unit] :
( ( B2
= ( F @ X ) )
=> ( member_nat @ B2 @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ) ).
% range_eqI
thf(fact_834_rangeI,axiom,
! [F: a > sum_sum_a_nat,X: a] : ( member_Sum_sum_a_nat @ ( F @ X ) @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) ) ).
% rangeI
thf(fact_835_rangeI,axiom,
! [F: nat > sum_sum_a_nat,X: nat] : ( member_Sum_sum_a_nat @ ( F @ X ) @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_836_rangeI,axiom,
! [F: nat > nat,X: nat] : ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ top_top_set_nat ) ) ).
% rangeI
thf(fact_837_rangeI,axiom,
! [F: literal > nat,X: literal] : ( member_nat @ ( F @ X ) @ ( image_literal_nat @ F @ top_top_set_literal ) ) ).
% rangeI
thf(fact_838_rangeI,axiom,
! [F: product_unit > nat,X: product_unit] : ( member_nat @ ( F @ X ) @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) ) ).
% rangeI
thf(fact_839_subset__UNIV,axiom,
! [A: set_nat] : ( ord_less_eq_set_nat @ A @ top_top_set_nat ) ).
% subset_UNIV
thf(fact_840_subset__UNIV,axiom,
! [A: set_literal] : ( ord_le7307670543136651348iteral @ A @ top_top_set_literal ) ).
% subset_UNIV
thf(fact_841_subset__UNIV,axiom,
! [A: set_Product_unit] : ( ord_le3507040750410214029t_unit @ A @ top_to1996260823553986621t_unit ) ).
% subset_UNIV
thf(fact_842_finite__Inl,axiom,
! [X6: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ X6 )
=> ( finite_finite_a @ ( vimage4607795094749595324_a_nat @ sum_Inl_a_nat @ X6 ) ) ) ).
% finite_Inl
thf(fact_843_Int__UNIV__right,axiom,
! [A: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ A @ top_to795618464972521135_a_nat )
= A ) ).
% Int_UNIV_right
thf(fact_844_Int__UNIV__right,axiom,
! [A: set_nat] :
( ( inf_inf_set_nat @ A @ top_top_set_nat )
= A ) ).
% Int_UNIV_right
thf(fact_845_Int__UNIV__right,axiom,
! [A: set_literal] :
( ( inf_inf_set_literal @ A @ top_top_set_literal )
= A ) ).
% Int_UNIV_right
thf(fact_846_Int__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ A @ top_to1996260823553986621t_unit )
= A ) ).
% Int_UNIV_right
thf(fact_847_Int__UNIV__left,axiom,
! [B: set_Sum_sum_a_nat] :
( ( inf_in7084830621192376909_a_nat @ top_to795618464972521135_a_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_848_Int__UNIV__left,axiom,
! [B: set_nat] :
( ( inf_inf_set_nat @ top_top_set_nat @ B )
= B ) ).
% Int_UNIV_left
thf(fact_849_Int__UNIV__left,axiom,
! [B: set_literal] :
( ( inf_inf_set_literal @ top_top_set_literal @ B )
= B ) ).
% Int_UNIV_left
thf(fact_850_Int__UNIV__left,axiom,
! [B: set_Product_unit] :
( ( inf_in4660618365625256667t_unit @ top_to1996260823553986621t_unit @ B )
= B ) ).
% Int_UNIV_left
thf(fact_851_Un__UNIV__right,axiom,
! [A: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ A @ top_to795618464972521135_a_nat )
= top_to795618464972521135_a_nat ) ).
% Un_UNIV_right
thf(fact_852_Un__UNIV__right,axiom,
! [A: set_nat] :
( ( sup_sup_set_nat @ A @ top_top_set_nat )
= top_top_set_nat ) ).
% Un_UNIV_right
thf(fact_853_Un__UNIV__right,axiom,
! [A: set_literal] :
( ( sup_sup_set_literal @ A @ top_top_set_literal )
= top_top_set_literal ) ).
% Un_UNIV_right
thf(fact_854_Un__UNIV__right,axiom,
! [A: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ A @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_right
thf(fact_855_Un__UNIV__left,axiom,
! [B: set_Sum_sum_a_nat] :
( ( sup_su6804446743777130803_a_nat @ top_to795618464972521135_a_nat @ B )
= top_to795618464972521135_a_nat ) ).
% Un_UNIV_left
thf(fact_856_Un__UNIV__left,axiom,
! [B: set_nat] :
( ( sup_sup_set_nat @ top_top_set_nat @ B )
= top_top_set_nat ) ).
% Un_UNIV_left
thf(fact_857_Un__UNIV__left,axiom,
! [B: set_literal] :
( ( sup_sup_set_literal @ top_top_set_literal @ B )
= top_top_set_literal ) ).
% Un_UNIV_left
thf(fact_858_Un__UNIV__left,axiom,
! [B: set_Product_unit] :
( ( sup_su793286257634532545t_unit @ top_to1996260823553986621t_unit @ B )
= top_to1996260823553986621t_unit ) ).
% Un_UNIV_left
thf(fact_859_finite__vimageD_H,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite_finite_a @ ( vimage4607795094749595324_a_nat @ F @ A ) )
=> ( ( ord_le1325389633284124927_a_nat @ A @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) )
=> ( finite502105017643426984_a_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_860_finite__vimageD_H,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ ( vimage6545432446589551483_a_nat @ F @ A ) )
=> ( ( ord_le1325389633284124927_a_nat @ A @ ( image_7142520692256960453_a_nat @ F @ top_to795618464972521135_a_nat ) )
=> ( finite502105017643426984_a_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_861_finite__vimageD_H,axiom,
! [F: nat > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( finite_finite_nat @ ( vimage3040984495076556338_a_nat @ F @ A ) )
=> ( ( ord_le1325389633284124927_a_nat @ A @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) )
=> ( finite502105017643426984_a_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_862_finite__vimageD_H,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( vimage_nat_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_863_finite__vimageD_H,axiom,
! [F: nat > literal,A: set_literal] :
( ( finite_finite_nat @ ( vimage_nat_literal @ F @ A ) )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_nat_literal @ F @ top_top_set_nat ) )
=> ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_vimageD'
thf(fact_864_finite__vimageD_H,axiom,
! [F: literal > nat,A: set_nat] :
( ( finite5847741373460823677iteral @ ( vimage_literal_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_literal_nat @ F @ top_top_set_literal ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_865_finite__vimageD_H,axiom,
! [F: literal > literal,A: set_literal] :
( ( finite5847741373460823677iteral @ ( vimage8238609917233974331iteral @ F @ A ) )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_8195128725298311301iteral @ F @ top_top_set_literal ) )
=> ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_vimageD'
thf(fact_866_finite__vimageD_H,axiom,
! [F: product_unit > nat,A: set_nat] :
( ( finite4290736615968046902t_unit @ ( vimage6253328473476588386it_nat @ F @ A ) )
=> ( ( ord_less_eq_set_nat @ A @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_vimageD'
thf(fact_867_finite__vimageD_H,axiom,
! [F: product_unit > literal,A: set_literal] :
( ( finite4290736615968046902t_unit @ ( vimage5137003825767189442iteral @ F @ A ) )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit ) )
=> ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_vimageD'
thf(fact_868_rremdups__set,axiom,
! [Xs: list_Sum_sum_a_nat] :
( ( set_Sum_sum_a_nat2 @ ( rremdu8304153113908149561_a_nat @ Xs ) )
= ( set_Sum_sum_a_nat2 @ Xs ) ) ).
% rremdups_set
thf(fact_869_image__set,axiom,
! [F: nat > nat,Xs: list_nat] :
( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_870_image__set,axiom,
! [F: a > sum_sum_a_nat,Xs: list_a] :
( ( image_7873763678140191238_a_nat @ F @ ( set_a2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_a_Sum_sum_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_871_image__set,axiom,
! [F: nat > sum_sum_a_nat,Xs: list_nat] :
( ( image_7293268710728258664_a_nat @ F @ ( set_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_na823391071729141993_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_872_image__set,axiom,
! [F: sum_sum_a_nat > sum_sum_a_nat,Xs: list_Sum_sum_a_nat] :
( ( image_7142520692256960453_a_nat @ F @ ( set_Sum_sum_a_nat2 @ Xs ) )
= ( set_Sum_sum_a_nat2 @ ( map_Su2790769393171190532_a_nat @ F @ Xs ) ) ) ).
% image_set
thf(fact_873_all__finite__subset__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_a] :
( ( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A ) )
=> ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_874_all__finite__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ! [B3: set_Sum_sum_a_nat] :
( ( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A ) )
=> ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_875_all__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A ) )
=> ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_876_all__finite__subset__image,axiom,
! [F: literal > nat,A: set_literal,P: set_nat > $o] :
( ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_literal_nat @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_literal] :
( ( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ A ) )
=> ( P @ ( image_literal_nat @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_877_all__finite__subset__image,axiom,
! [F: nat > literal,A: set_nat,P: set_literal > $o] :
( ( ! [B3: set_literal] :
( ( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ ( image_nat_literal @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_nat] :
( ( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A ) )
=> ( P @ ( image_nat_literal @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_878_all__finite__subset__image,axiom,
! [F: literal > literal,A: set_literal,P: set_literal > $o] :
( ( ! [B3: set_literal] :
( ( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ ( image_8195128725298311301iteral @ F @ A ) ) )
=> ( P @ B3 ) ) )
= ( ! [B3: set_literal] :
( ( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ A ) )
=> ( P @ ( image_8195128725298311301iteral @ F @ B3 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_879_ex__finite__subset__image,axiom,
! [F: a > sum_sum_a_nat,A: set_a,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7873763678140191238_a_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_a] :
( ( finite_finite_a @ B3 )
& ( ord_less_eq_set_a @ B3 @ A )
& ( P @ ( image_7873763678140191238_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_880_ex__finite__subset__image,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,P: set_Sum_sum_a_nat > $o] :
( ( ? [B3: set_Sum_sum_a_nat] :
( ( finite502105017643426984_a_nat @ B3 )
& ( ord_le1325389633284124927_a_nat @ B3 @ ( image_7293268710728258664_a_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A )
& ( P @ ( image_7293268710728258664_a_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_881_ex__finite__subset__image,axiom,
! [F: nat > nat,A: set_nat,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_nat_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A )
& ( P @ ( image_nat_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_882_ex__finite__subset__image,axiom,
! [F: literal > nat,A: set_literal,P: set_nat > $o] :
( ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ ( image_literal_nat @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_literal] :
( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ A )
& ( P @ ( image_literal_nat @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_883_ex__finite__subset__image,axiom,
! [F: nat > literal,A: set_nat,P: set_literal > $o] :
( ( ? [B3: set_literal] :
( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ ( image_nat_literal @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_nat] :
( ( finite_finite_nat @ B3 )
& ( ord_less_eq_set_nat @ B3 @ A )
& ( P @ ( image_nat_literal @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_884_ex__finite__subset__image,axiom,
! [F: literal > literal,A: set_literal,P: set_literal > $o] :
( ( ? [B3: set_literal] :
( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ ( image_8195128725298311301iteral @ F @ A ) )
& ( P @ B3 ) ) )
= ( ? [B3: set_literal] :
( ( finite5847741373460823677iteral @ B3 )
& ( ord_le7307670543136651348iteral @ B3 @ A )
& ( P @ ( image_8195128725298311301iteral @ F @ B3 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_885_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat,A: set_a] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ? [C3: set_a] :
( ( ord_less_eq_set_a @ C3 @ A )
& ( finite_finite_a @ C3 )
& ( B
= ( image_7873763678140191238_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_886_finite__subset__image,axiom,
! [B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat,A: set_nat] :
( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_7293268710728258664_a_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_887_finite__subset__image,axiom,
! [B: set_nat,F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_888_finite__subset__image,axiom,
! [B: set_nat,F: literal > nat,A: set_literal] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ B @ ( image_literal_nat @ F @ A ) )
=> ? [C3: set_literal] :
( ( ord_le7307670543136651348iteral @ C3 @ A )
& ( finite5847741373460823677iteral @ C3 )
& ( B
= ( image_literal_nat @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_889_finite__subset__image,axiom,
! [B: set_literal,F: nat > literal,A: set_nat] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_nat_literal @ F @ A ) )
=> ? [C3: set_nat] :
( ( ord_less_eq_set_nat @ C3 @ A )
& ( finite_finite_nat @ C3 )
& ( B
= ( image_nat_literal @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_890_finite__subset__image,axiom,
! [B: set_literal,F: literal > literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_8195128725298311301iteral @ F @ A ) )
=> ? [C3: set_literal] :
( ( ord_le7307670543136651348iteral @ C3 @ A )
& ( finite5847741373460823677iteral @ C3 )
& ( B
= ( image_8195128725298311301iteral @ F @ C3 ) ) ) ) ) ).
% finite_subset_image
thf(fact_891_finite__surj,axiom,
! [A: set_a,B: set_Sum_sum_a_nat,F: a > sum_sum_a_nat] :
( ( finite_finite_a @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( finite502105017643426984_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_892_finite__surj,axiom,
! [A: set_nat,B: set_Sum_sum_a_nat,F: nat > sum_sum_a_nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_le1325389633284124927_a_nat @ B @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( finite502105017643426984_a_nat @ B ) ) ) ).
% finite_surj
thf(fact_893_finite__surj,axiom,
! [A: set_nat,B: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_894_finite__surj,axiom,
! [A: set_nat,B: set_literal,F: nat > literal] :
( ( finite_finite_nat @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_nat_literal @ F @ A ) )
=> ( finite5847741373460823677iteral @ B ) ) ) ).
% finite_surj
thf(fact_895_finite__surj,axiom,
! [A: set_literal,B: set_nat,F: literal > nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_less_eq_set_nat @ B @ ( image_literal_nat @ F @ A ) )
=> ( finite_finite_nat @ B ) ) ) ).
% finite_surj
thf(fact_896_finite__surj,axiom,
! [A: set_literal,B: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( ord_le7307670543136651348iteral @ B @ ( image_8195128725298311301iteral @ F @ A ) )
=> ( finite5847741373460823677iteral @ B ) ) ) ).
% finite_surj
thf(fact_897_card__subset__eq,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ( finite410649719033368117t_unit @ A )
= ( finite410649719033368117t_unit @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_898_card__subset__eq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ( finite_card_nat @ A )
= ( finite_card_nat @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_899_card__subset__eq,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ( ( finite_card_literal @ A )
= ( finite_card_literal @ B ) )
=> ( A = B ) ) ) ) ).
% card_subset_eq
thf(fact_900_infinite__arbitrarily__large,axiom,
! [A: set_Product_unit,N2: nat] :
( ~ ( finite4290736615968046902t_unit @ A )
=> ? [B8: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B8 )
& ( ( finite410649719033368117t_unit @ B8 )
= N2 )
& ( ord_le3507040750410214029t_unit @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_901_infinite__arbitrarily__large,axiom,
! [A: set_nat,N2: nat] :
( ~ ( finite_finite_nat @ A )
=> ? [B8: set_nat] :
( ( finite_finite_nat @ B8 )
& ( ( finite_card_nat @ B8 )
= N2 )
& ( ord_less_eq_set_nat @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_902_infinite__arbitrarily__large,axiom,
! [A: set_literal,N2: nat] :
( ~ ( finite5847741373460823677iteral @ A )
=> ? [B8: set_literal] :
( ( finite5847741373460823677iteral @ B8 )
& ( ( finite_card_literal @ B8 )
= N2 )
& ( ord_le7307670543136651348iteral @ B8 @ A ) ) ) ).
% infinite_arbitrarily_large
thf(fact_903_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_nat,R: nat > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B9: product_unit] :
( ( member_Product_unit @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: product_unit] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_Product_unit @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_904_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_literal,R: literal > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A6: literal] :
( ( member_literal @ A6 @ A )
=> ? [B9: product_unit] :
( ( member_Product_unit @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: literal,A22: literal,B6: product_unit] :
( ( member_literal @ A1 @ A )
=> ( ( member_literal @ A22 @ A )
=> ( ( member_Product_unit @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_905_card__le__if__inj__on__rel,axiom,
! [B: set_Product_unit,A: set_Product_unit,R: product_unit > product_unit > $o] :
( ( finite4290736615968046902t_unit @ B )
=> ( ! [A6: product_unit] :
( ( member_Product_unit @ A6 @ A )
=> ? [B9: product_unit] :
( ( member_Product_unit @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B6: product_unit] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_Product_unit @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_906_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_nat,R: nat > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: nat] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_907_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_literal,R: literal > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: literal] :
( ( member_literal @ A6 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: literal,A22: literal,B6: nat] :
( ( member_literal @ A1 @ A )
=> ( ( member_literal @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_908_card__le__if__inj__on__rel,axiom,
! [B: set_nat,A: set_Product_unit,R: product_unit > nat > $o] :
( ( finite_finite_nat @ B )
=> ( ! [A6: product_unit] :
( ( member_Product_unit @ A6 @ A )
=> ? [B9: nat] :
( ( member_nat @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B6: nat] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_nat @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_909_card__le__if__inj__on__rel,axiom,
! [B: set_literal,A: set_nat,R: nat > literal > $o] :
( ( finite5847741373460823677iteral @ B )
=> ( ! [A6: nat] :
( ( member_nat @ A6 @ A )
=> ? [B9: literal] :
( ( member_literal @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: nat,A22: nat,B6: literal] :
( ( member_nat @ A1 @ A )
=> ( ( member_nat @ A22 @ A )
=> ( ( member_literal @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_910_card__le__if__inj__on__rel,axiom,
! [B: set_literal,A: set_literal,R: literal > literal > $o] :
( ( finite5847741373460823677iteral @ B )
=> ( ! [A6: literal] :
( ( member_literal @ A6 @ A )
=> ? [B9: literal] :
( ( member_literal @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: literal,A22: literal,B6: literal] :
( ( member_literal @ A1 @ A )
=> ( ( member_literal @ A22 @ A )
=> ( ( member_literal @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_911_card__le__if__inj__on__rel,axiom,
! [B: set_literal,A: set_Product_unit,R: product_unit > literal > $o] :
( ( finite5847741373460823677iteral @ B )
=> ( ! [A6: product_unit] :
( ( member_Product_unit @ A6 @ A )
=> ? [B9: literal] :
( ( member_literal @ B9 @ B )
& ( R @ A6 @ B9 ) ) )
=> ( ! [A1: product_unit,A22: product_unit,B6: literal] :
( ( member_Product_unit @ A1 @ A )
=> ( ( member_Product_unit @ A22 @ A )
=> ( ( member_literal @ B6 @ B )
=> ( ( R @ A1 @ B6 )
=> ( ( R @ A22 @ B6 )
=> ( A1 = A22 ) ) ) ) ) )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_literal @ B ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_912_PlusE,axiom,
! [U: sum_sum_nat_nat,A: set_nat,B: set_nat] :
( ( member8583185029347631382at_nat @ U @ ( sum_Plus_nat_nat @ A @ B ) )
=> ( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ( U
!= ( sum_Inl_nat_nat @ X2 ) ) )
=> ~ ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( U
!= ( sum_Inr_nat_nat @ Y4 ) ) ) ) ) ).
% PlusE
thf(fact_913_PlusE,axiom,
! [U: sum_sum_a_nat,A: set_a,B: set_nat] :
( ( member_Sum_sum_a_nat @ U @ ( sum_Plus_a_nat @ A @ B ) )
=> ( ! [X2: a] :
( ( member_a @ X2 @ A )
=> ( U
!= ( sum_Inl_a_nat @ X2 ) ) )
=> ~ ! [Y4: nat] :
( ( member_nat @ Y4 @ B )
=> ( U
!= ( sum_Inr_nat_a @ Y4 ) ) ) ) ) ).
% PlusE
thf(fact_914_range__subsetD,axiom,
! [F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat,I: a] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_915_range__subsetD,axiom,
! [F: nat > sum_sum_a_nat,B: set_Sum_sum_a_nat,I: nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) @ B )
=> ( member_Sum_sum_a_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_916_range__subsetD,axiom,
! [F: nat > nat,B: set_nat,I: nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ top_top_set_nat ) @ B )
=> ( member_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_917_range__subsetD,axiom,
! [F: literal > nat,B: set_nat,I: literal] :
( ( ord_less_eq_set_nat @ ( image_literal_nat @ F @ top_top_set_literal ) @ B )
=> ( member_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_918_range__subsetD,axiom,
! [F: product_unit > nat,B: set_nat,I: product_unit] :
( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) @ B )
=> ( member_nat @ ( F @ I ) @ B ) ) ).
% range_subsetD
thf(fact_919_UNIV__sum,axiom,
( top_to795618464972521135_a_nat
= ( sup_su6804446743777130803_a_nat @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ top_top_set_a ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ top_top_set_nat ) ) ) ).
% UNIV_sum
thf(fact_920_UNIV__sum,axiom,
( top_to6661820994512907621at_nat
= ( sup_su3567568935942035937at_nat @ ( image_678696785212003926at_nat @ sum_Inl_nat_nat @ top_top_set_nat ) @ ( image_678696785212003926at_nat @ sum_Inr_nat_nat @ top_top_set_nat ) ) ) ).
% UNIV_sum
thf(fact_921_UNIV__sum,axiom,
( top_to148093990134820907iteral
= ( sup_su5939970013262617775iteral @ ( image_948809277473826276iteral @ sum_Inl_nat_literal @ top_top_set_nat ) @ ( image_505656761235946540iteral @ sum_Inr_literal_nat @ top_top_set_literal ) ) ) ).
% UNIV_sum
thf(fact_922_UNIV__sum,axiom,
( top_to5465250082899874788t_unit
= ( sup_su4295710226244385384t_unit @ ( image_5493879627330158621t_unit @ sum_In8759356770566287949t_unit @ top_top_set_nat ) @ ( image_5632832758920211948t_unit @ sum_In6291552823763399529it_nat @ top_to1996260823553986621t_unit ) ) ) ).
% UNIV_sum
thf(fact_923_UNIV__sum,axiom,
( top_to7291364169502081925al_nat
= ( sup_su3859868155775102985al_nat @ ( image_5193637770761886854al_nat @ sum_Inl_literal_nat @ top_top_set_literal ) @ ( image_5636790286999766590al_nat @ sum_Inr_nat_literal @ top_top_set_nat ) ) ) ).
% UNIV_sum
thf(fact_924_UNIV__sum,axiom,
( top_to8459027286199865867iteral
= ( sup_su7822573100775623815iteral @ ( image_8650502459363062708iteral @ sum_In8673369501829128942iteral @ top_top_set_literal ) @ ( image_8650502459363062708iteral @ sum_In6960570357259124212iteral @ top_top_set_literal ) ) ) ).
% UNIV_sum
thf(fact_925_UNIV__sum,axiom,
( top_to6093211048104912324t_unit
= ( sup_su3450248338741798464t_unit @ ( image_6350320061251687917t_unit @ sum_In8972096284949572135t_unit @ top_top_set_literal ) @ ( image_7063812555127738022t_unit @ sum_In4586030566214162427iteral @ top_to1996260823553986621t_unit ) ) ) ).
% UNIV_sum
thf(fact_926_UNIV__sum,axiom,
( top_to2894617605782473790it_nat
= ( sup_su1725077749126984386it_nat @ ( image_5707273064416686918it_nat @ sum_In904822588899520495it_nat @ top_to1996260823553986621t_unit ) @ ( image_5568319932826633591it_nat @ sum_In4922714968575391175t_unit @ top_top_set_nat ) ) ) ).
% UNIV_sum
thf(fact_927_UNIV__sum,axiom,
( top_to8707194323715505426iteral
= ( sup_su6064231614352391566iteral @ ( image_1445046369150594292iteral @ sum_In8061226877302236661iteral @ top_to1996260823553986621t_unit ) @ ( image_731553875274544187iteral @ sum_In5496899973861497901t_unit @ top_top_set_literal ) ) ) ).
% UNIV_sum
thf(fact_928_UNIV__sum,axiom,
( top_to2771918933716375115t_unit
= ( sup_su9179690136711093447t_unit @ ( image_2037928093122670381t_unit @ sum_In8043524862774375086t_unit @ top_to1996260823553986621t_unit ) @ ( image_2037928093122670381t_unit @ sum_In1458446571380145076t_unit @ top_to1996260823553986621t_unit ) ) ) ).
% UNIV_sum
thf(fact_929_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_875570014554754200it_nat @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_930_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > literal] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( image_5876984745897992460iteral @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_931_card__image__le,axiom,
! [A: set_Product_unit,F: product_unit > product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_405062704495631173t_unit @ F @ A ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ).
% card_image_le
thf(fact_932_card__image__le,axiom,
! [A: set_nat,F: nat > nat] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_nat_nat @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_933_card__image__le,axiom,
! [A: set_nat,F: nat > literal] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( image_nat_literal @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_934_card__image__le,axiom,
! [A: set_nat,F: nat > product_unit] :
( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_8730104196221521654t_unit @ F @ A ) ) @ ( finite_card_nat @ A ) ) ) ).
% card_image_le
thf(fact_935_card__image__le,axiom,
! [A: set_literal,F: literal > nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( image_literal_nat @ F @ A ) ) @ ( finite_card_literal @ A ) ) ) ).
% card_image_le
thf(fact_936_card__image__le,axiom,
! [A: set_literal,F: literal > literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( image_8195128725298311301iteral @ F @ A ) ) @ ( finite_card_literal @ A ) ) ) ).
% card_image_le
thf(fact_937_card__image__le,axiom,
! [A: set_literal,F: literal > product_unit] :
( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( image_6787854153545327934t_unit @ F @ A ) ) @ ( finite_card_literal @ A ) ) ) ).
% card_image_le
thf(fact_938_card__image__le,axiom,
! [A: set_a,F: a > sum_sum_a_nat] :
( ( finite_finite_a @ A )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) ) @ ( finite_card_a @ A ) ) ) ).
% card_image_le
thf(fact_939_card__mono,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) ) ) ) ).
% card_mono
thf(fact_940_card__mono,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) ) ) ) ).
% card_mono
thf(fact_941_card__mono,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) ) ) ) ).
% card_mono
thf(fact_942_card__seteq,axiom,
! [B: set_Product_unit,A: set_Product_unit] :
( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_le3507040750410214029t_unit @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ B ) @ ( finite410649719033368117t_unit @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_943_card__seteq,axiom,
! [B: set_nat,A: set_nat] :
( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ B ) @ ( finite_card_nat @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_944_card__seteq,axiom,
! [B: set_literal,A: set_literal] :
( ( finite5847741373460823677iteral @ B )
=> ( ( ord_le7307670543136651348iteral @ A @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ B ) @ ( finite_card_literal @ A ) )
=> ( A = B ) ) ) ) ).
% card_seteq
thf(fact_945_exists__subset__between,axiom,
! [A: set_Product_unit,N2: nat,C2: set_Product_unit] :
( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ C2 ) )
=> ( ( ord_le3507040750410214029t_unit @ A @ C2 )
=> ( ( finite4290736615968046902t_unit @ C2 )
=> ? [B8: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ A @ B8 )
& ( ord_le3507040750410214029t_unit @ B8 @ C2 )
& ( ( finite410649719033368117t_unit @ B8 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_946_exists__subset__between,axiom,
! [A: set_nat,N2: nat,C2: set_nat] :
( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ C2 ) )
=> ( ( ord_less_eq_set_nat @ A @ C2 )
=> ( ( finite_finite_nat @ C2 )
=> ? [B8: set_nat] :
( ( ord_less_eq_set_nat @ A @ B8 )
& ( ord_less_eq_set_nat @ B8 @ C2 )
& ( ( finite_card_nat @ B8 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_947_exists__subset__between,axiom,
! [A: set_literal,N2: nat,C2: set_literal] :
( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ ( finite_card_literal @ C2 ) )
=> ( ( ord_le7307670543136651348iteral @ A @ C2 )
=> ( ( finite5847741373460823677iteral @ C2 )
=> ? [B8: set_literal] :
( ( ord_le7307670543136651348iteral @ A @ B8 )
& ( ord_le7307670543136651348iteral @ B8 @ C2 )
& ( ( finite_card_literal @ B8 )
= N2 ) ) ) ) ) ) ).
% exists_subset_between
thf(fact_948_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_Product_unit] :
( ( ord_less_eq_nat @ N2 @ ( finite410649719033368117t_unit @ S ) )
=> ~ ! [T4: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ T4 @ S )
=> ( ( ( finite410649719033368117t_unit @ T4 )
= N2 )
=> ~ ( finite4290736615968046902t_unit @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_949_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_nat] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_nat @ S ) )
=> ~ ! [T4: set_nat] :
( ( ord_less_eq_set_nat @ T4 @ S )
=> ( ( ( finite_card_nat @ T4 )
= N2 )
=> ~ ( finite_finite_nat @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_950_obtain__subset__with__card__n,axiom,
! [N2: nat,S: set_literal] :
( ( ord_less_eq_nat @ N2 @ ( finite_card_literal @ S ) )
=> ~ ! [T4: set_literal] :
( ( ord_le7307670543136651348iteral @ T4 @ S )
=> ( ( ( finite_card_literal @ T4 )
= N2 )
=> ~ ( finite5847741373460823677iteral @ T4 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_951_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_Product_unit,C2: nat] :
( ! [G3: set_Product_unit] :
( ( ord_le3507040750410214029t_unit @ G3 @ F2 )
=> ( ( finite4290736615968046902t_unit @ G3 )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ G3 ) @ C2 ) ) )
=> ( ( finite4290736615968046902t_unit @ F2 )
& ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_952_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_nat,C2: nat] :
( ! [G3: set_nat] :
( ( ord_less_eq_set_nat @ G3 @ F2 )
=> ( ( finite_finite_nat @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_nat @ G3 ) @ C2 ) ) )
=> ( ( finite_finite_nat @ F2 )
& ( ord_less_eq_nat @ ( finite_card_nat @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_953_finite__if__finite__subsets__card__bdd,axiom,
! [F2: set_literal,C2: nat] :
( ! [G3: set_literal] :
( ( ord_le7307670543136651348iteral @ G3 @ F2 )
=> ( ( finite5847741373460823677iteral @ G3 )
=> ( ord_less_eq_nat @ ( finite_card_literal @ G3 ) @ C2 ) ) )
=> ( ( finite5847741373460823677iteral @ F2 )
& ( ord_less_eq_nat @ ( finite_card_literal @ F2 ) @ C2 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_954_size__neq__size__imp__neq,axiom,
! [X: list_Sum_sum_a_nat,Y: list_Sum_sum_a_nat] :
( ( ( size_s5283204784079214577_a_nat @ X )
!= ( size_s5283204784079214577_a_nat @ Y ) )
=> ( X != Y ) ) ).
% size_neq_size_imp_neq
thf(fact_955_le__refl,axiom,
! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).
% le_refl
thf(fact_956_le__trans,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ J @ K )
=> ( ord_less_eq_nat @ I @ K ) ) ) ).
% le_trans
thf(fact_957_eq__imp__le,axiom,
! [M4: nat,N2: nat] :
( ( M4 = N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% eq_imp_le
thf(fact_958_le__antisym,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( ord_less_eq_nat @ N2 @ M4 )
=> ( M4 = N2 ) ) ) ).
% le_antisym
thf(fact_959_nat__le__linear,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
| ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% nat_le_linear
thf(fact_960_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y4: nat] :
( ( P @ Y4 )
=> ( ord_less_eq_nat @ Y4 @ B2 ) )
=> ? [X2: nat] :
( ( P @ X2 )
& ! [Y7: nat] :
( ( P @ Y7 )
=> ( ord_less_eq_nat @ Y7 @ X2 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_961_finite__option__UNIV,axiom,
( ( finite5523153139673422903on_nat @ top_to8920198386146353926on_nat )
= ( finite_finite_nat @ top_top_set_nat ) ) ).
% finite_option_UNIV
thf(fact_962_finite__option__UNIV,axiom,
( ( finite5071707688241699267iteral @ top_to8248435444729185354iteral )
= ( finite5847741373460823677iteral @ top_top_set_literal ) ) ).
% finite_option_UNIV
thf(fact_963_finite__option__UNIV,axiom,
( ( finite1445617369574913404t_unit @ top_to2690860209552263555t_unit )
= ( finite4290736615968046902t_unit @ top_to1996260823553986621t_unit ) ) ).
% finite_option_UNIV
thf(fact_964_vimage__subsetD,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_nat @ F @ B ) @ A )
=> ( ord_less_eq_set_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_965_vimage__subsetD,axiom,
! [F: nat > literal,B: set_literal,A: set_nat] :
( ( ( image_nat_literal @ F @ top_top_set_nat )
= top_top_set_literal )
=> ( ( ord_less_eq_set_nat @ ( vimage_nat_literal @ F @ B ) @ A )
=> ( ord_le7307670543136651348iteral @ B @ ( image_nat_literal @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_966_vimage__subsetD,axiom,
! [F: nat > product_unit,B: set_Product_unit,A: set_nat] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ ( vimage4884490618288580032t_unit @ F @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_8730104196221521654t_unit @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_967_vimage__subsetD,axiom,
! [F: literal > nat,B: set_nat,A: set_literal] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
=> ( ( ord_le7307670543136651348iteral @ ( vimage_literal_nat @ F @ B ) @ A )
=> ( ord_less_eq_set_nat @ B @ ( image_literal_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_968_vimage__subsetD,axiom,
! [F: literal > literal,B: set_literal,A: set_literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ( ( ord_le7307670543136651348iteral @ ( vimage8238609917233974331iteral @ F @ B ) @ A )
=> ( ord_le7307670543136651348iteral @ B @ ( image_8195128725298311301iteral @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_969_vimage__subsetD,axiom,
! [F: literal > product_unit,B: set_Product_unit,A: set_literal] :
( ( ( image_6787854153545327934t_unit @ F @ top_top_set_literal )
= top_to1996260823553986621t_unit )
=> ( ( ord_le7307670543136651348iteral @ ( vimage6047873233414524916t_unit @ F @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_6787854153545327934t_unit @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_970_vimage__subsetD,axiom,
! [F: product_unit > nat,B: set_nat,A: set_Product_unit] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( ord_le3507040750410214029t_unit @ ( vimage6253328473476588386it_nat @ F @ B ) @ A )
=> ( ord_less_eq_set_nat @ B @ ( image_875570014554754200it_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_971_vimage__subsetD,axiom,
! [F: product_unit > literal,B: set_literal,A: set_Product_unit] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
=> ( ( ord_le3507040750410214029t_unit @ ( vimage5137003825767189442iteral @ F @ B ) @ A )
=> ( ord_le7307670543136651348iteral @ B @ ( image_5876984745897992460iteral @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_972_vimage__subsetD,axiom,
! [F: product_unit > product_unit,B: set_Product_unit,A: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( ord_le3507040750410214029t_unit @ ( vimage7995052115951654139t_unit @ F @ B ) @ A )
=> ( ord_le3507040750410214029t_unit @ B @ ( image_405062704495631173t_unit @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_973_vimage__subsetD,axiom,
! [F: a > sum_sum_a_nat,B: set_Sum_sum_a_nat,A: set_a] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
=> ( ( ord_less_eq_set_a @ ( vimage4607795094749595324_a_nat @ F @ B ) @ A )
=> ( ord_le1325389633284124927_a_nat @ B @ ( image_7873763678140191238_a_nat @ F @ A ) ) ) ) ).
% vimage_subsetD
thf(fact_974_surj__image__vimage__eq,axiom,
! [F: nat > nat,A: set_nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ( ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_975_surj__image__vimage__eq,axiom,
! [F: nat > literal,A: set_literal] :
( ( ( image_nat_literal @ F @ top_top_set_nat )
= top_top_set_literal )
=> ( ( image_nat_literal @ F @ ( vimage_nat_literal @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_976_surj__image__vimage__eq,axiom,
! [F: nat > product_unit,A: set_Product_unit] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ( ( image_8730104196221521654t_unit @ F @ ( vimage4884490618288580032t_unit @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_977_surj__image__vimage__eq,axiom,
! [F: literal > nat,A: set_nat] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
=> ( ( image_literal_nat @ F @ ( vimage_literal_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_978_surj__image__vimage__eq,axiom,
! [F: literal > literal,A: set_literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ( ( image_8195128725298311301iteral @ F @ ( vimage8238609917233974331iteral @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_979_surj__image__vimage__eq,axiom,
! [F: literal > product_unit,A: set_Product_unit] :
( ( ( image_6787854153545327934t_unit @ F @ top_top_set_literal )
= top_to1996260823553986621t_unit )
=> ( ( image_6787854153545327934t_unit @ F @ ( vimage6047873233414524916t_unit @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_980_surj__image__vimage__eq,axiom,
! [F: product_unit > nat,A: set_nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ( ( image_875570014554754200it_nat @ F @ ( vimage6253328473476588386it_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_981_surj__image__vimage__eq,axiom,
! [F: product_unit > literal,A: set_literal] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
=> ( ( image_5876984745897992460iteral @ F @ ( vimage5137003825767189442iteral @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_982_surj__image__vimage__eq,axiom,
! [F: product_unit > product_unit,A: set_Product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ( ( image_405062704495631173t_unit @ F @ ( vimage7995052115951654139t_unit @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_983_surj__image__vimage__eq,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
=> ( ( image_7873763678140191238_a_nat @ F @ ( vimage4607795094749595324_a_nat @ F @ A ) )
= A ) ) ).
% surj_image_vimage_eq
thf(fact_984_iso__tuple__UNIV__I,axiom,
! [X: nat] : ( member_nat @ X @ top_top_set_nat ) ).
% iso_tuple_UNIV_I
thf(fact_985_iso__tuple__UNIV__I,axiom,
! [X: literal] : ( member_literal @ X @ top_top_set_literal ) ).
% iso_tuple_UNIV_I
thf(fact_986_iso__tuple__UNIV__I,axiom,
! [X: product_unit] : ( member_Product_unit @ X @ top_to1996260823553986621t_unit ) ).
% iso_tuple_UNIV_I
thf(fact_987_infinite__UNIV,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV
thf(fact_988_surjD,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_989_surjD,axiom,
! [F: nat > literal,Y: literal] :
( ( ( image_nat_literal @ F @ top_top_set_nat )
= top_top_set_literal )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_990_surjD,axiom,
! [F: nat > product_unit,Y: product_unit] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ? [X2: nat] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_991_surjD,axiom,
! [F: literal > nat,Y: nat] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_992_surjD,axiom,
! [F: literal > literal,Y: literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_993_surjD,axiom,
! [F: literal > product_unit,Y: product_unit] :
( ( ( image_6787854153545327934t_unit @ F @ top_top_set_literal )
= top_to1996260823553986621t_unit )
=> ? [X2: literal] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_994_surjD,axiom,
! [F: product_unit > nat,Y: nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_995_surjD,axiom,
! [F: product_unit > literal,Y: literal] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_996_surjD,axiom,
! [F: product_unit > product_unit,Y: product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ? [X2: product_unit] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_997_surjD,axiom,
! [F: a > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
=> ? [X2: a] :
( Y
= ( F @ X2 ) ) ) ).
% surjD
thf(fact_998_infinite__UNIV__nat,axiom,
~ ( finite_finite_nat @ top_top_set_nat ) ).
% infinite_UNIV_nat
thf(fact_999_top__set__def,axiom,
( top_top_set_nat
= ( collect_nat @ top_top_nat_o ) ) ).
% top_set_def
thf(fact_1000_top__set__def,axiom,
( top_top_set_literal
= ( collect_literal @ top_top_literal_o ) ) ).
% top_set_def
thf(fact_1001_top__set__def,axiom,
( top_to1996260823553986621t_unit
= ( collect_Product_unit @ top_to2465898995584390880unit_o ) ) ).
% top_set_def
thf(fact_1002_arb__element,axiom,
! [Y5: set_nat] :
( ( finite_finite_nat @ Y5 )
=> ? [X2: nat] :
~ ( member_nat @ X2 @ Y5 ) ) ).
% arb_element
thf(fact_1003_surj__def,axiom,
! [F: nat > nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X4: nat] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1004_surj__def,axiom,
! [F: nat > literal] :
( ( ( image_nat_literal @ F @ top_top_set_nat )
= top_top_set_literal )
= ( ! [Y3: literal] :
? [X4: nat] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1005_surj__def,axiom,
! [F: nat > product_unit] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
= ( ! [Y3: product_unit] :
? [X4: nat] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1006_surj__def,axiom,
! [F: literal > nat] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X4: literal] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1007_surj__def,axiom,
! [F: literal > literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
= ( ! [Y3: literal] :
? [X4: literal] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1008_surj__def,axiom,
! [F: literal > product_unit] :
( ( ( image_6787854153545327934t_unit @ F @ top_top_set_literal )
= top_to1996260823553986621t_unit )
= ( ! [Y3: product_unit] :
? [X4: literal] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1009_surj__def,axiom,
! [F: product_unit > nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
= ( ! [Y3: nat] :
? [X4: product_unit] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1010_surj__def,axiom,
! [F: product_unit > literal] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
= ( ! [Y3: literal] :
? [X4: product_unit] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1011_surj__def,axiom,
! [F: product_unit > product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
= ( ! [Y3: product_unit] :
? [X4: product_unit] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1012_surj__def,axiom,
! [F: a > sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
= ( ! [Y3: sum_sum_a_nat] :
? [X4: a] :
( Y3
= ( F @ X4 ) ) ) ) ).
% surj_def
thf(fact_1013_surjI,axiom,
! [G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_nat @ G @ top_top_set_nat )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1014_surjI,axiom,
! [G: nat > literal,F: literal > nat] :
( ! [X2: literal] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_nat_literal @ G @ top_top_set_nat )
= top_top_set_literal ) ) ).
% surjI
thf(fact_1015_surjI,axiom,
! [G: nat > product_unit,F: product_unit > nat] :
( ! [X2: product_unit] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_8730104196221521654t_unit @ G @ top_top_set_nat )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_1016_surjI,axiom,
! [G: literal > nat,F: nat > literal] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_literal_nat @ G @ top_top_set_literal )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1017_surjI,axiom,
! [G: literal > literal,F: literal > literal] :
( ! [X2: literal] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_8195128725298311301iteral @ G @ top_top_set_literal )
= top_top_set_literal ) ) ).
% surjI
thf(fact_1018_surjI,axiom,
! [G: literal > product_unit,F: product_unit > literal] :
( ! [X2: product_unit] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_6787854153545327934t_unit @ G @ top_top_set_literal )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_1019_surjI,axiom,
! [G: product_unit > nat,F: nat > product_unit] :
( ! [X2: nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_875570014554754200it_nat @ G @ top_to1996260823553986621t_unit )
= top_top_set_nat ) ) ).
% surjI
thf(fact_1020_surjI,axiom,
! [G: product_unit > literal,F: literal > product_unit] :
( ! [X2: literal] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_5876984745897992460iteral @ G @ top_to1996260823553986621t_unit )
= top_top_set_literal ) ) ).
% surjI
thf(fact_1021_surjI,axiom,
! [G: product_unit > product_unit,F: product_unit > product_unit] :
( ! [X2: product_unit] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_405062704495631173t_unit @ G @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit ) ) ).
% surjI
thf(fact_1022_surjI,axiom,
! [G: a > sum_sum_a_nat,F: sum_sum_a_nat > a] :
( ! [X2: sum_sum_a_nat] :
( ( G @ ( F @ X2 ) )
= X2 )
=> ( ( image_7873763678140191238_a_nat @ G @ top_top_set_a )
= top_to795618464972521135_a_nat ) ) ).
% surjI
thf(fact_1023_surjE,axiom,
! [F: nat > nat,Y: nat] :
( ( ( image_nat_nat @ F @ top_top_set_nat )
= top_top_set_nat )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1024_surjE,axiom,
! [F: nat > literal,Y: literal] :
( ( ( image_nat_literal @ F @ top_top_set_nat )
= top_top_set_literal )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1025_surjE,axiom,
! [F: nat > product_unit,Y: product_unit] :
( ( ( image_8730104196221521654t_unit @ F @ top_top_set_nat )
= top_to1996260823553986621t_unit )
=> ~ ! [X2: nat] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1026_surjE,axiom,
! [F: literal > nat,Y: nat] :
( ( ( image_literal_nat @ F @ top_top_set_literal )
= top_top_set_nat )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1027_surjE,axiom,
! [F: literal > literal,Y: literal] :
( ( ( image_8195128725298311301iteral @ F @ top_top_set_literal )
= top_top_set_literal )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1028_surjE,axiom,
! [F: literal > product_unit,Y: product_unit] :
( ( ( image_6787854153545327934t_unit @ F @ top_top_set_literal )
= top_to1996260823553986621t_unit )
=> ~ ! [X2: literal] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1029_surjE,axiom,
! [F: product_unit > nat,Y: nat] :
( ( ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit )
= top_top_set_nat )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1030_surjE,axiom,
! [F: product_unit > literal,Y: literal] :
( ( ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit )
= top_top_set_literal )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1031_surjE,axiom,
! [F: product_unit > product_unit,Y: product_unit] :
( ( ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit )
= top_to1996260823553986621t_unit )
=> ~ ! [X2: product_unit] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1032_surjE,axiom,
! [F: a > sum_sum_a_nat,Y: sum_sum_a_nat] :
( ( ( image_7873763678140191238_a_nat @ F @ top_top_set_a )
= top_to795618464972521135_a_nat )
=> ~ ! [X2: a] :
( Y
!= ( F @ X2 ) ) ) ).
% surjE
thf(fact_1033_card__vimage__inj__on__le,axiom,
! [F: literal > product_unit,D2: set_literal,A: set_Product_unit] :
( ( inj_on4267703138521210538t_unit @ F @ D2 )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( inf_inf_set_literal @ ( vimage6047873233414524916t_unit @ F @ A ) @ D2 ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1034_card__vimage__inj__on__le,axiom,
! [F: product_unit > product_unit,D2: set_Product_unit,A: set_Product_unit] :
( ( inj_on8151373323710067377t_unit @ F @ D2 )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ ( vimage7995052115951654139t_unit @ F @ A ) @ D2 ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1035_card__vimage__inj__on__le,axiom,
! [F: literal > nat,D2: set_literal,A: set_nat] :
( ( inj_on_literal_nat @ F @ D2 )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( inf_inf_set_literal @ ( vimage_literal_nat @ F @ A ) @ D2 ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1036_card__vimage__inj__on__le,axiom,
! [F: product_unit > nat,D2: set_Product_unit,A: set_nat] :
( ( inj_on8430439091780834860it_nat @ F @ D2 )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ ( vimage6253328473476588386it_nat @ F @ A ) @ D2 ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1037_card__vimage__inj__on__le,axiom,
! [F: literal > literal,D2: set_literal,A: set_literal] :
( ( inj_on602069361295035377iteral @ F @ D2 )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_literal @ ( inf_inf_set_literal @ ( vimage8238609917233974331iteral @ F @ A ) @ D2 ) ) @ ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1038_card__vimage__inj__on__le,axiom,
! [F: product_unit > literal,D2: set_Product_unit,A: set_literal] :
( ( inj_on3356833730873875064iteral @ F @ D2 )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ ( inf_in4660618365625256667t_unit @ ( vimage5137003825767189442iteral @ F @ A ) @ D2 ) ) @ ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1039_card__vimage__inj__on__le,axiom,
! [F: nat > product_unit,D2: set_nat,A: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F @ D2 )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage4884490618288580032t_unit @ F @ A ) @ D2 ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1040_card__vimage__inj__on__le,axiom,
! [F: nat > nat,D2: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ D2 )
=> ( ( finite_finite_nat @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage_nat_nat @ F @ A ) @ D2 ) ) @ ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1041_card__vimage__inj__on__le,axiom,
! [F: nat > literal,D2: set_nat,A: set_literal] :
( ( inj_on_nat_literal @ F @ D2 )
=> ( ( finite5847741373460823677iteral @ A )
=> ( ord_less_eq_nat @ ( finite_card_nat @ ( inf_inf_set_nat @ ( vimage_nat_literal @ F @ A ) @ D2 ) ) @ ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1042_card__vimage__inj__on__le,axiom,
! [F: sum_sum_a_nat > product_unit,D2: set_Sum_sum_a_nat,A: set_Product_unit] :
( ( inj_on4315315934791609919t_unit @ F @ D2 )
=> ( ( finite4290736615968046902t_unit @ A )
=> ( ord_less_eq_nat @ ( finite6080979521523705895_a_nat @ ( inf_in7084830621192376909_a_nat @ ( vimage2477388680732255113t_unit @ F @ A ) @ D2 ) ) @ ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_1043_card__vimage__inj,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ord_less_eq_set_nat @ A @ ( image_nat_nat @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1044_card__vimage__inj,axiom,
! [F: nat > literal,A: set_literal] :
( ( inj_on_nat_literal @ F @ top_top_set_nat )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_nat_literal @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage_nat_literal @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1045_card__vimage__inj,axiom,
! [F: nat > product_unit,A: set_Product_unit] :
( ( inj_on7061601236592826506t_unit @ F @ top_top_set_nat )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_8730104196221521654t_unit @ F @ top_top_set_nat ) )
=> ( ( finite_card_nat @ ( vimage4884490618288580032t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1046_card__vimage__inj,axiom,
! [F: literal > nat,A: set_nat] :
( ( inj_on_literal_nat @ F @ top_top_set_literal )
=> ( ( ord_less_eq_set_nat @ A @ ( image_literal_nat @ F @ top_top_set_literal ) )
=> ( ( finite_card_literal @ ( vimage_literal_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1047_card__vimage__inj,axiom,
! [F: literal > literal,A: set_literal] :
( ( inj_on602069361295035377iteral @ F @ top_top_set_literal )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_8195128725298311301iteral @ F @ top_top_set_literal ) )
=> ( ( finite_card_literal @ ( vimage8238609917233974331iteral @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1048_card__vimage__inj,axiom,
! [F: literal > product_unit,A: set_Product_unit] :
( ( inj_on4267703138521210538t_unit @ F @ top_top_set_literal )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_6787854153545327934t_unit @ F @ top_top_set_literal ) )
=> ( ( finite_card_literal @ ( vimage6047873233414524916t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1049_card__vimage__inj,axiom,
! [F: product_unit > nat,A: set_nat] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( ord_less_eq_set_nat @ A @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) )
=> ( ( finite410649719033368117t_unit @ ( vimage6253328473476588386it_nat @ F @ A ) )
= ( finite_card_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1050_card__vimage__inj,axiom,
! [F: product_unit > literal,A: set_literal] :
( ( inj_on3356833730873875064iteral @ F @ top_to1996260823553986621t_unit )
=> ( ( ord_le7307670543136651348iteral @ A @ ( image_5876984745897992460iteral @ F @ top_to1996260823553986621t_unit ) )
=> ( ( finite410649719033368117t_unit @ ( vimage5137003825767189442iteral @ F @ A ) )
= ( finite_card_literal @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1051_card__vimage__inj,axiom,
! [F: product_unit > product_unit,A: set_Product_unit] :
( ( inj_on8151373323710067377t_unit @ F @ top_to1996260823553986621t_unit )
=> ( ( ord_le3507040750410214029t_unit @ A @ ( image_405062704495631173t_unit @ F @ top_to1996260823553986621t_unit ) )
=> ( ( finite410649719033368117t_unit @ ( vimage7995052115951654139t_unit @ F @ A ) )
= ( finite410649719033368117t_unit @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1052_card__vimage__inj,axiom,
! [F: a > sum_sum_a_nat,A: set_Sum_sum_a_nat] :
( ( inj_on3092108040974901106_a_nat @ F @ top_top_set_a )
=> ( ( ord_le1325389633284124927_a_nat @ A @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) )
=> ( ( finite_card_a @ ( vimage4607795094749595324_a_nat @ F @ A ) )
= ( finite6080979521523705895_a_nat @ A ) ) ) ) ).
% card_vimage_inj
thf(fact_1053_card__le__inj,axiom,
! [A: set_Product_unit,B: set_Product_unit] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ? [F3: product_unit > product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_405062704495631173t_unit @ F3 @ A ) @ B )
& ( inj_on8151373323710067377t_unit @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1054_card__le__inj,axiom,
! [A: set_Product_unit,B: set_nat] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_nat @ B ) )
=> ? [F3: product_unit > nat] :
( ( ord_less_eq_set_nat @ ( image_875570014554754200it_nat @ F3 @ A ) @ B )
& ( inj_on8430439091780834860it_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1055_card__le__inj,axiom,
! [A: set_Product_unit,B: set_literal] :
( ( finite4290736615968046902t_unit @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_eq_nat @ ( finite410649719033368117t_unit @ A ) @ ( finite_card_literal @ B ) )
=> ? [F3: product_unit > literal] :
( ( ord_le7307670543136651348iteral @ ( image_5876984745897992460iteral @ F3 @ A ) @ B )
& ( inj_on3356833730873875064iteral @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1056_card__le__inj,axiom,
! [A: set_nat,B: set_Product_unit] :
( ( finite_finite_nat @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ? [F3: nat > product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_8730104196221521654t_unit @ F3 @ A ) @ B )
& ( inj_on7061601236592826506t_unit @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1057_card__le__inj,axiom,
! [A: set_nat,B: set_nat] :
( ( finite_finite_nat @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_nat @ B ) )
=> ? [F3: nat > nat] :
( ( ord_less_eq_set_nat @ ( image_nat_nat @ F3 @ A ) @ B )
& ( inj_on_nat_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1058_card__le__inj,axiom,
! [A: set_nat,B: set_literal] :
( ( finite_finite_nat @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_nat @ A ) @ ( finite_card_literal @ B ) )
=> ? [F3: nat > literal] :
( ( ord_le7307670543136651348iteral @ ( image_nat_literal @ F3 @ A ) @ B )
& ( inj_on_nat_literal @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1059_card__le__inj,axiom,
! [A: set_literal,B: set_Product_unit] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite4290736615968046902t_unit @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite410649719033368117t_unit @ B ) )
=> ? [F3: literal > product_unit] :
( ( ord_le3507040750410214029t_unit @ ( image_6787854153545327934t_unit @ F3 @ A ) @ B )
& ( inj_on4267703138521210538t_unit @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1060_card__le__inj,axiom,
! [A: set_literal,B: set_nat] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite_finite_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_nat @ B ) )
=> ? [F3: literal > nat] :
( ( ord_less_eq_set_nat @ ( image_literal_nat @ F3 @ A ) @ B )
& ( inj_on_literal_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1061_card__le__inj,axiom,
! [A: set_literal,B: set_literal] :
( ( finite5847741373460823677iteral @ A )
=> ( ( finite5847741373460823677iteral @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_literal @ A ) @ ( finite_card_literal @ B ) )
=> ? [F3: literal > literal] :
( ( ord_le7307670543136651348iteral @ ( image_8195128725298311301iteral @ F3 @ A ) @ B )
& ( inj_on602069361295035377iteral @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1062_card__le__inj,axiom,
! [A: set_a,B: set_Sum_sum_a_nat] :
( ( finite_finite_a @ A )
=> ( ( finite502105017643426984_a_nat @ B )
=> ( ( ord_less_eq_nat @ ( finite_card_a @ A ) @ ( finite6080979521523705895_a_nat @ B ) )
=> ? [F3: a > sum_sum_a_nat] :
( ( ord_le1325389633284124927_a_nat @ ( image_7873763678140191238_a_nat @ F3 @ A ) @ B )
& ( inj_on3092108040974901106_a_nat @ F3 @ A ) ) ) ) ) ).
% card_le_inj
thf(fact_1063_inj__mapI,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat ) ) ).
% inj_mapI
thf(fact_1064_inj__map,axiom,
! [F: nat > nat] :
( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
= ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_map
thf(fact_1065_inj__map__eq__map,axiom,
! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( map_nat_nat @ F @ Xs )
= ( map_nat_nat @ F @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_1066_inj__on__subset,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( ord_less_eq_set_nat @ B @ A )
=> ( inj_on_nat_nat @ F @ B ) ) ) ).
% inj_on_subset
thf(fact_1067_subset__inj__on,axiom,
! [F: nat > nat,B: set_nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( inj_on_nat_nat @ F @ A ) ) ) ).
% subset_inj_on
thf(fact_1068_inj__mapD,axiom,
! [F: nat > nat] :
( ( inj_on3049792774292151987st_nat @ ( map_nat_nat @ F ) @ top_top_set_list_nat )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% inj_mapD
thf(fact_1069_injD,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
=> ( X = Y ) ) ) ).
% injD
thf(fact_1070_injI,axiom,
! [F: nat > nat] :
( ! [X2: nat,Y4: nat] :
( ( ( F @ X2 )
= ( F @ Y4 ) )
=> ( X2 = Y4 ) )
=> ( inj_on_nat_nat @ F @ top_top_set_nat ) ) ).
% injI
thf(fact_1071_inj__eq,axiom,
! [F: nat > nat,X: nat,Y: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( F @ X )
= ( F @ Y ) )
= ( X = Y ) ) ) ).
% inj_eq
thf(fact_1072_inj__def,axiom,
! [F: nat > nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
= ( ! [X4: nat,Y3: nat] :
( ( ( F @ X4 )
= ( F @ Y3 ) )
=> ( X4 = Y3 ) ) ) ) ).
% inj_def
thf(fact_1073_inj__on__Int,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( ( inj_on_nat_nat @ F @ A )
| ( inj_on_nat_nat @ F @ B ) )
=> ( inj_on_nat_nat @ F @ ( inf_inf_set_nat @ A @ B ) ) ) ).
% inj_on_Int
thf(fact_1074_inj__on__image__iff,axiom,
! [A: set_nat,G: nat > nat,F: nat > nat] :
( ! [X2: nat] :
( ( member_nat @ X2 @ A )
=> ! [Xa2: nat] :
( ( member_nat @ Xa2 @ A )
=> ( ( ( G @ ( F @ X2 ) )
= ( G @ ( F @ Xa2 ) ) )
= ( ( G @ X2 )
= ( G @ Xa2 ) ) ) ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( ( inj_on_nat_nat @ G @ ( image_nat_nat @ F @ A ) )
= ( inj_on_nat_nat @ G @ A ) ) ) ) ).
% inj_on_image_iff
thf(fact_1075_inj__Inl,axiom,
! [A: set_a] : ( inj_on3092108040974901106_a_nat @ sum_Inl_a_nat @ A ) ).
% inj_Inl
thf(fact_1076_inj__Inr,axiom,
! [A: set_nat] : ( inj_on4348161877322679292_a_nat @ sum_Inr_nat_a @ A ) ).
% inj_Inr
thf(fact_1077_range__ex1__eq,axiom,
! [F: a > sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( inj_on3092108040974901106_a_nat @ F @ top_top_set_a )
=> ( ( member_Sum_sum_a_nat @ B2 @ ( image_7873763678140191238_a_nat @ F @ top_top_set_a ) )
= ( ? [X4: a] :
( ( B2
= ( F @ X4 ) )
& ! [Y3: a] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1078_range__ex1__eq,axiom,
! [F: nat > sum_sum_a_nat,B2: sum_sum_a_nat] :
( ( inj_on4348161877322679292_a_nat @ F @ top_top_set_nat )
=> ( ( member_Sum_sum_a_nat @ B2 @ ( image_7293268710728258664_a_nat @ F @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B2
= ( F @ X4 ) )
& ! [Y3: nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1079_range__ex1__eq,axiom,
! [F: nat > nat,B2: nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ B2 @ ( image_nat_nat @ F @ top_top_set_nat ) )
= ( ? [X4: nat] :
( ( B2
= ( F @ X4 ) )
& ! [Y3: nat] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1080_range__ex1__eq,axiom,
! [F: literal > nat,B2: nat] :
( ( inj_on_literal_nat @ F @ top_top_set_literal )
=> ( ( member_nat @ B2 @ ( image_literal_nat @ F @ top_top_set_literal ) )
= ( ? [X4: literal] :
( ( B2
= ( F @ X4 ) )
& ! [Y3: literal] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1081_range__ex1__eq,axiom,
! [F: product_unit > nat,B2: nat] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( member_nat @ B2 @ ( image_875570014554754200it_nat @ F @ top_to1996260823553986621t_unit ) )
= ( ? [X4: product_unit] :
( ( B2
= ( F @ X4 ) )
& ! [Y3: product_unit] :
( ( B2
= ( F @ Y3 ) )
=> ( Y3 = X4 ) ) ) ) ) ) ).
% range_ex1_eq
thf(fact_1082_inj__image__eq__iff,axiom,
! [F: a > sum_sum_a_nat,A: set_a,B: set_a] :
( ( inj_on3092108040974901106_a_nat @ F @ top_top_set_a )
=> ( ( ( image_7873763678140191238_a_nat @ F @ A )
= ( image_7873763678140191238_a_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_1083_inj__image__eq__iff,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat,B: set_nat] :
( ( inj_on4348161877322679292_a_nat @ F @ top_top_set_nat )
=> ( ( ( image_7293268710728258664_a_nat @ F @ A )
= ( image_7293268710728258664_a_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_1084_inj__image__eq__iff,axiom,
! [F: nat > nat,A: set_nat,B: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( ( image_nat_nat @ F @ A )
= ( image_nat_nat @ F @ B ) )
= ( A = B ) ) ) ).
% inj_image_eq_iff
thf(fact_1085_inj__image__mem__iff,axiom,
! [F: a > sum_sum_a_nat,A2: a,A: set_a] :
( ( inj_on3092108040974901106_a_nat @ F @ top_top_set_a )
=> ( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( member_a @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1086_inj__image__mem__iff,axiom,
! [F: nat > sum_sum_a_nat,A2: nat,A: set_nat] :
( ( inj_on4348161877322679292_a_nat @ F @ top_top_set_nat )
=> ( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1087_inj__image__mem__iff,axiom,
! [F: nat > nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ top_top_set_nat )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1088_inj__image__mem__iff,axiom,
! [F: literal > nat,A2: literal,A: set_literal] :
( ( inj_on_literal_nat @ F @ top_top_set_literal )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_literal_nat @ F @ A ) )
= ( member_literal @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1089_inj__image__mem__iff,axiom,
! [F: product_unit > nat,A2: product_unit,A: set_Product_unit] :
( ( inj_on8430439091780834860it_nat @ F @ top_to1996260823553986621t_unit )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_875570014554754200it_nat @ F @ A ) )
= ( member_Product_unit @ A2 @ A ) ) ) ).
% inj_image_mem_iff
thf(fact_1090_finite__image__iff,axiom,
! [F: a > sum_sum_a_nat,A: set_a] :
( ( inj_on3092108040974901106_a_nat @ F @ A )
=> ( ( finite502105017643426984_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) )
= ( finite_finite_a @ A ) ) ) ).
% finite_image_iff
thf(fact_1091_finite__image__iff,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat] :
( ( inj_on4348161877322679292_a_nat @ F @ A )
=> ( ( finite502105017643426984_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1092_finite__image__iff,axiom,
! [F: nat > nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1093_finite__image__iff,axiom,
! [F: literal > nat,A: set_literal] :
( ( inj_on_literal_nat @ F @ A )
=> ( ( finite_finite_nat @ ( image_literal_nat @ F @ A ) )
= ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_image_iff
thf(fact_1094_finite__image__iff,axiom,
! [F: nat > literal,A: set_nat] :
( ( inj_on_nat_literal @ F @ A )
=> ( ( finite5847741373460823677iteral @ ( image_nat_literal @ F @ A ) )
= ( finite_finite_nat @ A ) ) ) ).
% finite_image_iff
thf(fact_1095_finite__image__iff,axiom,
! [F: literal > literal,A: set_literal] :
( ( inj_on602069361295035377iteral @ F @ A )
=> ( ( finite5847741373460823677iteral @ ( image_8195128725298311301iteral @ F @ A ) )
= ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_image_iff
thf(fact_1096_finite__imageD,axiom,
! [F: a > sum_sum_a_nat,A: set_a] :
( ( finite502105017643426984_a_nat @ ( image_7873763678140191238_a_nat @ F @ A ) )
=> ( ( inj_on3092108040974901106_a_nat @ F @ A )
=> ( finite_finite_a @ A ) ) ) ).
% finite_imageD
thf(fact_1097_finite__imageD,axiom,
! [F: nat > sum_sum_a_nat,A: set_nat] :
( ( finite502105017643426984_a_nat @ ( image_7293268710728258664_a_nat @ F @ A ) )
=> ( ( inj_on4348161877322679292_a_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1098_finite__imageD,axiom,
! [F: nat > nat,A: set_nat] :
( ( finite_finite_nat @ ( image_nat_nat @ F @ A ) )
=> ( ( inj_on_nat_nat @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1099_finite__imageD,axiom,
! [F: literal > nat,A: set_literal] :
( ( finite_finite_nat @ ( image_literal_nat @ F @ A ) )
=> ( ( inj_on_literal_nat @ F @ A )
=> ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_imageD
thf(fact_1100_finite__imageD,axiom,
! [F: nat > literal,A: set_nat] :
( ( finite5847741373460823677iteral @ ( image_nat_literal @ F @ A ) )
=> ( ( inj_on_nat_literal @ F @ A )
=> ( finite_finite_nat @ A ) ) ) ).
% finite_imageD
thf(fact_1101_finite__imageD,axiom,
! [F: literal > literal,A: set_literal] :
( ( finite5847741373460823677iteral @ ( image_8195128725298311301iteral @ F @ A ) )
=> ( ( inj_on602069361295035377iteral @ F @ A )
=> ( finite5847741373460823677iteral @ A ) ) ) ).
% finite_imageD
thf(fact_1102_inj__on__image__mem__iff,axiom,
! [F: nat > sum_sum_a_nat,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on4348161877322679292_a_nat @ F @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_Sum_sum_a_nat @ ( F @ A2 ) @ ( image_7293268710728258664_a_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1103_inj__on__image__mem__iff,axiom,
! [F: nat > nat,B: set_nat,A2: nat,A: set_nat] :
( ( inj_on_nat_nat @ F @ B )
=> ( ( member_nat @ A2 @ B )
=> ( ( ord_less_eq_set_nat @ A @ B )
=> ( ( member_nat @ ( F @ A2 ) @ ( image_nat_nat @ F @ A ) )
= ( member_nat @ A2 @ A ) ) ) ) ) ).
% inj_on_image_mem_iff
thf(fact_1104_diff__diff__cancel,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ N2 )
=> ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_1105_min__diff,axiom,
! [M4: nat,I: nat,N2: nat] :
( ( ord_min_nat @ ( minus_minus_nat @ M4 @ I ) @ ( minus_minus_nat @ N2 @ I ) )
= ( minus_minus_nat @ ( ord_min_nat @ M4 @ N2 ) @ I ) ) ).
% min_diff
thf(fact_1106_diff__le__mono2,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M4 ) ) ) ).
% diff_le_mono2
thf(fact_1107_le__diff__iff_H,axiom,
! [A2: nat,C: nat,B2: nat] :
( ( ord_less_eq_nat @ A2 @ C )
=> ( ( ord_less_eq_nat @ B2 @ C )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A2 ) @ ( minus_minus_nat @ C @ B2 ) )
= ( ord_less_eq_nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1108_diff__le__self,axiom,
! [M4: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ N2 ) @ M4 ) ).
% diff_le_self
thf(fact_1109_diff__le__mono,axiom,
! [M4: nat,N2: nat,L: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).
% diff_le_mono
thf(fact_1110_Nat_Odiff__diff__eq,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( minus_minus_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1111_le__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ord_less_eq_nat @ ( minus_minus_nat @ M4 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ) ) ).
% le_diff_iff
thf(fact_1112_eq__diff__iff,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ K @ M4 )
=> ( ( ord_less_eq_nat @ K @ N2 )
=> ( ( ( minus_minus_nat @ M4 @ K )
= ( minus_minus_nat @ N2 @ K ) )
= ( M4 = N2 ) ) ) ) ).
% eq_diff_iff
thf(fact_1113_diff__commute,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).
% diff_commute
thf(fact_1114_diff__mult__distrib2,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( times_times_nat @ K @ ( minus_minus_nat @ M4 @ N2 ) )
= ( minus_minus_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% diff_mult_distrib2
thf(fact_1115_diff__mult__distrib,axiom,
! [M4: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K )
= ( minus_minus_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1116_nat__mult__min__left,axiom,
! [M4: nat,N2: nat,Q2: nat] :
( ( times_times_nat @ ( ord_min_nat @ M4 @ N2 ) @ Q2 )
= ( ord_min_nat @ ( times_times_nat @ M4 @ Q2 ) @ ( times_times_nat @ N2 @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_1117_nat__mult__min__right,axiom,
! [M4: nat,N2: nat,Q2: nat] :
( ( times_times_nat @ M4 @ ( ord_min_nat @ N2 @ Q2 ) )
= ( ord_min_nat @ ( times_times_nat @ M4 @ N2 ) @ ( times_times_nat @ M4 @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_1118_mult__le__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ).
% mult_le_mono2
thf(fact_1119_mult__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ).
% mult_le_mono1
thf(fact_1120_mult__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1121_le__square,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ).
% le_square
thf(fact_1122_le__cube,axiom,
! [M4: nat] : ( ord_less_eq_nat @ M4 @ ( times_times_nat @ M4 @ ( times_times_nat @ M4 @ M4 ) ) ) ).
% le_cube
thf(fact_1123_nat__add__left__cancel__le,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% nat_add_left_cancel_le
thf(fact_1124_diff__diff__left,axiom,
! [I: nat,J: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
= ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).
% diff_diff_left
thf(fact_1125_Nat_Odiff__diff__right,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1126_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1127_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1128_add__mult__distrib,axiom,
! [M4: nat,N2: nat,K: nat] :
( ( times_times_nat @ ( plus_plus_nat @ M4 @ N2 ) @ K )
= ( plus_plus_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).
% add_mult_distrib
thf(fact_1129_add__mult__distrib2,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( times_times_nat @ K @ ( plus_plus_nat @ M4 @ N2 ) )
= ( plus_plus_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) ) ) ).
% add_mult_distrib2
thf(fact_1130_add__leE,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ~ ( ( ord_less_eq_nat @ M4 @ N2 )
=> ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).
% add_leE
thf(fact_1131_le__add1,axiom,
! [N2: nat,M4: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M4 ) ) ).
% le_add1
thf(fact_1132_le__add2,axiom,
! [N2: nat,M4: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M4 @ N2 ) ) ).
% le_add2
thf(fact_1133_add__leD1,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% add_leD1
thf(fact_1134_add__leD2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( plus_plus_nat @ M4 @ K ) @ N2 )
=> ( ord_less_eq_nat @ K @ N2 ) ) ).
% add_leD2
thf(fact_1135_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq_nat @ K @ L )
=> ? [N4: nat] :
( L
= ( plus_plus_nat @ K @ N4 ) ) ) ).
% le_Suc_ex
thf(fact_1136_add__le__mono,axiom,
! [I: nat,J: nat,K: nat,L: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ K @ L )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_1137_add__le__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).
% add_le_mono1
thf(fact_1138_trans__le__add1,axiom,
! [I: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M4 ) ) ) ).
% trans_le_add1
thf(fact_1139_trans__le__add2,axiom,
! [I: nat,J: nat,M4: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M4 @ J ) ) ) ).
% trans_le_add2
thf(fact_1140_nat__le__iff__add,axiom,
( ord_less_eq_nat
= ( ^ [M3: nat,N5: nat] :
? [K2: nat] :
( N5
= ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1141_Nat_Odiff__cancel,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% Nat.diff_cancel
thf(fact_1142_diff__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% diff_cancel2
thf(fact_1143_diff__add__inverse,axiom,
! [N2: nat,M4: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M4 ) @ N2 )
= M4 ) ).
% diff_add_inverse
thf(fact_1144_diff__add__inverse2,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ ( plus_plus_nat @ M4 @ N2 ) @ N2 )
= M4 ) ).
% diff_add_inverse2
thf(fact_1145_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( minus_minus_nat @ J @ I )
= K )
= ( J
= ( plus_plus_nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1146_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
= ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1147_Nat_Odiff__add__assoc,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
= ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1148_Nat_Ole__diff__conv2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1149_le__diff__conv,axiom,
! [J: nat,K: nat,I: nat] :
( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
= ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1150_nat__eq__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M4 )
= N2 ) ) ) ).
% nat_eq_add_iff1
thf(fact_1151_nat__diff__add__eq2,axiom,
! [I: nat,J: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1152_nat__diff__add__eq1,axiom,
! [J: nat,I: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% nat_diff_add_eq1
thf(fact_1153_nat__le__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ M4 @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1154_nat__le__add__iff1,axiom,
! [J: nat,I: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ J @ I )
=> ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M4 ) @ N2 ) ) ) ).
% nat_le_add_iff1
thf(fact_1155_nat__eq__add__iff2,axiom,
! [I: nat,J: nat,U: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ I @ J )
=> ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M4 )
= ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
= ( M4
= ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1156_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A2 ) ).
% bot_nat_0.extremum
thf(fact_1157_le0,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% le0
thf(fact_1158_Nat_Oadd__0__right,axiom,
! [M4: nat] :
( ( plus_plus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% Nat.add_0_right
thf(fact_1159_add__is__0,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
& ( N2 = zero_zero_nat ) ) ) ).
% add_is_0
thf(fact_1160_diff__0__eq__0,axiom,
! [N2: nat] :
( ( minus_minus_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% diff_0_eq_0
thf(fact_1161_diff__self__eq__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ M4 )
= zero_zero_nat ) ).
% diff_self_eq_0
thf(fact_1162_mult__is__0,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ( M4 = zero_zero_nat )
| ( N2 = zero_zero_nat ) ) ) ).
% mult_is_0
thf(fact_1163_mult__0__right,axiom,
! [M4: nat] :
( ( times_times_nat @ M4 @ zero_zero_nat )
= zero_zero_nat ) ).
% mult_0_right
thf(fact_1164_mult__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ K @ M4 )
= ( times_times_nat @ K @ N2 ) )
= ( ( M4 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel1
thf(fact_1165_mult__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ K )
= ( times_times_nat @ N2 @ K ) )
= ( ( M4 = N2 )
| ( K = zero_zero_nat ) ) ) ).
% mult_cancel2
thf(fact_1166_lessThan__0,axiom,
( ( set_ord_lessThan_nat @ zero_zero_nat )
= bot_bot_set_nat ) ).
% lessThan_0
thf(fact_1167_min__0L,axiom,
! [N2: nat] :
( ( ord_min_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% min_0L
thf(fact_1168_min__0R,axiom,
! [N2: nat] :
( ( ord_min_nat @ N2 @ zero_zero_nat )
= zero_zero_nat ) ).
% min_0R
thf(fact_1169_diff__is__0__eq_H,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat ) ) ).
% diff_is_0_eq'
thf(fact_1170_diff__is__0__eq,axiom,
! [M4: nat,N2: nat] :
( ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% diff_is_0_eq
thf(fact_1171_card__nat,axiom,
( ( finite_card_nat @ top_top_set_nat )
= zero_zero_nat ) ).
% card_nat
thf(fact_1172_card__literal,axiom,
( ( finite_card_literal @ top_top_set_literal )
= zero_zero_nat ) ).
% card_literal
thf(fact_1173_add__eq__self__zero,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= M4 )
=> ( N2 = zero_zero_nat ) ) ).
% add_eq_self_zero
thf(fact_1174_plus__nat_Oadd__0,axiom,
! [N2: nat] :
( ( plus_plus_nat @ zero_zero_nat @ N2 )
= N2 ) ).
% plus_nat.add_0
thf(fact_1175_mult__0,axiom,
! [N2: nat] :
( ( times_times_nat @ zero_zero_nat @ N2 )
= zero_zero_nat ) ).
% mult_0
thf(fact_1176_le__0__eq,axiom,
! [N2: nat] :
( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
= ( N2 = zero_zero_nat ) ) ).
% le_0_eq
thf(fact_1177_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
=> ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_1178_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq_nat @ A2 @ zero_zero_nat )
= ( A2 = zero_zero_nat ) ) ).
% bot_nat_0.extremum_unique
thf(fact_1179_less__eq__nat_Osimps_I1_J,axiom,
! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).
% less_eq_nat.simps(1)
thf(fact_1180_minus__nat_Odiff__0,axiom,
! [M4: nat] :
( ( minus_minus_nat @ M4 @ zero_zero_nat )
= M4 ) ).
% minus_nat.diff_0
thf(fact_1181_bot__nat__def,axiom,
bot_bot_nat = zero_zero_nat ).
% bot_nat_def
thf(fact_1182_diffs0__imp__equal,axiom,
! [M4: nat,N2: nat] :
( ( ( minus_minus_nat @ M4 @ N2 )
= zero_zero_nat )
=> ( ( ( minus_minus_nat @ N2 @ M4 )
= zero_zero_nat )
=> ( M4 = N2 ) ) ) ).
% diffs0_imp_equal
thf(fact_1183_lessThan__empty__iff,axiom,
! [N2: nat] :
( ( ( set_ord_lessThan_nat @ N2 )
= bot_bot_set_nat )
= ( N2 = zero_zero_nat ) ) ).
% lessThan_empty_iff
thf(fact_1184_diff__add__0,axiom,
! [N2: nat,M4: nat] :
( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M4 ) )
= zero_zero_nat ) ).
% diff_add_0
thf(fact_1185_nat_Oinject,axiom,
! [X22: nat,Y2: nat] :
( ( ( suc @ X22 )
= ( suc @ Y2 ) )
= ( X22 = Y2 ) ) ).
% nat.inject
thf(fact_1186_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_1187_Suc__le__mono,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M4 ) )
= ( ord_less_eq_nat @ N2 @ M4 ) ) ).
% Suc_le_mono
thf(fact_1188_add__Suc__right,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ M4 @ ( suc @ N2 ) )
= ( suc @ ( plus_plus_nat @ M4 @ N2 ) ) ) ).
% add_Suc_right
thf(fact_1189_Suc__diff__diff,axiom,
! [M4: nat,N2: nat,K: nat] :
( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M4 ) @ N2 ) @ ( suc @ K ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ N2 ) @ K ) ) ).
% Suc_diff_diff
thf(fact_1190_diff__Suc__Suc,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
= ( minus_minus_nat @ M4 @ N2 ) ) ).
% diff_Suc_Suc
thf(fact_1191_min__Suc__Suc,axiom,
! [M4: nat,N2: nat] :
( ( ord_min_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
= ( suc @ ( ord_min_nat @ M4 @ N2 ) ) ) ).
% min_Suc_Suc
thf(fact_1192_one__eq__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( times_times_nat @ M4 @ N2 ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1193_mult__eq__1__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1194_mult__Suc__right,axiom,
! [M4: nat,N2: nat] :
( ( times_times_nat @ M4 @ ( suc @ N2 ) )
= ( plus_plus_nat @ M4 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% mult_Suc_right
thf(fact_1195_one__le__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M4 @ N2 ) )
= ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M4 )
& ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).
% one_le_mult_iff
thf(fact_1196_diff__Suc__diff__eq2,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I )
= ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1197_diff__Suc__diff__eq1,axiom,
! [K: nat,J: nat,I: nat] :
( ( ord_less_eq_nat @ K @ J )
=> ( ( minus_minus_nat @ I @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
= ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1198_one__is__add,axiom,
! [M4: nat,N2: nat] :
( ( ( suc @ zero_zero_nat )
= ( plus_plus_nat @ M4 @ N2 ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% one_is_add
thf(fact_1199_add__is__1,axiom,
! [M4: nat,N2: nat] :
( ( ( plus_plus_nat @ M4 @ N2 )
= ( suc @ zero_zero_nat ) )
= ( ( ( M4
= ( suc @ zero_zero_nat ) )
& ( N2 = zero_zero_nat ) )
| ( ( M4 = zero_zero_nat )
& ( N2
= ( suc @ zero_zero_nat ) ) ) ) ) ).
% add_is_1
thf(fact_1200_Suc__mult__le__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_mult_le_cancel1
thf(fact_1201_mult__Suc,axiom,
! [M4: nat,N2: nat] :
( ( times_times_nat @ ( suc @ M4 ) @ N2 )
= ( plus_plus_nat @ N2 @ ( times_times_nat @ M4 @ N2 ) ) ) ).
% mult_Suc
thf(fact_1202_zero__notin__Suc__image,axiom,
! [A: set_nat] :
~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).
% zero_notin_Suc_image
thf(fact_1203_Suc__mult__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ( times_times_nat @ ( suc @ K ) @ M4 )
= ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( M4 = N2 ) ) ).
% Suc_mult_cancel1
thf(fact_1204_not0__implies__Suc,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
=> ? [M2: nat] :
( N2
= ( suc @ M2 ) ) ) ).
% not0_implies_Suc
thf(fact_1205_Zero__not__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_not_Suc
thf(fact_1206_Zero__neq__Suc,axiom,
! [M4: nat] :
( zero_zero_nat
!= ( suc @ M4 ) ) ).
% Zero_neq_Suc
thf(fact_1207_Suc__neq__Zero,axiom,
! [M4: nat] :
( ( suc @ M4 )
!= zero_zero_nat ) ).
% Suc_neq_Zero
thf(fact_1208_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ zero_zero_nat ) ) ) ).
% zero_induct
thf(fact_1209_diff__induct,axiom,
! [P: nat > nat > $o,M4: nat,N2: nat] :
( ! [X2: nat] : ( P @ X2 @ zero_zero_nat )
=> ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
=> ( ! [X2: nat,Y4: nat] :
( ( P @ X2 @ Y4 )
=> ( P @ ( suc @ X2 ) @ ( suc @ Y4 ) ) )
=> ( P @ M4 @ N2 ) ) ) ) ).
% diff_induct
thf(fact_1210_nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ zero_zero_nat )
=> ( ! [N4: nat] :
( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) )
=> ( P @ N2 ) ) ) ).
% nat_induct
thf(fact_1211_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero_nat )
=> ~ ! [Nat3: nat] :
( Y
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_1212_nat_OdiscI,axiom,
! [Nat: nat,X22: nat] :
( ( Nat
= ( suc @ X22 ) )
=> ( Nat != zero_zero_nat ) ) ).
% nat.discI
thf(fact_1213_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( zero_zero_nat
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_1214_old_Onat_Odistinct_I2_J,axiom,
! [Nat2: nat] :
( ( suc @ Nat2 )
!= zero_zero_nat ) ).
% old.nat.distinct(2)
thf(fact_1215_nat_Odistinct_I1_J,axiom,
! [X22: nat] :
( zero_zero_nat
!= ( suc @ X22 ) ) ).
% nat.distinct(1)
thf(fact_1216_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N4: nat] :
( ( P @ ( suc @ N4 ) )
=> ( P @ N4 ) )
=> ( P @ ( minus_minus_nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_1217_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_1218_n__not__Suc__n,axiom,
! [N2: nat] :
( N2
!= ( suc @ N2 ) ) ).
% n_not_Suc_n
thf(fact_1219_inj__Suc,axiom,
! [N: set_nat] : ( inj_on_nat_nat @ suc @ N ) ).
% inj_Suc
thf(fact_1220_transitive__stepwise__le,axiom,
! [M4: nat,N2: nat,R2: nat > nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ! [X2: nat] : ( R2 @ X2 @ X2 )
=> ( ! [X2: nat,Y4: nat,Z2: nat] :
( ( R2 @ X2 @ Y4 )
=> ( ( R2 @ Y4 @ Z2 )
=> ( R2 @ X2 @ Z2 ) ) )
=> ( ! [N4: nat] : ( R2 @ N4 @ ( suc @ N4 ) )
=> ( R2 @ M4 @ N2 ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_1221_nat__induct__at__least,axiom,
! [M4: nat,N2: nat,P: nat > $o] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ( P @ M4 )
=> ( ! [N4: nat] :
( ( ord_less_eq_nat @ M4 @ N4 )
=> ( ( P @ N4 )
=> ( P @ ( suc @ N4 ) ) ) )
=> ( P @ N2 ) ) ) ) ).
% nat_induct_at_least
thf(fact_1222_full__nat__induct,axiom,
! [P: nat > $o,N2: nat] :
( ! [N4: nat] :
( ! [M5: nat] :
( ( ord_less_eq_nat @ ( suc @ M5 ) @ N4 )
=> ( P @ M5 ) )
=> ( P @ N4 ) )
=> ( P @ N2 ) ) ).
% full_nat_induct
thf(fact_1223_not__less__eq__eq,axiom,
! [M4: nat,N2: nat] :
( ( ~ ( ord_less_eq_nat @ M4 @ N2 ) )
= ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 ) ) ).
% not_less_eq_eq
thf(fact_1224_Suc__n__not__le__n,axiom,
! [N2: nat] :
~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).
% Suc_n_not_le_n
thf(fact_1225_le__Suc__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
= ( ( ord_less_eq_nat @ M4 @ N2 )
| ( M4
= ( suc @ N2 ) ) ) ) ).
% le_Suc_eq
thf(fact_1226_Suc__le__D,axiom,
! [N2: nat,M6: nat] :
( ( ord_less_eq_nat @ ( suc @ N2 ) @ M6 )
=> ? [M2: nat] :
( M6
= ( suc @ M2 ) ) ) ).
% Suc_le_D
thf(fact_1227_le__SucI,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ N2 )
=> ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) ) ) ).
% le_SucI
thf(fact_1228_le__SucE,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ M4 @ ( suc @ N2 ) )
=> ( ~ ( ord_less_eq_nat @ M4 @ N2 )
=> ( M4
= ( suc @ N2 ) ) ) ) ).
% le_SucE
thf(fact_1229_Suc__leD,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ M4 ) @ N2 )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ).
% Suc_leD
thf(fact_1230_Suc__diff__le,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_eq_nat @ N2 @ M4 )
=> ( ( minus_minus_nat @ ( suc @ M4 ) @ N2 )
= ( suc @ ( minus_minus_nat @ M4 @ N2 ) ) ) ) ).
% Suc_diff_le
thf(fact_1231_nat__arith_Osuc1,axiom,
! [A: nat,K: nat,A2: nat] :
( ( A
= ( plus_plus_nat @ K @ A2 ) )
=> ( ( suc @ A )
= ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1232_add__Suc,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N2 )
= ( suc @ ( plus_plus_nat @ M4 @ N2 ) ) ) ).
% add_Suc
thf(fact_1233_add__Suc__shift,axiom,
! [M4: nat,N2: nat] :
( ( plus_plus_nat @ ( suc @ M4 ) @ N2 )
= ( plus_plus_nat @ M4 @ ( suc @ N2 ) ) ) ).
% add_Suc_shift
thf(fact_1234_card__UNIV__unit,axiom,
( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
= one_one_nat ) ).
% card_UNIV_unit
thf(fact_1235_nat__mult__eq__1__iff,axiom,
! [M4: nat,N2: nat] :
( ( ( times_times_nat @ M4 @ N2 )
= one_one_nat )
= ( ( M4 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1236_nat__1__eq__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( one_one_nat
= ( times_times_nat @ M4 @ N2 ) )
= ( ( M4 = one_one_nat )
& ( N2 = one_one_nat ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1237_diff__Suc__1,axiom,
! [N2: nat] :
( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
= N2 ) ).
% diff_Suc_1
thf(fact_1238_mult__eq__self__implies__10,axiom,
! [M4: nat,N2: nat] :
( ( M4
= ( times_times_nat @ M4 @ N2 ) )
=> ( ( N2 = one_one_nat )
| ( M4 = zero_zero_nat ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1239_One__nat__def,axiom,
( one_one_nat
= ( suc @ zero_zero_nat ) ) ).
% One_nat_def
thf(fact_1240_diff__Suc__eq__diff__pred,axiom,
! [M4: nat,N2: nat] :
( ( minus_minus_nat @ M4 @ ( suc @ N2 ) )
= ( minus_minus_nat @ ( minus_minus_nat @ M4 @ one_one_nat ) @ N2 ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_1241_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).
% Suc_eq_plus1
thf(fact_1242_plus__1__eq__Suc,axiom,
( ( plus_plus_nat @ one_one_nat )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1243_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus_nat @ one_one_nat ) ) ).
% Suc_eq_plus1_left
thf(fact_1244_nat__mult__1__right,axiom,
! [N2: nat] :
( ( times_times_nat @ N2 @ one_one_nat )
= N2 ) ).
% nat_mult_1_right
thf(fact_1245_nat__mult__1,axiom,
! [N2: nat] :
( ( times_times_nat @ one_one_nat @ N2 )
= N2 ) ).
% nat_mult_1
thf(fact_1246_add__eq__if,axiom,
( plus_plus_nat
= ( ^ [M3: nat,N5: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1247_mult__eq__if,axiom,
( times_times_nat
= ( ^ [M3: nat,N5: nat] : ( if_nat @ ( M3 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N5 @ ( times_times_nat @ ( minus_minus_nat @ M3 @ one_one_nat ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1248_infinite__UNIV__literal,axiom,
~ ( finite5847741373460823677iteral @ top_top_set_literal ) ).
% infinite_UNIV_literal
thf(fact_1249_nat__one__le__power,axiom,
! [I: nat,N2: nat] :
( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I )
=> ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I @ N2 ) ) ) ).
% nat_one_le_power
thf(fact_1250_less__nat__zero__code,axiom,
! [N2: nat] :
~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).
% less_nat_zero_code
thf(fact_1251_neq0__conv,axiom,
! [N2: nat] :
( ( N2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).
% neq0_conv
thf(fact_1252_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2 != zero_zero_nat )
= ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_1253_lessI,axiom,
! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).
% lessI
thf(fact_1254_Suc__mono,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ N2 )
=> ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) ) ) ).
% Suc_mono
thf(fact_1255_Suc__less__eq,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ ( suc @ M4 ) @ ( suc @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_less_eq
thf(fact_1256_nat__add__left__cancel__less,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( plus_plus_nat @ K @ M4 ) @ ( plus_plus_nat @ K @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% nat_add_left_cancel_less
thf(fact_1257_zero__less__Suc,axiom,
! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).
% zero_less_Suc
thf(fact_1258_less__Suc0,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
= ( N2 = zero_zero_nat ) ) ).
% less_Suc0
thf(fact_1259_add__gr__0,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M4 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
| ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% add_gr_0
thf(fact_1260_zero__less__diff,axiom,
! [N2: nat,M4: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M4 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% zero_less_diff
thf(fact_1261_less__one,axiom,
! [N2: nat] :
( ( ord_less_nat @ N2 @ one_one_nat )
= ( N2 = zero_zero_nat ) ) ).
% less_one
thf(fact_1262_nat__0__less__mult__iff,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M4 @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ M4 )
& ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1263_mult__less__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
& ( ord_less_nat @ M4 @ N2 ) ) ) ).
% mult_less_cancel2
thf(fact_1264_Suc__pred,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
= N2 ) ) ).
% Suc_pred
thf(fact_1265_nat__mult__le__cancel__disj,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ K @ M4 ) @ ( times_times_nat @ K @ N2 ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1266_mult__le__cancel2,axiom,
! [M4: nat,K: nat,N2: nat] :
( ( ord_less_eq_nat @ ( times_times_nat @ M4 @ K ) @ ( times_times_nat @ N2 @ K ) )
= ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_eq_nat @ M4 @ N2 ) ) ) ).
% mult_le_cancel2
thf(fact_1267_Suc__diff__1,axiom,
! [N2: nat] :
( ( ord_less_nat @ zero_zero_nat @ N2 )
=> ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
= N2 ) ) ).
% Suc_diff_1
thf(fact_1268_less__imp__add__positive,axiom,
! [I: nat,J: nat] :
( ( ord_less_nat @ I @ J )
=> ? [K3: nat] :
( ( ord_less_nat @ zero_zero_nat @ K3 )
& ( ( plus_plus_nat @ I @ K3 )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_1269_mult__less__mono1,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ I @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1270_mult__less__mono2,axiom,
! [I: nat,J: nat,K: nat] :
( ( ord_less_nat @ I @ J )
=> ( ( ord_less_nat @ zero_zero_nat @ K )
=> ( ord_less_nat @ ( times_times_nat @ K @ I ) @ ( times_times_nat @ K @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1271_Suc__mult__less__cancel1,axiom,
! [K: nat,M4: nat,N2: nat] :
( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M4 ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
= ( ord_less_nat @ M4 @ N2 ) ) ).
% Suc_mult_less_cancel1
thf(fact_1272_ex__least__nat__le,axiom,
! [P: nat > $o,N2: nat] :
( ( P @ N2 )
=> ( ~ ( P @ zero_zero_nat )
=> ? [K3: nat] :
( ( ord_less_eq_nat @ K3 @ N2 )
& ! [I2: nat] :
( ( ord_less_nat @ I2 @ K3 )
=> ~ ( P @ I2 ) )
& ( P @ K3 ) ) ) ) ).
% ex_least_nat_le
thf(fact_1273_less__Suc__eq__0__disj,axiom,
! [M4: nat,N2: nat] :
( ( ord_less_nat @ M4 @ ( suc @ N2 ) )
= ( ( M4 = zero_zero_nat )
| ? [J2: nat] :
( ( M4
= ( suc @ J2 ) )
& ( ord_less_nat @ J2 @ N2 ) ) ) ) ).
% less_Suc_eq_0_disj
% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
! [X: nat,Y: nat] :
( ( if_nat @ $true @ X @ Y )
= X ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( set_Sum_sum_a_nat2 @ xs )
= ( sup_su6804446743777130803_a_nat @ ( inf_in7084830621192376909_a_nat @ ( set_Sum_sum_a_nat2 @ xs ) @ ( image_7873763678140191238_a_nat @ sum_Inl_a_nat @ ad ) ) @ ( image_7293268710728258664_a_nat @ sum_Inr_nat_a @ ( set_ord_lessThan_nat @ ( ord_min_nat @ ( size_s5283204784079214577_a_nat @ xs ) @ ( finite_card_nat @ ( vimage3040984495076556338_a_nat @ sum_Inr_nat_a @ ( set_Sum_sum_a_nat2 @ xs ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------