%------------------------------------------------------------------------------
% 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    : Multiset_Ordering_NPC/0002_Multiset_Ordering_in_NP/prob_00534_023423__13702982_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1847 ( 630 unt; 570 typ;   0 def)
%            Number of atoms       : 3868 (1694 equ;   0 cnn)
%            Maximal formula atoms :  105 (   3 avg)
%            Number of connectives : 14850 ( 282   ~;  47   |; 377   &;12612   @)
%                                         (   0 <=>;1532  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   26 (   7 avg)
%            Number of types       :  106 ( 105 usr)
%            Number of type conns  : 1612 (1612   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  468 ( 465 usr;  39 con; 0-7 aty)
%            Number of variables   : 4615 ( 534   ^;3846   !; 235   ?;4615   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:29:39.349
%------------------------------------------------------------------------------
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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__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__Ochar,type,
    char: $tType ).

thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

thf(ty_n_tf__a,type,
    a: $tType ).

% Explicit typings (465)
thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    minus_1245740275806398586pVar_o: ( multis3193088007478089820ropVar > $o ) > ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Nat__Onat_M_Eo_J,type,
    minus_minus_nat_o: ( nat > $o ) > ( nat > $o ) > nat > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_M_Eo_J,type,
    minus_1780674721092442863et_a_o: ( produc6518373309651786023iset_a > $o ) > ( produc6518373309651786023iset_a > $o ) > produc6518373309651786023iset_a > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
    minus_4793868396798367471_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > product_prod_a_a > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001_062_Itf__a_M_Eo_J,type,
    minus_minus_a_o: ( a > $o ) > ( a > $o ) > a > $o ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    minus_287902380581138057ropVar: multis193838646080965794ropVar > multis193838646080965794ropVar > multis193838646080965794ropVar ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    minus_8522176038001411705et_nat: multiset_nat > multiset_nat > multiset_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    minus_9211619285413683668iset_a: multis4781622204924204013iset_a > multis4781622204924204013iset_a > multis4781622204924204013iset_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    minus_7313152569058643924od_a_a: multis501812127501805293od_a_a > multis501812127501805293od_a_a > multis501812127501805293od_a_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_Itf__a_J,type,
    minus_3765977307040488491iset_a: multiset_a > multiset_a > multiset_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    minus_2621998875110806563ropVar: set_Mu5658373669842198716ropVar > set_Mu5658373669842198716ropVar > set_Mu5658373669842198716ropVar ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
    minus_minus_set_nat: set_nat > set_nat > set_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    minus_2461365523287052142iset_a: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    minus_6817036919807184750od_a_a: set_Product_prod_a_a > set_Product_prod_a_a > set_Product_prod_a_a ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_Itf__a_J,type,
    minus_minus_set_a: set_a > set_a > set_a ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
    plus_p6738641960240532161iset_a: multiset_multiset_a > multiset_multiset_a > multiset_multiset_a ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    plus_p1339091108561003481ropVar: multis193838646080965794ropVar > multis193838646080965794ropVar > multis193838646080965794ropVar ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    plus_p6334493942879108393et_nat: multiset_nat > multiset_nat > multiset_nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    plus_p1053901119268525860iset_a: multis4781622204924204013iset_a > multis4781622204924204013iset_a > multis4781622204924204013iset_a ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    plus_p6967097483979887908od_a_a: multis501812127501805293od_a_a > multis501812127501805293od_a_a > multis501812127501805293od_a_a ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_Itf__a_J,type,
    plus_plus_multiset_a: multiset_a > multiset_a > multiset_a ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Multiset__Omultiset_Itf__a_J_J,type,
    zero_z7799948378220188993iset_a: multiset_multiset_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    zero_z6226653793019358809ropVar: multis193838646080965794ropVar ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    zero_z7348594199698428585et_nat: multiset_nat ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    zero_z4591712717340762532iset_a: multis4781622204924204013iset_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    zero_z3024502163831676836od_a_a: multis501812127501805293od_a_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_Itf__a_J,type,
    zero_zero_multiset_a: multiset_a ).

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset____Ordering____in____NP__OPropVar,type,
    groups6371087482465774614ropVar: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > ( multis3193088007478089820ropVar > multiset_nat ) > set_Mu5658373669842198716ropVar > multiset_nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Nat__Onat,type,
    groups5394990218802192790at_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > ( nat > multiset_nat ) > set_nat > multiset_nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
    groups3729319811690247393iset_a: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > ( produc6518373309651786023iset_a > multiset_nat ) > set_Pr79727621955416071iset_a > multiset_nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    groups8557384953281641185od_a_a: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > ( product_prod_a_a > multiset_nat ) > set_Product_prod_a_a > multiset_nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_It__Nat__Onat_J_001tf__a,type,
    groups6763380508735453880_nat_a: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > ( a > multiset_nat ) > set_a > multiset_nat ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_Itf__a_J_001t__Multiset____Ordering____in____NP__OPropVar,type,
    groups3121060555026902844ropVar: ( multiset_a > multiset_a > multiset_a ) > multiset_a > ( multis3193088007478089820ropVar > multiset_a ) > set_Mu5658373669842198716ropVar > multiset_a ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_Itf__a_J_001t__Nat__Onat,type,
    groups9157842243955967216_a_nat: ( multiset_a > multiset_a > multiset_a ) > multiset_a > ( nat > multiset_a ) > set_nat > multiset_a ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_Itf__a_J_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
    groups6473759573426450439iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > ( produc6518373309651786023iset_a > multiset_a ) > set_Pr79727621955416071iset_a > multiset_a ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_Itf__a_J_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    groups1455199411266135047od_a_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > ( product_prod_a_a > multiset_a ) > set_Product_prod_a_a > multiset_a ).

thf(sy_c_Groups__Big_Ocomm__monoid__add_Osum_001t__Multiset__Omultiset_Itf__a_J_001tf__a,type,
    groups409869659785034206et_a_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > ( a > multiset_a ) > set_a > multiset_a ).

thf(sy_c_Groups__List_Omonoid__add_Osum__list_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    groups2887787882517827221et_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > list_multiset_nat > multiset_nat ).

thf(sy_c_Groups__List_Omonoid__add_Osum__list_001t__Multiset__Omultiset_Itf__a_J,type,
    groups778166972783649551iset_a: ( multiset_a > multiset_a > multiset_a ) > multiset_a > list_multiset_a > multiset_a ).

thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
    the_nat: ( nat > $o ) > nat ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
    the_Pr1459549734356548398iset_a: ( produc6518373309651786023iset_a > $o ) > produc6518373309651786023iset_a ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    the_Product_prod_a_a: ( product_prod_a_a > $o ) > product_prod_a_a ).

thf(sy_c_HOL_Oundefined_001_Eo,type,
    undefined_o: $o ).

thf(sy_c_If_001t__Nat__Onat,type,
    if_nat: $o > nat > nat > nat ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Multiset__Omultiset_Itf__a_J_M_062_It__Multiset__Omultiset_Itf__a_J_M_Eo_J_J,type,
    sup_su6917467955185334722et_a_o: ( multiset_a > multiset_a > $o ) > ( multiset_a > multiset_a > $o ) > multiset_a > multiset_a > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    sup_su5656099892504388685pVar_o: ( multis3193088007478089820ropVar > $o ) > ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Nat__Onat_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_M_Eo_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001_062_It__Product____Type__Oprod_Itf__a_Mtf__a_J_M_Eo_J,type,
    sup_su1640154476453168578_a_a_o: ( product_prod_a_a > $o ) > ( product_prod_a_a > $o ) > product_prod_a_a > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_062_Itf__a_M_Eo_J_J,type,
    sup_sup_a_a_o: ( a > a > $o ) > ( a > a > $o ) > a > a > $o ).

thf(sy_c_Lattices_Osup__class_Osup_001_062_Itf__a_M_Eo_J,type,
    sup_sup_a_o: ( a > $o ) > ( a > $o ) > a > $o ).

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__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    sup_su1925787886501348827iset_a: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a ).

thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J_J,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_Itf__a_J,type,
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thf(sy_c_List_Oappend_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
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thf(sy_c_List_Oappend_001tf__a,type,
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thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
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thf(sy_c_List_Odistinct_001tf__a,type,
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thf(sy_c_List_Odrop_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member6519385997651362275od_a_a: produc2100646355360341836od_a_a > set_Pr5009853057886717698od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__a_J,type,
    member8962352052110095674_nat_a: product_prod_nat_a > set_Pr4193341848836149977_nat_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Nat__Onat_J,type,
    member4256827664654810685_a_nat: produc2962902410472220326_a_nat > set_Pr487246842938036828_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_J,type,
    member1562461016398818064iset_a: produc8554440688619196647iset_a > set_Pr3059988933816007879iset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Nat__Onat_J,type,
    member1268613965720311869_a_nat: produc6073246360284067238_a_nat > set_Pr8287382171553239132_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member6330455413206600464od_a_a: produc3498347346309940967od_a_a > set_Pr8600417178894128327od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__a_J_Mtf__a_J,type,
    member2829916517802586983_a_a_a: produc3802892049952890430_a_a_a > set_Pr8876520727511657886_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    member5951705825339765678ropVar: produc6590503114050168709ropVar > set_Pr1287879138905681509ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__a_J_J,type,
    member3071122053849602553od_a_a: produc4044097585999906000od_a_a > set_Pr5530083903271594800od_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_v_NS,type,
    ns: set_Product_prod_a_a ).

thf(sy_v_S,type,
    s: set_Product_prod_a_a ).

thf(sy_v_Xs1____,type,
    xs1: list_a ).

thf(sy_v_Xs2____,type,
    xs2: list_a ).

thf(sy_v_Ys1____,type,
    ys1: list_a ).

thf(sy_v_Ys2____,type,
    ys2: list_a ).

thf(sy_v_cns,type,
    cns: nat > nat > $o ).

thf(sy_v_cs,type,
    cs: nat > nat > $o ).

thf(sy_v_i__of__j2____,type,
    i_of_j2: nat > nat ).

thf(sy_v_j____,type,
    j: nat ).

thf(sy_v_m,type,
    m: nat ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_pos__of____,type,
    pos_of: list_nat > nat > nat ).

thf(sy_v_s,type,
    s2: nat > a ).

thf(sy_v_t,type,
    t: nat > a ).

thf(sy_v_v_H____,type,
    v: multis3193088007478089820ropVar > $o ).

thf(sy_v_v____,type,
    v2: multis3193088007478089820ropVar > $o ).

thf(sy_v_xs1____,type,
    xs12: list_nat ).

thf(sy_v_xs2____,type,
    xs22: list_nat ).

thf(sy_v_ys1____,type,
    ys12: list_nat ).

thf(sy_v_ys2____,type,
    ys22: list_nat ).

% Relevant facts (1268)
thf(fact_0_j1,axiom,
    member_nat @ j @ ( set_nat2 @ ys12 ) ).

% j1
thf(fact_1_mem,axiom,
    member_nat @ ( nth_nat @ xs12 @ ( pos_of @ ys12 @ j ) ) @ ( set_nat2 @ xs12 ) ).

% mem
thf(fact_2_PropVar_Oinject_I1_J,axiom,
    ! [X11: nat,X12: nat,Y11: nat,Y12: nat] :
      ( ( ( multis387687052011358179_Gamma @ X11 @ X12 )
        = ( multis387687052011358179_Gamma @ Y11 @ Y12 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 ) ) ) ).

% PropVar.inject(1)
thf(fact_3_p__ys_I2_J,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ys12 ) )
     => ( ( nth_nat @ ys12 @ ( pos_of @ ys12 @ X ) )
        = X ) ) ).

% p_ys(2)
thf(fact_4_p__xs_I2_J,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ xs12 ) )
     => ( ( nth_nat @ xs12 @ ( pos_of @ xs12 @ X ) )
        = X ) ) ).

% p_xs(2)
thf(fact_5_yj,axiom,
    ( ( nth_nat @ ys12 @ ( pos_of @ ys12 @ j ) )
    = j ) ).

% yj
thf(fact_6_j__ys2_I1_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys22 ) )
     => ( member_nat @ ( i_of_j2 @ J ) @ ( set_nat2 @ xs22 ) ) ) ).

% j_ys2(1)
thf(fact_7_j__ys2_I3_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys22 ) )
     => ( v2 @ ( multis387687052011358179_Gamma @ ( i_of_j2 @ J ) @ J ) ) ) ).

% j_ys2(3)
thf(fact_8__092_060open_062v_H_A_092_060equiv_062_A_092_060lambda_062x_O_Acase_Ax_Aof_AGamma_Ai_Aj_A_092_060Rightarrow_062_Ai_A_092_060in_062_Aset_Axs1_A_092_060and_062_Aj_A_092_060in_062_Aset_Ays1_A_092_060and_062_Ai_A_061_Axs1_A_B_Apos__of_Ays1_Aj_A_092_060or_062_Ai_A_092_060in_062_Aset_Axs2_A_092_060and_062_Aj_A_092_060in_062_Aset_Ays2_A_092_060and_062_Ai_A_061_Ai__of__j2_Aj_A_124_AEpsilon_Ai_A_092_060Rightarrow_062_Ai_A_092_060in_062_Aset_Axs1_092_060close_062,axiom,
    ( v
    = ( multis8641292176345589959pVar_o
      @ ^ [I: nat,J2: nat] :
          ( ( ( member_nat @ I @ ( set_nat2 @ xs12 ) )
            & ( member_nat @ J2 @ ( set_nat2 @ ys12 ) )
            & ( I
              = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ J2 ) ) ) )
          | ( ( member_nat @ I @ ( set_nat2 @ xs22 ) )
            & ( member_nat @ J2 @ ( set_nat2 @ ys22 ) )
            & ( I
              = ( i_of_j2 @ J2 ) ) ) )
      @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs12 ) )
      @ ^ [A: nat,B: nat] : undefined_o
      @ ^ [A: nat,B: nat] : undefined_o
      @ ^ [A: nat,B: nat] : undefined_o
      @ ^ [A: nat,B: nat] : undefined_o ) ) ).

% \<open>v' \<equiv> \<lambda>x. case x of Gamma i j \<Rightarrow> i \<in> set xs1 \<and> j \<in> set ys1 \<and> i = xs1 ! pos_of ys1 j \<or> i \<in> set xs2 \<and> j \<in> set ys2 \<and> i = i_of_j2 j | Epsilon i \<Rightarrow> i \<in> set xs1\<close>
thf(fact_9_cns,axiom,
    cns @ ( nth_nat @ xs12 @ ( pos_of @ ys12 @ j ) ) @ j ).

% cns
thf(fact_10_PropVar_Osimps_I37_J,axiom,
    ! [F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,X11: nat,X12: nat] :
      ( ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = ( F1 @ X11 @ X12 ) ) ).

% PropVar.simps(37)
thf(fact_11_PropVar_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,PropVar: multis3193088007478089820ropVar] :
      ( ( H @ ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ PropVar ) )
      = ( multis8641292176345589959pVar_o
        @ ^ [X1: nat,X2: nat] : ( H @ ( F1 @ X1 @ X2 ) )
        @ ^ [X3: nat] : ( H @ ( F2 @ X3 ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F3 @ X1 @ X2 ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F4 @ X1 @ X2 ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F5 @ X1 @ X2 ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F6 @ X1 @ X2 ) )
        @ PropVar ) ) ).

% PropVar.case_distrib
thf(fact_12__092_060open_062v_A_092_060equiv_062_A_092_060lambda_062x_O_Acase_Ax_Aof_AAuxZeroJI_Ai_Aj_A_092_060Rightarrow_062_A_092_060not_062_ABex_A_Iset_A_Idrop_Ai_A_Imap_A_I_092_060lambda_062i_O_AGamma_Ai_Aj_J_A_0910_O_O_060n_093_J_J_J_Av_H_A_124_AAuxOneJI_Ai_Aj_A_092_060Rightarrow_062_A_092_060exists_062_By_O_Ay_A_092_060in_062_Aset_A_Idrop_Ai_A_Imap_A_I_092_060lambda_062i_O_AGamma_Ai_Aj_J_A_0910_O_O_060n_093_J_J_A_092_060and_062_Av_H_Ay_A_124_AAuxZeroIJ_Ai_Aj_A_092_060Rightarrow_062_A_092_060not_062_ABex_A_Iset_A_Idrop_Aj_A_Imap_A_IGamma_Ai_J_A_0910_O_O_060m_093_J_J_J_Av_H_A_124_AAuxOneIJ_Ai_Aj_A_092_060Rightarrow_062_A_092_060exists_062_By_O_Ay_A_092_060in_062_Aset_A_Idrop_Aj_A_Imap_A_IGamma_Ai_J_A_0910_O_O_060m_093_J_J_A_092_060and_062_Av_H_Ay_A_124_A___A_092_060Rightarrow_062_Av_H_Ax_092_060close_062,axiom,
    ( v2
    = ( ^ [X3: multis3193088007478089820ropVar] :
          ( multis8641292176345589959pVar_o
          @ ^ [Nat1: nat,Nat2: nat] : ( v @ X3 )
          @ ^ [Nat: nat] : ( v @ X3 )
          @ ^ [I: nat,J2: nat] :
              ~ ? [Y: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y
                    @ ( set_Mu146256361055586289ropVar
                      @ ( drop_M2293212096379261346ropVar @ I
                        @ ( map_na2466558649495876678ropVar
                          @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                          @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                  & ( v @ Y ) )
          @ ^ [I: nat,J2: nat] :
            ? [Y: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Y
                @ ( set_Mu146256361055586289ropVar
                  @ ( drop_M2293212096379261346ropVar @ I
                    @ ( map_na2466558649495876678ropVar
                      @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                      @ ( upt @ zero_zero_nat @ n ) ) ) ) )
              & ( v @ Y )
              & ! [Z: multis3193088007478089820ropVar] :
                  ( ( ( member8421201611525043589ropVar @ Z
                      @ ( set_Mu146256361055586289ropVar
                        @ ( drop_M2293212096379261346ropVar @ I
                          @ ( map_na2466558649495876678ropVar
                            @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                            @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                    & ( v @ Z ) )
                 => ( Z = Y ) ) )
          @ ^ [I: nat,J2: nat] :
              ~ ? [Y: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                  & ( v @ Y ) )
          @ ^ [I: nat,J2: nat] :
            ? [Y: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
              & ( v @ Y )
              & ! [Z: multis3193088007478089820ropVar] :
                  ( ( ( member8421201611525043589ropVar @ Z @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                    & ( v @ Z ) )
                 => ( Z = Y ) ) )
          @ X3 ) ) ) ).

% \<open>v \<equiv> \<lambda>x. case x of AuxZeroJI i j \<Rightarrow> \<not> Bex (set (drop i (map (\<lambda>i. Gamma i j) [0..<n]))) v' | AuxOneJI i j \<Rightarrow> \<exists>!y. y \<in> set (drop i (map (\<lambda>i. Gamma i j) [0..<n])) \<and> v' y | AuxZeroIJ i j \<Rightarrow> \<not> Bex (set (drop j (map (Gamma i) [0..<m]))) v' | AuxOneIJ i j \<Rightarrow> \<exists>!y. y \<in> set (drop j (map (Gamma i) [0..<m])) \<and> v' y | _ \<Rightarrow> v' x\<close>
thf(fact_13_v_H__def,axiom,
    ! [X: multis3193088007478089820ropVar] :
      ( ( v @ X )
      = ( multis8641292176345589959pVar_o
        @ ^ [I: nat,J2: nat] :
            ( ( ( member_nat @ I @ ( set_nat2 @ xs12 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys12 ) )
              & ( I
                = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ J2 ) ) ) )
            | ( ( member_nat @ I @ ( set_nat2 @ xs22 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys22 ) )
              & ( I
                = ( i_of_j2 @ J2 ) ) ) )
        @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs12 ) )
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ X ) ) ).

% v'_def
thf(fact_14_j__ys2_I2_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys22 ) )
     => ( cs @ ( i_of_j2 @ J ) @ J ) ) ).

% j_ys2(2)
thf(fact_15_v__def,axiom,
    ! [X: multis3193088007478089820ropVar] :
      ( ( v2 @ X )
      = ( multis8641292176345589959pVar_o
        @ ^ [Nat1: nat,Nat2: nat] : ( v @ X )
        @ ^ [Nat: nat] : ( v @ X )
        @ ^ [I: nat,J2: nat] :
            ~ ? [X3: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X3
                  @ ( set_Mu146256361055586289ropVar
                    @ ( drop_M2293212096379261346ropVar @ I
                      @ ( map_na2466558649495876678ropVar
                        @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                        @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                & ( v @ X3 ) )
        @ ^ [I: nat,J2: nat] :
          ? [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3
              @ ( set_Mu146256361055586289ropVar
                @ ( drop_M2293212096379261346ropVar @ I
                  @ ( map_na2466558649495876678ropVar
                    @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                    @ ( upt @ zero_zero_nat @ n ) ) ) ) )
            & ( v @ X3 )
            & ! [Y: multis3193088007478089820ropVar] :
                ( ( ( member8421201611525043589ropVar @ Y
                    @ ( set_Mu146256361055586289ropVar
                      @ ( drop_M2293212096379261346ropVar @ I
                        @ ( map_na2466558649495876678ropVar
                          @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                          @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                  & ( v @ Y ) )
               => ( Y = X3 ) ) )
        @ ^ [I: nat,J2: nat] :
            ~ ? [X3: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                & ( v @ X3 ) )
        @ ^ [I: nat,J2: nat] :
          ? [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
            & ( v @ X3 )
            & ! [Y: multis3193088007478089820ropVar] :
                ( ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                  & ( v @ Y ) )
               => ( Y = X3 ) ) )
        @ X ) ) ).

% v_def
thf(fact_16_drop0,axiom,
    ( ( drop_M2293212096379261346ropVar @ zero_zero_nat )
    = ( ^ [X3: list_M6740372356978808802ropVar] : X3 ) ) ).

% drop0
thf(fact_17_drop0,axiom,
    ( ( drop_nat @ zero_zero_nat )
    = ( ^ [X3: list_nat] : X3 ) ) ).

% drop0
thf(fact_18_map__eq__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Xs ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_19_map__eq__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_20_map__eq__conv,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Xs ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) ) ) ) ).

% map_eq_conv
thf(fact_21_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X3: nat] : X3 )
    = ( ^ [Xs2: list_nat] : Xs2 ) ) ).

% map_ident
thf(fact_22__092_060open_062i__of__j2_A_092_060equiv_062_A_092_060lambda_062j_O_ASOME_Ai_O_Ai_A_092_060in_062_Aset_Axs2_A_092_060and_062_Acs_Ai_Aj_092_060close_062,axiom,
    ( i_of_j2
    = ( ^ [J2: nat] :
          ( fChoice_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ ( set_nat2 @ xs22 ) )
              & ( cs @ I @ J2 ) ) ) ) ) ).

% \<open>i_of_j2 \<equiv> \<lambda>j. SOME i. i \<in> set xs2 \<and> cs i j\<close>
thf(fact_23_pos_I2_J,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( ( nth_nat @ Xs @ ( pos_of @ Xs @ X ) )
          = X ) ) ) ).

% pos(2)
thf(fact_24_xs,axiom,
    ( ( mset_nat @ ( upt @ zero_zero_nat @ n ) )
    = ( plus_p6334493942879108393et_nat @ ( mset_nat @ xs12 ) @ ( mset_nat @ xs22 ) ) ) ).

% xs
thf(fact_25_ys,axiom,
    ( ( mset_nat @ ( upt @ zero_zero_nat @ m ) )
    = ( plus_p6334493942879108393et_nat @ ( mset_nat @ ys12 ) @ ( mset_nat @ ys22 ) ) ) ).

% ys
thf(fact_26__092_060open_062distinct_Ays1_092_060close_062,axiom,
    distinct_nat @ ys12 ).

% \<open>distinct ys1\<close>
thf(fact_27__092_060open_062distinct_Axs1_092_060close_062,axiom,
    distinct_nat @ xs12 ).

% \<open>distinct xs1\<close>
thf(fact_28_add__left__cancel,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A2 @ B2 )
        = ( plus_p6334493942879108393et_nat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_29_add__left__cancel,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_30_add__left__cancel,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A2 @ B2 )
        = ( plus_plus_multiset_a @ A2 @ C ) )
      = ( B2 = C ) ) ).

% add_left_cancel
thf(fact_31_add__right__cancel,axiom,
    ! [B2: multiset_nat,A2: multiset_nat,C: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ B2 @ A2 )
        = ( plus_p6334493942879108393et_nat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_32_add__right__cancel,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_33_add__right__cancel,axiom,
    ! [B2: multiset_a,A2: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ B2 @ A2 )
        = ( plus_plus_multiset_a @ C @ A2 ) )
      = ( B2 = C ) ) ).

% add_right_cancel
thf(fact_34_add__0,axiom,
    ! [A2: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_35_add__0,axiom,
    ! [A2: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
      = A2 ) ).

% add_0
thf(fact_36_add__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% add_0
thf(fact_37_zero__eq__add__iff__both__eq__0,axiom,
    ! [X: nat,Y2: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X @ Y2 ) )
      = ( ( X = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_38_add__eq__0__iff__both__eq__0,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ( plus_plus_nat @ X @ Y2 )
        = zero_zero_nat )
      = ( ( X = zero_zero_nat )
        & ( Y2 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_39_add__cancel__right__right,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( A2
        = ( plus_p6334493942879108393et_nat @ A2 @ B2 ) )
      = ( B2 = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_right_right
thf(fact_40_add__cancel__right__right,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( A2
        = ( plus_plus_multiset_a @ A2 @ B2 ) )
      = ( B2 = zero_zero_multiset_a ) ) ).

% add_cancel_right_right
thf(fact_41_add__cancel__right__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ A2 @ B2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_42_add__cancel__right__left,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( A2
        = ( plus_p6334493942879108393et_nat @ B2 @ A2 ) )
      = ( B2 = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_right_left
thf(fact_43_add__cancel__right__left,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( A2
        = ( plus_plus_multiset_a @ B2 @ A2 ) )
      = ( B2 = zero_zero_multiset_a ) ) ).

% add_cancel_right_left
thf(fact_44_add__cancel__right__left,axiom,
    ! [A2: nat,B2: nat] :
      ( ( A2
        = ( plus_plus_nat @ B2 @ A2 ) )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_45_add__cancel__left__right,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_left_right
thf(fact_46_add__cancel__left__right,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_multiset_a ) ) ).

% add_cancel_left_right
thf(fact_47_add__cancel__left__right,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_48_add__cancel__left__left,axiom,
    ! [B2: multiset_nat,A2: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_left_left
thf(fact_49_add__cancel__left__left,axiom,
    ! [B2: multiset_a,A2: multiset_a] :
      ( ( ( plus_plus_multiset_a @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_multiset_a ) ) ).

% add_cancel_left_left
thf(fact_50_add__cancel__left__left,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = A2 )
      = ( B2 = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_51_add_Oright__neutral,axiom,
    ! [A2: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ A2 @ zero_z7348594199698428585et_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_52_add_Oright__neutral,axiom,
    ! [A2: multiset_a] :
      ( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
      = A2 ) ).

% add.right_neutral
thf(fact_53_add_Oright__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.right_neutral
thf(fact_54_i__of__j2__def,axiom,
    ! [J: nat] :
      ( ( i_of_j2 @ J )
      = ( fChoice_nat
        @ ^ [I: nat] :
            ( ( member_nat @ I @ ( set_nat2 @ xs22 ) )
            & ( cs @ I @ J ) ) ) ) ).

% i_of_j2_def
thf(fact_55_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ C )
      = ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_56_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_57_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ C )
      = ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_58_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: multiset_nat,J: multiset_nat,K2: multiset_nat,L: multiset_nat] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( ( plus_p6334493942879108393et_nat @ I2 @ K2 )
        = ( plus_p6334493942879108393et_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_59_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K2 = L ) )
     => ( ( plus_plus_nat @ I2 @ K2 )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_60_group__cancel_Oadd1,axiom,
    ! [A3: multiset_nat,K2: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
      ( ( A3
        = ( plus_p6334493942879108393et_nat @ K2 @ A2 ) )
     => ( ( plus_p6334493942879108393et_nat @ A3 @ B2 )
        = ( plus_p6334493942879108393et_nat @ K2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_61_group__cancel_Oadd1,axiom,
    ! [A3: nat,K2: nat,A2: nat,B2: nat] :
      ( ( A3
        = ( plus_plus_nat @ K2 @ A2 ) )
     => ( ( plus_plus_nat @ A3 @ B2 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_62_group__cancel_Oadd1,axiom,
    ! [A3: multiset_a,K2: multiset_a,A2: multiset_a,B2: multiset_a] :
      ( ( A3
        = ( plus_plus_multiset_a @ K2 @ A2 ) )
     => ( ( plus_plus_multiset_a @ A3 @ B2 )
        = ( plus_plus_multiset_a @ K2 @ ( plus_plus_multiset_a @ A2 @ B2 ) ) ) ) ).

% group_cancel.add1
thf(fact_63_group__cancel_Oadd2,axiom,
    ! [B3: multiset_nat,K2: multiset_nat,B2: multiset_nat,A2: multiset_nat] :
      ( ( B3
        = ( plus_p6334493942879108393et_nat @ K2 @ B2 ) )
     => ( ( plus_p6334493942879108393et_nat @ A2 @ B3 )
        = ( plus_p6334493942879108393et_nat @ K2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_64_group__cancel_Oadd2,axiom,
    ! [B3: nat,K2: nat,B2: nat,A2: nat] :
      ( ( B3
        = ( plus_plus_nat @ K2 @ B2 ) )
     => ( ( plus_plus_nat @ A2 @ B3 )
        = ( plus_plus_nat @ K2 @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_65_group__cancel_Oadd2,axiom,
    ! [B3: multiset_a,K2: multiset_a,B2: multiset_a,A2: multiset_a] :
      ( ( B3
        = ( plus_plus_multiset_a @ K2 @ B2 ) )
     => ( ( plus_plus_multiset_a @ A2 @ B3 )
        = ( plus_plus_multiset_a @ K2 @ ( plus_plus_multiset_a @ A2 @ B2 ) ) ) ) ).

% group_cancel.add2
thf(fact_66_add_Oassoc,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ C )
      = ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_67_add_Oassoc,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ C )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_68_add_Oassoc,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ C )
      = ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).

% add.assoc
thf(fact_69_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_70_mem__Collect__eq,axiom,
    ! [A2: multis3193088007478089820ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( member8421201611525043589ropVar @ A2 @ ( collec8658135508005558727ropVar @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_71_mem__Collect__eq,axiom,
    ! [A2: product_prod_a_a,P: product_prod_a_a > $o] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( collec3336397797384452498od_a_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_72_mem__Collect__eq,axiom,
    ! [A2: a,P: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_73_mem__Collect__eq,axiom,
    ! [A2: produc6518373309651786023iset_a,P: produc6518373309651786023iset_a > $o] :
      ( ( member5199237121806060112iset_a @ A2 @ ( collec6344740439120713362iset_a @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_74_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X3: nat] : ( member_nat @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_75_Collect__mem__eq,axiom,
    ! [A3: set_Mu5658373669842198716ropVar] :
      ( ( collec8658135508005558727ropVar
        @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_76_Collect__mem__eq,axiom,
    ! [A3: set_Product_prod_a_a] :
      ( ( collec3336397797384452498od_a_a
        @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_77_Collect__mem__eq,axiom,
    ! [A3: set_a] :
      ( ( collect_a
        @ ^ [X3: a] : ( member_a @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_78_Collect__mem__eq,axiom,
    ! [A3: set_Pr79727621955416071iset_a] :
      ( ( collec6344740439120713362iset_a
        @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_79_add_Ocommute,axiom,
    ( plus_p6334493942879108393et_nat
    = ( ^ [A: multiset_nat,B: multiset_nat] : ( plus_p6334493942879108393et_nat @ B @ A ) ) ) ).

% add.commute
thf(fact_80_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A: nat,B: nat] : ( plus_plus_nat @ B @ A ) ) ) ).

% add.commute
thf(fact_81_add_Ocommute,axiom,
    ( plus_plus_multiset_a
    = ( ^ [A: multiset_a,B: multiset_a] : ( plus_plus_multiset_a @ B @ A ) ) ) ).

% add.commute
thf(fact_82_add_Oleft__commute,axiom,
    ! [B2: multiset_nat,A2: multiset_nat,C: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A2 @ C ) )
      = ( plus_p6334493942879108393et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_83_add_Oleft__commute,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( plus_plus_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) )
      = ( plus_plus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_84_add_Oleft__commute,axiom,
    ! [B2: multiset_a,A2: multiset_a,C: multiset_a] :
      ( ( plus_plus_multiset_a @ B2 @ ( plus_plus_multiset_a @ A2 @ C ) )
      = ( plus_plus_multiset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).

% add.left_commute
thf(fact_85_add__left__imp__eq,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A2 @ B2 )
        = ( plus_p6334493942879108393et_nat @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_86_add__left__imp__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ( plus_plus_nat @ A2 @ B2 )
        = ( plus_plus_nat @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_87_add__left__imp__eq,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A2 @ B2 )
        = ( plus_plus_multiset_a @ A2 @ C ) )
     => ( B2 = C ) ) ).

% add_left_imp_eq
thf(fact_88_add__right__imp__eq,axiom,
    ! [B2: multiset_nat,A2: multiset_nat,C: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ B2 @ A2 )
        = ( plus_p6334493942879108393et_nat @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_89_add__right__imp__eq,axiom,
    ! [B2: nat,A2: nat,C: nat] :
      ( ( ( plus_plus_nat @ B2 @ A2 )
        = ( plus_plus_nat @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_90_add__right__imp__eq,axiom,
    ! [B2: multiset_a,A2: multiset_a,C: multiset_a] :
      ( ( ( plus_plus_multiset_a @ B2 @ A2 )
        = ( plus_plus_multiset_a @ C @ A2 ) )
     => ( B2 = C ) ) ).

% add_right_imp_eq
thf(fact_91_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_92_add_Ocomm__neutral,axiom,
    ! [A2: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ A2 @ zero_z7348594199698428585et_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_93_add_Ocomm__neutral,axiom,
    ! [A2: multiset_a] :
      ( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
      = A2 ) ).

% add.comm_neutral
thf(fact_94_add_Ocomm__neutral,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% add.comm_neutral
thf(fact_95_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_96_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_97_comm__monoid__add__class_Oadd__0,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A2 )
      = A2 ) ).

% comm_monoid_add_class.add_0
thf(fact_98_distinct__upt,axiom,
    ! [I2: nat,J: nat] : ( distinct_nat @ ( upt @ I2 @ J ) ) ).

% distinct_upt
thf(fact_99_distinct__drop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,I2: nat] :
      ( ( distin2675581014522817237ropVar @ Xs )
     => ( distin2675581014522817237ropVar @ ( drop_M2293212096379261346ropVar @ I2 @ Xs ) ) ) ).

% distinct_drop
thf(fact_100_distinct__drop,axiom,
    ! [Xs: list_nat,I2: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( drop_nat @ I2 @ Xs ) ) ) ).

% distinct_drop
thf(fact_101_zero__reorient,axiom,
    ! [X: nat] :
      ( ( zero_zero_nat = X )
      = ( X = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_102_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X3: nat] : X3
        @ T )
      = T ) ).

% list.map_ident
thf(fact_103_ex__map__conv,axiom,
    ! [Ys: list_nat,F: nat > nat] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_nat_nat @ F @ Xs2 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ? [Y: nat] :
                ( X3
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_104_ex__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_na2466558649495876678ropVar @ F @ Xs2 ) ) )
      = ( ! [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Ys ) )
           => ? [Y: nat] :
                ( X3
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_105_ex__map__conv,axiom,
    ! [Ys: list_a,F: nat > a] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_nat_a @ F @ Xs2 ) ) )
      = ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys ) )
           => ? [Y: nat] :
                ( X3
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_106_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ Xs )
          = ( map_na2466558649495876678ropVar @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_107_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_nat_nat @ F @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_108_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > a,G: nat > a] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_nat_a @ F @ Xs )
          = ( map_nat_a @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_109_map__idI,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,F: product_prod_a_a > product_prod_a_a] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Pr7904243085458786820od_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_110_map__idI,axiom,
    ! [Xs: list_P6988024003011616557iset_a,F: produc6518373309651786023iset_a > produc6518373309651786023iset_a] :
      ( ! [X4: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ X4 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Pr5905680810515755012iset_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_111_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_112_map__idI,axiom,
    ! [Xs: list_M6740372356978808802ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Mu2265489581466636612ropVar @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_113_map__idI,axiom,
    ! [Xs: list_a,F: a > a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_114_map__ext,axiom,
    ! [Xs: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Xs ) ) ) ).

% map_ext
thf(fact_115_map__ext,axiom,
    ! [Xs: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_116_map__ext,axiom,
    ! [Xs: list_nat,F: nat > a,G: nat > a] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Xs ) ) ) ).

% map_ext
thf(fact_117_list_Omap__ident__strong,axiom,
    ! [T: list_P1396940483166286381od_a_a,F: product_prod_a_a > product_prod_a_a] :
      ( ! [Z2: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ Z2 @ ( set_Product_prod_a_a2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_Pr7904243085458786820od_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_118_list_Omap__ident__strong,axiom,
    ! [T: list_P6988024003011616557iset_a,F: produc6518373309651786023iset_a > produc6518373309651786023iset_a] :
      ( ! [Z2: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ Z2 @ ( set_Pr5550698219922361532iset_a @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_Pr5905680810515755012iset_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_119_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_nat_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_120_list_Omap__ident__strong,axiom,
    ! [T: list_M6740372356978808802ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ! [Z2: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ Z2 @ ( set_Mu146256361055586289ropVar @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_Mu2265489581466636612ropVar @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_121_list_Omap__ident__strong,axiom,
    ! [T: list_a,F: a > a] :
      ( ! [Z2: a] :
          ( ( member_a @ Z2 @ ( set_a2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_122_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa: list_nat,F: nat > multis3193088007478089820ropVar,Fa: nat > multis3193088007478089820ropVar] :
      ( ! [Z2: nat,Za: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_na2466558649495876678ropVar @ F @ X )
          = ( map_na2466558649495876678ropVar @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_123_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
      ( ! [Z2: nat,Za: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_124_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa: list_nat,F: nat > a,Fa: nat > a] :
      ( ! [Z2: nat,Za: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_nat_a @ F @ X )
          = ( map_nat_a @ Fa @ Xa ) )
       => ( X = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_125_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_na2466558649495876678ropVar @ F @ X )
        = ( map_na2466558649495876678ropVar @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_126_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_nat_nat @ F @ X )
        = ( map_nat_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_127_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > a,G: nat > a] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_nat_a @ F @ X )
        = ( map_nat_a @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_128_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ( X = Ya )
     => ( ! [Z2: nat] :
            ( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ X )
          = ( map_na2466558649495876678ropVar @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_129_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
      ( ( X = Ya )
     => ( ! [Z2: nat] :
            ( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_130_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > a,G: nat > a] :
      ( ( X = Ya )
     => ( ! [Z2: nat] :
            ( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_nat_a @ F @ X )
          = ( map_nat_a @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_131_drop__0,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ zero_zero_nat @ Xs )
      = Xs ) ).

% drop_0
thf(fact_132_drop__0,axiom,
    ! [Xs: list_nat] :
      ( ( drop_nat @ zero_zero_nat @ Xs )
      = Xs ) ).

% drop_0
thf(fact_133_in__set__dropD,axiom,
    ! [X: product_prod_a_a,N: nat,Xs: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( drop_P8456769997282094189od_a_a @ N @ Xs ) ) )
     => ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_134_in__set__dropD,axiom,
    ! [X: produc6518373309651786023iset_a,N: nat,Xs: list_P6988024003011616557iset_a] :
      ( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ ( drop_P694518542769080941iset_a @ N @ Xs ) ) )
     => ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).

% in_set_dropD
thf(fact_135_in__set__dropD,axiom,
    ! [X: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_136_in__set__dropD,axiom,
    ! [X: multis3193088007478089820ropVar,N: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) ) )
     => ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% in_set_dropD
thf(fact_137_in__set__dropD,axiom,
    ! [X: a,N: nat,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ ( drop_a @ N @ Xs ) ) )
     => ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_138_drop__map,axiom,
    ! [N: nat,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( map_Mu2265489581466636612ropVar @ F @ Xs ) )
      = ( map_Mu2265489581466636612ropVar @ F @ ( drop_M2293212096379261346ropVar @ N @ Xs ) ) ) ).

% drop_map
thf(fact_139_drop__map,axiom,
    ! [N: nat,F: multis3193088007478089820ropVar > nat,Xs: list_M6740372356978808802ropVar] :
      ( ( drop_nat @ N @ ( map_Mu1970517791673442536ar_nat @ F @ Xs ) )
      = ( map_Mu1970517791673442536ar_nat @ F @ ( drop_M2293212096379261346ropVar @ N @ Xs ) ) ) ).

% drop_map
thf(fact_140_drop__map,axiom,
    ! [N: nat,F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( map_na2466558649495876678ropVar @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_141_drop__map,axiom,
    ! [N: nat,F: nat > nat,Xs: list_nat] :
      ( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_142_drop__map,axiom,
    ! [N: nat,F: nat > a,Xs: list_nat] :
      ( ( drop_a @ N @ ( map_nat_a @ F @ Xs ) )
      = ( map_nat_a @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_143_dist__xs,axiom,
    distinct_nat @ ( append_nat @ xs12 @ xs22 ) ).

% dist_xs
thf(fact_144_some__equality,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( P @ A2 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A2 ) )
       => ( ( fChoice_nat @ P )
          = A2 ) ) ) ).

% some_equality
thf(fact_145_some__eq__trivial,axiom,
    ! [X: nat] :
      ( ( fChoice_nat
        @ ^ [Y: nat] : ( Y = X ) )
      = X ) ).

% some_eq_trivial
thf(fact_146_some__sym__eq__trivial,axiom,
    ! [X: nat] :
      ( ( fChoice_nat
        @ ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 )
          @ X ) )
      = X ) ).

% some_sym_eq_trivial
thf(fact_147_dist__ys,axiom,
    distinct_nat @ ( append_nat @ ys12 @ ys22 ) ).

% dist_ys
thf(fact_148_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_M6740372356978808802ropVar,Y2: list_M6740372356978808802ropVar] :
      ( ( distin2675581014522817237ropVar @ X )
     => ( ( distin2675581014522817237ropVar @ Y2 )
       => ( ( ( set_Mu146256361055586289ropVar @ X )
            = ( set_Mu146256361055586289ropVar @ Y2 ) )
          = ( ( mset_M6581516532424202993ropVar @ X )
            = ( mset_M6581516532424202993ropVar @ Y2 ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_149_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( distinct_nat @ X )
     => ( ( distinct_nat @ Y2 )
       => ( ( ( set_nat2 @ X )
            = ( set_nat2 @ Y2 ) )
          = ( ( mset_nat @ X )
            = ( mset_nat @ Y2 ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_150_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_a,Y2: list_a] :
      ( ( distinct_a @ X )
     => ( ( distinct_a @ Y2 )
       => ( ( ( set_a2 @ X )
            = ( set_a2 @ Y2 ) )
          = ( ( mset_a @ X )
            = ( mset_a @ Y2 ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_151_mset__map__split,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys1: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
      ( ( ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys1 ) @ ( mset_M6581516532424202993ropVar @ Ys2 ) ) )
     => ? [Zs1: list_nat,Zs2: list_nat] :
          ( ( ( mset_nat @ Xs )
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
          & ( Ys1
            = ( map_na2466558649495876678ropVar @ F @ Zs1 ) )
          & ( Ys2
            = ( map_na2466558649495876678ropVar @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_152_mset__map__split,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys1: list_nat,Ys2: list_nat] :
      ( ( ( mset_nat @ ( map_nat_nat @ F @ Xs ) )
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys2 ) ) )
     => ? [Zs1: list_nat,Zs2: list_nat] :
          ( ( ( mset_nat @ Xs )
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
          & ( Ys1
            = ( map_nat_nat @ F @ Zs1 ) )
          & ( Ys2
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_153_mset__map__split,axiom,
    ! [F: a > nat,Xs: list_a,Ys1: list_nat,Ys2: list_nat] :
      ( ( ( mset_nat @ ( map_a_nat @ F @ Xs ) )
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys2 ) ) )
     => ? [Zs1: list_a,Zs2: list_a] :
          ( ( ( mset_a @ Xs )
            = ( plus_plus_multiset_a @ ( mset_a @ Zs1 ) @ ( mset_a @ Zs2 ) ) )
          & ( Ys1
            = ( map_a_nat @ F @ Zs1 ) )
          & ( Ys2
            = ( map_a_nat @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_154_mset__map__split,axiom,
    ! [F: nat > a,Xs: list_nat,Ys1: list_a,Ys2: list_a] :
      ( ( ( mset_a @ ( map_nat_a @ F @ Xs ) )
        = ( plus_plus_multiset_a @ ( mset_a @ Ys1 ) @ ( mset_a @ Ys2 ) ) )
     => ? [Zs1: list_nat,Zs2: list_nat] :
          ( ( ( mset_nat @ Xs )
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
          & ( Ys1
            = ( map_nat_a @ F @ Zs1 ) )
          & ( Ys2
            = ( map_nat_a @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_155_mset__map__split,axiom,
    ! [F: a > a,Xs: list_a,Ys1: list_a,Ys2: list_a] :
      ( ( ( mset_a @ ( map_a_a @ F @ Xs ) )
        = ( plus_plus_multiset_a @ ( mset_a @ Ys1 ) @ ( mset_a @ Ys2 ) ) )
     => ? [Zs1: list_a,Zs2: list_a] :
          ( ( ( mset_a @ Xs )
            = ( plus_plus_multiset_a @ ( mset_a @ Zs1 ) @ ( mset_a @ Zs2 ) ) )
          & ( Ys1
            = ( map_a_a @ F @ Zs1 ) )
          & ( Ys2
            = ( map_a_a @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_156_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X: multiset_nat,Y2: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ X @ Y2 )
        = zero_z7348594199698428585et_nat )
      = ( ( X = zero_z7348594199698428585et_nat )
        & ( Y2 = zero_z7348594199698428585et_nat ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_157_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X: multiset_a,Y2: multiset_a] :
      ( ( ( plus_plus_multiset_a @ X @ Y2 )
        = zero_zero_multiset_a )
      = ( ( X = zero_zero_multiset_a )
        & ( Y2 = zero_zero_multiset_a ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_158_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X: multiset_nat,Y2: multiset_nat] :
      ( ( zero_z7348594199698428585et_nat
        = ( plus_p6334493942879108393et_nat @ X @ Y2 ) )
      = ( ( X = zero_z7348594199698428585et_nat )
        & ( Y2 = zero_z7348594199698428585et_nat ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_159_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X: multiset_a,Y2: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ X @ Y2 ) )
      = ( ( X = zero_zero_multiset_a )
        & ( Y2 = zero_zero_multiset_a ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_160_empty__eq__union,axiom,
    ! [M: multiset_nat,N2: multiset_nat] :
      ( ( zero_z7348594199698428585et_nat
        = ( plus_p6334493942879108393et_nat @ M @ N2 ) )
      = ( ( M = zero_z7348594199698428585et_nat )
        & ( N2 = zero_z7348594199698428585et_nat ) ) ) ).

% empty_eq_union
thf(fact_161_empty__eq__union,axiom,
    ! [M: multiset_a,N2: multiset_a] :
      ( ( zero_zero_multiset_a
        = ( plus_plus_multiset_a @ M @ N2 ) )
      = ( ( M = zero_zero_multiset_a )
        & ( N2 = zero_zero_multiset_a ) ) ) ).

% empty_eq_union
thf(fact_162_append_Oassoc,axiom,
    ! [A2: list_nat,B2: list_nat,C: list_nat] :
      ( ( append_nat @ ( append_nat @ A2 @ B2 ) @ C )
      = ( append_nat @ A2 @ ( append_nat @ B2 @ C ) ) ) ).

% append.assoc
thf(fact_163_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_164_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_165_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_166_drop__upt,axiom,
    ! [M2: nat,I2: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I2 @ J ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_167_drop__drop,axiom,
    ! [N: nat,M2: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( drop_M2293212096379261346ropVar @ M2 @ Xs ) )
      = ( drop_M2293212096379261346ropVar @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ).

% drop_drop
thf(fact_168_drop__drop,axiom,
    ! [N: nat,M2: nat,Xs: list_nat] :
      ( ( drop_nat @ N @ ( drop_nat @ M2 @ Xs ) )
      = ( drop_nat @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ).

% drop_drop
thf(fact_169_union__eq__empty,axiom,
    ! [M: multiset_nat,N2: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ M @ N2 )
        = zero_z7348594199698428585et_nat )
      = ( ( M = zero_z7348594199698428585et_nat )
        & ( N2 = zero_z7348594199698428585et_nat ) ) ) ).

% union_eq_empty
thf(fact_170_union__eq__empty,axiom,
    ! [M: multiset_a,N2: multiset_a] :
      ( ( ( plus_plus_multiset_a @ M @ N2 )
        = zero_zero_multiset_a )
      = ( ( M = zero_zero_multiset_a )
        & ( N2 = zero_zero_multiset_a ) ) ) ).

% union_eq_empty
thf(fact_171_map__append,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append2684835082924936439ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ F @ Ys ) ) ) ).

% map_append
thf(fact_172_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_173_map__append,axiom,
    ! [F: nat > a,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_a @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_a @ ( map_nat_a @ F @ Xs ) @ ( map_nat_a @ F @ Ys ) ) ) ).

% map_append
thf(fact_174_mset__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( mset_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ ( mset_nat @ Ys ) ) ) ).

% mset_append
thf(fact_175_mset__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( mset_a @ ( append_a @ Xs @ Ys ) )
      = ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ ( mset_a @ Ys ) ) ) ).

% mset_append
thf(fact_176_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_177_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_178_append__eq__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( ( append2684835082924936439ropVar @ Ys @ Zs )
        = ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_na2466558649495876678ropVar @ F @ Us2 ) )
            & ( Zs
              = ( map_na2466558649495876678ropVar @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_179_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_180_append__eq__map__conv,axiom,
    ! [Ys: list_a,Zs: list_a,F: nat > a,Xs: list_nat] :
      ( ( ( append_a @ Ys @ Zs )
        = ( map_nat_a @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_a @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_a @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_181_map__eq__append__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( append2684835082924936439ropVar @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_na2466558649495876678ropVar @ F @ Us2 ) )
            & ( Zs
              = ( map_na2466558649495876678ropVar @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_182_map__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_183_map__eq__append__conv,axiom,
    ! [F: nat > a,Xs: list_nat,Ys: list_a,Zs: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( append_a @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_a @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_a @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_184_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_185_multi__union__self__other__eq,axiom,
    ! [A3: multiset_nat,X5: multiset_nat,Y4: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A3 @ X5 )
        = ( plus_p6334493942879108393et_nat @ A3 @ Y4 ) )
     => ( X5 = Y4 ) ) ).

% multi_union_self_other_eq
thf(fact_186_multi__union__self__other__eq,axiom,
    ! [A3: multiset_a,X5: multiset_a,Y4: multiset_a] :
      ( ( ( plus_plus_multiset_a @ A3 @ X5 )
        = ( plus_plus_multiset_a @ A3 @ Y4 ) )
     => ( X5 = Y4 ) ) ).

% multi_union_self_other_eq
thf(fact_187_union__right__cancel,axiom,
    ! [M: multiset_nat,K3: multiset_nat,N2: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ M @ K3 )
        = ( plus_p6334493942879108393et_nat @ N2 @ K3 ) )
      = ( M = N2 ) ) ).

% union_right_cancel
thf(fact_188_union__right__cancel,axiom,
    ! [M: multiset_a,K3: multiset_a,N2: multiset_a] :
      ( ( ( plus_plus_multiset_a @ M @ K3 )
        = ( plus_plus_multiset_a @ N2 @ K3 ) )
      = ( M = N2 ) ) ).

% union_right_cancel
thf(fact_189_union__left__cancel,axiom,
    ! [K3: multiset_nat,M: multiset_nat,N2: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ K3 @ M )
        = ( plus_p6334493942879108393et_nat @ K3 @ N2 ) )
      = ( M = N2 ) ) ).

% union_left_cancel
thf(fact_190_union__left__cancel,axiom,
    ! [K3: multiset_a,M: multiset_a,N2: multiset_a] :
      ( ( ( plus_plus_multiset_a @ K3 @ M )
        = ( plus_plus_multiset_a @ K3 @ N2 ) )
      = ( M = N2 ) ) ).

% union_left_cancel
thf(fact_191_union__commute,axiom,
    ( plus_p6334493942879108393et_nat
    = ( ^ [M3: multiset_nat,N3: multiset_nat] : ( plus_p6334493942879108393et_nat @ N3 @ M3 ) ) ) ).

% union_commute
thf(fact_192_union__commute,axiom,
    ( plus_plus_multiset_a
    = ( ^ [M3: multiset_a,N3: multiset_a] : ( plus_plus_multiset_a @ N3 @ M3 ) ) ) ).

% union_commute
thf(fact_193_union__lcomm,axiom,
    ! [M: multiset_nat,N2: multiset_nat,K3: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ M @ ( plus_p6334493942879108393et_nat @ N2 @ K3 ) )
      = ( plus_p6334493942879108393et_nat @ N2 @ ( plus_p6334493942879108393et_nat @ M @ K3 ) ) ) ).

% union_lcomm
thf(fact_194_union__lcomm,axiom,
    ! [M: multiset_a,N2: multiset_a,K3: multiset_a] :
      ( ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N2 @ K3 ) )
      = ( plus_plus_multiset_a @ N2 @ ( plus_plus_multiset_a @ M @ K3 ) ) ) ).

% union_lcomm
thf(fact_195_union__assoc,axiom,
    ! [M: multiset_nat,N2: multiset_nat,K3: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ M @ N2 ) @ K3 )
      = ( plus_p6334493942879108393et_nat @ M @ ( plus_p6334493942879108393et_nat @ N2 @ K3 ) ) ) ).

% union_assoc
thf(fact_196_union__assoc,axiom,
    ! [M: multiset_a,N2: multiset_a,K3: multiset_a] :
      ( ( plus_plus_multiset_a @ ( plus_plus_multiset_a @ M @ N2 ) @ K3 )
      = ( plus_plus_multiset_a @ M @ ( plus_plus_multiset_a @ N2 @ K3 ) ) ) ).

% union_assoc
thf(fact_197_empty__neutral_I1_J,axiom,
    ! [X: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ X )
      = X ) ).

% empty_neutral(1)
thf(fact_198_empty__neutral_I1_J,axiom,
    ! [X: multiset_a] :
      ( ( plus_plus_multiset_a @ zero_zero_multiset_a @ X )
      = X ) ).

% empty_neutral(1)
thf(fact_199_empty__neutral_I2_J,axiom,
    ! [X: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ X @ zero_z7348594199698428585et_nat )
      = X ) ).

% empty_neutral(2)
thf(fact_200_empty__neutral_I2_J,axiom,
    ! [X: multiset_a] :
      ( ( plus_plus_multiset_a @ X @ zero_zero_multiset_a )
      = X ) ).

% empty_neutral(2)
thf(fact_201_ex__mset,axiom,
    ! [X5: multiset_nat] :
    ? [Xs3: list_nat] :
      ( ( mset_nat @ Xs3 )
      = X5 ) ).

% ex_mset
thf(fact_202_ex__mset,axiom,
    ! [X5: multiset_a] :
    ? [Xs3: list_a] :
      ( ( mset_a @ Xs3 )
      = X5 ) ).

% ex_mset
thf(fact_203_some__eq__imp,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( ( fChoice_nat @ P )
        = A2 )
     => ( ( P @ B2 )
       => ( P @ A2 ) ) ) ).

% some_eq_imp
thf(fact_204_tfl__some,axiom,
    ! [P2: nat > $o,X6: nat] :
      ( ( P2 @ X6 )
     => ( P2 @ ( fChoice_nat @ P2 ) ) ) ).

% tfl_some
thf(fact_205_Eps__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( fChoice_nat @ P )
        = ( fChoice_nat @ Q ) ) ) ).

% Eps_cong
thf(fact_206_someI,axiom,
    ! [P: nat > $o,X: nat] :
      ( ( P @ X )
     => ( P @ ( fChoice_nat @ P ) ) ) ).

% someI
thf(fact_207_some1__equality,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y5: nat] :
              ( ( P @ Y5 )
             => ( Y5 = X6 ) ) )
     => ( ( P @ A2 )
       => ( ( fChoice_nat @ P )
          = A2 ) ) ) ).

% some1_equality
thf(fact_208_some__eq__ex,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X7: nat] : ( P @ X7 ) ) ) ).

% some_eq_ex
thf(fact_209_someI2__bex,axiom,
    ! [A3: set_Mu5658373669842198716ropVar,P: multis3193088007478089820ropVar > $o,Q: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: multis3193088007478089820ropVar] :
            ( ( ( member8421201611525043589ropVar @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoic539809714188224432ropVar
            @ ^ [X3: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X3 @ A3 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_210_someI2__bex,axiom,
    ! [A3: set_Product_prod_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ? [X6: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: product_prod_a_a] :
            ( ( ( member1426531477525435216od_a_a @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoic4124218645493772411od_a_a
            @ ^ [X3: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ X3 @ A3 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_211_someI2__bex,axiom,
    ! [A3: set_a,P: a > $o,Q: a > $o] :
      ( ? [X6: a] :
          ( ( member_a @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: a] :
            ( ( ( member_a @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoice_a
            @ ^ [X3: a] :
                ( ( member_a @ X3 @ A3 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_212_someI2__bex,axiom,
    ! [A3: set_Pr79727621955416071iset_a,P: produc6518373309651786023iset_a > $o,Q: produc6518373309651786023iset_a > $o] :
      ( ? [X6: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: produc6518373309651786023iset_a] :
            ( ( ( member5199237121806060112iset_a @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoic1152277584823490683iset_a
            @ ^ [X3: produc6518373309651786023iset_a] :
                ( ( member5199237121806060112iset_a @ X3 @ A3 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_213_someI2__bex,axiom,
    ! [A3: set_nat,P: nat > $o,Q: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ A3 )
          & ( P @ X6 ) )
     => ( ! [X4: nat] :
            ( ( ( member_nat @ X4 @ A3 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoice_nat
            @ ^ [X3: nat] :
                ( ( member_nat @ X3 @ A3 )
                & ( P @ X3 ) ) ) ) ) ) ).

% someI2_bex
thf(fact_214_someI2__ex,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( fChoice_nat @ P ) ) ) ) ).

% someI2_ex
thf(fact_215_someI__ex,axiom,
    ! [P: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( P @ ( fChoice_nat @ P ) ) ) ).

% someI_ex
thf(fact_216_someI2,axiom,
    ! [P: nat > $o,A2: nat,Q: nat > $o] :
      ( ( P @ A2 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( fChoice_nat @ P ) ) ) ) ).

% someI2
thf(fact_217_mset__eq__setD,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( mset_M6581516532424202993ropVar @ Xs )
        = ( mset_M6581516532424202993ropVar @ Ys ) )
     => ( ( set_Mu146256361055586289ropVar @ Xs )
        = ( set_Mu146256361055586289ropVar @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_218_mset__eq__setD,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( set_nat2 @ Xs )
        = ( set_nat2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_219_mset__eq__setD,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( mset_a @ Xs )
        = ( mset_a @ Ys ) )
     => ( ( set_a2 @ Xs )
        = ( set_a2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_220_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( distinct_nat @ Xs )
        = ( distinct_nat @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_221_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( mset_a @ Xs )
        = ( mset_a @ Ys ) )
     => ( ( distinct_a @ Xs )
        = ( distinct_a @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_222_add__is__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_223_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_224_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_225_add__0__iff,axiom,
    ! [B2: nat,A2: nat] :
      ( ( B2
        = ( plus_plus_nat @ B2 @ A2 ) )
      = ( A2 = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_226_verit__sum__simplify,axiom,
    ! [A2: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ A2 @ zero_z7348594199698428585et_nat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_227_verit__sum__simplify,axiom,
    ! [A2: multiset_a] :
      ( ( plus_plus_multiset_a @ A2 @ zero_zero_multiset_a )
      = A2 ) ).

% verit_sum_simplify
thf(fact_228_verit__sum__simplify,axiom,
    ! [A2: nat] :
      ( ( plus_plus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% verit_sum_simplify
thf(fact_229_PropVar_Osize__gen_I1_J,axiom,
    ! [X11: nat,X12: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(1)
thf(fact_230_subset__mset_Osum__list__eq__0__iff,axiom,
    ! [Ns: list_multiset_nat] :
      ( ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Ns )
        = zero_z7348594199698428585et_nat )
      = ( ! [X3: multiset_nat] :
            ( ( member_multiset_nat @ X3 @ ( set_multiset_nat2 @ Ns ) )
           => ( X3 = zero_z7348594199698428585et_nat ) ) ) ) ).

% subset_mset.sum_list_eq_0_iff
thf(fact_231_subset__mset_Osum__list__eq__0__iff,axiom,
    ! [Ns: list_multiset_a] :
      ( ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Ns )
        = zero_zero_multiset_a )
      = ( ! [X3: multiset_a] :
            ( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Ns ) )
           => ( X3 = zero_zero_multiset_a ) ) ) ) ).

% subset_mset.sum_list_eq_0_iff
thf(fact_232_verit__sko__forall__indirect2,axiom,
    ! [X: nat,P: nat > $o,P3: nat > $o] :
      ( ( X
        = ( fChoice_nat
          @ ^ [X3: nat] :
              ~ ( P @ X3 ) ) )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
            = ( P3 @ X4 ) )
       => ( ( ! [X7: nat] : ( P3 @ X7 ) )
          = ( P @ X ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_233_verit__sko__forall__indirect,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( X
        = ( fChoice_nat
          @ ^ [X3: nat] :
              ~ ( P @ X3 ) ) )
     => ( ( ! [X7: nat] : ( P @ X7 ) )
        = ( P @ X ) ) ) ).

% verit_sko_forall_indirect
thf(fact_234_add__eq__self__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = M2 )
     => ( N = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_235_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_236_verit__sko__ex_H,axiom,
    ! [P: nat > $o,A3: $o] :
      ( ( ( P @ ( fChoice_nat @ P ) )
        = A3 )
     => ( ( ? [X7: nat] : ( P @ X7 ) )
        = A3 ) ) ).

% verit_sko_ex'
thf(fact_237_verit__sko__forall,axiom,
    ( ( ^ [P4: nat > $o] :
        ! [X8: nat] : ( P4 @ X8 ) )
    = ( ^ [P5: nat > $o] :
          ( P5
          @ ( fChoice_nat
            @ ^ [X3: nat] :
                ~ ( P5 @ X3 ) ) ) ) ) ).

% verit_sko_forall
thf(fact_238_verit__sko__forall_H,axiom,
    ! [P: nat > $o,A3: $o] :
      ( ( ( P
          @ ( fChoice_nat
            @ ^ [X3: nat] :
                ~ ( P @ X3 ) ) )
        = A3 )
     => ( ( ! [X7: nat] : ( P @ X7 ) )
        = A3 ) ) ).

% verit_sko_forall'
thf(fact_239_verit__sko__forall_H_H,axiom,
    ! [B3: nat,A3: nat,P: nat > $o] :
      ( ( B3 = A3 )
     => ( ( ( fChoice_nat @ P )
          = A3 )
        = ( ( fChoice_nat @ P )
          = B3 ) ) ) ).

% verit_sko_forall''
thf(fact_240_verit__sko__ex__indirect,axiom,
    ! [X: nat,P: nat > $o] :
      ( ( X
        = ( fChoice_nat @ P ) )
     => ( ( ? [X7: nat] : ( P @ X7 ) )
        = ( P @ X ) ) ) ).

% verit_sko_ex_indirect
thf(fact_241_verit__sko__ex__indirect2,axiom,
    ! [X: nat,P: nat > $o,P3: nat > $o] :
      ( ( X
        = ( fChoice_nat @ P ) )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
            = ( P3 @ X4 ) )
       => ( ( ? [X7: nat] : ( P3 @ X7 ) )
          = ( P @ X ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_242_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A2: nat,B2: nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( P @ A4 @ B4 )
          = ( P @ B4 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B4: nat] :
              ( ( P @ A4 @ B4 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B4 ) ) )
         => ( P @ A2 @ B2 ) ) ) ) ).

% Euclid_induct
thf(fact_243_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
    ! [Xs: list_multiset_nat] :
      ( ! [X4: multiset_nat] :
          ( ( member_multiset_nat @ X4 @ ( set_multiset_nat2 @ Xs ) )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X4 ) )
     => ( ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs )
          = zero_z7348594199698428585et_nat )
        = ( ! [X3: multiset_nat] :
              ( ( member_multiset_nat @ X3 @ ( set_multiset_nat2 @ Xs ) )
             => ( X3 = zero_z7348594199698428585et_nat ) ) ) ) ) ).

% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_244_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
    ! [Xs: list_multiset_a] :
      ( ! [X4: multiset_a] :
          ( ( member_multiset_a @ X4 @ ( set_multiset_a2 @ Xs ) )
         => ( subseteq_mset_a @ zero_zero_multiset_a @ X4 ) )
     => ( ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs )
          = zero_zero_multiset_a )
        = ( ! [X3: multiset_a] :
              ( ( member_multiset_a @ X3 @ ( set_multiset_a2 @ Xs ) )
             => ( X3 = zero_zero_multiset_a ) ) ) ) ) ).

% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_245_subset__mset_Omember__le__sum__list,axiom,
    ! [X: multiset_nat,Xs: list_multiset_nat] :
      ( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
     => ( subseteq_mset_nat @ X @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) ) ) ).

% subset_mset.member_le_sum_list
thf(fact_246_subset__mset_Omember__le__sum__list,axiom,
    ! [X: multiset_a,Xs: list_multiset_a] :
      ( ( member_multiset_a @ X @ ( set_multiset_a2 @ Xs ) )
     => ( subseteq_mset_a @ X @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) ) ) ).

% subset_mset.member_le_sum_list
thf(fact_247_subset__mset_Osum__list__nonpos,axiom,
    ! [Xs: list_multiset_nat] :
      ( ! [X4: multiset_nat] :
          ( ( member_multiset_nat @ X4 @ ( set_multiset_nat2 @ Xs ) )
         => ( subseteq_mset_nat @ X4 @ zero_z7348594199698428585et_nat ) )
     => ( subseteq_mset_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.sum_list_nonpos
thf(fact_248_subset__mset_Osum__list__nonpos,axiom,
    ! [Xs: list_multiset_a] :
      ( ! [X4: multiset_a] :
          ( ( member_multiset_a @ X4 @ ( set_multiset_a2 @ Xs ) )
         => ( subseteq_mset_a @ X4 @ zero_zero_multiset_a ) )
     => ( subseteq_mset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) @ zero_zero_multiset_a ) ) ).

% subset_mset.sum_list_nonpos
thf(fact_249_subset__mset_Osum__list__nonneg,axiom,
    ! [Xs: list_multiset_nat] :
      ( ! [X4: multiset_nat] :
          ( ( member_multiset_nat @ X4 @ ( set_multiset_nat2 @ Xs ) )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X4 ) )
     => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) ) ) ).

% subset_mset.sum_list_nonneg
thf(fact_250_subset__mset_Osum__list__nonneg,axiom,
    ! [Xs: list_multiset_a] :
      ( ! [X4: multiset_a] :
          ( ( member_multiset_a @ X4 @ ( set_multiset_a2 @ Xs ) )
         => ( subseteq_mset_a @ zero_zero_multiset_a @ X4 ) )
     => ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) ) ) ).

% subset_mset.sum_list_nonneg
thf(fact_251_map__eq__map__tailrec,axiom,
    map_na2466558649495876678ropVar = map_ta2313973780298087292ropVar ).

% map_eq_map_tailrec
thf(fact_252_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_253_map__eq__map__tailrec,axiom,
    map_nat_a = map_tailrec_nat_a ).

% map_eq_map_tailrec
thf(fact_254_un__xs,axiom,
    ( ( sup_sup_set_nat @ ( set_nat2 @ xs12 ) @ ( set_nat2 @ xs22 ) )
    = ( set_ord_lessThan_nat @ n ) ) ).

% un_xs
thf(fact_255_un__ys,axiom,
    ( ( sup_sup_set_nat @ ( set_nat2 @ ys12 ) @ ( set_nat2 @ ys22 ) )
    = ( set_ord_lessThan_nat @ m ) ) ).

% un_ys
thf(fact_256_mset__subset__eq__mono__add__right__cancel,axiom,
    ! [A3: multiset_nat,C2: multiset_nat,B3: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ B3 @ C2 ) )
      = ( subseteq_mset_nat @ A3 @ B3 ) ) ).

% mset_subset_eq_mono_add_right_cancel
thf(fact_257_mset__subset__eq__mono__add__right__cancel,axiom,
    ! [A3: multiset_a,C2: multiset_a,B3: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B3 @ C2 ) )
      = ( subseteq_mset_a @ A3 @ B3 ) ) ).

% mset_subset_eq_mono_add_right_cancel
thf(fact_258_mset__subset__eq__mono__add__left__cancel,axiom,
    ! [C2: multiset_nat,A3: multiset_nat,B3: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C2 @ A3 ) @ ( plus_p6334493942879108393et_nat @ C2 @ B3 ) )
      = ( subseteq_mset_nat @ A3 @ B3 ) ) ).

% mset_subset_eq_mono_add_left_cancel
thf(fact_259_mset__subset__eq__mono__add__left__cancel,axiom,
    ! [C2: multiset_a,A3: multiset_a,B3: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C2 @ A3 ) @ ( plus_plus_multiset_a @ C2 @ B3 ) )
      = ( subseteq_mset_a @ A3 @ B3 ) ) ).

% mset_subset_eq_mono_add_left_cancel
thf(fact_260_subset__mset_Oadd__le__cancel__right,axiom,
    ! [A2: multiset_nat,C: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ C ) )
      = ( subseteq_mset_nat @ A2 @ B2 ) ) ).

% subset_mset.add_le_cancel_right
thf(fact_261_subset__mset_Oadd__le__cancel__right,axiom,
    ! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) )
      = ( subseteq_mset_a @ A2 @ B2 ) ) ).

% subset_mset.add_le_cancel_right
thf(fact_262_subset__mset_Oadd__le__cancel__left,axiom,
    ! [C: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A2 ) @ ( plus_p6334493942879108393et_nat @ C @ B2 ) )
      = ( subseteq_mset_nat @ A2 @ B2 ) ) ).

% subset_mset.add_le_cancel_left
thf(fact_263_subset__mset_Oadd__le__cancel__left,axiom,
    ! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) )
      = ( subseteq_mset_a @ A2 @ B2 ) ) ).

% subset_mset.add_le_cancel_left
thf(fact_264_set__append,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( set_Mu146256361055586289ropVar @ ( append2684835082924936439ropVar @ Xs @ Ys ) )
      = ( sup_su4102146785206298256ropVar @ ( set_Mu146256361055586289ropVar @ Xs ) @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ).

% set_append
thf(fact_265_set__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( append_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_append
thf(fact_266_set__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).

% set_append
thf(fact_267_subset__mset_Ole__add__same__cancel2,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ A2 ) )
      = ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B2 ) ) ).

% subset_mset.le_add_same_cancel2
thf(fact_268_subset__mset_Ole__add__same__cancel2,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ A2 ) )
      = ( subseteq_mset_a @ zero_zero_multiset_a @ B2 ) ) ).

% subset_mset.le_add_same_cancel2
thf(fact_269_subset__mset_Ole__add__same__cancel1,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) )
      = ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B2 ) ) ).

% subset_mset.le_add_same_cancel1
thf(fact_270_subset__mset_Ole__add__same__cancel1,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B2 ) )
      = ( subseteq_mset_a @ zero_zero_multiset_a @ B2 ) ) ).

% subset_mset.le_add_same_cancel1
thf(fact_271_subset__mset_Oadd__le__same__cancel2,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ B2 )
      = ( subseteq_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.add_le_same_cancel2
thf(fact_272_subset__mset_Oadd__le__same__cancel2,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ B2 )
      = ( subseteq_mset_a @ A2 @ zero_zero_multiset_a ) ) ).

% subset_mset.add_le_same_cancel2
thf(fact_273_subset__mset_Oadd__le__same__cancel1,axiom,
    ! [B2: multiset_nat,A2: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ B2 @ A2 ) @ B2 )
      = ( subseteq_mset_nat @ A2 @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.add_le_same_cancel1
thf(fact_274_subset__mset_Oadd__le__same__cancel1,axiom,
    ! [B2: multiset_a,A2: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ B2 @ A2 ) @ B2 )
      = ( subseteq_mset_a @ A2 @ zero_zero_multiset_a ) ) ).

% subset_mset.add_le_same_cancel1
thf(fact_275_set__union,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( set_Mu146256361055586289ropVar @ ( union_4893538870499498070ropVar @ Xs @ Ys ) )
      = ( sup_su4102146785206298256ropVar @ ( set_Mu146256361055586289ropVar @ Xs ) @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ).

% set_union
thf(fact_276_set__union,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( union_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_union
thf(fact_277_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_278_mset__subset__eq__exists__conv,axiom,
    ( subseteq_mset_nat
    = ( ^ [A5: multiset_nat,B5: multiset_nat] :
        ? [C3: multiset_nat] :
          ( B5
          = ( plus_p6334493942879108393et_nat @ A5 @ C3 ) ) ) ) ).

% mset_subset_eq_exists_conv
thf(fact_279_mset__subset__eq__exists__conv,axiom,
    ( subseteq_mset_a
    = ( ^ [A5: multiset_a,B5: multiset_a] :
        ? [C3: multiset_a] :
          ( B5
          = ( plus_plus_multiset_a @ A5 @ C3 ) ) ) ) ).

% mset_subset_eq_exists_conv
thf(fact_280_mset__subset__eq__add__right,axiom,
    ! [B3: multiset_nat,A3: multiset_nat] : ( subseteq_mset_nat @ B3 @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) ) ).

% mset_subset_eq_add_right
thf(fact_281_mset__subset__eq__add__right,axiom,
    ! [B3: multiset_a,A3: multiset_a] : ( subseteq_mset_a @ B3 @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ).

% mset_subset_eq_add_right
thf(fact_282_mset__subset__eq__mono__add,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,C2: multiset_nat,D: multiset_nat] :
      ( ( subseteq_mset_nat @ A3 @ B3 )
     => ( ( subseteq_mset_nat @ C2 @ D )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ B3 @ D ) ) ) ) ).

% mset_subset_eq_mono_add
thf(fact_283_mset__subset__eq__mono__add,axiom,
    ! [A3: multiset_a,B3: multiset_a,C2: multiset_a,D: multiset_a] :
      ( ( subseteq_mset_a @ A3 @ B3 )
     => ( ( subseteq_mset_a @ C2 @ D )
       => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B3 @ D ) ) ) ) ).

% mset_subset_eq_mono_add
thf(fact_284_mset__subset__eq__add__left,axiom,
    ! [A3: multiset_nat,B3: multiset_nat] : ( subseteq_mset_nat @ A3 @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) ) ).

% mset_subset_eq_add_left
thf(fact_285_mset__subset__eq__add__left,axiom,
    ! [A3: multiset_a,B3: multiset_a] : ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ A3 @ B3 ) ) ).

% mset_subset_eq_add_left
thf(fact_286_subset__mset_Oadd__le__imp__le__right,axiom,
    ! [A2: multiset_nat,C: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ C ) )
     => ( subseteq_mset_nat @ A2 @ B2 ) ) ).

% subset_mset.add_le_imp_le_right
thf(fact_287_subset__mset_Oadd__le__imp__le__right,axiom,
    ! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) )
     => ( subseteq_mset_a @ A2 @ B2 ) ) ).

% subset_mset.add_le_imp_le_right
thf(fact_288_subset__mset_Oadd__le__imp__le__left,axiom,
    ! [C: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A2 ) @ ( plus_p6334493942879108393et_nat @ C @ B2 ) )
     => ( subseteq_mset_nat @ A2 @ B2 ) ) ).

% subset_mset.add_le_imp_le_left
thf(fact_289_subset__mset_Oadd__le__imp__le__left,axiom,
    ! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) )
     => ( subseteq_mset_a @ A2 @ B2 ) ) ).

% subset_mset.add_le_imp_le_left
thf(fact_290_subset__mset_Oadd__right__mono,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).

% subset_mset.add_right_mono
thf(fact_291_subset__mset_Oadd__right__mono,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).

% subset_mset.add_right_mono
thf(fact_292_subset__mset_Oadd__left__mono,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A2 ) @ ( plus_p6334493942879108393et_nat @ C @ B2 ) ) ) ).

% subset_mset.add_left_mono
thf(fact_293_subset__mset_Oadd__left__mono,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) ) ) ).

% subset_mset.add_left_mono
thf(fact_294_subset__mset_Ole__iff__add,axiom,
    ( subseteq_mset_nat
    = ( ^ [A: multiset_nat,B: multiset_nat] :
        ? [C4: multiset_nat] :
          ( B
          = ( plus_p6334493942879108393et_nat @ A @ C4 ) ) ) ) ).

% subset_mset.le_iff_add
thf(fact_295_subset__mset_Ole__iff__add,axiom,
    ( subseteq_mset_a
    = ( ^ [A: multiset_a,B: multiset_a] :
        ? [C4: multiset_a] :
          ( B
          = ( plus_plus_multiset_a @ A @ C4 ) ) ) ) ).

% subset_mset.le_iff_add
thf(fact_296_subset__mset_Oless__eqE,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ~ ! [C5: multiset_nat] :
            ( B2
           != ( plus_p6334493942879108393et_nat @ A2 @ C5 ) ) ) ).

% subset_mset.less_eqE
thf(fact_297_subset__mset_Oless__eqE,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ~ ! [C5: multiset_a] :
            ( B2
           != ( plus_plus_multiset_a @ A2 @ C5 ) ) ) ).

% subset_mset.less_eqE
thf(fact_298_subset__mset_Oadd__mono,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat,D2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( subseteq_mset_nat @ C @ D2 )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ D2 ) ) ) ) ).

% subset_mset.add_mono
thf(fact_299_subset__mset_Oadd__mono,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a,D2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( subseteq_mset_a @ C @ D2 )
       => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ D2 ) ) ) ) ).

% subset_mset.add_mono
thf(fact_300_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
    ! [X: multiset_nat,Y2: multiset_nat] :
      ( ( subseteq_mset_nat @ X @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ Y2 @ zero_z7348594199698428585et_nat )
       => ( ( ( plus_p6334493942879108393et_nat @ X @ Y2 )
            = zero_z7348594199698428585et_nat )
          = ( ( X = zero_z7348594199698428585et_nat )
            & ( Y2 = zero_z7348594199698428585et_nat ) ) ) ) ) ).

% subset_mset.add_nonpos_eq_0_iff
thf(fact_301_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
    ! [X: multiset_a,Y2: multiset_a] :
      ( ( subseteq_mset_a @ X @ zero_zero_multiset_a )
     => ( ( subseteq_mset_a @ Y2 @ zero_zero_multiset_a )
       => ( ( ( plus_plus_multiset_a @ X @ Y2 )
            = zero_zero_multiset_a )
          = ( ( X = zero_zero_multiset_a )
            & ( Y2 = zero_zero_multiset_a ) ) ) ) ) ).

% subset_mset.add_nonpos_eq_0_iff
thf(fact_302_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
    ! [X: multiset_nat,Y2: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X )
     => ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ Y2 )
       => ( ( ( plus_p6334493942879108393et_nat @ X @ Y2 )
            = zero_z7348594199698428585et_nat )
          = ( ( X = zero_z7348594199698428585et_nat )
            & ( Y2 = zero_z7348594199698428585et_nat ) ) ) ) ) ).

% subset_mset.add_nonneg_eq_0_iff
thf(fact_303_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
    ! [X: multiset_a,Y2: multiset_a] :
      ( ( subseteq_mset_a @ zero_zero_multiset_a @ X )
     => ( ( subseteq_mset_a @ zero_zero_multiset_a @ Y2 )
       => ( ( ( plus_plus_multiset_a @ X @ Y2 )
            = zero_zero_multiset_a )
          = ( ( X = zero_zero_multiset_a )
            & ( Y2 = zero_zero_multiset_a ) ) ) ) ) ).

% subset_mset.add_nonneg_eq_0_iff
thf(fact_304_subset__mset_Oadd__nonpos__nonpos,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ B2 @ zero_z7348594199698428585et_nat )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ zero_z7348594199698428585et_nat ) ) ) ).

% subset_mset.add_nonpos_nonpos
thf(fact_305_subset__mset_Oadd__nonpos__nonpos,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
     => ( ( subseteq_mset_a @ B2 @ zero_zero_multiset_a )
       => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ zero_zero_multiset_a ) ) ) ).

% subset_mset.add_nonpos_nonpos
thf(fact_306_subset__mset_Oadd__nonneg__nonneg,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A2 )
     => ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B2 )
       => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) ) ) ) ).

% subset_mset.add_nonneg_nonneg
thf(fact_307_subset__mset_Oadd__nonneg__nonneg,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
     => ( ( subseteq_mset_a @ zero_zero_multiset_a @ B2 )
       => ( subseteq_mset_a @ zero_zero_multiset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) ) ) ) ).

% subset_mset.add_nonneg_nonneg
thf(fact_308_subset__mset_Oadd__increasing2,axiom,
    ! [C: multiset_nat,B2: multiset_nat,A2: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ C )
     => ( ( subseteq_mset_nat @ B2 @ A2 )
       => ( subseteq_mset_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A2 @ C ) ) ) ) ).

% subset_mset.add_increasing2
thf(fact_309_subset__mset_Oadd__increasing2,axiom,
    ! [C: multiset_a,B2: multiset_a,A2: multiset_a] :
      ( ( subseteq_mset_a @ zero_zero_multiset_a @ C )
     => ( ( subseteq_mset_a @ B2 @ A2 )
       => ( subseteq_mset_a @ B2 @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).

% subset_mset.add_increasing2
thf(fact_310_subset__mset_Oadd__decreasing2,axiom,
    ! [C: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ C @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ A2 @ B2 )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ B2 ) ) ) ).

% subset_mset.add_decreasing2
thf(fact_311_subset__mset_Oadd__decreasing2,axiom,
    ! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ C @ zero_zero_multiset_a )
     => ( ( subseteq_mset_a @ A2 @ B2 )
       => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ B2 ) ) ) ).

% subset_mset.add_decreasing2
thf(fact_312_subset__mset_Oadd__increasing,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A2 )
     => ( ( subseteq_mset_nat @ B2 @ C )
       => ( subseteq_mset_nat @ B2 @ ( plus_p6334493942879108393et_nat @ A2 @ C ) ) ) ) ).

% subset_mset.add_increasing
thf(fact_313_subset__mset_Oadd__increasing,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ zero_zero_multiset_a @ A2 )
     => ( ( subseteq_mset_a @ B2 @ C )
       => ( subseteq_mset_a @ B2 @ ( plus_plus_multiset_a @ A2 @ C ) ) ) ) ).

% subset_mset.add_increasing
thf(fact_314_subset__mset_Oadd__decreasing,axiom,
    ! [A2: multiset_nat,C: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ C @ B2 )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ B2 ) ) ) ).

% subset_mset.add_decreasing
thf(fact_315_subset__mset_Oadd__decreasing,axiom,
    ! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ zero_zero_multiset_a )
     => ( ( subseteq_mset_a @ C @ B2 )
       => ( subseteq_mset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ B2 ) ) ) ).

% subset_mset.add_decreasing
thf(fact_316_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N4: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N4 ) ) ) ) ).

% atLeast_upt
thf(fact_317_lessThan__eq__iff,axiom,
    ! [X: nat,Y2: nat] :
      ( ( ( set_ord_lessThan_nat @ X )
        = ( set_ord_lessThan_nat @ Y2 ) )
      = ( X = Y2 ) ) ).

% lessThan_eq_iff
thf(fact_318_Un__iff,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ ( sup_su4102146785206298256ropVar @ A3 @ B3 ) )
      = ( ( member8421201611525043589ropVar @ C @ A3 )
        | ( member8421201611525043589ropVar @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_319_Un__iff,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B3 ) )
      = ( ( member1426531477525435216od_a_a @ C @ A3 )
        | ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_320_Un__iff,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
      = ( ( member_a @ C @ A3 )
        | ( member_a @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_321_Un__iff,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B3 ) )
      = ( ( member5199237121806060112iset_a @ C @ A3 )
        | ( member5199237121806060112iset_a @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_322_Un__iff,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( ( member_nat @ C @ A3 )
        | ( member_nat @ C @ B3 ) ) ) ).

% Un_iff
thf(fact_323_UnCI,axiom,
    ! [C: multis3193088007478089820ropVar,B3: set_Mu5658373669842198716ropVar,A3: set_Mu5658373669842198716ropVar] :
      ( ( ~ ( member8421201611525043589ropVar @ C @ B3 )
       => ( member8421201611525043589ropVar @ C @ A3 ) )
     => ( member8421201611525043589ropVar @ C @ ( sup_su4102146785206298256ropVar @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_324_UnCI,axiom,
    ! [C: product_prod_a_a,B3: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
      ( ( ~ ( member1426531477525435216od_a_a @ C @ B3 )
       => ( member1426531477525435216od_a_a @ C @ A3 ) )
     => ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_325_UnCI,axiom,
    ! [C: a,B3: set_a,A3: set_a] :
      ( ( ~ ( member_a @ C @ B3 )
       => ( member_a @ C @ A3 ) )
     => ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_326_UnCI,axiom,
    ! [C: produc6518373309651786023iset_a,B3: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
      ( ( ~ ( member5199237121806060112iset_a @ C @ B3 )
       => ( member5199237121806060112iset_a @ C @ A3 ) )
     => ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_327_UnCI,axiom,
    ! [C: nat,B3: set_nat,A3: set_nat] :
      ( ( ~ ( member_nat @ C @ B3 )
       => ( member_nat @ C @ A3 ) )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_328_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_329_sup__left__idem,axiom,
    ! [X: set_nat,Y2: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
      = ( sup_sup_set_nat @ X @ Y2 ) ) ).

% sup_left_idem
thf(fact_330_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_331_sup__idem,axiom,
    ! [X: set_nat] :
      ( ( sup_sup_set_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_332_sup_Oidem,axiom,
    ! [A2: set_nat] :
      ( ( sup_sup_set_nat @ A2 @ A2 )
      = A2 ) ).

% sup.idem
thf(fact_333_sup__set__def,axiom,
    ( sup_su4102146785206298256ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
          ( collec8658135508005558727ropVar
          @ ( sup_su5656099892504388685pVar_o
            @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ A5 )
            @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_334_sup__set__def,axiom,
    ( sup_su3048258781599657691od_a_a
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
          ( collec3336397797384452498od_a_a
          @ ( sup_su1640154476453168578_a_a_o
            @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A5 )
            @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_335_sup__set__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ( sup_sup_a_o
            @ ^ [X3: a] : ( member_a @ X3 @ A5 )
            @ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_336_sup__set__def,axiom,
    ( sup_su1925787886501348827iset_a
    = ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
          ( collec6344740439120713362iset_a
          @ ( sup_su2013817538080420546et_a_o
            @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ A5 )
            @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_337_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
            @ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_338_inf__sup__aci_I8_J,axiom,
    ! [X: set_nat,Y2: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ X @ Y2 ) )
      = ( sup_sup_set_nat @ X @ Y2 ) ) ).

% inf_sup_aci(8)
thf(fact_339_inf__sup__aci_I7_J,axiom,
    ! [X: set_nat,Y2: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z4 ) )
      = ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z4 ) ) ) ).

% inf_sup_aci(7)
thf(fact_340_inf__sup__aci_I6_J,axiom,
    ! [X: set_nat,Y2: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z4 )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z4 ) ) ) ).

% inf_sup_aci(6)
thf(fact_341_inf__sup__aci_I5_J,axiom,
    ( sup_sup_set_nat
    = ( ^ [X3: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X3 ) ) ) ).

% inf_sup_aci(5)
thf(fact_342_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_343_sup__assoc,axiom,
    ! [X: set_nat,Y2: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ X @ Y2 ) @ Z4 )
      = ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z4 ) ) ) ).

% sup_assoc
thf(fact_344_sup_Ocommute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A: set_nat,B: set_nat] : ( sup_sup_set_nat @ B @ A ) ) ) ).

% sup.commute
thf(fact_345_sup__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [X3: set_nat,Y: set_nat] : ( sup_sup_set_nat @ Y @ X3 ) ) ) ).

% sup_commute
thf(fact_346_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_347_sup__left__commute,axiom,
    ! [X: set_nat,Y2: set_nat,Z4: set_nat] :
      ( ( sup_sup_set_nat @ X @ ( sup_sup_set_nat @ Y2 @ Z4 ) )
      = ( sup_sup_set_nat @ Y2 @ ( sup_sup_set_nat @ X @ Z4 ) ) ) ).

% sup_left_commute
thf(fact_348_UnE,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ ( sup_su4102146785206298256ropVar @ A3 @ B3 ) )
     => ( ~ ( member8421201611525043589ropVar @ C @ A3 )
       => ( member8421201611525043589ropVar @ C @ B3 ) ) ) ).

% UnE
thf(fact_349_UnE,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B3 ) )
     => ( ~ ( member1426531477525435216od_a_a @ C @ A3 )
       => ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% UnE
thf(fact_350_UnE,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) )
     => ( ~ ( member_a @ C @ A3 )
       => ( member_a @ C @ B3 ) ) ) ).

% UnE
thf(fact_351_UnE,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B3 ) )
     => ( ~ ( member5199237121806060112iset_a @ C @ A3 )
       => ( member5199237121806060112iset_a @ C @ B3 ) ) ) ).

% UnE
thf(fact_352_UnE,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) )
     => ( ~ ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B3 ) ) ) ).

% UnE
thf(fact_353_UnI1,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ A3 )
     => ( member8421201611525043589ropVar @ C @ ( sup_su4102146785206298256ropVar @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_354_UnI1,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ A3 )
     => ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_355_UnI1,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ A3 )
     => ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_356_UnI1,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ A3 )
     => ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_357_UnI1,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_358_UnI2,axiom,
    ! [C: multis3193088007478089820ropVar,B3: set_Mu5658373669842198716ropVar,A3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ B3 )
     => ( member8421201611525043589ropVar @ C @ ( sup_su4102146785206298256ropVar @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_359_UnI2,axiom,
    ! [C: product_prod_a_a,B3: set_Product_prod_a_a,A3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ B3 )
     => ( member1426531477525435216od_a_a @ C @ ( sup_su3048258781599657691od_a_a @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_360_UnI2,axiom,
    ! [C: a,B3: set_a,A3: set_a] :
      ( ( member_a @ C @ B3 )
     => ( member_a @ C @ ( sup_sup_set_a @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_361_UnI2,axiom,
    ! [C: produc6518373309651786023iset_a,B3: set_Pr79727621955416071iset_a,A3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ B3 )
     => ( member5199237121806060112iset_a @ C @ ( sup_su1925787886501348827iset_a @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_362_UnI2,axiom,
    ! [C: nat,B3: set_nat,A3: set_nat] :
      ( ( member_nat @ C @ B3 )
     => ( member_nat @ C @ ( sup_sup_set_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_363_bex__Un,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
            & ( P @ X3 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
            & ( P @ X3 ) )
        | ? [X3: nat] :
            ( ( member_nat @ X3 @ B3 )
            & ( P @ X3 ) ) ) ) ).

% bex_Un
thf(fact_364_ball__Un,axiom,
    ! [A3: set_nat,B3: set_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
           => ( P @ X3 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ A3 )
           => ( P @ X3 ) )
        & ! [X3: nat] :
            ( ( member_nat @ X3 @ B3 )
           => ( P @ X3 ) ) ) ) ).

% ball_Un
thf(fact_365_Un__assoc,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C2 )
      = ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C2 ) ) ) ).

% Un_assoc
thf(fact_366_Un__absorb,axiom,
    ! [A3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ A3 )
      = A3 ) ).

% Un_absorb
thf(fact_367_Un__commute,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] : ( sup_sup_set_nat @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_368_Un__left__absorb,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ A3 @ B3 ) )
      = ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_left_absorb
thf(fact_369_Un__left__commute,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( sup_sup_set_nat @ B3 @ C2 ) )
      = ( sup_sup_set_nat @ B3 @ ( sup_sup_set_nat @ A3 @ C2 ) ) ) ).

% Un_left_commute
thf(fact_370_Bex__def,axiom,
    ( bex_nat
    = ( ^ [A5: set_nat,P5: nat > $o] :
        ? [X3: nat] :
          ( ( member_nat @ X3 @ A5 )
          & ( P5 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_371_Bex__def,axiom,
    ( bex_Mu7070592493510950844ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,P5: multis3193088007478089820ropVar > $o] :
        ? [X3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X3 @ A5 )
          & ( P5 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_372_Bex__def,axiom,
    ( bex_Product_prod_a_a
    = ( ^ [A5: set_Product_prod_a_a,P5: product_prod_a_a > $o] :
        ? [X3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X3 @ A5 )
          & ( P5 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_373_Bex__def,axiom,
    ( bex_a
    = ( ^ [A5: set_a,P5: a > $o] :
        ? [X3: a] :
          ( ( member_a @ X3 @ A5 )
          & ( P5 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_374_Bex__def,axiom,
    ( bex_Pr5870656025513623687iset_a
    = ( ^ [A5: set_Pr79727621955416071iset_a,P5: produc6518373309651786023iset_a > $o] :
        ? [X3: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ X3 @ A5 )
          & ( P5 @ X3 ) ) ) ) ).

% Bex_def
thf(fact_375_Un__def,axiom,
    ( sup_su4102146785206298256ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
          ( collec8658135508005558727ropVar
          @ ^ [X3: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X3 @ A5 )
              | ( member8421201611525043589ropVar @ X3 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_376_Un__def,axiom,
    ( sup_su3048258781599657691od_a_a
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
          ( collec3336397797384452498od_a_a
          @ ^ [X3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X3 @ A5 )
              | ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_377_Un__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A5 )
              | ( member_a @ X3 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_378_Un__def,axiom,
    ( sup_su1925787886501348827iset_a
    = ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
          ( collec6344740439120713362iset_a
          @ ^ [X3: produc6518373309651786023iset_a] :
              ( ( member5199237121806060112iset_a @ X3 @ A5 )
              | ( member5199237121806060112iset_a @ X3 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_379_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ A5 )
              | ( member_nat @ X3 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_380_Collect__disj__eq,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( collect_nat
        @ ^ [X3: nat] :
            ( ( P @ X3 )
            | ( Q @ X3 ) ) )
      = ( sup_sup_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_381_mset__set__upto__eq__mset__upto,axiom,
    ! [N: nat] :
      ( ( mset_set_nat @ ( set_ord_lessThan_nat @ N ) )
      = ( mset_nat @ ( upt @ zero_zero_nat @ N ) ) ) ).

% mset_set_upto_eq_mset_upto
thf(fact_382_subset__mset_Osum__mono,axiom,
    ! [K3: set_nat,F: nat > multiset_nat,G: nat > multiset_nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K3 )
         => ( subseteq_mset_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_nat @ ( groups5394990218802192790at_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ K3 ) @ ( groups5394990218802192790at_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_383_subset__mset_Osum__mono,axiom,
    ! [K3: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > multiset_nat,G: multis3193088007478089820ropVar > multiset_nat] :
      ( ! [I3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ I3 @ K3 )
         => ( subseteq_mset_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_nat @ ( groups6371087482465774614ropVar @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ K3 ) @ ( groups6371087482465774614ropVar @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_384_subset__mset_Osum__mono,axiom,
    ! [K3: set_Product_prod_a_a,F: product_prod_a_a > multiset_nat,G: product_prod_a_a > multiset_nat] :
      ( ! [I3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ I3 @ K3 )
         => ( subseteq_mset_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_nat @ ( groups8557384953281641185od_a_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ K3 ) @ ( groups8557384953281641185od_a_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_385_subset__mset_Osum__mono,axiom,
    ! [K3: set_a,F: a > multiset_nat,G: a > multiset_nat] :
      ( ! [I3: a] :
          ( ( member_a @ I3 @ K3 )
         => ( subseteq_mset_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_nat @ ( groups6763380508735453880_nat_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ K3 ) @ ( groups6763380508735453880_nat_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_386_subset__mset_Osum__mono,axiom,
    ! [K3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_nat,G: produc6518373309651786023iset_a > multiset_nat] :
      ( ! [I3: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ I3 @ K3 )
         => ( subseteq_mset_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_nat @ ( groups3729319811690247393iset_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ K3 ) @ ( groups3729319811690247393iset_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_387_subset__mset_Osum__mono,axiom,
    ! [K3: set_nat,F: nat > multiset_a,G: nat > multiset_a] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K3 )
         => ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_a @ ( groups9157842243955967216_a_nat @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ K3 ) @ ( groups9157842243955967216_a_nat @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_388_subset__mset_Osum__mono,axiom,
    ! [K3: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > multiset_a,G: multis3193088007478089820ropVar > multiset_a] :
      ( ! [I3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ I3 @ K3 )
         => ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_a @ ( groups3121060555026902844ropVar @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ K3 ) @ ( groups3121060555026902844ropVar @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_389_subset__mset_Osum__mono,axiom,
    ! [K3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a,G: product_prod_a_a > multiset_a] :
      ( ! [I3: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ I3 @ K3 )
         => ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ K3 ) @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_390_subset__mset_Osum__mono,axiom,
    ! [K3: set_a,F: a > multiset_a,G: a > multiset_a] :
      ( ! [I3: a] :
          ( ( member_a @ I3 @ K3 )
         => ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ K3 ) @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_391_subset__mset_Osum__mono,axiom,
    ! [K3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a,G: produc6518373309651786023iset_a > multiset_a] :
      ( ! [I3: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ I3 @ K3 )
         => ( subseteq_mset_a @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( subseteq_mset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ K3 ) @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ G @ K3 ) ) ) ).

% subset_mset.sum_mono
thf(fact_392_sup__Un__eq,axiom,
    ! [R: set_Mu5658373669842198716ropVar,S: set_Mu5658373669842198716ropVar] :
      ( ( sup_su5656099892504388685pVar_o
        @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ R )
        @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ S ) )
      = ( ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ ( sup_su4102146785206298256ropVar @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_393_sup__Un__eq,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( sup_su1640154476453168578_a_a_o
        @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ R )
        @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ S ) )
      = ( ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ ( sup_su3048258781599657691od_a_a @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_394_sup__Un__eq,axiom,
    ! [R: set_a,S: set_a] :
      ( ( sup_sup_a_o
        @ ^ [X3: a] : ( member_a @ X3 @ R )
        @ ^ [X3: a] : ( member_a @ X3 @ S ) )
      = ( ^ [X3: a] : ( member_a @ X3 @ ( sup_sup_set_a @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_395_sup__Un__eq,axiom,
    ! [R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( sup_su2013817538080420546et_a_o
        @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ R )
        @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ S ) )
      = ( ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ ( sup_su1925787886501348827iset_a @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_396_sup__Un__eq,axiom,
    ! [R: set_nat,S: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X3: nat] : ( member_nat @ X3 @ R )
        @ ^ [X3: nat] : ( member_nat @ X3 @ S ) )
      = ( ^ [X3: nat] : ( member_nat @ X3 @ ( sup_sup_set_nat @ R @ S ) ) ) ) ).

% sup_Un_eq
thf(fact_397_map__nth,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( map_na2466558649495876678ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s1943289338788136526ropVar @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_398_map__nth,axiom,
    ! [Xs: list_nat] :
      ( ( map_nat_nat @ ( nth_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_399_map__nth,axiom,
    ! [Xs: list_a] :
      ( ( map_nat_a @ ( nth_a @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_400_length__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( size_s1943289338788136526ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_401_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_402_length__map,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( size_size_list_nat @ ( map_a_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_403_length__map,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( size_size_list_a @ ( map_nat_a @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_404_length__map,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( size_size_list_a @ ( map_a_a @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_405_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs2: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs2 ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs2 ) )
        = ( ( Xs = Ys )
          & ( Us = Vs2 ) ) ) ) ).

% append_eq_append_conv
thf(fact_406_append__eq__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a,Us: list_a,Vs2: list_a] :
      ( ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        | ( ( size_size_list_a @ Us )
          = ( size_size_list_a @ Vs2 ) ) )
     => ( ( ( append_a @ Xs @ Us )
          = ( append_a @ Ys @ Vs2 ) )
        = ( ( Xs = Ys )
          & ( Us = Vs2 ) ) ) ) ).

% append_eq_append_conv
thf(fact_407_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_408_length__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( size_size_list_a @ ( append_a @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ ( size_size_list_a @ Ys ) ) ) ).

% length_append
thf(fact_409_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_410_nth__append__length__plus,axiom,
    ! [Xs: list_a,Ys: list_a,N: nat] :
      ( ( nth_a @ ( append_a @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_a @ Xs ) @ N ) )
      = ( nth_a @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_411_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_412_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_a] :
      ( ( size_size_list_a @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_413_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_414_neq__if__length__neq,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
       != ( size_size_list_a @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_415_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y2: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y2 ) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_416_size__neq__size__imp__neq,axiom,
    ! [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] :
      ( ( ( size_s6253272723116879048ropVar @ X )
       != ( size_s6253272723116879048ropVar @ Y2 ) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_417_size__neq__size__imp__neq,axiom,
    ! [X: list_a,Y2: list_a] :
      ( ( ( size_size_list_a @ X )
       != ( size_size_list_a @ Y2 ) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_418_size__neq__size__imp__neq,axiom,
    ! [X: char,Y2: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y2 ) )
     => ( X != Y2 ) ) ).

% size_neq_size_imp_neq
thf(fact_419_map__eq__imp__length__eq,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_420_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_421_map__eq__imp__length__eq,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_422_map__eq__imp__length__eq,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: a > multis3193088007478089820ropVar,Ys: list_a] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_a_7060686498508945698ropVar @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_423_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_a_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_424_map__eq__imp__length__eq,axiom,
    ! [F: nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_a_a @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_425_map__eq__imp__length__eq,axiom,
    ! [F: a > multis3193088007478089820ropVar,Xs: list_a,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_a_7060686498508945698ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_426_map__eq__imp__length__eq,axiom,
    ! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( ( map_a_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_427_map__eq__imp__length__eq,axiom,
    ! [F: a > a,Xs: list_a,G: nat > a,Ys: list_nat] :
      ( ( ( map_a_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_428_mset__eq__length,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% mset_eq_length
thf(fact_429_mset__eq__length,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( mset_a @ Xs )
        = ( mset_a @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% mset_eq_length
thf(fact_430_subset__mset_Omax__add__distrib__left,axiom,
    ! [X: multiset_nat,Y2: multiset_nat,Z4: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( max_multiset_nat @ subseteq_mset_nat @ X @ Y2 ) @ Z4 )
      = ( max_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X @ Z4 ) @ ( plus_p6334493942879108393et_nat @ Y2 @ Z4 ) ) ) ).

% subset_mset.max_add_distrib_left
thf(fact_431_subset__mset_Omax__add__distrib__left,axiom,
    ! [X: multiset_a,Y2: multiset_a,Z4: multiset_a] :
      ( ( plus_plus_multiset_a @ ( max_multiset_a @ subseteq_mset_a @ X @ Y2 ) @ Z4 )
      = ( max_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X @ Z4 ) @ ( plus_plus_multiset_a @ Y2 @ Z4 ) ) ) ).

% subset_mset.max_add_distrib_left
thf(fact_432_subset__mset_Omin__add__distrib__left,axiom,
    ! [X: multiset_nat,Y2: multiset_nat,Z4: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( min_multiset_nat @ subseteq_mset_nat @ X @ Y2 ) @ Z4 )
      = ( min_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X @ Z4 ) @ ( plus_p6334493942879108393et_nat @ Y2 @ Z4 ) ) ) ).

% subset_mset.min_add_distrib_left
thf(fact_433_subset__mset_Omin__add__distrib__left,axiom,
    ! [X: multiset_a,Y2: multiset_a,Z4: multiset_a] :
      ( ( plus_plus_multiset_a @ ( min_multiset_a @ subseteq_mset_a @ X @ Y2 ) @ Z4 )
      = ( min_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X @ Z4 ) @ ( plus_plus_multiset_a @ Y2 @ Z4 ) ) ) ).

% subset_mset.min_add_distrib_left
thf(fact_434_subset__mset_Omax__add__distrib__right,axiom,
    ! [X: multiset_nat,Y2: multiset_nat,Z4: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ X @ ( max_multiset_nat @ subseteq_mset_nat @ Y2 @ Z4 ) )
      = ( max_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X @ Y2 ) @ ( plus_p6334493942879108393et_nat @ X @ Z4 ) ) ) ).

% subset_mset.max_add_distrib_right
thf(fact_435_subset__mset_Omax__add__distrib__right,axiom,
    ! [X: multiset_a,Y2: multiset_a,Z4: multiset_a] :
      ( ( plus_plus_multiset_a @ X @ ( max_multiset_a @ subseteq_mset_a @ Y2 @ Z4 ) )
      = ( max_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X @ Y2 ) @ ( plus_plus_multiset_a @ X @ Z4 ) ) ) ).

% subset_mset.max_add_distrib_right
thf(fact_436_subset__mset_Omin__add__distrib__right,axiom,
    ! [X: multiset_nat,Y2: multiset_nat,Z4: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ X @ ( min_multiset_nat @ subseteq_mset_nat @ Y2 @ Z4 ) )
      = ( min_multiset_nat @ subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ X @ Y2 ) @ ( plus_p6334493942879108393et_nat @ X @ Z4 ) ) ) ).

% subset_mset.min_add_distrib_right
thf(fact_437_subset__mset_Omin__add__distrib__right,axiom,
    ! [X: multiset_a,Y2: multiset_a,Z4: multiset_a] :
      ( ( plus_plus_multiset_a @ X @ ( min_multiset_a @ subseteq_mset_a @ Y2 @ Z4 ) )
      = ( min_multiset_a @ subseteq_mset_a @ ( plus_plus_multiset_a @ X @ Y2 ) @ ( plus_plus_multiset_a @ X @ Z4 ) ) ) ).

% subset_mset.min_add_distrib_right
thf(fact_438_mset__set__set,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( distin2675581014522817237ropVar @ Xs )
     => ( ( mset_s697675537654382723ropVar @ ( set_Mu146256361055586289ropVar @ Xs ) )
        = ( mset_M6581516532424202993ropVar @ Xs ) ) ) ).

% mset_set_set
thf(fact_439_mset__set__set,axiom,
    ! [Xs: list_a] :
      ( ( distinct_a @ Xs )
     => ( ( mset_set_a @ ( set_a2 @ Xs ) )
        = ( mset_a @ Xs ) ) ) ).

% mset_set_set
thf(fact_440_mset__set__set,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( ( mset_set_nat @ ( set_nat2 @ Xs ) )
        = ( mset_nat @ Xs ) ) ) ).

% mset_set_set
thf(fact_441_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_nat,F: nat > multiset_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( F @ X4 ) ) )
     => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups5394990218802192790at_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_442_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > multiset_nat] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ A3 )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( F @ X4 ) ) )
     => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups6371087482465774614ropVar @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_443_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_nat] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A3 )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( F @ X4 ) ) )
     => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups8557384953281641185od_a_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_444_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_a,F: a > multiset_nat] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A3 )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( F @ X4 ) ) )
     => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups6763380508735453880_nat_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_445_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_nat] :
      ( ! [X4: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ X4 @ A3 )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( F @ X4 ) ) )
     => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups3729319811690247393iset_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_446_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_nat,F: nat > multiset_a] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X4 ) ) )
     => ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups9157842243955967216_a_nat @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_447_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > multiset_a] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ A3 )
         => ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X4 ) ) )
     => ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups3121060555026902844ropVar @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_448_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A3 )
         => ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X4 ) ) )
     => ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_449_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_a,F: a > multiset_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A3 )
         => ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X4 ) ) )
     => ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_450_subset__mset_Osum__nonneg,axiom,
    ! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a] :
      ( ! [X4: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ X4 @ A3 )
         => ( subseteq_mset_a @ zero_zero_multiset_a @ ( F @ X4 ) ) )
     => ( subseteq_mset_a @ zero_zero_multiset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) ) ) ).

% subset_mset.sum_nonneg
thf(fact_451_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_nat,F: nat > multiset_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( subseteq_mset_nat @ ( F @ X4 ) @ zero_z7348594199698428585et_nat ) )
     => ( subseteq_mset_nat @ ( groups5394990218802192790at_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.sum_nonpos
thf(fact_452_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > multiset_nat] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ A3 )
         => ( subseteq_mset_nat @ ( F @ X4 ) @ zero_z7348594199698428585et_nat ) )
     => ( subseteq_mset_nat @ ( groups6371087482465774614ropVar @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.sum_nonpos
thf(fact_453_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_nat] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A3 )
         => ( subseteq_mset_nat @ ( F @ X4 ) @ zero_z7348594199698428585et_nat ) )
     => ( subseteq_mset_nat @ ( groups8557384953281641185od_a_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.sum_nonpos
thf(fact_454_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_a,F: a > multiset_nat] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A3 )
         => ( subseteq_mset_nat @ ( F @ X4 ) @ zero_z7348594199698428585et_nat ) )
     => ( subseteq_mset_nat @ ( groups6763380508735453880_nat_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.sum_nonpos
thf(fact_455_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_nat] :
      ( ! [X4: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ X4 @ A3 )
         => ( subseteq_mset_nat @ ( F @ X4 ) @ zero_z7348594199698428585et_nat ) )
     => ( subseteq_mset_nat @ ( groups3729319811690247393iset_a @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ F @ A3 ) @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.sum_nonpos
thf(fact_456_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_nat,F: nat > multiset_a] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A3 )
         => ( subseteq_mset_a @ ( F @ X4 ) @ zero_zero_multiset_a ) )
     => ( subseteq_mset_a @ ( groups9157842243955967216_a_nat @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).

% subset_mset.sum_nonpos
thf(fact_457_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > multiset_a] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ A3 )
         => ( subseteq_mset_a @ ( F @ X4 ) @ zero_zero_multiset_a ) )
     => ( subseteq_mset_a @ ( groups3121060555026902844ropVar @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).

% subset_mset.sum_nonpos
thf(fact_458_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_Product_prod_a_a,F: product_prod_a_a > multiset_a] :
      ( ! [X4: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X4 @ A3 )
         => ( subseteq_mset_a @ ( F @ X4 ) @ zero_zero_multiset_a ) )
     => ( subseteq_mset_a @ ( groups1455199411266135047od_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).

% subset_mset.sum_nonpos
thf(fact_459_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_a,F: a > multiset_a] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ A3 )
         => ( subseteq_mset_a @ ( F @ X4 ) @ zero_zero_multiset_a ) )
     => ( subseteq_mset_a @ ( groups409869659785034206et_a_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).

% subset_mset.sum_nonpos
thf(fact_460_subset__mset_Osum__nonpos,axiom,
    ! [A3: set_Pr79727621955416071iset_a,F: produc6518373309651786023iset_a > multiset_a] :
      ( ! [X4: produc6518373309651786023iset_a] :
          ( ( member5199237121806060112iset_a @ X4 @ A3 )
         => ( subseteq_mset_a @ ( F @ X4 ) @ zero_zero_multiset_a ) )
     => ( subseteq_mset_a @ ( groups6473759573426450439iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ F @ A3 ) @ zero_zero_multiset_a ) ) ).

% subset_mset.sum_nonpos
thf(fact_461__092_060open_062pos__of_Ays1_Aj_A_060_Alength_Ays1_092_060close_062,axiom,
    ord_less_nat @ ( pos_of @ ys12 @ j ) @ ( size_size_list_nat @ ys12 ) ).

% \<open>pos_of ys1 j < length ys1\<close>
thf(fact_462__092_060open_062_092_060exists_062_Bi_O_Ai_A_060_Alength_Ays1_A_092_060and_062_Ays1_A_B_Ai_A_061_Aj_092_060close_062,axiom,
    ? [X4: nat] :
      ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ ys12 ) )
      & ( ( nth_nat @ ys12 @ X4 )
        = j )
      & ! [Y6: nat] :
          ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ ys12 ) )
            & ( ( nth_nat @ ys12 @ Y6 )
              = j ) )
         => ( Y6 = X4 ) ) ) ).

% \<open>\<exists>!i. i < length ys1 \<and> ys1 ! i = j\<close>
thf(fact_463_p__xs_I1_J,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ xs12 ) )
     => ( ord_less_nat @ ( pos_of @ xs12 @ X ) @ ( size_size_list_nat @ xs12 ) ) ) ).

% p_xs(1)
thf(fact_464_p__ys_I1_J,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ys12 ) )
     => ( ord_less_nat @ ( pos_of @ ys12 @ X ) @ ( size_size_list_nat @ ys12 ) ) ) ).

% p_ys(1)
thf(fact_465_pos_I1_J,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( ord_less_nat @ ( pos_of @ Xs @ X ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% pos(1)
thf(fact_466_p__xs_I3_J,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ xs12 ) )
     => ? [X4: nat] :
          ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ xs12 ) )
          & ( ( nth_nat @ xs12 @ X4 )
            = X )
          & ! [Y6: nat] :
              ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ xs12 ) )
                & ( ( nth_nat @ xs12 @ Y6 )
                  = X ) )
             => ( Y6 = X4 ) ) ) ) ).

% p_xs(3)
thf(fact_467_pos_I3_J,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X4 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y6 )
                    = X ) )
               => ( Y6 = X4 ) ) ) ) ) ).

% pos(3)
thf(fact_468_p__ys_I3_J,axiom,
    ! [X: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ys12 ) )
     => ? [X4: nat] :
          ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ ys12 ) )
          & ( ( nth_nat @ ys12 @ X4 )
            = X )
          & ! [Y6: nat] :
              ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ ys12 ) )
                & ( ( nth_nat @ ys12 @ Y6 )
                  = X ) )
             => ( Y6 = X4 ) ) ) ) ).

% p_ys(3)
thf(fact_469_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_470_add__less__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_left
thf(fact_471_add__less__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_cancel_right
thf(fact_472_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_473_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_474_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_475_nat__add__left__cancel__less,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_476_size__union,axiom,
    ! [M: multiset_nat,N2: multiset_nat] :
      ( ( size_s5917832649809541300et_nat @ ( plus_p6334493942879108393et_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( size_s5917832649809541300et_nat @ M ) @ ( size_s5917832649809541300et_nat @ N2 ) ) ) ).

% size_union
thf(fact_477_size__union,axiom,
    ! [M: multiset_a,N2: multiset_a] :
      ( ( size_size_multiset_a @ ( plus_plus_multiset_a @ M @ N2 ) )
      = ( plus_plus_nat @ ( size_size_multiset_a @ M ) @ ( size_size_multiset_a @ N2 ) ) ) ).

% size_union
thf(fact_478_lessThan__iff,axiom,
    ! [I2: nat,K2: nat] :
      ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K2 ) )
      = ( ord_less_nat @ I2 @ K2 ) ) ).

% lessThan_iff
thf(fact_479_size__mset,axiom,
    ! [Xs: list_nat] :
      ( ( size_s5917832649809541300et_nat @ ( mset_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% size_mset
thf(fact_480_size__mset,axiom,
    ! [Xs: list_a] :
      ( ( size_size_multiset_a @ ( mset_a @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% size_mset
thf(fact_481_add__less__same__cancel1,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B2 @ A2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_482_add__less__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = ( ord_less_nat @ A2 @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_483_less__add__same__cancel1,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel1
thf(fact_484_less__add__same__cancel2,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ ( plus_plus_nat @ B2 @ A2 ) )
      = ( ord_less_nat @ zero_zero_nat @ B2 ) ) ).

% less_add_same_cancel2
thf(fact_485_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_486__092_060open_062pos__of_A_092_060equiv_062_A_092_060lambda_062xs_Ai_O_ATHE_Ap_O_Ap_A_060_Alength_Axs_A_092_060and_062_Axs_A_B_Ap_A_061_Ai_092_060close_062,axiom,
    ( pos_of
    = ( ^ [Xs2: list_nat,I: nat] :
          ( the_nat
          @ ^ [P6: nat] :
              ( ( ord_less_nat @ P6 @ ( size_size_list_nat @ Xs2 ) )
              & ( ( nth_nat @ Xs2 @ P6 )
                = I ) ) ) ) ) ).

% \<open>pos_of \<equiv> \<lambda>xs i. THE p. p < length xs \<and> xs ! p = i\<close>
thf(fact_487_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Mu4537670848183934179ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_488_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_489_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > a] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_a @ ( map_nat_a @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_490_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_nat @ ( map_a_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_491_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_a @ ( map_a_a @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_492_nth__upt,axiom,
    ! [I2: nat,K2: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J )
     => ( ( nth_nat @ ( upt @ I2 @ J ) @ K2 )
        = ( plus_plus_nat @ I2 @ K2 ) ) ) ).

% nth_upt
thf(fact_493_linorder__neqE__nat,axiom,
    ! [X: nat,Y2: nat] :
      ( ( X != Y2 )
     => ( ~ ( ord_less_nat @ X @ Y2 )
       => ( ord_less_nat @ Y2 @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_494_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N5: nat] :
          ( ~ ( P @ N5 )
         => ? [M4: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
              & ~ ( P @ M4 ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_495_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N5: nat] :
          ( ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N5 )
             => ( P @ M4 ) )
         => ( P @ N5 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_496_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_497_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_498_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_499_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_500_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_501_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_502_lessThan__strict__subset__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M2 ) @ ( set_ord_lessThan_nat @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% lessThan_strict_subset_iff
thf(fact_503_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_504_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_505_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_506_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_507_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_508_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_nat @ K2 @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_509_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( K2 = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_510_add__strict__mono,axiom,
    ! [A2: nat,B2: nat,C: nat,D2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ C @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_511_add__strict__left__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) ) ) ).

% add_strict_left_mono
thf(fact_512_add__strict__right__mono,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% add_strict_right_mono
thf(fact_513_add__less__imp__less__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_left
thf(fact_514_add__less__imp__less__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
     => ( ord_less_nat @ A2 @ B2 ) ) ).

% add_less_imp_less_right
thf(fact_515_sup_Ostrict__coboundedI2,axiom,
    ! [C: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ C @ B2 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_516_sup_Ostrict__coboundedI2,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ C @ B2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI2
thf(fact_517_sup_Ostrict__coboundedI1,axiom,
    ! [C: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ C @ A2 )
     => ( ord_less_set_nat @ C @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_518_sup_Ostrict__coboundedI1,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ C @ A2 )
     => ( ord_less_nat @ C @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% sup.strict_coboundedI1
thf(fact_519_sup_Ostrict__order__iff,axiom,
    ( ord_less_set_nat
    = ( ^ [B: set_nat,A: set_nat] :
          ( ( A
            = ( sup_sup_set_nat @ A @ B ) )
          & ( A != B ) ) ) ) ).

% sup.strict_order_iff
thf(fact_520_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B: nat,A: nat] :
          ( ( A
            = ( sup_sup_nat @ A @ B ) )
          & ( A != B ) ) ) ) ).

% sup.strict_order_iff
thf(fact_521_sup_Ostrict__boundedE,axiom,
    ! [B2: set_nat,C: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ ( sup_sup_set_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_set_nat @ B2 @ A2 )
         => ~ ( ord_less_set_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_522_sup_Ostrict__boundedE,axiom,
    ! [B2: nat,C: nat,A2: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B2 @ C ) @ A2 )
     => ~ ( ( ord_less_nat @ B2 @ A2 )
         => ~ ( ord_less_nat @ C @ A2 ) ) ) ).

% sup.strict_boundedE
thf(fact_523_sup_Oabsorb4,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_524_sup_Oabsorb4,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = B2 ) ) ).

% sup.absorb4
thf(fact_525_sup_Oabsorb3,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ B2 @ A2 )
     => ( ( sup_sup_set_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_526_sup_Oabsorb3,axiom,
    ! [B2: nat,A2: nat] :
      ( ( ord_less_nat @ B2 @ A2 )
     => ( ( sup_sup_nat @ A2 @ B2 )
        = A2 ) ) ).

% sup.absorb3
thf(fact_527_less__supI2,axiom,
    ! [X: set_nat,B2: set_nat,A2: set_nat] :
      ( ( ord_less_set_nat @ X @ B2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_528_less__supI2,axiom,
    ! [X: nat,B2: nat,A2: nat] :
      ( ( ord_less_nat @ X @ B2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI2
thf(fact_529_less__supI1,axiom,
    ! [X: set_nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ X @ A2 )
     => ( ord_less_set_nat @ X @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_530_less__supI1,axiom,
    ! [X: nat,A2: nat,B2: nat] :
      ( ( ord_less_nat @ X @ A2 )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A2 @ B2 ) ) ) ).

% less_supI1
thf(fact_531_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N5: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N5 )
           => ( ~ ( P @ N5 )
             => ? [M4: nat] :
                  ( ( ord_less_nat @ M4 @ N5 )
                  & ~ ( P @ M4 ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_532_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_533_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_534_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_535_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_536_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_537_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_538_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_539_length__induct,axiom,
    ! [P: list_a > $o,Xs: list_a] :
      ( ! [Xs3: list_a] :
          ( ! [Ys3: list_a] :
              ( ( ord_less_nat @ ( size_size_list_a @ Ys3 ) @ ( size_size_list_a @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_540_add__lessD1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K2 )
     => ( ord_less_nat @ I2 @ K2 ) ) ).

% add_lessD1
thf(fact_541_add__less__mono,axiom,
    ! [I2: nat,J: nat,K2: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ K2 @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_542_not__add__less1,axiom,
    ! [I2: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).

% not_add_less1
thf(fact_543_not__add__less2,axiom,
    ! [J: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_544_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% add_less_mono1
thf(fact_545_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_546_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_547_less__add__eq__less,axiom,
    ! [K2: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K2 @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K2 @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_548_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U: nat] :
          ( collect_nat
          @ ^ [X3: nat] : ( ord_less_nat @ X3 @ U ) ) ) ) ).

% lessThan_def
thf(fact_549_add__neg__neg,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ zero_zero_nat )
     => ( ( ord_less_nat @ B2 @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A2 @ B2 ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_550_add__pos__pos,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ zero_zero_nat @ B2 )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A2 @ B2 ) ) ) ) ).

% add_pos_pos
thf(fact_551_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A2: nat,B2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ~ ! [C5: nat] :
            ( ( B2
              = ( plus_plus_nat @ A2 @ C5 ) )
           => ( C5 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_552_pos__add__strict,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A2 )
     => ( ( ord_less_nat @ B2 @ C )
       => ( ord_less_nat @ B2 @ ( plus_plus_nat @ A2 @ C ) ) ) ) ).

% pos_add_strict
thf(fact_553_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ? [K4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K4 )
          & ( ( plus_plus_nat @ I2 @ K4 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_554_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_nat,Z3: list_nat] : ( Y3 = Z3 ) )
    = ( ^ [Xs2: list_nat,Ys4: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys4 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I )
                = ( nth_nat @ Ys4 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_555_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y3: list_a,Z3: list_a] : ( Y3 = Z3 ) )
    = ( ^ [Xs2: list_a,Ys4: list_a] :
          ( ( ( size_size_list_a @ Xs2 )
            = ( size_size_list_a @ Ys4 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
             => ( ( nth_a @ Xs2 @ I )
                = ( nth_a @ Ys4 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_556_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X7: nat] : ( P @ I @ X7 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_557_Skolem__list__nth,axiom,
    ! [K2: nat,P: nat > a > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K2 )
           => ? [X7: a] : ( P @ I @ X7 ) ) )
      = ( ? [Xs2: list_a] :
            ( ( ( size_size_list_a @ Xs2 )
              = K2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ( P @ I @ ( nth_a @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_558_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_559_nth__equalityI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
           => ( ( nth_a @ Xs @ I3 )
              = ( nth_a @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_560_length__pos__if__in__set,axiom,
    ! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3885678630836030617od_a_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_561_length__pos__if__in__set,axiom,
    ! [X: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
      ( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s7238940367606365593iset_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_562_length__pos__if__in__set,axiom,
    ! [X: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s1943289338788136526ropVar @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_563_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_564_length__pos__if__in__set,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_565_all__set__conv__all__nth,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ! [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1943289338788136526ropVar @ Xs ) )
           => ( P @ ( nth_Mu4537670848183934179ropVar @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_566_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_567_all__set__conv__all__nth,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ( P @ X3 ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
           => ( P @ ( nth_a @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_568_all__nth__imp__all__set,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,P: product_prod_a_a > $o,X: product_prod_a_a] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Xs ) )
         => ( P @ ( nth_Product_prod_a_a @ Xs @ I3 ) ) )
     => ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_569_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6988024003011616557iset_a,P: produc6518373309651786023iset_a > $o,X: produc6518373309651786023iset_a] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Xs ) )
         => ( P @ ( nth_Pr1079437516805851566iset_a @ Xs @ I3 ) ) )
     => ( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_570_all__nth__imp__all__set,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o,X: multis3193088007478089820ropVar] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Xs ) )
         => ( P @ ( nth_Mu4537670848183934179ropVar @ Xs @ I3 ) ) )
     => ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_571_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I3 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_572_all__nth__imp__all__set,axiom,
    ! [Xs: list_a,P: a > $o,X: a] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
         => ( P @ ( nth_a @ Xs @ I3 ) ) )
     => ( ( member_a @ X @ ( set_a2 @ Xs ) )
       => ( P @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_573_in__set__conv__nth,axiom,
    ! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s3885678630836030617od_a_a @ Xs ) )
            & ( ( nth_Product_prod_a_a @ Xs @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_574_in__set__conv__nth,axiom,
    ! [X: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
      ( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7238940367606365593iset_a @ Xs ) )
            & ( ( nth_Pr1079437516805851566iset_a @ Xs @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_575_in__set__conv__nth,axiom,
    ! [X: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1943289338788136526ropVar @ Xs ) )
            & ( ( nth_Mu4537670848183934179ropVar @ Xs @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_576_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_577_in__set__conv__nth,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
            & ( ( nth_a @ Xs @ I )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_578_list__ball__nth,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ord_less_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( ! [X4: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_Mu4537670848183934179ropVar @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_579_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_580_list__ball__nth,axiom,
    ! [N: nat,Xs: list_a,P: a > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_581_nth__mem,axiom,
    ! [N: nat,Xs: list_P1396940483166286381od_a_a] :
      ( ( ord_less_nat @ N @ ( size_s3885678630836030617od_a_a @ Xs ) )
     => ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Xs @ N ) @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_582_nth__mem,axiom,
    ! [N: nat,Xs: list_P6988024003011616557iset_a] :
      ( ( ord_less_nat @ N @ ( size_s7238940367606365593iset_a @ Xs ) )
     => ( member5199237121806060112iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs @ N ) @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).

% nth_mem
thf(fact_583_nth__mem,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( ord_less_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( member8421201611525043589ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ N ) @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% nth_mem
thf(fact_584_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_585_nth__mem,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_586_map__equality__iff,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_587_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_588_map__equality__iff,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_589_map__equality__iff,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: a > multis3193088007478089820ropVar,Ys: list_a] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_a_7060686498508945698ropVar @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_a @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_590_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: a > nat,Ys: list_a] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_a_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_a @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_591_map__equality__iff,axiom,
    ! [F: nat > a,Xs: list_nat,G: a > a,Ys: list_a] :
      ( ( ( map_nat_a @ F @ Xs )
        = ( map_a_a @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_a @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_592_map__equality__iff,axiom,
    ! [F: a > multis3193088007478089820ropVar,Xs: list_a,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_a_7060686498508945698ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_593_map__equality__iff,axiom,
    ! [F: a > nat,Xs: list_a,G: nat > nat,Ys: list_nat] :
      ( ( ( map_a_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_594_map__equality__iff,axiom,
    ! [F: a > a,Xs: list_a,G: nat > a,Ys: list_nat] :
      ( ( ( map_a_a @ F @ Xs )
        = ( map_nat_a @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_595_boolean__algebra__cancel_Osup1,axiom,
    ! [A3: set_nat,K2: set_nat,A2: set_nat,B2: set_nat] :
      ( ( A3
        = ( sup_sup_set_nat @ K2 @ A2 ) )
     => ( ( sup_sup_set_nat @ A3 @ B2 )
        = ( sup_sup_set_nat @ K2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup1
thf(fact_596_boolean__algebra__cancel_Osup2,axiom,
    ! [B3: set_nat,K2: set_nat,B2: set_nat,A2: set_nat] :
      ( ( B3
        = ( sup_sup_set_nat @ K2 @ B2 ) )
     => ( ( sup_sup_set_nat @ A2 @ B3 )
        = ( sup_sup_set_nat @ K2 @ ( sup_sup_set_nat @ A2 @ B2 ) ) ) ) ).

% boolean_algebra_cancel.sup2
thf(fact_597_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs2: list_nat] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( I != J2 )
               => ( ( nth_nat @ Xs2 @ I )
                 != ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_598_distinct__conv__nth,axiom,
    ( distinct_a
    = ( ^ [Xs2: list_a] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_a @ Xs2 ) )
             => ( ( I != J2 )
               => ( ( nth_a @ Xs2 @ I )
                 != ( nth_a @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_599_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I2: nat,J: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Xs @ J ) )
            = ( I2 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_600_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_a,I2: nat,J: nat] :
      ( ( distinct_a @ Xs )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
         => ( ( ( nth_a @ Xs @ I2 )
              = ( nth_a @ Xs @ J ) )
            = ( I2 = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_601_distinct__Ex1,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
      ( ( distin132333870042060960od_a_a @ Xs )
     => ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s3885678630836030617od_a_a @ Xs ) )
            & ( ( nth_Product_prod_a_a @ Xs @ X4 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_s3885678630836030617od_a_a @ Xs ) )
                  & ( ( nth_Product_prod_a_a @ Xs @ Y6 )
                    = X ) )
               => ( Y6 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_602_distinct__Ex1,axiom,
    ! [Xs: list_P6988024003011616557iset_a,X: produc6518373309651786023iset_a] :
      ( ( distin8319736171526766496iset_a @ Xs )
     => ( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s7238940367606365593iset_a @ Xs ) )
            & ( ( nth_Pr1079437516805851566iset_a @ Xs @ X4 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_s7238940367606365593iset_a @ Xs ) )
                  & ( ( nth_Pr1079437516805851566iset_a @ Xs @ Y6 )
                    = X ) )
               => ( Y6 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_603_distinct__Ex1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,X: multis3193088007478089820ropVar] :
      ( ( distin2675581014522817237ropVar @ Xs )
     => ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s1943289338788136526ropVar @ Xs ) )
            & ( ( nth_Mu4537670848183934179ropVar @ Xs @ X4 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_s1943289338788136526ropVar @ Xs ) )
                  & ( ( nth_Mu4537670848183934179ropVar @ Xs @ Y6 )
                    = X ) )
               => ( Y6 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_604_distinct__Ex1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X4 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y6 )
                    = X ) )
               => ( Y6 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_605_distinct__Ex1,axiom,
    ! [Xs: list_a,X: a] :
      ( ( distinct_a @ Xs )
     => ( ( member_a @ X @ ( set_a2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_size_list_a @ Xs ) )
            & ( ( nth_a @ Xs @ X4 )
              = X )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_a @ Xs ) )
                  & ( ( nth_a @ Xs @ Y6 )
                    = X ) )
               => ( Y6 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_606_PropVar_Osize_I7_J,axiom,
    ! [X11: nat,X12: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = zero_zero_nat ) ).

% PropVar.size(7)
thf(fact_607_subset__mset_Oelem__le__sum__list,axiom,
    ! [K2: nat,Ns: list_multiset_nat] :
      ( ( ord_less_nat @ K2 @ ( size_s6386657463320973636et_nat @ Ns ) )
     => ( subseteq_mset_nat @ ( nth_multiset_nat @ Ns @ K2 ) @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Ns ) ) ) ).

% subset_mset.elem_le_sum_list
thf(fact_608_subset__mset_Oelem__le__sum__list,axiom,
    ! [K2: nat,Ns: list_multiset_a] :
      ( ( ord_less_nat @ K2 @ ( size_s3003218658903709366iset_a @ Ns ) )
     => ( subseteq_mset_a @ ( nth_multiset_a @ Ns @ K2 ) @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Ns ) ) ) ).

% subset_mset.elem_le_sum_list
thf(fact_609_pj_H,axiom,
    ord_less_nat @ ( pos_of @ ys12 @ j ) @ ( size_size_list_a @ xs1 ) ).

% pj'
thf(fact_610_pj,axiom,
    ord_less_nat @ ( pos_of @ ys12 @ j ) @ ( size_size_list_a @ ys1 ) ).

% pj
thf(fact_611_pos__of__def,axiom,
    ! [Xs: list_nat,I2: nat] :
      ( ( pos_of @ Xs @ I2 )
      = ( the_nat
        @ ^ [P6: nat] :
            ( ( ord_less_nat @ P6 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ P6 )
              = I2 ) ) ) ) ).

% pos_of_def
thf(fact_612_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C4: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_613__092_060open_062Xs1_A_B_Apos__of_Ays1_Aj_A_061_As_A_Ixs1_A_B_Apos__of_Ays1_Aj_J_092_060close_062,axiom,
    ( ( nth_a @ xs1 @ ( pos_of @ ys12 @ j ) )
    = ( s2 @ ( nth_nat @ xs12 @ ( pos_of @ ys12 @ j ) ) ) ) ).

% \<open>Xs1 ! pos_of ys1 j = s (xs1 ! pos_of ys1 j)\<close>
thf(fact_614_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_615_len,axiom,
    ( ( size_size_list_a @ xs1 )
    = ( size_size_list_a @ ys1 ) ) ).

% len
thf(fact_616_xs1,axiom,
    ( xs1
    = ( map_nat_a @ s2 @ xs12 ) ) ).

% xs1
thf(fact_617_xs2,axiom,
    ( xs2
    = ( map_nat_a @ s2 @ xs22 ) ) ).

% xs2
thf(fact_618_eq1,axiom,
    ( ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) )
    = ( plus_plus_multiset_a @ ( mset_a @ xs1 ) @ ( mset_a @ xs2 ) ) ) ).

% eq1
thf(fact_619_ys1,axiom,
    ( ys1
    = ( map_nat_a @ t @ ys12 ) ) ).

% ys1
thf(fact_620_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_nat_o
          @ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
          @ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_621_less__set__def,axiom,
    ( ord_le8651388201640276840ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
          ( ord_le1780504345282606453pVar_o
          @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ A5 )
          @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_622_less__set__def,axiom,
    ( ord_le6819997720685908915od_a_a
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
          ( ord_le3684186632072642282_a_a_o
          @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A5 )
          @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_623_less__set__def,axiom,
    ( ord_less_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( ord_less_a_o
          @ ^ [X3: a] : ( member_a @ X3 @ A5 )
          @ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_624_less__set__def,axiom,
    ( ord_le4018519769653605043iset_a
    = ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
          ( ord_le5648499418131129322et_a_o
          @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ A5 )
          @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ B5 ) ) ) ) ).

% less_set_def
thf(fact_625_psubsetD,axiom,
    ! [A3: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ( ( member_nat @ C @ A3 )
       => ( member_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_626_psubsetD,axiom,
    ! [A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar,C: multis3193088007478089820ropVar] :
      ( ( ord_le8651388201640276840ropVar @ A3 @ B3 )
     => ( ( member8421201611525043589ropVar @ C @ A3 )
       => ( member8421201611525043589ropVar @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_627_psubsetD,axiom,
    ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a,C: product_prod_a_a] :
      ( ( ord_le6819997720685908915od_a_a @ A3 @ B3 )
     => ( ( member1426531477525435216od_a_a @ C @ A3 )
       => ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_628_psubsetD,axiom,
    ! [A3: set_a,B3: set_a,C: a] :
      ( ( ord_less_set_a @ A3 @ B3 )
     => ( ( member_a @ C @ A3 )
       => ( member_a @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_629_psubsetD,axiom,
    ! [A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a,C: produc6518373309651786023iset_a] :
      ( ( ord_le4018519769653605043iset_a @ A3 @ B3 )
     => ( ( member5199237121806060112iset_a @ C @ A3 )
       => ( member5199237121806060112iset_a @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_630_union__less__mono,axiom,
    ! [A3: multiset_nat,C2: multiset_nat,B3: multiset_nat,D: multiset_nat] :
      ( ( ord_le5777773500796000884et_nat @ A3 @ C2 )
     => ( ( ord_le5777773500796000884et_nat @ B3 @ D )
       => ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) @ ( plus_p6334493942879108393et_nat @ C2 @ D ) ) ) ) ).

% union_less_mono
thf(fact_631_union__le__mono2,axiom,
    ! [B3: multiset_nat,D: multiset_nat,C2: multiset_nat] :
      ( ( ord_le5777773500796000884et_nat @ B3 @ D )
     => ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ C2 @ B3 ) @ ( plus_p6334493942879108393et_nat @ C2 @ D ) ) ) ).

% union_le_mono2
thf(fact_632_union__le__mono1,axiom,
    ! [B3: multiset_nat,D: multiset_nat,C2: multiset_nat] :
      ( ( ord_le5777773500796000884et_nat @ B3 @ D )
     => ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ B3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ D @ C2 ) ) ) ).

% union_le_mono1
thf(fact_633__092_060open_062_092_060exists_062zs1_Azs2_O_Amset_A_0910_O_O_060n_093_A_061_Amset_Azs1_A_L_Amset_Azs2_A_092_060and_062_AXs1_A_061_Amap_As_Azs1_A_092_060and_062_AXs2_A_061_Amap_As_Azs2_092_060close_062,axiom,
    ? [Zs1: list_nat,Zs2: list_nat] :
      ( ( ( mset_nat @ ( upt @ zero_zero_nat @ n ) )
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
      & ( xs1
        = ( map_nat_a @ s2 @ Zs1 ) )
      & ( xs2
        = ( map_nat_a @ s2 @ Zs2 ) ) ) ).

% \<open>\<exists>zs1 zs2. mset [0..<n] = mset zs1 + mset zs2 \<and> Xs1 = map s zs1 \<and> Xs2 = map s zs2\<close>
thf(fact_634__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs1_Axs2_O_A_092_060lbrakk_062mset_A_0910_O_O_060n_093_A_061_Amset_Axs1_A_L_Amset_Axs2_059_AXs1_A_061_Amap_As_Axs1_059_AXs2_A_061_Amap_As_Axs2_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Xs12: list_nat,Xs22: list_nat] :
        ( ( ( mset_nat @ ( upt @ zero_zero_nat @ n ) )
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs22 ) ) )
       => ( ( xs1
            = ( map_nat_a @ s2 @ Xs12 ) )
         => ( xs2
           != ( map_nat_a @ s2 @ Xs22 ) ) ) ) ).

% \<open>\<And>thesis. (\<And>xs1 xs2. \<lbrakk>mset [0..<n] = mset xs1 + mset xs2; Xs1 = map s xs1; Xs2 = map s xs2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_635__092_060open_062Ys1_A_B_Apos__of_Ays1_Aj_A_061_At_Aj_092_060close_062,axiom,
    ( ( nth_a @ ys1 @ ( pos_of @ ys12 @ j ) )
    = ( t @ j ) ) ).

% \<open>Ys1 ! pos_of ys1 j = t j\<close>
thf(fact_636_the__sym__eq__trivial,axiom,
    ! [X: nat] :
      ( ( the_nat
        @ ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 )
          @ X ) )
      = X ) ).

% the_sym_eq_trivial
thf(fact_637_the__eq__trivial,axiom,
    ! [A2: nat] :
      ( ( the_nat
        @ ^ [X3: nat] : ( X3 = A2 ) )
      = A2 ) ).

% the_eq_trivial
thf(fact_638_ys2,axiom,
    ( ys2
    = ( map_nat_a @ t @ ys22 ) ) ).

% ys2
thf(fact_639_the__equality,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( P @ A2 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A2 ) )
       => ( ( the_nat @ P )
          = A2 ) ) ) ).

% the_equality
thf(fact_640_eq2,axiom,
    ( ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) )
    = ( plus_plus_multiset_a @ ( mset_a @ ys1 ) @ ( mset_a @ ys2 ) ) ) ).

% eq2
thf(fact_641_theI,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( P @ A2 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A2 ) )
       => ( P @ ( the_nat @ P ) ) ) ) ).

% theI
thf(fact_642_theI_H,axiom,
    ! [P: nat > $o] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y5: nat] :
              ( ( P @ Y5 )
             => ( Y5 = X6 ) ) )
     => ( P @ ( the_nat @ P ) ) ) ).

% theI'
thf(fact_643_theI2,axiom,
    ! [P: nat > $o,A2: nat,Q: nat > $o] :
      ( ( P @ A2 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A2 ) )
       => ( ! [X4: nat] :
              ( ( P @ X4 )
             => ( Q @ X4 ) )
         => ( Q @ ( the_nat @ P ) ) ) ) ) ).

% theI2
thf(fact_644_If__def,axiom,
    ( if_nat
    = ( ^ [P5: $o,X3: nat,Y: nat] :
          ( the_nat
          @ ^ [Z: nat] :
              ( ( P5
               => ( Z = X3 ) )
              & ( ~ P5
               => ( Z = Y ) ) ) ) ) ) ).

% If_def
thf(fact_645_the1I2,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y5: nat] :
              ( ( P @ Y5 )
             => ( Y5 = X6 ) ) )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( the_nat @ P ) ) ) ) ).

% the1I2
thf(fact_646_the1__equality,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y5: nat] :
              ( ( P @ Y5 )
             => ( Y5 = X6 ) ) )
     => ( ( P @ A2 )
       => ( ( the_nat @ P )
          = A2 ) ) ) ).

% the1_equality
thf(fact_647__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ys1_Ays2_O_A_092_060lbrakk_062mset_A_0910_O_O_060m_093_A_061_Amset_Ays1_A_L_Amset_Ays2_059_AYs1_A_061_Amap_At_Ays1_059_AYs2_A_061_Amap_At_Ays2_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Ys12: list_nat,Ys22: list_nat] :
        ( ( ( mset_nat @ ( upt @ zero_zero_nat @ m ) )
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys22 ) ) )
       => ( ( ys1
            = ( map_nat_a @ t @ Ys12 ) )
         => ( ys2
           != ( map_nat_a @ t @ Ys22 ) ) ) ) ).

% \<open>\<And>thesis. (\<And>ys1 ys2. \<lbrakk>mset [0..<m] = mset ys1 + mset ys2; Ys1 = map t ys1; Ys2 = map t ys2\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_648__092_060open_062_092_060exists_062zs1_Azs2_O_Amset_A_0910_O_O_060m_093_A_061_Amset_Azs1_A_L_Amset_Azs2_A_092_060and_062_AYs1_A_061_Amap_At_Azs1_A_092_060and_062_AYs2_A_061_Amap_At_Azs2_092_060close_062,axiom,
    ? [Zs1: list_nat,Zs2: list_nat] :
      ( ( ( mset_nat @ ( upt @ zero_zero_nat @ m ) )
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
      & ( ys1
        = ( map_nat_a @ t @ Zs1 ) )
      & ( ys2
        = ( map_nat_a @ t @ Zs2 ) ) ) ).

% \<open>\<exists>zs1 zs2. mset [0..<m] = mset zs1 + mset zs2 \<and> Ys1 = map t zs1 \<and> Ys2 = map t zs2\<close>
thf(fact_649_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C4: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_650__092_060open_062_IXs1_A_B_Apos__of_Ays1_Aj_M_AYs1_A_B_Apos__of_Ays1_Aj_J_A_092_060in_062_ANS_092_060close_062,axiom,
    member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ xs1 @ ( pos_of @ ys12 @ j ) ) @ ( nth_a @ ys1 @ ( pos_of @ ys12 @ j ) ) ) @ ns ).

% \<open>(Xs1 ! pos_of ys1 j, Ys1 ! pos_of ys1 j) \<in> NS\<close>
thf(fact_651__092_060open_062_092_060exists_062xs1_Axs2_Ays1_Ays2_O_Amset_A_Imap_As_A_0910_O_O_060n_093_J_A_061_Amset_Axs1_A_L_Amset_Axs2_A_092_060and_062_Amset_A_Imap_At_A_0910_O_O_060m_093_J_A_061_Amset_Ays1_A_L_Amset_Ays2_A_092_060and_062_Alength_Axs1_A_061_Alength_Ays1_A_092_060and_062_A_I_092_060forall_062i_060length_Ays1_O_A_Ixs1_A_B_Ai_M_Ays1_A_B_Ai_J_A_092_060in_062_ANS_J_A_092_060and_062_A_I_092_060forall_062y_092_060in_062set_Ays2_O_A_092_060exists_062x_092_060in_062set_Axs2_O_A_Ix_M_Ay_J_A_092_060in_062_AS_J_092_060close_062,axiom,
    ? [Xs12: list_a,Xs22: list_a] :
      ( ( ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) )
        = ( plus_plus_multiset_a @ ( mset_a @ Xs12 ) @ ( mset_a @ Xs22 ) ) )
      & ? [Ys12: list_a,Ys22: list_a] :
          ( ( ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) )
            = ( plus_plus_multiset_a @ ( mset_a @ Ys12 ) @ ( mset_a @ Ys22 ) ) )
          & ( ( size_size_list_a @ Xs12 )
            = ( size_size_list_a @ Ys12 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys12 ) )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs12 @ I4 ) @ ( nth_a @ Ys12 @ I4 ) ) @ ns ) )
          & ! [X6: a] :
              ( ( member_a @ X6 @ ( set_a2 @ Ys22 ) )
             => ? [Xa2: a] :
                  ( ( member_a @ Xa2 @ ( set_a2 @ Xs22 ) )
                  & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ s ) ) ) ) ) ).

% \<open>\<exists>xs1 xs2 ys1 ys2. mset (map s [0..<n]) = mset xs1 + mset xs2 \<and> mset (map t [0..<m]) = mset ys1 + mset ys2 \<and> length xs1 = length ys1 \<and> (\<forall>i<length ys1. (xs1 ! i, ys1 ! i) \<in> NS) \<and> (\<forall>y\<in>set ys2. \<exists>x\<in>set xs2. (x, y) \<in> S)\<close>
thf(fact_652_cs,axiom,
    ! [I2: nat,J: nat] :
      ( ( cs @ I2 @ J )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( s2 @ I2 ) @ ( t @ J ) ) @ s ) ) ).

% cs
thf(fact_653_S,axiom,
    ! [Y2: a] :
      ( ( member_a @ Y2 @ ( set_a2 @ ys2 ) )
     => ? [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ xs2 ) )
          & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ s ) ) ) ).

% S
thf(fact_654_NS,axiom,
    ! [I2: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ ys1 ) )
     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ xs1 @ I2 ) @ ( nth_a @ ys1 @ I2 ) ) @ ns ) ) ).

% NS
thf(fact_655_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ( ^ [X3: a,Y: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R ) )
        = ( ^ [X3: a,Y: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_656_pred__equals__eq2,axiom,
    ! [R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( ( ^ [X3: multiset_a,Y: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R ) )
        = ( ^ [X3: multiset_a,Y: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_657_sup__Un__eq2,axiom,
    ! [R: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( sup_sup_a_a_o
        @ ^ [X3: a,Y: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
        @ ^ [X3: a,Y: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ S ) )
      = ( ^ [X3: a,Y: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( sup_su3048258781599657691od_a_a @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_658_sup__Un__eq2,axiom,
    ! [R: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( sup_su6917467955185334722et_a_o
        @ ^ [X3: multiset_a,Y: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R )
        @ ^ [X3: multiset_a,Y: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ S ) )
      = ( ^ [X3: multiset_a,Y: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( sup_su1925787886501348827iset_a @ R @ S ) ) ) ) ).

% sup_Un_eq2
thf(fact_659_assms,axiom,
    member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) ) @ ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) ) ) @ ( multis4023584895248362547_ext_a @ ns @ s ) ).

% assms
thf(fact_660_ne,axiom,
    ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) ) @ ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) ) ) @ ( multiset_s_mul_ext_a @ ns @ s ) )
   => ( xs2 != nil_a ) ) ).

% ne
thf(fact_661__092_060open_062_Imset_A_Imap_As_A_0910_O_O_060n_093_J_M_Amset_A_Imap_At_A_0910_O_O_060m_093_J_J_A_092_060in_062_As__mul__ext_ANS_AS_A_092_060Longrightarrow_062_A_092_060exists_062xs1_Axs2_Ays1_Ays2_O_Amset_A_Imap_As_A_0910_O_O_060n_093_J_A_061_Amset_Axs1_A_L_Amset_Axs2_A_092_060and_062_Amset_A_Imap_At_A_0910_O_O_060m_093_J_A_061_Amset_Ays1_A_L_Amset_Ays2_A_092_060and_062_Alength_Axs1_A_061_Alength_Ays1_A_092_060and_062_Axs2_A_092_060noteq_062_A_091_093_A_092_060and_062_A_I_092_060forall_062i_060length_Ays1_O_A_Ixs1_A_B_Ai_M_Ays1_A_B_Ai_J_A_092_060in_062_ANS_J_A_092_060and_062_A_I_092_060forall_062y_092_060in_062set_Ays2_O_A_092_060exists_062x_092_060in_062set_Axs2_O_A_Ix_M_Ay_J_A_092_060in_062_AS_J_092_060close_062,axiom,
    ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) ) @ ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) ) ) @ ( multiset_s_mul_ext_a @ ns @ s ) )
   => ? [Xs12: list_a,Xs22: list_a] :
        ( ( ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) )
          = ( plus_plus_multiset_a @ ( mset_a @ Xs12 ) @ ( mset_a @ Xs22 ) ) )
        & ? [Ys12: list_a,Ys22: list_a] :
            ( ( ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) )
              = ( plus_plus_multiset_a @ ( mset_a @ Ys12 ) @ ( mset_a @ Ys22 ) ) )
            & ( ( size_size_list_a @ Xs12 )
              = ( size_size_list_a @ Ys12 ) )
            & ( Xs22 != nil_a )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys12 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs12 @ I4 ) @ ( nth_a @ Ys12 @ I4 ) ) @ ns ) )
            & ! [X6: a] :
                ( ( member_a @ X6 @ ( set_a2 @ Ys22 ) )
               => ? [Xa2: a] :
                    ( ( member_a @ Xa2 @ ( set_a2 @ Xs22 ) )
                    & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ s ) ) ) ) ) ) ).

% \<open>(mset (map s [0..<n]), mset (map t [0..<m])) \<in> s_mul_ext NS S \<Longrightarrow> \<exists>xs1 xs2 ys1 ys2. mset (map s [0..<n]) = mset xs1 + mset xs2 \<and> mset (map t [0..<m]) = mset ys1 + mset ys2 \<and> length xs1 = length ys1 \<and> xs2 \<noteq> [] \<and> (\<forall>i<length ys1. (xs1 ! i, ys1 ! i) \<in> NS) \<and> (\<forall>y\<in>set ys2. \<exists>x\<in>set xs2. (x, y) \<in> S)\<close>
thf(fact_662__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Xs1_AXs2_AYs1_AYs2_O_A_092_060lbrakk_062mset_A_Imap_As_A_0910_O_O_060n_093_J_A_061_Amset_AXs1_A_L_Amset_AXs2_059_Amset_A_Imap_At_A_0910_O_O_060m_093_J_A_061_Amset_AYs1_A_L_Amset_AYs2_059_Alength_AXs1_A_061_Alength_AYs1_059_A_Imset_A_Imap_As_A_0910_O_O_060n_093_J_M_Amset_A_Imap_At_A_0910_O_O_060m_093_J_J_A_092_060in_062_As__mul__ext_ANS_AS_A_092_060Longrightarrow_062_AXs2_A_092_060noteq_062_A_091_093_059_A_092_060And_062i_O_Ai_A_060_Alength_AYs1_A_092_060Longrightarrow_062_A_IXs1_A_B_Ai_M_AYs1_A_B_Ai_J_A_092_060in_062_ANS_059_A_092_060And_062y_O_Ay_A_092_060in_062_Aset_AYs2_A_092_060Longrightarrow_062_A_092_060exists_062x_092_060in_062set_AXs2_O_A_Ix_M_Ay_J_A_092_060in_062_AS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Xs13: list_a,Xs23: list_a] :
        ( ( ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) )
          = ( plus_plus_multiset_a @ ( mset_a @ Xs13 ) @ ( mset_a @ Xs23 ) ) )
       => ! [Ys13: list_a,Ys23: list_a] :
            ( ( ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) )
              = ( plus_plus_multiset_a @ ( mset_a @ Ys13 ) @ ( mset_a @ Ys23 ) ) )
           => ( ( ( size_size_list_a @ Xs13 )
                = ( size_size_list_a @ Ys13 ) )
             => ( ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ ( map_nat_a @ s2 @ ( upt @ zero_zero_nat @ n ) ) ) @ ( mset_a @ ( map_nat_a @ t @ ( upt @ zero_zero_nat @ m ) ) ) ) @ ( multiset_s_mul_ext_a @ ns @ s ) )
                 => ( Xs23 != nil_a ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys13 ) )
                     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs13 @ I4 ) @ ( nth_a @ Ys13 @ I4 ) ) @ ns ) )
                 => ~ ! [Y6: a] :
                        ( ( member_a @ Y6 @ ( set_a2 @ Ys23 ) )
                       => ? [X4: a] :
                            ( ( member_a @ X4 @ ( set_a2 @ Xs23 ) )
                            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y6 ) @ s ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>Xs1 Xs2 Ys1 Ys2. \<lbrakk>mset (map s [0..<n]) = mset Xs1 + mset Xs2; mset (map t [0..<m]) = mset Ys1 + mset Ys2; length Xs1 = length Ys1; (mset (map s [0..<n]), mset (map t [0..<m])) \<in> s_mul_ext NS S \<Longrightarrow> Xs2 \<noteq> []; \<And>i. i < length Ys1 \<Longrightarrow> (Xs1 ! i, Ys1 ! i) \<in> NS; \<And>y. y \<in> set Ys2 \<Longrightarrow> \<exists>x\<in>set Xs2. (x, y) \<in> S\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_663_ns__mul__ext__elim,axiom,
    ! [Xs: multis193838646080965794ropVar,Ys: multis193838646080965794ropVar,NS: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis738276037714153361ropVar @ NS @ S ) )
     => ? [Xs12: list_M6740372356978808802ropVar,Xs22: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs12 ) @ ( mset_M6581516532424202993ropVar @ Xs22 ) ) )
          & ? [Ys12: list_M6740372356978808802ropVar,Ys22: list_M6740372356978808802ropVar] :
              ( ( Ys
                = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys12 ) @ ( mset_M6581516532424202993ropVar @ Ys22 ) ) )
              & ( ( size_s1943289338788136526ropVar @ Xs12 )
                = ( size_s1943289338788136526ropVar @ Ys12 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s1943289338788136526ropVar @ Ys12 ) )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs12 @ I4 ) @ ( nth_Mu4537670848183934179ropVar @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Ys22 ) )
                 => ? [Xa2: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Xs22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_664_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_multiset_a,Ys: multiset_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Xs @ Ys ) @ ( multis8112516946959054073iset_a @ NS @ S ) )
     => ? [Xs12: list_multiset_a,Xs22: list_multiset_a] :
          ( ( Xs
            = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs12 ) @ ( mset_multiset_a @ Xs22 ) ) )
          & ? [Ys12: list_multiset_a,Ys22: list_multiset_a] :
              ( ( Ys
                = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys12 ) @ ( mset_multiset_a @ Ys22 ) ) )
              & ( ( size_s3003218658903709366iset_a @ Xs12 )
                = ( size_s3003218658903709366iset_a @ Ys12 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s3003218658903709366iset_a @ Ys12 ) )
                 => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs12 @ I4 ) @ ( nth_multiset_a @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: multiset_a] :
                  ( ( member_multiset_a @ X6 @ ( set_multiset_a2 @ Ys22 ) )
                 => ? [Xa2: multiset_a] :
                      ( ( member_multiset_a @ Xa2 @ ( set_multiset_a2 @ Xs22 ) )
                      & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_665_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S ) )
     => ? [Xs12: list_nat,Xs22: list_nat] :
          ( ( Xs
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs22 ) ) )
          & ? [Ys12: list_nat,Ys22: list_nat] :
              ( ( Ys
                = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys22 ) ) )
              & ( ( size_size_list_nat @ Xs12 )
                = ( size_size_list_nat @ Ys12 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys12 ) )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs12 @ I4 ) @ ( nth_nat @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: nat] :
                  ( ( member_nat @ X6 @ ( set_nat2 @ Ys22 ) )
                 => ? [Xa2: nat] :
                      ( ( member_nat @ Xa2 @ ( set_nat2 @ Xs22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_666_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_a,Ys: multiset_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multis4023584895248362547_ext_a @ NS @ S ) )
     => ? [Xs12: list_a,Xs22: list_a] :
          ( ( Xs
            = ( plus_plus_multiset_a @ ( mset_a @ Xs12 ) @ ( mset_a @ Xs22 ) ) )
          & ? [Ys12: list_a,Ys22: list_a] :
              ( ( Ys
                = ( plus_plus_multiset_a @ ( mset_a @ Ys12 ) @ ( mset_a @ Ys22 ) ) )
              & ( ( size_size_list_a @ Xs12 )
                = ( size_size_list_a @ Ys12 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys12 ) )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs12 @ I4 ) @ ( nth_a @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: a] :
                  ( ( member_a @ X6 @ ( set_a2 @ Ys22 ) )
                 => ? [Xa2: a] :
                      ( ( member_a @ Xa2 @ ( set_a2 @ Xs22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_667_map__is__Nil__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( ( map_a_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_668_map__is__Nil__conv,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( ( map_a_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_669_map__is__Nil__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = nil_Mu4132399263956683874ropVar )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_670_map__is__Nil__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_671_map__is__Nil__conv,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( ( map_nat_a @ F @ Xs )
        = nil_a )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_672_Nil__is__map__conv,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( nil_a
        = ( map_a_a @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_673_Nil__is__map__conv,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( nil_nat
        = ( map_a_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_674_Nil__is__map__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( nil_Mu4132399263956683874ropVar
        = ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_675_Nil__is__map__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_676_Nil__is__map__conv,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( nil_a
        = ( map_nat_a @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_677_list_Omap__disc__iff,axiom,
    ! [F: a > a,A2: list_a] :
      ( ( ( map_a_a @ F @ A2 )
        = nil_a )
      = ( A2 = nil_a ) ) ).

% list.map_disc_iff
thf(fact_678_list_Omap__disc__iff,axiom,
    ! [F: a > nat,A2: list_a] :
      ( ( ( map_a_nat @ F @ A2 )
        = nil_nat )
      = ( A2 = nil_a ) ) ).

% list.map_disc_iff
thf(fact_679_list_Omap__disc__iff,axiom,
    ! [F: nat > multis3193088007478089820ropVar,A2: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ A2 )
        = nil_Mu4132399263956683874ropVar )
      = ( A2 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_680_list_Omap__disc__iff,axiom,
    ! [F: nat > nat,A2: list_nat] :
      ( ( ( map_nat_nat @ F @ A2 )
        = nil_nat )
      = ( A2 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_681_list_Omap__disc__iff,axiom,
    ! [F: nat > a,A2: list_nat] :
      ( ( ( map_nat_a @ F @ A2 )
        = nil_a )
      = ( A2 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_682_append__is__Nil__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = nil_a )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% append_is_Nil_conv
thf(fact_683_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_684_Nil__is__append__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_a
        = ( append_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        & ( Ys = nil_a ) ) ) ).

% Nil_is_append_conv
thf(fact_685_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_686_self__append__conv2,axiom,
    ! [Y2: list_a,Xs: list_a] :
      ( ( Y2
        = ( append_a @ Xs @ Y2 ) )
      = ( Xs = nil_a ) ) ).

% self_append_conv2
thf(fact_687_self__append__conv2,axiom,
    ! [Y2: list_nat,Xs: list_nat] :
      ( ( Y2
        = ( append_nat @ Xs @ Y2 ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_688_append__self__conv2,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_a ) ) ).

% append_self_conv2
thf(fact_689_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_690_self__append__conv,axiom,
    ! [Y2: list_a,Ys: list_a] :
      ( ( Y2
        = ( append_a @ Y2 @ Ys ) )
      = ( Ys = nil_a ) ) ).

% self_append_conv
thf(fact_691_self__append__conv,axiom,
    ! [Y2: list_nat,Ys: list_nat] :
      ( ( Y2
        = ( append_nat @ Y2 @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_692_append__self__conv,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( append_a @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_a ) ) ).

% append_self_conv
thf(fact_693_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_694_append__Nil2,axiom,
    ! [Xs: list_a] :
      ( ( append_a @ Xs @ nil_a )
      = Xs ) ).

% append_Nil2
thf(fact_695_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_696_append_Oright__neutral,axiom,
    ! [A2: list_a] :
      ( ( append_a @ A2 @ nil_a )
      = A2 ) ).

% append.right_neutral
thf(fact_697_append_Oright__neutral,axiom,
    ! [A2: list_nat] :
      ( ( append_nat @ A2 @ nil_nat )
      = A2 ) ).

% append.right_neutral
thf(fact_698_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_699_length__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_a ) ) ).

% length_0_conv
thf(fact_700_mset__zero__iff,axiom,
    ! [X: list_nat] :
      ( ( ( mset_nat @ X )
        = zero_z7348594199698428585et_nat )
      = ( X = nil_nat ) ) ).

% mset_zero_iff
thf(fact_701_mset__zero__iff,axiom,
    ! [X: list_a] :
      ( ( ( mset_a @ X )
        = zero_zero_multiset_a )
      = ( X = nil_a ) ) ).

% mset_zero_iff
thf(fact_702_mset__zero__iff__right,axiom,
    ! [X: list_nat] :
      ( ( zero_z7348594199698428585et_nat
        = ( mset_nat @ X ) )
      = ( X = nil_nat ) ) ).

% mset_zero_iff_right
thf(fact_703_mset__zero__iff__right,axiom,
    ! [X: list_a] :
      ( ( zero_zero_multiset_a
        = ( mset_a @ X ) )
      = ( X = nil_a ) ) ).

% mset_zero_iff_right
thf(fact_704_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_705_length__greater__0__conv,axiom,
    ! [Xs: list_a] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) )
      = ( Xs != nil_a ) ) ).

% length_greater_0_conv
thf(fact_706_ns__mul__ext__NS__union__S,axiom,
    ! [NS: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( multis4023584895248362547_ext_a @ ( sup_su3048258781599657691od_a_a @ NS @ S ) @ S )
      = ( multis4023584895248362547_ext_a @ NS @ S ) ) ).

% ns_mul_ext_NS_union_S
thf(fact_707_list_Osimps_I8_J,axiom,
    ! [F: a > a] :
      ( ( map_a_a @ F @ nil_a )
      = nil_a ) ).

% list.simps(8)
thf(fact_708_list_Osimps_I8_J,axiom,
    ! [F: a > nat] :
      ( ( map_a_nat @ F @ nil_a )
      = nil_nat ) ).

% list.simps(8)
thf(fact_709_list_Osimps_I8_J,axiom,
    ! [F: nat > multis3193088007478089820ropVar] :
      ( ( map_na2466558649495876678ropVar @ F @ nil_nat )
      = nil_Mu4132399263956683874ropVar ) ).

% list.simps(8)
thf(fact_710_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_711_list_Osimps_I8_J,axiom,
    ! [F: nat > a] :
      ( ( map_nat_a @ F @ nil_nat )
      = nil_a ) ).

% list.simps(8)
thf(fact_712_eq__Nil__appendI,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_a @ nil_a @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_713_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_714_append_Oleft__neutral,axiom,
    ! [A2: list_a] :
      ( ( append_a @ nil_a @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_715_append_Oleft__neutral,axiom,
    ! [A2: list_nat] :
      ( ( append_nat @ nil_nat @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_716_append__Nil,axiom,
    ! [Ys: list_a] :
      ( ( append_a @ nil_a @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_717_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_718_distinct_Osimps_I1_J,axiom,
    distinct_a @ nil_a ).

% distinct.simps(1)
thf(fact_719_distinct_Osimps_I1_J,axiom,
    distinct_nat @ nil_nat ).

% distinct.simps(1)
thf(fact_720_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_M2293212096379261346ropVar @ N @ nil_Mu4132399263956683874ropVar )
      = nil_Mu4132399263956683874ropVar ) ).

% drop_Nil
thf(fact_721_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_a @ N @ nil_a )
      = nil_a ) ).

% drop_Nil
thf(fact_722_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_nat @ N @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_723_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_724_list_Osize_I3_J,axiom,
    ( ( size_size_list_a @ nil_a )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_725_mset_Osimps_I1_J,axiom,
    ( ( mset_nat @ nil_nat )
    = zero_z7348594199698428585et_nat ) ).

% mset.simps(1)
thf(fact_726_mset_Osimps_I1_J,axiom,
    ( ( mset_a @ nil_a )
    = zero_zero_multiset_a ) ).

% mset.simps(1)
thf(fact_727_s__mul__ext__intro,axiom,
    ! [Xs: multis501812127501805293od_a_a,Xs1: list_P1396940483166286381od_a_a,Xs24: list_P1396940483166286381od_a_a,Ys: multis501812127501805293od_a_a,Ys1: list_P1396940483166286381od_a_a,Ys2: list_P1396940483166286381od_a_a,NS: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
      ( ( Xs
        = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs1 ) @ ( mset_P3709024939784119484od_a_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys1 ) @ ( mset_P3709024939784119484od_a_a @ Ys2 ) ) )
       => ( ( ( size_s3885678630836030617od_a_a @ Xs1 )
            = ( size_s3885678630836030617od_a_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys1 ) )
               => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs1 @ I3 ) @ ( nth_Product_prod_a_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ( Xs24 != nil_Product_prod_a_a )
             => ( ! [Y5: product_prod_a_a] :
                    ( ( member1426531477525435216od_a_a @ Y5 @ ( set_Product_prod_a_a2 @ Ys2 ) )
                   => ? [X6: product_prod_a_a] :
                        ( ( member1426531477525435216od_a_a @ X6 @ ( set_Product_prod_a_a2 @ Xs24 ) )
                        & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X6 @ Y5 ) @ S ) ) )
               => ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Xs @ Ys ) @ ( multis6523615221270609726od_a_a @ NS @ S ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_728_s__mul__ext__intro,axiom,
    ! [Xs: multis4781622204924204013iset_a,Xs1: list_P6988024003011616557iset_a,Xs24: list_P6988024003011616557iset_a,Ys: multis4781622204924204013iset_a,Ys1: list_P6988024003011616557iset_a,Ys2: list_P6988024003011616557iset_a,NS: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
      ( ( Xs
        = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs1 ) @ ( mset_P2771936761501114300iset_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys1 ) @ ( mset_P2771936761501114300iset_a @ Ys2 ) ) )
       => ( ( ( size_s7238940367606365593iset_a @ Xs1 )
            = ( size_s7238940367606365593iset_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys1 ) )
               => ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs1 @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ( Xs24 != nil_Pr2722442113885406509iset_a )
             => ( ! [Y5: produc6518373309651786023iset_a] :
                    ( ( member5199237121806060112iset_a @ Y5 @ ( set_Pr5550698219922361532iset_a @ Ys2 ) )
                   => ? [X6: produc6518373309651786023iset_a] :
                        ( ( member5199237121806060112iset_a @ X6 @ ( set_Pr5550698219922361532iset_a @ Xs24 ) )
                        & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X6 @ Y5 ) @ S ) ) )
               => ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Xs @ Ys ) @ ( multis6482401367445528638iset_a @ NS @ S ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_729_s__mul__ext__intro,axiom,
    ! [Xs: multis193838646080965794ropVar,Xs1: list_M6740372356978808802ropVar,Xs24: list_M6740372356978808802ropVar,Ys: multis193838646080965794ropVar,Ys1: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,NS: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( Xs
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs1 ) @ ( mset_M6581516532424202993ropVar @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys1 ) @ ( mset_M6581516532424202993ropVar @ Ys2 ) ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs1 )
            = ( size_s1943289338788136526ropVar @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Ys1 ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs1 @ I3 ) @ ( nth_Mu4537670848183934179ropVar @ Ys1 @ I3 ) ) @ NS ) )
           => ( ( Xs24 != nil_Mu4132399263956683874ropVar )
             => ( ! [Y5: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ Y5 @ ( set_Mu146256361055586289ropVar @ Ys2 ) )
                   => ? [X6: multis3193088007478089820ropVar] :
                        ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs24 ) )
                        & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X6 @ Y5 ) @ S ) ) )
               => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis4691071231837107059ropVar @ NS @ S ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_730_s__mul__ext__intro,axiom,
    ! [Xs: multiset_multiset_a,Xs1: list_multiset_a,Xs24: list_multiset_a,Ys: multiset_multiset_a,Ys1: list_multiset_a,Ys2: list_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( Xs
        = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs1 ) @ ( mset_multiset_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys1 ) @ ( mset_multiset_a @ Ys2 ) ) )
       => ( ( ( size_s3003218658903709366iset_a @ Xs1 )
            = ( size_s3003218658903709366iset_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys1 ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs1 @ I3 ) @ ( nth_multiset_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ( Xs24 != nil_multiset_a )
             => ( ! [Y5: multiset_a] :
                    ( ( member_multiset_a @ Y5 @ ( set_multiset_a2 @ Ys2 ) )
                   => ? [X6: multiset_a] :
                        ( ( member_multiset_a @ X6 @ ( set_multiset_a2 @ Xs24 ) )
                        & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y5 ) @ S ) ) )
               => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Xs @ Ys ) @ ( multis7166037729339985883iset_a @ NS @ S ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_731_s__mul__ext__intro,axiom,
    ! [Xs: multiset_nat,Xs1: list_nat,Xs24: list_nat,Ys: multiset_nat,Ys1: list_nat,Ys2: list_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( Xs
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs1 ) @ ( mset_nat @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys2 ) ) )
       => ( ( ( size_size_list_nat @ Xs1 )
            = ( size_size_list_nat @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys1 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I3 ) @ ( nth_nat @ Ys1 @ I3 ) ) @ NS ) )
           => ( ( Xs24 != nil_nat )
             => ( ! [Y5: nat] :
                    ( ( member_nat @ Y5 @ ( set_nat2 @ Ys2 ) )
                   => ? [X6: nat] :
                        ( ( member_nat @ X6 @ ( set_nat2 @ Xs24 ) )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ S ) ) )
               => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_732_s__mul__ext__intro,axiom,
    ! [Xs: multiset_a,Xs1: list_a,Xs24: list_a,Ys: multiset_a,Ys1: list_a,Ys2: list_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( Xs
        = ( plus_plus_multiset_a @ ( mset_a @ Xs1 ) @ ( mset_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_plus_multiset_a @ ( mset_a @ Ys1 ) @ ( mset_a @ Ys2 ) ) )
       => ( ( ( size_size_list_a @ Xs1 )
            = ( size_size_list_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys1 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs1 @ I3 ) @ ( nth_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ( Xs24 != nil_a )
             => ( ! [Y5: a] :
                    ( ( member_a @ Y5 @ ( set_a2 @ Ys2 ) )
                   => ? [X6: a] :
                        ( ( member_a @ X6 @ ( set_a2 @ Xs24 ) )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y5 ) @ S ) ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multiset_s_mul_ext_a @ NS @ S ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_733_s__mul__ext__elim,axiom,
    ! [Xs: multis193838646080965794ropVar,Ys: multis193838646080965794ropVar,NS: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis4691071231837107059ropVar @ NS @ S ) )
     => ? [Xs12: list_M6740372356978808802ropVar,Xs22: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs12 ) @ ( mset_M6581516532424202993ropVar @ Xs22 ) ) )
          & ? [Ys12: list_M6740372356978808802ropVar,Ys22: list_M6740372356978808802ropVar] :
              ( ( Ys
                = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys12 ) @ ( mset_M6581516532424202993ropVar @ Ys22 ) ) )
              & ( ( size_s1943289338788136526ropVar @ Xs12 )
                = ( size_s1943289338788136526ropVar @ Ys12 ) )
              & ( Xs22 != nil_Mu4132399263956683874ropVar )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s1943289338788136526ropVar @ Ys12 ) )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs12 @ I4 ) @ ( nth_Mu4537670848183934179ropVar @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Ys22 ) )
                 => ? [Xa2: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Xs22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_734_s__mul__ext__elim,axiom,
    ! [Xs: multiset_multiset_a,Ys: multiset_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Xs @ Ys ) @ ( multis7166037729339985883iset_a @ NS @ S ) )
     => ? [Xs12: list_multiset_a,Xs22: list_multiset_a] :
          ( ( Xs
            = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs12 ) @ ( mset_multiset_a @ Xs22 ) ) )
          & ? [Ys12: list_multiset_a,Ys22: list_multiset_a] :
              ( ( Ys
                = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys12 ) @ ( mset_multiset_a @ Ys22 ) ) )
              & ( ( size_s3003218658903709366iset_a @ Xs12 )
                = ( size_s3003218658903709366iset_a @ Ys12 ) )
              & ( Xs22 != nil_multiset_a )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s3003218658903709366iset_a @ Ys12 ) )
                 => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs12 @ I4 ) @ ( nth_multiset_a @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: multiset_a] :
                  ( ( member_multiset_a @ X6 @ ( set_multiset_a2 @ Ys22 ) )
                 => ? [Xa2: multiset_a] :
                      ( ( member_multiset_a @ Xa2 @ ( set_multiset_a2 @ Xs22 ) )
                      & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_735_s__mul__ext__elim,axiom,
    ! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S ) )
     => ? [Xs12: list_nat,Xs22: list_nat] :
          ( ( Xs
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs22 ) ) )
          & ? [Ys12: list_nat,Ys22: list_nat] :
              ( ( Ys
                = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys22 ) ) )
              & ( ( size_size_list_nat @ Xs12 )
                = ( size_size_list_nat @ Ys12 ) )
              & ( Xs22 != nil_nat )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys12 ) )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs12 @ I4 ) @ ( nth_nat @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: nat] :
                  ( ( member_nat @ X6 @ ( set_nat2 @ Ys22 ) )
                 => ? [Xa2: nat] :
                      ( ( member_nat @ Xa2 @ ( set_nat2 @ Xs22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_736_s__mul__ext__elim,axiom,
    ! [Xs: multiset_a,Ys: multiset_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multiset_s_mul_ext_a @ NS @ S ) )
     => ? [Xs12: list_a,Xs22: list_a] :
          ( ( Xs
            = ( plus_plus_multiset_a @ ( mset_a @ Xs12 ) @ ( mset_a @ Xs22 ) ) )
          & ? [Ys12: list_a,Ys22: list_a] :
              ( ( Ys
                = ( plus_plus_multiset_a @ ( mset_a @ Ys12 ) @ ( mset_a @ Ys22 ) ) )
              & ( ( size_size_list_a @ Xs12 )
                = ( size_size_list_a @ Ys12 ) )
              & ( Xs22 != nil_a )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys12 ) )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs12 @ I4 ) @ ( nth_a @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: a] :
                  ( ( member_a @ X6 @ ( set_a2 @ Ys22 ) )
                 => ? [Xa2: a] :
                      ( ( member_a @ Xa2 @ ( set_a2 @ Xs22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ S ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_737_ns__mul__ext__intro,axiom,
    ! [Xs: multis501812127501805293od_a_a,Xs1: list_P1396940483166286381od_a_a,Xs24: list_P1396940483166286381od_a_a,Ys: multis501812127501805293od_a_a,Ys1: list_P1396940483166286381od_a_a,Ys2: list_P1396940483166286381od_a_a,NS: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a] :
      ( ( Xs
        = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs1 ) @ ( mset_P3709024939784119484od_a_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys1 ) @ ( mset_P3709024939784119484od_a_a @ Ys2 ) ) )
       => ( ( ( size_s3885678630836030617od_a_a @ Xs1 )
            = ( size_s3885678630836030617od_a_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys1 ) )
               => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs1 @ I3 ) @ ( nth_Product_prod_a_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ! [Y5: product_prod_a_a] :
                  ( ( member1426531477525435216od_a_a @ Y5 @ ( set_Product_prod_a_a2 @ Ys2 ) )
                 => ? [X6: product_prod_a_a] :
                      ( ( member1426531477525435216od_a_a @ X6 @ ( set_Product_prod_a_a2 @ Xs24 ) )
                      & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X6 @ Y5 ) @ S ) ) )
             => ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ Xs @ Ys ) @ ( multis1178328011278088540od_a_a @ NS @ S ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_738_ns__mul__ext__intro,axiom,
    ! [Xs: multis4781622204924204013iset_a,Xs1: list_P6988024003011616557iset_a,Xs24: list_P6988024003011616557iset_a,Ys: multis4781622204924204013iset_a,Ys1: list_P6988024003011616557iset_a,Ys2: list_P6988024003011616557iset_a,NS: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a] :
      ( ( Xs
        = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs1 ) @ ( mset_P2771936761501114300iset_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys1 ) @ ( mset_P2771936761501114300iset_a @ Ys2 ) ) )
       => ( ( ( size_s7238940367606365593iset_a @ Xs1 )
            = ( size_s7238940367606365593iset_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys1 ) )
               => ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs1 @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ! [Y5: produc6518373309651786023iset_a] :
                  ( ( member5199237121806060112iset_a @ Y5 @ ( set_Pr5550698219922361532iset_a @ Ys2 ) )
                 => ? [X6: produc6518373309651786023iset_a] :
                      ( ( member5199237121806060112iset_a @ X6 @ ( set_Pr5550698219922361532iset_a @ Xs24 ) )
                      & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X6 @ Y5 ) @ S ) ) )
             => ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ Xs @ Ys ) @ ( multis2244641948333282396iset_a @ NS @ S ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_739_ns__mul__ext__intro,axiom,
    ! [Xs: multis193838646080965794ropVar,Xs1: list_M6740372356978808802ropVar,Xs24: list_M6740372356978808802ropVar,Ys: multis193838646080965794ropVar,Ys1: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,NS: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( Xs
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs1 ) @ ( mset_M6581516532424202993ropVar @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys1 ) @ ( mset_M6581516532424202993ropVar @ Ys2 ) ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs1 )
            = ( size_s1943289338788136526ropVar @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Ys1 ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs1 @ I3 ) @ ( nth_Mu4537670848183934179ropVar @ Ys1 @ I3 ) ) @ NS ) )
           => ( ! [Y5: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y5 @ ( set_Mu146256361055586289ropVar @ Ys2 ) )
                 => ? [X6: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs24 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X6 @ Y5 ) @ S ) ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis738276037714153361ropVar @ NS @ S ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_740_ns__mul__ext__intro,axiom,
    ! [Xs: multiset_multiset_a,Xs1: list_multiset_a,Xs24: list_multiset_a,Ys: multiset_multiset_a,Ys1: list_multiset_a,Ys2: list_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( Xs
        = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs1 ) @ ( mset_multiset_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys1 ) @ ( mset_multiset_a @ Ys2 ) ) )
       => ( ( ( size_s3003218658903709366iset_a @ Xs1 )
            = ( size_s3003218658903709366iset_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys1 ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs1 @ I3 ) @ ( nth_multiset_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ! [Y5: multiset_a] :
                  ( ( member_multiset_a @ Y5 @ ( set_multiset_a2 @ Ys2 ) )
                 => ? [X6: multiset_a] :
                      ( ( member_multiset_a @ X6 @ ( set_multiset_a2 @ Xs24 ) )
                      & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X6 @ Y5 ) @ S ) ) )
             => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ Xs @ Ys ) @ ( multis8112516946959054073iset_a @ NS @ S ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_741_ns__mul__ext__intro,axiom,
    ! [Xs: multiset_nat,Xs1: list_nat,Xs24: list_nat,Ys: multiset_nat,Ys1: list_nat,Ys2: list_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( Xs
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs1 ) @ ( mset_nat @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys2 ) ) )
       => ( ( ( size_size_list_nat @ Xs1 )
            = ( size_size_list_nat @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys1 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I3 ) @ ( nth_nat @ Ys1 @ I3 ) ) @ NS ) )
           => ( ! [Y5: nat] :
                  ( ( member_nat @ Y5 @ ( set_nat2 @ Ys2 ) )
                 => ? [X6: nat] :
                      ( ( member_nat @ X6 @ ( set_nat2 @ Xs24 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y5 ) @ S ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_742_ns__mul__ext__intro,axiom,
    ! [Xs: multiset_a,Xs1: list_a,Xs24: list_a,Ys: multiset_a,Ys1: list_a,Ys2: list_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( Xs
        = ( plus_plus_multiset_a @ ( mset_a @ Xs1 ) @ ( mset_a @ Xs24 ) ) )
     => ( ( Ys
          = ( plus_plus_multiset_a @ ( mset_a @ Ys1 ) @ ( mset_a @ Ys2 ) ) )
       => ( ( ( size_size_list_a @ Xs1 )
            = ( size_size_list_a @ Ys1 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys1 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs1 @ I3 ) @ ( nth_a @ Ys1 @ I3 ) ) @ NS ) )
           => ( ! [Y5: a] :
                  ( ( member_a @ Y5 @ ( set_a2 @ Ys2 ) )
                 => ? [X6: a] :
                      ( ( member_a @ X6 @ ( set_a2 @ Xs24 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X6 @ Y5 ) @ S ) ) )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multis4023584895248362547_ext_a @ NS @ S ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_743_all__ns__ns__mul__ext,axiom,
    ! [As: list_multiset_a,Bs: list_multiset_a,Ns: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
      ( ( ( size_s3003218658903709366iset_a @ As )
        = ( size_s3003218658903709366iset_a @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Bs ) )
           => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ As @ I3 ) @ ( nth_multiset_a @ Bs @ I3 ) ) @ Ns ) )
       => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( mset_multiset_a @ As ) @ ( mset_multiset_a @ Bs ) ) @ ( multis8112516946959054073iset_a @ Ns @ S2 ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_744_all__ns__ns__mul__ext,axiom,
    ! [As: list_nat,Bs: list_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ Ns ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As ) @ ( mset_nat @ Bs ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_745_all__ns__ns__mul__ext,axiom,
    ! [As: list_a,Bs: list_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( ( size_size_list_a @ As )
        = ( size_size_list_a @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Bs ) )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) @ Ns ) )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As ) @ ( mset_a @ Bs ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_746_s__mul__ext__ne__extend__left,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,C2: multiset_nat,W: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( A3 != zero_z7348594199698428585et_nat )
     => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ B3 @ C2 ) @ ( multis8831328596575508315xt_nat @ W @ S ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) @ C2 ) @ ( multis4215214144601311993xt_nat @ W @ S ) ) ) ) ).

% s_mul_ext_ne_extend_left
thf(fact_747_s__mul__ext__ne__extend__left,axiom,
    ! [A3: multiset_a,B3: multiset_a,C2: multiset_a,W: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( A3 != zero_zero_multiset_a )
     => ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B3 @ C2 ) @ ( multis4023584895248362547_ext_a @ W @ S ) )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ C2 ) @ ( multiset_s_mul_ext_a @ W @ S ) ) ) ) ).

% s_mul_ext_ne_extend_left
thf(fact_748_s__ns__mul__ext__union__compat,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,C2: multiset_nat,D: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) )
     => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ C2 @ D ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ B3 @ D ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ).

% s_ns_mul_ext_union_compat
thf(fact_749_s__ns__mul__ext__union__compat,axiom,
    ! [A3: multiset_a,B3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a,C2: multiset_a,D: multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) )
     => ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C2 @ D ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B3 @ D ) ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ).

% s_ns_mul_ext_union_compat
thf(fact_750_s__mul__ext__extend__left,axiom,
    ! [B3: multiset_nat,C2: multiset_nat,W: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,A3: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ B3 @ C2 ) @ ( multis4215214144601311993xt_nat @ W @ S ) )
     => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) @ C2 ) @ ( multis4215214144601311993xt_nat @ W @ S ) ) ) ).

% s_mul_ext_extend_left
thf(fact_751_s__mul__ext__extend__left,axiom,
    ! [B3: multiset_a,C2: multiset_a,W: set_Product_prod_a_a,S: set_Product_prod_a_a,A3: multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B3 @ C2 ) @ ( multiset_s_mul_ext_a @ W @ S ) )
     => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ C2 ) @ ( multiset_s_mul_ext_a @ W @ S ) ) ) ).

% s_mul_ext_extend_left
thf(fact_752_ns__ns__mul__ext__union__compat,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,C2: multiset_nat,D: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) )
     => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ C2 @ D ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ B3 @ D ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) ) ) ) ).

% ns_ns_mul_ext_union_compat
thf(fact_753_ns__ns__mul__ext__union__compat,axiom,
    ! [A3: multiset_a,B3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a,C2: multiset_a,D: multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
     => ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C2 @ D ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B3 @ D ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) ) ) ) ).

% ns_ns_mul_ext_union_compat
thf(fact_754_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_755_ns__mul__ext__bottom,axiom,
    ! [A3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ zero_zero_multiset_a ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) ) ).

% ns_mul_ext_bottom
thf(fact_756_ns__mul__ext__bottom2,axiom,
    ! [A3: multiset_a,B3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
     => ( ( B3 != zero_zero_multiset_a )
       => ( A3 != zero_zero_multiset_a ) ) ) ).

% ns_mul_ext_bottom2
thf(fact_757_ns__mul__ext__bottom__uniqueness,axiom,
    ! [A3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ A3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
     => ( A3 = zero_zero_multiset_a ) ) ).

% ns_mul_ext_bottom_uniqueness
thf(fact_758_s__mul__ext__bottom,axiom,
    ! [A3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( A3 != zero_zero_multiset_a )
     => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ zero_zero_multiset_a ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ).

% s_mul_ext_bottom
thf(fact_759_s__mul__ext__bottom__strict,axiom,
    ! [A3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ zero_zero_multiset_a @ A3 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ).

% s_mul_ext_bottom_strict
thf(fact_760_s__mul__extI__old,axiom,
    ! [A3: multis193838646080965794ropVar,Xs: list_M6740372356978808802ropVar,A22: multis193838646080965794ropVar,B3: multis193838646080965794ropVar,Ys: list_M6740372356978808802ropVar,B22: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( A3
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys ) @ B22 ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs )
            = ( size_s1943289338788136526ropVar @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Ys ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ I3 ) @ ( nth_Mu4537670848183934179ropVar @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z6226653793019358809ropVar )
             => ( ! [B4: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ B4 @ ( set_ms1147286257884310937ropVar @ B22 ) )
                   => ? [A6: multis3193088007478089820ropVar] :
                        ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A22 ) )
                        & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B4 ) @ S2 ) ) )
               => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A3 @ B3 ) @ ( multis4691071231837107059ropVar @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_761_s__mul__extI__old,axiom,
    ! [A3: multis501812127501805293od_a_a,Xs: list_P1396940483166286381od_a_a,A22: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,Ys: list_P1396940483166286381od_a_a,B22: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
      ( ( A3
        = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys ) @ B22 ) )
       => ( ( ( size_s3885678630836030617od_a_a @ Xs )
            = ( size_s3885678630836030617od_a_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys ) )
               => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs @ I3 ) @ ( nth_Product_prod_a_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z3024502163831676836od_a_a )
             => ( ! [B4: product_prod_a_a] :
                    ( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ B22 ) )
                   => ? [A6: product_prod_a_a] :
                        ( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A22 ) )
                        & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B4 ) @ S2 ) ) )
               => ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B3 ) @ ( multis6523615221270609726od_a_a @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_762_s__mul__extI__old,axiom,
    ! [A3: multis4781622204924204013iset_a,Xs: list_P6988024003011616557iset_a,A22: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,Ys: list_P6988024003011616557iset_a,B22: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S2: set_Pr3059988933816007879iset_a] :
      ( ( A3
        = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys ) @ B22 ) )
       => ( ( ( size_s7238940367606365593iset_a @ Xs )
            = ( size_s7238940367606365593iset_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys ) )
               => ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z4591712717340762532iset_a )
             => ( ! [B4: produc6518373309651786023iset_a] :
                    ( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ B22 ) )
                   => ? [A6: produc6518373309651786023iset_a] :
                        ( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A22 ) )
                        & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B4 ) @ S2 ) ) )
               => ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B3 ) @ ( multis6482401367445528638iset_a @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_763_s__mul__extI__old,axiom,
    ! [A3: multiset_multiset_a,Xs: list_multiset_a,A22: multiset_multiset_a,B3: multiset_multiset_a,Ys: list_multiset_a,B22: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
      ( ( A3
        = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys ) @ B22 ) )
       => ( ( ( size_s3003218658903709366iset_a @ Xs )
            = ( size_s3003218658903709366iset_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I3 ) @ ( nth_multiset_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z7799948378220188993iset_a )
             => ( ! [B4: multiset_a] :
                    ( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ B22 ) )
                   => ? [A6: multiset_a] :
                        ( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A22 ) )
                        & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B4 ) @ S2 ) ) )
               => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B3 ) @ ( multis7166037729339985883iset_a @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_764_s__mul__extI__old,axiom,
    ! [A3: multiset_nat,Xs: list_nat,A22: multiset_nat,B3: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z7348594199698428585et_nat )
             => ( ! [B4: nat] :
                    ( ( member_nat @ B4 @ ( set_mset_nat @ B22 ) )
                   => ? [A6: nat] :
                        ( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B4 ) @ S2 ) ) )
               => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_765_s__mul__extI__old,axiom,
    ! [A3: multiset_a,Xs: list_a,A22: multiset_a,B3: multiset_a,Ys: list_a,B22: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( A3
        = ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_plus_multiset_a @ ( mset_a @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_a @ Xs )
            = ( size_size_list_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_zero_multiset_a )
             => ( ! [B4: a] :
                    ( ( member_a @ B4 @ ( set_mset_a @ B22 ) )
                   => ? [A6: a] :
                        ( ( member_a @ A6 @ ( set_mset_a @ A22 ) )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B4 ) @ S2 ) ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_766_ns__mul__extI__old,axiom,
    ! [A3: multis193838646080965794ropVar,Xs: list_M6740372356978808802ropVar,A22: multis193838646080965794ropVar,B3: multis193838646080965794ropVar,Ys: list_M6740372356978808802ropVar,B22: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( A3
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys ) @ B22 ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs )
            = ( size_s1943289338788136526ropVar @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Ys ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ I3 ) @ ( nth_Mu4537670848183934179ropVar @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B4: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ B4 @ ( set_ms1147286257884310937ropVar @ B22 ) )
                 => ? [A6: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B4 ) @ S2 ) ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A3 @ B3 ) @ ( multis738276037714153361ropVar @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_767_ns__mul__extI__old,axiom,
    ! [A3: multis501812127501805293od_a_a,Xs: list_P1396940483166286381od_a_a,A22: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,Ys: list_P1396940483166286381od_a_a,B22: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
      ( ( A3
        = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p6967097483979887908od_a_a @ ( mset_P3709024939784119484od_a_a @ Ys ) @ B22 ) )
       => ( ( ( size_s3885678630836030617od_a_a @ Xs )
            = ( size_s3885678630836030617od_a_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3885678630836030617od_a_a @ Ys ) )
               => ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ ( nth_Product_prod_a_a @ Xs @ I3 ) @ ( nth_Product_prod_a_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B4: product_prod_a_a] :
                  ( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ B22 ) )
                 => ? [A6: product_prod_a_a] :
                      ( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A22 ) )
                      & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B3 ) @ ( multis1178328011278088540od_a_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_768_ns__mul__extI__old,axiom,
    ! [A3: multis4781622204924204013iset_a,Xs: list_P6988024003011616557iset_a,A22: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,Ys: list_P6988024003011616557iset_a,B22: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S2: set_Pr3059988933816007879iset_a] :
      ( ( A3
        = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p1053901119268525860iset_a @ ( mset_P2771936761501114300iset_a @ Ys ) @ B22 ) )
       => ( ( ( size_s7238940367606365593iset_a @ Xs )
            = ( size_s7238940367606365593iset_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s7238940367606365593iset_a @ Ys ) )
               => ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ ( nth_Pr1079437516805851566iset_a @ Xs @ I3 ) @ ( nth_Pr1079437516805851566iset_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B4: produc6518373309651786023iset_a] :
                  ( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ B22 ) )
                 => ? [A6: produc6518373309651786023iset_a] :
                      ( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A22 ) )
                      & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B3 ) @ ( multis2244641948333282396iset_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_769_ns__mul__extI__old,axiom,
    ! [A3: multiset_multiset_a,Xs: list_multiset_a,A22: multiset_multiset_a,B3: multiset_multiset_a,Ys: list_multiset_a,B22: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
      ( ( A3
        = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p6738641960240532161iset_a @ ( mset_multiset_a @ Ys ) @ B22 ) )
       => ( ( ( size_s3003218658903709366iset_a @ Xs )
            = ( size_s3003218658903709366iset_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Ys ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I3 ) @ ( nth_multiset_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B4: multiset_a] :
                  ( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ B22 ) )
                 => ? [A6: multiset_a] :
                      ( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A22 ) )
                      & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B3 ) @ ( multis8112516946959054073iset_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_770_ns__mul__extI__old,axiom,
    ! [A3: multiset_nat,Xs: list_nat,A22: multiset_nat,B3: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( set_mset_nat @ B22 ) )
                 => ? [A6: nat] :
                      ( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B4 ) @ S2 ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_771_ns__mul__extI__old,axiom,
    ! [A3: multiset_a,Xs: list_a,A22: multiset_a,B3: multiset_a,Ys: list_a,B22: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( A3
        = ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_plus_multiset_a @ ( mset_a @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_a @ Xs )
            = ( size_size_list_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B4: a] :
                  ( ( member_a @ B4 @ ( set_mset_a @ B22 ) )
                 => ? [A6: a] :
                      ( ( member_a @ A6 @ ( set_mset_a @ A22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_772_transpose__rectangle,axiom,
    ! [Xs: list_l3041241048860003048ropVar,N: nat] :
      ( ( ( Xs = nil_li2787573280958550248ropVar )
       => ( N = zero_zero_nat ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s9082923797619839828ropVar @ Xs ) )
           => ( ( size_s1943289338788136526ropVar @ ( nth_li1789610779111697641ropVar @ Xs @ I3 ) )
              = N ) )
       => ( ( transp2038539320897558508ropVar @ Xs )
          = ( map_na5381558106008680140ropVar
            @ ^ [I: nat] :
                ( map_na2466558649495876678ropVar
                @ ^ [J2: nat] : ( nth_Mu4537670848183934179ropVar @ ( nth_li1789610779111697641ropVar @ Xs @ J2 ) @ I )
                @ ( upt @ zero_zero_nat @ ( size_s9082923797619839828ropVar @ Xs ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% transpose_rectangle
thf(fact_773_transpose__rectangle,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ( ( Xs = nil_list_nat )
       => ( N = zero_zero_nat ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I3 ) )
              = N ) )
       => ( ( transpose_nat @ Xs )
          = ( map_nat_list_nat
            @ ^ [I: nat] :
                ( map_nat_nat
                @ ^ [J2: nat] : ( nth_nat @ ( nth_list_nat @ Xs @ J2 ) @ I )
                @ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% transpose_rectangle
thf(fact_774_transpose__rectangle,axiom,
    ! [Xs: list_list_a,N: nat] :
      ( ( ( Xs = nil_list_a )
       => ( N = zero_zero_nat ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s349497388124573686list_a @ Xs ) )
           => ( ( size_size_list_a @ ( nth_list_a @ Xs @ I3 ) )
              = N ) )
       => ( ( transpose_a @ Xs )
          = ( map_nat_list_a
            @ ^ [I: nat] :
                ( map_nat_a
                @ ^ [J2: nat] : ( nth_a @ ( nth_list_a @ Xs @ J2 ) @ I )
                @ ( upt @ zero_zero_nat @ ( size_s349497388124573686list_a @ Xs ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% transpose_rectangle
thf(fact_775_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_multiset_a,Bs: list_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( ( size_s3003218658903709366iset_a @ As )
        = ( size_s3003218658903709366iset_a @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Bs ) )
           => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ As @ I3 ) @ ( nth_multiset_a @ Bs @ I3 ) ) @ NS ) )
       => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As @ Bs )
          @ ( collec5227239471539426194iset_a
            @ ( produc5519185254700049772et_a_o
              @ ^ [As2: list_multiset_a,Bs2: list_multiset_a] : ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( mset_multiset_a @ As2 ) @ ( mset_multiset_a @ Bs2 ) ) @ ( multis8112516946959054073iset_a @ NS @ S ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_776_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ NS ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs )
          @ ( collec1570431334306492044st_nat
            @ ( produc6382697403170970154_nat_o
              @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_777_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_a,Bs: list_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( ( size_size_list_a @ As )
        = ( size_size_list_a @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Bs ) )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) @ NS ) )
       => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As @ Bs )
          @ ( collec943055143889122450list_a
            @ ( produc8172378796822260076st_a_o
              @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multis4023584895248362547_ext_a @ NS @ S ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_778_Eps__case__prod__eq,axiom,
    ! [X: a,Y2: a] :
      ( ( fChoic4124218645493772411od_a_a
        @ ( produc6436628058953941356_a_a_o
          @ ^ [X9: a,Y7: a] :
              ( ( X = X9 )
              & ( Y2 = Y7 ) ) ) )
      = ( product_Pair_a_a @ X @ Y2 ) ) ).

% Eps_case_prod_eq
thf(fact_779_Eps__case__prod__eq,axiom,
    ! [X: multiset_a,Y2: multiset_a] :
      ( ( fChoic1152277584823490683iset_a
        @ ( produc3007774963387989868et_a_o
          @ ^ [X9: multiset_a,Y7: multiset_a] :
              ( ( X = X9 )
              & ( Y2 = Y7 ) ) ) )
      = ( produc654756711066625303iset_a @ X @ Y2 ) ) ).

% Eps_case_prod_eq
thf(fact_780_set__mset__union,axiom,
    ! [M: multiset_a,N2: multiset_a] :
      ( ( set_mset_a @ ( plus_plus_multiset_a @ M @ N2 ) )
      = ( sup_sup_set_a @ ( set_mset_a @ M ) @ ( set_mset_a @ N2 ) ) ) ).

% set_mset_union
thf(fact_781_set__mset__union,axiom,
    ! [M: multiset_nat,N2: multiset_nat] :
      ( ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ M @ N2 ) )
      = ( sup_sup_set_nat @ ( set_mset_nat @ M ) @ ( set_mset_nat @ N2 ) ) ) ).

% set_mset_union
thf(fact_782_set__mset__mset,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( set_ms1147286257884310937ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) )
      = ( set_Mu146256361055586289ropVar @ Xs ) ) ).

% set_mset_mset
thf(fact_783_set__mset__mset,axiom,
    ! [Xs: list_nat] :
      ( ( set_mset_nat @ ( mset_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_mset_mset
thf(fact_784_set__mset__mset,axiom,
    ! [Xs: list_a] :
      ( ( set_mset_a @ ( mset_a @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% set_mset_mset
thf(fact_785_mset__subset__eqD,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,X: nat] :
      ( ( subseteq_mset_nat @ A3 @ B3 )
     => ( ( member_nat @ X @ ( set_mset_nat @ A3 ) )
       => ( member_nat @ X @ ( set_mset_nat @ B3 ) ) ) ) ).

% mset_subset_eqD
thf(fact_786_mset__subset__eqD,axiom,
    ! [A3: multis193838646080965794ropVar,B3: multis193838646080965794ropVar,X: multis3193088007478089820ropVar] :
      ( ( subset828532044049478595ropVar @ A3 @ B3 )
     => ( ( member8421201611525043589ropVar @ X @ ( set_ms1147286257884310937ropVar @ A3 ) )
       => ( member8421201611525043589ropVar @ X @ ( set_ms1147286257884310937ropVar @ B3 ) ) ) ) ).

% mset_subset_eqD
thf(fact_787_mset__subset__eqD,axiom,
    ! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,X: product_prod_a_a] :
      ( ( subset4108007524207251342od_a_a @ A3 @ B3 )
     => ( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ A3 ) )
       => ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ).

% mset_subset_eqD
thf(fact_788_mset__subset__eqD,axiom,
    ! [A3: multiset_a,B3: multiset_a,X: a] :
      ( ( subseteq_mset_a @ A3 @ B3 )
     => ( ( member_a @ X @ ( set_mset_a @ A3 ) )
       => ( member_a @ X @ ( set_mset_a @ B3 ) ) ) ) ).

% mset_subset_eqD
thf(fact_789_mset__subset__eqD,axiom,
    ! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,X: produc6518373309651786023iset_a] :
      ( ( subset6292090195676340366iset_a @ A3 @ B3 )
     => ( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ A3 ) )
       => ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ).

% mset_subset_eqD
thf(fact_790_multiset__nonemptyE,axiom,
    ! [A3: multiset_nat] :
      ( ( A3 != zero_z7348594199698428585et_nat )
     => ~ ! [X4: nat] :
            ~ ( member_nat @ X4 @ ( set_mset_nat @ A3 ) ) ) ).

% multiset_nonemptyE
thf(fact_791_multiset__nonemptyE,axiom,
    ! [A3: multis193838646080965794ropVar] :
      ( ( A3 != zero_z6226653793019358809ropVar )
     => ~ ! [X4: multis3193088007478089820ropVar] :
            ~ ( member8421201611525043589ropVar @ X4 @ ( set_ms1147286257884310937ropVar @ A3 ) ) ) ).

% multiset_nonemptyE
thf(fact_792_multiset__nonemptyE,axiom,
    ! [A3: multis501812127501805293od_a_a] :
      ( ( A3 != zero_z3024502163831676836od_a_a )
     => ~ ! [X4: product_prod_a_a] :
            ~ ( member1426531477525435216od_a_a @ X4 @ ( set_ms119794563918438244od_a_a @ A3 ) ) ) ).

% multiset_nonemptyE
thf(fact_793_multiset__nonemptyE,axiom,
    ! [A3: multiset_a] :
      ( ( A3 != zero_zero_multiset_a )
     => ~ ! [X4: a] :
            ~ ( member_a @ X4 @ ( set_mset_a @ A3 ) ) ) ).

% multiset_nonemptyE
thf(fact_794_multiset__nonemptyE,axiom,
    ! [A3: multis4781622204924204013iset_a] :
      ( ( A3 != zero_z4591712717340762532iset_a )
     => ~ ! [X4: produc6518373309651786023iset_a] :
            ~ ( member5199237121806060112iset_a @ X4 @ ( set_ms2379723421352415844iset_a @ A3 ) ) ) ).

% multiset_nonemptyE
thf(fact_795_union__iff,axiom,
    ! [A2: multis3193088007478089820ropVar,A3: multis193838646080965794ropVar,B3: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A2 @ ( set_ms1147286257884310937ropVar @ ( plus_p1339091108561003481ropVar @ A3 @ B3 ) ) )
      = ( ( member8421201611525043589ropVar @ A2 @ ( set_ms1147286257884310937ropVar @ A3 ) )
        | ( member8421201611525043589ropVar @ A2 @ ( set_ms1147286257884310937ropVar @ B3 ) ) ) ) ).

% union_iff
thf(fact_796_union__iff,axiom,
    ! [A2: product_prod_a_a,A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ B3 ) ) )
      = ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ A3 ) )
        | ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ B3 ) ) ) ) ).

% union_iff
thf(fact_797_union__iff,axiom,
    ! [A2: produc6518373309651786023iset_a,A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a] :
      ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ B3 ) ) )
      = ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ A3 ) )
        | ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ B3 ) ) ) ) ).

% union_iff
thf(fact_798_union__iff,axiom,
    ! [A2: nat,A3: multiset_nat,B3: multiset_nat] :
      ( ( member_nat @ A2 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) ) )
      = ( ( member_nat @ A2 @ ( set_mset_nat @ A3 ) )
        | ( member_nat @ A2 @ ( set_mset_nat @ B3 ) ) ) ) ).

% union_iff
thf(fact_799_union__iff,axiom,
    ! [A2: a,A3: multiset_a,B3: multiset_a] :
      ( ( member_a @ A2 @ ( set_mset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) ) )
      = ( ( member_a @ A2 @ ( set_mset_a @ A3 ) )
        | ( member_a @ A2 @ ( set_mset_a @ B3 ) ) ) ) ).

% union_iff
thf(fact_800_split__paired__Eps,axiom,
    ( fChoic4124218645493772411od_a_a
    = ( ^ [P5: product_prod_a_a > $o] :
          ( fChoic4124218645493772411od_a_a
          @ ( produc6436628058953941356_a_a_o
            @ ^ [A: a,B: a] : ( P5 @ ( product_Pair_a_a @ A @ B ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_801_split__paired__Eps,axiom,
    ( fChoic1152277584823490683iset_a
    = ( ^ [P5: produc6518373309651786023iset_a > $o] :
          ( fChoic1152277584823490683iset_a
          @ ( produc3007774963387989868et_a_o
            @ ^ [A: multiset_a,B: multiset_a] : ( P5 @ ( produc654756711066625303iset_a @ A @ B ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_802_transpose__empty,axiom,
    ! [Xs: list_list_a] :
      ( ( ( transpose_a @ Xs )
        = nil_list_a )
      = ( ! [X3: list_a] :
            ( ( member_list_a @ X3 @ ( set_list_a2 @ Xs ) )
           => ( X3 = nil_a ) ) ) ) ).

% transpose_empty
thf(fact_803_transpose__empty,axiom,
    ! [Xs: list_list_nat] :
      ( ( ( transpose_nat @ Xs )
        = nil_list_nat )
      = ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xs ) )
           => ( X3 = nil_nat ) ) ) ) ).

% transpose_empty
thf(fact_804_transpose__map__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_list_nat] :
      ( ( transp2038539320897558508ropVar @ ( map_li5946103070670478908ropVar @ ( map_na2466558649495876678ropVar @ F ) @ Xs ) )
      = ( map_li5946103070670478908ropVar @ ( map_na2466558649495876678ropVar @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_805_transpose__map__map,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) )
      = ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_806_transpose__map__map,axiom,
    ! [F: nat > a,Xs: list_list_nat] :
      ( ( transpose_a @ ( map_list_nat_list_a @ ( map_nat_a @ F ) @ Xs ) )
      = ( map_list_nat_list_a @ ( map_nat_a @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_807_in__multiset__in__set,axiom,
    ! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ ( mset_P3709024939784119484od_a_a @ Xs ) ) )
      = ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_808_in__multiset__in__set,axiom,
    ! [X: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
      ( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ ( mset_P2771936761501114300iset_a @ Xs ) ) )
      = ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_809_in__multiset__in__set,axiom,
    ! [X: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X @ ( set_ms1147286257884310937ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) ) )
      = ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_810_in__multiset__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
      = ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_811_in__multiset__in__set,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
      = ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_812_mset__set__set__mset__msubset,axiom,
    ! [A3: multiset_nat] : ( subseteq_mset_nat @ ( mset_set_nat @ ( set_mset_nat @ A3 ) ) @ A3 ) ).

% mset_set_set_mset_msubset
thf(fact_813_nth__mem__mset,axiom,
    ! [I2: nat,Ls: list_M6740372356978808802ropVar] :
      ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Ls ) )
     => ( member8421201611525043589ropVar @ ( nth_Mu4537670848183934179ropVar @ Ls @ I2 ) @ ( set_ms1147286257884310937ropVar @ ( mset_M6581516532424202993ropVar @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_814_nth__mem__mset,axiom,
    ! [I2: nat,Ls: list_P1396940483166286381od_a_a] :
      ( ( ord_less_nat @ I2 @ ( size_s3885678630836030617od_a_a @ Ls ) )
     => ( member1426531477525435216od_a_a @ ( nth_Product_prod_a_a @ Ls @ I2 ) @ ( set_ms119794563918438244od_a_a @ ( mset_P3709024939784119484od_a_a @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_815_nth__mem__mset,axiom,
    ! [I2: nat,Ls: list_P6988024003011616557iset_a] :
      ( ( ord_less_nat @ I2 @ ( size_s7238940367606365593iset_a @ Ls ) )
     => ( member5199237121806060112iset_a @ ( nth_Pr1079437516805851566iset_a @ Ls @ I2 ) @ ( set_ms2379723421352415844iset_a @ ( mset_P2771936761501114300iset_a @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_816_nth__mem__mset,axiom,
    ! [I2: nat,Ls: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ls ) )
     => ( member_nat @ ( nth_nat @ Ls @ I2 ) @ ( set_mset_nat @ ( mset_nat @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_817_nth__mem__mset,axiom,
    ! [I2: nat,Ls: list_a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ls ) )
     => ( member_a @ ( nth_a @ Ls @ I2 ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_818_all__s__s__mul__ext,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,S2: set_Pr1261947904930325089at_nat,Ns: set_Pr1261947904930325089at_nat] :
      ( ( A3 != zero_z7348594199698428585et_nat )
     => ( ! [B4: nat] :
            ( ( member_nat @ B4 @ ( set_mset_nat @ B3 ) )
           => ? [A6: nat] :
                ( ( member_nat @ A6 @ ( set_mset_nat @ A3 ) )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B4 ) @ S2 ) ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_819_all__s__s__mul__ext,axiom,
    ! [A3: multis193838646080965794ropVar,B3: multis193838646080965794ropVar,S2: set_Pr7799659934693882759ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( A3 != zero_z6226653793019358809ropVar )
     => ( ! [B4: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ B4 @ ( set_ms1147286257884310937ropVar @ B3 ) )
           => ? [A6: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A3 ) )
                & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B4 ) @ S2 ) ) )
       => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A3 @ B3 ) @ ( multis4691071231837107059ropVar @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_820_all__s__s__mul__ext,axiom,
    ! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,S2: set_Pr8600417178894128327od_a_a,Ns: set_Pr8600417178894128327od_a_a] :
      ( ( A3 != zero_z3024502163831676836od_a_a )
     => ( ! [B4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ B3 ) )
           => ? [A6: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A3 ) )
                & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B4 ) @ S2 ) ) )
       => ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B3 ) @ ( multis6523615221270609726od_a_a @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_821_all__s__s__mul__ext,axiom,
    ! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,S2: set_Pr3059988933816007879iset_a,Ns: set_Pr3059988933816007879iset_a] :
      ( ( A3 != zero_z4591712717340762532iset_a )
     => ( ! [B4: produc6518373309651786023iset_a] :
            ( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ B3 ) )
           => ? [A6: produc6518373309651786023iset_a] :
                ( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A3 ) )
                & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B4 ) @ S2 ) ) )
       => ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B3 ) @ ( multis6482401367445528638iset_a @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_822_all__s__s__mul__ext,axiom,
    ! [A3: multiset_multiset_a,B3: multiset_multiset_a,S2: set_Pr79727621955416071iset_a,Ns: set_Pr79727621955416071iset_a] :
      ( ( A3 != zero_z7799948378220188993iset_a )
     => ( ! [B4: multiset_a] :
            ( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ B3 ) )
           => ? [A6: multiset_a] :
                ( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A3 ) )
                & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B4 ) @ S2 ) ) )
       => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B3 ) @ ( multis7166037729339985883iset_a @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_823_all__s__s__mul__ext,axiom,
    ! [A3: multiset_a,B3: multiset_a,S2: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
      ( ( A3 != zero_zero_multiset_a )
     => ( ! [B4: a] :
            ( ( member_a @ B4 @ ( set_mset_a @ B3 ) )
           => ? [A6: a] :
                ( ( member_a @ A6 @ ( set_mset_a @ A3 ) )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B4 ) @ S2 ) ) )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_824_ns__mul__ext__point,axiom,
    ! [As: multiset_nat,Bs: multiset_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,B2: nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ As @ Bs ) @ ( multis8831328596575508315xt_nat @ NS @ S ) )
     => ( ( member_nat @ B2 @ ( set_mset_nat @ Bs ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_mset_nat @ As ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ B2 ) @ ( sup_su6327502436637775413at_nat @ NS @ S ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_825_ns__mul__ext__point,axiom,
    ! [As: multis193838646080965794ropVar,Bs: multis193838646080965794ropVar,NS: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar,B2: multis3193088007478089820ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ As @ Bs ) @ ( multis738276037714153361ropVar @ NS @ S ) )
     => ( ( member8421201611525043589ropVar @ B2 @ ( set_ms1147286257884310937ropVar @ Bs ) )
       => ? [X4: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X4 @ ( set_ms1147286257884310937ropVar @ As ) )
            & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ B2 ) @ ( sup_su5823796184441390427ropVar @ NS @ S ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_826_ns__mul__ext__point,axiom,
    ! [As: multis501812127501805293od_a_a,Bs: multis501812127501805293od_a_a,NS: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a,B2: product_prod_a_a] :
      ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ As @ Bs ) @ ( multis1178328011278088540od_a_a @ NS @ S ) )
     => ( ( member1426531477525435216od_a_a @ B2 @ ( set_ms119794563918438244od_a_a @ Bs ) )
       => ? [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ ( set_ms119794563918438244od_a_a @ As ) )
            & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ B2 ) @ ( sup_su8193676293155882651od_a_a @ NS @ S ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_827_ns__mul__ext__point,axiom,
    ! [As: multis4781622204924204013iset_a,Bs: multis4781622204924204013iset_a,NS: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a,B2: produc6518373309651786023iset_a] :
      ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ As @ Bs ) @ ( multis2244641948333282396iset_a @ NS @ S ) )
     => ( ( member5199237121806060112iset_a @ B2 @ ( set_ms2379723421352415844iset_a @ Bs ) )
       => ? [X4: produc6518373309651786023iset_a] :
            ( ( member5199237121806060112iset_a @ X4 @ ( set_ms2379723421352415844iset_a @ As ) )
            & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X4 @ B2 ) @ ( sup_su3115323614739818139iset_a @ NS @ S ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_828_ns__mul__ext__point,axiom,
    ! [As: multiset_multiset_a,Bs: multiset_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,B2: multiset_a] :
      ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ As @ Bs ) @ ( multis8112516946959054073iset_a @ NS @ S ) )
     => ( ( member_multiset_a @ B2 @ ( set_mset_multiset_a @ Bs ) )
       => ? [X4: multiset_a] :
            ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ As ) )
            & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ B2 ) @ ( sup_su1925787886501348827iset_a @ NS @ S ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_829_ns__mul__ext__point,axiom,
    ! [As: multiset_a,Bs: multiset_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a,B2: a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ As @ Bs ) @ ( multis4023584895248362547_ext_a @ NS @ S ) )
     => ( ( member_a @ B2 @ ( set_mset_a @ Bs ) )
       => ? [X4: a] :
            ( ( member_a @ X4 @ ( set_mset_a @ As ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ B2 ) @ ( sup_su3048258781599657691od_a_a @ NS @ S ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_830_s__mul__ext__point,axiom,
    ! [As: multiset_nat,Bs: multiset_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,B2: nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ As @ Bs ) @ ( multis4215214144601311993xt_nat @ NS @ S ) )
     => ( ( member_nat @ B2 @ ( set_mset_nat @ Bs ) )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_mset_nat @ As ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ B2 ) @ ( sup_su6327502436637775413at_nat @ NS @ S ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_831_s__mul__ext__point,axiom,
    ! [As: multis193838646080965794ropVar,Bs: multis193838646080965794ropVar,NS: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar,B2: multis3193088007478089820ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ As @ Bs ) @ ( multis4691071231837107059ropVar @ NS @ S ) )
     => ( ( member8421201611525043589ropVar @ B2 @ ( set_ms1147286257884310937ropVar @ Bs ) )
       => ? [X4: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X4 @ ( set_ms1147286257884310937ropVar @ As ) )
            & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ B2 ) @ ( sup_su5823796184441390427ropVar @ NS @ S ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_832_s__mul__ext__point,axiom,
    ! [As: multis501812127501805293od_a_a,Bs: multis501812127501805293od_a_a,NS: set_Pr8600417178894128327od_a_a,S: set_Pr8600417178894128327od_a_a,B2: product_prod_a_a] :
      ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ As @ Bs ) @ ( multis6523615221270609726od_a_a @ NS @ S ) )
     => ( ( member1426531477525435216od_a_a @ B2 @ ( set_ms119794563918438244od_a_a @ Bs ) )
       => ? [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ ( set_ms119794563918438244od_a_a @ As ) )
            & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ X4 @ B2 ) @ ( sup_su8193676293155882651od_a_a @ NS @ S ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_833_s__mul__ext__point,axiom,
    ! [As: multis4781622204924204013iset_a,Bs: multis4781622204924204013iset_a,NS: set_Pr3059988933816007879iset_a,S: set_Pr3059988933816007879iset_a,B2: produc6518373309651786023iset_a] :
      ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ As @ Bs ) @ ( multis6482401367445528638iset_a @ NS @ S ) )
     => ( ( member5199237121806060112iset_a @ B2 @ ( set_ms2379723421352415844iset_a @ Bs ) )
       => ? [X4: produc6518373309651786023iset_a] :
            ( ( member5199237121806060112iset_a @ X4 @ ( set_ms2379723421352415844iset_a @ As ) )
            & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ X4 @ B2 ) @ ( sup_su3115323614739818139iset_a @ NS @ S ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_834_s__mul__ext__point,axiom,
    ! [As: multiset_multiset_a,Bs: multiset_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a,B2: multiset_a] :
      ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ As @ Bs ) @ ( multis7166037729339985883iset_a @ NS @ S ) )
     => ( ( member_multiset_a @ B2 @ ( set_mset_multiset_a @ Bs ) )
       => ? [X4: multiset_a] :
            ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ As ) )
            & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ B2 ) @ ( sup_su1925787886501348827iset_a @ NS @ S ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_835_s__mul__ext__point,axiom,
    ! [As: multiset_a,Bs: multiset_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a,B2: a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ As @ Bs ) @ ( multiset_s_mul_ext_a @ NS @ S ) )
     => ( ( member_a @ B2 @ ( set_mset_a @ Bs ) )
       => ? [X4: a] :
            ( ( member_a @ X4 @ ( set_mset_a @ As ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ B2 ) @ ( sup_su3048258781599657691od_a_a @ NS @ S ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_836_mul__ext__list_Ons__map,axiom,
    ! [S: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ S )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
     => ( ! [A4: nat,B4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ NS )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S ) ) ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_837_mul__ext__list_Ons__map,axiom,
    ! [S: set_Pr79727621955416071iset_a,F: multiset_a > multiset_a,NS: set_Pr79727621955416071iset_a,As: list_multiset_a,Bs: list_multiset_a] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ S )
         => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
     => ( ! [A4: multiset_a,B4: multiset_a] :
            ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ NS )
           => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
       => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As @ Bs )
            @ ( collec5227239471539426194iset_a
              @ ( produc5519185254700049772et_a_o
                @ ^ [As2: list_multiset_a,Bs2: list_multiset_a] : ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( mset_multiset_a @ As2 ) @ ( mset_multiset_a @ Bs2 ) ) @ ( multis8112516946959054073iset_a @ NS @ S ) ) ) ) )
         => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F @ As ) @ ( map_mu8355158918406241476iset_a @ F @ Bs ) )
            @ ( collec5227239471539426194iset_a
              @ ( produc5519185254700049772et_a_o
                @ ^ [As2: list_multiset_a,Bs2: list_multiset_a] : ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( mset_multiset_a @ As2 ) @ ( mset_multiset_a @ Bs2 ) ) @ ( multis8112516946959054073iset_a @ NS @ S ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_838_mul__ext__list_Ons__map,axiom,
    ! [S: set_Product_prod_a_a,F: a > a,NS: set_Product_prod_a_a,As: list_a,Bs: list_a] :
      ( ! [A4: a,B4: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ S )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
     => ( ! [A4: a,B4: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ NS )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
       => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As @ Bs )
            @ ( collec943055143889122450list_a
              @ ( produc8172378796822260076st_a_o
                @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multis4023584895248362547_ext_a @ NS @ S ) ) ) ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F @ As ) @ ( map_a_a @ F @ Bs ) )
            @ ( collec943055143889122450list_a
              @ ( produc8172378796822260076st_a_o
                @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multis4023584895248362547_ext_a @ NS @ S ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_839_mul__ext__list_Os__map,axiom,
    ! [S: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ S )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
     => ( ! [A4: nat,B4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ NS )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S ) ) ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_840_mul__ext__list_Os__map,axiom,
    ! [S: set_Pr79727621955416071iset_a,F: multiset_a > multiset_a,NS: set_Pr79727621955416071iset_a,As: list_multiset_a,Bs: list_multiset_a] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ S )
         => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
     => ( ! [A4: multiset_a,B4: multiset_a] :
            ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ NS )
           => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
       => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As @ Bs )
            @ ( collec5227239471539426194iset_a
              @ ( produc5519185254700049772et_a_o
                @ ^ [As2: list_multiset_a,Bs2: list_multiset_a] : ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( mset_multiset_a @ As2 ) @ ( mset_multiset_a @ Bs2 ) ) @ ( multis7166037729339985883iset_a @ NS @ S ) ) ) ) )
         => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F @ As ) @ ( map_mu8355158918406241476iset_a @ F @ Bs ) )
            @ ( collec5227239471539426194iset_a
              @ ( produc5519185254700049772et_a_o
                @ ^ [As2: list_multiset_a,Bs2: list_multiset_a] : ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( mset_multiset_a @ As2 ) @ ( mset_multiset_a @ Bs2 ) ) @ ( multis7166037729339985883iset_a @ NS @ S ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_841_mul__ext__list_Os__map,axiom,
    ! [S: set_Product_prod_a_a,F: a > a,NS: set_Product_prod_a_a,As: list_a,Bs: list_a] :
      ( ! [A4: a,B4: a] :
          ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ S )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
     => ( ! [A4: a,B4: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ NS )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
       => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As @ Bs )
            @ ( collec943055143889122450list_a
              @ ( produc8172378796822260076st_a_o
                @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multiset_s_mul_ext_a @ NS @ S ) ) ) ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F @ As ) @ ( map_a_a @ F @ Bs ) )
            @ ( collec943055143889122450list_a
              @ ( produc8172378796822260076st_a_o
                @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multiset_s_mul_ext_a @ NS @ S ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_842_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multis193838646080965794ropVar,B3: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar,C2: multis193838646080965794ropVar,D: multis193838646080965794ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A3 @ B3 ) @ ( multis738276037714153361ropVar @ Ns @ S2 ) )
     => ( ( C2 != zero_z6226653793019358809ropVar )
       => ( ! [D3: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ D3 @ ( set_ms1147286257884310937ropVar @ D ) )
             => ? [C6: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ C6 @ ( set_ms1147286257884310937ropVar @ C2 ) )
                  & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ C6 @ D3 ) @ S2 ) ) )
         => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( plus_p1339091108561003481ropVar @ A3 @ C2 ) @ ( plus_p1339091108561003481ropVar @ B3 @ D ) ) @ ( multis4691071231837107059ropVar @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_843_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a,C2: multis501812127501805293od_a_a,D: multis501812127501805293od_a_a] :
      ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B3 ) @ ( multis1178328011278088540od_a_a @ Ns @ S2 ) )
     => ( ( C2 != zero_z3024502163831676836od_a_a )
       => ( ! [D3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ D3 @ ( set_ms119794563918438244od_a_a @ D ) )
             => ? [C6: product_prod_a_a] :
                  ( ( member1426531477525435216od_a_a @ C6 @ ( set_ms119794563918438244od_a_a @ C2 ) )
                  & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ C6 @ D3 ) @ S2 ) ) )
         => ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ ( plus_p6967097483979887908od_a_a @ A3 @ C2 ) @ ( plus_p6967097483979887908od_a_a @ B3 @ D ) ) @ ( multis6523615221270609726od_a_a @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_844_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S2: set_Pr3059988933816007879iset_a,C2: multis4781622204924204013iset_a,D: multis4781622204924204013iset_a] :
      ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B3 ) @ ( multis2244641948333282396iset_a @ Ns @ S2 ) )
     => ( ( C2 != zero_z4591712717340762532iset_a )
       => ( ! [D3: produc6518373309651786023iset_a] :
              ( ( member5199237121806060112iset_a @ D3 @ ( set_ms2379723421352415844iset_a @ D ) )
             => ? [C6: produc6518373309651786023iset_a] :
                  ( ( member5199237121806060112iset_a @ C6 @ ( set_ms2379723421352415844iset_a @ C2 ) )
                  & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ C6 @ D3 ) @ S2 ) ) )
         => ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ ( plus_p1053901119268525860iset_a @ A3 @ C2 ) @ ( plus_p1053901119268525860iset_a @ B3 @ D ) ) @ ( multis6482401367445528638iset_a @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_845_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multiset_multiset_a,B3: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a,C2: multiset_multiset_a,D: multiset_multiset_a] :
      ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B3 ) @ ( multis8112516946959054073iset_a @ Ns @ S2 ) )
     => ( ( C2 != zero_z7799948378220188993iset_a )
       => ( ! [D3: multiset_a] :
              ( ( member_multiset_a @ D3 @ ( set_mset_multiset_a @ D ) )
             => ? [C6: multiset_a] :
                  ( ( member_multiset_a @ C6 @ ( set_mset_multiset_a @ C2 ) )
                  & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ C6 @ D3 ) @ S2 ) ) )
         => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ ( plus_p6738641960240532161iset_a @ A3 @ C2 ) @ ( plus_p6738641960240532161iset_a @ B3 @ D ) ) @ ( multis7166037729339985883iset_a @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_846_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,C2: multiset_nat,D: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) )
     => ( ( C2 != zero_z7348594199698428585et_nat )
       => ( ! [D3: nat] :
              ( ( member_nat @ D3 @ ( set_mset_nat @ D ) )
             => ? [C6: nat] :
                  ( ( member_nat @ C6 @ ( set_mset_nat @ C2 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C6 @ D3 ) @ S2 ) ) )
         => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ B3 @ D ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_847_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multiset_a,B3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a,C2: multiset_a,D: multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
     => ( ( C2 != zero_zero_multiset_a )
       => ( ! [D3: a] :
              ( ( member_a @ D3 @ ( set_mset_a @ D ) )
             => ? [C6: a] :
                  ( ( member_a @ C6 @ ( set_mset_a @ C2 ) )
                  & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ C6 @ D3 ) @ S2 ) ) )
         => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B3 @ D ) ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_848_case__prodI,axiom,
    ! [F: a > a > $o,A2: a,B2: a] :
      ( ( F @ A2 @ B2 )
     => ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_849_case__prodI,axiom,
    ! [F: multiset_a > multiset_a > $o,A2: multiset_a,B2: multiset_a] :
      ( ( F @ A2 @ B2 )
     => ( produc3007774963387989868et_a_o @ F @ ( produc654756711066625303iset_a @ A2 @ B2 ) ) ) ).

% case_prodI
thf(fact_850_case__prodI2,axiom,
    ! [P7: product_prod_a_a,C: a > a > $o] :
      ( ! [A4: a,B4: a] :
          ( ( P7
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc6436628058953941356_a_a_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_851_case__prodI2,axiom,
    ! [P7: produc6518373309651786023iset_a,C: multiset_a > multiset_a > $o] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( P7
            = ( produc654756711066625303iset_a @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc3007774963387989868et_a_o @ C @ P7 ) ) ).

% case_prodI2
thf(fact_852_The__split__eq,axiom,
    ! [X: a,Y2: a] :
      ( ( the_Product_prod_a_a
        @ ( produc6436628058953941356_a_a_o
          @ ^ [X9: a,Y7: a] :
              ( ( X = X9 )
              & ( Y2 = Y7 ) ) ) )
      = ( product_Pair_a_a @ X @ Y2 ) ) ).

% The_split_eq
thf(fact_853_The__split__eq,axiom,
    ! [X: multiset_a,Y2: multiset_a] :
      ( ( the_Pr1459549734356548398iset_a
        @ ( produc3007774963387989868et_a_o
          @ ^ [X9: multiset_a,Y7: multiset_a] :
              ( ( X = X9 )
              & ( Y2 = Y7 ) ) ) )
      = ( produc654756711066625303iset_a @ X @ Y2 ) ) ).

% The_split_eq
thf(fact_854_mem__case__prodI,axiom,
    ! [Z4: nat,C: a > a > set_nat,A2: a,B2: a] :
      ( ( member_nat @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member_nat @ Z4 @ ( produc153843693180602034et_nat @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_855_mem__case__prodI,axiom,
    ! [Z4: multis3193088007478089820ropVar,C: a > a > set_Mu5658373669842198716ropVar,A2: a,B2: a] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc827557285005739152ropVar @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_856_mem__case__prodI,axiom,
    ! [Z4: product_prod_a_a,C: a > a > set_Product_prod_a_a,A2: a,B2: a] :
      ( ( member1426531477525435216od_a_a @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member1426531477525435216od_a_a @ Z4 @ ( produc5766521956407364827od_a_a @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_857_mem__case__prodI,axiom,
    ! [Z4: a,C: a > a > set_a,A2: a,B2: a] :
      ( ( member_a @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z4 @ ( produc9217457822752978994_set_a @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_858_mem__case__prodI,axiom,
    ! [Z4: produc6518373309651786023iset_a,C: a > a > set_Pr79727621955416071iset_a,A2: a,B2: a] :
      ( ( member5199237121806060112iset_a @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member5199237121806060112iset_a @ Z4 @ ( produc6070977564805178331iset_a @ C @ ( product_Pair_a_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_859_mem__case__prodI,axiom,
    ! [Z4: nat,C: multiset_a > multiset_a > set_nat,A2: multiset_a,B2: multiset_a] :
      ( ( member_nat @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member_nat @ Z4 @ ( produc8854979857967171762et_nat @ C @ ( produc654756711066625303iset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_860_mem__case__prodI,axiom,
    ! [Z4: multis3193088007478089820ropVar,C: multiset_a > multiset_a > set_Mu5658373669842198716ropVar,A2: multiset_a,B2: multiset_a] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc1593905761737884304ropVar @ C @ ( produc654756711066625303iset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_861_mem__case__prodI,axiom,
    ! [Z4: product_prod_a_a,C: multiset_a > multiset_a > set_Product_prod_a_a,A2: multiset_a,B2: multiset_a] :
      ( ( member1426531477525435216od_a_a @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member1426531477525435216od_a_a @ Z4 @ ( produc5649935134530259675od_a_a @ C @ ( produc654756711066625303iset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_862_mem__case__prodI,axiom,
    ! [Z4: a,C: multiset_a > multiset_a > set_a,A2: multiset_a,B2: multiset_a] :
      ( ( member_a @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member_a @ Z4 @ ( produc2023483729548817970_set_a @ C @ ( produc654756711066625303iset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_863_mem__case__prodI,axiom,
    ! [Z4: produc6518373309651786023iset_a,C: multiset_a > multiset_a > set_Pr79727621955416071iset_a,A2: multiset_a,B2: multiset_a] :
      ( ( member5199237121806060112iset_a @ Z4 @ ( C @ A2 @ B2 ) )
     => ( member5199237121806060112iset_a @ Z4 @ ( produc505406162287731163iset_a @ C @ ( produc654756711066625303iset_a @ A2 @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_864_mem__case__prodI2,axiom,
    ! [P7: product_prod_a_a,Z4: nat,C: a > a > set_nat] :
      ( ! [A4: a,B4: a] :
          ( ( P7
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member_nat @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member_nat @ Z4 @ ( produc153843693180602034et_nat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_865_mem__case__prodI2,axiom,
    ! [P7: product_prod_a_a,Z4: multis3193088007478089820ropVar,C: a > a > set_Mu5658373669842198716ropVar] :
      ( ! [A4: a,B4: a] :
          ( ( P7
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member8421201611525043589ropVar @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc827557285005739152ropVar @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_866_mem__case__prodI2,axiom,
    ! [P7: product_prod_a_a,Z4: product_prod_a_a,C: a > a > set_Product_prod_a_a] :
      ( ! [A4: a,B4: a] :
          ( ( P7
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member1426531477525435216od_a_a @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member1426531477525435216od_a_a @ Z4 @ ( produc5766521956407364827od_a_a @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_867_mem__case__prodI2,axiom,
    ! [P7: product_prod_a_a,Z4: a,C: a > a > set_a] :
      ( ! [A4: a,B4: a] :
          ( ( P7
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member_a @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member_a @ Z4 @ ( produc9217457822752978994_set_a @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_868_mem__case__prodI2,axiom,
    ! [P7: product_prod_a_a,Z4: produc6518373309651786023iset_a,C: a > a > set_Pr79727621955416071iset_a] :
      ( ! [A4: a,B4: a] :
          ( ( P7
            = ( product_Pair_a_a @ A4 @ B4 ) )
         => ( member5199237121806060112iset_a @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member5199237121806060112iset_a @ Z4 @ ( produc6070977564805178331iset_a @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_869_mem__case__prodI2,axiom,
    ! [P7: produc6518373309651786023iset_a,Z4: nat,C: multiset_a > multiset_a > set_nat] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( P7
            = ( produc654756711066625303iset_a @ A4 @ B4 ) )
         => ( member_nat @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member_nat @ Z4 @ ( produc8854979857967171762et_nat @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_870_mem__case__prodI2,axiom,
    ! [P7: produc6518373309651786023iset_a,Z4: multis3193088007478089820ropVar,C: multiset_a > multiset_a > set_Mu5658373669842198716ropVar] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( P7
            = ( produc654756711066625303iset_a @ A4 @ B4 ) )
         => ( member8421201611525043589ropVar @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc1593905761737884304ropVar @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_871_mem__case__prodI2,axiom,
    ! [P7: produc6518373309651786023iset_a,Z4: product_prod_a_a,C: multiset_a > multiset_a > set_Product_prod_a_a] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( P7
            = ( produc654756711066625303iset_a @ A4 @ B4 ) )
         => ( member1426531477525435216od_a_a @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member1426531477525435216od_a_a @ Z4 @ ( produc5649935134530259675od_a_a @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_872_mem__case__prodI2,axiom,
    ! [P7: produc6518373309651786023iset_a,Z4: a,C: multiset_a > multiset_a > set_a] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( P7
            = ( produc654756711066625303iset_a @ A4 @ B4 ) )
         => ( member_a @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member_a @ Z4 @ ( produc2023483729548817970_set_a @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_873_mem__case__prodI2,axiom,
    ! [P7: produc6518373309651786023iset_a,Z4: produc6518373309651786023iset_a,C: multiset_a > multiset_a > set_Pr79727621955416071iset_a] :
      ( ! [A4: multiset_a,B4: multiset_a] :
          ( ( P7
            = ( produc654756711066625303iset_a @ A4 @ B4 ) )
         => ( member5199237121806060112iset_a @ Z4 @ ( C @ A4 @ B4 ) ) )
     => ( member5199237121806060112iset_a @ Z4 @ ( produc505406162287731163iset_a @ C @ P7 ) ) ) ).

% mem_case_prodI2
thf(fact_874_case__prodE,axiom,
    ! [C: a > a > $o,P7: product_prod_a_a] :
      ( ( produc6436628058953941356_a_a_o @ C @ P7 )
     => ~ ! [X4: a,Y5: a] :
            ( ( P7
              = ( product_Pair_a_a @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_875_case__prodE,axiom,
    ! [C: multiset_a > multiset_a > $o,P7: produc6518373309651786023iset_a] :
      ( ( produc3007774963387989868et_a_o @ C @ P7 )
     => ~ ! [X4: multiset_a,Y5: multiset_a] :
            ( ( P7
              = ( produc654756711066625303iset_a @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_876_case__prodD,axiom,
    ! [F: a > a > $o,A2: a,B2: a] :
      ( ( produc6436628058953941356_a_a_o @ F @ ( product_Pair_a_a @ A2 @ B2 ) )
     => ( F @ A2 @ B2 ) ) ).

% case_prodD
thf(fact_877_case__prodD,axiom,
    ! [F: multiset_a > multiset_a > $o,A2: multiset_a,B2: multiset_a] :
      ( ( produc3007774963387989868et_a_o @ F @ ( produc654756711066625303iset_a @ A2 @ B2 ) )
     => ( F @ A2 @ B2 ) ) ).

% case_prodD
thf(fact_878_mul__ext__list_Olist__order__extension__axioms,axiom,
    ( list_l792762466888043652on_nat
    @ ^ [S3: set_Pr1261947904930325089at_nat,Ns2: set_Pr1261947904930325089at_nat] :
        ( collec1570431334306492044st_nat
        @ ( produc6382697403170970154_nat_o
          @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis4215214144601311993xt_nat @ Ns2 @ S3 ) ) ) )
    @ ^ [S3: set_Pr1261947904930325089at_nat,Ns2: set_Pr1261947904930325089at_nat] :
        ( collec1570431334306492044st_nat
        @ ( produc6382697403170970154_nat_o
          @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis8831328596575508315xt_nat @ Ns2 @ S3 ) ) ) ) ) ).

% mul_ext_list.list_order_extension_axioms
thf(fact_879_mul__ext__list_Olist__order__extension__axioms,axiom,
    ( list_l5019108106899435338sion_a
    @ ^ [S3: set_Product_prod_a_a,Ns2: set_Product_prod_a_a] :
        ( collec943055143889122450list_a
        @ ( produc8172378796822260076st_a_o
          @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multiset_s_mul_ext_a @ Ns2 @ S3 ) ) ) )
    @ ^ [S3: set_Product_prod_a_a,Ns2: set_Product_prod_a_a] :
        ( collec943055143889122450list_a
        @ ( produc8172378796822260076st_a_o
          @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multis4023584895248362547_ext_a @ Ns2 @ S3 ) ) ) ) ) ).

% mul_ext_list.list_order_extension_axioms
thf(fact_880_subset__mset_Osum__mset__0__iff,axiom,
    ! [M: multis1201202736280713200et_nat] :
      ( ( ( comm_m5787568287065167983et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ M )
        = zero_z7348594199698428585et_nat )
      = ( ! [X3: multiset_nat] :
            ( ( member_multiset_nat @ X3 @ ( set_ms4188662328148412963et_nat @ M ) )
           => ( X3 = zero_z7348594199698428585et_nat ) ) ) ) ).

% subset_mset.sum_mset_0_iff
thf(fact_881_subset__mset_Osum__mset__0__iff,axiom,
    ! [M: multiset_multiset_a] :
      ( ( ( comm_m1977238983017651125iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ M )
        = zero_zero_multiset_a )
      = ( ! [X3: multiset_a] :
            ( ( member_multiset_a @ X3 @ ( set_mset_multiset_a @ M ) )
           => ( X3 = zero_zero_multiset_a ) ) ) ) ).

% subset_mset.sum_mset_0_iff
thf(fact_882_nth__transpose,axiom,
    ! [I2: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) ) )
     => ( ( nth_list_nat @ ( transpose_nat @ Xs ) @ I2 )
        = ( map_list_nat_nat
          @ ^ [Xs2: list_nat] : ( nth_nat @ Xs2 @ I2 )
          @ ( filter_list_nat
            @ ^ [Ys4: list_nat] : ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys4 ) )
            @ Xs ) ) ) ) ).

% nth_transpose
thf(fact_883_nth__transpose,axiom,
    ! [I2: nat,Xs: list_list_a] :
      ( ( ord_less_nat @ I2 @ ( size_s349497388124573686list_a @ ( transpose_a @ Xs ) ) )
     => ( ( nth_list_a @ ( transpose_a @ Xs ) @ I2 )
        = ( map_list_a_a
          @ ^ [Xs2: list_a] : ( nth_a @ Xs2 @ I2 )
          @ ( filter_list_a
            @ ^ [Ys4: list_a] : ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys4 ) )
            @ Xs ) ) ) ) ).

% nth_transpose
thf(fact_884_s__mul__ext__self__extend__left,axiom,
    ! [A3: multiset_nat,W: set_Pr1261947904930325089at_nat,B3: multiset_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( A3 != zero_z7348594199698428585et_nat )
     => ( ( locally_refl_nat @ W @ B3 )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) @ B3 ) @ ( multis4215214144601311993xt_nat @ W @ S ) ) ) ) ).

% s_mul_ext_self_extend_left
thf(fact_885_s__mul__ext__self__extend__left,axiom,
    ! [A3: multiset_a,W: set_Product_prod_a_a,B3: multiset_a,S: set_Product_prod_a_a] :
      ( ( A3 != zero_zero_multiset_a )
     => ( ( locally_refl_a @ W @ B3 )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ B3 ) @ ( multiset_s_mul_ext_a @ W @ S ) ) ) ) ).

% s_mul_ext_self_extend_left
thf(fact_886_filter__True,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( filter_nat @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_887_filter__True,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ( P @ X4 ) )
     => ( ( filter8895658007216384985ropVar @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_888_filter__True,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( filter_a @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_889_filter__append,axiom,
    ! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( filter_nat @ P @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys ) ) ) ).

% filter_append
thf(fact_890_set__filter,axiom,
    ! [P: product_prod_a_a > $o,Xs: list_P1396940483166286381od_a_a] :
      ( ( set_Product_prod_a_a2 @ ( filter7581757530173611428od_a_a @ P @ Xs ) )
      = ( collec3336397797384452498od_a_a
        @ ^ [X3: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X3 @ ( set_Product_prod_a_a2 @ Xs ) )
            & ( P @ X3 ) ) ) ) ).

% set_filter
thf(fact_891_set__filter,axiom,
    ! [P: produc6518373309651786023iset_a > $o,Xs: list_P6988024003011616557iset_a] :
      ( ( set_Pr5550698219922361532iset_a @ ( filter6941754759161614500iset_a @ P @ Xs ) )
      = ( collec6344740439120713362iset_a
        @ ^ [X3: produc6518373309651786023iset_a] :
            ( ( member5199237121806060112iset_a @ X3 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
            & ( P @ X3 ) ) ) ) ).

% set_filter
thf(fact_892_set__filter,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
      = ( collect_nat
        @ ^ [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
            & ( P @ X3 ) ) ) ) ).

% set_filter
thf(fact_893_set__filter,axiom,
    ! [P: multis3193088007478089820ropVar > $o,Xs: list_M6740372356978808802ropVar] :
      ( ( set_Mu146256361055586289ropVar @ ( filter8895658007216384985ropVar @ P @ Xs ) )
      = ( collec8658135508005558727ropVar
        @ ^ [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs ) )
            & ( P @ X3 ) ) ) ) ).

% set_filter
thf(fact_894_set__filter,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( set_a2 @ ( filter_a @ P @ Xs ) )
      = ( collect_a
        @ ^ [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
            & ( P @ X3 ) ) ) ) ).

% set_filter
thf(fact_895_filter__False,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ~ ( P @ X4 ) )
     => ( ( filter_nat @ P @ Xs )
        = nil_nat ) ) ).

% filter_False
thf(fact_896_filter__False,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( P @ X4 ) )
     => ( ( filter8895658007216384985ropVar @ P @ Xs )
        = nil_Mu4132399263956683874ropVar ) ) ).

% filter_False
thf(fact_897_filter__False,axiom,
    ! [Xs: list_a,P: a > $o] :
      ( ! [X4: a] :
          ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
         => ~ ( P @ X4 ) )
     => ( ( filter_a @ P @ Xs )
        = nil_a ) ) ).

% filter_False
thf(fact_898_filter__cong,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,P: product_prod_a_a > $o,Q: product_prod_a_a > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter7581757530173611428od_a_a @ P @ Xs )
          = ( filter7581757530173611428od_a_a @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_899_filter__cong,axiom,
    ! [Xs: list_P6988024003011616557iset_a,Ys: list_P6988024003011616557iset_a,P: produc6518373309651786023iset_a > $o,Q: produc6518373309651786023iset_a > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: produc6518373309651786023iset_a] :
            ( ( member5199237121806060112iset_a @ X4 @ ( set_Pr5550698219922361532iset_a @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter6941754759161614500iset_a @ P @ Xs )
          = ( filter6941754759161614500iset_a @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_900_filter__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter_nat @ P @ Xs )
          = ( filter_nat @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_901_filter__cong,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o,Q: multis3193088007478089820ropVar > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter8895658007216384985ropVar @ P @ Xs )
          = ( filter8895658007216384985ropVar @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_902_filter__cong,axiom,
    ! [Xs: list_a,Ys: list_a,P: a > $o,Q: a > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( filter_a @ P @ Xs )
          = ( filter_a @ Q @ Ys ) ) ) ) ).

% filter_cong
thf(fact_903_filter__id__conv,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( ( filter_nat @ P @ Xs )
        = Xs )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P @ X3 ) ) ) ) ).

% filter_id_conv
thf(fact_904_filter__id__conv,axiom,
    ! [P: multis3193088007478089820ropVar > $o,Xs: list_M6740372356978808802ropVar] :
      ( ( ( filter8895658007216384985ropVar @ P @ Xs )
        = Xs )
      = ( ! [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs ) )
           => ( P @ X3 ) ) ) ) ).

% filter_id_conv
thf(fact_905_filter__id__conv,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( ( filter_a @ P @ Xs )
        = Xs )
      = ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ( P @ X3 ) ) ) ) ).

% filter_id_conv
thf(fact_906_distinct__filter,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( filter_nat @ P @ Xs ) ) ) ).

% distinct_filter
thf(fact_907_filter_Osimps_I1_J,axiom,
    ! [P: a > $o] :
      ( ( filter_a @ P @ nil_a )
      = nil_a ) ).

% filter.simps(1)
thf(fact_908_filter_Osimps_I1_J,axiom,
    ! [P: nat > $o] :
      ( ( filter_nat @ P @ nil_nat )
      = nil_nat ) ).

% filter.simps(1)
thf(fact_909_filter__empty__conv,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( ( filter_nat @ P @ Xs )
        = nil_nat )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% filter_empty_conv
thf(fact_910_filter__empty__conv,axiom,
    ! [P: multis3193088007478089820ropVar > $o,Xs: list_M6740372356978808802ropVar] :
      ( ( ( filter8895658007216384985ropVar @ P @ Xs )
        = nil_Mu4132399263956683874ropVar )
      = ( ! [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% filter_empty_conv
thf(fact_911_filter__empty__conv,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( ( filter_a @ P @ Xs )
        = nil_a )
      = ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% filter_empty_conv
thf(fact_912_empty__filter__conv,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( nil_nat
        = ( filter_nat @ P @ Xs ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% empty_filter_conv
thf(fact_913_empty__filter__conv,axiom,
    ! [P: multis3193088007478089820ropVar > $o,Xs: list_M6740372356978808802ropVar] :
      ( ( nil_Mu4132399263956683874ropVar
        = ( filter8895658007216384985ropVar @ P @ Xs ) )
      = ( ! [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% empty_filter_conv
thf(fact_914_empty__filter__conv,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( nil_a
        = ( filter_a @ P @ Xs ) )
      = ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ~ ( P @ X3 ) ) ) ) ).

% empty_filter_conv
thf(fact_915_distinct__map__filter,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,P: nat > $o] :
      ( ( distin2675581014522817237ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
     => ( distin2675581014522817237ropVar @ ( map_na2466558649495876678ropVar @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filter
thf(fact_916_distinct__map__filter,axiom,
    ! [F: nat > nat,Xs: list_nat,P: nat > $o] :
      ( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
     => ( distinct_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filter
thf(fact_917_distinct__map__filter,axiom,
    ! [F: nat > a,Xs: list_nat,P: nat > $o] :
      ( ( distinct_a @ ( map_nat_a @ F @ Xs ) )
     => ( distinct_a @ ( map_nat_a @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filter
thf(fact_918_sum__length__filter__compl,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( plus_plus_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) )
        @ ( size_size_list_nat
          @ ( filter_nat
            @ ^ [X3: nat] :
                ~ ( P @ X3 )
            @ Xs ) ) )
      = ( size_size_list_nat @ Xs ) ) ).

% sum_length_filter_compl
thf(fact_919_sum__length__filter__compl,axiom,
    ! [P: a > $o,Xs: list_a] :
      ( ( plus_plus_nat @ ( size_size_list_a @ ( filter_a @ P @ Xs ) )
        @ ( size_size_list_a
          @ ( filter_a
            @ ^ [X3: a] :
                ~ ( P @ X3 )
            @ Xs ) ) )
      = ( size_size_list_a @ Xs ) ) ).

% sum_length_filter_compl
thf(fact_920_mset__eq__length__filter,axiom,
    ! [Xs: list_nat,Ys: list_nat,Z4: nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( size_size_list_nat
          @ ( filter_nat
            @ ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 )
              @ Z4 )
            @ Xs ) )
        = ( size_size_list_nat
          @ ( filter_nat
            @ ( ^ [Y3: nat,Z3: nat] : ( Y3 = Z3 )
              @ Z4 )
            @ Ys ) ) ) ) ).

% mset_eq_length_filter
thf(fact_921_mset__eq__length__filter,axiom,
    ! [Xs: list_a,Ys: list_a,Z4: a] :
      ( ( ( mset_a @ Xs )
        = ( mset_a @ Ys ) )
     => ( ( size_size_list_a
          @ ( filter_a
            @ ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 )
              @ Z4 )
            @ Xs ) )
        = ( size_size_list_a
          @ ( filter_a
            @ ( ^ [Y3: a,Z3: a] : ( Y3 = Z3 )
              @ Z4 )
            @ Ys ) ) ) ) ).

% mset_eq_length_filter
thf(fact_922_length__filter__less,axiom,
    ! [X: product_prod_a_a,Xs: list_P1396940483166286381od_a_a,P: product_prod_a_a > $o] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
     => ( ~ ( P @ X )
       => ( ord_less_nat @ ( size_s3885678630836030617od_a_a @ ( filter7581757530173611428od_a_a @ P @ Xs ) ) @ ( size_s3885678630836030617od_a_a @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_923_length__filter__less,axiom,
    ! [X: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a,P: produc6518373309651786023iset_a > $o] :
      ( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) )
     => ( ~ ( P @ X )
       => ( ord_less_nat @ ( size_s7238940367606365593iset_a @ ( filter6941754759161614500iset_a @ P @ Xs ) ) @ ( size_s7238940367606365593iset_a @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_924_length__filter__less,axiom,
    ! [X: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ( ~ ( P @ X )
       => ( ord_less_nat @ ( size_s1943289338788136526ropVar @ ( filter8895658007216384985ropVar @ P @ Xs ) ) @ ( size_s1943289338788136526ropVar @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_925_length__filter__less,axiom,
    ! [X: nat,Xs: list_nat,P: nat > $o] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ~ ( P @ X )
       => ( ord_less_nat @ ( size_size_list_nat @ ( filter_nat @ P @ Xs ) ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_926_length__filter__less,axiom,
    ! [X: a,Xs: list_a,P: a > $o] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ~ ( P @ X )
       => ( ord_less_nat @ ( size_size_list_a @ ( filter_a @ P @ Xs ) ) @ ( size_size_list_a @ Xs ) ) ) ) ).

% length_filter_less
thf(fact_927_mset__map__filter,axiom,
    ! [V: nat > multis3193088007478089820ropVar,C: nat > $o,T: list_nat] :
      ( ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ V @ ( filter_nat @ C @ T ) ) )
        @ ( mset_M6581516532424202993ropVar
          @ ( map_na2466558649495876678ropVar @ V
            @ ( filter_nat
              @ ^ [E: nat] :
                  ~ ( C @ E )
              @ T ) ) ) )
      = ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ V @ T ) ) ) ).

% mset_map_filter
thf(fact_928_mset__map__filter,axiom,
    ! [V: nat > nat,C: nat > $o,T: list_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( mset_nat @ ( map_nat_nat @ V @ ( filter_nat @ C @ T ) ) )
        @ ( mset_nat
          @ ( map_nat_nat @ V
            @ ( filter_nat
              @ ^ [E: nat] :
                  ~ ( C @ E )
              @ T ) ) ) )
      = ( mset_nat @ ( map_nat_nat @ V @ T ) ) ) ).

% mset_map_filter
thf(fact_929_mset__map__filter,axiom,
    ! [V: nat > a,C: nat > $o,T: list_nat] :
      ( ( plus_plus_multiset_a @ ( mset_a @ ( map_nat_a @ V @ ( filter_nat @ C @ T ) ) )
        @ ( mset_a
          @ ( map_nat_a @ V
            @ ( filter_nat
              @ ^ [E: nat] :
                  ~ ( C @ E )
              @ T ) ) ) )
      = ( mset_a @ ( map_nat_a @ V @ T ) ) ) ).

% mset_map_filter
thf(fact_930_ns__mul__ext__union__compat,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,C2: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) )
     => ( ( locally_refl_nat @ Ns @ C2 )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ B3 @ C2 ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) ) ) ) ).

% ns_mul_ext_union_compat
thf(fact_931_ns__mul__ext__union__compat,axiom,
    ! [A3: multiset_a,B3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a,C2: multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
     => ( ( locally_refl_a @ Ns @ C2 )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B3 @ C2 ) ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) ) ) ) ).

% ns_mul_ext_union_compat
thf(fact_932_s__mul__ext__union__compat,axiom,
    ! [A3: multiset_nat,B3: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,C2: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) )
     => ( ( locally_refl_nat @ Ns @ C2 )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ ( plus_p6334493942879108393et_nat @ B3 @ C2 ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ).

% s_mul_ext_union_compat
thf(fact_933_s__mul__ext__union__compat,axiom,
    ! [A3: multiset_a,B3: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a,C2: multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) )
     => ( ( locally_refl_a @ Ns @ C2 )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ ( plus_plus_multiset_a @ B3 @ C2 ) ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ).

% s_mul_ext_union_compat
thf(fact_934_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,Ns_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,As: list_multiset_a,Bs: list_multiset_a,NS: set_Pr79727621955416071iset_a,S: set_Pr79727621955416071iset_a] :
      ( ( list_l1107371739660948112iset_a @ S_list @ Ns_list )
     => ( ( ( size_s3003218658903709366iset_a @ As )
          = ( size_s3003218658903709366iset_a @ Bs ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3003218658903709366iset_a @ Bs ) )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ As @ I3 ) @ ( nth_multiset_a @ Bs @ I3 ) ) @ NS ) )
         => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As @ Bs ) @ ( Ns_list @ S @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_935_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ( ( size_size_list_nat @ As )
          = ( size_size_list_nat @ Bs ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ NS ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_936_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,Ns_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,As: list_a,Bs: list_a,NS: set_Product_prod_a_a,S: set_Product_prod_a_a] :
      ( ( list_l5019108106899435338sion_a @ S_list @ Ns_list )
     => ( ( ( size_size_list_a @ As )
          = ( size_size_list_a @ Bs ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Bs ) )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As @ I3 ) @ ( nth_a @ Bs @ I3 ) ) @ NS ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As @ Bs ) @ ( Ns_list @ S @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_937_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,S: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ! [A4: nat,B4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ S )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
       => ( ! [A4: nat,B4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ NS )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( S_list @ S @ NS ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) ) @ ( S_list @ S @ NS ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_938_list__order__extension_Os__map,axiom,
    ! [S_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,Ns_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,S: set_Product_prod_a_a,F: a > a,NS: set_Product_prod_a_a,As: list_a,Bs: list_a] :
      ( ( list_l5019108106899435338sion_a @ S_list @ Ns_list )
     => ( ! [A4: a,B4: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ S )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
       => ( ! [A4: a,B4: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ NS )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
         => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As @ Bs ) @ ( S_list @ S @ NS ) )
           => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F @ As ) @ ( map_a_a @ F @ Bs ) ) @ ( S_list @ S @ NS ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_939_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,Ns_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,S: set_Pr79727621955416071iset_a,F: multiset_a > multiset_a,NS: set_Pr79727621955416071iset_a,As: list_multiset_a,Bs: list_multiset_a] :
      ( ( list_l1107371739660948112iset_a @ S_list @ Ns_list )
     => ( ! [A4: multiset_a,B4: multiset_a] :
            ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ S )
           => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
       => ( ! [A4: multiset_a,B4: multiset_a] :
              ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ NS )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
         => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As @ Bs ) @ ( S_list @ S @ NS ) )
           => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F @ As ) @ ( map_mu8355158918406241476iset_a @ F @ Bs ) ) @ ( S_list @ S @ NS ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_940_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,S: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ! [A4: nat,B4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ S )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
       => ( ! [A4: nat,B4: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B4 ) @ NS )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S @ NS ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) ) @ ( Ns_list @ S @ NS ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_941_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,Ns_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,S: set_Product_prod_a_a,F: a > a,NS: set_Product_prod_a_a,As: list_a,Bs: list_a] :
      ( ( list_l5019108106899435338sion_a @ S_list @ Ns_list )
     => ( ! [A4: a,B4: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ S )
           => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
       => ( ! [A4: a,B4: a] :
              ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ B4 ) @ NS )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
         => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As @ Bs ) @ ( Ns_list @ S @ NS ) )
           => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F @ As ) @ ( map_a_a @ F @ Bs ) ) @ ( Ns_list @ S @ NS ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_942_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,Ns_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,S: set_Pr79727621955416071iset_a,F: multiset_a > multiset_a,NS: set_Pr79727621955416071iset_a,As: list_multiset_a,Bs: list_multiset_a] :
      ( ( list_l1107371739660948112iset_a @ S_list @ Ns_list )
     => ( ! [A4: multiset_a,B4: multiset_a] :
            ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ S )
           => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ S ) )
       => ( ! [A4: multiset_a,B4: multiset_a] :
              ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ NS )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F @ A4 ) @ ( F @ B4 ) ) @ NS ) )
         => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As @ Bs ) @ ( Ns_list @ S @ NS ) )
           => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F @ As ) @ ( map_mu8355158918406241476iset_a @ F @ Bs ) ) @ ( Ns_list @ S @ NS ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_943_case__prod__Pair__iden,axiom,
    ! [P7: product_prod_a_a] :
      ( ( produc408267641121961211od_a_a @ product_Pair_a_a @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_944_case__prod__Pair__iden,axiom,
    ! [P7: produc6518373309651786023iset_a] :
      ( ( produc7821597265843428091iset_a @ produc654756711066625303iset_a @ P7 )
      = P7 ) ).

% case_prod_Pair_iden
thf(fact_945_lr__trans__r,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A3: multiset_nat,B3: multiset_nat] :
      ( ( locally_refl_nat @ R @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) )
     => ( locally_refl_nat @ R @ B3 ) ) ).

% lr_trans_r
thf(fact_946_lr__trans__r,axiom,
    ! [R: set_Product_prod_a_a,A3: multiset_a,B3: multiset_a] :
      ( ( locally_refl_a @ R @ ( plus_plus_multiset_a @ A3 @ B3 ) )
     => ( locally_refl_a @ R @ B3 ) ) ).

% lr_trans_r
thf(fact_947_lr__trans__l,axiom,
    ! [R: set_Pr1261947904930325089at_nat,A3: multiset_nat,B3: multiset_nat] :
      ( ( locally_refl_nat @ R @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) )
     => ( locally_refl_nat @ R @ A3 ) ) ).

% lr_trans_l
thf(fact_948_lr__trans__l,axiom,
    ! [R: set_Product_prod_a_a,A3: multiset_a,B3: multiset_a] :
      ( ( locally_refl_a @ R @ ( plus_plus_multiset_a @ A3 @ B3 ) )
     => ( locally_refl_a @ R @ A3 ) ) ).

% lr_trans_l
thf(fact_949_in__mset__fold__plus__iff,axiom,
    ! [X: multis3193088007478089820ropVar,M: multis193838646080965794ropVar,NN: multis7892729734460161640ropVar] :
      ( ( member8421201611525043589ropVar @ X @ ( set_ms1147286257884310937ropVar @ ( fold_m7486329858681358475ropVar @ plus_p1339091108561003481ropVar @ M @ NN ) ) )
      = ( ( member8421201611525043589ropVar @ X @ ( set_ms1147286257884310937ropVar @ M ) )
        | ? [N3: multis193838646080965794ropVar] :
            ( ( member1575826910415056715ropVar @ N3 @ ( set_ms4465608862170883423ropVar @ NN ) )
            & ( member8421201611525043589ropVar @ X @ ( set_ms1147286257884310937ropVar @ N3 ) ) ) ) ) ).

% in_mset_fold_plus_iff
thf(fact_950_in__mset__fold__plus__iff,axiom,
    ! [X: product_prod_a_a,M: multis501812127501805293od_a_a,NN: multis5935151272263360051od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ ( fold_m3854860582760972491od_a_a @ plus_p6967097483979887908od_a_a @ M @ NN ) ) )
      = ( ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ M ) )
        | ? [N3: multis501812127501805293od_a_a] :
            ( ( member7700929758516679318od_a_a @ N3 @ ( set_ms8253844250382261418od_a_a @ NN ) )
            & ( member1426531477525435216od_a_a @ X @ ( set_ms119794563918438244od_a_a @ N3 ) ) ) ) ) ).

% in_mset_fold_plus_iff
thf(fact_951_in__mset__fold__plus__iff,axiom,
    ! [X: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,NN: multis7221419791219231795iset_a] :
      ( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ ( fold_m7087023614847598283iset_a @ plus_p1053901119268525860iset_a @ M @ NN ) ) )
      = ( ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ M ) )
        | ? [N3: multis4781622204924204013iset_a] :
            ( ( member2597461214246177430iset_a @ N3 @ ( set_ms5327031259215843498iset_a @ NN ) )
            & ( member5199237121806060112iset_a @ X @ ( set_ms2379723421352415844iset_a @ N3 ) ) ) ) ) ).

% in_mset_fold_plus_iff
thf(fact_952_in__mset__fold__plus__iff,axiom,
    ! [X: nat,M: multiset_nat,NN: multis1201202736280713200et_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ ( fold_m1829410296857755981et_nat @ plus_p6334493942879108393et_nat @ M @ NN ) ) )
      = ( ( member_nat @ X @ ( set_mset_nat @ M ) )
        | ? [N3: multiset_nat] :
            ( ( member_multiset_nat @ N3 @ ( set_ms4188662328148412963et_nat @ NN ) )
            & ( member_nat @ X @ ( set_mset_nat @ N3 ) ) ) ) ) ).

% in_mset_fold_plus_iff
thf(fact_953_in__mset__fold__plus__iff,axiom,
    ! [X: a,M: multiset_a,NN: multiset_multiset_a] :
      ( ( member_a @ X @ ( set_mset_a @ ( fold_m6601649825673331723iset_a @ plus_plus_multiset_a @ M @ NN ) ) )
      = ( ( member_a @ X @ ( set_mset_a @ M ) )
        | ? [N3: multiset_a] :
            ( ( member_multiset_a @ N3 @ ( set_mset_multiset_a @ NN ) )
            & ( member_a @ X @ ( set_mset_a @ N3 ) ) ) ) ) ).

% in_mset_fold_plus_iff
thf(fact_954_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_955_in__set__product__lists__length,axiom,
    ! [Xs: list_a,Xss: list_list_a] :
      ( ( member_list_a @ Xs @ ( set_list_a2 @ ( product_lists_a @ Xss ) ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_s349497388124573686list_a @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_956_map__upt__eqI,axiom,
    ! [Xs: list_M6740372356978808802ropVar,N: nat,M2: nat,F: nat > multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( minus_minus_nat @ N @ M2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Xs ) )
           => ( ( nth_Mu4537670848183934179ropVar @ Xs @ I3 )
              = ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ ( upt @ M2 @ N ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_957_map__upt__eqI,axiom,
    ! [Xs: list_nat,N: nat,M2: nat,F: nat > nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( minus_minus_nat @ N @ M2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) )
       => ( ( map_nat_nat @ F @ ( upt @ M2 @ N ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_958_map__upt__eqI,axiom,
    ! [Xs: list_a,N: nat,M2: nat,F: nat > a] :
      ( ( ( size_size_list_a @ Xs )
        = ( minus_minus_nat @ N @ M2 ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
           => ( ( nth_a @ Xs @ I3 )
              = ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) )
       => ( ( map_nat_a @ F @ ( upt @ M2 @ N ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_959_list__order__extension_Ointro,axiom,
    ! [S_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,Ns_list: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a] :
      ( ! [S4: set_Pr79727621955416071iset_a,NS2: set_Pr79727621955416071iset_a] :
          ( ( order_898045247964865234iset_a @ S4 @ NS2 )
         => ( order_8753316321938898392iset_a @ ( S_list @ S4 @ NS2 ) @ ( Ns_list @ S4 @ NS2 ) ) )
     => ( ! [S4: set_Pr79727621955416071iset_a,F7: multiset_a > multiset_a,NS2: set_Pr79727621955416071iset_a,As3: list_multiset_a,Bs3: list_multiset_a] :
            ( ! [A6: multiset_a,B6: multiset_a] :
                ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B6 ) @ S4 )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ S4 ) )
           => ( ! [A6: multiset_a,B6: multiset_a] :
                  ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B6 ) @ NS2 )
                 => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ NS2 ) )
             => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As3 @ Bs3 ) @ ( S_list @ S4 @ NS2 ) )
               => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F7 @ As3 ) @ ( map_mu8355158918406241476iset_a @ F7 @ Bs3 ) ) @ ( S_list @ S4 @ NS2 ) ) ) ) )
       => ( ! [S4: set_Pr79727621955416071iset_a,F7: multiset_a > multiset_a,NS2: set_Pr79727621955416071iset_a,As3: list_multiset_a,Bs3: list_multiset_a] :
              ( ! [A6: multiset_a,B6: multiset_a] :
                  ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B6 ) @ S4 )
                 => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ S4 ) )
             => ( ! [A6: multiset_a,B6: multiset_a] :
                    ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B6 ) @ NS2 )
                   => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ NS2 ) )
               => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS2 ) )
                 => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F7 @ As3 ) @ ( map_mu8355158918406241476iset_a @ F7 @ Bs3 ) ) @ ( Ns_list @ S4 @ NS2 ) ) ) ) )
         => ( ! [As3: list_multiset_a,Bs3: list_multiset_a,NS2: set_Pr79727621955416071iset_a,S4: set_Pr79727621955416071iset_a] :
                ( ( ( size_s3003218658903709366iset_a @ As3 )
                  = ( size_s3003218658903709366iset_a @ Bs3 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s3003218658903709366iset_a @ Bs3 ) )
                     => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ As3 @ I4 ) @ ( nth_multiset_a @ Bs3 @ I4 ) ) @ NS2 ) )
                 => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS2 ) ) ) )
           => ( list_l1107371739660948112iset_a @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_960_list__order__extension_Ointro,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
      ( ! [S4: set_Pr1261947904930325089at_nat,NS2: set_Pr1261947904930325089at_nat] :
          ( ( order_7802623214671921858ir_nat @ S4 @ NS2 )
         => ( order_2144575240049343698st_nat @ ( S_list @ S4 @ NS2 ) @ ( Ns_list @ S4 @ NS2 ) ) )
     => ( ! [S4: set_Pr1261947904930325089at_nat,F7: nat > nat,NS2: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
            ( ! [A6: nat,B6: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ S4 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ S4 ) )
           => ( ! [A6: nat,B6: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ NS2 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ NS2 ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( S_list @ S4 @ NS2 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F7 @ As3 ) @ ( map_nat_nat @ F7 @ Bs3 ) ) @ ( S_list @ S4 @ NS2 ) ) ) ) )
       => ( ! [S4: set_Pr1261947904930325089at_nat,F7: nat > nat,NS2: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
              ( ! [A6: nat,B6: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ S4 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ S4 ) )
             => ( ! [A6: nat,B6: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ NS2 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ NS2 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS2 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F7 @ As3 ) @ ( map_nat_nat @ F7 @ Bs3 ) ) @ ( Ns_list @ S4 @ NS2 ) ) ) ) )
         => ( ! [As3: list_nat,Bs3: list_nat,NS2: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
                ( ( ( size_size_list_nat @ As3 )
                  = ( size_size_list_nat @ Bs3 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Bs3 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As3 @ I4 ) @ ( nth_nat @ Bs3 @ I4 ) ) @ NS2 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS2 ) ) ) )
           => ( list_l792762466888043652on_nat @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_961_list__order__extension_Ointro,axiom,
    ! [S_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,Ns_list: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a] :
      ( ! [S4: set_Product_prod_a_a,NS2: set_Product_prod_a_a] :
          ( ( order_6408410059812637836pair_a @ S4 @ NS2 )
         => ( order_131025672941931026list_a @ ( S_list @ S4 @ NS2 ) @ ( Ns_list @ S4 @ NS2 ) ) )
     => ( ! [S4: set_Product_prod_a_a,F7: a > a,NS2: set_Product_prod_a_a,As3: list_a,Bs3: list_a] :
            ( ! [A6: a,B6: a] :
                ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B6 ) @ S4 )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ S4 ) )
           => ( ! [A6: a,B6: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B6 ) @ NS2 )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ NS2 ) )
             => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As3 @ Bs3 ) @ ( S_list @ S4 @ NS2 ) )
               => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F7 @ As3 ) @ ( map_a_a @ F7 @ Bs3 ) ) @ ( S_list @ S4 @ NS2 ) ) ) ) )
       => ( ! [S4: set_Product_prod_a_a,F7: a > a,NS2: set_Product_prod_a_a,As3: list_a,Bs3: list_a] :
              ( ! [A6: a,B6: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B6 ) @ S4 )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ S4 ) )
             => ( ! [A6: a,B6: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B6 ) @ NS2 )
                   => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F7 @ A6 ) @ ( F7 @ B6 ) ) @ NS2 ) )
               => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS2 ) )
                 => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F7 @ As3 ) @ ( map_a_a @ F7 @ Bs3 ) ) @ ( Ns_list @ S4 @ NS2 ) ) ) ) )
         => ( ! [As3: list_a,Bs3: list_a,NS2: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
                ( ( ( size_size_list_a @ As3 )
                  = ( size_size_list_a @ Bs3 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Bs3 ) )
                     => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As3 @ I4 ) @ ( nth_a @ Bs3 @ I4 ) ) @ NS2 ) )
                 => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As3 @ Bs3 ) @ ( Ns_list @ S4 @ NS2 ) ) ) )
           => ( list_l5019108106899435338sion_a @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_962_list__order__extension__def,axiom,
    ( list_l1107371739660948112iset_a
    = ( ^ [S_list2: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a,Ns_list2: set_Pr79727621955416071iset_a > set_Pr79727621955416071iset_a > set_Pr2598276052810994439iset_a] :
          ( ! [S5: set_Pr79727621955416071iset_a,NS3: set_Pr79727621955416071iset_a] :
              ( ( order_898045247964865234iset_a @ S5 @ NS3 )
             => ( order_8753316321938898392iset_a @ ( S_list2 @ S5 @ NS3 ) @ ( Ns_list2 @ S5 @ NS3 ) ) )
          & ! [S5: set_Pr79727621955416071iset_a,F8: multiset_a > multiset_a,NS3: set_Pr79727621955416071iset_a,As2: list_multiset_a,Bs2: list_multiset_a] :
              ( ! [A: multiset_a,B: multiset_a] :
                  ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ S5 )
                 => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ S5 ) )
             => ( ! [A: multiset_a,B: multiset_a] :
                    ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ NS3 )
                   => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ NS3 ) )
               => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As2 @ Bs2 ) @ ( S_list2 @ S5 @ NS3 ) )
                 => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F8 @ As2 ) @ ( map_mu8355158918406241476iset_a @ F8 @ Bs2 ) ) @ ( S_list2 @ S5 @ NS3 ) ) ) ) )
          & ! [S5: set_Pr79727621955416071iset_a,F8: multiset_a > multiset_a,NS3: set_Pr79727621955416071iset_a,As2: list_multiset_a,Bs2: list_multiset_a] :
              ( ! [A: multiset_a,B: multiset_a] :
                  ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ S5 )
                 => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ S5 ) )
             => ( ! [A: multiset_a,B: multiset_a] :
                    ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A @ B ) @ NS3 )
                   => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ NS3 ) )
               => ( ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS3 ) )
                 => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ ( map_mu8355158918406241476iset_a @ F8 @ As2 ) @ ( map_mu8355158918406241476iset_a @ F8 @ Bs2 ) ) @ ( Ns_list2 @ S5 @ NS3 ) ) ) ) )
          & ! [As2: list_multiset_a,Bs2: list_multiset_a,NS3: set_Pr79727621955416071iset_a,S5: set_Pr79727621955416071iset_a] :
              ( ( ( size_s3003218658903709366iset_a @ As2 )
                = ( size_s3003218658903709366iset_a @ Bs2 ) )
             => ( ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( size_s3003218658903709366iset_a @ Bs2 ) )
                   => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( nth_multiset_a @ As2 @ I ) @ ( nth_multiset_a @ Bs2 @ I ) ) @ NS3 ) )
               => ( member1640862112313989456iset_a @ ( produc1434922032272282391iset_a @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS3 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_963_list__order__extension__def,axiom,
    ( list_l792762466888043652on_nat
    = ( ^ [S_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
          ( ! [S5: set_Pr1261947904930325089at_nat,NS3: set_Pr1261947904930325089at_nat] :
              ( ( order_7802623214671921858ir_nat @ S5 @ NS3 )
             => ( order_2144575240049343698st_nat @ ( S_list2 @ S5 @ NS3 ) @ ( Ns_list2 @ S5 @ NS3 ) ) )
          & ! [S5: set_Pr1261947904930325089at_nat,F8: nat > nat,NS3: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
              ( ! [A: nat,B: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S5 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A ) @ ( F8 @ B ) ) @ S5 ) )
             => ( ! [A: nat,B: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ NS3 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A ) @ ( F8 @ B ) ) @ NS3 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( S_list2 @ S5 @ NS3 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F8 @ As2 ) @ ( map_nat_nat @ F8 @ Bs2 ) ) @ ( S_list2 @ S5 @ NS3 ) ) ) ) )
          & ! [S5: set_Pr1261947904930325089at_nat,F8: nat > nat,NS3: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
              ( ! [A: nat,B: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ S5 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A ) @ ( F8 @ B ) ) @ S5 ) )
             => ( ! [A: nat,B: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ NS3 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A ) @ ( F8 @ B ) ) @ NS3 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS3 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F8 @ As2 ) @ ( map_nat_nat @ F8 @ Bs2 ) ) @ ( Ns_list2 @ S5 @ NS3 ) ) ) ) )
          & ! [As2: list_nat,Bs2: list_nat,NS3: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
              ( ( ( size_size_list_nat @ As2 )
                = ( size_size_list_nat @ Bs2 ) )
             => ( ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( size_size_list_nat @ Bs2 ) )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As2 @ I ) @ ( nth_nat @ Bs2 @ I ) ) @ NS3 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS3 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_964_list__order__extension__def,axiom,
    ( list_l5019108106899435338sion_a
    = ( ^ [S_list2: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a,Ns_list2: set_Product_prod_a_a > set_Product_prod_a_a > set_Pr4048851178543822343list_a] :
          ( ! [S5: set_Product_prod_a_a,NS3: set_Product_prod_a_a] :
              ( ( order_6408410059812637836pair_a @ S5 @ NS3 )
             => ( order_131025672941931026list_a @ ( S_list2 @ S5 @ NS3 ) @ ( Ns_list2 @ S5 @ NS3 ) ) )
          & ! [S5: set_Product_prod_a_a,F8: a > a,NS3: set_Product_prod_a_a,As2: list_a,Bs2: list_a] :
              ( ! [A: a,B: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ S5 )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ S5 ) )
             => ( ! [A: a,B: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ NS3 )
                   => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ NS3 ) )
               => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As2 @ Bs2 ) @ ( S_list2 @ S5 @ NS3 ) )
                 => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F8 @ As2 ) @ ( map_a_a @ F8 @ Bs2 ) ) @ ( S_list2 @ S5 @ NS3 ) ) ) ) )
          & ! [S5: set_Product_prod_a_a,F8: a > a,NS3: set_Product_prod_a_a,As2: list_a,Bs2: list_a] :
              ( ! [A: a,B: a] :
                  ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ S5 )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ S5 ) )
             => ( ! [A: a,B: a] :
                    ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B ) @ NS3 )
                   => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( F8 @ A ) @ ( F8 @ B ) ) @ NS3 ) )
               => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS3 ) )
                 => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ ( map_a_a @ F8 @ As2 ) @ ( map_a_a @ F8 @ Bs2 ) ) @ ( Ns_list2 @ S5 @ NS3 ) ) ) ) )
          & ! [As2: list_a,Bs2: list_a,NS3: set_Product_prod_a_a,S5: set_Product_prod_a_a] :
              ( ( ( size_size_list_a @ As2 )
                = ( size_size_list_a @ Bs2 ) )
             => ( ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( size_size_list_a @ Bs2 ) )
                   => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ As2 @ I ) @ ( nth_a @ Bs2 @ I ) ) @ NS3 ) )
               => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS3 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_965_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_966_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_967_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_968_add__diff__cancel__left,axiom,
    ! [C: multiset_nat,A2: multiset_nat,B2: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ A2 ) @ ( plus_p6334493942879108393et_nat @ C @ B2 ) )
      = ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_969_add__diff__cancel__left,axiom,
    ! [C: multiset_a,A2: multiset_a,B2: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ ( plus_plus_multiset_a @ C @ B2 ) )
      = ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_970_add__diff__cancel__left,axiom,
    ! [C: nat,A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A2 ) @ ( plus_plus_nat @ C @ B2 ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_left
thf(fact_971_add__diff__cancel__left_H,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_972_add__diff__cancel__left_H,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_973_add__diff__cancel__left_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ A2 )
      = B2 ) ).

% add_diff_cancel_left'
thf(fact_974_add__diff__cancel__right,axiom,
    ! [A2: multiset_nat,C: multiset_nat,B2: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C ) @ ( plus_p6334493942879108393et_nat @ B2 @ C ) )
      = ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_975_add__diff__cancel__right,axiom,
    ! [A2: multiset_a,C: multiset_a,B2: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ C ) @ ( plus_plus_multiset_a @ B2 @ C ) )
      = ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_976_add__diff__cancel__right,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ C ) @ ( plus_plus_nat @ B2 @ C ) )
      = ( minus_minus_nat @ A2 @ B2 ) ) ).

% add_diff_cancel_right
thf(fact_977_add__diff__cancel__right_H,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_978_add__diff__cancel__right_H,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_979_add__diff__cancel__right_H,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A2 @ B2 ) @ B2 )
      = A2 ) ).

% add_diff_cancel_right'
thf(fact_980_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_981_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_982_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K2 ) ) ) ).

% diff_diff_left
thf(fact_983_diff__add__zero,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ A2 @ B2 ) )
      = zero_z7348594199698428585et_nat ) ).

% diff_add_zero
thf(fact_984_diff__add__zero,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ A2 @ B2 ) )
      = zero_zero_multiset_a ) ).

% diff_add_zero
thf(fact_985_diff__add__zero,axiom,
    ! [A2: nat,B2: nat] :
      ( ( minus_minus_nat @ A2 @ ( plus_plus_nat @ A2 @ B2 ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_986_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_987_length__drop,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( size_s1943289338788136526ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) )
      = ( minus_minus_nat @ ( size_s1943289338788136526ropVar @ Xs ) @ N ) ) ).

% length_drop
thf(fact_988_length__drop,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% length_drop
thf(fact_989_length__drop,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( size_size_list_a @ ( drop_a @ N @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_a @ Xs ) @ N ) ) ).

% length_drop
thf(fact_990_length__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J ) )
      = ( minus_minus_nat @ J @ I2 ) ) ).

% length_upt
thf(fact_991_drop__append,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( append2684835082924936439ropVar @ Xs @ Ys ) )
      = ( append2684835082924936439ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) @ ( drop_M2293212096379261346ropVar @ ( minus_minus_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_992_drop__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_993_drop__append,axiom,
    ! [N: nat,Xs: list_a,Ys: list_a] :
      ( ( drop_a @ N @ ( append_a @ Xs @ Ys ) )
      = ( append_a @ ( drop_a @ N @ Xs ) @ ( drop_a @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_994_add__implies__diff,axiom,
    ! [C: multiset_nat,B2: multiset_nat,A2: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_8522176038001411705et_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_995_add__implies__diff,axiom,
    ! [C: multiset_a,B2: multiset_a,A2: multiset_a] :
      ( ( ( plus_plus_multiset_a @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_3765977307040488491iset_a @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_996_add__implies__diff,axiom,
    ! [C: nat,B2: nat,A2: nat] :
      ( ( ( plus_plus_nat @ C @ B2 )
        = A2 )
     => ( C
        = ( minus_minus_nat @ A2 @ B2 ) ) ) ).

% add_implies_diff
thf(fact_997_diff__diff__eq,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A2 @ B2 ) @ C )
      = ( minus_8522176038001411705et_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_998_diff__diff__eq,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ A2 @ B2 ) @ C )
      = ( minus_3765977307040488491iset_a @ A2 @ ( plus_plus_multiset_a @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_999_diff__diff__eq,axiom,
    ! [A2: nat,B2: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C )
      = ( minus_minus_nat @ A2 @ ( plus_plus_nat @ B2 @ C ) ) ) ).

% diff_diff_eq
thf(fact_1000_diff__commute,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K2 ) @ J ) ) ).

% diff_commute
thf(fact_1001_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C: nat,B2: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C ) @ B2 )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B2 ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1002_Nat_Odiff__cancel,axiom,
    ! [K2: nat,M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K2 @ M2 ) @ ( plus_plus_nat @ K2 @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% Nat.diff_cancel
thf(fact_1003_diff__cancel2,axiom,
    ! [M2: nat,K2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K2 ) @ ( plus_plus_nat @ N @ K2 ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_cancel2
thf(fact_1004_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_1005_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_1006_less__imp__diff__less,axiom,
    ! [J: nat,K2: nat,N: nat] :
      ( ( ord_less_nat @ J @ K2 )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K2 ) ) ).

% less_imp_diff_less
thf(fact_1007_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_1008_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M2 )
          = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_1009_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_1010_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A2: nat,B2: nat] :
      ( ~ ( ord_less_nat @ A2 @ B2 )
     => ( ( plus_plus_nat @ B2 @ ( minus_minus_nat @ A2 @ B2 ) )
        = A2 ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_1011_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_1012_diff__add__0,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1013_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K2: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K2 ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K2 ) @ J ) ) ).

% less_diff_conv
thf(fact_1014_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_1015_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ~ ( ( ( ord_less_nat @ A2 @ B2 )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B2 @ D4 ) )
                & ~ ( P @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1016_nat__diff__split,axiom,
    ! [P: nat > $o,A2: nat,B2: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B2 ) )
      = ( ( ( ord_less_nat @ A2 @ B2 )
         => ( P @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A2
              = ( plus_plus_nat @ B2 @ D4 ) )
           => ( P @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_1017_nth__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1018_nth__append,axiom,
    ! [N: nat,Xs: list_a,Ys: list_a] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
       => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
          = ( nth_a @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
       => ( ( nth_a @ ( append_a @ Xs @ Ys ) @ N )
          = ( nth_a @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_a @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1019_distinct__product__lists,axiom,
    ! [Xss: list_list_nat] :
      ( ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss ) )
         => ( distinct_nat @ X4 ) )
     => ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_1020_mul__ext__list_Oextension,axiom,
    ! [S: set_Pr1261947904930325089at_nat,NS: set_Pr1261947904930325089at_nat] :
      ( ( order_7802623214671921858ir_nat @ S @ NS )
     => ( order_2144575240049343698st_nat
        @ ( collec1570431334306492044st_nat
          @ ( produc6382697403170970154_nat_o
            @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S ) ) ) )
        @ ( collec1570431334306492044st_nat
          @ ( produc6382697403170970154_nat_o
            @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S ) ) ) ) ) ) ).

% mul_ext_list.extension
thf(fact_1021_mul__ext__list_Oextension,axiom,
    ! [S: set_Product_prod_a_a,NS: set_Product_prod_a_a] :
      ( ( order_6408410059812637836pair_a @ S @ NS )
     => ( order_131025672941931026list_a
        @ ( collec943055143889122450list_a
          @ ( produc8172378796822260076st_a_o
            @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multiset_s_mul_ext_a @ NS @ S ) ) ) )
        @ ( collec943055143889122450list_a
          @ ( produc8172378796822260076st_a_o
            @ ^ [As2: list_a,Bs2: list_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( mset_a @ As2 ) @ ( mset_a @ Bs2 ) ) @ ( multis4023584895248362547_ext_a @ NS @ S ) ) ) ) ) ) ).

% mul_ext_list.extension
thf(fact_1022_nth__map__upt,axiom,
    ! [I2: nat,N: nat,M2: nat,F: nat > multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N @ M2 ) )
     => ( ( nth_Mu4537670848183934179ropVar @ ( map_na2466558649495876678ropVar @ F @ ( upt @ M2 @ N ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_1023_nth__map__upt,axiom,
    ! [I2: nat,N: nat,M2: nat,F: nat > nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N @ M2 ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M2 @ N ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_1024_nth__map__upt,axiom,
    ! [I2: nat,N: nat,M2: nat,F: nat > a] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N @ M2 ) )
     => ( ( nth_a @ ( map_nat_a @ F @ ( upt @ M2 @ N ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_1025_s__mul__ext__IdE,axiom,
    ! [M: multiset_multiset_a,N2: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
      ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N2 ) @ ( multis7166037729339985883iset_a @ id_multiset_a @ R ) )
     => ~ ! [X10: multiset_multiset_a] :
            ( ( X10 != zero_z7799948378220188993iset_a )
           => ! [Y8: multiset_multiset_a,Z5: multiset_multiset_a] :
                ( ( M
                  = ( plus_p6738641960240532161iset_a @ X10 @ Z5 ) )
               => ( ( N2
                    = ( plus_p6738641960240532161iset_a @ Y8 @ Z5 ) )
                 => ~ ! [X6: multiset_a] :
                        ( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ Y8 ) )
                       => ? [Xa2: multiset_a] :
                            ( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ X10 ) )
                            & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ) ).

% s_mul_ext_IdE
thf(fact_1026_s__mul__ext__IdE,axiom,
    ! [M: multiset_nat,N2: multiset_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M @ N2 ) @ ( multis4215214144601311993xt_nat @ id_nat @ R ) )
     => ~ ! [X10: multiset_nat] :
            ( ( X10 != zero_z7348594199698428585et_nat )
           => ! [Y8: multiset_nat,Z5: multiset_nat] :
                ( ( M
                  = ( plus_p6334493942879108393et_nat @ X10 @ Z5 ) )
               => ( ( N2
                    = ( plus_p6334493942879108393et_nat @ Y8 @ Z5 ) )
                 => ~ ! [X6: nat] :
                        ( ( member_nat @ X6 @ ( set_mset_nat @ Y8 ) )
                       => ? [Xa2: nat] :
                            ( ( member_nat @ Xa2 @ ( set_mset_nat @ X10 ) )
                            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ) ).

% s_mul_ext_IdE
thf(fact_1027_s__mul__ext__IdE,axiom,
    ! [M: multiset_a,N2: multiset_a,R: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N2 ) @ ( multiset_s_mul_ext_a @ id_a @ R ) )
     => ~ ! [X10: multiset_a] :
            ( ( X10 != zero_zero_multiset_a )
           => ! [Y8: multiset_a,Z5: multiset_a] :
                ( ( M
                  = ( plus_plus_multiset_a @ X10 @ Z5 ) )
               => ( ( N2
                    = ( plus_plus_multiset_a @ Y8 @ Z5 ) )
                 => ~ ! [X6: a] :
                        ( ( member_a @ X6 @ ( set_mset_a @ Y8 ) )
                       => ? [Xa2: a] :
                            ( ( member_a @ Xa2 @ ( set_mset_a @ X10 ) )
                            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ) ).

% s_mul_ext_IdE
thf(fact_1028_s__mul__ext__IdI,axiom,
    ! [X5: multiset_multiset_a,M: multiset_multiset_a,Z6: multiset_multiset_a,N2: multiset_multiset_a,Y4: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
      ( ( X5 != zero_z7799948378220188993iset_a )
     => ( ( M
          = ( plus_p6738641960240532161iset_a @ X5 @ Z6 ) )
       => ( ( N2
            = ( plus_p6738641960240532161iset_a @ Y4 @ Z6 ) )
         => ( ! [X4: multiset_a] :
                ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ Y4 ) )
               => ? [Xa3: multiset_a] :
                    ( ( member_multiset_a @ Xa3 @ ( set_mset_multiset_a @ X5 ) )
                    & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa3 @ X4 ) @ R ) ) )
           => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N2 ) @ ( multis7166037729339985883iset_a @ id_multiset_a @ R ) ) ) ) ) ) ).

% s_mul_ext_IdI
thf(fact_1029_s__mul__ext__IdI,axiom,
    ! [X5: multiset_nat,M: multiset_nat,Z6: multiset_nat,N2: multiset_nat,Y4: multiset_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( X5 != zero_z7348594199698428585et_nat )
     => ( ( M
          = ( plus_p6334493942879108393et_nat @ X5 @ Z6 ) )
       => ( ( N2
            = ( plus_p6334493942879108393et_nat @ Y4 @ Z6 ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( set_mset_nat @ Y4 ) )
               => ? [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ ( set_mset_nat @ X5 ) )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X4 ) @ R ) ) )
           => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M @ N2 ) @ ( multis4215214144601311993xt_nat @ id_nat @ R ) ) ) ) ) ) ).

% s_mul_ext_IdI
thf(fact_1030_s__mul__ext__IdI,axiom,
    ! [X5: multiset_a,M: multiset_a,Z6: multiset_a,N2: multiset_a,Y4: multiset_a,R: set_Product_prod_a_a] :
      ( ( X5 != zero_zero_multiset_a )
     => ( ( M
          = ( plus_plus_multiset_a @ X5 @ Z6 ) )
       => ( ( N2
            = ( plus_plus_multiset_a @ Y4 @ Z6 ) )
         => ( ! [X4: a] :
                ( ( member_a @ X4 @ ( set_mset_a @ Y4 ) )
               => ? [Xa3: a] :
                    ( ( member_a @ Xa3 @ ( set_mset_a @ X5 ) )
                    & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X4 ) @ R ) ) )
           => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N2 ) @ ( multiset_s_mul_ext_a @ id_a @ R ) ) ) ) ) ) ).

% s_mul_ext_IdI
thf(fact_1031_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1032_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_a,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_a @ Xs ) )
     => ( ( nth_Pr2476257081389315164_nat_a @ ( enumerate_a @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_a @ ( plus_plus_nat @ N @ M2 ) @ ( nth_a @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1033_Un__Diff__cancel,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( sup_sup_set_nat @ A3 @ ( minus_minus_set_nat @ B3 @ A3 ) )
      = ( sup_sup_set_nat @ A3 @ B3 ) ) ).

% Un_Diff_cancel
thf(fact_1034_Un__Diff__cancel2,axiom,
    ! [B3: set_nat,A3: set_nat] :
      ( ( sup_sup_set_nat @ ( minus_minus_set_nat @ B3 @ A3 ) @ A3 )
      = ( sup_sup_set_nat @ B3 @ A3 ) ) ).

% Un_Diff_cancel2
thf(fact_1035_diff__diff__add__mset,axiom,
    ! [M: multiset_nat,N2: multiset_nat,P: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M @ N2 ) @ P )
      = ( minus_8522176038001411705et_nat @ M @ ( plus_p6334493942879108393et_nat @ N2 @ P ) ) ) ).

% diff_diff_add_mset
thf(fact_1036_diff__diff__add__mset,axiom,
    ! [M: multiset_a,N2: multiset_a,P: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ N2 ) @ P )
      = ( minus_3765977307040488491iset_a @ M @ ( plus_plus_multiset_a @ N2 @ P ) ) ) ).

% diff_diff_add_mset
thf(fact_1037_subset__mset_Oadd__diff__assoc,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( plus_p6334493942879108393et_nat @ C @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ B2 ) @ A2 ) ) ) ).

% subset_mset.add_diff_assoc
thf(fact_1038_subset__mset_Oadd__diff__assoc,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
        = ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B2 ) @ A2 ) ) ) ).

% subset_mset.add_diff_assoc
thf(fact_1039_subset__mset_Oadd__diff__assoc2,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) @ C )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B2 @ C ) @ A2 ) ) ) ).

% subset_mset.add_diff_assoc2
thf(fact_1040_subset__mset_Oadd__diff__assoc2,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) @ C )
        = ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B2 @ C ) @ A2 ) ) ) ).

% subset_mset.add_diff_assoc2
thf(fact_1041_mset__subset__eq__multiset__union__diff__commute,axiom,
    ! [B3: multiset_nat,A3: multiset_nat,C2: multiset_nat] :
      ( ( subseteq_mset_nat @ B3 @ A3 )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ A3 @ B3 ) @ C2 )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ C2 ) @ B3 ) ) ) ).

% mset_subset_eq_multiset_union_diff_commute
thf(fact_1042_mset__subset__eq__multiset__union__diff__commute,axiom,
    ! [B3: multiset_a,A3: multiset_a,C2: multiset_a] :
      ( ( subseteq_mset_a @ B3 @ A3 )
     => ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ A3 @ B3 ) @ C2 )
        = ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ C2 ) @ B3 ) ) ) ).

% mset_subset_eq_multiset_union_diff_commute
thf(fact_1043_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_a @ N @ nil_a )
      = nil_Pr1417316670369895453_nat_a ) ).

% enumerate_simps(1)
thf(fact_1044_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_1045_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_1046_length__enumerate,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( size_s243904063682394823_nat_a @ ( enumerate_a @ N @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_enumerate
thf(fact_1047_in__diffD,axiom,
    ! [A2: nat,M: multiset_nat,N2: multiset_nat] :
      ( ( member_nat @ A2 @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M @ N2 ) ) )
     => ( member_nat @ A2 @ ( set_mset_nat @ M ) ) ) ).

% in_diffD
thf(fact_1048_in__diffD,axiom,
    ! [A2: multis3193088007478089820ropVar,M: multis193838646080965794ropVar,N2: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A2 @ ( set_ms1147286257884310937ropVar @ ( minus_287902380581138057ropVar @ M @ N2 ) ) )
     => ( member8421201611525043589ropVar @ A2 @ ( set_ms1147286257884310937ropVar @ M ) ) ) ).

% in_diffD
thf(fact_1049_in__diffD,axiom,
    ! [A2: product_prod_a_a,M: multis501812127501805293od_a_a,N2: multis501812127501805293od_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ ( minus_7313152569058643924od_a_a @ M @ N2 ) ) )
     => ( member1426531477525435216od_a_a @ A2 @ ( set_ms119794563918438244od_a_a @ M ) ) ) ).

% in_diffD
thf(fact_1050_in__diffD,axiom,
    ! [A2: a,M: multiset_a,N2: multiset_a] :
      ( ( member_a @ A2 @ ( set_mset_a @ ( minus_3765977307040488491iset_a @ M @ N2 ) ) )
     => ( member_a @ A2 @ ( set_mset_a @ M ) ) ) ).

% in_diffD
thf(fact_1051_in__diffD,axiom,
    ! [A2: produc6518373309651786023iset_a,M: multis4781622204924204013iset_a,N2: multis4781622204924204013iset_a] :
      ( ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ ( minus_9211619285413683668iset_a @ M @ N2 ) ) )
     => ( member5199237121806060112iset_a @ A2 @ ( set_ms2379723421352415844iset_a @ M ) ) ) ).

% in_diffD
thf(fact_1052_Un__Diff,axiom,
    ! [A3: set_nat,B3: set_nat,C2: set_nat] :
      ( ( minus_minus_set_nat @ ( sup_sup_set_nat @ A3 @ B3 ) @ C2 )
      = ( sup_sup_set_nat @ ( minus_minus_set_nat @ A3 @ C2 ) @ ( minus_minus_set_nat @ B3 @ C2 ) ) ) ).

% Un_Diff
thf(fact_1053_Multiset_Odiff__add,axiom,
    ! [M: multiset_nat,N2: multiset_nat,Q: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ M @ ( plus_p6334493942879108393et_nat @ N2 @ Q ) )
      = ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M @ N2 ) @ Q ) ) ).

% Multiset.diff_add
thf(fact_1054_Multiset_Odiff__add,axiom,
    ! [M: multiset_a,N2: multiset_a,Q: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ M @ ( plus_plus_multiset_a @ N2 @ Q ) )
      = ( minus_3765977307040488491iset_a @ ( minus_3765977307040488491iset_a @ M @ N2 ) @ Q ) ) ).

% Multiset.diff_add
thf(fact_1055_diff__union__cancelL,axiom,
    ! [N2: multiset_nat,M: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_union_cancelL
thf(fact_1056_diff__union__cancelL,axiom,
    ! [N2: multiset_a,M: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ N2 @ M ) @ N2 )
      = M ) ).

% diff_union_cancelL
thf(fact_1057_diff__union__cancelR,axiom,
    ! [M: multiset_nat,N2: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_union_cancelR
thf(fact_1058_diff__union__cancelR,axiom,
    ! [M: multiset_a,N2: multiset_a] :
      ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ M @ N2 ) @ N2 )
      = M ) ).

% diff_union_cancelR
thf(fact_1059_psubset__imp__ex__mem,axiom,
    ! [A3: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A3 @ B3 )
     => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1060_psubset__imp__ex__mem,axiom,
    ! [A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( ord_le8651388201640276840ropVar @ A3 @ B3 )
     => ? [B4: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ B4 @ ( minus_2621998875110806563ropVar @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1061_psubset__imp__ex__mem,axiom,
    ! [A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( ord_le6819997720685908915od_a_a @ A3 @ B3 )
     => ? [B4: product_prod_a_a] : ( member1426531477525435216od_a_a @ B4 @ ( minus_6817036919807184750od_a_a @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1062_psubset__imp__ex__mem,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ord_less_set_a @ A3 @ B3 )
     => ? [B4: a] : ( member_a @ B4 @ ( minus_minus_set_a @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1063_psubset__imp__ex__mem,axiom,
    ! [A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( ord_le4018519769653605043iset_a @ A3 @ B3 )
     => ? [B4: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ B4 @ ( minus_2461365523287052142iset_a @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1064_ns__mul__ext__Id__eq,axiom,
    ! [R: set_Product_prod_a_a] :
      ( ( multis4023584895248362547_ext_a @ id_a @ R )
      = ( sup_su1925787886501348827iset_a @ ( multiset_s_mul_ext_a @ id_a @ R ) @ id_multiset_a ) ) ).

% ns_mul_ext_Id_eq
thf(fact_1065_multiset__diff__union__assoc,axiom,
    ! [C2: multiset_nat,B3: multiset_nat,A3: multiset_nat] :
      ( ( subseteq_mset_nat @ C2 @ B3 )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A3 @ B3 ) @ C2 )
        = ( plus_p6334493942879108393et_nat @ A3 @ ( minus_8522176038001411705et_nat @ B3 @ C2 ) ) ) ) ).

% multiset_diff_union_assoc
thf(fact_1066_multiset__diff__union__assoc,axiom,
    ! [C2: multiset_a,B3: multiset_a,A3: multiset_a] :
      ( ( subseteq_mset_a @ C2 @ B3 )
     => ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ A3 @ B3 ) @ C2 )
        = ( plus_plus_multiset_a @ A3 @ ( minus_3765977307040488491iset_a @ B3 @ C2 ) ) ) ) ).

% multiset_diff_union_assoc
thf(fact_1067_subset__eq__diff__conv,axiom,
    ! [A3: multiset_nat,C2: multiset_nat,B3: multiset_nat] :
      ( ( subseteq_mset_nat @ ( minus_8522176038001411705et_nat @ A3 @ C2 ) @ B3 )
      = ( subseteq_mset_nat @ A3 @ ( plus_p6334493942879108393et_nat @ B3 @ C2 ) ) ) ).

% subset_eq_diff_conv
thf(fact_1068_subset__eq__diff__conv,axiom,
    ! [A3: multiset_a,C2: multiset_a,B3: multiset_a] :
      ( ( subseteq_mset_a @ ( minus_3765977307040488491iset_a @ A3 @ C2 ) @ B3 )
      = ( subseteq_mset_a @ A3 @ ( plus_plus_multiset_a @ B3 @ C2 ) ) ) ).

% subset_eq_diff_conv
thf(fact_1069_subset__mset_Ole__imp__diff__is__add,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( subseteq_mset_nat @ A2 @ B2 )
       => ( ( ( minus_8522176038001411705et_nat @ B2 @ A2 )
            = C )
          = ( B2
            = ( plus_p6334493942879108393et_nat @ C @ A2 ) ) ) ) ) ).

% subset_mset.le_imp_diff_is_add
thf(fact_1070_subset__mset_Ole__imp__diff__is__add,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( subseteq_mset_a @ A2 @ B2 )
       => ( ( ( minus_3765977307040488491iset_a @ B2 @ A2 )
            = C )
          = ( B2
            = ( plus_plus_multiset_a @ C @ A2 ) ) ) ) ) ).

% subset_mset.le_imp_diff_is_add
thf(fact_1071_subset__mset_Oadd__diff__inverse,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( plus_p6334493942879108393et_nat @ A2 @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) )
        = B2 ) ) ).

% subset_mset.add_diff_inverse
thf(fact_1072_subset__mset_Oadd__diff__inverse,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( plus_plus_multiset_a @ A2 @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
        = B2 ) ) ).

% subset_mset.add_diff_inverse
thf(fact_1073_subset__mset_Odiff__diff__right,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( minus_8522176038001411705et_nat @ C @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ A2 ) @ B2 ) ) ) ).

% subset_mset.diff_diff_right
thf(fact_1074_subset__mset_Odiff__diff__right,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( minus_3765977307040488491iset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
        = ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ B2 ) ) ) ).

% subset_mset.diff_diff_right
thf(fact_1075_subset__mset_Odiff__add__assoc2,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B2 @ C ) @ A2 )
        = ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) @ C ) ) ) ).

% subset_mset.diff_add_assoc2
thf(fact_1076_subset__mset_Odiff__add__assoc2,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B2 @ C ) @ A2 )
        = ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) @ C ) ) ) ).

% subset_mset.diff_add_assoc2
thf(fact_1077_subset__mset_Odiff__add__assoc,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C @ B2 ) @ A2 )
        = ( plus_p6334493942879108393et_nat @ C @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) ) ) ) ).

% subset_mset.diff_add_assoc
thf(fact_1078_subset__mset_Odiff__add__assoc,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ C @ B2 ) @ A2 )
        = ( plus_plus_multiset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) ) ) ) ).

% subset_mset.diff_add_assoc
thf(fact_1079_subset__mset_Ole__diff__conv2,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( subseteq_mset_nat @ C @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) )
        = ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C @ A2 ) @ B2 ) ) ) ).

% subset_mset.le_diff_conv2
thf(fact_1080_subset__mset_Ole__diff__conv2,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) )
        = ( subseteq_mset_a @ ( plus_plus_multiset_a @ C @ A2 ) @ B2 ) ) ) ).

% subset_mset.le_diff_conv2
thf(fact_1081_subset__mset_Ole__add__diff,axiom,
    ! [A2: multiset_nat,B2: multiset_nat,C: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( subseteq_mset_nat @ C @ ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B2 @ C ) @ A2 ) ) ) ).

% subset_mset.le_add_diff
thf(fact_1082_subset__mset_Ole__add__diff,axiom,
    ! [A2: multiset_a,B2: multiset_a,C: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( subseteq_mset_a @ C @ ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ B2 @ C ) @ A2 ) ) ) ).

% subset_mset.le_add_diff
thf(fact_1083_subset__mset_Odiff__add,axiom,
    ! [A2: multiset_nat,B2: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B2 )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% subset_mset.diff_add
thf(fact_1084_subset__mset_Odiff__add,axiom,
    ! [A2: multiset_a,B2: multiset_a] :
      ( ( subseteq_mset_a @ A2 @ B2 )
     => ( ( plus_plus_multiset_a @ ( minus_3765977307040488491iset_a @ B2 @ A2 ) @ A2 )
        = B2 ) ) ).

% subset_mset.diff_add
thf(fact_1085_subseteq__mset__imp__ns__mul__ext__Id,axiom,
    ! [A3: multiset_a,B3: multiset_a,R: set_Product_prod_a_a] :
      ( ( subseteq_mset_a @ A3 @ B3 )
     => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ B3 @ A3 ) @ ( multis4023584895248362547_ext_a @ id_a @ R ) ) ) ).

% subseteq_mset_imp_ns_mul_ext_Id
thf(fact_1086_reflcl__set__eq,axiom,
    ! [R2: set_Product_prod_a_a] :
      ( ( sup_sup_a_a_o
        @ ^ [X3: a,Y: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R2 )
        @ ^ [Y3: a,Z3: a] : ( Y3 = Z3 ) )
      = ( ^ [X3: a,Y: a] : ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( sup_su3048258781599657691od_a_a @ R2 @ id_a ) ) ) ) ).

% reflcl_set_eq
thf(fact_1087_reflcl__set__eq,axiom,
    ! [R2: set_Pr79727621955416071iset_a] :
      ( ( sup_su6917467955185334722et_a_o
        @ ^ [X3: multiset_a,Y: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ R2 )
        @ ^ [Y3: multiset_a,Z3: multiset_a] : ( Y3 = Z3 ) )
      = ( ^ [X3: multiset_a,Y: multiset_a] : ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X3 @ Y ) @ ( sup_su1925787886501348827iset_a @ R2 @ id_multiset_a ) ) ) ) ).

% reflcl_set_eq
thf(fact_1088_enumerate__map__upt,axiom,
    ! [N: nat,F: nat > multis3193088007478089820ropVar,M2: nat] :
      ( ( enumer7962022052585343041ropVar @ N @ ( map_na2466558649495876678ropVar @ F @ ( upt @ N @ M2 ) ) )
      = ( map_na8756600662017154455ropVar
        @ ^ [K: nat] : ( produc1720817013404193843ropVar @ K @ ( F @ K ) )
        @ ( upt @ N @ M2 ) ) ) ).

% enumerate_map_upt
thf(fact_1089_enumerate__map__upt,axiom,
    ! [N: nat,F: nat > nat,M2: nat] :
      ( ( enumerate_nat @ N @ ( map_nat_nat @ F @ ( upt @ N @ M2 ) ) )
      = ( map_na7298421622053143531at_nat
        @ ^ [K: nat] : ( product_Pair_nat_nat @ K @ ( F @ K ) )
        @ ( upt @ N @ M2 ) ) ) ).

% enumerate_map_upt
thf(fact_1090_enumerate__map__upt,axiom,
    ! [N: nat,F: nat > a,M2: nat] :
      ( ( enumerate_a @ N @ ( map_nat_a @ F @ ( upt @ N @ M2 ) ) )
      = ( map_na7918145567968794425_nat_a
        @ ^ [K: nat] : ( product_Pair_nat_a @ K @ ( F @ K ) )
        @ ( upt @ N @ M2 ) ) ) ).

% enumerate_map_upt
thf(fact_1091_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_1092_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_a,Ys: list_a] :
      ( ( enumerate_a @ N @ ( append_a @ Xs @ Ys ) )
      = ( append1694031006427026248_nat_a @ ( enumerate_a @ N @ Xs ) @ ( enumerate_a @ ( plus_plus_nat @ N @ ( size_size_list_a @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_1093_ns__mul__ext__IdI,axiom,
    ! [M: multiset_multiset_a,X5: multiset_multiset_a,Z6: multiset_multiset_a,N2: multiset_multiset_a,Y4: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
      ( ( M
        = ( plus_p6738641960240532161iset_a @ X5 @ Z6 ) )
     => ( ( N2
          = ( plus_p6738641960240532161iset_a @ Y4 @ Z6 ) )
       => ( ! [X4: multiset_a] :
              ( ( member_multiset_a @ X4 @ ( set_mset_multiset_a @ Y4 ) )
             => ? [Xa3: multiset_a] :
                  ( ( member_multiset_a @ Xa3 @ ( set_mset_multiset_a @ X5 ) )
                  & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa3 @ X4 ) @ R ) ) )
         => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N2 ) @ ( multis8112516946959054073iset_a @ id_multiset_a @ R ) ) ) ) ) ).

% ns_mul_ext_IdI
thf(fact_1094_ns__mul__ext__IdI,axiom,
    ! [M: multiset_nat,X5: multiset_nat,Z6: multiset_nat,N2: multiset_nat,Y4: multiset_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( M
        = ( plus_p6334493942879108393et_nat @ X5 @ Z6 ) )
     => ( ( N2
          = ( plus_p6334493942879108393et_nat @ Y4 @ Z6 ) )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( set_mset_nat @ Y4 ) )
             => ? [Xa3: nat] :
                  ( ( member_nat @ Xa3 @ ( set_mset_nat @ X5 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa3 @ X4 ) @ R ) ) )
         => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M @ N2 ) @ ( multis8831328596575508315xt_nat @ id_nat @ R ) ) ) ) ) ).

% ns_mul_ext_IdI
thf(fact_1095_ns__mul__ext__IdI,axiom,
    ! [M: multiset_a,X5: multiset_a,Z6: multiset_a,N2: multiset_a,Y4: multiset_a,R: set_Product_prod_a_a] :
      ( ( M
        = ( plus_plus_multiset_a @ X5 @ Z6 ) )
     => ( ( N2
          = ( plus_plus_multiset_a @ Y4 @ Z6 ) )
       => ( ! [X4: a] :
              ( ( member_a @ X4 @ ( set_mset_a @ Y4 ) )
             => ? [Xa3: a] :
                  ( ( member_a @ Xa3 @ ( set_mset_a @ X5 ) )
                  & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa3 @ X4 ) @ R ) ) )
         => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N2 ) @ ( multis4023584895248362547_ext_a @ id_a @ R ) ) ) ) ) ).

% ns_mul_ext_IdI
thf(fact_1096_ns__mul__ext__IdE,axiom,
    ! [M: multiset_multiset_a,N2: multiset_multiset_a,R: set_Pr79727621955416071iset_a] :
      ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ M @ N2 ) @ ( multis8112516946959054073iset_a @ id_multiset_a @ R ) )
     => ~ ! [X10: multiset_multiset_a,Y8: multiset_multiset_a,Z5: multiset_multiset_a] :
            ( ( M
              = ( plus_p6738641960240532161iset_a @ X10 @ Z5 ) )
           => ( ( N2
                = ( plus_p6738641960240532161iset_a @ Y8 @ Z5 ) )
             => ~ ! [X6: multiset_a] :
                    ( ( member_multiset_a @ X6 @ ( set_mset_multiset_a @ Y8 ) )
                   => ? [Xa2: multiset_a] :
                        ( ( member_multiset_a @ Xa2 @ ( set_mset_multiset_a @ X10 ) )
                        & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ).

% ns_mul_ext_IdE
thf(fact_1097_ns__mul__ext__IdE,axiom,
    ! [M: multiset_nat,N2: multiset_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M @ N2 ) @ ( multis8831328596575508315xt_nat @ id_nat @ R ) )
     => ~ ! [X10: multiset_nat,Y8: multiset_nat,Z5: multiset_nat] :
            ( ( M
              = ( plus_p6334493942879108393et_nat @ X10 @ Z5 ) )
           => ( ( N2
                = ( plus_p6334493942879108393et_nat @ Y8 @ Z5 ) )
             => ~ ! [X6: nat] :
                    ( ( member_nat @ X6 @ ( set_mset_nat @ Y8 ) )
                   => ? [Xa2: nat] :
                        ( ( member_nat @ Xa2 @ ( set_mset_nat @ X10 ) )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ).

% ns_mul_ext_IdE
thf(fact_1098_ns__mul__ext__IdE,axiom,
    ! [M: multiset_a,N2: multiset_a,R: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ M @ N2 ) @ ( multis4023584895248362547_ext_a @ id_a @ R ) )
     => ~ ! [X10: multiset_a,Y8: multiset_a,Z5: multiset_a] :
            ( ( M
              = ( plus_plus_multiset_a @ X10 @ Z5 ) )
           => ( ( N2
                = ( plus_plus_multiset_a @ Y8 @ Z5 ) )
             => ~ ! [X6: a] :
                    ( ( member_a @ X6 @ ( set_mset_a @ Y8 ) )
                   => ? [Xa2: a] :
                        ( ( member_a @ Xa2 @ ( set_mset_a @ X10 ) )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X6 ) @ R ) ) ) ) ) ) ).

% ns_mul_ext_IdE
thf(fact_1099_subset__mset_Osum__list__update,axiom,
    ! [K2: nat,Xs: list_multiset_nat,X: multiset_nat] :
      ( ( ord_less_nat @ K2 @ ( size_s6386657463320973636et_nat @ Xs ) )
     => ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ ( list_u3438943574295160626et_nat @ Xs @ K2 @ X ) )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) @ X ) @ ( nth_multiset_nat @ Xs @ K2 ) ) ) ) ).

% subset_mset.sum_list_update
thf(fact_1100_subset__mset_Osum__list__update,axiom,
    ! [K2: nat,Xs: list_multiset_a,X: multiset_a] :
      ( ( ord_less_nat @ K2 @ ( size_s3003218658903709366iset_a @ Xs ) )
     => ( ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ ( list_u788720842672076722iset_a @ Xs @ K2 @ X ) )
        = ( minus_3765977307040488491iset_a @ ( plus_plus_multiset_a @ ( groups778166972783649551iset_a @ plus_plus_multiset_a @ zero_zero_multiset_a @ Xs ) @ X ) @ ( nth_multiset_a @ Xs @ K2 ) ) ) ) ).

% subset_mset.sum_list_update
thf(fact_1101_nth__zip,axiom,
    ! [I2: nat,Xs: list_multiset_a,Ys: list_multiset_a] :
      ( ( ord_less_nat @ I2 @ ( size_s3003218658903709366iset_a @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_s3003218658903709366iset_a @ Ys ) )
       => ( ( nth_Pr1079437516805851566iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) @ I2 )
          = ( produc654756711066625303iset_a @ ( nth_multiset_a @ Xs @ I2 ) @ ( nth_multiset_a @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_1102_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I2 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_1103_nth__zip,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
       => ( ( nth_Pr2476257081389315164_nat_a @ ( zip_nat_a @ Xs @ Ys ) @ I2 )
          = ( product_Pair_nat_a @ ( nth_nat @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_1104_nth__zip,axiom,
    ! [I2: nat,Xs: list_a,Ys: list_nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr8461465654520414006_a_nat @ ( zip_a_nat @ Xs @ Ys ) @ I2 )
          = ( product_Pair_a_nat @ ( nth_a @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_1105_nth__zip,axiom,
    ! [I2: nat,Xs: list_a,Ys: list_a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
       => ( ( nth_Product_prod_a_a @ ( zip_a_a @ Xs @ Ys ) @ I2 )
          = ( product_Pair_a_a @ ( nth_a @ Xs @ I2 ) @ ( nth_a @ Ys @ I2 ) ) ) ) ) ).

% nth_zip
thf(fact_1106_s__mul__extI,axiom,
    ! [A3: multis193838646080965794ropVar,A1: multis193838646080965794ropVar,A22: multis193838646080965794ropVar,B3: multis193838646080965794ropVar,B1: multis193838646080965794ropVar,B22: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( A3
        = ( plus_p1339091108561003481ropVar @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p1339091108561003481ropVar @ B1 @ B22 ) )
       => ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A1 @ B1 ) @ ( multis9091339148773653984ropVar @ Ns ) )
         => ( ( A22 != zero_z6226653793019358809ropVar )
           => ( ! [B4: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ B4 @ ( set_ms1147286257884310937ropVar @ B22 ) )
                 => ? [A6: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B4 ) @ S2 ) ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A3 @ B3 ) @ ( multis4691071231837107059ropVar @ Ns @ S2 ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_1107_s__mul__extI,axiom,
    ! [A3: multis501812127501805293od_a_a,A1: multis501812127501805293od_a_a,A22: multis501812127501805293od_a_a,B3: multis501812127501805293od_a_a,B1: multis501812127501805293od_a_a,B22: multis501812127501805293od_a_a,Ns: set_Pr8600417178894128327od_a_a,S2: set_Pr8600417178894128327od_a_a] :
      ( ( A3
        = ( plus_p6967097483979887908od_a_a @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p6967097483979887908od_a_a @ B1 @ B22 ) )
       => ( ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A1 @ B1 ) @ ( multis1821517007024690859od_a_a @ Ns ) )
         => ( ( A22 != zero_z3024502163831676836od_a_a )
           => ( ! [B4: product_prod_a_a] :
                  ( ( member1426531477525435216od_a_a @ B4 @ ( set_ms119794563918438244od_a_a @ B22 ) )
                 => ? [A6: product_prod_a_a] :
                      ( ( member1426531477525435216od_a_a @ A6 @ ( set_ms119794563918438244od_a_a @ A22 ) )
                      & ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member1567053331504362128od_a_a @ ( produc6961845650964105687od_a_a @ A3 @ B3 ) @ ( multis6523615221270609726od_a_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_1108_s__mul__extI,axiom,
    ! [A3: multis4781622204924204013iset_a,A1: multis4781622204924204013iset_a,A22: multis4781622204924204013iset_a,B3: multis4781622204924204013iset_a,B1: multis4781622204924204013iset_a,B22: multis4781622204924204013iset_a,Ns: set_Pr3059988933816007879iset_a,S2: set_Pr3059988933816007879iset_a] :
      ( ( A3
        = ( plus_p1053901119268525860iset_a @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p1053901119268525860iset_a @ B1 @ B22 ) )
       => ( ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A1 @ B1 ) @ ( multis3430030276581740203iset_a @ Ns ) )
         => ( ( A22 != zero_z4591712717340762532iset_a )
           => ( ! [B4: produc6518373309651786023iset_a] :
                  ( ( member5199237121806060112iset_a @ B4 @ ( set_ms2379723421352415844iset_a @ B22 ) )
                 => ? [A6: produc6518373309651786023iset_a] :
                      ( ( member5199237121806060112iset_a @ A6 @ ( set_ms2379723421352415844iset_a @ A22 ) )
                      & ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member8574442277963079824iset_a @ ( produc1739639915645390807iset_a @ A3 @ B3 ) @ ( multis6482401367445528638iset_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_1109_s__mul__extI,axiom,
    ! [A3: multiset_multiset_a,A1: multiset_multiset_a,A22: multiset_multiset_a,B3: multiset_multiset_a,B1: multiset_multiset_a,B22: multiset_multiset_a,Ns: set_Pr79727621955416071iset_a,S2: set_Pr79727621955416071iset_a] :
      ( ( A3
        = ( plus_p6738641960240532161iset_a @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p6738641960240532161iset_a @ B1 @ B22 ) )
       => ( ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A1 @ B1 ) @ ( multis451435756853798856iset_a @ Ns ) )
         => ( ( A22 != zero_z7799948378220188993iset_a )
           => ( ! [B4: multiset_a] :
                  ( ( member_multiset_a @ B4 @ ( set_mset_multiset_a @ B22 ) )
                 => ? [A6: multiset_a] :
                      ( ( member_multiset_a @ A6 @ ( set_mset_multiset_a @ A22 ) )
                      & ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member1345422370561994576iset_a @ ( produc8381460346935720215iset_a @ A3 @ B3 ) @ ( multis7166037729339985883iset_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_1110_s__mul__extI,axiom,
    ! [A3: multiset_nat,A1: multiset_nat,A22: multiset_nat,B3: multiset_nat,B1: multiset_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( A3
        = ( plus_p6334493942879108393et_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p6334493942879108393et_nat @ B1 @ B22 ) )
       => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A1 @ B1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
         => ( ( A22 != zero_z7348594199698428585et_nat )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( set_mset_nat @ B22 ) )
                 => ? [A6: nat] :
                      ( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B4 ) @ S2 ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A3 @ B3 ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_1111_s__mul__extI,axiom,
    ! [A3: multiset_a,A1: multiset_a,A22: multiset_a,B3: multiset_a,B1: multiset_a,B22: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( A3
        = ( plus_plus_multiset_a @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_plus_multiset_a @ B1 @ B22 ) )
       => ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A1 @ B1 ) @ ( multis2373687813370458242ltpw_a @ Ns ) )
         => ( ( A22 != zero_zero_multiset_a )
           => ( ! [B4: a] :
                  ( ( member_a @ B4 @ ( set_mset_a @ B22 ) )
                 => ? [A6: a] :
                      ( ( member_a @ A6 @ ( set_mset_a @ A22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ B4 ) @ S2 ) ) )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A3 @ B3 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_1112_DiffI,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ A3 )
     => ( ~ ( member_nat @ C @ B3 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_1113_DiffI,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ A3 )
     => ( ~ ( member8421201611525043589ropVar @ C @ B3 )
       => ( member8421201611525043589ropVar @ C @ ( minus_2621998875110806563ropVar @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_1114_DiffI,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ A3 )
     => ( ~ ( member1426531477525435216od_a_a @ C @ B3 )
       => ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_1115_DiffI,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ A3 )
     => ( ~ ( member_a @ C @ B3 )
       => ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_1116_DiffI,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ A3 )
     => ( ~ ( member5199237121806060112iset_a @ C @ B3 )
       => ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_1117_Diff__iff,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
      = ( ( member_nat @ C @ A3 )
        & ~ ( member_nat @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_1118_Diff__iff,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ ( minus_2621998875110806563ropVar @ A3 @ B3 ) )
      = ( ( member8421201611525043589ropVar @ C @ A3 )
        & ~ ( member8421201611525043589ropVar @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_1119_Diff__iff,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B3 ) )
      = ( ( member1426531477525435216od_a_a @ C @ A3 )
        & ~ ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_1120_Diff__iff,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
      = ( ( member_a @ C @ A3 )
        & ~ ( member_a @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_1121_Diff__iff,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B3 ) )
      = ( ( member5199237121806060112iset_a @ C @ A3 )
        & ~ ( member5199237121806060112iset_a @ C @ B3 ) ) ) ).

% Diff_iff
thf(fact_1122_list__update__nonempty,axiom,
    ! [Xs: list_a,K2: nat,X: a] :
      ( ( ( list_update_a @ Xs @ K2 @ X )
        = nil_a )
      = ( Xs = nil_a ) ) ).

% list_update_nonempty
thf(fact_1123_list__update__nonempty,axiom,
    ! [Xs: list_nat,K2: nat,X: nat] :
      ( ( ( list_update_nat @ Xs @ K2 @ X )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_1124_length__list__update,axiom,
    ! [Xs: list_nat,I2: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I2 @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_1125_length__list__update,axiom,
    ! [Xs: list_a,I2: nat,X: a] :
      ( ( size_size_list_a @ ( list_update_a @ Xs @ I2 @ X ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_list_update
thf(fact_1126_list__update__id,axiom,
    ! [Xs: list_nat,I2: nat] :
      ( ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ I2 ) )
      = Xs ) ).

% list_update_id
thf(fact_1127_list__update__id,axiom,
    ! [Xs: list_a,I2: nat] :
      ( ( list_update_a @ Xs @ I2 @ ( nth_a @ Xs @ I2 ) )
      = Xs ) ).

% list_update_id
thf(fact_1128_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I2 != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1129_nth__list__update__neq,axiom,
    ! [I2: nat,J: nat,Xs: list_a,X: a] :
      ( ( I2 != J )
     => ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X ) @ J )
        = ( nth_a @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_1130_drop__update__cancel,axiom,
    ! [N: nat,M2: nat,Xs: list_M6740372356978808802ropVar,X: multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( drop_M2293212096379261346ropVar @ M2 @ ( list_u5207168704986282698ropVar @ Xs @ N @ X ) )
        = ( drop_M2293212096379261346ropVar @ M2 @ Xs ) ) ) ).

% drop_update_cancel
thf(fact_1131_drop__update__cancel,axiom,
    ! [N: nat,M2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( drop_nat @ M2 @ ( list_update_nat @ Xs @ N @ X ) )
        = ( drop_nat @ M2 @ Xs ) ) ) ).

% drop_update_cancel
thf(fact_1132_Nil__eq__zip__iff,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( nil_Product_prod_a_a
        = ( zip_a_a @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        | ( Ys = nil_a ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1133_Nil__eq__zip__iff,axiom,
    ! [Xs: list_a,Ys: list_nat] :
      ( ( nil_Pr7402525243500994295_a_nat
        = ( zip_a_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_a )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1134_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_a] :
      ( ( nil_Pr1417316670369895453_nat_a
        = ( zip_nat_a @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_a ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1135_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1136_zip__eq__Nil__iff,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( zip_a_a @ Xs @ Ys )
        = nil_Product_prod_a_a )
      = ( ( Xs = nil_a )
        | ( Ys = nil_a ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1137_zip__eq__Nil__iff,axiom,
    ! [Xs: list_a,Ys: list_nat] :
      ( ( ( zip_a_nat @ Xs @ Ys )
        = nil_Pr7402525243500994295_a_nat )
      = ( ( Xs = nil_a )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1138_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_a] :
      ( ( ( zip_nat_a @ Xs @ Ys )
        = nil_Pr1417316670369895453_nat_a )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_a ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1139_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1140_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1141_nth__list__update__eq,axiom,
    ! [I2: nat,Xs: list_a,X: a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X ) @ I2 )
        = X ) ) ).

% nth_list_update_eq
thf(fact_1142_zip__append,axiom,
    ! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs2: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs2 ) )
        = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs2 ) ) ) ) ).

% zip_append
thf(fact_1143_zip__append,axiom,
    ! [Xs: list_nat,Us: list_a,Ys: list_nat,Vs2: list_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Us ) )
     => ( ( zip_nat_a @ ( append_nat @ Xs @ Ys ) @ ( append_a @ Us @ Vs2 ) )
        = ( append1694031006427026248_nat_a @ ( zip_nat_a @ Xs @ Us ) @ ( zip_nat_a @ Ys @ Vs2 ) ) ) ) ).

% zip_append
thf(fact_1144_zip__append,axiom,
    ! [Xs: list_a,Us: list_nat,Ys: list_a,Vs2: list_nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_a_nat @ ( append_a @ Xs @ Ys ) @ ( append_nat @ Us @ Vs2 ) )
        = ( append7679239579558125090_a_nat @ ( zip_a_nat @ Xs @ Us ) @ ( zip_a_nat @ Ys @ Vs2 ) ) ) ) ).

% zip_append
thf(fact_1145_zip__append,axiom,
    ! [Xs: list_a,Us: list_a,Ys: list_a,Vs2: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Us ) )
     => ( ( zip_a_a @ ( append_a @ Xs @ Ys ) @ ( append_a @ Us @ Vs2 ) )
        = ( append5335208819046833346od_a_a @ ( zip_a_a @ Xs @ Us ) @ ( zip_a_a @ Ys @ Vs2 ) ) ) ) ).

% zip_append
thf(fact_1146_set__swap,axiom,
    ! [I2: nat,Xs: list_M6740372356978808802ropVar,J: nat] :
      ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s1943289338788136526ropVar @ Xs ) )
       => ( ( set_Mu146256361055586289ropVar @ ( list_u5207168704986282698ropVar @ ( list_u5207168704986282698ropVar @ Xs @ I2 @ ( nth_Mu4537670848183934179ropVar @ Xs @ J ) ) @ J @ ( nth_Mu4537670848183934179ropVar @ Xs @ I2 ) ) )
          = ( set_Mu146256361055586289ropVar @ Xs ) ) ) ) ).

% set_swap
thf(fact_1147_set__swap,axiom,
    ! [I2: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I2 ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_1148_set__swap,axiom,
    ! [I2: nat,Xs: list_a,J: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
       => ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I2 @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I2 ) ) )
          = ( set_a2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_1149_distinct__swap,axiom,
    ! [I2: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I2 @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I2 ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_1150_distinct__swap,axiom,
    ! [I2: nat,Xs: list_a,J: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
       => ( ( distinct_a @ ( list_update_a @ ( list_update_a @ Xs @ I2 @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I2 ) ) )
          = ( distinct_a @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_1151_DiffE,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
     => ~ ( ( member_nat @ C @ A3 )
         => ( member_nat @ C @ B3 ) ) ) ).

% DiffE
thf(fact_1152_DiffE,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ ( minus_2621998875110806563ropVar @ A3 @ B3 ) )
     => ~ ( ( member8421201611525043589ropVar @ C @ A3 )
         => ( member8421201611525043589ropVar @ C @ B3 ) ) ) ).

% DiffE
thf(fact_1153_DiffE,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B3 ) )
     => ~ ( ( member1426531477525435216od_a_a @ C @ A3 )
         => ( member1426531477525435216od_a_a @ C @ B3 ) ) ) ).

% DiffE
thf(fact_1154_DiffE,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
     => ~ ( ( member_a @ C @ A3 )
         => ( member_a @ C @ B3 ) ) ) ).

% DiffE
thf(fact_1155_DiffE,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B3 ) )
     => ~ ( ( member5199237121806060112iset_a @ C @ A3 )
         => ( member5199237121806060112iset_a @ C @ B3 ) ) ) ).

% DiffE
thf(fact_1156_DiffD1,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
     => ( member_nat @ C @ A3 ) ) ).

% DiffD1
thf(fact_1157_DiffD1,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ ( minus_2621998875110806563ropVar @ A3 @ B3 ) )
     => ( member8421201611525043589ropVar @ C @ A3 ) ) ).

% DiffD1
thf(fact_1158_DiffD1,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B3 ) )
     => ( member1426531477525435216od_a_a @ C @ A3 ) ) ).

% DiffD1
thf(fact_1159_DiffD1,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
     => ( member_a @ C @ A3 ) ) ).

% DiffD1
thf(fact_1160_DiffD1,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B3 ) )
     => ( member5199237121806060112iset_a @ C @ A3 ) ) ).

% DiffD1
thf(fact_1161_DiffD2,axiom,
    ! [C: nat,A3: set_nat,B3: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A3 @ B3 ) )
     => ~ ( member_nat @ C @ B3 ) ) ).

% DiffD2
thf(fact_1162_DiffD2,axiom,
    ! [C: multis3193088007478089820ropVar,A3: set_Mu5658373669842198716ropVar,B3: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C @ ( minus_2621998875110806563ropVar @ A3 @ B3 ) )
     => ~ ( member8421201611525043589ropVar @ C @ B3 ) ) ).

% DiffD2
thf(fact_1163_DiffD2,axiom,
    ! [C: product_prod_a_a,A3: set_Product_prod_a_a,B3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ C @ ( minus_6817036919807184750od_a_a @ A3 @ B3 ) )
     => ~ ( member1426531477525435216od_a_a @ C @ B3 ) ) ).

% DiffD2
thf(fact_1164_DiffD2,axiom,
    ! [C: a,A3: set_a,B3: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A3 @ B3 ) )
     => ~ ( member_a @ C @ B3 ) ) ).

% DiffD2
thf(fact_1165_DiffD2,axiom,
    ! [C: produc6518373309651786023iset_a,A3: set_Pr79727621955416071iset_a,B3: set_Pr79727621955416071iset_a] :
      ( ( member5199237121806060112iset_a @ C @ ( minus_2461365523287052142iset_a @ A3 @ B3 ) )
     => ~ ( member5199237121806060112iset_a @ C @ B3 ) ) ).

% DiffD2
thf(fact_1166_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X3: nat] :
              ( ( member_nat @ X3 @ A5 )
              & ~ ( member_nat @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1167_set__diff__eq,axiom,
    ( minus_2621998875110806563ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
          ( collec8658135508005558727ropVar
          @ ^ [X3: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X3 @ A5 )
              & ~ ( member8421201611525043589ropVar @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1168_set__diff__eq,axiom,
    ( minus_6817036919807184750od_a_a
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
          ( collec3336397797384452498od_a_a
          @ ^ [X3: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X3 @ A5 )
              & ~ ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1169_set__diff__eq,axiom,
    ( minus_minus_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ^ [X3: a] :
              ( ( member_a @ X3 @ A5 )
              & ~ ( member_a @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1170_set__diff__eq,axiom,
    ( minus_2461365523287052142iset_a
    = ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
          ( collec6344740439120713362iset_a
          @ ^ [X3: produc6518373309651786023iset_a] :
              ( ( member5199237121806060112iset_a @ X3 @ A5 )
              & ~ ( member5199237121806060112iset_a @ X3 @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1171_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X3: nat] : ( member_nat @ X3 @ A5 )
            @ ^ [X3: nat] : ( member_nat @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_1172_minus__set__def,axiom,
    ( minus_2621998875110806563ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
          ( collec8658135508005558727ropVar
          @ ( minus_1245740275806398586pVar_o
            @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ A5 )
            @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_1173_minus__set__def,axiom,
    ( minus_6817036919807184750od_a_a
    = ( ^ [A5: set_Product_prod_a_a,B5: set_Product_prod_a_a] :
          ( collec3336397797384452498od_a_a
          @ ( minus_4793868396798367471_a_a_o
            @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ A5 )
            @ ^ [X3: product_prod_a_a] : ( member1426531477525435216od_a_a @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_1174_minus__set__def,axiom,
    ( minus_minus_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ( minus_minus_a_o
            @ ^ [X3: a] : ( member_a @ X3 @ A5 )
            @ ^ [X3: a] : ( member_a @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_1175_minus__set__def,axiom,
    ( minus_2461365523287052142iset_a
    = ( ^ [A5: set_Pr79727621955416071iset_a,B5: set_Pr79727621955416071iset_a] :
          ( collec6344740439120713362iset_a
          @ ( minus_1780674721092442863et_a_o
            @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ A5 )
            @ ^ [X3: produc6518373309651786023iset_a] : ( member5199237121806060112iset_a @ X3 @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_1176_zip__same__conv__map,axiom,
    ! [Xs: list_a] :
      ( ( zip_a_a @ Xs @ Xs )
      = ( map_a_7860052162900579309od_a_a
        @ ^ [X3: a] : ( product_Pair_a_a @ X3 @ X3 )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_1177_zip__same__conv__map,axiom,
    ! [Xs: list_multiset_a] :
      ( ( zip_mu5258324667929623607iset_a @ Xs @ Xs )
      = ( map_mu8477655255319028135iset_a
        @ ^ [X3: multiset_a] : ( produc654756711066625303iset_a @ X3 @ X3 )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_1178_zip__update,axiom,
    ! [Xs: list_a,I2: nat,X: a,Ys: list_a,Y2: a] :
      ( ( zip_a_a @ ( list_update_a @ Xs @ I2 @ X ) @ ( list_update_a @ Ys @ I2 @ Y2 ) )
      = ( list_u4967751623001292181od_a_a @ ( zip_a_a @ Xs @ Ys ) @ I2 @ ( product_Pair_a_a @ X @ Y2 ) ) ) ).

% zip_update
thf(fact_1179_zip__update,axiom,
    ! [Xs: list_multiset_a,I2: nat,X: multiset_a,Ys: list_multiset_a,Y2: multiset_a] :
      ( ( zip_mu5258324667929623607iset_a @ ( list_u788720842672076722iset_a @ Xs @ I2 @ X ) @ ( list_u788720842672076722iset_a @ Ys @ I2 @ Y2 ) )
      = ( list_u2687166788783756181iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) @ I2 @ ( produc654756711066625303iset_a @ X @ Y2 ) ) ) ).

% zip_update
thf(fact_1180_map__update,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,K2: nat,Y2: nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( list_update_nat @ Xs @ K2 @ Y2 ) )
      = ( list_u5207168704986282698ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ K2 @ ( F @ Y2 ) ) ) ).

% map_update
thf(fact_1181_map__update,axiom,
    ! [F: nat > nat,Xs: list_nat,K2: nat,Y2: nat] :
      ( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K2 @ Y2 ) )
      = ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K2 @ ( F @ Y2 ) ) ) ).

% map_update
thf(fact_1182_map__update,axiom,
    ! [F: nat > a,Xs: list_nat,K2: nat,Y2: nat] :
      ( ( map_nat_a @ F @ ( list_update_nat @ Xs @ K2 @ Y2 ) )
      = ( list_update_a @ ( map_nat_a @ F @ Xs ) @ K2 @ ( F @ Y2 ) ) ) ).

% map_update
thf(fact_1183_list__update_Osimps_I1_J,axiom,
    ! [I2: nat,V: a] :
      ( ( list_update_a @ nil_a @ I2 @ V )
      = nil_a ) ).

% list_update.simps(1)
thf(fact_1184_list__update_Osimps_I1_J,axiom,
    ! [I2: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I2 @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_1185_list__update__code_I1_J,axiom,
    ! [I2: nat,Y2: a] :
      ( ( list_update_a @ nil_a @ I2 @ Y2 )
      = nil_a ) ).

% list_update_code(1)
thf(fact_1186_list__update__code_I1_J,axiom,
    ! [I2: nat,Y2: nat] :
      ( ( list_update_nat @ nil_nat @ I2 @ Y2 )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_1187_drop__zip,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( drop_P7722043279991717613ropVar @ N @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) )
      = ( zip_Mu2706791923038522295ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) @ ( drop_M2293212096379261346ropVar @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_1188_drop__zip,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,Ys: list_nat] :
      ( ( drop_P2520701842345426197ar_nat @ N @ ( zip_Mu6626014532916887989ar_nat @ Xs @ Ys ) )
      = ( zip_Mu6626014532916887989ar_nat @ ( drop_M2293212096379261346ropVar @ N @ Xs ) @ ( drop_nat @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_1189_drop__zip,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_M6740372356978808802ropVar] :
      ( ( drop_P4751389034305084731ropVar @ N @ ( zip_na7122055390739322131ropVar @ Xs @ Ys ) )
      = ( zip_na7122055390739322131ropVar @ ( drop_nat @ N @ Xs ) @ ( drop_M2293212096379261346ropVar @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_1190_drop__zip,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_P8868858903918902087at_nat @ N @ ( zip_nat_nat @ Xs @ Ys ) )
      = ( zip_nat_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_1191_set__zip__rightD,axiom,
    ! [X: a,Y2: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1192_set__zip__rightD,axiom,
    ! [X: multiset_a,Y2: multiset_a,Xs: list_multiset_a,Ys: list_multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y2 ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) )
     => ( member_multiset_a @ Y2 @ ( set_multiset_a2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1193_set__zip__leftD,axiom,
    ! [X: a,Y2: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1194_set__zip__leftD,axiom,
    ! [X: multiset_a,Y2: multiset_a,Xs: list_multiset_a,Ys: list_multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y2 ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) )
     => ( member_multiset_a @ X @ ( set_multiset_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1195_in__set__zipE,axiom,
    ! [X: nat,Y2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1196_in__set__zipE,axiom,
    ! [X: nat,Y2: multis3193088007478089820ropVar,Xs: list_nat,Ys: list_M6740372356978808802ropVar] :
      ( ( member1824086966109417880ropVar @ ( produc1720817013404193843ropVar @ X @ Y2 ) @ ( set_Pr1649027773232293932ropVar @ ( zip_na7122055390739322131ropVar @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1197_in__set__zipE,axiom,
    ! [X: nat,Y2: a,Xs: list_nat,Ys: list_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y2 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1198_in__set__zipE,axiom,
    ! [X: multis3193088007478089820ropVar,Y2: nat,Xs: list_M6740372356978808802ropVar,Ys: list_nat] :
      ( ( member8816771811004535154ar_nat @ ( produc1224776155581759701ar_nat @ X @ Y2 ) @ ( set_Pr8641712618127411206ar_nat @ ( zip_Mu6626014532916887989ar_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1199_in__set__zipE,axiom,
    ! [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y2 ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) )
     => ~ ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1200_in__set__zipE,axiom,
    ! [X: multis3193088007478089820ropVar,Y2: a,Xs: list_M6740372356978808802ropVar,Ys: list_a] :
      ( ( member3180940680879922546pVar_a @ ( produc3140662905487486265pVar_a @ X @ Y2 ) @ ( set_Pr1829365067786544094pVar_a @ ( zip_Mu1062270463881110617pVar_a @ Xs @ Ys ) ) )
     => ~ ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1201_in__set__zipE,axiom,
    ! [X: a,Y2: nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y2 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1202_in__set__zipE,axiom,
    ! [X: a,Y2: multis3193088007478089820ropVar,Xs: list_a,Ys: list_M6740372356978808802ropVar] :
      ( ( member5951705825339765678ropVar @ ( produc3383013678355495541ropVar @ X @ Y2 ) @ ( set_Pr4600130212246387226ropVar @ ( zip_a_1304621236749119893ropVar @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ~ ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1203_in__set__zipE,axiom,
    ! [X: a,Y2: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a @ X @ ( set_a2 @ Xs ) )
         => ~ ( member_a @ Y2 @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1204_in__set__zipE,axiom,
    ! [X: product_prod_a_a,Y2: nat,Xs: list_P1396940483166286381od_a_a,Ys: list_nat] :
      ( ( member1268613965720311869_a_nat @ ( produc6483765539767234976_a_nat @ X @ Y2 ) @ ( set_Pr6407419057626374865_a_nat @ ( zip_Pr4385798588157923968_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
         => ~ ( member_nat @ Y2 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1205_zip__same,axiom,
    ! [A2: product_prod_a_a,B2: product_prod_a_a,Xs: list_P1396940483166286381od_a_a] :
      ( ( member6330455413206600464od_a_a @ ( produc7886510207707329367od_a_a @ A2 @ B2 ) @ ( set_Pr6493056779096584572od_a_a @ ( zip_Pr1329156589218441847od_a_a @ Xs @ Xs ) ) )
      = ( ( member1426531477525435216od_a_a @ A2 @ ( set_Product_prod_a_a2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1206_zip__same,axiom,
    ! [A2: produc6518373309651786023iset_a,B2: produc6518373309651786023iset_a,Xs: list_P6988024003011616557iset_a] :
      ( ( member1562461016398818064iset_a @ ( produc2591098542590733143iset_a @ A2 @ B2 ) @ ( set_Pr3130690497449335164iset_a @ ( zip_Pr8115149548399026807iset_a @ Xs @ Xs ) ) )
      = ( ( member5199237121806060112iset_a @ A2 @ ( set_Pr5550698219922361532iset_a @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1207_zip__same,axiom,
    ! [A2: nat,B2: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A2 @ ( set_nat2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1208_zip__same,axiom,
    ! [A2: multis3193088007478089820ropVar,B2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A2 @ B2 ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Xs ) ) )
      = ( ( member8421201611525043589ropVar @ A2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1209_zip__same,axiom,
    ! [A2: a,B2: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
      = ( ( member_a @ A2 @ ( set_a2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1210_zip__same,axiom,
    ! [A2: multiset_a,B2: multiset_a,Xs: list_multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A2 @ B2 ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Xs ) ) )
      = ( ( member_multiset_a @ A2 @ ( set_multiset_a2 @ Xs ) )
        & ( A2 = B2 ) ) ) ).

% zip_same
thf(fact_1211_ex__mset__zip__left,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xs4: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( mset_nat @ Xs4 )
          = ( mset_nat @ Xs ) )
       => ? [Ys5: list_nat] :
            ( ( ( size_size_list_nat @ Ys5 )
              = ( size_size_list_nat @ Xs4 ) )
            & ( ( mset_P6383711406899277590at_nat @ ( zip_nat_nat @ Xs4 @ Ys5 ) )
              = ( mset_P6383711406899277590at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_1212_ex__mset__zip__left,axiom,
    ! [Xs: list_nat,Ys: list_a,Xs4: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( mset_nat @ Xs4 )
          = ( mset_nat @ Xs ) )
       => ? [Ys5: list_a] :
            ( ( ( size_size_list_a @ Ys5 )
              = ( size_size_list_nat @ Xs4 ) )
            & ( ( mset_P3740496182239305934_nat_a @ ( zip_nat_a @ Xs4 @ Ys5 ) )
              = ( mset_P3740496182239305934_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_1213_ex__mset__zip__left,axiom,
    ! [Xs: list_a,Ys: list_nat,Xs4: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( mset_a @ Xs4 )
          = ( mset_a @ Xs ) )
       => ? [Ys5: list_nat] :
            ( ( ( size_size_list_nat @ Ys5 )
              = ( size_size_list_a @ Xs4 ) )
            & ( ( mset_P502332718515628968_a_nat @ ( zip_a_nat @ Xs4 @ Ys5 ) )
              = ( mset_P502332718515628968_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_1214_ex__mset__zip__left,axiom,
    ! [Xs: list_a,Ys: list_a,Xs4: list_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( ( mset_a @ Xs4 )
          = ( mset_a @ Xs ) )
       => ? [Ys5: list_a] :
            ( ( ( size_size_list_a @ Ys5 )
              = ( size_size_list_a @ Xs4 ) )
            & ( ( mset_P3709024939784119484od_a_a @ ( zip_a_a @ Xs4 @ Ys5 ) )
              = ( mset_P3709024939784119484od_a_a @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_1215_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr6810726163584182713ropVar @ ( produc1416661995297908400ropVar @ H ) @ ( zip_Mu2706791923038522295ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1216_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr6582868597391307571ar_nat @ ( produc7431473831949406332ar_nat @ H ) @ ( zip_Mu2706791923038522295ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1217_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > multis3193088007478089820ropVar > a,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr2353050391483111003pVar_a @ ( produc7971905830127696210pVar_a @ H ) @ ( zip_Mu2706791923038522295ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_nat_a
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1218_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > nat > multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr193826012314142715ropVar @ ( produc7927514689771840474ropVar @ H ) @ ( zip_Mu6626014532916887989ar_nat @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1219_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > nat > nat,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr1531560619341599857at_nat @ ( produc2601766963269458130at_nat @ H ) @ ( zip_Mu6626014532916887989ar_nat @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1220_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > nat > a,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr7584183961379211421_nat_a @ ( produc6515848207870728828_nat_a @ H ) @ ( zip_Mu6626014532916887989ar_nat @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_a
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1221_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > a > multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > a] :
      ( ( map_Pr7588290962900681367ropVar @ ( produc8214256602995705486ropVar @ H ) @ ( zip_Mu1062270463881110617pVar_a @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_a @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1222_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > a > nat,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > a] :
      ( ( map_Pr755273659744948949_a_nat @ ( produc2483916706692442398_a_nat @ H ) @ ( zip_Mu1062270463881110617pVar_a @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_a @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1223_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > a > a,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > a] :
      ( ( map_Pr151777675724692281ar_a_a @ ( produc7434819305959638320ar_a_a @ H ) @ ( zip_Mu1062270463881110617pVar_a @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_a @ G @ Xs ) ) )
      = ( map_nat_a
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1224_map2__map__map,axiom,
    ! [H: nat > multis3193088007478089820ropVar > multis3193088007478089820ropVar,F: nat > nat,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr1062706098098207061ropVar @ ( produc222539320033202172ropVar @ H ) @ ( zip_na7122055390739322131ropVar @ ( map_nat_nat @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X3: nat] : ( H @ ( F @ X3 ) @ ( G @ X3 ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_1225_zip__commute,axiom,
    ( zip_a_a
    = ( ^ [Xs2: list_a,Ys4: list_a] :
          ( map_Pr7904243085458786820od_a_a
          @ ( produc408267641121961211od_a_a
            @ ^ [X3: a,Y: a] : ( product_Pair_a_a @ Y @ X3 ) )
          @ ( zip_a_a @ Ys4 @ Xs2 ) ) ) ) ).

% zip_commute
thf(fact_1226_zip__commute,axiom,
    ( zip_mu5258324667929623607iset_a
    = ( ^ [Xs2: list_multiset_a,Ys4: list_multiset_a] :
          ( map_Pr5905680810515755012iset_a
          @ ( produc7821597265843428091iset_a
            @ ^ [X3: multiset_a,Y: multiset_a] : ( produc654756711066625303iset_a @ Y @ X3 ) )
          @ ( zip_mu5258324667929623607iset_a @ Ys4 @ Xs2 ) ) ) ) ).

% zip_commute
thf(fact_1227_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_nat,X: multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ~ ! [Y5: nat] :
              ~ ( member8816771811004535154ar_nat @ ( produc1224776155581759701ar_nat @ X @ Y5 ) @ ( set_Pr8641712618127411206ar_nat @ ( zip_Mu6626014532916887989ar_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1228_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_a,X: multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ~ ! [Y5: a] :
              ~ ( member3180940680879922546pVar_a @ ( produc3140662905487486265pVar_a @ X @ Y5 ) @ ( set_Pr1829365067786544094pVar_a @ ( zip_Mu1062270463881110617pVar_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1229_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ~ ! [Y5: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y5 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1230_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_a,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ~ ! [Y5: a] :
              ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X @ Y5 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1231_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_nat,X: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_a @ X @ ( set_a2 @ Xs ) )
       => ~ ! [Y5: nat] :
              ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X @ Y5 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1232_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_a,X: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ X @ ( set_a2 @ Xs ) )
       => ~ ! [Y5: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y5 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1233_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_multiset_a,Ys: list_multiset_a,X: multiset_a] :
      ( ( ( size_s3003218658903709366iset_a @ Xs )
        = ( size_s3003218658903709366iset_a @ Ys ) )
     => ( ( member_multiset_a @ X @ ( set_multiset_a2 @ Xs ) )
       => ~ ! [Y5: multiset_a] :
              ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X @ Y5 ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1234_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,Ys: list_nat,X: product_prod_a_a] :
      ( ( ( size_s3885678630836030617od_a_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
       => ~ ! [Y5: nat] :
              ~ ( member1268613965720311869_a_nat @ ( produc6483765539767234976_a_nat @ X @ Y5 ) @ ( set_Pr6407419057626374865_a_nat @ ( zip_Pr4385798588157923968_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1235_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P1396940483166286381od_a_a,Ys: list_a,X: product_prod_a_a] :
      ( ( ( size_s3885678630836030617od_a_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs ) )
       => ~ ! [Y5: a] :
              ~ ( member2829916517802586983_a_a_a @ ( produc5692536848587512110_a_a_a @ X @ Y5 ) @ ( set_Pr6767713769399854547_a_a_a @ ( zip_Pr9109025954503487566_a_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1236_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P6988024003011616557iset_a,Ys: list_nat,X: produc6518373309651786023iset_a] :
      ( ( ( size_s7238940367606365593iset_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ Xs ) )
       => ~ ! [Y5: nat] :
              ~ ( member4256827664654810685_a_nat @ ( produc3833396812392740256_a_nat @ X @ Y5 ) @ ( set_Pr7251121483129668817_a_nat @ ( zip_Pr5738322098899826304_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1237_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_M6740372356978808802ropVar,Y2: multis3193088007478089820ropVar] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1943289338788136526ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member1824086966109417880ropVar @ ( produc1720817013404193843ropVar @ X4 @ Y2 ) @ ( set_Pr1649027773232293932ropVar @ ( zip_na7122055390739322131ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1238_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Y2: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1239_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_a,Y2: a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X4 @ Y2 ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1240_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_M6740372356978808802ropVar,Y2: multis3193088007478089820ropVar] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s1943289338788136526ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member5951705825339765678ropVar @ ( produc3383013678355495541ropVar @ X4 @ Y2 ) @ ( set_Pr4600130212246387226ropVar @ ( zip_a_1304621236749119893ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1241_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_nat,Y2: nat] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y2 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X4 @ Y2 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1242_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_a,Y2: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a @ Y2 @ ( set_a2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1243_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_multiset_a,Ys: list_multiset_a,Y2: multiset_a] :
      ( ( ( size_s3003218658903709366iset_a @ Xs )
        = ( size_s3003218658903709366iset_a @ Ys ) )
     => ( ( member_multiset_a @ Y2 @ ( set_multiset_a2 @ Ys ) )
       => ~ ! [X4: multiset_a] :
              ~ ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ X4 @ Y2 ) @ ( set_Pr5550698219922361532iset_a @ ( zip_mu5258324667929623607iset_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1244_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P1396940483166286381od_a_a,Y2: product_prod_a_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3885678630836030617od_a_a @ Ys ) )
     => ( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member6519385997651362275od_a_a @ ( produc7026408565941641214od_a_a @ X4 @ Y2 ) @ ( set_Pr2434819052702649463od_a_a @ ( zip_na4928441614332330206od_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1245_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_P1396940483166286381od_a_a,Y2: product_prod_a_a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3885678630836030617od_a_a @ Ys ) )
     => ( ( member1426531477525435216od_a_a @ Y2 @ ( set_Product_prod_a_a2 @ Ys ) )
       => ~ ! [X4: a] :
              ~ ( member3071122053849602553od_a_a @ ( produc431845341423274048od_a_a @ X4 @ Y2 ) @ ( set_Pr7008919305446870117od_a_a @ ( zip_a_3848334447339249504od_a_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1246_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P6988024003011616557iset_a,Y2: produc6518373309651786023iset_a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s7238940367606365593iset_a @ Ys ) )
     => ( ( member5199237121806060112iset_a @ Y2 @ ( set_Pr5550698219922361532iset_a @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member2488034671137206499iset_a @ ( produc6784301187465429246iset_a @ X4 @ Y2 ) @ ( set_Pr5482328489612064631iset_a @ ( zip_na8689226473972515294iset_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1247_set__update__memI,axiom,
    ! [N: nat,Xs: list_P1396940483166286381od_a_a,X: product_prod_a_a] :
      ( ( ord_less_nat @ N @ ( size_s3885678630836030617od_a_a @ Xs ) )
     => ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ ( list_u4967751623001292181od_a_a @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1248_set__update__memI,axiom,
    ! [N: nat,Xs: list_P6988024003011616557iset_a,X: produc6518373309651786023iset_a] :
      ( ( ord_less_nat @ N @ ( size_s7238940367606365593iset_a @ Xs ) )
     => ( member5199237121806060112iset_a @ X @ ( set_Pr5550698219922361532iset_a @ ( list_u2687166788783756181iset_a @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1249_set__update__memI,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,X: multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ ( list_u5207168704986282698ropVar @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1250_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1251_set__update__memI,axiom,
    ! [N: nat,Xs: list_a,X: a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_1252_list__update__append1,axiom,
    ! [I2: nat,Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I2 @ X )
        = ( append_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1253_list__update__append1,axiom,
    ! [I2: nat,Xs: list_a,Ys: list_a,X: a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( list_update_a @ ( append_a @ Xs @ Ys ) @ I2 @ X )
        = ( append_a @ ( list_update_a @ Xs @ I2 @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1254_nth__list__update,axiom,
    ! [I2: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I2 = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
            = X ) )
        & ( ( I2 != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I2 @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1255_nth__list__update,axiom,
    ! [I2: nat,Xs: list_a,J: nat,X: a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( ( I2 = J )
         => ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X ) @ J )
            = X ) )
        & ( ( I2 != J )
         => ( ( nth_a @ ( list_update_a @ Xs @ I2 @ X ) @ J )
            = ( nth_a @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_1256_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I2 @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1257_list__update__same__conv,axiom,
    ! [I2: nat,Xs: list_a,X: a] :
      ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
     => ( ( ( list_update_a @ Xs @ I2 @ X )
          = Xs )
        = ( ( nth_a @ Xs @ I2 )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_1258_zip__map1,axiom,
    ! [F: nat > a,Xs: list_nat,Ys: list_a] :
      ( ( zip_a_a @ ( map_nat_a @ F @ Xs ) @ Ys )
      = ( map_Pr4215426277517063678od_a_a
        @ ( produc892607601009605285od_a_a
          @ ^ [X3: nat] : ( product_Pair_a_a @ ( F @ X3 ) ) )
        @ ( zip_nat_a @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_1259_zip__map2,axiom,
    ! [Xs: list_a,F: nat > a,Ys: list_nat] :
      ( ( zip_a_a @ Xs @ ( map_nat_a @ F @ Ys ) )
      = ( map_Pr1850710106576109476od_a_a
        @ ( produc6848318051473230467od_a_a
          @ ^ [X3: a,Y: nat] : ( product_Pair_a_a @ X3 @ ( F @ Y ) ) )
        @ ( zip_a_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_1260_zip__map__map,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X3: nat,Y: nat] : ( product_Pair_nat_nat @ ( F @ X3 ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_1261_zip__map__map,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( zip_nat_a @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_a @ G @ Ys ) )
      = ( map_Pr7092700573198925024_nat_a
        @ ( produc7553103386638219761_nat_a
          @ ^ [X3: nat,Y: nat] : ( product_Pair_nat_a @ ( F @ X3 ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_1262_zip__map__map,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( zip_a_1304621236749119893ropVar @ ( map_nat_a @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( map_Pr6548382628697326984ropVar
        @ ( produc3556211537524544695ropVar
          @ ^ [X3: nat,Y: nat] : ( produc3383013678355495541ropVar @ ( F @ X3 ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_1263_zip__map__map,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( zip_a_nat @ ( map_nat_a @ F @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr3854537109475248058_a_nat
        @ ( produc4314939922914542795_a_nat
          @ ^ [X3: nat,Y: nat] : ( product_Pair_a_nat @ ( F @ X3 ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_1264_zip__map__map,axiom,
    ! [F: nat > a,Xs: list_nat,G: nat > a,Ys: list_nat] :
      ( ( zip_a_a @ ( map_nat_a @ F @ Xs ) @ ( map_nat_a @ G @ Ys ) )
      = ( map_Pr3160916253582813482od_a_a
        @ ( produc5281920919252274777od_a_a
          @ ^ [X3: nat,Y: nat] : ( product_Pair_a_a @ ( F @ X3 ) @ ( G @ Y ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_1265_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1266_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J )
        & ( I2 = X )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_1267_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( hd_nat @ ( upt @ I2 @ J ) )
        = I2 ) ) ).

% hd_upt

% Helper facts (8)
thf(help_fChoice_1_1_fChoice_001tf__a_T,axiom,
    ! [P: a > $o] :
      ( ( P @ ( fChoice_a @ P ) )
      = ( ? [X7: a] : ( P @ X7 ) ) ) ).

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,Y2: nat] :
      ( ( if_nat @ $false @ X @ Y2 )
      = Y2 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X: nat,Y2: nat] :
      ( ( if_nat @ $true @ X @ Y2 )
      = X ) ).

thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X7: nat] : ( P @ X7 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_Itf__a_Mtf__a_J_T,axiom,
    ! [P: product_prod_a_a > $o] :
      ( ( P @ ( fChoic4124218645493772411od_a_a @ P ) )
      = ( ? [X7: product_prod_a_a] : ( P @ X7 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Multiset____Ordering____in____NP__OPropVar_T,axiom,
    ! [P: multis3193088007478089820ropVar > $o] :
      ( ( P @ ( fChoic539809714188224432ropVar @ P ) )
      = ( ? [X7: multis3193088007478089820ropVar] : ( P @ X7 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J_T,axiom,
    ! [P: produc6518373309651786023iset_a > $o] :
      ( ( P @ ( fChoic1152277584823490683iset_a @ P ) )
      = ( ? [X7: produc6518373309651786023iset_a] : ( P @ X7 ) ) ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( multis8641292176345589959pVar_o
    @ ^ [Nat1: nat,Nat2: nat] :
        ( multis8641292176345589959pVar_o
        @ ^ [I: nat,J2: nat] :
            ( ( ( member_nat @ I @ ( set_nat2 @ xs12 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys12 ) )
              & ( I
                = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ J2 ) ) ) )
            | ( ( member_nat @ I @ ( set_nat2 @ xs22 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys22 ) )
              & ( I
                = ( i_of_j2 @ J2 ) ) ) )
        @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs12 ) )
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ( multis387687052011358179_Gamma @ ( nth_nat @ xs12 @ ( pos_of @ ys12 @ j ) ) @ j ) )
    @ ^ [Nat: nat] :
        ( multis8641292176345589959pVar_o
        @ ^ [I: nat,J2: nat] :
            ( ( ( member_nat @ I @ ( set_nat2 @ xs12 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys12 ) )
              & ( I
                = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ J2 ) ) ) )
            | ( ( member_nat @ I @ ( set_nat2 @ xs22 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys22 ) )
              & ( I
                = ( i_of_j2 @ J2 ) ) ) )
        @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs12 ) )
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ^ [A: nat,B: nat] : undefined_o
        @ ( multis387687052011358179_Gamma @ ( nth_nat @ xs12 @ ( pos_of @ ys12 @ j ) ) @ j ) )
    @ ^ [I: nat,J2: nat] :
        ~ ? [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3
              @ ( set_Mu146256361055586289ropVar
                @ ( drop_M2293212096379261346ropVar @ I
                  @ ( map_na2466558649495876678ropVar
                    @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                    @ ( upt @ zero_zero_nat @ n ) ) ) ) )
            & ( multis8641292176345589959pVar_o
              @ ^ [K: nat,L2: nat] :
                  ( ( ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                    & ( member_nat @ L2 @ ( set_nat2 @ ys12 ) )
                    & ( K
                      = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ L2 ) ) ) )
                  | ( ( member_nat @ K @ ( set_nat2 @ xs22 ) )
                    & ( member_nat @ L2 @ ( set_nat2 @ ys22 ) )
                    & ( K
                      = ( i_of_j2 @ L2 ) ) ) )
              @ ^ [K: nat] : ( member_nat @ K @ ( set_nat2 @ xs12 ) )
              @ ^ [A: nat,B: nat] : undefined_o
              @ ^ [A: nat,B: nat] : undefined_o
              @ ^ [A: nat,B: nat] : undefined_o
              @ ^ [A: nat,B: nat] : undefined_o
              @ X3 ) )
    @ ^ [I: nat,J2: nat] :
      ? [X3: multis3193088007478089820ropVar] :
        ( ( member8421201611525043589ropVar @ X3
          @ ( set_Mu146256361055586289ropVar
            @ ( drop_M2293212096379261346ropVar @ I
              @ ( map_na2466558649495876678ropVar
                @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                @ ( upt @ zero_zero_nat @ n ) ) ) ) )
        & ( multis8641292176345589959pVar_o
          @ ^ [K: nat,L2: nat] :
              ( ( ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                & ( member_nat @ L2 @ ( set_nat2 @ ys12 ) )
                & ( K
                  = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ L2 ) ) ) )
              | ( ( member_nat @ K @ ( set_nat2 @ xs22 ) )
                & ( member_nat @ L2 @ ( set_nat2 @ ys22 ) )
                & ( K
                  = ( i_of_j2 @ L2 ) ) ) )
          @ ^ [K: nat] : ( member_nat @ K @ ( set_nat2 @ xs12 ) )
          @ ^ [A: nat,B: nat] : undefined_o
          @ ^ [A: nat,B: nat] : undefined_o
          @ ^ [A: nat,B: nat] : undefined_o
          @ ^ [A: nat,B: nat] : undefined_o
          @ X3 )
        & ! [Y: multis3193088007478089820ropVar] :
            ( ( ( member8421201611525043589ropVar @ Y
                @ ( set_Mu146256361055586289ropVar
                  @ ( drop_M2293212096379261346ropVar @ I
                    @ ( map_na2466558649495876678ropVar
                      @ ^ [K: nat] : ( multis387687052011358179_Gamma @ K @ J2 )
                      @ ( upt @ zero_zero_nat @ n ) ) ) ) )
              & ( multis8641292176345589959pVar_o
                @ ^ [K: nat,L2: nat] :
                    ( ( ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                      & ( member_nat @ L2 @ ( set_nat2 @ ys12 ) )
                      & ( K
                        = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ L2 ) ) ) )
                    | ( ( member_nat @ K @ ( set_nat2 @ xs22 ) )
                      & ( member_nat @ L2 @ ( set_nat2 @ ys22 ) )
                      & ( K
                        = ( i_of_j2 @ L2 ) ) ) )
                @ ^ [K: nat] : ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                @ ^ [A: nat,B: nat] : undefined_o
                @ ^ [A: nat,B: nat] : undefined_o
                @ ^ [A: nat,B: nat] : undefined_o
                @ ^ [A: nat,B: nat] : undefined_o
                @ Y ) )
           => ( Y = X3 ) ) )
    @ ^ [I: nat,J2: nat] :
        ~ ? [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
            & ( multis8641292176345589959pVar_o
              @ ^ [K: nat,L2: nat] :
                  ( ( ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                    & ( member_nat @ L2 @ ( set_nat2 @ ys12 ) )
                    & ( K
                      = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ L2 ) ) ) )
                  | ( ( member_nat @ K @ ( set_nat2 @ xs22 ) )
                    & ( member_nat @ L2 @ ( set_nat2 @ ys22 ) )
                    & ( K
                      = ( i_of_j2 @ L2 ) ) ) )
              @ ^ [K: nat] : ( member_nat @ K @ ( set_nat2 @ xs12 ) )
              @ ^ [A: nat,B: nat] : undefined_o
              @ ^ [A: nat,B: nat] : undefined_o
              @ ^ [A: nat,B: nat] : undefined_o
              @ ^ [A: nat,B: nat] : undefined_o
              @ X3 ) )
    @ ^ [I: nat,J2: nat] :
      ? [X3: multis3193088007478089820ropVar] :
        ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
        & ( multis8641292176345589959pVar_o
          @ ^ [K: nat,L2: nat] :
              ( ( ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                & ( member_nat @ L2 @ ( set_nat2 @ ys12 ) )
                & ( K
                  = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ L2 ) ) ) )
              | ( ( member_nat @ K @ ( set_nat2 @ xs22 ) )
                & ( member_nat @ L2 @ ( set_nat2 @ ys22 ) )
                & ( K
                  = ( i_of_j2 @ L2 ) ) ) )
          @ ^ [K: nat] : ( member_nat @ K @ ( set_nat2 @ xs12 ) )
          @ ^ [A: nat,B: nat] : undefined_o
          @ ^ [A: nat,B: nat] : undefined_o
          @ ^ [A: nat,B: nat] : undefined_o
          @ ^ [A: nat,B: nat] : undefined_o
          @ X3 )
        & ! [Y: multis3193088007478089820ropVar] :
            ( ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
              & ( multis8641292176345589959pVar_o
                @ ^ [K: nat,L2: nat] :
                    ( ( ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                      & ( member_nat @ L2 @ ( set_nat2 @ ys12 ) )
                      & ( K
                        = ( nth_nat @ xs12 @ ( pos_of @ ys12 @ L2 ) ) ) )
                    | ( ( member_nat @ K @ ( set_nat2 @ xs22 ) )
                      & ( member_nat @ L2 @ ( set_nat2 @ ys22 ) )
                      & ( K
                        = ( i_of_j2 @ L2 ) ) ) )
                @ ^ [K: nat] : ( member_nat @ K @ ( set_nat2 @ xs12 ) )
                @ ^ [A: nat,B: nat] : undefined_o
                @ ^ [A: nat,B: nat] : undefined_o
                @ ^ [A: nat,B: nat] : undefined_o
                @ ^ [A: nat,B: nat] : undefined_o
                @ Y ) )
           => ( Y = X3 ) ) )
    @ ( multis387687052011358179_Gamma @ ( nth_nat @ xs12 @ ( pos_of @ ys12 @ j ) ) @ j ) ) ).

%------------------------------------------------------------------------------