TPTP Problem File: SLH0733^1.p

%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Eval_FO/0005_Ailamazyan/prob_00233_008672__15509456_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1948 ( 533 unt; 660 typ;   0 def)
%            Number of atoms       : 4333 (1882 equ;   0 cnn)
%            Maximal formula atoms :   18 (   3 avg)
%            Number of connectives : 18181 ( 891   ~;  90   |; 392   &;14525   @)
%                                         (   0 <=>;2283  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :  140 ( 139 usr)
%            Number of type conns  : 2263 (2263   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  524 ( 521 usr;  18 con; 0-5 aty)
%            Number of variables   : 4882 ( 199   ^;4462   !; 221   ?;4882   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-19 13:05:41.558
%------------------------------------------------------------------------------
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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 (521)
thf(sy_c_Ailamazyan_Oad__agr__list_001tf__a_001tf__b,type,
    ad_agr_list_a_b: set_a > list_Sum_sum_a_b > list_Sum_sum_a_b > $o ).

thf(sy_c_Ailamazyan_Oad__agr__sets_001tf__a_001tf__b,type,
    ad_agr_sets_a_b: set_nat > set_nat > set_a > ( nat > sum_sum_a_b ) > ( nat > sum_sum_a_b ) > $o ).

thf(sy_c_Ailamazyan_Oad__equiv__list_001tf__a_001tf__b,type,
    ad_equiv_list_a_b: set_a > list_Sum_sum_a_b > list_Sum_sum_a_b > $o ).

thf(sy_c_Ailamazyan_Oad__equiv__pair_001tf__a_001tf__b,type,
    ad_equiv_pair_a_b: set_a > produc4970237071031658413um_a_b > $o ).

thf(sy_c_Ailamazyan_Oad__equiv__pair__rel_001tf__a_001tf__b,type,
    ad_equ6444359169260110102el_a_b: produc2771778909502576118um_a_b > produc2771778909502576118um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv_001tf__a_001tf__b,type,
    sp_equiv_a_b: ( nat > sum_sum_a_b ) > ( nat > sum_sum_a_b ) > set_nat > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__list_001tf__a_001tf__b,type,
    sp_equiv_list_a_b: list_Sum_sum_a_b > list_Sum_sum_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair_001t__Nat__Onat_001t__Nat__Onat,type,
    sp_equ885134019442699404at_nat: product_prod_nat_nat > product_prod_nat_nat > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    sp_equ3763899520926221297um_a_b: produc7779921308951573458um_a_b > produc7779921308951573458um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    sp_equ5014097358103830626um_a_b: produc5919649888572472791um_a_b > produc5919649888572472791um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    sp_equ785082339516739170_b_nat: produc6336967135452217047_b_nat > produc6336967135452217047_b_nat > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    sp_equ110388464097044123um_a_b: produc5987322259985443452um_a_b > produc5987322259985443452um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    sp_equ4404609506204868152um_a_b: produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair__rel_001t__Nat__Onat_001t__Nat__Onat,type,
    sp_equ364710361814684337at_nat: produc859450856879609959at_nat > produc859450856879609959at_nat > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair__rel_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    sp_equ537355538419398540um_a_b: produc8017504567453546117um_a_b > produc8017504567453546117um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair__rel_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    sp_equ7033894156223228039um_a_b: produc4234450811433635303um_a_b > produc4234450811433635303um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair__rel_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    sp_equ2804879137636136583_b_nat: produc7960547251881905639_b_nat > produc7960547251881905639_b_nat > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair__rel_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    sp_equ8848776977153521718um_a_b: produc6755615443351954905um_a_b > produc6755615443351954905um_a_b > $o ).

thf(sy_c_Ailamazyan_Osp__equiv__pair__rel_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    sp_equ3929279581461973085um_a_b: produc3571035918378404711um_a_b > produc3571035918378404711um_a_b > $o ).

thf(sy_c_AssocList_Odelete__aux_001t__Nat__Onat_001t__Nat__Onat,type,
    delete_aux_nat_nat: nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_AssocList_Odelete__aux_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    delete3348065510246518057um_a_b: nat > list_P5598968436384720482um_a_b > list_P5598968436384720482um_a_b ).

thf(sy_c_AssocList_Odelete__aux_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    delete5312104020516564266um_a_b: nat > list_P6497659457863146077um_a_b > list_P6497659457863146077um_a_b ).

thf(sy_c_AssocList_Odelete__aux_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    delete1083089001929472810_b_nat: sum_sum_a_b > list_P667245297671117661_b_nat > list_P667245297671117661_b_nat ).

thf(sy_c_AssocList_Odelete__aux_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    delete1161692237630136275um_a_b: sum_sum_a_b > list_P6716435105863944716um_a_b > list_P6716435105863944716um_a_b ).

thf(sy_c_AssocList_Odelete__aux_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    delete6941486676214758144um_a_b: sum_sum_a_b > list_P911570089447184179um_a_b > list_P911570089447184179um_a_b ).

thf(sy_c_AssocList_Odelete__aux__rel_001t__Nat__Onat_001t__Nat__Onat,type,
    delete4291352608092917737at_nat: produc8472197452120411308at_nat > produc8472197452120411308at_nat > $o ).

thf(sy_c_AssocList_Odelete__aux__rel_001t__Nat__Onat_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    delete8508285360664163412um_a_b: produc7925846991739328147um_a_b > produc7925846991739328147um_a_b > $o ).

thf(sy_c_AssocList_Odelete__aux__rel_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    delete7852130366313534911um_a_b: produc8393995592613569922um_a_b > produc8393995592613569922um_a_b > $o ).

thf(sy_c_AssocList_Odelete__aux__rel_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    delete3623115347726443455_b_nat: produc8787139405198277676_b_nat > produc8787139405198277676_b_nat > $o ).

thf(sy_c_AssocList_Odelete__aux__rel_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    delete532881961126511870um_a_b: produc8659213103137179411um_a_b > produc8659213103137179411um_a_b > $o ).

thf(sy_c_AssocList_Odelete__aux__rel_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    delete2169469174080775573um_a_b: produc6661094113385059586um_a_b > produc6661094113385059586um_a_b > $o ).

thf(sy_c_AssocList_Oupdate__with__aux_001t__Nat__Onat_001t__Nat__Onat,type,
    update8447963936283049648at_nat: nat > nat > ( nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_AssocList_Oupdate__with__aux_001t__Nat__Onat_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    update3120058914886408838um_a_b: nat > sum_sum_a_b > ( nat > nat ) > list_P667245297671117661_b_nat > list_P667245297671117661_b_nat ).

thf(sy_c_AssocList_Oupdate__with__aux_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_001t__Nat__Onat,type,
    update1964132014378800879_b_nat: produc4970237071031658413um_a_b > nat > ( produc4970237071031658413um_a_b > produc4970237071031658413um_a_b ) > list_P5598968436384720482um_a_b > list_P5598968436384720482um_a_b ).

thf(sy_c_AssocList_Oupdate__with__aux_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    update2011012084792080581um_a_b: produc4970237071031658413um_a_b > sum_sum_a_b > ( produc4970237071031658413um_a_b > produc4970237071031658413um_a_b ) > list_P6716435105863944716um_a_b > list_P6716435105863944716um_a_b ).

thf(sy_c_AssocList_Oupdate__with__aux_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    update8114415933154093190_b_nat: sum_sum_a_b > nat > ( sum_sum_a_b > sum_sum_a_b ) > list_P6497659457863146077um_a_b > list_P6497659457863146077um_a_b ).

thf(sy_c_AssocList_Oupdate__with__aux_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    update7109155850483007580um_a_b: sum_sum_a_b > sum_sum_a_b > ( sum_sum_a_b > sum_sum_a_b ) > list_P911570089447184179um_a_b > list_P911570089447184179um_a_b ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Nat__Onat,type,
    bNF_Ca968750328013420230at_nat: set_Pr1261947904930325089at_nat > ( nat > nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Nat__Onat_001t__Set__Oset_Itf__a_J,type,
    bNF_Ca350241719529149160_set_a: set_Pr1261947904930325089at_nat > ( nat > set_a ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Nat__Onat,type,
    bNF_Ca8277740611438156956_b_nat: set_Pr7446880501794416397um_a_b > ( sum_sum_a_b > nat ) > $o ).

thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain_001t__Sum____Type__Osum_Itf__a_Mtf__b_J_001t__Set__Oset_Itf__a_J,type,
    bNF_Ca26987310307872402_set_a: set_Pr7446880501794416397um_a_b > ( sum_sum_a_b > set_a ) > $o ).

thf(sy_c_Finite__Set_OFpow_001t__Nat__Onat,type,
    finite_Fpow_nat: set_nat > set_set_nat ).

thf(sy_c_Finite__Set_OFpow_001tf__a,type,
    finite_Fpow_a: set_a > set_set_a ).

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

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__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
    if_list_nat: $o > list_nat > list_nat > list_nat ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    if_lis9186351972506106189at_nat: $o > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_J,type,
    if_lis4596758700692144156um_a_b: $o > list_P5598968436384720482um_a_b > list_P5598968436384720482um_a_b > list_P5598968436384720482um_a_b ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
    if_lis8991567957713360163um_a_b: $o > list_P6497659457863146077um_a_b > list_P6497659457863146077um_a_b > list_P6497659457863146077um_a_b ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J,type,
    if_lis3161153797521331747_b_nat: $o > list_P667245297671117661_b_nat > list_P667245297671117661_b_nat > list_P667245297671117661_b_nat ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
    if_lis2231744601426488825um_a_b: $o > list_P911570089447184179um_a_b > list_P911570089447184179um_a_b > list_P911570089447184179um_a_b ).

thf(sy_c_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    if_list_Sum_sum_a_b: $o > list_Sum_sum_a_b > list_Sum_sum_a_b > list_Sum_sum_a_b ).

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,
    append985823374593552924at_nat: list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    append7648079923110641138_b_nat: list_P667245297671117661_b_nat > list_P667245297671117661_b_nat > list_P667245297671117661_b_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
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thf(sy_c_List_Ocoset_001t__Nat__Onat,type,
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thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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    coset_1753088348400206111um_a_b: list_P5598968436384720482um_a_b > set_Pr5956926064383532552um_a_b ).

thf(sy_c_List_Ocoset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
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thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Nat__Onat_J,type,
    member8290135809484590976_b_nat: produc6336967135452217047_b_nat > set_Pr7634312471471294263_b_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
    member152508769716563603um_a_b: produc5987322259985443452um_a_b > set_Pr4696577323406728626um_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    member6480126262776431190um_a_b: produc4970237071031658413um_a_b > set_Pr7446880501794416397um_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    member2643936169264416010at_nat: set_Pr1261947904930325089at_nat > set_se7855581050983116737at_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_J,type,
    member8475297673577648159um_a_b: set_Pr5956926064383532552um_a_b > set_se4364611324521144638um_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
    member1123962445896381152um_a_b: set_Pr4241354594808546871um_a_b > set_se5282520812936825495um_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J,type,
    member4516920322559128544_b_nat: set_Pr7634312471471294263_b_nat > set_se5886930887450119063_b_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J,type,
    member5262719511116389814um_a_b: set_Pr7446880501794416397um_a_b > set_se8991359590818079341um_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J,type,
    member4060935254435997939um_a_b: set_Sum_sum_a_b > set_set_Sum_sum_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_Itf__a_Mtf__b_J,type,
    member_Sum_sum_a_b: sum_sum_a_b > set_Sum_sum_a_b > $o ).

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

thf(sy_v_X,type,
    x: set_a ).

thf(sy_v_Y,type,
    y: set_a ).

thf(sy_v_a____,type,
    a2: sum_sum_a_b ).

thf(sy_v_b____,type,
    b: sum_sum_a_b ).

thf(sy_v_xs,type,
    xs: list_Sum_sum_a_b ).

thf(sy_v_ys,type,
    ys: list_Sum_sum_a_b ).

% Relevant facts (1270)
thf(fact_0_assms_I1_J,axiom,
    ord_less_eq_set_a @ y @ x ).

% assms(1)
thf(fact_1_assms_I2_J,axiom,
    ad_agr_list_a_b @ y @ ys @ xs ).

% assms(2)
thf(fact_2_assms_I4_J,axiom,
    ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_b @ sum_Inl_a_b @ ( set_Sum_sum_a_b2 @ ys ) ) @ y ).

% assms(4)
thf(fact_3_assms_I3_J,axiom,
    ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_b @ sum_Inl_a_b @ ( set_Sum_sum_a_b2 @ xs ) ) @ y ).

% assms(3)
thf(fact_4_sp__equiv__pair_Ocases,axiom,
    ! [X: produc6755615443351954905um_a_b] :
      ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b,A2: sum_sum_a_b,B2: produc4970237071031658413um_a_b] :
          ( X
         != ( produc4166429159020078609um_a_b @ ( produc7989310782275393198um_a_b @ A @ B ) @ ( produc7989310782275393198um_a_b @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_5_sp__equiv__pair_Ocases,axiom,
    ! [X: produc7960547251881905639_b_nat] :
      ~ ! [A: sum_sum_a_b,B: nat,A2: sum_sum_a_b,B2: nat] :
          ( X
         != ( produc490312431154969431_b_nat @ ( produc9094079831559073295_b_nat @ A @ B ) @ ( produc9094079831559073295_b_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_6_sp__equiv__pair_Ocases,axiom,
    ! [X: produc8017504567453546117um_a_b] :
      ~ ! [A: nat,B: produc4970237071031658413um_a_b,A2: nat,B2: produc4970237071031658413um_a_b] :
          ( X
         != ( produc8504789778075864509um_a_b @ ( produc6016411306719553028um_a_b @ A @ B ) @ ( produc6016411306719553028um_a_b @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_7_sp__equiv__pair_Ocases,axiom,
    ! [X: produc4234450811433635303um_a_b] :
      ~ ! [A: nat,B: sum_sum_a_b,A2: nat,B2: sum_sum_a_b] :
          ( X
         != ( produc8963228657344478039um_a_b @ ( produc4099722813291388943um_a_b @ A @ B ) @ ( produc4099722813291388943um_a_b @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_8_sp__equiv__pair_Ocases,axiom,
    ! [X: produc859450856879609959at_nat] :
      ~ ! [A: nat,B: nat,A2: nat,B2: nat] :
          ( X
         != ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_9_sp__equiv__pair_Ocases,axiom,
    ! [X: produc3571035918378404711um_a_b] :
      ~ ! [A: sum_sum_a_b,B: sum_sum_a_b,A2: sum_sum_a_b,B2: sum_sum_a_b] :
          ( X
         != ( produc1529531236301088727um_a_b @ ( produc8374529264883863525um_a_b @ A @ B ) @ ( produc8374529264883863525um_a_b @ A2 @ B2 ) ) ) ).

% sp_equiv_pair.cases
thf(fact_10_set__zip__comm,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: sum_sum_a_b,Xs: list_P911570089447184179um_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member1422665382716164761um_a_b @ ( produc5264852919545901052um_a_b @ X @ Y ) @ ( set_Pr176988522575013677um_a_b @ ( zip_Pr6112709887408914652um_a_b @ Xs @ Ys ) ) )
     => ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ Y @ X ) @ ( set_Pr8130203946430188327um_a_b @ ( zip_Su8837167750138406798um_a_b @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_11_set__zip__comm,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: nat,Xs: list_P911570089447184179um_a_b,Ys: list_nat] :
      ( ( member6152736929284073923_b_nat @ ( produc8698646781263173670_b_nat @ X @ Y ) @ ( set_Pr276058860415286871_b_nat @ ( zip_Pr5051627046265366790_b_nat @ Xs @ Ys ) ) )
     => ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ Y @ X ) @ ( set_Pr7399620718385288829um_a_b @ ( zip_na2369391571721746148um_a_b @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_12_set__zip__comm,axiom,
    ! [X: sum_sum_a_b,Y: produc4970237071031658413um_a_b,Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y ) @ ( set_Pr8130203946430188327um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) )
     => ( member1422665382716164761um_a_b @ ( produc5264852919545901052um_a_b @ Y @ X ) @ ( set_Pr176988522575013677um_a_b @ ( zip_Pr6112709887408914652um_a_b @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_13_set__zip__comm,axiom,
    ! [X: sum_sum_a_b,Y: nat,Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y ) @ ( set_Pr176305205332118508_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) )
     => ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ Y @ X ) @ ( set_Pr8982359995307150060um_a_b @ ( zip_nat_Sum_sum_a_b @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_14_set__zip__comm,axiom,
    ! [X: nat,Y: produc4970237071031658413um_a_b,Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y ) @ ( set_Pr7399620718385288829um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) )
     => ( member6152736929284073923_b_nat @ ( produc8698646781263173670_b_nat @ Y @ X ) @ ( set_Pr276058860415286871_b_nat @ ( zip_Pr5051627046265366790_b_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_15_set__zip__comm,axiom,
    ! [X: nat,Y: sum_sum_a_b,Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y ) @ ( set_Pr8982359995307150060um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) )
     => ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ Y @ X ) @ ( set_Pr176305205332118508_b_nat @ ( zip_Sum_sum_a_b_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_16_set__zip__comm,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ X ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_17_set__zip__comm,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) )
     => ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ Y @ X ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Ys @ Xs ) ) ) ) ).

% set_zip_comm
thf(fact_18_ad__equiv__pair__comm,axiom,
    ! [X2: set_a,A3: sum_sum_a_b,A4: sum_sum_a_b] :
      ( ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ A3 @ A4 ) )
      = ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ A4 @ A3 ) ) ) ).

% ad_equiv_pair_comm
thf(fact_19_ad__equiv__pair__refl,axiom,
    ! [X2: set_a,A3: sum_sum_a_b] : ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ A3 @ A3 ) ) ).

% ad_equiv_pair_refl
thf(fact_20_zip__same,axiom,
    ! [A3: produc6336967135452217047_b_nat,B3: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( member681441685064050448_b_nat @ ( produc490312431154969431_b_nat @ A3 @ B3 ) @ ( set_Pr7155044192068831612_b_nat @ ( zip_Pr4045231663290569335_b_nat @ Xs @ Xs ) ) )
      = ( ( member8290135809484590976_b_nat @ A3 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
        & ( A3 = B3 ) ) ) ).

% zip_same
thf(fact_21_zip__same,axiom,
    ! [A3: produc7779921308951573458um_a_b,B3: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( member885238345412010542um_a_b @ ( produc8504789778075864509um_a_b @ A3 @ B3 ) @ ( set_Pr2621926141165544090um_a_b @ ( zip_Pr3111714058699715805um_a_b @ Xs @ Xs ) ) )
      = ( ( member4052926750399300073um_a_b @ A3 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
        & ( A3 = B3 ) ) ) ).

% zip_same
thf(fact_22_zip__same,axiom,
    ! [A3: produc5919649888572472791um_a_b,B3: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( member6178717281470555920um_a_b @ ( produc8963228657344478039um_a_b @ A3 @ B3 ) @ ( set_Pr3428947751620561276um_a_b @ ( zip_Pr3294775852625302135um_a_b @ Xs @ Xs ) ) )
      = ( ( member7872818562604846720um_a_b @ A3 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
        & ( A3 = B3 ) ) ) ).

% zip_same
thf(fact_23_zip__same,axiom,
    ! [A3: product_prod_nat_nat,B3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A3 @ B3 ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
      = ( ( member8440522571783428010at_nat @ A3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( A3 = B3 ) ) ) ).

% zip_same
thf(fact_24_zip__same,axiom,
    ! [A3: nat,B3: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A3 @ ( set_nat2 @ Xs ) )
        & ( A3 = B3 ) ) ) ).

% zip_same
thf(fact_25_zip__same,axiom,
    ! [A3: produc4970237071031658413um_a_b,B3: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ A3 @ B3 ) @ ( set_Pr7343516712102431484um_a_b @ ( zip_Pr2791044230619283191um_a_b @ Xs @ Xs ) ) )
      = ( ( member6480126262776431190um_a_b @ A3 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
        & ( A3 = B3 ) ) ) ).

% zip_same
thf(fact_26_zip__same,axiom,
    ! [A3: sum_sum_a_b,B3: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ A3 @ B3 ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Xs ) ) )
      = ( ( member_Sum_sum_a_b @ A3 @ ( set_Sum_sum_a_b2 @ Xs ) )
        & ( A3 = B3 ) ) ) ).

% zip_same
thf(fact_27_in__set__zipE,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_28_in__set__zipE,axiom,
    ! [X: nat,Y: sum_sum_a_b,Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y ) @ ( set_Pr8982359995307150060um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_29_in__set__zipE,axiom,
    ! [X: sum_sum_a_b,Y: nat,Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y ) @ ( set_Pr176305205332118508_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_30_in__set__zipE,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
         => ~ ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_31_in__set__zipE,axiom,
    ! [X: nat,Y: produc4970237071031658413um_a_b,Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y ) @ ( set_Pr7399620718385288829um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
         => ~ ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_32_in__set__zipE,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: nat,Xs: list_P911570089447184179um_a_b,Ys: list_nat] :
      ( ( member6152736929284073923_b_nat @ ( produc8698646781263173670_b_nat @ X @ Y ) @ ( set_Pr276058860415286871_b_nat @ ( zip_Pr5051627046265366790_b_nat @ Xs @ Ys ) ) )
     => ~ ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_33_in__set__zipE,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: sum_sum_a_b,Xs: list_P911570089447184179um_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member1422665382716164761um_a_b @ ( produc5264852919545901052um_a_b @ X @ Y ) @ ( set_Pr176988522575013677um_a_b @ ( zip_Pr6112709887408914652um_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
         => ~ ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_34_in__set__zipE,axiom,
    ! [X: sum_sum_a_b,Y: produc4970237071031658413um_a_b,Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y ) @ ( set_Pr8130203946430188327um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
         => ~ ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_35_in__set__zipE,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ X @ Y ) @ ( set_Pr7343516712102431484um_a_b @ ( zip_Pr2791044230619283191um_a_b @ Xs @ Ys ) ) )
     => ~ ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
         => ~ ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_36_in__set__zipE,axiom,
    ! [X: product_prod_nat_nat,Y: nat,Xs: list_P6011104703257516679at_nat,Ys: list_nat] :
      ( ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X @ Y ) @ ( set_Pr7836445846575771563at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_37_set__zip__leftD,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) )
     => ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_38_set__zip__leftD,axiom,
    ! [X: sum_sum_a_b,Y: produc4970237071031658413um_a_b,Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y ) @ ( set_Pr8130203946430188327um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) )
     => ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_39_set__zip__leftD,axiom,
    ! [X: sum_sum_a_b,Y: nat,Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y ) @ ( set_Pr176305205332118508_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) )
     => ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_40_set__zip__leftD,axiom,
    ! [X: nat,Y: produc4970237071031658413um_a_b,Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y ) @ ( set_Pr7399620718385288829um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_41_set__zip__leftD,axiom,
    ! [X: nat,Y: sum_sum_a_b,Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y ) @ ( set_Pr8982359995307150060um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_42_set__zip__leftD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_43_set__zip__rightD,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) )
     => ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_44_set__zip__rightD,axiom,
    ! [X: sum_sum_a_b,Y: produc4970237071031658413um_a_b,Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y ) @ ( set_Pr8130203946430188327um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) )
     => ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_45_set__zip__rightD,axiom,
    ! [X: sum_sum_a_b,Y: nat,Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y ) @ ( set_Pr176305205332118508_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_46_set__zip__rightD,axiom,
    ! [X: nat,Y: produc4970237071031658413um_a_b,Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y ) @ ( set_Pr7399620718385288829um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) )
     => ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_47_set__zip__rightD,axiom,
    ! [X: nat,Y: sum_sum_a_b,Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y ) @ ( set_Pr8982359995307150060um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) )
     => ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_48_set__zip__rightD,axiom,
    ! [X: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_49_ad__equiv__pair__mono,axiom,
    ! [X2: set_a,Y2: set_a,A3: sum_sum_a_b,A4: sum_sum_a_b] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( ad_equiv_pair_a_b @ Y2 @ ( produc8374529264883863525um_a_b @ A3 @ A4 ) )
       => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ A3 @ A4 ) ) ) ) ).

% ad_equiv_pair_mono
thf(fact_50_prod_Oinject,axiom,
    ! [X1: sum_sum_a_b,X22: sum_sum_a_b,Y1: sum_sum_a_b,Y22: sum_sum_a_b] :
      ( ( ( produc8374529264883863525um_a_b @ X1 @ X22 )
        = ( produc8374529264883863525um_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_51_prod_Oinject,axiom,
    ! [X1: sum_sum_a_b,X22: produc4970237071031658413um_a_b,Y1: sum_sum_a_b,Y22: produc4970237071031658413um_a_b] :
      ( ( ( produc7989310782275393198um_a_b @ X1 @ X22 )
        = ( produc7989310782275393198um_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_52_prod_Oinject,axiom,
    ! [X1: sum_sum_a_b,X22: nat,Y1: sum_sum_a_b,Y22: nat] :
      ( ( ( produc9094079831559073295_b_nat @ X1 @ X22 )
        = ( produc9094079831559073295_b_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_53_prod_Oinject,axiom,
    ! [X1: nat,X22: produc4970237071031658413um_a_b,Y1: nat,Y22: produc4970237071031658413um_a_b] :
      ( ( ( produc6016411306719553028um_a_b @ X1 @ X22 )
        = ( produc6016411306719553028um_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_54_prod_Oinject,axiom,
    ! [X1: nat,X22: sum_sum_a_b,Y1: nat,Y22: sum_sum_a_b] :
      ( ( ( produc4099722813291388943um_a_b @ X1 @ X22 )
        = ( produc4099722813291388943um_a_b @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_55_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_56_old_Oprod_Oinject,axiom,
    ! [A3: sum_sum_a_b,B3: sum_sum_a_b,A4: sum_sum_a_b,B4: sum_sum_a_b] :
      ( ( ( produc8374529264883863525um_a_b @ A3 @ B3 )
        = ( produc8374529264883863525um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_57_old_Oprod_Oinject,axiom,
    ! [A3: sum_sum_a_b,B3: produc4970237071031658413um_a_b,A4: sum_sum_a_b,B4: produc4970237071031658413um_a_b] :
      ( ( ( produc7989310782275393198um_a_b @ A3 @ B3 )
        = ( produc7989310782275393198um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_58_old_Oprod_Oinject,axiom,
    ! [A3: sum_sum_a_b,B3: nat,A4: sum_sum_a_b,B4: nat] :
      ( ( ( produc9094079831559073295_b_nat @ A3 @ B3 )
        = ( produc9094079831559073295_b_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_59_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: produc4970237071031658413um_a_b,A4: nat,B4: produc4970237071031658413um_a_b] :
      ( ( ( produc6016411306719553028um_a_b @ A3 @ B3 )
        = ( produc6016411306719553028um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_60_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: sum_sum_a_b,A4: nat,B4: sum_sum_a_b] :
      ( ( ( produc4099722813291388943um_a_b @ A3 @ B3 )
        = ( produc4099722813291388943um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_61_old_Oprod_Oinject,axiom,
    ! [A3: nat,B3: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A3 @ B3 )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        & ( B3 = B4 ) ) ) ).

% old.prod.inject
thf(fact_62_ad__equiv__list__def,axiom,
    ( ad_equiv_list_a_b
    = ( ^ [X3: set_a,Xs2: list_Sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
          ( ( ( size_s5949480688510517322um_a_b @ Xs2 )
            = ( size_s5949480688510517322um_a_b @ Ys2 ) )
          & ! [X4: produc4970237071031658413um_a_b] :
              ( ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs2 @ Ys2 ) ) )
             => ( ad_equiv_pair_a_b @ X3 @ X4 ) ) ) ) ) ).

% ad_equiv_list_def
thf(fact_63_ad__agr__list__comm,axiom,
    ! [X2: set_a,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( ad_agr_list_a_b @ X2 @ Xs @ Ys )
     => ( ad_agr_list_a_b @ X2 @ Ys @ Xs ) ) ).

% ad_agr_list_comm
thf(fact_64_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4970237071031658413um_a_b] :
      ~ ! [A: sum_sum_a_b,B: sum_sum_a_b] :
          ( Y
         != ( produc8374529264883863525um_a_b @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_65_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5987322259985443452um_a_b] :
      ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] :
          ( Y
         != ( produc7989310782275393198um_a_b @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_66_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6336967135452217047_b_nat] :
      ~ ! [A: sum_sum_a_b,B: nat] :
          ( Y
         != ( produc9094079831559073295_b_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_67_old_Oprod_Oexhaust,axiom,
    ! [Y: produc7779921308951573458um_a_b] :
      ~ ! [A: nat,B: produc4970237071031658413um_a_b] :
          ( Y
         != ( produc6016411306719553028um_a_b @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_68_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5919649888572472791um_a_b] :
      ~ ! [A: nat,B: sum_sum_a_b] :
          ( Y
         != ( produc4099722813291388943um_a_b @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_69_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A: nat,B: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A @ B ) ) ).

% old.prod.exhaust
thf(fact_70_surj__pair,axiom,
    ! [P: produc4970237071031658413um_a_b] :
    ? [X5: sum_sum_a_b,Y3: sum_sum_a_b] :
      ( P
      = ( produc8374529264883863525um_a_b @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_71_surj__pair,axiom,
    ! [P: produc5987322259985443452um_a_b] :
    ? [X5: sum_sum_a_b,Y3: produc4970237071031658413um_a_b] :
      ( P
      = ( produc7989310782275393198um_a_b @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_72_surj__pair,axiom,
    ! [P: produc6336967135452217047_b_nat] :
    ? [X5: sum_sum_a_b,Y3: nat] :
      ( P
      = ( produc9094079831559073295_b_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_73_surj__pair,axiom,
    ! [P: produc7779921308951573458um_a_b] :
    ? [X5: nat,Y3: produc4970237071031658413um_a_b] :
      ( P
      = ( produc6016411306719553028um_a_b @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_74_surj__pair,axiom,
    ! [P: produc5919649888572472791um_a_b] :
    ? [X5: nat,Y3: sum_sum_a_b] :
      ( P
      = ( produc4099722813291388943um_a_b @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_75_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X5: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X5 @ Y3 ) ) ).

% surj_pair
thf(fact_76_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_77_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_78_ad__agr__list__mono,axiom,
    ! [X2: set_a,Y2: set_a,Ys: list_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( ord_less_eq_set_a @ X2 @ Y2 )
     => ( ( ad_agr_list_a_b @ Y2 @ Ys @ Xs )
       => ( ad_agr_list_a_b @ X2 @ Ys @ Xs ) ) ) ).

% ad_agr_list_mono
thf(fact_79_ad__equiv__pair_Ocases,axiom,
    ! [X: produc2771778909502576118um_a_b] :
      ~ ! [X6: set_a,A: sum_sum_a_b,A2: sum_sum_a_b] :
          ( X
         != ( produc1385271121253346790um_a_b @ X6 @ ( produc8374529264883863525um_a_b @ A @ A2 ) ) ) ).

% ad_equiv_pair.cases
thf(fact_80_subset__code_I1_J,axiom,
    ! [Xs: list_P911570089447184179um_a_b,B5: set_Pr7446880501794416397um_a_b] :
      ( ( ord_le1431929631403617965um_a_b @ ( set_Pr2436422074441463490um_a_b @ Xs ) @ B5 )
      = ( ! [X4: produc4970237071031658413um_a_b] :
            ( ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
           => ( member6480126262776431190um_a_b @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_81_subset__code_I1_J,axiom,
    ! [Xs: list_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( ord_le9019793522827316924um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ B5 )
      = ( ! [X4: sum_sum_a_b] :
            ( ( member_Sum_sum_a_b @ X4 @ ( set_Sum_sum_a_b2 @ Xs ) )
           => ( member_Sum_sum_a_b @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_82_subset__code_I1_J,axiom,
    ! [Xs: list_P667245297671117661_b_nat,B5: set_Pr7634312471471294263_b_nat] :
      ( ( ord_le8626435567010533079_b_nat @ ( set_Pr176305205332118508_b_nat @ Xs ) @ B5 )
      = ( ! [X4: produc6336967135452217047_b_nat] :
            ( ( member8290135809484590976_b_nat @ X4 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
           => ( member8290135809484590976_b_nat @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_83_subset__code_I1_J,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,B5: set_Pr5956926064383532552um_a_b] :
      ( ( ord_le5806309049788436072um_a_b @ ( set_Pr7399620718385288829um_a_b @ Xs ) @ B5 )
      = ( ! [X4: produc7779921308951573458um_a_b] :
            ( ( member4052926750399300073um_a_b @ X4 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
           => ( member4052926750399300073um_a_b @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_84_subset__code_I1_J,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,B5: set_Pr4241354594808546871um_a_b] :
      ( ( ord_le5233477690347785687um_a_b @ ( set_Pr8982359995307150060um_a_b @ Xs ) @ B5 )
      = ( ! [X4: produc5919649888572472791um_a_b] :
            ( ( member7872818562604846720um_a_b @ X4 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
           => ( member7872818562604846720um_a_b @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_85_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B5 )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_86_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B5 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_87_subset__code_I1_J,axiom,
    ! [Xs: list_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B5 )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( member_a @ X4 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_88_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b,Y: sum_sum_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s5949480688510517322um_a_b @ Ys ) )
     => ( ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Ys ) )
       => ~ ! [X5: sum_sum_a_b] :
              ~ ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ Y ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_89_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b,Y: produc4970237071031658413um_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s8239069388891432863um_a_b @ Ys ) )
     => ( ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Ys ) )
       => ~ ! [X5: sum_sum_a_b] :
              ~ ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X5 @ Y ) @ ( set_Pr8130203946430188327um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_90_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_nat,Y: nat] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X5: sum_sum_a_b] :
              ~ ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X5 @ Y ) @ ( set_Pr176305205332118508_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_91_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P667245297671117661_b_nat,Y: produc6336967135452217047_b_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1016725329725912009_b_nat @ Ys ) )
     => ( ( member8290135809484590976_b_nat @ Y @ ( set_Pr176305205332118508_b_nat @ Ys ) )
       => ~ ! [X5: nat] :
              ~ ( member5911503655529155091_b_nat @ ( produc1707233295301429038_b_nat @ X5 @ Y ) @ ( set_Pr7814343867551897255_b_nat @ ( zip_na7279855810519521294_b_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_92_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P5598968436384720482um_a_b,Y: produc7779921308951573458um_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s4067164177703034614um_a_b @ Ys ) )
     => ( ( member4052926750399300073um_a_b @ Y @ ( set_Pr7399620718385288829um_a_b @ Ys ) )
       => ~ ! [X5: nat] :
              ~ ( member82873394083643436um_a_b @ ( produc2221959834898418875um_a_b @ X5 @ Y ) @ ( set_Pr6556475901088424600um_a_b @ ( zip_na5776879067034018779um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_93_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P6497659457863146077um_a_b,Y: produc5919649888572472791um_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6847139489917940425um_a_b @ Ys ) )
     => ( ( member7872818562604846720um_a_b @ Y @ ( set_Pr8982359995307150060um_a_b @ Ys ) )
       => ~ ! [X5: nat] :
              ~ ( member2518545778866407699um_a_b @ ( produc1289916048421684782um_a_b @ X5 @ Y ) @ ( set_Pr4421385990889149863um_a_b @ ( zip_na6862538563639777038um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_94_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P6011104703257516679at_nat,Y: product_prod_nat_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5460976970255530739at_nat @ Ys ) )
     => ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Ys ) )
       => ~ ! [X5: nat] :
              ~ ( member2223272150424702269at_nat @ ( produc487386426758144856at_nat @ X5 @ Y ) @ ( set_Pr6710958862608470481at_nat @ ( zip_na1006125974040638520at_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_95_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P911570089447184179um_a_b,Y: produc4970237071031658413um_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8239069388891432863um_a_b @ Ys ) )
     => ( ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Ys ) )
       => ~ ! [X5: nat] :
              ~ ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X5 @ Y ) @ ( set_Pr7399620718385288829um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_96_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_b,Y: sum_sum_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5949480688510517322um_a_b @ Ys ) )
     => ( ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Ys ) )
       => ~ ! [X5: nat] :
              ~ ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X5 @ Y ) @ ( set_Pr8982359995307150060um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_97_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X5: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_98_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 ) )
       => ~ ! [Y3: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_99_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_nat,X: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member3348759134392003351at_nat @ ( produc6350711070570205562at_nat @ X @ Y3 ) @ ( set_Pr7836445846575771563at_nat @ ( zip_Pr6869450617852699226at_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_100_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_nat,X: sum_sum_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y3 ) @ ( set_Pr176305205332118508_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_101_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_b,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5949480688510517322um_a_b @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: sum_sum_a_b] :
              ~ ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y3 ) @ ( set_Pr8982359995307150060um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_102_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b,X: sum_sum_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s5949480688510517322um_a_b @ Ys ) )
     => ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
       => ~ ! [Y3: sum_sum_a_b] :
              ~ ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y3 ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_103_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P667245297671117661_b_nat,Ys: list_nat,X: produc6336967135452217047_b_nat] :
      ( ( ( size_s1016725329725912009_b_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member7142433613139807469at_nat @ ( produc7083902330551678800at_nat @ X @ Y3 ) @ ( set_Pr9045273825162549633at_nat @ ( zip_Pr3433152808914995248at_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_104_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,Ys: list_nat,X: produc5919649888572472791um_a_b] :
      ( ( ( size_s6847139489917940425um_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member3436140821934296045_b_nat @ ( produc3377609539346167376_b_nat @ X @ Y3 ) @ ( set_Pr5338981033957038209_b_nat @ ( zip_Pr8950232054564259632_b_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_105_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P911570089447184179um_a_b,Ys: list_nat,X: produc4970237071031658413um_a_b] :
      ( ( ( size_s8239069388891432863um_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member6152736929284073923_b_nat @ ( produc8698646781263173670_b_nat @ X @ Y3 ) @ ( set_Pr276058860415286871_b_nat @ ( zip_Pr5051627046265366790_b_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_106_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_P911570089447184179um_a_b,X: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8239069388891432863um_a_b @ Ys ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: produc4970237071031658413um_a_b] :
              ~ ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y3 ) @ ( set_Pr7399620718385288829um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_107_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b,X: sum_sum_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s8239069388891432863um_a_b @ Ys ) )
     => ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
       => ~ ! [Y3: produc4970237071031658413um_a_b] :
              ~ ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y3 ) @ ( set_Pr8130203946430188327um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_108_prod__induct3,axiom,
    ! [P2: produc5987322259985443452um_a_b > $o,X: produc5987322259985443452um_a_b] :
      ( ! [A: sum_sum_a_b,B: sum_sum_a_b,C: sum_sum_a_b] : ( P2 @ ( produc7989310782275393198um_a_b @ A @ ( produc8374529264883863525um_a_b @ B @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_109_prod__induct3,axiom,
    ! [P2: produc7779921308951573458um_a_b > $o,X: produc7779921308951573458um_a_b] :
      ( ! [A: nat,B: sum_sum_a_b,C: sum_sum_a_b] : ( P2 @ ( produc6016411306719553028um_a_b @ A @ ( produc8374529264883863525um_a_b @ B @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_110_prod__cases3,axiom,
    ! [Y: produc5987322259985443452um_a_b] :
      ~ ! [A: sum_sum_a_b,B: sum_sum_a_b,C: sum_sum_a_b] :
          ( Y
         != ( produc7989310782275393198um_a_b @ A @ ( produc8374529264883863525um_a_b @ B @ C ) ) ) ).

% prod_cases3
thf(fact_111_prod__cases3,axiom,
    ! [Y: produc7779921308951573458um_a_b] :
      ~ ! [A: nat,B: sum_sum_a_b,C: sum_sum_a_b] :
          ( Y
         != ( produc6016411306719553028um_a_b @ A @ ( produc8374529264883863525um_a_b @ B @ C ) ) ) ).

% prod_cases3
thf(fact_112_Pair__inject,axiom,
    ! [A3: sum_sum_a_b,B3: sum_sum_a_b,A4: sum_sum_a_b,B4: sum_sum_a_b] :
      ( ( ( produc8374529264883863525um_a_b @ A3 @ B3 )
        = ( produc8374529264883863525um_a_b @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_113_Pair__inject,axiom,
    ! [A3: sum_sum_a_b,B3: produc4970237071031658413um_a_b,A4: sum_sum_a_b,B4: produc4970237071031658413um_a_b] :
      ( ( ( produc7989310782275393198um_a_b @ A3 @ B3 )
        = ( produc7989310782275393198um_a_b @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_114_Pair__inject,axiom,
    ! [A3: sum_sum_a_b,B3: nat,A4: sum_sum_a_b,B4: nat] :
      ( ( ( produc9094079831559073295_b_nat @ A3 @ B3 )
        = ( produc9094079831559073295_b_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_115_Pair__inject,axiom,
    ! [A3: nat,B3: produc4970237071031658413um_a_b,A4: nat,B4: produc4970237071031658413um_a_b] :
      ( ( ( produc6016411306719553028um_a_b @ A3 @ B3 )
        = ( produc6016411306719553028um_a_b @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_116_Pair__inject,axiom,
    ! [A3: nat,B3: sum_sum_a_b,A4: nat,B4: sum_sum_a_b] :
      ( ( ( produc4099722813291388943um_a_b @ A3 @ B3 )
        = ( produc4099722813291388943um_a_b @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_117_Pair__inject,axiom,
    ! [A3: nat,B3: nat,A4: nat,B4: nat] :
      ( ( ( product_Pair_nat_nat @ A3 @ B3 )
        = ( product_Pair_nat_nat @ A4 @ B4 ) )
     => ~ ( ( A3 = A4 )
         => ( B3 != B4 ) ) ) ).

% Pair_inject
thf(fact_118_prod__cases,axiom,
    ! [P2: produc4970237071031658413um_a_b > $o,P: produc4970237071031658413um_a_b] :
      ( ! [A: sum_sum_a_b,B: sum_sum_a_b] : ( P2 @ ( produc8374529264883863525um_a_b @ A @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_119_prod__cases,axiom,
    ! [P2: produc5987322259985443452um_a_b > $o,P: produc5987322259985443452um_a_b] :
      ( ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] : ( P2 @ ( produc7989310782275393198um_a_b @ A @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_120_prod__cases,axiom,
    ! [P2: produc6336967135452217047_b_nat > $o,P: produc6336967135452217047_b_nat] :
      ( ! [A: sum_sum_a_b,B: nat] : ( P2 @ ( produc9094079831559073295_b_nat @ A @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_121_prod__cases,axiom,
    ! [P2: produc7779921308951573458um_a_b > $o,P: produc7779921308951573458um_a_b] :
      ( ! [A: nat,B: produc4970237071031658413um_a_b] : ( P2 @ ( produc6016411306719553028um_a_b @ A @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_122_prod__cases,axiom,
    ! [P2: produc5919649888572472791um_a_b > $o,P: produc5919649888572472791um_a_b] :
      ( ! [A: nat,B: sum_sum_a_b] : ( P2 @ ( produc4099722813291388943um_a_b @ A @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_123_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A: nat,B: nat] : ( P2 @ ( product_Pair_nat_nat @ A @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_124_mem__Collect__eq,axiom,
    ! [A3: produc4970237071031658413um_a_b,P2: produc4970237071031658413um_a_b > $o] :
      ( ( member6480126262776431190um_a_b @ A3 @ ( collec2013886903173073304um_a_b @ P2 ) )
      = ( P2 @ A3 ) ) ).

% mem_Collect_eq
thf(fact_125_mem__Collect__eq,axiom,
    ! [A3: nat,P2: nat > $o] :
      ( ( member_nat @ A3 @ ( collect_nat @ P2 ) )
      = ( P2 @ A3 ) ) ).

% mem_Collect_eq
thf(fact_126_mem__Collect__eq,axiom,
    ! [A3: produc6336967135452217047_b_nat,P2: produc6336967135452217047_b_nat > $o] :
      ( ( member8290135809484590976_b_nat @ A3 @ ( collec5634107928093745858_b_nat @ P2 ) )
      = ( P2 @ A3 ) ) ).

% mem_Collect_eq
thf(fact_127_mem__Collect__eq,axiom,
    ! [A3: produc7779921308951573458um_a_b,P2: produc7779921308951573458um_a_b > $o] :
      ( ( member4052926750399300073um_a_b @ A3 @ ( collec3936109633321590823um_a_b @ P2 ) )
      = ( P2 @ A3 ) ) ).

% mem_Collect_eq
thf(fact_128_mem__Collect__eq,axiom,
    ! [A3: produc5919649888572472791um_a_b,P2: produc5919649888572472791um_a_b > $o] :
      ( ( member7872818562604846720um_a_b @ A3 @ ( collec5216790681214001602um_a_b @ P2 ) )
      = ( P2 @ A3 ) ) ).

% mem_Collect_eq
thf(fact_129_mem__Collect__eq,axiom,
    ! [A3: product_prod_nat_nat,P2: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A3 @ ( collec3392354462482085612at_nat @ P2 ) )
      = ( P2 @ A3 ) ) ).

% mem_Collect_eq
thf(fact_130_mem__Collect__eq,axiom,
    ! [A3: sum_sum_a_b,P2: sum_sum_a_b > $o] :
      ( ( member_Sum_sum_a_b @ A3 @ ( collect_Sum_sum_a_b @ P2 ) )
      = ( P2 @ A3 ) ) ).

% mem_Collect_eq
thf(fact_131_Collect__mem__eq,axiom,
    ! [A5: set_Pr7446880501794416397um_a_b] :
      ( ( collec2013886903173073304um_a_b
        @ ^ [X4: produc4970237071031658413um_a_b] : ( member6480126262776431190um_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_132_Collect__mem__eq,axiom,
    ! [A5: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_133_Collect__mem__eq,axiom,
    ! [A5: set_Pr7634312471471294263_b_nat] :
      ( ( collec5634107928093745858_b_nat
        @ ^ [X4: produc6336967135452217047_b_nat] : ( member8290135809484590976_b_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_134_Collect__mem__eq,axiom,
    ! [A5: set_Pr5956926064383532552um_a_b] :
      ( ( collec3936109633321590823um_a_b
        @ ^ [X4: produc7779921308951573458um_a_b] : ( member4052926750399300073um_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_135_Collect__mem__eq,axiom,
    ! [A5: set_Pr4241354594808546871um_a_b] :
      ( ( collec5216790681214001602um_a_b
        @ ^ [X4: produc5919649888572472791um_a_b] : ( member7872818562604846720um_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_136_Collect__mem__eq,axiom,
    ! [A5: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_137_Collect__mem__eq,axiom,
    ! [A5: set_Sum_sum_a_b] :
      ( ( collect_Sum_sum_a_b
        @ ^ [X4: sum_sum_a_b] : ( member_Sum_sum_a_b @ X4 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_138_vimageI,axiom,
    ! [F: nat > nat,A3: nat,B3: nat,B5: set_nat] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member_nat @ B3 @ B5 )
       => ( member_nat @ A3 @ ( vimage_nat_nat @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_139_vimageI,axiom,
    ! [F: product_prod_nat_nat > nat,A3: product_prod_nat_nat,B3: nat,B5: set_nat] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member_nat @ B3 @ B5 )
       => ( member8440522571783428010at_nat @ A3 @ ( vimage4653281326611754070at_nat @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_140_vimageI,axiom,
    ! [F: sum_sum_a_b > nat,A3: sum_sum_a_b,B3: nat,B5: set_nat] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member_nat @ B3 @ B5 )
       => ( member_Sum_sum_a_b @ A3 @ ( vimage1823869771417566771_b_nat @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_141_vimageI,axiom,
    ! [F: nat > product_prod_nat_nat,A3: nat,B3: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member8440522571783428010at_nat @ B3 @ B5 )
       => ( member_nat @ A3 @ ( vimage8013328719654469172at_nat @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_142_vimageI,axiom,
    ! [F: nat > sum_sum_a_b,A3: nat,B3: sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member_Sum_sum_a_b @ B3 @ B5 )
       => ( member_nat @ A3 @ ( vimage6052884790004658227um_a_b @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_143_vimageI,axiom,
    ! [F: a > sum_sum_a_b,A3: a,B3: sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member_Sum_sum_a_b @ B3 @ B5 )
       => ( member_a @ A3 @ ( vimage_a_Sum_sum_a_b @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_144_vimageI,axiom,
    ! [F: produc6336967135452217047_b_nat > nat,A3: produc6336967135452217047_b_nat,B3: nat,B5: set_nat] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member_nat @ B3 @ B5 )
       => ( member8290135809484590976_b_nat @ A3 @ ( vimage8879000288251783724at_nat @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_145_vimageI,axiom,
    ! [F: produc5919649888572472791um_a_b > nat,A3: produc5919649888572472791um_a_b,B3: nat,B5: set_nat] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member_nat @ B3 @ B5 )
       => ( member7872818562604846720um_a_b @ A3 @ ( vimage5172707497046272300_b_nat @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_146_vimageI,axiom,
    ! [F: nat > produc6336967135452217047_b_nat,A3: nat,B3: produc6336967135452217047_b_nat,B5: set_Pr7634312471471294263_b_nat] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member8290135809484590976_b_nat @ B3 @ B5 )
       => ( member_nat @ A3 @ ( vimage3502331253001533962_b_nat @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_147_vimageI,axiom,
    ! [F: nat > produc5919649888572472791um_a_b,A3: nat,B3: produc5919649888572472791um_a_b,B5: set_Pr4241354594808546871um_a_b] :
      ( ( ( F @ A3 )
        = B3 )
     => ( ( member7872818562604846720um_a_b @ B3 @ B5 )
       => ( member_nat @ A3 @ ( vimage3085014006121789706um_a_b @ F @ B5 ) ) ) ) ).

% vimageI
thf(fact_148_vimage__eq,axiom,
    ! [A3: nat,F: nat > nat,B5: set_nat] :
      ( ( member_nat @ A3 @ ( vimage_nat_nat @ F @ B5 ) )
      = ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_149_vimage__eq,axiom,
    ! [A3: nat,F: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A3 @ ( vimage8013328719654469172at_nat @ F @ B5 ) )
      = ( member8440522571783428010at_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_150_vimage__eq,axiom,
    ! [A3: nat,F: nat > sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( member_nat @ A3 @ ( vimage6052884790004658227um_a_b @ F @ B5 ) )
      = ( member_Sum_sum_a_b @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_151_vimage__eq,axiom,
    ! [A3: product_prod_nat_nat,F: product_prod_nat_nat > nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ A3 @ ( vimage4653281326611754070at_nat @ F @ B5 ) )
      = ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_152_vimage__eq,axiom,
    ! [A3: sum_sum_a_b,F: sum_sum_a_b > nat,B5: set_nat] :
      ( ( member_Sum_sum_a_b @ A3 @ ( vimage1823869771417566771_b_nat @ F @ B5 ) )
      = ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_153_vimage__eq,axiom,
    ! [A3: a,F: a > sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( member_a @ A3 @ ( vimage_a_Sum_sum_a_b @ F @ B5 ) )
      = ( member_Sum_sum_a_b @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_154_vimage__eq,axiom,
    ! [A3: nat,F: nat > produc6336967135452217047_b_nat,B5: set_Pr7634312471471294263_b_nat] :
      ( ( member_nat @ A3 @ ( vimage3502331253001533962_b_nat @ F @ B5 ) )
      = ( member8290135809484590976_b_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_155_vimage__eq,axiom,
    ! [A3: nat,F: nat > produc5919649888572472791um_a_b,B5: set_Pr4241354594808546871um_a_b] :
      ( ( member_nat @ A3 @ ( vimage3085014006121789706um_a_b @ F @ B5 ) )
      = ( member7872818562604846720um_a_b @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_156_vimage__eq,axiom,
    ! [A3: produc6336967135452217047_b_nat,F: produc6336967135452217047_b_nat > nat,B5: set_nat] :
      ( ( member8290135809484590976_b_nat @ A3 @ ( vimage8879000288251783724at_nat @ F @ B5 ) )
      = ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_157_vimage__eq,axiom,
    ! [A3: produc5919649888572472791um_a_b,F: produc5919649888572472791um_a_b > nat,B5: set_nat] :
      ( ( member7872818562604846720um_a_b @ A3 @ ( vimage5172707497046272300_b_nat @ F @ B5 ) )
      = ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimage_eq
thf(fact_158_sum_Oinject_I1_J,axiom,
    ! [X1: a,Y1: a] :
      ( ( ( sum_Inl_a_b @ X1 )
        = ( sum_Inl_a_b @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_159_old_Osum_Oinject_I1_J,axiom,
    ! [A3: a,A4: a] :
      ( ( ( sum_Inl_a_b @ A3 )
        = ( sum_Inl_a_b @ A4 ) )
      = ( A3 = A4 ) ) ).

% old.sum.inject(1)
thf(fact_160_ad__agr__list__def,axiom,
    ( ad_agr_list_a_b
    = ( ^ [X3: set_a,Xs2: list_Sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
          ( ( ( size_s5949480688510517322um_a_b @ Xs2 )
            = ( size_s5949480688510517322um_a_b @ Ys2 ) )
          & ( ad_equiv_list_a_b @ X3 @ Xs2 @ Ys2 )
          & ( sp_equiv_list_a_b @ Xs2 @ Ys2 ) ) ) ) ).

% ad_agr_list_def
thf(fact_161_subsetI,axiom,
    ! [A5: set_Pr7446880501794416397um_a_b,B5: set_Pr7446880501794416397um_a_b] :
      ( ! [X5: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X5 @ A5 )
         => ( member6480126262776431190um_a_b @ X5 @ B5 ) )
     => ( ord_le1431929631403617965um_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_162_subsetI,axiom,
    ! [A5: set_nat,B5: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( member_nat @ X5 @ B5 ) )
     => ( ord_less_eq_set_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_163_subsetI,axiom,
    ! [A5: set_Pr7634312471471294263_b_nat,B5: set_Pr7634312471471294263_b_nat] :
      ( ! [X5: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X5 @ A5 )
         => ( member8290135809484590976_b_nat @ X5 @ B5 ) )
     => ( ord_le8626435567010533079_b_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_164_subsetI,axiom,
    ! [A5: set_Pr5956926064383532552um_a_b,B5: set_Pr5956926064383532552um_a_b] :
      ( ! [X5: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X5 @ A5 )
         => ( member4052926750399300073um_a_b @ X5 @ B5 ) )
     => ( ord_le5806309049788436072um_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_165_subsetI,axiom,
    ! [A5: set_Pr4241354594808546871um_a_b,B5: set_Pr4241354594808546871um_a_b] :
      ( ! [X5: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X5 @ A5 )
         => ( member7872818562604846720um_a_b @ X5 @ B5 ) )
     => ( ord_le5233477690347785687um_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_166_subsetI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A5 )
         => ( member8440522571783428010at_nat @ X5 @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_167_subsetI,axiom,
    ! [A5: set_Sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ! [X5: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X5 @ A5 )
         => ( member_Sum_sum_a_b @ X5 @ B5 ) )
     => ( ord_le9019793522827316924um_a_b @ A5 @ B5 ) ) ).

% subsetI
thf(fact_168_subsetI,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ! [X5: a] :
          ( ( member_a @ X5 @ A5 )
         => ( member_a @ X5 @ B5 ) )
     => ( ord_less_eq_set_a @ A5 @ B5 ) ) ).

% subsetI
thf(fact_169_subset__antisym,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( ord_less_eq_set_a @ B5 @ A5 )
       => ( A5 = B5 ) ) ) ).

% subset_antisym
thf(fact_170_order__refl,axiom,
    ! [X: set_a] : ( ord_less_eq_set_a @ X @ X ) ).

% order_refl
thf(fact_171_order__refl,axiom,
    ! [X: nat] : ( ord_less_eq_nat @ X @ X ) ).

% order_refl
thf(fact_172_dual__order_Orefl,axiom,
    ! [A3: set_a] : ( ord_less_eq_set_a @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_173_dual__order_Orefl,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ A3 @ A3 ) ).

% dual_order.refl
thf(fact_174_vimage__mono,axiom,
    ! [A5: set_Sum_sum_a_b,B5: set_Sum_sum_a_b,F: a > sum_sum_a_b] :
      ( ( ord_le9019793522827316924um_a_b @ A5 @ B5 )
     => ( ord_less_eq_set_a @ ( vimage_a_Sum_sum_a_b @ F @ A5 ) @ ( vimage_a_Sum_sum_a_b @ F @ B5 ) ) ) ).

% vimage_mono
thf(fact_175_vimage__mono,axiom,
    ! [A5: set_a,B5: set_a,F: a > a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ord_less_eq_set_a @ ( vimage_a_a @ F @ A5 ) @ ( vimage_a_a @ F @ B5 ) ) ) ).

% vimage_mono
thf(fact_176_order__antisym__conv,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ( ( ord_less_eq_set_a @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_177_order__antisym__conv,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% order_antisym_conv
thf(fact_178_linorder__le__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_le_cases
thf(fact_179_ord__le__eq__subst,axiom,
    ! [A3: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_180_ord__le__eq__subst,axiom,
    ! [A3: set_a,B3: set_a,F: set_a > nat,C2: nat] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_181_ord__le__eq__subst,axiom,
    ! [A3: nat,B3: nat,F: nat > set_a,C2: set_a] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_182_ord__le__eq__subst,axiom,
    ! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_183_ord__eq__le__subst,axiom,
    ! [A3: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_184_ord__eq__le__subst,axiom,
    ! [A3: nat,F: set_a > nat,B3: set_a,C2: set_a] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_185_ord__eq__le__subst,axiom,
    ! [A3: set_a,F: nat > set_a,B3: nat,C2: nat] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_186_ord__eq__le__subst,axiom,
    ! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_187_linorder__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_linear
thf(fact_188_order__eq__refl,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( X = Y )
     => ( ord_less_eq_set_a @ X @ Y ) ) ).

% order_eq_refl
thf(fact_189_order__eq__refl,axiom,
    ! [X: nat,Y: nat] :
      ( ( X = Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_eq_refl
thf(fact_190_order__subst2,axiom,
    ! [A3: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_191_order__subst2,axiom,
    ! [A3: set_a,B3: set_a,F: set_a > nat,C2: nat] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_192_order__subst2,axiom,
    ! [A3: nat,B3: nat,F: nat > set_a,C2: set_a] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_193_order__subst2,axiom,
    ! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_194_order__subst1,axiom,
    ! [A3: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_195_order__subst1,axiom,
    ! [A3: set_a,F: nat > set_a,B3: nat,C2: nat] :
      ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_196_order__subst1,axiom,
    ! [A3: nat,F: set_a > nat,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_197_order__subst1,axiom,
    ! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_198_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
    = ( ^ [A6: set_a,B6: set_a] :
          ( ( ord_less_eq_set_a @ A6 @ B6 )
          & ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_199_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_200_antisym,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% antisym
thf(fact_201_antisym,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ A3 )
       => ( A3 = B3 ) ) ) ).

% antisym
thf(fact_202_dual__order_Otrans,axiom,
    ! [B3: set_a,A3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A3 )
     => ( ( ord_less_eq_set_a @ C2 @ B3 )
       => ( ord_less_eq_set_a @ C2 @ A3 ) ) ) ).

% dual_order.trans
thf(fact_203_dual__order_Otrans,axiom,
    ! [B3: nat,A3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( ord_less_eq_nat @ C2 @ B3 )
       => ( ord_less_eq_nat @ C2 @ A3 ) ) ) ).

% dual_order.trans
thf(fact_204_dual__order_Oantisym,axiom,
    ! [B3: set_a,A3: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A3 )
     => ( ( ord_less_eq_set_a @ A3 @ B3 )
       => ( A3 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_205_dual__order_Oantisym,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( ord_less_eq_nat @ A3 @ B3 )
       => ( A3 = B3 ) ) ) ).

% dual_order.antisym
thf(fact_206_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
    = ( ^ [A6: set_a,B6: set_a] :
          ( ( ord_less_eq_set_a @ B6 @ A6 )
          & ( ord_less_eq_set_a @ A6 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_207_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_208_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A3: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( ( ord_less_eq_nat @ A @ B )
         => ( P2 @ A @ B ) )
     => ( ! [A: nat,B: nat] :
            ( ( P2 @ B @ A )
           => ( P2 @ A @ B ) )
       => ( P2 @ A3 @ B3 ) ) ) ).

% linorder_wlog
thf(fact_209_order__trans,axiom,
    ! [X: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z2 )
       => ( ord_less_eq_set_a @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_210_order__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_eq_nat @ X @ Z2 ) ) ) ).

% order_trans
thf(fact_211_order_Otrans,axiom,
    ! [A3: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).

% order.trans
thf(fact_212_order_Otrans,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).

% order.trans
thf(fact_213_order__antisym,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_214_order__antisym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ X )
       => ( X = Y ) ) ) ).

% order_antisym
thf(fact_215_ord__le__eq__trans,axiom,
    ! [A3: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_216_ord__le__eq__trans,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_217_ord__eq__le__trans,axiom,
    ! [A3: set_a,B3: set_a,C2: set_a] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ord_less_eq_set_a @ A3 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_218_ord__eq__le__trans,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( A3 = B3 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ord_less_eq_nat @ A3 @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_219_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y5 )
          & ( ord_less_eq_set_a @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_220_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : ( Y4 = Z ) )
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_221_le__cases3,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y @ X )
         => ~ ( ord_less_eq_nat @ X @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X )
                 => ~ ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_222_nle__le,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ~ ( ord_less_eq_nat @ A3 @ B3 ) )
      = ( ( ord_less_eq_nat @ B3 @ A3 )
        & ( B3 != A3 ) ) ) ).

% nle_le
thf(fact_223_Collect__mono__iff,axiom,
    ! [P2: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) )
      = ( ! [X4: a] :
            ( ( P2 @ X4 )
           => ( Q @ X4 ) ) ) ) ).

% Collect_mono_iff
thf(fact_224_set__eq__subset,axiom,
    ( ( ^ [Y4: set_a,Z: set_a] : ( Y4 = Z ) )
    = ( ^ [A7: set_a,B7: set_a] :
          ( ( ord_less_eq_set_a @ A7 @ B7 )
          & ( ord_less_eq_set_a @ B7 @ A7 ) ) ) ) ).

% set_eq_subset
thf(fact_225_subset__trans,axiom,
    ! [A5: set_a,B5: set_a,C3: set_a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( ord_less_eq_set_a @ B5 @ C3 )
       => ( ord_less_eq_set_a @ A5 @ C3 ) ) ) ).

% subset_trans
thf(fact_226_Collect__mono,axiom,
    ! [P2: a > $o,Q: a > $o] :
      ( ! [X5: a] :
          ( ( P2 @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_a @ ( collect_a @ P2 ) @ ( collect_a @ Q ) ) ) ).

% Collect_mono
thf(fact_227_subset__refl,axiom,
    ! [A5: set_a] : ( ord_less_eq_set_a @ A5 @ A5 ) ).

% subset_refl
thf(fact_228_subset__iff,axiom,
    ( ord_le1431929631403617965um_a_b
    = ( ^ [A7: set_Pr7446880501794416397um_a_b,B7: set_Pr7446880501794416397um_a_b] :
        ! [T: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ T @ A7 )
         => ( member6480126262776431190um_a_b @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_229_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
        ! [T: nat] :
          ( ( member_nat @ T @ A7 )
         => ( member_nat @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_230_subset__iff,axiom,
    ( ord_le8626435567010533079_b_nat
    = ( ^ [A7: set_Pr7634312471471294263_b_nat,B7: set_Pr7634312471471294263_b_nat] :
        ! [T: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ T @ A7 )
         => ( member8290135809484590976_b_nat @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_231_subset__iff,axiom,
    ( ord_le5806309049788436072um_a_b
    = ( ^ [A7: set_Pr5956926064383532552um_a_b,B7: set_Pr5956926064383532552um_a_b] :
        ! [T: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ T @ A7 )
         => ( member4052926750399300073um_a_b @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_232_subset__iff,axiom,
    ( ord_le5233477690347785687um_a_b
    = ( ^ [A7: set_Pr4241354594808546871um_a_b,B7: set_Pr4241354594808546871um_a_b] :
        ! [T: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ T @ A7 )
         => ( member7872818562604846720um_a_b @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_233_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
        ! [T: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T @ A7 )
         => ( member8440522571783428010at_nat @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_234_subset__iff,axiom,
    ( ord_le9019793522827316924um_a_b
    = ( ^ [A7: set_Sum_sum_a_b,B7: set_Sum_sum_a_b] :
        ! [T: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ T @ A7 )
         => ( member_Sum_sum_a_b @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_235_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A7: set_a,B7: set_a] :
        ! [T: a] :
          ( ( member_a @ T @ A7 )
         => ( member_a @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_236_Set_OequalityD2,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_a @ B5 @ A5 ) ) ).

% Set.equalityD2
thf(fact_237_equalityD1,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( A5 = B5 )
     => ( ord_less_eq_set_a @ A5 @ B5 ) ) ).

% equalityD1
thf(fact_238_subset__eq,axiom,
    ( ord_le1431929631403617965um_a_b
    = ( ^ [A7: set_Pr7446880501794416397um_a_b,B7: set_Pr7446880501794416397um_a_b] :
        ! [X4: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X4 @ A7 )
         => ( member6480126262776431190um_a_b @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_239_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A7: set_nat,B7: set_nat] :
        ! [X4: nat] :
          ( ( member_nat @ X4 @ A7 )
         => ( member_nat @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_240_subset__eq,axiom,
    ( ord_le8626435567010533079_b_nat
    = ( ^ [A7: set_Pr7634312471471294263_b_nat,B7: set_Pr7634312471471294263_b_nat] :
        ! [X4: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X4 @ A7 )
         => ( member8290135809484590976_b_nat @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_241_subset__eq,axiom,
    ( ord_le5806309049788436072um_a_b
    = ( ^ [A7: set_Pr5956926064383532552um_a_b,B7: set_Pr5956926064383532552um_a_b] :
        ! [X4: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X4 @ A7 )
         => ( member4052926750399300073um_a_b @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_242_subset__eq,axiom,
    ( ord_le5233477690347785687um_a_b
    = ( ^ [A7: set_Pr4241354594808546871um_a_b,B7: set_Pr4241354594808546871um_a_b] :
        ! [X4: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X4 @ A7 )
         => ( member7872818562604846720um_a_b @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_243_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A7: set_Pr1261947904930325089at_nat,B7: set_Pr1261947904930325089at_nat] :
        ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A7 )
         => ( member8440522571783428010at_nat @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_244_subset__eq,axiom,
    ( ord_le9019793522827316924um_a_b
    = ( ^ [A7: set_Sum_sum_a_b,B7: set_Sum_sum_a_b] :
        ! [X4: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X4 @ A7 )
         => ( member_Sum_sum_a_b @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_245_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A7: set_a,B7: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A7 )
         => ( member_a @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_246_equalityE,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( A5 = B5 )
     => ~ ( ( ord_less_eq_set_a @ A5 @ B5 )
         => ~ ( ord_less_eq_set_a @ B5 @ A5 ) ) ) ).

% equalityE
thf(fact_247_subsetD,axiom,
    ! [A5: set_Pr7446880501794416397um_a_b,B5: set_Pr7446880501794416397um_a_b,C2: produc4970237071031658413um_a_b] :
      ( ( ord_le1431929631403617965um_a_b @ A5 @ B5 )
     => ( ( member6480126262776431190um_a_b @ C2 @ A5 )
       => ( member6480126262776431190um_a_b @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_248_subsetD,axiom,
    ! [A5: set_nat,B5: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat @ C2 @ A5 )
       => ( member_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_249_subsetD,axiom,
    ! [A5: set_Pr7634312471471294263_b_nat,B5: set_Pr7634312471471294263_b_nat,C2: produc6336967135452217047_b_nat] :
      ( ( ord_le8626435567010533079_b_nat @ A5 @ B5 )
     => ( ( member8290135809484590976_b_nat @ C2 @ A5 )
       => ( member8290135809484590976_b_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_250_subsetD,axiom,
    ! [A5: set_Pr5956926064383532552um_a_b,B5: set_Pr5956926064383532552um_a_b,C2: produc7779921308951573458um_a_b] :
      ( ( ord_le5806309049788436072um_a_b @ A5 @ B5 )
     => ( ( member4052926750399300073um_a_b @ C2 @ A5 )
       => ( member4052926750399300073um_a_b @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_251_subsetD,axiom,
    ! [A5: set_Pr4241354594808546871um_a_b,B5: set_Pr4241354594808546871um_a_b,C2: produc5919649888572472791um_a_b] :
      ( ( ord_le5233477690347785687um_a_b @ A5 @ B5 )
     => ( ( member7872818562604846720um_a_b @ C2 @ A5 )
       => ( member7872818562604846720um_a_b @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_252_subsetD,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,C2: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
     => ( ( member8440522571783428010at_nat @ C2 @ A5 )
       => ( member8440522571783428010at_nat @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_253_subsetD,axiom,
    ! [A5: set_Sum_sum_a_b,B5: set_Sum_sum_a_b,C2: sum_sum_a_b] :
      ( ( ord_le9019793522827316924um_a_b @ A5 @ B5 )
     => ( ( member_Sum_sum_a_b @ C2 @ A5 )
       => ( member_Sum_sum_a_b @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_254_subsetD,axiom,
    ! [A5: set_a,B5: set_a,C2: a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( member_a @ C2 @ A5 )
       => ( member_a @ C2 @ B5 ) ) ) ).

% subsetD
thf(fact_255_in__mono,axiom,
    ! [A5: set_Pr7446880501794416397um_a_b,B5: set_Pr7446880501794416397um_a_b,X: produc4970237071031658413um_a_b] :
      ( ( ord_le1431929631403617965um_a_b @ A5 @ B5 )
     => ( ( member6480126262776431190um_a_b @ X @ A5 )
       => ( member6480126262776431190um_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_256_in__mono,axiom,
    ! [A5: set_nat,B5: set_nat,X: nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ( member_nat @ X @ A5 )
       => ( member_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_257_in__mono,axiom,
    ! [A5: set_Pr7634312471471294263_b_nat,B5: set_Pr7634312471471294263_b_nat,X: produc6336967135452217047_b_nat] :
      ( ( ord_le8626435567010533079_b_nat @ A5 @ B5 )
     => ( ( member8290135809484590976_b_nat @ X @ A5 )
       => ( member8290135809484590976_b_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_258_in__mono,axiom,
    ! [A5: set_Pr5956926064383532552um_a_b,B5: set_Pr5956926064383532552um_a_b,X: produc7779921308951573458um_a_b] :
      ( ( ord_le5806309049788436072um_a_b @ A5 @ B5 )
     => ( ( member4052926750399300073um_a_b @ X @ A5 )
       => ( member4052926750399300073um_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_259_in__mono,axiom,
    ! [A5: set_Pr4241354594808546871um_a_b,B5: set_Pr4241354594808546871um_a_b,X: produc5919649888572472791um_a_b] :
      ( ( ord_le5233477690347785687um_a_b @ A5 @ B5 )
     => ( ( member7872818562604846720um_a_b @ X @ A5 )
       => ( member7872818562604846720um_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_260_in__mono,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A5 @ B5 )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( member8440522571783428010at_nat @ X @ B5 ) ) ) ).

% in_mono
thf(fact_261_in__mono,axiom,
    ! [A5: set_Sum_sum_a_b,B5: set_Sum_sum_a_b,X: sum_sum_a_b] :
      ( ( ord_le9019793522827316924um_a_b @ A5 @ B5 )
     => ( ( member_Sum_sum_a_b @ X @ A5 )
       => ( member_Sum_sum_a_b @ X @ B5 ) ) ) ).

% in_mono
thf(fact_262_in__mono,axiom,
    ! [A5: set_a,B5: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ( member_a @ X @ A5 )
       => ( member_a @ X @ B5 ) ) ) ).

% in_mono
thf(fact_263_Inl__inject,axiom,
    ! [X: a,Y: a] :
      ( ( ( sum_Inl_a_b @ X )
        = ( sum_Inl_a_b @ Y ) )
     => ( X = Y ) ) ).

% Inl_inject
thf(fact_264_vimage__Collect,axiom,
    ! [P2: sum_sum_a_b > $o,F: a > sum_sum_a_b,Q: a > $o] :
      ( ! [X5: a] :
          ( ( P2 @ ( F @ X5 ) )
          = ( Q @ X5 ) )
     => ( ( vimage_a_Sum_sum_a_b @ F @ ( collect_Sum_sum_a_b @ P2 ) )
        = ( collect_a @ Q ) ) ) ).

% vimage_Collect
thf(fact_265_vimageI2,axiom,
    ! [F: nat > nat,A3: nat,A5: set_nat] :
      ( ( member_nat @ ( F @ A3 ) @ A5 )
     => ( member_nat @ A3 @ ( vimage_nat_nat @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_266_vimageI2,axiom,
    ! [F: product_prod_nat_nat > nat,A3: product_prod_nat_nat,A5: set_nat] :
      ( ( member_nat @ ( F @ A3 ) @ A5 )
     => ( member8440522571783428010at_nat @ A3 @ ( vimage4653281326611754070at_nat @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_267_vimageI2,axiom,
    ! [F: sum_sum_a_b > nat,A3: sum_sum_a_b,A5: set_nat] :
      ( ( member_nat @ ( F @ A3 ) @ A5 )
     => ( member_Sum_sum_a_b @ A3 @ ( vimage1823869771417566771_b_nat @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_268_vimageI2,axiom,
    ! [F: nat > product_prod_nat_nat,A3: nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ ( F @ A3 ) @ A5 )
     => ( member_nat @ A3 @ ( vimage8013328719654469172at_nat @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_269_vimageI2,axiom,
    ! [F: nat > sum_sum_a_b,A3: nat,A5: set_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ ( F @ A3 ) @ A5 )
     => ( member_nat @ A3 @ ( vimage6052884790004658227um_a_b @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_270_vimageI2,axiom,
    ! [F: a > sum_sum_a_b,A3: a,A5: set_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ ( F @ A3 ) @ A5 )
     => ( member_a @ A3 @ ( vimage_a_Sum_sum_a_b @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_271_vimageI2,axiom,
    ! [F: produc6336967135452217047_b_nat > nat,A3: produc6336967135452217047_b_nat,A5: set_nat] :
      ( ( member_nat @ ( F @ A3 ) @ A5 )
     => ( member8290135809484590976_b_nat @ A3 @ ( vimage8879000288251783724at_nat @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_272_vimageI2,axiom,
    ! [F: produc5919649888572472791um_a_b > nat,A3: produc5919649888572472791um_a_b,A5: set_nat] :
      ( ( member_nat @ ( F @ A3 ) @ A5 )
     => ( member7872818562604846720um_a_b @ A3 @ ( vimage5172707497046272300_b_nat @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_273_vimageI2,axiom,
    ! [F: nat > produc6336967135452217047_b_nat,A3: nat,A5: set_Pr7634312471471294263_b_nat] :
      ( ( member8290135809484590976_b_nat @ ( F @ A3 ) @ A5 )
     => ( member_nat @ A3 @ ( vimage3502331253001533962_b_nat @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_274_vimageI2,axiom,
    ! [F: nat > produc5919649888572472791um_a_b,A3: nat,A5: set_Pr4241354594808546871um_a_b] :
      ( ( member7872818562604846720um_a_b @ ( F @ A3 ) @ A5 )
     => ( member_nat @ A3 @ ( vimage3085014006121789706um_a_b @ F @ A5 ) ) ) ).

% vimageI2
thf(fact_275_vimageE,axiom,
    ! [A3: nat,F: nat > nat,B5: set_nat] :
      ( ( member_nat @ A3 @ ( vimage_nat_nat @ F @ B5 ) )
     => ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_276_vimageE,axiom,
    ! [A3: nat,F: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A3 @ ( vimage8013328719654469172at_nat @ F @ B5 ) )
     => ( member8440522571783428010at_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_277_vimageE,axiom,
    ! [A3: nat,F: nat > sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( member_nat @ A3 @ ( vimage6052884790004658227um_a_b @ F @ B5 ) )
     => ( member_Sum_sum_a_b @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_278_vimageE,axiom,
    ! [A3: product_prod_nat_nat,F: product_prod_nat_nat > nat,B5: set_nat] :
      ( ( member8440522571783428010at_nat @ A3 @ ( vimage4653281326611754070at_nat @ F @ B5 ) )
     => ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_279_vimageE,axiom,
    ! [A3: sum_sum_a_b,F: sum_sum_a_b > nat,B5: set_nat] :
      ( ( member_Sum_sum_a_b @ A3 @ ( vimage1823869771417566771_b_nat @ F @ B5 ) )
     => ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_280_vimageE,axiom,
    ! [A3: a,F: a > sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( member_a @ A3 @ ( vimage_a_Sum_sum_a_b @ F @ B5 ) )
     => ( member_Sum_sum_a_b @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_281_vimageE,axiom,
    ! [A3: nat,F: nat > produc6336967135452217047_b_nat,B5: set_Pr7634312471471294263_b_nat] :
      ( ( member_nat @ A3 @ ( vimage3502331253001533962_b_nat @ F @ B5 ) )
     => ( member8290135809484590976_b_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_282_vimageE,axiom,
    ! [A3: nat,F: nat > produc5919649888572472791um_a_b,B5: set_Pr4241354594808546871um_a_b] :
      ( ( member_nat @ A3 @ ( vimage3085014006121789706um_a_b @ F @ B5 ) )
     => ( member7872818562604846720um_a_b @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_283_vimageE,axiom,
    ! [A3: produc6336967135452217047_b_nat,F: produc6336967135452217047_b_nat > nat,B5: set_nat] :
      ( ( member8290135809484590976_b_nat @ A3 @ ( vimage8879000288251783724at_nat @ F @ B5 ) )
     => ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_284_vimageE,axiom,
    ! [A3: produc5919649888572472791um_a_b,F: produc5919649888572472791um_a_b > nat,B5: set_nat] :
      ( ( member7872818562604846720um_a_b @ A3 @ ( vimage5172707497046272300_b_nat @ F @ B5 ) )
     => ( member_nat @ ( F @ A3 ) @ B5 ) ) ).

% vimageE
thf(fact_285_vimageD,axiom,
    ! [A3: nat,F: nat > nat,A5: set_nat] :
      ( ( member_nat @ A3 @ ( vimage_nat_nat @ F @ A5 ) )
     => ( member_nat @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_286_vimageD,axiom,
    ! [A3: nat,F: nat > product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member_nat @ A3 @ ( vimage8013328719654469172at_nat @ F @ A5 ) )
     => ( member8440522571783428010at_nat @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_287_vimageD,axiom,
    ! [A3: nat,F: nat > sum_sum_a_b,A5: set_Sum_sum_a_b] :
      ( ( member_nat @ A3 @ ( vimage6052884790004658227um_a_b @ F @ A5 ) )
     => ( member_Sum_sum_a_b @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_288_vimageD,axiom,
    ! [A3: product_prod_nat_nat,F: product_prod_nat_nat > nat,A5: set_nat] :
      ( ( member8440522571783428010at_nat @ A3 @ ( vimage4653281326611754070at_nat @ F @ A5 ) )
     => ( member_nat @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_289_vimageD,axiom,
    ! [A3: sum_sum_a_b,F: sum_sum_a_b > nat,A5: set_nat] :
      ( ( member_Sum_sum_a_b @ A3 @ ( vimage1823869771417566771_b_nat @ F @ A5 ) )
     => ( member_nat @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_290_vimageD,axiom,
    ! [A3: a,F: a > sum_sum_a_b,A5: set_Sum_sum_a_b] :
      ( ( member_a @ A3 @ ( vimage_a_Sum_sum_a_b @ F @ A5 ) )
     => ( member_Sum_sum_a_b @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_291_vimageD,axiom,
    ! [A3: nat,F: nat > produc6336967135452217047_b_nat,A5: set_Pr7634312471471294263_b_nat] :
      ( ( member_nat @ A3 @ ( vimage3502331253001533962_b_nat @ F @ A5 ) )
     => ( member8290135809484590976_b_nat @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_292_vimageD,axiom,
    ! [A3: nat,F: nat > produc5919649888572472791um_a_b,A5: set_Pr4241354594808546871um_a_b] :
      ( ( member_nat @ A3 @ ( vimage3085014006121789706um_a_b @ F @ A5 ) )
     => ( member7872818562604846720um_a_b @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_293_vimageD,axiom,
    ! [A3: produc6336967135452217047_b_nat,F: produc6336967135452217047_b_nat > nat,A5: set_nat] :
      ( ( member8290135809484590976_b_nat @ A3 @ ( vimage8879000288251783724at_nat @ F @ A5 ) )
     => ( member_nat @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_294_vimageD,axiom,
    ! [A3: produc5919649888572472791um_a_b,F: produc5919649888572472791um_a_b > nat,A5: set_nat] :
      ( ( member7872818562604846720um_a_b @ A3 @ ( vimage5172707497046272300_b_nat @ F @ A5 ) )
     => ( member_nat @ ( F @ A3 ) @ A5 ) ) ).

% vimageD
thf(fact_295_subset__vimage__iff,axiom,
    ! [A5: set_a,F: a > produc4970237071031658413um_a_b,B5: set_Pr7446880501794416397um_a_b] :
      ( ( ord_less_eq_set_a @ A5 @ ( vimage1386989498120282986um_a_b @ F @ B5 ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A5 )
           => ( member6480126262776431190um_a_b @ ( F @ X4 ) @ B5 ) ) ) ) ).

% subset_vimage_iff
thf(fact_296_subset__vimage__iff,axiom,
    ! [A5: set_a,F: a > nat,B5: set_nat] :
      ( ( ord_less_eq_set_a @ A5 @ ( vimage_a_nat @ F @ B5 ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A5 )
           => ( member_nat @ ( F @ X4 ) @ B5 ) ) ) ) ).

% subset_vimage_iff
thf(fact_297_subset__vimage__iff,axiom,
    ! [A5: set_a,F: a > produc6336967135452217047_b_nat,B5: set_Pr7634312471471294263_b_nat] :
      ( ( ord_less_eq_set_a @ A5 @ ( vimage8301957965227796628_b_nat @ F @ B5 ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A5 )
           => ( member8290135809484590976_b_nat @ ( F @ X4 ) @ B5 ) ) ) ) ).

% subset_vimage_iff
thf(fact_298_subset__vimage__iff,axiom,
    ! [A5: set_a,F: a > produc7779921308951573458um_a_b,B5: set_Pr5956926064383532552um_a_b] :
      ( ( ord_less_eq_set_a @ A5 @ ( vimage2972767995635513301um_a_b @ F @ B5 ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A5 )
           => ( member4052926750399300073um_a_b @ ( F @ X4 ) @ B5 ) ) ) ) ).

% subset_vimage_iff
thf(fact_299_subset__vimage__iff,axiom,
    ! [A5: set_a,F: a > produc5919649888572472791um_a_b,B5: set_Pr4241354594808546871um_a_b] :
      ( ( ord_less_eq_set_a @ A5 @ ( vimage7884640718348052372um_a_b @ F @ B5 ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A5 )
           => ( member7872818562604846720um_a_b @ ( F @ X4 ) @ B5 ) ) ) ) ).

% subset_vimage_iff
thf(fact_300_subset__vimage__iff,axiom,
    ! [A5: set_a,F: a > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_set_a @ A5 @ ( vimage5677430894802790590at_nat @ F @ B5 ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A5 )
           => ( member8440522571783428010at_nat @ ( F @ X4 ) @ B5 ) ) ) ) ).

% subset_vimage_iff
thf(fact_301_subset__vimage__iff,axiom,
    ! [A5: set_a,F: a > sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ( ord_less_eq_set_a @ A5 @ ( vimage_a_Sum_sum_a_b @ F @ B5 ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A5 )
           => ( member_Sum_sum_a_b @ ( F @ X4 ) @ B5 ) ) ) ) ).

% subset_vimage_iff
thf(fact_302_sp__equiv__list__def,axiom,
    ( sp_equiv_list_a_b
    = ( ^ [Xs2: list_Sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
          ( ( ( size_s5949480688510517322um_a_b @ Xs2 )
            = ( size_s5949480688510517322um_a_b @ Ys2 ) )
          & ( pairwi757434828799223448um_a_b @ sp_equ4404609506204868152um_a_b @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs2 @ Ys2 ) ) ) ) ) ) ).

% sp_equiv_list_def
thf(fact_303_ad__equiv__list__link,axiom,
    ! [Ns: list_P911570089447184179um_a_b,X2: set_a,Sigma: produc4970237071031658413um_a_b > sum_sum_a_b,Tau: produc4970237071031658413um_a_b > sum_sum_a_b] :
      ( ( ! [X4: produc4970237071031658413um_a_b] :
            ( ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ Ns ) )
           => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ ( Sigma @ X4 ) @ ( Tau @ X4 ) ) ) ) )
      = ( ad_equiv_list_a_b @ X2 @ ( map_Pr8306842558415043215um_a_b @ Sigma @ Ns ) @ ( map_Pr8306842558415043215um_a_b @ Tau @ Ns ) ) ) ).

% ad_equiv_list_link
thf(fact_304_ad__equiv__list__link,axiom,
    ! [Ns: list_Sum_sum_a_b,X2: set_a,Sigma: sum_sum_a_b > sum_sum_a_b,Tau: sum_sum_a_b > sum_sum_a_b] :
      ( ( ! [X4: sum_sum_a_b] :
            ( ( member_Sum_sum_a_b @ X4 @ ( set_Sum_sum_a_b2 @ Ns ) )
           => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ ( Sigma @ X4 ) @ ( Tau @ X4 ) ) ) ) )
      = ( ad_equiv_list_a_b @ X2 @ ( map_Su3007332696077240082um_a_b @ Sigma @ Ns ) @ ( map_Su3007332696077240082um_a_b @ Tau @ Ns ) ) ) ).

% ad_equiv_list_link
thf(fact_305_ad__equiv__list__link,axiom,
    ! [Ns: list_P667245297671117661_b_nat,X2: set_a,Sigma: produc6336967135452217047_b_nat > sum_sum_a_b,Tau: produc6336967135452217047_b_nat > sum_sum_a_b] :
      ( ( ! [X4: produc6336967135452217047_b_nat] :
            ( ( member8290135809484590976_b_nat @ X4 @ ( set_Pr176305205332118508_b_nat @ Ns ) )
           => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ ( Sigma @ X4 ) @ ( Tau @ X4 ) ) ) ) )
      = ( ad_equiv_list_a_b @ X2 @ ( map_Pr1993257472740102073um_a_b @ Sigma @ Ns ) @ ( map_Pr1993257472740102073um_a_b @ Tau @ Ns ) ) ) ).

% ad_equiv_list_link
thf(fact_306_ad__equiv__list__link,axiom,
    ! [Ns: list_P5598968436384720482um_a_b,X2: set_a,Sigma: produc7779921308951573458um_a_b > sum_sum_a_b,Tau: produc7779921308951573458um_a_b > sum_sum_a_b] :
      ( ( ! [X4: produc7779921308951573458um_a_b] :
            ( ( member4052926750399300073um_a_b @ X4 @ ( set_Pr7399620718385288829um_a_b @ Ns ) )
           => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ ( Sigma @ X4 ) @ ( Tau @ X4 ) ) ) ) )
      = ( ad_equiv_list_a_b @ X2 @ ( map_Pr6728900482313898430um_a_b @ Sigma @ Ns ) @ ( map_Pr6728900482313898430um_a_b @ Tau @ Ns ) ) ) ).

% ad_equiv_list_link
thf(fact_307_ad__equiv__list__link,axiom,
    ! [Ns: list_P6497659457863146077um_a_b,X2: set_a,Sigma: produc5919649888572472791um_a_b > sum_sum_a_b,Tau: produc5919649888572472791um_a_b > sum_sum_a_b] :
      ( ( ! [X4: produc5919649888572472791um_a_b] :
            ( ( member7872818562604846720um_a_b @ X4 @ ( set_Pr8982359995307150060um_a_b @ Ns ) )
           => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ ( Sigma @ X4 ) @ ( Tau @ X4 ) ) ) ) )
      = ( ad_equiv_list_a_b @ X2 @ ( map_Pr2204973106654727865um_a_b @ Sigma @ Ns ) @ ( map_Pr2204973106654727865um_a_b @ Tau @ Ns ) ) ) ).

% ad_equiv_list_link
thf(fact_308_ad__equiv__list__link,axiom,
    ! [Ns: list_P6011104703257516679at_nat,X2: set_a,Sigma: product_prod_nat_nat > sum_sum_a_b,Tau: product_prod_nat_nat > sum_sum_a_b] :
      ( ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ns ) )
           => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ ( Sigma @ X4 ) @ ( Tau @ X4 ) ) ) ) )
      = ( ad_equiv_list_a_b @ X2 @ ( map_Pr607579787586854883um_a_b @ Sigma @ Ns ) @ ( map_Pr607579787586854883um_a_b @ Tau @ Ns ) ) ) ).

% ad_equiv_list_link
thf(fact_309_ad__equiv__list__link,axiom,
    ! [Ns: list_nat,X2: set_a,Sigma: nat > sum_sum_a_b,Tau: nat > sum_sum_a_b] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ns ) )
           => ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ ( Sigma @ X4 ) @ ( Tau @ X4 ) ) ) ) )
      = ( ad_equiv_list_a_b @ X2 @ ( map_nat_Sum_sum_a_b @ Sigma @ Ns ) @ ( map_nat_Sum_sum_a_b @ Tau @ Ns ) ) ) ).

% ad_equiv_list_link
thf(fact_310_same__fstI,axiom,
    ! [P2: sum_sum_a_b > $o,X: sum_sum_a_b,Y6: produc4970237071031658413um_a_b,Y: produc4970237071031658413um_a_b,R: sum_sum_a_b > set_Pr8399320699097828935um_a_b] :
      ( ( P2 @ X )
     => ( ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ Y6 @ Y ) @ ( R @ X ) )
       => ( member4727105268299851138um_a_b @ ( produc4166429159020078609um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y6 ) @ ( produc7989310782275393198um_a_b @ X @ Y ) ) @ ( same_f9045959835080170777um_a_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_311_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y6: produc4970237071031658413um_a_b,Y: produc4970237071031658413um_a_b,R: nat > set_Pr8399320699097828935um_a_b] :
      ( ( P2 @ X )
     => ( ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ Y6 @ Y ) @ ( R @ X ) )
       => ( member885238345412010542um_a_b @ ( produc8504789778075864509um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y6 ) @ ( produc6016411306719553028um_a_b @ X @ Y ) ) @ ( same_f6102919435884752495um_a_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_312_same__fstI,axiom,
    ! [P2: sum_sum_a_b > $o,X: sum_sum_a_b,Y6: sum_sum_a_b,Y: sum_sum_a_b,R: sum_sum_a_b > set_Pr7446880501794416397um_a_b] :
      ( ( P2 @ X )
     => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ Y6 @ Y ) @ ( R @ X ) )
       => ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y6 ) @ ( produc8374529264883863525um_a_b @ X @ Y ) ) @ ( same_f5633148067467159866um_a_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_313_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y6: sum_sum_a_b,Y: sum_sum_a_b,R: nat > set_Pr7446880501794416397um_a_b] :
      ( ( P2 @ X )
     => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ Y6 @ Y ) @ ( R @ X ) )
       => ( member6178717281470555920um_a_b @ ( produc8963228657344478039um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y6 ) @ ( produc4099722813291388943um_a_b @ X @ Y ) ) @ ( same_f4421999330425108324um_a_b @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_314_same__fstI,axiom,
    ! [P2: sum_sum_a_b > $o,X: sum_sum_a_b,Y6: nat,Y: nat,R: sum_sum_a_b > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R @ X ) )
       => ( member681441685064050448_b_nat @ ( produc490312431154969431_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y6 ) @ ( produc9094079831559073295_b_nat @ X @ Y ) ) @ ( same_f192984311838016868_b_nat @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_315_same__fstI,axiom,
    ! [P2: nat > $o,X: nat,Y6: nat,Y: nat,R: nat > set_Pr1261947904930325089at_nat] :
      ( ( P2 @ X )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y6 @ Y ) @ ( R @ X ) )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X @ Y6 ) @ ( product_Pair_nat_nat @ X @ Y ) ) @ ( same_fst_nat_nat @ P2 @ R ) ) ) ) ).

% same_fstI
thf(fact_316_relChain__def,axiom,
    ( bNF_Ca26987310307872402_set_a
    = ( ^ [R2: set_Pr7446880501794416397um_a_b,As: sum_sum_a_b > set_a] :
        ! [I: sum_sum_a_b,J: sum_sum_a_b] :
          ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ I @ J ) @ R2 )
         => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_317_relChain__def,axiom,
    ( bNF_Ca350241719529149160_set_a
    = ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > set_a] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_set_a @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_318_relChain__def,axiom,
    ( bNF_Ca8277740611438156956_b_nat
    = ( ^ [R2: set_Pr7446880501794416397um_a_b,As: sum_sum_a_b > nat] :
        ! [I: sum_sum_a_b,J: sum_sum_a_b] :
          ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ I @ J ) @ R2 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_319_relChain__def,axiom,
    ( bNF_Ca968750328013420230at_nat
    = ( ^ [R2: set_Pr1261947904930325089at_nat,As: nat > nat] :
        ! [I: nat,J: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ I @ J ) @ R2 )
         => ( ord_less_eq_nat @ ( As @ I ) @ ( As @ J ) ) ) ) ) ).

% relChain_def
thf(fact_320_ad__equiv__pair_Oelims_I3_J,axiom,
    ! [X: set_a,Xa: produc4970237071031658413um_a_b] :
      ( ~ ( ad_equiv_pair_a_b @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,A2: sum_sum_a_b] :
            ( ( Xa
              = ( produc8374529264883863525um_a_b @ A @ A2 ) )
           => ( ( ( member_Sum_sum_a_b @ A @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
               => ( A = A2 ) )
              & ( ( member_Sum_sum_a_b @ A2 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
               => ( A = A2 ) ) ) ) ) ).

% ad_equiv_pair.elims(3)
thf(fact_321_ad__equiv__pair_Oelims_I2_J,axiom,
    ! [X: set_a,Xa: produc4970237071031658413um_a_b] :
      ( ( ad_equiv_pair_a_b @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,A2: sum_sum_a_b] :
            ( ( Xa
              = ( produc8374529264883863525um_a_b @ A @ A2 ) )
           => ~ ( ( ( member_Sum_sum_a_b @ A @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                 => ( A = A2 ) )
                & ( ( member_Sum_sum_a_b @ A2 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                 => ( A = A2 ) ) ) ) ) ).

% ad_equiv_pair.elims(2)
thf(fact_322_ad__equiv__pair_Oelims_I1_J,axiom,
    ! [X: set_a,Xa: produc4970237071031658413um_a_b,Y: $o] :
      ( ( ( ad_equiv_pair_a_b @ X @ Xa )
        = Y )
     => ~ ! [A: sum_sum_a_b,A2: sum_sum_a_b] :
            ( ( Xa
              = ( produc8374529264883863525um_a_b @ A @ A2 ) )
           => ( Y
              = ( ~ ( ( ( member_Sum_sum_a_b @ A @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                     => ( A = A2 ) )
                    & ( ( member_Sum_sum_a_b @ A2 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                     => ( A = A2 ) ) ) ) ) ) ) ).

% ad_equiv_pair.elims(1)
thf(fact_323_ad__equiv__pair_Osimps,axiom,
    ! [X2: set_a,A3: sum_sum_a_b,A4: sum_sum_a_b] :
      ( ( ad_equiv_pair_a_b @ X2 @ ( produc8374529264883863525um_a_b @ A3 @ A4 ) )
      = ( ( ( member_Sum_sum_a_b @ A3 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X2 ) )
         => ( A3 = A4 ) )
        & ( ( member_Sum_sum_a_b @ A4 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X2 ) )
         => ( A3 = A4 ) ) ) ) ).

% ad_equiv_pair.simps
thf(fact_324_image__eqI,axiom,
    ! [B3: nat,F: nat > nat,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member_nat @ B3 @ ( image_nat_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_325_image__eqI,axiom,
    ! [B3: product_prod_nat_nat,F: nat > product_prod_nat_nat,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member8440522571783428010at_nat @ B3 @ ( image_5846123807819985514at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_326_image__eqI,axiom,
    ! [B3: sum_sum_a_b,F: nat > sum_sum_a_b,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member_Sum_sum_a_b @ B3 @ ( image_95481336540478333um_a_b @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_327_image__eqI,axiom,
    ! [B3: nat,F: product_prod_nat_nat > nat,X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( member_nat @ B3 @ ( image_2486076414777270412at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_328_image__eqI,axiom,
    ! [B3: nat,F: sum_sum_a_b > nat,X: sum_sum_a_b,A5: set_Sum_sum_a_b] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_Sum_sum_a_b @ X @ A5 )
       => ( member_nat @ B3 @ ( image_5089838354808162685_b_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_329_image__eqI,axiom,
    ! [B3: produc6336967135452217047_b_nat,F: nat > produc6336967135452217047_b_nat,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member8290135809484590976_b_nat @ B3 @ ( image_7767039860428352064_b_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_330_image__eqI,axiom,
    ! [B3: produc5919649888572472791um_a_b,F: nat > produc5919649888572472791um_a_b,X: nat,A5: set_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member_nat @ X @ A5 )
       => ( member7872818562604846720um_a_b @ B3 @ ( image_7349722613548607808um_a_b @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_331_image__eqI,axiom,
    ! [B3: nat,F: produc6336967135452217047_b_nat > nat,X: produc6336967135452217047_b_nat,A5: set_Pr7634312471471294263_b_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member8290135809484590976_b_nat @ X @ A5 )
       => ( member_nat @ B3 @ ( image_3920336858823826018at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_332_image__eqI,axiom,
    ! [B3: nat,F: produc5919649888572472791um_a_b > nat,X: produc5919649888572472791um_a_b,A5: set_Pr4241354594808546871um_a_b] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member7872818562604846720um_a_b @ X @ A5 )
       => ( member_nat @ B3 @ ( image_214044067618314594_b_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_333_image__eqI,axiom,
    ! [B3: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat,X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( B3
        = ( F @ X ) )
     => ( ( member8440522571783428010at_nat @ X @ A5 )
       => ( member8440522571783428010at_nat @ B3 @ ( image_5168914502847457605at_nat @ F @ A5 ) ) ) ) ).

% image_eqI
thf(fact_334_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_335_list_Oset__map,axiom,
    ! [F: nat > nat,V: list_nat] :
      ( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
      = ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_336_list_Oset__map,axiom,
    ! [F: nat > sum_sum_a_b,V: list_nat] :
      ( ( set_Sum_sum_a_b2 @ ( map_nat_Sum_sum_a_b @ F @ V ) )
      = ( image_95481336540478333um_a_b @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_337_list_Oset__map,axiom,
    ! [F: nat > product_prod_nat_nat,V: list_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( map_na7298421622053143531at_nat @ F @ V ) )
      = ( image_5846123807819985514at_nat @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_338_list_Oset__map,axiom,
    ! [F: sum_sum_a_b > nat,V: list_Sum_sum_a_b] :
      ( ( set_nat2 @ ( map_Sum_sum_a_b_nat @ F @ V ) )
      = ( image_5089838354808162685_b_nat @ F @ ( set_Sum_sum_a_b2 @ V ) ) ) ).

% list.set_map
thf(fact_339_list_Oset__map,axiom,
    ! [F: product_prod_nat_nat > nat,V: list_P6011104703257516679at_nat] :
      ( ( set_nat2 @ ( map_Pr3938374229010428429at_nat @ F @ V ) )
      = ( image_2486076414777270412at_nat @ F @ ( set_Pr5648618587558075414at_nat @ V ) ) ) ).

% list.set_map
thf(fact_340_list_Oset__map,axiom,
    ! [F: sum_sum_a_b > sum_sum_a_b,V: list_Sum_sum_a_b] :
      ( ( set_Sum_sum_a_b2 @ ( map_Su3007332696077240082um_a_b @ F @ V ) )
      = ( image_3358989901043947347um_a_b @ F @ ( set_Sum_sum_a_b2 @ V ) ) ) ).

% list.set_map
thf(fact_341_list_Oset__map,axiom,
    ! [F: product_prod_nat_nat > sum_sum_a_b,V: list_P6011104703257516679at_nat] :
      ( ( set_Sum_sum_a_b2 @ ( map_Pr607579787586854883um_a_b @ F @ V ) )
      = ( image_7145616961338119266um_a_b @ F @ ( set_Pr5648618587558075414at_nat @ V ) ) ) ).

% list.set_map
thf(fact_342_list_Oset__map,axiom,
    ! [F: nat > produc6336967135452217047_b_nat,V: list_nat] :
      ( ( set_Pr176305205332118508_b_nat @ ( map_na1229002686677087681_b_nat @ F @ V ) )
      = ( image_7767039860428352064_b_nat @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_343_list_Oset__map,axiom,
    ! [F: nat > produc5919649888572472791um_a_b,V: list_nat] :
      ( ( set_Pr8982359995307150060um_a_b @ ( map_na811685439797343425um_a_b @ F @ V ) )
      = ( image_7349722613548607808um_a_b @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_344_list_Oset__map,axiom,
    ! [F: sum_sum_a_b > product_prod_nat_nat,V: list_Sum_sum_a_b] :
      ( ( set_Pr5648618587558075414at_nat @ ( map_Su599239160917965461at_nat @ F @ V ) )
      = ( image_7137276334669229844at_nat @ F @ ( set_Sum_sum_a_b2 @ V ) ) ) ).

% list.set_map
thf(fact_345_pairwise__imageI,axiom,
    ! [A5: set_nat,F: nat > nat,P2: nat > nat > $o] :
      ( ! [X5: nat,Y3: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( ( member_nat @ Y3 @ A5 )
           => ( ( X5 != Y3 )
             => ( ( ( F @ X5 )
                 != ( F @ Y3 ) )
               => ( P2 @ ( F @ X5 ) @ ( F @ Y3 ) ) ) ) ) )
     => ( pairwise_nat @ P2 @ ( image_nat_nat @ F @ A5 ) ) ) ).

% pairwise_imageI
thf(fact_346_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: nat,F: nat > nat] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_nat_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_347_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: product_prod_nat_nat,F: nat > product_prod_nat_nat] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member8440522571783428010at_nat @ B3 @ ( image_5846123807819985514at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_348_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: sum_sum_a_b,F: nat > sum_sum_a_b] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_Sum_sum_a_b @ B3 @ ( image_95481336540478333um_a_b @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_349_rev__image__eqI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B3: nat,F: product_prod_nat_nat > nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_2486076414777270412at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_350_rev__image__eqI,axiom,
    ! [X: sum_sum_a_b,A5: set_Sum_sum_a_b,B3: nat,F: sum_sum_a_b > nat] :
      ( ( member_Sum_sum_a_b @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_5089838354808162685_b_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_351_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: produc6336967135452217047_b_nat,F: nat > produc6336967135452217047_b_nat] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member8290135809484590976_b_nat @ B3 @ ( image_7767039860428352064_b_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_352_rev__image__eqI,axiom,
    ! [X: nat,A5: set_nat,B3: produc5919649888572472791um_a_b,F: nat > produc5919649888572472791um_a_b] :
      ( ( member_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member7872818562604846720um_a_b @ B3 @ ( image_7349722613548607808um_a_b @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_353_rev__image__eqI,axiom,
    ! [X: produc6336967135452217047_b_nat,A5: set_Pr7634312471471294263_b_nat,B3: nat,F: produc6336967135452217047_b_nat > nat] :
      ( ( member8290135809484590976_b_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_3920336858823826018at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_354_rev__image__eqI,axiom,
    ! [X: produc5919649888572472791um_a_b,A5: set_Pr4241354594808546871um_a_b,B3: nat,F: produc5919649888572472791um_a_b > nat] :
      ( ( member7872818562604846720um_a_b @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member_nat @ B3 @ ( image_214044067618314594_b_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_355_rev__image__eqI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( ( B3
          = ( F @ X ) )
       => ( member8440522571783428010at_nat @ B3 @ ( image_5168914502847457605at_nat @ F @ A5 ) ) ) ) ).

% rev_image_eqI
thf(fact_356_ball__imageD,axiom,
    ! [F: nat > nat,A5: set_nat,P2: nat > $o] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( image_nat_nat @ F @ A5 ) )
         => ( P2 @ X5 ) )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ A5 )
         => ( P2 @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_357_image__cong,axiom,
    ! [M: set_nat,N2: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M = N2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ N2 )
           => ( ( F @ X5 )
              = ( G @ X5 ) ) )
       => ( ( image_nat_nat @ F @ M )
          = ( image_nat_nat @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_358_bex__imageD,axiom,
    ! [F: nat > nat,A5: set_nat,P2: nat > $o] :
      ( ? [X7: nat] :
          ( ( member_nat @ X7 @ ( image_nat_nat @ F @ A5 ) )
          & ( P2 @ X7 ) )
     => ? [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
          & ( P2 @ ( F @ X5 ) ) ) ) ).

% bex_imageD
thf(fact_359_pairwiseI,axiom,
    ! [S: set_Pr7446880501794416397um_a_b,R: produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > $o] :
      ( ! [X5: produc4970237071031658413um_a_b,Y3: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X5 @ S )
         => ( ( member6480126262776431190um_a_b @ Y3 @ S )
           => ( ( X5 != Y3 )
             => ( R @ X5 @ Y3 ) ) ) )
     => ( pairwi757434828799223448um_a_b @ R @ S ) ) ).

% pairwiseI
thf(fact_360_pairwiseI,axiom,
    ! [S: set_nat,R: nat > nat > $o] :
      ( ! [X5: nat,Y3: nat] :
          ( ( member_nat @ X5 @ S )
         => ( ( member_nat @ Y3 @ S )
           => ( ( X5 != Y3 )
             => ( R @ X5 @ Y3 ) ) ) )
     => ( pairwise_nat @ R @ S ) ) ).

% pairwiseI
thf(fact_361_pairwiseI,axiom,
    ! [S: set_Pr7634312471471294263_b_nat,R: produc6336967135452217047_b_nat > produc6336967135452217047_b_nat > $o] :
      ( ! [X5: produc6336967135452217047_b_nat,Y3: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X5 @ S )
         => ( ( member8290135809484590976_b_nat @ Y3 @ S )
           => ( ( X5 != Y3 )
             => ( R @ X5 @ Y3 ) ) ) )
     => ( pairwi4739247019612192706_b_nat @ R @ S ) ) ).

% pairwiseI
thf(fact_362_pairwiseI,axiom,
    ! [S: set_Pr5956926064383532552um_a_b,R: produc7779921308951573458um_a_b > produc7779921308951573458um_a_b > $o] :
      ( ! [X5: produc7779921308951573458um_a_b,Y3: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X5 @ S )
         => ( ( member4052926750399300073um_a_b @ Y3 @ S )
           => ( ( X5 != Y3 )
             => ( R @ X5 @ Y3 ) ) ) )
     => ( pairwi6301539524301505319um_a_b @ R @ S ) ) ).

% pairwiseI
thf(fact_363_pairwiseI,axiom,
    ! [S: set_Pr4241354594808546871um_a_b,R: produc5919649888572472791um_a_b > produc5919649888572472791um_a_b > $o] :
      ( ! [X5: produc5919649888572472791um_a_b,Y3: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X5 @ S )
         => ( ( member7872818562604846720um_a_b @ Y3 @ S )
           => ( ( X5 != Y3 )
             => ( R @ X5 @ Y3 ) ) ) )
     => ( pairwi4321929772732448450um_a_b @ R @ S ) ) ).

% pairwiseI
thf(fact_364_pairwiseI,axiom,
    ! [S: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ S )
         => ( ( member8440522571783428010at_nat @ Y3 @ S )
           => ( ( X5 != Y3 )
             => ( R @ X5 @ Y3 ) ) ) )
     => ( pairwi6357175311820755948at_nat @ R @ S ) ) ).

% pairwiseI
thf(fact_365_pairwiseI,axiom,
    ! [S: set_Sum_sum_a_b,R: sum_sum_a_b > sum_sum_a_b > $o] :
      ( ! [X5: sum_sum_a_b,Y3: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X5 @ S )
         => ( ( member_Sum_sum_a_b @ Y3 @ S )
           => ( ( X5 != Y3 )
             => ( R @ X5 @ Y3 ) ) ) )
     => ( pairwise_Sum_sum_a_b @ R @ S ) ) ).

% pairwiseI
thf(fact_366_pairwiseD,axiom,
    ! [R: produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > $o,S: set_Pr7446880501794416397um_a_b,X: produc4970237071031658413um_a_b,Y: produc4970237071031658413um_a_b] :
      ( ( pairwi757434828799223448um_a_b @ R @ S )
     => ( ( member6480126262776431190um_a_b @ X @ S )
       => ( ( member6480126262776431190um_a_b @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_367_pairwiseD,axiom,
    ! [R: nat > nat > $o,S: set_nat,X: nat,Y: nat] :
      ( ( pairwise_nat @ R @ S )
     => ( ( member_nat @ X @ S )
       => ( ( member_nat @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_368_pairwiseD,axiom,
    ! [R: produc6336967135452217047_b_nat > produc6336967135452217047_b_nat > $o,S: set_Pr7634312471471294263_b_nat,X: produc6336967135452217047_b_nat,Y: produc6336967135452217047_b_nat] :
      ( ( pairwi4739247019612192706_b_nat @ R @ S )
     => ( ( member8290135809484590976_b_nat @ X @ S )
       => ( ( member8290135809484590976_b_nat @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_369_pairwiseD,axiom,
    ! [R: produc7779921308951573458um_a_b > produc7779921308951573458um_a_b > $o,S: set_Pr5956926064383532552um_a_b,X: produc7779921308951573458um_a_b,Y: produc7779921308951573458um_a_b] :
      ( ( pairwi6301539524301505319um_a_b @ R @ S )
     => ( ( member4052926750399300073um_a_b @ X @ S )
       => ( ( member4052926750399300073um_a_b @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_370_pairwiseD,axiom,
    ! [R: produc5919649888572472791um_a_b > produc5919649888572472791um_a_b > $o,S: set_Pr4241354594808546871um_a_b,X: produc5919649888572472791um_a_b,Y: produc5919649888572472791um_a_b] :
      ( ( pairwi4321929772732448450um_a_b @ R @ S )
     => ( ( member7872818562604846720um_a_b @ X @ S )
       => ( ( member7872818562604846720um_a_b @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_371_pairwiseD,axiom,
    ! [R: product_prod_nat_nat > product_prod_nat_nat > $o,S: set_Pr1261947904930325089at_nat,X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( pairwi6357175311820755948at_nat @ R @ S )
     => ( ( member8440522571783428010at_nat @ X @ S )
       => ( ( member8440522571783428010at_nat @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_372_pairwiseD,axiom,
    ! [R: sum_sum_a_b > sum_sum_a_b > $o,S: set_Sum_sum_a_b,X: sum_sum_a_b,Y: sum_sum_a_b] :
      ( ( pairwise_Sum_sum_a_b @ R @ S )
     => ( ( member_Sum_sum_a_b @ X @ S )
       => ( ( member_Sum_sum_a_b @ Y @ S )
         => ( ( X != Y )
           => ( R @ X @ Y ) ) ) ) ) ).

% pairwiseD
thf(fact_373_image__iff,axiom,
    ! [Z2: nat,F: nat > nat,A5: set_nat] :
      ( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A5 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A5 )
            & ( Z2
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_374_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_375_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > product_prod_nat_nat] :
      ( ( member_nat @ X @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ X ) @ ( image_5846123807819985514at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_376_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > sum_sum_a_b] :
      ( ( member_nat @ X @ A5 )
     => ( member_Sum_sum_a_b @ ( F @ X ) @ ( image_95481336540478333um_a_b @ F @ A5 ) ) ) ).

% imageI
thf(fact_377_imageI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_2486076414777270412at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_378_imageI,axiom,
    ! [X: sum_sum_a_b,A5: set_Sum_sum_a_b,F: sum_sum_a_b > nat] :
      ( ( member_Sum_sum_a_b @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_5089838354808162685_b_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_379_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > produc6336967135452217047_b_nat] :
      ( ( member_nat @ X @ A5 )
     => ( member8290135809484590976_b_nat @ ( F @ X ) @ ( image_7767039860428352064_b_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_380_imageI,axiom,
    ! [X: nat,A5: set_nat,F: nat > produc5919649888572472791um_a_b] :
      ( ( member_nat @ X @ A5 )
     => ( member7872818562604846720um_a_b @ ( F @ X ) @ ( image_7349722613548607808um_a_b @ F @ A5 ) ) ) ).

% imageI
thf(fact_381_imageI,axiom,
    ! [X: produc6336967135452217047_b_nat,A5: set_Pr7634312471471294263_b_nat,F: produc6336967135452217047_b_nat > nat] :
      ( ( member8290135809484590976_b_nat @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_3920336858823826018at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_382_imageI,axiom,
    ! [X: produc5919649888572472791um_a_b,A5: set_Pr4241354594808546871um_a_b,F: produc5919649888572472791um_a_b > nat] :
      ( ( member7872818562604846720um_a_b @ X @ A5 )
     => ( member_nat @ ( F @ X ) @ ( image_214044067618314594_b_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_383_imageI,axiom,
    ! [X: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ A5 )
     => ( member8440522571783428010at_nat @ ( F @ X ) @ ( image_5168914502847457605at_nat @ F @ A5 ) ) ) ).

% imageI
thf(fact_384_image__set,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
      = ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_385_image__set,axiom,
    ! [F: sum_sum_a_b > nat,Xs: list_Sum_sum_a_b] :
      ( ( image_5089838354808162685_b_nat @ F @ ( set_Sum_sum_a_b2 @ Xs ) )
      = ( set_nat2 @ ( map_Sum_sum_a_b_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_386_image__set,axiom,
    ! [F: product_prod_nat_nat > nat,Xs: list_P6011104703257516679at_nat] :
      ( ( image_2486076414777270412at_nat @ F @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( set_nat2 @ ( map_Pr3938374229010428429at_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_387_image__set,axiom,
    ! [F: nat > sum_sum_a_b,Xs: list_nat] :
      ( ( image_95481336540478333um_a_b @ F @ ( set_nat2 @ Xs ) )
      = ( set_Sum_sum_a_b2 @ ( map_nat_Sum_sum_a_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_388_image__set,axiom,
    ! [F: nat > product_prod_nat_nat,Xs: list_nat] :
      ( ( image_5846123807819985514at_nat @ F @ ( set_nat2 @ Xs ) )
      = ( set_Pr5648618587558075414at_nat @ ( map_na7298421622053143531at_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_389_image__set,axiom,
    ! [F: sum_sum_a_b > sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( image_3358989901043947347um_a_b @ F @ ( set_Sum_sum_a_b2 @ Xs ) )
      = ( set_Sum_sum_a_b2 @ ( map_Su3007332696077240082um_a_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_390_image__set,axiom,
    ! [F: sum_sum_a_b > product_prod_nat_nat,Xs: list_Sum_sum_a_b] :
      ( ( image_7137276334669229844at_nat @ F @ ( set_Sum_sum_a_b2 @ Xs ) )
      = ( set_Pr5648618587558075414at_nat @ ( map_Su599239160917965461at_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_391_image__set,axiom,
    ! [F: produc6336967135452217047_b_nat > nat,Xs: list_P667245297671117661_b_nat] :
      ( ( image_3920336858823826018at_nat @ F @ ( set_Pr176305205332118508_b_nat @ Xs ) )
      = ( set_nat2 @ ( map_Pr6605671721927337443at_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_392_image__set,axiom,
    ! [F: produc5919649888572472791um_a_b > nat,Xs: list_P6497659457863146077um_a_b] :
      ( ( image_214044067618314594_b_nat @ F @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
      = ( set_nat2 @ ( map_Pr2899378930721826019_b_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_393_image__set,axiom,
    ! [F: product_prod_nat_nat > sum_sum_a_b,Xs: list_P6011104703257516679at_nat] :
      ( ( image_7145616961338119266um_a_b @ F @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( set_Sum_sum_a_b2 @ ( map_Pr607579787586854883um_a_b @ F @ Xs ) ) ) ).

% image_set
thf(fact_394_map__idI,axiom,
    ! [Xs: list_P911570089447184179um_a_b,F: produc4970237071031658413um_a_b > produc4970237071031658413um_a_b] :
      ( ! [X5: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X5 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
         => ( ( F @ X5 )
            = X5 ) )
     => ( ( map_Pr528701375478819716um_a_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_395_map__idI,axiom,
    ! [Xs: list_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b] :
      ( ! [X5: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X5 @ ( set_Sum_sum_a_b2 @ Xs ) )
         => ( ( F @ X5 )
            = X5 ) )
     => ( ( map_Su3007332696077240082um_a_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_396_map__idI,axiom,
    ! [Xs: list_P667245297671117661_b_nat,F: produc6336967135452217047_b_nat > produc6336967135452217047_b_nat] :
      ( ! [X5: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X5 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
         => ( ( F @ X5 )
            = X5 ) )
     => ( ( map_Pr3785055891335715588_b_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_397_map__idI,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,F: produc7779921308951573458um_a_b > produc7779921308951573458um_a_b] :
      ( ! [X5: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X5 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
         => ( ( F @ X5 )
            = X5 ) )
     => ( ( map_Pr3913663300537039338um_a_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_398_map__idI,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,F: produc5919649888572472791um_a_b > produc5919649888572472791um_a_b] :
      ( ! [X5: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X5 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
         => ( ( F @ X5 )
            = X5 ) )
     => ( ( map_Pr3034600080670448388um_a_b @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_399_map__idI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ( F @ X5 )
            = X5 ) )
     => ( ( map_Pr8058819605623181956at_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_400_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X5 )
            = X5 ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_401_list_Omap__ident__strong,axiom,
    ! [T2: list_P911570089447184179um_a_b,F: produc4970237071031658413um_a_b > produc4970237071031658413um_a_b] :
      ( ! [Z3: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ Z3 @ ( set_Pr2436422074441463490um_a_b @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr528701375478819716um_a_b @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_402_list_Omap__ident__strong,axiom,
    ! [T2: list_Sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b] :
      ( ! [Z3: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ Z3 @ ( set_Sum_sum_a_b2 @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Su3007332696077240082um_a_b @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_403_list_Omap__ident__strong,axiom,
    ! [T2: list_P667245297671117661_b_nat,F: produc6336967135452217047_b_nat > produc6336967135452217047_b_nat] :
      ( ! [Z3: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ Z3 @ ( set_Pr176305205332118508_b_nat @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr3785055891335715588_b_nat @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_404_list_Omap__ident__strong,axiom,
    ! [T2: list_P5598968436384720482um_a_b,F: produc7779921308951573458um_a_b > produc7779921308951573458um_a_b] :
      ( ! [Z3: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ Z3 @ ( set_Pr7399620718385288829um_a_b @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr3913663300537039338um_a_b @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_405_list_Omap__ident__strong,axiom,
    ! [T2: list_P6497659457863146077um_a_b,F: produc5919649888572472791um_a_b > produc5919649888572472791um_a_b] :
      ( ! [Z3: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ Z3 @ ( set_Pr8982359995307150060um_a_b @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr3034600080670448388um_a_b @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_406_list_Omap__ident__strong,axiom,
    ! [T2: list_P6011104703257516679at_nat,F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [Z3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ Z3 @ ( set_Pr5648618587558075414at_nat @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr8058819605623181956at_nat @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_407_list_Omap__ident__strong,axiom,
    ! [T2: list_nat,F: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ T2 ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_nat_nat @ F @ T2 )
        = T2 ) ) ).

% list.map_ident_strong
thf(fact_408_image__mono,axiom,
    ! [A5: set_nat,B5: set_nat,F: nat > nat] :
      ( ( ord_less_eq_set_nat @ A5 @ B5 )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ ( image_nat_nat @ F @ B5 ) ) ) ).

% image_mono
thf(fact_409_image__mono,axiom,
    ! [A5: set_a,B5: set_a,F: a > a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A5 ) @ ( image_a_a @ F @ B5 ) ) ) ).

% image_mono
thf(fact_410_image__subsetI,axiom,
    ! [A5: set_nat,F: nat > nat,B5: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( member_nat @ ( F @ X5 ) @ B5 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_411_image__subsetI,axiom,
    ! [A5: set_nat,F: nat > a,B5: set_a] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( member_a @ ( F @ X5 ) @ B5 ) )
     => ( ord_less_eq_set_a @ ( image_nat_a @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_412_image__subsetI,axiom,
    ! [A5: set_nat,F: nat > product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( member8440522571783428010at_nat @ ( F @ X5 ) @ B5 ) )
     => ( ord_le3146513528884898305at_nat @ ( image_5846123807819985514at_nat @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_413_image__subsetI,axiom,
    ! [A5: set_nat,F: nat > sum_sum_a_b,B5: set_Sum_sum_a_b] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( member_Sum_sum_a_b @ ( F @ X5 ) @ B5 ) )
     => ( ord_le9019793522827316924um_a_b @ ( image_95481336540478333um_a_b @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_414_image__subsetI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat,B5: set_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A5 )
         => ( member_nat @ ( F @ X5 ) @ B5 ) )
     => ( ord_less_eq_set_nat @ ( image_2486076414777270412at_nat @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_415_image__subsetI,axiom,
    ! [A5: set_Sum_sum_a_b,F: sum_sum_a_b > nat,B5: set_nat] :
      ( ! [X5: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X5 @ A5 )
         => ( member_nat @ ( F @ X5 ) @ B5 ) )
     => ( ord_less_eq_set_nat @ ( image_5089838354808162685_b_nat @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_416_image__subsetI,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > a,B5: set_a] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A5 )
         => ( member_a @ ( F @ X5 ) @ B5 ) )
     => ( ord_less_eq_set_a @ ( image_5995000214508162370_nat_a @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_417_image__subsetI,axiom,
    ! [A5: set_Sum_sum_a_b,F: sum_sum_a_b > a,B5: set_a] :
      ( ! [X5: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X5 @ A5 )
         => ( member_a @ ( F @ X5 ) @ B5 ) )
     => ( ord_less_eq_set_a @ ( image_Sum_sum_a_b_a @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_418_image__subsetI,axiom,
    ! [A5: set_nat,F: nat > produc6336967135452217047_b_nat,B5: set_Pr7634312471471294263_b_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( member8290135809484590976_b_nat @ ( F @ X5 ) @ B5 ) )
     => ( ord_le8626435567010533079_b_nat @ ( image_7767039860428352064_b_nat @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_419_image__subsetI,axiom,
    ! [A5: set_nat,F: nat > produc5919649888572472791um_a_b,B5: set_Pr4241354594808546871um_a_b] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A5 )
         => ( member7872818562604846720um_a_b @ ( F @ X5 ) @ B5 ) )
     => ( ord_le5233477690347785687um_a_b @ ( image_7349722613548607808um_a_b @ F @ A5 ) @ B5 ) ) ).

% image_subsetI
thf(fact_420_subset__imageE,axiom,
    ! [B5: set_nat,F: nat > nat,A5: set_nat] :
      ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A5 ) )
     => ~ ! [C4: set_nat] :
            ( ( ord_less_eq_set_nat @ C4 @ A5 )
           => ( B5
             != ( image_nat_nat @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_421_subset__imageE,axiom,
    ! [B5: set_a,F: a > a,A5: set_a] :
      ( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A5 ) )
     => ~ ! [C4: set_a] :
            ( ( ord_less_eq_set_a @ C4 @ A5 )
           => ( B5
             != ( image_a_a @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_422_image__subset__iff,axiom,
    ! [F: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ B5 )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A5 )
           => ( member_nat @ ( F @ X4 ) @ B5 ) ) ) ) ).

% image_subset_iff
thf(fact_423_subset__image__iff,axiom,
    ! [B5: set_nat,F: nat > nat,A5: set_nat] :
      ( ( ord_less_eq_set_nat @ B5 @ ( image_nat_nat @ F @ A5 ) )
      = ( ? [AA: set_nat] :
            ( ( ord_less_eq_set_nat @ AA @ A5 )
            & ( B5
              = ( image_nat_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_424_subset__image__iff,axiom,
    ! [B5: set_a,F: a > a,A5: set_a] :
      ( ( ord_less_eq_set_a @ B5 @ ( image_a_a @ F @ A5 ) )
      = ( ? [AA: set_a] :
            ( ( ord_less_eq_set_a @ AA @ A5 )
            & ( B5
              = ( image_a_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_425_pairwise__mono,axiom,
    ! [P2: a > a > $o,A5: set_a,Q: a > a > $o,B5: set_a] :
      ( ( pairwise_a @ P2 @ A5 )
     => ( ! [X5: a,Y3: a] :
            ( ( P2 @ X5 @ Y3 )
           => ( Q @ X5 @ Y3 ) )
       => ( ( ord_less_eq_set_a @ B5 @ A5 )
         => ( pairwise_a @ Q @ B5 ) ) ) ) ).

% pairwise_mono
thf(fact_426_pairwise__subset,axiom,
    ! [P2: a > a > $o,S: set_a,T3: set_a] :
      ( ( pairwise_a @ P2 @ S )
     => ( ( ord_less_eq_set_a @ T3 @ S )
       => ( pairwise_a @ P2 @ T3 ) ) ) ).

% pairwise_subset
thf(fact_427_image__vimage__subset,axiom,
    ! [F: nat > nat,A5: set_nat] : ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ ( vimage_nat_nat @ F @ A5 ) ) @ A5 ) ).

% image_vimage_subset
thf(fact_428_image__vimage__subset,axiom,
    ! [F: a > sum_sum_a_b,A5: set_Sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ F @ ( vimage_a_Sum_sum_a_b @ F @ A5 ) ) @ A5 ) ).

% image_vimage_subset
thf(fact_429_image__subset__iff__subset__vimage,axiom,
    ! [F: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ B5 )
      = ( ord_less_eq_set_nat @ A5 @ ( vimage_nat_nat @ F @ B5 ) ) ) ).

% image_subset_iff_subset_vimage
thf(fact_430_image__subset__iff__subset__vimage,axiom,
    ! [F: a > sum_sum_a_b,A5: set_a,B5: set_Sum_sum_a_b] :
      ( ( ord_le9019793522827316924um_a_b @ ( image_a_Sum_sum_a_b @ F @ A5 ) @ B5 )
      = ( ord_less_eq_set_a @ A5 @ ( vimage_a_Sum_sum_a_b @ F @ B5 ) ) ) ).

% image_subset_iff_subset_vimage
thf(fact_431_image__subset__iff__subset__vimage,axiom,
    ! [F: a > a,A5: set_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ ( image_a_a @ F @ A5 ) @ B5 )
      = ( ord_less_eq_set_a @ A5 @ ( vimage_a_a @ F @ B5 ) ) ) ).

% image_subset_iff_subset_vimage
thf(fact_432_sp__equiv__pair_Osimps,axiom,
    ! [A3: sum_sum_a_b,B3: sum_sum_a_b,A4: sum_sum_a_b,B4: sum_sum_a_b] :
      ( ( sp_equ4404609506204868152um_a_b @ ( produc8374529264883863525um_a_b @ A3 @ B3 ) @ ( produc8374529264883863525um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        = ( B3 = B4 ) ) ) ).

% sp_equiv_pair.simps
thf(fact_433_sp__equiv__pair_Osimps,axiom,
    ! [A3: sum_sum_a_b,B3: produc4970237071031658413um_a_b,A4: sum_sum_a_b,B4: produc4970237071031658413um_a_b] :
      ( ( sp_equ110388464097044123um_a_b @ ( produc7989310782275393198um_a_b @ A3 @ B3 ) @ ( produc7989310782275393198um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        = ( B3 = B4 ) ) ) ).

% sp_equiv_pair.simps
thf(fact_434_sp__equiv__pair_Osimps,axiom,
    ! [A3: sum_sum_a_b,B3: nat,A4: sum_sum_a_b,B4: nat] :
      ( ( sp_equ785082339516739170_b_nat @ ( produc9094079831559073295_b_nat @ A3 @ B3 ) @ ( produc9094079831559073295_b_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        = ( B3 = B4 ) ) ) ).

% sp_equiv_pair.simps
thf(fact_435_sp__equiv__pair_Osimps,axiom,
    ! [A3: nat,B3: produc4970237071031658413um_a_b,A4: nat,B4: produc4970237071031658413um_a_b] :
      ( ( sp_equ3763899520926221297um_a_b @ ( produc6016411306719553028um_a_b @ A3 @ B3 ) @ ( produc6016411306719553028um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        = ( B3 = B4 ) ) ) ).

% sp_equiv_pair.simps
thf(fact_436_sp__equiv__pair_Osimps,axiom,
    ! [A3: nat,B3: sum_sum_a_b,A4: nat,B4: sum_sum_a_b] :
      ( ( sp_equ5014097358103830626um_a_b @ ( produc4099722813291388943um_a_b @ A3 @ B3 ) @ ( produc4099722813291388943um_a_b @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        = ( B3 = B4 ) ) ) ).

% sp_equiv_pair.simps
thf(fact_437_sp__equiv__pair_Osimps,axiom,
    ! [A3: nat,B3: nat,A4: nat,B4: nat] :
      ( ( sp_equ885134019442699404at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ A4 @ B4 ) )
      = ( ( A3 = A4 )
        = ( B3 = B4 ) ) ) ).

% sp_equiv_pair.simps
thf(fact_438_sp__equiv__pair_Oelims_I1_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Xa: produc4970237071031658413um_a_b,Y: $o] :
      ( ( ( sp_equ4404609506204868152um_a_b @ X @ Xa )
        = Y )
     => ~ ! [A: sum_sum_a_b,B: sum_sum_a_b] :
            ( ( X
              = ( produc8374529264883863525um_a_b @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: sum_sum_a_b] :
                ( ( Xa
                  = ( produc8374529264883863525um_a_b @ A2 @ B2 ) )
               => ( Y
                  = ( ( A = A2 )
                    = ( B != B2 ) ) ) ) ) ) ).

% sp_equiv_pair.elims(1)
thf(fact_439_sp__equiv__pair_Oelims_I1_J,axiom,
    ! [X: produc5987322259985443452um_a_b,Xa: produc5987322259985443452um_a_b,Y: $o] :
      ( ( ( sp_equ110388464097044123um_a_b @ X @ Xa )
        = Y )
     => ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] :
            ( ( X
              = ( produc7989310782275393198um_a_b @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: produc4970237071031658413um_a_b] :
                ( ( Xa
                  = ( produc7989310782275393198um_a_b @ A2 @ B2 ) )
               => ( Y
                  = ( ( A = A2 )
                    = ( B != B2 ) ) ) ) ) ) ).

% sp_equiv_pair.elims(1)
thf(fact_440_sp__equiv__pair_Oelims_I1_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Xa: produc6336967135452217047_b_nat,Y: $o] :
      ( ( ( sp_equ785082339516739170_b_nat @ X @ Xa )
        = Y )
     => ~ ! [A: sum_sum_a_b,B: nat] :
            ( ( X
              = ( produc9094079831559073295_b_nat @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: nat] :
                ( ( Xa
                  = ( produc9094079831559073295_b_nat @ A2 @ B2 ) )
               => ( Y
                  = ( ( A = A2 )
                    = ( B != B2 ) ) ) ) ) ) ).

% sp_equiv_pair.elims(1)
thf(fact_441_sp__equiv__pair_Oelims_I1_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Xa: produc7779921308951573458um_a_b,Y: $o] :
      ( ( ( sp_equ3763899520926221297um_a_b @ X @ Xa )
        = Y )
     => ~ ! [A: nat,B: produc4970237071031658413um_a_b] :
            ( ( X
              = ( produc6016411306719553028um_a_b @ A @ B ) )
           => ! [A2: nat,B2: produc4970237071031658413um_a_b] :
                ( ( Xa
                  = ( produc6016411306719553028um_a_b @ A2 @ B2 ) )
               => ( Y
                  = ( ( A = A2 )
                    = ( B != B2 ) ) ) ) ) ) ).

% sp_equiv_pair.elims(1)
thf(fact_442_sp__equiv__pair_Oelims_I1_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Xa: produc5919649888572472791um_a_b,Y: $o] :
      ( ( ( sp_equ5014097358103830626um_a_b @ X @ Xa )
        = Y )
     => ~ ! [A: nat,B: sum_sum_a_b] :
            ( ( X
              = ( produc4099722813291388943um_a_b @ A @ B ) )
           => ! [A2: nat,B2: sum_sum_a_b] :
                ( ( Xa
                  = ( produc4099722813291388943um_a_b @ A2 @ B2 ) )
               => ( Y
                  = ( ( A = A2 )
                    = ( B != B2 ) ) ) ) ) ) ).

% sp_equiv_pair.elims(1)
thf(fact_443_sp__equiv__pair_Oelims_I1_J,axiom,
    ! [X: product_prod_nat_nat,Xa: product_prod_nat_nat,Y: $o] :
      ( ( ( sp_equ885134019442699404at_nat @ X @ Xa )
        = Y )
     => ~ ! [A: nat,B: nat] :
            ( ( X
              = ( product_Pair_nat_nat @ A @ B ) )
           => ! [A2: nat,B2: nat] :
                ( ( Xa
                  = ( product_Pair_nat_nat @ A2 @ B2 ) )
               => ( Y
                  = ( ( A = A2 )
                    = ( B != B2 ) ) ) ) ) ) ).

% sp_equiv_pair.elims(1)
thf(fact_444_sp__equiv__pair_Oelims_I2_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Xa: produc4970237071031658413um_a_b] :
      ( ( sp_equ4404609506204868152um_a_b @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,B: sum_sum_a_b] :
            ( ( X
              = ( produc8374529264883863525um_a_b @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: sum_sum_a_b] :
                ( ( Xa
                  = ( produc8374529264883863525um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B != B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(2)
thf(fact_445_sp__equiv__pair_Oelims_I2_J,axiom,
    ! [X: produc5987322259985443452um_a_b,Xa: produc5987322259985443452um_a_b] :
      ( ( sp_equ110388464097044123um_a_b @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] :
            ( ( X
              = ( produc7989310782275393198um_a_b @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: produc4970237071031658413um_a_b] :
                ( ( Xa
                  = ( produc7989310782275393198um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B != B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(2)
thf(fact_446_sp__equiv__pair_Oelims_I2_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Xa: produc6336967135452217047_b_nat] :
      ( ( sp_equ785082339516739170_b_nat @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,B: nat] :
            ( ( X
              = ( produc9094079831559073295_b_nat @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: nat] :
                ( ( Xa
                  = ( produc9094079831559073295_b_nat @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B != B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(2)
thf(fact_447_sp__equiv__pair_Oelims_I2_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Xa: produc7779921308951573458um_a_b] :
      ( ( sp_equ3763899520926221297um_a_b @ X @ Xa )
     => ~ ! [A: nat,B: produc4970237071031658413um_a_b] :
            ( ( X
              = ( produc6016411306719553028um_a_b @ A @ B ) )
           => ! [A2: nat,B2: produc4970237071031658413um_a_b] :
                ( ( Xa
                  = ( produc6016411306719553028um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B != B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(2)
thf(fact_448_sp__equiv__pair_Oelims_I2_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Xa: produc5919649888572472791um_a_b] :
      ( ( sp_equ5014097358103830626um_a_b @ X @ Xa )
     => ~ ! [A: nat,B: sum_sum_a_b] :
            ( ( X
              = ( produc4099722813291388943um_a_b @ A @ B ) )
           => ! [A2: nat,B2: sum_sum_a_b] :
                ( ( Xa
                  = ( produc4099722813291388943um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B != B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(2)
thf(fact_449_sp__equiv__pair_Oelims_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( ( sp_equ885134019442699404at_nat @ X @ Xa )
     => ~ ! [A: nat,B: nat] :
            ( ( X
              = ( product_Pair_nat_nat @ A @ B ) )
           => ! [A2: nat,B2: nat] :
                ( ( Xa
                  = ( product_Pair_nat_nat @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B != B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(2)
thf(fact_450_sp__equiv__pair_Oelims_I3_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Xa: produc4970237071031658413um_a_b] :
      ( ~ ( sp_equ4404609506204868152um_a_b @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,B: sum_sum_a_b] :
            ( ( X
              = ( produc8374529264883863525um_a_b @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: sum_sum_a_b] :
                ( ( Xa
                  = ( produc8374529264883863525um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B = B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(3)
thf(fact_451_sp__equiv__pair_Oelims_I3_J,axiom,
    ! [X: produc5987322259985443452um_a_b,Xa: produc5987322259985443452um_a_b] :
      ( ~ ( sp_equ110388464097044123um_a_b @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] :
            ( ( X
              = ( produc7989310782275393198um_a_b @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: produc4970237071031658413um_a_b] :
                ( ( Xa
                  = ( produc7989310782275393198um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B = B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(3)
thf(fact_452_sp__equiv__pair_Oelims_I3_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Xa: produc6336967135452217047_b_nat] :
      ( ~ ( sp_equ785082339516739170_b_nat @ X @ Xa )
     => ~ ! [A: sum_sum_a_b,B: nat] :
            ( ( X
              = ( produc9094079831559073295_b_nat @ A @ B ) )
           => ! [A2: sum_sum_a_b,B2: nat] :
                ( ( Xa
                  = ( produc9094079831559073295_b_nat @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B = B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(3)
thf(fact_453_sp__equiv__pair_Oelims_I3_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Xa: produc7779921308951573458um_a_b] :
      ( ~ ( sp_equ3763899520926221297um_a_b @ X @ Xa )
     => ~ ! [A: nat,B: produc4970237071031658413um_a_b] :
            ( ( X
              = ( produc6016411306719553028um_a_b @ A @ B ) )
           => ! [A2: nat,B2: produc4970237071031658413um_a_b] :
                ( ( Xa
                  = ( produc6016411306719553028um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B = B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(3)
thf(fact_454_sp__equiv__pair_Oelims_I3_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Xa: produc5919649888572472791um_a_b] :
      ( ~ ( sp_equ5014097358103830626um_a_b @ X @ Xa )
     => ~ ! [A: nat,B: sum_sum_a_b] :
            ( ( X
              = ( produc4099722813291388943um_a_b @ A @ B ) )
           => ! [A2: nat,B2: sum_sum_a_b] :
                ( ( Xa
                  = ( produc4099722813291388943um_a_b @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B = B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(3)
thf(fact_455_sp__equiv__pair_Oelims_I3_J,axiom,
    ! [X: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( ~ ( sp_equ885134019442699404at_nat @ X @ Xa )
     => ~ ! [A: nat,B: nat] :
            ( ( X
              = ( product_Pair_nat_nat @ A @ B ) )
           => ! [A2: nat,B2: nat] :
                ( ( Xa
                  = ( product_Pair_nat_nat @ A2 @ B2 ) )
               => ( ( A = A2 )
                  = ( B = B2 ) ) ) ) ) ).

% sp_equiv_pair.elims(3)
thf(fact_456_all__subset__image,axiom,
    ! [F: nat > nat,A5: set_nat,P2: set_nat > $o] :
      ( ( ! [B7: set_nat] :
            ( ( ord_less_eq_set_nat @ B7 @ ( image_nat_nat @ F @ A5 ) )
           => ( P2 @ B7 ) ) )
      = ( ! [B7: set_nat] :
            ( ( ord_less_eq_set_nat @ B7 @ A5 )
           => ( P2 @ ( image_nat_nat @ F @ B7 ) ) ) ) ) ).

% all_subset_image
thf(fact_457_all__subset__image,axiom,
    ! [F: a > a,A5: set_a,P2: set_a > $o] :
      ( ( ! [B7: set_a] :
            ( ( ord_less_eq_set_a @ B7 @ ( image_a_a @ F @ A5 ) )
           => ( P2 @ B7 ) ) )
      = ( ! [B7: set_a] :
            ( ( ord_less_eq_set_a @ B7 @ A5 )
           => ( P2 @ ( image_a_a @ F @ B7 ) ) ) ) ) ).

% all_subset_image
thf(fact_458_ad__equiv__pair_Opelims_I1_J,axiom,
    ! [X: set_a,Xa: produc4970237071031658413um_a_b,Y: $o] :
      ( ( ( ad_equiv_pair_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P484073618832398463um_a_b @ ad_equ6444359169260110102el_a_b @ ( produc1385271121253346790um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,A2: sum_sum_a_b] :
              ( ( Xa
                = ( produc8374529264883863525um_a_b @ A @ A2 ) )
             => ( ( Y
                  = ( ( ( member_Sum_sum_a_b @ A @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                     => ( A = A2 ) )
                    & ( ( member_Sum_sum_a_b @ A2 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                     => ( A = A2 ) ) ) )
               => ~ ( accp_P484073618832398463um_a_b @ ad_equ6444359169260110102el_a_b @ ( produc1385271121253346790um_a_b @ X @ ( produc8374529264883863525um_a_b @ A @ A2 ) ) ) ) ) ) ) ).

% ad_equiv_pair.pelims(1)
thf(fact_459_ad__equiv__pair_Opelims_I2_J,axiom,
    ! [X: set_a,Xa: produc4970237071031658413um_a_b] :
      ( ( ad_equiv_pair_a_b @ X @ Xa )
     => ( ( accp_P484073618832398463um_a_b @ ad_equ6444359169260110102el_a_b @ ( produc1385271121253346790um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,A2: sum_sum_a_b] :
              ( ( Xa
                = ( produc8374529264883863525um_a_b @ A @ A2 ) )
             => ( ( accp_P484073618832398463um_a_b @ ad_equ6444359169260110102el_a_b @ ( produc1385271121253346790um_a_b @ X @ ( produc8374529264883863525um_a_b @ A @ A2 ) ) )
               => ~ ( ( ( member_Sum_sum_a_b @ A @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                     => ( A = A2 ) )
                    & ( ( member_Sum_sum_a_b @ A2 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                     => ( A = A2 ) ) ) ) ) ) ) ).

% ad_equiv_pair.pelims(2)
thf(fact_460_ad__equiv__pair_Opelims_I3_J,axiom,
    ! [X: set_a,Xa: produc4970237071031658413um_a_b] :
      ( ~ ( ad_equiv_pair_a_b @ X @ Xa )
     => ( ( accp_P484073618832398463um_a_b @ ad_equ6444359169260110102el_a_b @ ( produc1385271121253346790um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,A2: sum_sum_a_b] :
              ( ( Xa
                = ( produc8374529264883863525um_a_b @ A @ A2 ) )
             => ( ( accp_P484073618832398463um_a_b @ ad_equ6444359169260110102el_a_b @ ( produc1385271121253346790um_a_b @ X @ ( produc8374529264883863525um_a_b @ A @ A2 ) ) )
               => ( ( ( member_Sum_sum_a_b @ A @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                   => ( A = A2 ) )
                  & ( ( member_Sum_sum_a_b @ A2 @ ( image_a_Sum_sum_a_b @ sum_Inl_a_b @ X ) )
                   => ( A = A2 ) ) ) ) ) ) ) ).

% ad_equiv_pair.pelims(3)
thf(fact_461_insort__insert__key__triv,axiom,
    ! [F: produc4970237071031658413um_a_b > nat,X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( member_nat @ ( F @ X ) @ ( image_8744169858129055544_b_nat @ F @ ( set_Pr2436422074441463490um_a_b @ Xs ) ) )
     => ( ( linord9135931367038690811_b_nat @ F @ X @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_462_insort__insert__key__triv,axiom,
    ! [F: sum_sum_a_b > nat,X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( member_nat @ ( F @ X ) @ ( image_5089838354808162685_b_nat @ F @ ( set_Sum_sum_a_b2 @ Xs ) ) )
     => ( ( linord1205179515702317178_b_nat @ F @ X @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_463_insort__insert__key__triv,axiom,
    ! [F: produc6336967135452217047_b_nat > nat,X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( member_nat @ ( F @ X ) @ ( image_3920336858823826018at_nat @ F @ ( set_Pr176305205332118508_b_nat @ Xs ) ) )
     => ( ( linord900385247953437477at_nat @ F @ X @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_464_insort__insert__key__triv,axiom,
    ! [F: produc7779921308951573458um_a_b > nat,X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( member_nat @ ( F @ X ) @ ( image_718544360348960937_b_nat @ F @ ( set_Pr7399620718385288829um_a_b @ Xs ) ) )
     => ( ( linord7646177846504673830_b_nat @ F @ X @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_465_insort__insert__key__triv,axiom,
    ! [F: produc5919649888572472791um_a_b > nat,X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( member_nat @ ( F @ X ) @ ( image_214044067618314594_b_nat @ F @ ( set_Pr8982359995307150060um_a_b @ Xs ) ) )
     => ( ( linord6417464493602701861_b_nat @ F @ X @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_466_insort__insert__key__triv,axiom,
    ! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member_nat @ ( F @ X ) @ ( image_2486076414777270412at_nat @ F @ ( set_Pr5648618587558075414at_nat @ Xs ) ) )
     => ( ( linord3394743409064221519at_nat @ F @ X @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_467_insort__insert__key__triv,axiom,
    ! [F: nat > nat,X: nat,Xs: list_nat] :
      ( ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
     => ( ( linord1921536354676448932at_nat @ F @ X @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_468_GreatestI2__order,axiom,
    ! [P2: set_a > $o,X: set_a,Q: set_a > $o] :
      ( ( P2 @ X )
     => ( ! [Y3: set_a] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_set_a @ Y3 @ X ) )
       => ( ! [X5: set_a] :
              ( ( P2 @ X5 )
             => ( ! [Y7: set_a] :
                    ( ( P2 @ Y7 )
                   => ( ord_less_eq_set_a @ Y7 @ X5 ) )
               => ( Q @ X5 ) ) )
         => ( Q @ ( order_Greatest_set_a @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_469_GreatestI2__order,axiom,
    ! [P2: nat > $o,X: nat,Q: nat > $o] :
      ( ( P2 @ X )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X ) )
       => ( ! [X5: nat] :
              ( ( P2 @ X5 )
             => ( ! [Y7: nat] :
                    ( ( P2 @ Y7 )
                   => ( ord_less_eq_nat @ Y7 @ X5 ) )
               => ( Q @ X5 ) ) )
         => ( Q @ ( order_Greatest_nat @ P2 ) ) ) ) ) ).

% GreatestI2_order
thf(fact_470_Greatest__equality,axiom,
    ! [P2: set_a > $o,X: set_a] :
      ( ( P2 @ X )
     => ( ! [Y3: set_a] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_set_a @ Y3 @ X ) )
       => ( ( order_Greatest_set_a @ P2 )
          = X ) ) ) ).

% Greatest_equality
thf(fact_471_Greatest__equality,axiom,
    ! [P2: nat > $o,X: nat] :
      ( ( P2 @ X )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ X ) )
       => ( ( order_Greatest_nat @ P2 )
          = X ) ) ) ).

% Greatest_equality
thf(fact_472_ad__agr__sets__def,axiom,
    ( ad_agr_sets_a_b
    = ( ^ [FV: set_nat,S2: set_nat,X3: set_a,Sigma2: nat > sum_sum_a_b,Tau2: nat > sum_sum_a_b] :
          ( ! [X4: nat] :
              ( ( member_nat @ X4 @ FV )
             => ( ad_equiv_pair_a_b @ X3 @ ( produc8374529264883863525um_a_b @ ( Sigma2 @ X4 ) @ ( Tau2 @ X4 ) ) ) )
          & ( sp_equiv_a_b @ Sigma2 @ Tau2 @ S2 ) ) ) ) ).

% ad_agr_sets_def
thf(fact_473_image__Fpow__mono,axiom,
    ! [F: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ B5 )
     => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( finite_Fpow_nat @ A5 ) ) @ ( finite_Fpow_nat @ B5 ) ) ) ).

% image_Fpow_mono
thf(fact_474_subset__code_I2_J,axiom,
    ! [A5: set_Pr7446880501794416397um_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( ord_le1431929631403617965um_a_b @ A5 @ ( coset_4334520101952487072um_a_b @ Ys ) )
      = ( ! [X4: produc4970237071031658413um_a_b] :
            ( ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ Ys ) )
           => ~ ( member6480126262776431190um_a_b @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_475_subset__code_I2_J,axiom,
    ! [A5: set_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( ord_le9019793522827316924um_a_b @ A5 @ ( coset_Sum_sum_a_b @ Ys ) )
      = ( ! [X4: sum_sum_a_b] :
            ( ( member_Sum_sum_a_b @ X4 @ ( set_Sum_sum_a_b2 @ Ys ) )
           => ~ ( member_Sum_sum_a_b @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_476_subset__code_I2_J,axiom,
    ! [A5: set_Pr7634312471471294263_b_nat,Ys: list_P667245297671117661_b_nat] :
      ( ( ord_le8626435567010533079_b_nat @ A5 @ ( coset_3799895497595635658_b_nat @ Ys ) )
      = ( ! [X4: produc6336967135452217047_b_nat] :
            ( ( member8290135809484590976_b_nat @ X4 @ ( set_Pr176305205332118508_b_nat @ Ys ) )
           => ~ ( member8290135809484590976_b_nat @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_477_subset__code_I2_J,axiom,
    ! [A5: set_Pr5956926064383532552um_a_b,Ys: list_P5598968436384720482um_a_b] :
      ( ( ord_le5806309049788436072um_a_b @ A5 @ ( coset_1753088348400206111um_a_b @ Ys ) )
      = ( ! [X4: produc7779921308951573458um_a_b] :
            ( ( member4052926750399300073um_a_b @ X4 @ ( set_Pr7399620718385288829um_a_b @ Ys ) )
           => ~ ( member4052926750399300073um_a_b @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_478_subset__code_I2_J,axiom,
    ! [A5: set_Pr4241354594808546871um_a_b,Ys: list_P6497659457863146077um_a_b] :
      ( ( ord_le5233477690347785687um_a_b @ A5 @ ( coset_3382578250715891402um_a_b @ Ys ) )
      = ( ! [X4: produc5919649888572472791um_a_b] :
            ( ( member7872818562604846720um_a_b @ X4 @ ( set_Pr8982359995307150060um_a_b @ Ys ) )
           => ~ ( member7872818562604846720um_a_b @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_479_subset__code_I2_J,axiom,
    ! [A5: set_Pr1261947904930325089at_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A5 @ ( coset_6809516720700967924at_nat @ Ys ) )
      = ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_480_subset__code_I2_J,axiom,
    ! [A5: set_nat,Ys: list_nat] :
      ( ( ord_less_eq_set_nat @ A5 @ ( coset_nat @ Ys ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ~ ( member_nat @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_481_subset__code_I2_J,axiom,
    ! [A5: set_a,Ys: list_a] :
      ( ( ord_less_eq_set_a @ A5 @ ( coset_a @ Ys ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Ys ) )
           => ~ ( member_a @ X4 @ A5 ) ) ) ) ).

% subset_code(2)
thf(fact_482_insort__insert__key__def,axiom,
    ( linord9135931367038690811_b_nat
    = ( ^ [F2: produc4970237071031658413um_a_b > nat,X4: produc4970237071031658413um_a_b,Xs2: list_P911570089447184179um_a_b] : ( if_lis2231744601426488825um_a_b @ ( member_nat @ ( F2 @ X4 ) @ ( image_8744169858129055544_b_nat @ F2 @ ( set_Pr2436422074441463490um_a_b @ Xs2 ) ) ) @ Xs2 @ ( linord1506269616935411906_b_nat @ F2 @ X4 @ Xs2 ) ) ) ) ).

% insort_insert_key_def
thf(fact_483_insort__insert__key__def,axiom,
    ( linord1205179515702317178_b_nat
    = ( ^ [F2: sum_sum_a_b > nat,X4: sum_sum_a_b,Xs2: list_Sum_sum_a_b] : ( if_list_Sum_sum_a_b @ ( member_nat @ ( F2 @ X4 ) @ ( image_5089838354808162685_b_nat @ F2 @ ( set_Sum_sum_a_b2 @ Xs2 ) ) ) @ Xs2 @ ( linord6900668025526808691_b_nat @ F2 @ X4 @ Xs2 ) ) ) ) ).

% insort_insert_key_def
thf(fact_484_insort__insert__key__def,axiom,
    ( linord900385247953437477at_nat
    = ( ^ [F2: produc6336967135452217047_b_nat > nat,X4: produc6336967135452217047_b_nat,Xs2: list_P667245297671117661_b_nat] : ( if_lis3161153797521331747_b_nat @ ( member_nat @ ( F2 @ X4 ) @ ( image_3920336858823826018at_nat @ F2 @ ( set_Pr176305205332118508_b_nat @ Xs2 ) ) ) @ Xs2 @ ( linord5985088777968003052at_nat @ F2 @ X4 @ Xs2 ) ) ) ) ).

% insort_insert_key_def
thf(fact_485_insort__insert__key__def,axiom,
    ( linord7646177846504673830_b_nat
    = ( ^ [F2: produc7779921308951573458um_a_b > nat,X4: produc7779921308951573458um_a_b,Xs2: list_P5598968436384720482um_a_b] : ( if_lis4596758700692144156um_a_b @ ( member_nat @ ( F2 @ X4 ) @ ( image_718544360348960937_b_nat @ F2 @ ( set_Pr7399620718385288829um_a_b @ Xs2 ) ) ) @ Xs2 @ ( linord7548316905644991135_b_nat @ F2 @ X4 @ Xs2 ) ) ) ) ).

% insort_insert_key_def
thf(fact_486_insort__insert__key__def,axiom,
    ( linord6417464493602701861_b_nat
    = ( ^ [F2: produc5919649888572472791um_a_b > nat,X4: produc5919649888572472791um_a_b,Xs2: list_P6497659457863146077um_a_b] : ( if_lis8991567957713360163um_a_b @ ( member_nat @ ( F2 @ X4 ) @ ( image_214044067618314594_b_nat @ F2 @ ( set_Pr8982359995307150060um_a_b @ Xs2 ) ) ) @ Xs2 @ ( linord2278795986762491628_b_nat @ F2 @ X4 @ Xs2 ) ) ) ) ).

% insort_insert_key_def
thf(fact_487_insort__insert__key__def,axiom,
    ( linord3394743409064221519at_nat
    = ( ^ [F2: product_prod_nat_nat > nat,X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] : ( if_lis9186351972506106189at_nat @ ( member_nat @ ( F2 @ X4 ) @ ( image_2486076414777270412at_nat @ F2 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) ) @ Xs2 @ ( linord2653521316007143958at_nat @ F2 @ X4 @ Xs2 ) ) ) ) ).

% insort_insert_key_def
thf(fact_488_insort__insert__key__def,axiom,
    ( linord1921536354676448932at_nat
    = ( ^ [F2: nat > nat,X4: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat @ ( F2 @ X4 ) @ ( image_nat_nat @ F2 @ ( set_nat2 @ Xs2 ) ) ) @ Xs2 @ ( linord8961336180081300637at_nat @ F2 @ X4 @ Xs2 ) ) ) ) ).

% insort_insert_key_def
thf(fact_489_insort__insert__insort__key,axiom,
    ! [F: produc4970237071031658413um_a_b > nat,X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ~ ( member_nat @ ( F @ X ) @ ( image_8744169858129055544_b_nat @ F @ ( set_Pr2436422074441463490um_a_b @ Xs ) ) )
     => ( ( linord9135931367038690811_b_nat @ F @ X @ Xs )
        = ( linord1506269616935411906_b_nat @ F @ X @ Xs ) ) ) ).

% insort_insert_insort_key
thf(fact_490_insort__insert__insort__key,axiom,
    ! [F: sum_sum_a_b > nat,X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ~ ( member_nat @ ( F @ X ) @ ( image_5089838354808162685_b_nat @ F @ ( set_Sum_sum_a_b2 @ Xs ) ) )
     => ( ( linord1205179515702317178_b_nat @ F @ X @ Xs )
        = ( linord6900668025526808691_b_nat @ F @ X @ Xs ) ) ) ).

% insort_insert_insort_key
thf(fact_491_insort__insert__insort__key,axiom,
    ! [F: produc6336967135452217047_b_nat > nat,X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ~ ( member_nat @ ( F @ X ) @ ( image_3920336858823826018at_nat @ F @ ( set_Pr176305205332118508_b_nat @ Xs ) ) )
     => ( ( linord900385247953437477at_nat @ F @ X @ Xs )
        = ( linord5985088777968003052at_nat @ F @ X @ Xs ) ) ) ).

% insort_insert_insort_key
thf(fact_492_insort__insert__insort__key,axiom,
    ! [F: produc7779921308951573458um_a_b > nat,X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ~ ( member_nat @ ( F @ X ) @ ( image_718544360348960937_b_nat @ F @ ( set_Pr7399620718385288829um_a_b @ Xs ) ) )
     => ( ( linord7646177846504673830_b_nat @ F @ X @ Xs )
        = ( linord7548316905644991135_b_nat @ F @ X @ Xs ) ) ) ).

% insort_insert_insort_key
thf(fact_493_insort__insert__insort__key,axiom,
    ! [F: produc5919649888572472791um_a_b > nat,X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ~ ( member_nat @ ( F @ X ) @ ( image_214044067618314594_b_nat @ F @ ( set_Pr8982359995307150060um_a_b @ Xs ) ) )
     => ( ( linord6417464493602701861_b_nat @ F @ X @ Xs )
        = ( linord2278795986762491628_b_nat @ F @ X @ Xs ) ) ) ).

% insort_insert_insort_key
thf(fact_494_insort__insert__insort__key,axiom,
    ! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member_nat @ ( F @ X ) @ ( image_2486076414777270412at_nat @ F @ ( set_Pr5648618587558075414at_nat @ Xs ) ) )
     => ( ( linord3394743409064221519at_nat @ F @ X @ Xs )
        = ( linord2653521316007143958at_nat @ F @ X @ Xs ) ) ) ).

% insort_insert_insort_key
thf(fact_495_insort__insert__insort__key,axiom,
    ! [F: nat > nat,X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
     => ( ( linord1921536354676448932at_nat @ F @ X @ Xs )
        = ( linord8961336180081300637at_nat @ F @ X @ Xs ) ) ) ).

% insort_insert_insort_key
thf(fact_496_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_497_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_498_Fpow__mono,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ord_le3724670747650509150_set_a @ ( finite_Fpow_a @ A5 ) @ ( finite_Fpow_a @ B5 ) ) ) ).

% Fpow_mono
thf(fact_499_sp__equiv__pair_Opelims_I1_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Xa: produc4970237071031658413um_a_b,Y: $o] :
      ( ( ( sp_equ4404609506204868152um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P8628657159683155312um_a_b @ sp_equ3929279581461973085um_a_b @ ( produc1529531236301088727um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: sum_sum_a_b] :
              ( ( X
                = ( produc8374529264883863525um_a_b @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: sum_sum_a_b] :
                  ( ( Xa
                    = ( produc8374529264883863525um_a_b @ A2 @ B2 ) )
                 => ( ( Y
                      = ( ( A = A2 )
                        = ( B = B2 ) ) )
                   => ~ ( accp_P8628657159683155312um_a_b @ sp_equ3929279581461973085um_a_b @ ( produc1529531236301088727um_a_b @ ( produc8374529264883863525um_a_b @ A @ B ) @ ( produc8374529264883863525um_a_b @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(1)
thf(fact_500_sp__equiv__pair_Opelims_I1_J,axiom,
    ! [X: produc5987322259985443452um_a_b,Xa: produc5987322259985443452um_a_b,Y: $o] :
      ( ( ( sp_equ110388464097044123um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P1406985159115915874um_a_b @ sp_equ8848776977153521718um_a_b @ ( produc4166429159020078609um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] :
              ( ( X
                = ( produc7989310782275393198um_a_b @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: produc4970237071031658413um_a_b] :
                  ( ( Xa
                    = ( produc7989310782275393198um_a_b @ A2 @ B2 ) )
                 => ( ( Y
                      = ( ( A = A2 )
                        = ( B = B2 ) ) )
                   => ~ ( accp_P1406985159115915874um_a_b @ sp_equ8848776977153521718um_a_b @ ( produc4166429159020078609um_a_b @ ( produc7989310782275393198um_a_b @ A @ B ) @ ( produc7989310782275393198um_a_b @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(1)
thf(fact_501_sp__equiv__pair_Opelims_I1_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Xa: produc6336967135452217047_b_nat,Y: $o] :
      ( ( ( sp_equ785082339516739170_b_nat @ X @ Xa )
        = Y )
     => ( ( accp_P3437629075677549552_b_nat @ sp_equ2804879137636136583_b_nat @ ( produc490312431154969431_b_nat @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: nat] :
              ( ( X
                = ( produc9094079831559073295_b_nat @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: nat] :
                  ( ( Xa
                    = ( produc9094079831559073295_b_nat @ A2 @ B2 ) )
                 => ( ( Y
                      = ( ( A = A2 )
                        = ( B = B2 ) ) )
                   => ~ ( accp_P3437629075677549552_b_nat @ sp_equ2804879137636136583_b_nat @ ( produc490312431154969431_b_nat @ ( produc9094079831559073295_b_nat @ A @ B ) @ ( produc9094079831559073295_b_nat @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(1)
thf(fact_502_sp__equiv__pair_Opelims_I1_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Xa: produc7779921308951573458um_a_b,Y: $o] :
      ( ( ( sp_equ3763899520926221297um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P8981828477233053454um_a_b @ sp_equ537355538419398540um_a_b @ ( produc8504789778075864509um_a_b @ X @ Xa ) )
       => ~ ! [A: nat,B: produc4970237071031658413um_a_b] :
              ( ( X
                = ( produc6016411306719553028um_a_b @ A @ B ) )
             => ! [A2: nat,B2: produc4970237071031658413um_a_b] :
                  ( ( Xa
                    = ( produc6016411306719553028um_a_b @ A2 @ B2 ) )
                 => ( ( Y
                      = ( ( A = A2 )
                        = ( B = B2 ) ) )
                   => ~ ( accp_P8981828477233053454um_a_b @ sp_equ537355538419398540um_a_b @ ( produc8504789778075864509um_a_b @ ( produc6016411306719553028um_a_b @ A @ B ) @ ( produc6016411306719553028um_a_b @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(1)
thf(fact_503_sp__equiv__pair_Opelims_I1_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Xa: produc5919649888572472791um_a_b,Y: $o] :
      ( ( ( sp_equ5014097358103830626um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P8934904672084055024um_a_b @ sp_equ7033894156223228039um_a_b @ ( produc8963228657344478039um_a_b @ X @ Xa ) )
       => ~ ! [A: nat,B: sum_sum_a_b] :
              ( ( X
                = ( produc4099722813291388943um_a_b @ A @ B ) )
             => ! [A2: nat,B2: sum_sum_a_b] :
                  ( ( Xa
                    = ( produc4099722813291388943um_a_b @ A2 @ B2 ) )
                 => ( ( Y
                      = ( ( A = A2 )
                        = ( B = B2 ) ) )
                   => ~ ( accp_P8934904672084055024um_a_b @ sp_equ7033894156223228039um_a_b @ ( produc8963228657344478039um_a_b @ ( produc4099722813291388943um_a_b @ A @ B ) @ ( produc4099722813291388943um_a_b @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(1)
thf(fact_504_sp__equiv__pair_Opelims_I1_J,axiom,
    ! [X: product_prod_nat_nat,Xa: product_prod_nat_nat,Y: $o] :
      ( ( ( sp_equ885134019442699404at_nat @ X @ Xa )
        = Y )
     => ( ( accp_P1267725715503270512at_nat @ sp_equ364710361814684337at_nat @ ( produc6161850002892822231at_nat @ X @ Xa ) )
       => ~ ! [A: nat,B: nat] :
              ( ( X
                = ( product_Pair_nat_nat @ A @ B ) )
             => ! [A2: nat,B2: nat] :
                  ( ( Xa
                    = ( product_Pair_nat_nat @ A2 @ B2 ) )
                 => ( ( Y
                      = ( ( A = A2 )
                        = ( B = B2 ) ) )
                   => ~ ( accp_P1267725715503270512at_nat @ sp_equ364710361814684337at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(1)
thf(fact_505_sp__equiv__pair_Opelims_I2_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Xa: produc4970237071031658413um_a_b] :
      ( ( sp_equ4404609506204868152um_a_b @ X @ Xa )
     => ( ( accp_P8628657159683155312um_a_b @ sp_equ3929279581461973085um_a_b @ ( produc1529531236301088727um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: sum_sum_a_b] :
              ( ( X
                = ( produc8374529264883863525um_a_b @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: sum_sum_a_b] :
                  ( ( Xa
                    = ( produc8374529264883863525um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P8628657159683155312um_a_b @ sp_equ3929279581461973085um_a_b @ ( produc1529531236301088727um_a_b @ ( produc8374529264883863525um_a_b @ A @ B ) @ ( produc8374529264883863525um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B != B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(2)
thf(fact_506_sp__equiv__pair_Opelims_I2_J,axiom,
    ! [X: produc5987322259985443452um_a_b,Xa: produc5987322259985443452um_a_b] :
      ( ( sp_equ110388464097044123um_a_b @ X @ Xa )
     => ( ( accp_P1406985159115915874um_a_b @ sp_equ8848776977153521718um_a_b @ ( produc4166429159020078609um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] :
              ( ( X
                = ( produc7989310782275393198um_a_b @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: produc4970237071031658413um_a_b] :
                  ( ( Xa
                    = ( produc7989310782275393198um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P1406985159115915874um_a_b @ sp_equ8848776977153521718um_a_b @ ( produc4166429159020078609um_a_b @ ( produc7989310782275393198um_a_b @ A @ B ) @ ( produc7989310782275393198um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B != B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(2)
thf(fact_507_sp__equiv__pair_Opelims_I2_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Xa: produc6336967135452217047_b_nat] :
      ( ( sp_equ785082339516739170_b_nat @ X @ Xa )
     => ( ( accp_P3437629075677549552_b_nat @ sp_equ2804879137636136583_b_nat @ ( produc490312431154969431_b_nat @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: nat] :
              ( ( X
                = ( produc9094079831559073295_b_nat @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: nat] :
                  ( ( Xa
                    = ( produc9094079831559073295_b_nat @ A2 @ B2 ) )
                 => ( ( accp_P3437629075677549552_b_nat @ sp_equ2804879137636136583_b_nat @ ( produc490312431154969431_b_nat @ ( produc9094079831559073295_b_nat @ A @ B ) @ ( produc9094079831559073295_b_nat @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B != B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(2)
thf(fact_508_sp__equiv__pair_Opelims_I2_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Xa: produc7779921308951573458um_a_b] :
      ( ( sp_equ3763899520926221297um_a_b @ X @ Xa )
     => ( ( accp_P8981828477233053454um_a_b @ sp_equ537355538419398540um_a_b @ ( produc8504789778075864509um_a_b @ X @ Xa ) )
       => ~ ! [A: nat,B: produc4970237071031658413um_a_b] :
              ( ( X
                = ( produc6016411306719553028um_a_b @ A @ B ) )
             => ! [A2: nat,B2: produc4970237071031658413um_a_b] :
                  ( ( Xa
                    = ( produc6016411306719553028um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P8981828477233053454um_a_b @ sp_equ537355538419398540um_a_b @ ( produc8504789778075864509um_a_b @ ( produc6016411306719553028um_a_b @ A @ B ) @ ( produc6016411306719553028um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B != B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(2)
thf(fact_509_sp__equiv__pair_Opelims_I2_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Xa: produc5919649888572472791um_a_b] :
      ( ( sp_equ5014097358103830626um_a_b @ X @ Xa )
     => ( ( accp_P8934904672084055024um_a_b @ sp_equ7033894156223228039um_a_b @ ( produc8963228657344478039um_a_b @ X @ Xa ) )
       => ~ ! [A: nat,B: sum_sum_a_b] :
              ( ( X
                = ( produc4099722813291388943um_a_b @ A @ B ) )
             => ! [A2: nat,B2: sum_sum_a_b] :
                  ( ( Xa
                    = ( produc4099722813291388943um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P8934904672084055024um_a_b @ sp_equ7033894156223228039um_a_b @ ( produc8963228657344478039um_a_b @ ( produc4099722813291388943um_a_b @ A @ B ) @ ( produc4099722813291388943um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B != B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(2)
thf(fact_510_sp__equiv__pair_Opelims_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( ( sp_equ885134019442699404at_nat @ X @ Xa )
     => ( ( accp_P1267725715503270512at_nat @ sp_equ364710361814684337at_nat @ ( produc6161850002892822231at_nat @ X @ Xa ) )
       => ~ ! [A: nat,B: nat] :
              ( ( X
                = ( product_Pair_nat_nat @ A @ B ) )
             => ! [A2: nat,B2: nat] :
                  ( ( Xa
                    = ( product_Pair_nat_nat @ A2 @ B2 ) )
                 => ( ( accp_P1267725715503270512at_nat @ sp_equ364710361814684337at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B != B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(2)
thf(fact_511_sp__equiv__pair_Opelims_I3_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Xa: produc4970237071031658413um_a_b] :
      ( ~ ( sp_equ4404609506204868152um_a_b @ X @ Xa )
     => ( ( accp_P8628657159683155312um_a_b @ sp_equ3929279581461973085um_a_b @ ( produc1529531236301088727um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: sum_sum_a_b] :
              ( ( X
                = ( produc8374529264883863525um_a_b @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: sum_sum_a_b] :
                  ( ( Xa
                    = ( produc8374529264883863525um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P8628657159683155312um_a_b @ sp_equ3929279581461973085um_a_b @ ( produc1529531236301088727um_a_b @ ( produc8374529264883863525um_a_b @ A @ B ) @ ( produc8374529264883863525um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B = B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(3)
thf(fact_512_sp__equiv__pair_Opelims_I3_J,axiom,
    ! [X: produc5987322259985443452um_a_b,Xa: produc5987322259985443452um_a_b] :
      ( ~ ( sp_equ110388464097044123um_a_b @ X @ Xa )
     => ( ( accp_P1406985159115915874um_a_b @ sp_equ8848776977153521718um_a_b @ ( produc4166429159020078609um_a_b @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: produc4970237071031658413um_a_b] :
              ( ( X
                = ( produc7989310782275393198um_a_b @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: produc4970237071031658413um_a_b] :
                  ( ( Xa
                    = ( produc7989310782275393198um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P1406985159115915874um_a_b @ sp_equ8848776977153521718um_a_b @ ( produc4166429159020078609um_a_b @ ( produc7989310782275393198um_a_b @ A @ B ) @ ( produc7989310782275393198um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B = B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(3)
thf(fact_513_sp__equiv__pair_Opelims_I3_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Xa: produc6336967135452217047_b_nat] :
      ( ~ ( sp_equ785082339516739170_b_nat @ X @ Xa )
     => ( ( accp_P3437629075677549552_b_nat @ sp_equ2804879137636136583_b_nat @ ( produc490312431154969431_b_nat @ X @ Xa ) )
       => ~ ! [A: sum_sum_a_b,B: nat] :
              ( ( X
                = ( produc9094079831559073295_b_nat @ A @ B ) )
             => ! [A2: sum_sum_a_b,B2: nat] :
                  ( ( Xa
                    = ( produc9094079831559073295_b_nat @ A2 @ B2 ) )
                 => ( ( accp_P3437629075677549552_b_nat @ sp_equ2804879137636136583_b_nat @ ( produc490312431154969431_b_nat @ ( produc9094079831559073295_b_nat @ A @ B ) @ ( produc9094079831559073295_b_nat @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B = B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(3)
thf(fact_514_sp__equiv__pair_Opelims_I3_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Xa: produc7779921308951573458um_a_b] :
      ( ~ ( sp_equ3763899520926221297um_a_b @ X @ Xa )
     => ( ( accp_P8981828477233053454um_a_b @ sp_equ537355538419398540um_a_b @ ( produc8504789778075864509um_a_b @ X @ Xa ) )
       => ~ ! [A: nat,B: produc4970237071031658413um_a_b] :
              ( ( X
                = ( produc6016411306719553028um_a_b @ A @ B ) )
             => ! [A2: nat,B2: produc4970237071031658413um_a_b] :
                  ( ( Xa
                    = ( produc6016411306719553028um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P8981828477233053454um_a_b @ sp_equ537355538419398540um_a_b @ ( produc8504789778075864509um_a_b @ ( produc6016411306719553028um_a_b @ A @ B ) @ ( produc6016411306719553028um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B = B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(3)
thf(fact_515_sp__equiv__pair_Opelims_I3_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Xa: produc5919649888572472791um_a_b] :
      ( ~ ( sp_equ5014097358103830626um_a_b @ X @ Xa )
     => ( ( accp_P8934904672084055024um_a_b @ sp_equ7033894156223228039um_a_b @ ( produc8963228657344478039um_a_b @ X @ Xa ) )
       => ~ ! [A: nat,B: sum_sum_a_b] :
              ( ( X
                = ( produc4099722813291388943um_a_b @ A @ B ) )
             => ! [A2: nat,B2: sum_sum_a_b] :
                  ( ( Xa
                    = ( produc4099722813291388943um_a_b @ A2 @ B2 ) )
                 => ( ( accp_P8934904672084055024um_a_b @ sp_equ7033894156223228039um_a_b @ ( produc8963228657344478039um_a_b @ ( produc4099722813291388943um_a_b @ A @ B ) @ ( produc4099722813291388943um_a_b @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B = B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(3)
thf(fact_516_sp__equiv__pair_Opelims_I3_J,axiom,
    ! [X: product_prod_nat_nat,Xa: product_prod_nat_nat] :
      ( ~ ( sp_equ885134019442699404at_nat @ X @ Xa )
     => ( ( accp_P1267725715503270512at_nat @ sp_equ364710361814684337at_nat @ ( produc6161850002892822231at_nat @ X @ Xa ) )
       => ~ ! [A: nat,B: nat] :
              ( ( X
                = ( product_Pair_nat_nat @ A @ B ) )
             => ! [A2: nat,B2: nat] :
                  ( ( Xa
                    = ( product_Pair_nat_nat @ A2 @ B2 ) )
                 => ( ( accp_P1267725715503270512at_nat @ sp_equ364710361814684337at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( product_Pair_nat_nat @ A2 @ B2 ) ) )
                   => ( ( A = A2 )
                      = ( B = B2 ) ) ) ) ) ) ) ).

% sp_equiv_pair.pelims(3)
thf(fact_517_in__lex__prod,axiom,
    ! [A3: sum_sum_a_b,B3: sum_sum_a_b,A4: sum_sum_a_b,B4: sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b,S3: set_Pr7446880501794416397um_a_b] :
      ( ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ ( produc8374529264883863525um_a_b @ A3 @ B3 ) @ ( produc8374529264883863525um_a_b @ A4 @ B4 ) ) @ ( lex_pr683455219244167165um_a_b @ R3 @ S3 ) )
      = ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ A3 @ A4 ) @ R3 )
        | ( ( A3 = A4 )
          & ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ B3 @ B4 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_518_in__lex__prod,axiom,
    ! [A3: sum_sum_a_b,B3: produc4970237071031658413um_a_b,A4: sum_sum_a_b,B4: produc4970237071031658413um_a_b,R3: set_Pr7446880501794416397um_a_b,S3: set_Pr8399320699097828935um_a_b] :
      ( ( member4727105268299851138um_a_b @ ( produc4166429159020078609um_a_b @ ( produc7989310782275393198um_a_b @ A3 @ B3 ) @ ( produc7989310782275393198um_a_b @ A4 @ B4 ) ) @ ( lex_pr6811497256589117590um_a_b @ R3 @ S3 ) )
      = ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ A3 @ A4 ) @ R3 )
        | ( ( A3 = A4 )
          & ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ B3 @ B4 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_519_in__lex__prod,axiom,
    ! [A3: sum_sum_a_b,B3: nat,A4: sum_sum_a_b,B4: nat,R3: set_Pr7446880501794416397um_a_b,S3: set_Pr1261947904930325089at_nat] :
      ( ( member681441685064050448_b_nat @ ( produc490312431154969431_b_nat @ ( produc9094079831559073295_b_nat @ A3 @ B3 ) @ ( produc9094079831559073295_b_nat @ A4 @ B4 ) ) @ ( lex_pr1485026810718412327_b_nat @ R3 @ S3 ) )
      = ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ A3 @ A4 ) @ R3 )
        | ( ( A3 = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B4 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_520_in__lex__prod,axiom,
    ! [A3: nat,B3: produc4970237071031658413um_a_b,A4: nat,B4: produc4970237071031658413um_a_b,R3: set_Pr1261947904930325089at_nat,S3: set_Pr8399320699097828935um_a_b] :
      ( ( member885238345412010542um_a_b @ ( produc8504789778075864509um_a_b @ ( produc6016411306719553028um_a_b @ A3 @ B3 ) @ ( produc6016411306719553028um_a_b @ A4 @ B4 ) ) @ ( lex_pr1794409810597562348um_a_b @ R3 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A4 ) @ R3 )
        | ( ( A3 = A4 )
          & ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ B3 @ B4 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_521_in__lex__prod,axiom,
    ! [A3: nat,B3: sum_sum_a_b,A4: nat,B4: sum_sum_a_b,R3: set_Pr1261947904930325089at_nat,S3: set_Pr7446880501794416397um_a_b] :
      ( ( member6178717281470555920um_a_b @ ( produc8963228657344478039um_a_b @ ( produc4099722813291388943um_a_b @ A3 @ B3 ) @ ( produc4099722813291388943um_a_b @ A4 @ B4 ) ) @ ( lex_pr5714041829305503783um_a_b @ R3 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A4 ) @ R3 )
        | ( ( A3 = A4 )
          & ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ B3 @ B4 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_522_in__lex__prod,axiom,
    ! [A3: nat,B3: nat,A4: nat,B4: nat,R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ ( product_Pair_nat_nat @ A4 @ B4 ) ) @ ( lex_prod_nat_nat @ R3 @ S3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A4 ) @ R3 )
        | ( ( A3 = A4 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ B4 ) @ S3 ) ) ) ) ).

% in_lex_prod
thf(fact_523_image__Pow__mono,axiom,
    ! [F: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A5 ) @ B5 )
     => ( ord_le6893508408891458716et_nat @ ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A5 ) ) @ ( pow_nat @ B5 ) ) ) ).

% image_Pow_mono
thf(fact_524_distinct__insort__key,axiom,
    ! [F: produc4970237071031658413um_a_b > nat,X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( distinct_nat @ ( map_Pr4285056615875073209_b_nat @ F @ ( linord1506269616935411906_b_nat @ F @ X @ Xs ) ) )
      = ( ~ ( member_nat @ ( F @ X ) @ ( image_8744169858129055544_b_nat @ F @ ( set_Pr2436422074441463490um_a_b @ Xs ) ) )
        & ( distinct_nat @ ( map_Pr4285056615875073209_b_nat @ F @ Xs ) ) ) ) ).

% distinct_insort_key
thf(fact_525_distinct__insort__key,axiom,
    ! [F: sum_sum_a_b > nat,X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( distinct_nat @ ( map_Sum_sum_a_b_nat @ F @ ( linord6900668025526808691_b_nat @ F @ X @ Xs ) ) )
      = ( ~ ( member_nat @ ( F @ X ) @ ( image_5089838354808162685_b_nat @ F @ ( set_Sum_sum_a_b2 @ Xs ) ) )
        & ( distinct_nat @ ( map_Sum_sum_a_b_nat @ F @ Xs ) ) ) ) ).

% distinct_insort_key
thf(fact_526_distinct__insort__key,axiom,
    ! [F: produc6336967135452217047_b_nat > nat,X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( distinct_nat @ ( map_Pr6605671721927337443at_nat @ F @ ( linord5985088777968003052at_nat @ F @ X @ Xs ) ) )
      = ( ~ ( member_nat @ ( F @ X ) @ ( image_3920336858823826018at_nat @ F @ ( set_Pr176305205332118508_b_nat @ Xs ) ) )
        & ( distinct_nat @ ( map_Pr6605671721927337443at_nat @ F @ Xs ) ) ) ) ).

% distinct_insort_key
thf(fact_527_distinct__insort__key,axiom,
    ! [F: produc7779921308951573458um_a_b > nat,X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( distinct_nat @ ( map_Pr218574505274605032_b_nat @ F @ ( linord7548316905644991135_b_nat @ F @ X @ Xs ) ) )
      = ( ~ ( member_nat @ ( F @ X ) @ ( image_718544360348960937_b_nat @ F @ ( set_Pr7399620718385288829um_a_b @ Xs ) ) )
        & ( distinct_nat @ ( map_Pr218574505274605032_b_nat @ F @ Xs ) ) ) ) ).

% distinct_insort_key
thf(fact_528_distinct__insort__key,axiom,
    ! [F: produc5919649888572472791um_a_b > nat,X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( distinct_nat @ ( map_Pr2899378930721826019_b_nat @ F @ ( linord2278795986762491628_b_nat @ F @ X @ Xs ) ) )
      = ( ~ ( member_nat @ ( F @ X ) @ ( image_214044067618314594_b_nat @ F @ ( set_Pr8982359995307150060um_a_b @ Xs ) ) )
        & ( distinct_nat @ ( map_Pr2899378930721826019_b_nat @ F @ Xs ) ) ) ) ).

% distinct_insort_key
thf(fact_529_distinct__insort__key,axiom,
    ! [F: product_prod_nat_nat > nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( distinct_nat @ ( map_Pr3938374229010428429at_nat @ F @ ( linord2653521316007143958at_nat @ F @ X @ Xs ) ) )
      = ( ~ ( member_nat @ ( F @ X ) @ ( image_2486076414777270412at_nat @ F @ ( set_Pr5648618587558075414at_nat @ Xs ) ) )
        & ( distinct_nat @ ( map_Pr3938374229010428429at_nat @ F @ Xs ) ) ) ) ).

% distinct_insort_key
thf(fact_530_distinct__insort__key,axiom,
    ! [F: nat > nat,X: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( map_nat_nat @ F @ ( linord8961336180081300637at_nat @ F @ X @ Xs ) ) )
      = ( ~ ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
        & ( distinct_nat @ ( map_nat_nat @ F @ Xs ) ) ) ) ).

% distinct_insort_key
thf(fact_531_Pow__iff,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( member_set_a @ A5 @ ( pow_a @ B5 ) )
      = ( ord_less_eq_set_a @ A5 @ B5 ) ) ).

% Pow_iff
thf(fact_532_PowI,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( member_set_a @ A5 @ ( pow_a @ B5 ) ) ) ).

% PowI
thf(fact_533_distinct__zipI1,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( distinct_Sum_sum_a_b @ Xs )
     => ( distin3760490539933216678um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_534_distinct__zipI1,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( distinct_Sum_sum_a_b @ Xs )
     => ( distin3366046370829352259um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_535_distinct__zipI1,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( distinct_Sum_sum_a_b @ Xs )
     => ( distin7255498516875091152_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_536_distinct__zipI1,axiom,
    ! [Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( distinct_nat @ Xs )
     => ( distin2733616231799517337um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_537_distinct__zipI1,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( distinct_nat @ Xs )
     => ( distin6838181269995346896um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_538_distinct__zipI1,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distin6923225563576452346at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI1
thf(fact_539_distinct__zipI2,axiom,
    ! [Ys: list_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( distinct_Sum_sum_a_b @ Ys )
     => ( distin3760490539933216678um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_540_distinct__zipI2,axiom,
    ! [Ys: list_P911570089447184179um_a_b,Xs: list_Sum_sum_a_b] :
      ( ( distin3760490539933216678um_a_b @ Ys )
     => ( distin3366046370829352259um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_541_distinct__zipI2,axiom,
    ! [Ys: list_nat,Xs: list_Sum_sum_a_b] :
      ( ( distinct_nat @ Ys )
     => ( distin7255498516875091152_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_542_distinct__zipI2,axiom,
    ! [Ys: list_P911570089447184179um_a_b,Xs: list_nat] :
      ( ( distin3760490539933216678um_a_b @ Ys )
     => ( distin2733616231799517337um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_543_distinct__zipI2,axiom,
    ! [Ys: list_Sum_sum_a_b,Xs: list_nat] :
      ( ( distinct_Sum_sum_a_b @ Ys )
     => ( distin6838181269995346896um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_544_distinct__zipI2,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( distinct_nat @ Ys )
     => ( distin6923225563576452346at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% distinct_zipI2
thf(fact_545_PowD,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( member_set_a @ A5 @ ( pow_a @ B5 ) )
     => ( ord_less_eq_set_a @ A5 @ B5 ) ) ).

% PowD
thf(fact_546_Pow__mono,axiom,
    ! [A5: set_a,B5: set_a] :
      ( ( ord_less_eq_set_a @ A5 @ B5 )
     => ( ord_le3724670747650509150_set_a @ ( pow_a @ A5 ) @ ( pow_a @ B5 ) ) ) ).

% Pow_mono
thf(fact_547_image__Pow__surj,axiom,
    ! [F: nat > nat,A5: set_nat,B5: set_nat] :
      ( ( ( image_nat_nat @ F @ A5 )
        = B5 )
     => ( ( image_7916887816326733075et_nat @ ( image_nat_nat @ F ) @ ( pow_nat @ A5 ) )
        = ( pow_nat @ B5 ) ) ) ).

% image_Pow_surj
thf(fact_548_subset__code_I3_J,axiom,
    ~ ( ord_le1431929631403617965um_a_b @ ( coset_4334520101952487072um_a_b @ nil_Pr8520116601857385651um_a_b ) @ ( set_Pr2436422074441463490um_a_b @ nil_Pr8520116601857385651um_a_b ) ) ).

% subset_code(3)
thf(fact_549_subset__code_I3_J,axiom,
    ~ ( ord_le9019793522827316924um_a_b @ ( coset_Sum_sum_a_b @ nil_Sum_sum_a_b ) @ ( set_Sum_sum_a_b2 @ nil_Sum_sum_a_b ) ) ).

% subset_code(3)
thf(fact_550_subset__code_I3_J,axiom,
    ~ ( ord_le8626435567010533079_b_nat @ ( coset_3799895497595635658_b_nat @ nil_Pr3750691640972353501_b_nat ) @ ( set_Pr176305205332118508_b_nat @ nil_Pr3750691640972353501_b_nat ) ) ).

% subset_code(3)
thf(fact_551_subset__code_I3_J,axiom,
    ~ ( ord_le5806309049788436072um_a_b @ ( coset_1753088348400206111um_a_b @ nil_Pr1090988463983433548um_a_b ) @ ( set_Pr7399620718385288829um_a_b @ nil_Pr1090988463983433548um_a_b ) ) ).

% subset_code(3)
thf(fact_552_subset__code_I3_J,axiom,
    ~ ( ord_le5233477690347785687um_a_b @ ( coset_3382578250715891402um_a_b @ nil_Pr3333374394092609245um_a_b ) @ ( set_Pr8982359995307150060um_a_b @ nil_Pr3333374394092609245um_a_b ) ) ).

% subset_code(3)
thf(fact_553_subset__code_I3_J,axiom,
    ~ ( ord_le3146513528884898305at_nat @ ( coset_6809516720700967924at_nat @ nil_Pr5478986624290739719at_nat ) @ ( set_Pr5648618587558075414at_nat @ nil_Pr5478986624290739719at_nat ) ) ).

% subset_code(3)
thf(fact_554_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_nat @ ( coset_nat @ nil_nat ) @ ( set_nat2 @ nil_nat ) ) ).

% subset_code(3)
thf(fact_555_subset__code_I3_J,axiom,
    ~ ( ord_less_eq_set_a @ ( coset_a @ nil_a ) @ ( set_a2 @ nil_a ) ) ).

% subset_code(3)
thf(fact_556_subseqs__powset,axiom,
    ! [Xs: list_P911570089447184179um_a_b] :
      ( ( image_6206480448382822303um_a_b @ set_Pr2436422074441463490um_a_b @ ( set_li6530523165260044232um_a_b @ ( subseq770281342586140802um_a_b @ Xs ) ) )
      = ( pow_Pr5647006400412165066um_a_b @ ( set_Pr2436422074441463490um_a_b @ Xs ) ) ) ).

% subseqs_powset
thf(fact_557_subseqs__powset,axiom,
    ! [Xs: list_Sum_sum_a_b] :
      ( ( image_3885185050486581529um_a_b @ set_Sum_sum_a_b2 @ ( set_list_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) )
      = ( pow_Sum_sum_a_b @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ).

% subseqs_powset
thf(fact_558_subseqs__powset,axiom,
    ! [Xs: list_P667245297671117661_b_nat] :
      ( ( image_8279935827611976479_b_nat @ set_Pr176305205332118508_b_nat @ ( set_li4576262994822683122_b_nat @ ( subseq5583020958151495084_b_nat @ Xs ) ) )
      = ( pow_Pr555540794441689332_b_nat @ ( set_Pr176305205332118508_b_nat @ Xs ) ) ) ).

% subseqs_powset
thf(fact_559_subseqs__powset,axiom,
    ! [Xs: list_P5598968436384720482um_a_b] :
      ( ( image_3456916899849611505um_a_b @ set_Pr7399620718385288829um_a_b @ ( set_li7179527725634549645um_a_b @ ( subseq353916544268657853um_a_b @ Xs ) ) )
      = ( pow_Pr1932833224159313781um_a_b @ ( set_Pr7399620718385288829um_a_b @ Xs ) ) ) ).

% subseqs_powset
thf(fact_560_subseqs__powset,axiom,
    ! [Xs: list_P6497659457863146077um_a_b] :
      ( ( image_5678317050538148639um_a_b @ set_Pr8982359995307150060um_a_b @ ( set_li1183305118159935730um_a_b @ ( subseq5165703711271750828um_a_b @ Xs ) ) )
      = ( pow_Pr138223547561945076um_a_b @ ( set_Pr8982359995307150060um_a_b @ Xs ) ) ) ).

% subseqs_powset
thf(fact_561_subseqs__powset,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( image_2829140658542241439at_nat @ set_Pr5648618587558075414at_nat @ ( set_li2962362208618583836at_nat @ ( subseq4535541509918465494at_nat @ Xs ) ) )
      = ( pow_Pr3466873192417945886at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% subseqs_powset
thf(fact_562_subseqs__powset,axiom,
    ! [Xs: list_nat] :
      ( ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
      = ( pow_nat @ ( set_nat2 @ Xs ) ) ) ).

% subseqs_powset
thf(fact_563_distinct__set__subseqs,axiom,
    ! [Xs: list_P911570089447184179um_a_b] :
      ( ( distin3760490539933216678um_a_b @ Xs )
     => ( distin1927142837123975942um_a_b @ ( map_li5300164894675075422um_a_b @ set_Pr2436422074441463490um_a_b @ ( subseq770281342586140802um_a_b @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_564_distinct__set__subseqs,axiom,
    ! [Xs: list_Sum_sum_a_b] :
      ( ( distinct_Sum_sum_a_b @ Xs )
     => ( distin8614596981818736547um_a_b @ ( map_li8802012604203463000um_a_b @ set_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_565_distinct__set__subseqs,axiom,
    ! [Xs: list_P667245297671117661_b_nat] :
      ( ( distin7255498516875091152_b_nat @ Xs )
     => ( distin2138710427895824688_b_nat @ ( map_li7732737647756021982_b_nat @ set_Pr176305205332118508_b_nat @ ( subseq5583020958151495084_b_nat @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_566_distinct__set__subseqs,axiom,
    ! [Xs: list_P5598968436384720482um_a_b] :
      ( ( distin2733616231799517337um_a_b @ Xs )
     => ( distin3393396725998943951um_a_b @ ( map_li3194960037626202160um_a_b @ set_Pr7399620718385288829um_a_b @ ( subseq353916544268657853um_a_b @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_567_distinct__set__subseqs,axiom,
    ! [Xs: list_P6497659457863146077um_a_b] :
      ( ( distin6838181269995346896um_a_b @ Xs )
     => ( distin7969124588087853104um_a_b @ ( map_li5131118870682194142um_a_b @ set_Pr8982359995307150060um_a_b @ ( subseq5165703711271750828um_a_b @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_568_distinct__set__subseqs,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( distin6923225563576452346at_nat @ Xs )
     => ( distin8719635941469336154at_nat @ ( map_li1611971986790279262at_nat @ set_Pr5648618587558075414at_nat @ ( subseq4535541509918465494at_nat @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_569_distinct__set__subseqs,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_set_nat @ ( map_list_nat_set_nat @ set_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).

% distinct_set_subseqs
thf(fact_570_insort__is__Cons,axiom,
    ! [Xs: list_P911570089447184179um_a_b,F: produc4970237071031658413um_a_b > nat,A3: produc4970237071031658413um_a_b] :
      ( ! [X5: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X5 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ ( F @ X5 ) ) )
     => ( ( linord1506269616935411906_b_nat @ F @ A3 @ Xs )
        = ( cons_P4211151491411527779um_a_b @ A3 @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_571_insort__is__Cons,axiom,
    ! [Xs: list_Sum_sum_a_b,F: sum_sum_a_b > nat,A3: sum_sum_a_b] :
      ( ! [X5: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X5 @ ( set_Sum_sum_a_b2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ ( F @ X5 ) ) )
     => ( ( linord6900668025526808691_b_nat @ F @ A3 @ Xs )
        = ( cons_Sum_sum_a_b @ A3 @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_572_insort__is__Cons,axiom,
    ! [Xs: list_P667245297671117661_b_nat,F: produc6336967135452217047_b_nat > nat,A3: produc6336967135452217047_b_nat] :
      ( ! [X5: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X5 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ ( F @ X5 ) ) )
     => ( ( linord5985088777968003052at_nat @ F @ A3 @ Xs )
        = ( cons_P4197526505774637965_b_nat @ A3 @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_573_insort__is__Cons,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,F: produc7779921308951573458um_a_b > nat,A3: produc7779921308951573458um_a_b] :
      ( ! [X5: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X5 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ ( F @ X5 ) ) )
     => ( ( linord7548316905644991135_b_nat @ F @ A3 @ Xs )
        = ( cons_P7543864023423915932um_a_b @ A3 @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_574_insort__is__Cons,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,F: produc5919649888572472791um_a_b > nat,A3: produc5919649888572472791um_a_b] :
      ( ! [X5: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X5 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ ( F @ X5 ) ) )
     => ( ( linord2278795986762491628_b_nat @ F @ A3 @ Xs )
        = ( cons_P3780209258894893709um_a_b @ A3 @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_575_insort__is__Cons,axiom,
    ! [Xs: list_P6011104703257516679at_nat,F: product_prod_nat_nat > nat,A3: product_prod_nat_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ ( F @ X5 ) ) )
     => ( ( linord2653521316007143958at_nat @ F @ A3 @ Xs )
        = ( cons_P6512896166579812791at_nat @ A3 @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_576_insort__is__Cons,axiom,
    ! [Xs: list_nat,F: nat > nat,A3: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( ord_less_eq_nat @ ( F @ A3 ) @ ( F @ X5 ) ) )
     => ( ( linord8961336180081300637at_nat @ F @ A3 @ Xs )
        = ( cons_nat @ A3 @ Xs ) ) ) ).

% insort_is_Cons
thf(fact_577_Sup_OSUP__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat,Sup: set_nat > nat] :
      ( ( A5 = B5 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B5 )
           => ( ( C3 @ X5 )
              = ( D @ X5 ) ) )
       => ( ( Sup @ ( image_nat_nat @ C3 @ A5 ) )
          = ( Sup @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_578_Inf_OINF__cong,axiom,
    ! [A5: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat,Inf: set_nat > nat] :
      ( ( A5 = B5 )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ B5 )
           => ( ( C3 @ X5 )
              = ( D @ X5 ) ) )
       => ( ( Inf @ ( image_nat_nat @ C3 @ A5 ) )
          = ( Inf @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_579_zip__eq__Nil__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( ( zip_Su8953288692215371525um_a_b @ Xs @ Ys )
        = nil_Pr8520116601857385651um_a_b )
      = ( ( Xs = nil_Sum_sum_a_b )
        | ( Ys = nil_Sum_sum_a_b ) ) ) ).

% zip_eq_Nil_iff
thf(fact_580_zip__eq__Nil__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( ( zip_Su8837167750138406798um_a_b @ Xs @ Ys )
        = nil_Pr7649374833807421942um_a_b )
      = ( ( Xs = nil_Sum_sum_a_b )
        | ( Ys = nil_Pr8520116601857385651um_a_b ) ) ) ).

% zip_eq_Nil_iff
thf(fact_581_zip__eq__Nil__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( ( zip_Sum_sum_a_b_nat @ Xs @ Ys )
        = nil_Pr3750691640972353501_b_nat )
      = ( ( Xs = nil_Sum_sum_a_b )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_582_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( ( zip_na2369391571721746148um_a_b @ Xs @ Ys )
        = nil_Pr1090988463983433548um_a_b )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr8520116601857385651um_a_b ) ) ) ).

% zip_eq_Nil_iff
thf(fact_583_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( ( zip_nat_Sum_sum_a_b @ Xs @ Ys )
        = nil_Pr3333374394092609245um_a_b )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Sum_sum_a_b ) ) ) ).

% zip_eq_Nil_iff
thf(fact_584_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_585_Nil__eq__zip__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( nil_Pr8520116601857385651um_a_b
        = ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_Sum_sum_a_b )
        | ( Ys = nil_Sum_sum_a_b ) ) ) ).

% Nil_eq_zip_iff
thf(fact_586_Nil__eq__zip__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( nil_Pr7649374833807421942um_a_b
        = ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_Sum_sum_a_b )
        | ( Ys = nil_Pr8520116601857385651um_a_b ) ) ) ).

% Nil_eq_zip_iff
thf(fact_587_Nil__eq__zip__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( nil_Pr3750691640972353501_b_nat
        = ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Sum_sum_a_b )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_588_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( nil_Pr1090988463983433548um_a_b
        = ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr8520116601857385651um_a_b ) ) ) ).

% Nil_eq_zip_iff
thf(fact_589_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( nil_Pr3333374394092609245um_a_b
        = ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Sum_sum_a_b ) ) ) ).

% Nil_eq_zip_iff
thf(fact_590_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_591_zip__Nil,axiom,
    ! [Ys: list_Sum_sum_a_b] :
      ( ( zip_Su8953288692215371525um_a_b @ nil_Sum_sum_a_b @ Ys )
      = nil_Pr8520116601857385651um_a_b ) ).

% zip_Nil
thf(fact_592_zip__Nil,axiom,
    ! [Ys: list_P911570089447184179um_a_b] :
      ( ( zip_Su8837167750138406798um_a_b @ nil_Sum_sum_a_b @ Ys )
      = nil_Pr7649374833807421942um_a_b ) ).

% zip_Nil
thf(fact_593_zip__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( zip_Sum_sum_a_b_nat @ nil_Sum_sum_a_b @ Ys )
      = nil_Pr3750691640972353501_b_nat ) ).

% zip_Nil
thf(fact_594_zip__Nil,axiom,
    ! [Ys: list_P911570089447184179um_a_b] :
      ( ( zip_na2369391571721746148um_a_b @ nil_nat @ Ys )
      = nil_Pr1090988463983433548um_a_b ) ).

% zip_Nil
thf(fact_595_zip__Nil,axiom,
    ! [Ys: list_Sum_sum_a_b] :
      ( ( zip_nat_Sum_sum_a_b @ nil_nat @ Ys )
      = nil_Pr3333374394092609245um_a_b ) ).

% zip_Nil
thf(fact_596_zip__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( zip_nat_nat @ nil_nat @ Ys )
      = nil_Pr5478986624290739719at_nat ) ).

% zip_Nil
thf(fact_597_zip__Cons__Cons,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b,Y: sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( zip_Su8953288692215371525um_a_b @ ( cons_Sum_sum_a_b @ X @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) )
      = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_598_zip__Cons__Cons,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b,Y: produc4970237071031658413um_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( zip_Su8837167750138406798um_a_b @ ( cons_Sum_sum_a_b @ X @ Xs ) @ ( cons_P4211151491411527779um_a_b @ Y @ Ys ) )
      = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y ) @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_599_zip__Cons__Cons,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b,Y: nat,Ys: list_nat] :
      ( ( zip_Sum_sum_a_b_nat @ ( cons_Sum_sum_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y ) @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_600_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: produc4970237071031658413um_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( zip_na2369391571721746148um_a_b @ ( cons_nat @ X @ Xs ) @ ( cons_P4211151491411527779um_a_b @ Y @ Ys ) )
      = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y ) @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_601_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( zip_nat_Sum_sum_a_b @ ( cons_nat @ X @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) )
      = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y ) @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_602_zip__Cons__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_603_zip_Osimps_I1_J,axiom,
    ! [Xs: list_Sum_sum_a_b] :
      ( ( zip_Su8953288692215371525um_a_b @ Xs @ nil_Sum_sum_a_b )
      = nil_Pr8520116601857385651um_a_b ) ).

% zip.simps(1)
thf(fact_604_zip_Osimps_I1_J,axiom,
    ! [Xs: list_Sum_sum_a_b] :
      ( ( zip_Su8837167750138406798um_a_b @ Xs @ nil_Pr8520116601857385651um_a_b )
      = nil_Pr7649374833807421942um_a_b ) ).

% zip.simps(1)
thf(fact_605_zip_Osimps_I1_J,axiom,
    ! [Xs: list_Sum_sum_a_b] :
      ( ( zip_Sum_sum_a_b_nat @ Xs @ nil_nat )
      = nil_Pr3750691640972353501_b_nat ) ).

% zip.simps(1)
thf(fact_606_zip_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( zip_na2369391571721746148um_a_b @ Xs @ nil_Pr8520116601857385651um_a_b )
      = nil_Pr1090988463983433548um_a_b ) ).

% zip.simps(1)
thf(fact_607_zip_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( zip_nat_Sum_sum_a_b @ Xs @ nil_Sum_sum_a_b )
      = nil_Pr3333374394092609245um_a_b ) ).

% zip.simps(1)
thf(fact_608_zip_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( zip_nat_nat @ Xs @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% zip.simps(1)
thf(fact_609_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P2: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P2 @ nil_nat @ nil_nat )
       => ( ! [X5: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys3 ) )
             => ( ( P2 @ Xs3 @ Ys3 )
               => ( P2 @ ( cons_nat @ X5 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_610_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P2: list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P2 @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X5: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat,Z3: nat,Zs2: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys3 ) )
               => ( ( ( size_size_list_nat @ Ys3 )
                    = ( size_size_list_nat @ Zs2 ) )
                 => ( ( P2 @ Xs3 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_nat @ X5 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z3 @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_611_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P2: list_nat > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P2 @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X5: nat,Xs3: list_nat,Y3: nat,Ys3: list_nat,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys3 ) )
                 => ( ( ( size_size_list_nat @ Ys3 )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P2 @ Xs3 @ Ys3 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_nat @ X5 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys3 ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_612_zip__eq__ConsE,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b,Xy: produc4970237071031658413um_a_b,Xys: list_P911570089447184179um_a_b] :
      ( ( ( zip_Su8953288692215371525um_a_b @ Xs @ Ys )
        = ( cons_P4211151491411527779um_a_b @ Xy @ Xys ) )
     => ~ ! [X5: sum_sum_a_b,Xs4: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ X5 @ Xs4 ) )
           => ! [Y3: sum_sum_a_b,Ys4: list_Sum_sum_a_b] :
                ( ( Ys
                  = ( cons_Sum_sum_a_b @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc8374529264883863525um_a_b @ X5 @ Y3 ) )
                 => ( Xys
                   != ( zip_Su8953288692215371525um_a_b @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_613_zip__eq__ConsE,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b,Xy: produc5987322259985443452um_a_b,Xys: list_P6716435105863944716um_a_b] :
      ( ( ( zip_Su8837167750138406798um_a_b @ Xs @ Ys )
        = ( cons_P2376077528734995014um_a_b @ Xy @ Xys ) )
     => ~ ! [X5: sum_sum_a_b,Xs4: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ X5 @ Xs4 ) )
           => ! [Y3: produc4970237071031658413um_a_b,Ys4: list_P911570089447184179um_a_b] :
                ( ( Ys
                  = ( cons_P4211151491411527779um_a_b @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc7989310782275393198um_a_b @ X5 @ Y3 ) )
                 => ( Xys
                   != ( zip_Su8837167750138406798um_a_b @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_614_zip__eq__ConsE,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_nat,Xy: produc6336967135452217047_b_nat,Xys: list_P667245297671117661_b_nat] :
      ( ( ( zip_Sum_sum_a_b_nat @ Xs @ Ys )
        = ( cons_P4197526505774637965_b_nat @ Xy @ Xys ) )
     => ~ ! [X5: sum_sum_a_b,Xs4: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ X5 @ Xs4 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc9094079831559073295_b_nat @ X5 @ Y3 ) )
                 => ( Xys
                   != ( zip_Sum_sum_a_b_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_615_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_P911570089447184179um_a_b,Xy: produc7779921308951573458um_a_b,Xys: list_P5598968436384720482um_a_b] :
      ( ( ( zip_na2369391571721746148um_a_b @ Xs @ Ys )
        = ( cons_P7543864023423915932um_a_b @ Xy @ Xys ) )
     => ~ ! [X5: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X5 @ Xs4 ) )
           => ! [Y3: produc4970237071031658413um_a_b,Ys4: list_P911570089447184179um_a_b] :
                ( ( Ys
                  = ( cons_P4211151491411527779um_a_b @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6016411306719553028um_a_b @ X5 @ Y3 ) )
                 => ( Xys
                   != ( zip_na2369391571721746148um_a_b @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_616_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_b,Xy: produc5919649888572472791um_a_b,Xys: list_P6497659457863146077um_a_b] :
      ( ( ( zip_nat_Sum_sum_a_b @ Xs @ Ys )
        = ( cons_P3780209258894893709um_a_b @ Xy @ Xys ) )
     => ~ ! [X5: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X5 @ Xs4 ) )
           => ! [Y3: sum_sum_a_b,Ys4: list_Sum_sum_a_b] :
                ( ( Ys
                  = ( cons_Sum_sum_a_b @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( produc4099722813291388943um_a_b @ X5 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_Sum_sum_a_b @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_617_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xy: product_prod_nat_nat,Xys: list_P6011104703257516679at_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = ( cons_P6512896166579812791at_nat @ Xy @ Xys ) )
     => ~ ! [X5: nat,Xs4: list_nat] :
            ( ( Xs
              = ( cons_nat @ X5 @ Xs4 ) )
           => ! [Y3: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X5 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs4 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_618_set__ConsD,axiom,
    ! [Y: produc4970237071031658413um_a_b,X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_619_set__ConsD,axiom,
    ! [Y: sum_sum_a_b,X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_620_set__ConsD,axiom,
    ! [Y: produc6336967135452217047_b_nat,X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( member8290135809484590976_b_nat @ Y @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member8290135809484590976_b_nat @ Y @ ( set_Pr176305205332118508_b_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_621_set__ConsD,axiom,
    ! [Y: produc7779921308951573458um_a_b,X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( member4052926750399300073um_a_b @ Y @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member4052926750399300073um_a_b @ Y @ ( set_Pr7399620718385288829um_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_622_set__ConsD,axiom,
    ! [Y: produc5919649888572472791um_a_b,X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( member7872818562604846720um_a_b @ Y @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member7872818562604846720um_a_b @ Y @ ( set_Pr8982359995307150060um_a_b @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_623_set__ConsD,axiom,
    ! [Y: product_prod_nat_nat,X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_624_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_625_list_Oset__cases,axiom,
    ! [E: produc4970237071031658413um_a_b,A3: list_P911570089447184179um_a_b] :
      ( ( member6480126262776431190um_a_b @ E @ ( set_Pr2436422074441463490um_a_b @ A3 ) )
     => ( ! [Z22: list_P911570089447184179um_a_b] :
            ( A3
           != ( cons_P4211151491411527779um_a_b @ E @ Z22 ) )
       => ~ ! [Z1: produc4970237071031658413um_a_b,Z22: list_P911570089447184179um_a_b] :
              ( ( A3
                = ( cons_P4211151491411527779um_a_b @ Z1 @ Z22 ) )
             => ~ ( member6480126262776431190um_a_b @ E @ ( set_Pr2436422074441463490um_a_b @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_626_list_Oset__cases,axiom,
    ! [E: sum_sum_a_b,A3: list_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ E @ ( set_Sum_sum_a_b2 @ A3 ) )
     => ( ! [Z22: list_Sum_sum_a_b] :
            ( A3
           != ( cons_Sum_sum_a_b @ E @ Z22 ) )
       => ~ ! [Z1: sum_sum_a_b,Z22: list_Sum_sum_a_b] :
              ( ( A3
                = ( cons_Sum_sum_a_b @ Z1 @ Z22 ) )
             => ~ ( member_Sum_sum_a_b @ E @ ( set_Sum_sum_a_b2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_627_list_Oset__cases,axiom,
    ! [E: produc6336967135452217047_b_nat,A3: list_P667245297671117661_b_nat] :
      ( ( member8290135809484590976_b_nat @ E @ ( set_Pr176305205332118508_b_nat @ A3 ) )
     => ( ! [Z22: list_P667245297671117661_b_nat] :
            ( A3
           != ( cons_P4197526505774637965_b_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc6336967135452217047_b_nat,Z22: list_P667245297671117661_b_nat] :
              ( ( A3
                = ( cons_P4197526505774637965_b_nat @ Z1 @ Z22 ) )
             => ~ ( member8290135809484590976_b_nat @ E @ ( set_Pr176305205332118508_b_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_628_list_Oset__cases,axiom,
    ! [E: produc7779921308951573458um_a_b,A3: list_P5598968436384720482um_a_b] :
      ( ( member4052926750399300073um_a_b @ E @ ( set_Pr7399620718385288829um_a_b @ A3 ) )
     => ( ! [Z22: list_P5598968436384720482um_a_b] :
            ( A3
           != ( cons_P7543864023423915932um_a_b @ E @ Z22 ) )
       => ~ ! [Z1: produc7779921308951573458um_a_b,Z22: list_P5598968436384720482um_a_b] :
              ( ( A3
                = ( cons_P7543864023423915932um_a_b @ Z1 @ Z22 ) )
             => ~ ( member4052926750399300073um_a_b @ E @ ( set_Pr7399620718385288829um_a_b @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_629_list_Oset__cases,axiom,
    ! [E: produc5919649888572472791um_a_b,A3: list_P6497659457863146077um_a_b] :
      ( ( member7872818562604846720um_a_b @ E @ ( set_Pr8982359995307150060um_a_b @ A3 ) )
     => ( ! [Z22: list_P6497659457863146077um_a_b] :
            ( A3
           != ( cons_P3780209258894893709um_a_b @ E @ Z22 ) )
       => ~ ! [Z1: produc5919649888572472791um_a_b,Z22: list_P6497659457863146077um_a_b] :
              ( ( A3
                = ( cons_P3780209258894893709um_a_b @ Z1 @ Z22 ) )
             => ~ ( member7872818562604846720um_a_b @ E @ ( set_Pr8982359995307150060um_a_b @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_630_list_Oset__cases,axiom,
    ! [E: product_prod_nat_nat,A3: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ A3 ) )
     => ( ! [Z22: list_P6011104703257516679at_nat] :
            ( A3
           != ( cons_P6512896166579812791at_nat @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_nat_nat,Z22: list_P6011104703257516679at_nat] :
              ( ( A3
                = ( cons_P6512896166579812791at_nat @ Z1 @ Z22 ) )
             => ~ ( member8440522571783428010at_nat @ E @ ( set_Pr5648618587558075414at_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_631_list_Oset__cases,axiom,
    ! [E: nat,A3: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A3 ) )
     => ( ! [Z22: list_nat] :
            ( A3
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A3
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_632_list_Oset__intros_I1_J,axiom,
    ! [X21: produc4970237071031658413um_a_b,X222: list_P911570089447184179um_a_b] : ( member6480126262776431190um_a_b @ X21 @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_633_list_Oset__intros_I1_J,axiom,
    ! [X21: sum_sum_a_b,X222: list_Sum_sum_a_b] : ( member_Sum_sum_a_b @ X21 @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_634_list_Oset__intros_I1_J,axiom,
    ! [X21: produc6336967135452217047_b_nat,X222: list_P667245297671117661_b_nat] : ( member8290135809484590976_b_nat @ X21 @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_635_list_Oset__intros_I1_J,axiom,
    ! [X21: produc7779921308951573458um_a_b,X222: list_P5598968436384720482um_a_b] : ( member4052926750399300073um_a_b @ X21 @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_636_list_Oset__intros_I1_J,axiom,
    ! [X21: produc5919649888572472791um_a_b,X222: list_P6497659457863146077um_a_b] : ( member7872818562604846720um_a_b @ X21 @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_637_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_nat_nat,X222: list_P6011104703257516679at_nat] : ( member8440522571783428010at_nat @ X21 @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_638_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_639_list_Oset__intros_I2_J,axiom,
    ! [Y: produc4970237071031658413um_a_b,X222: list_P911570089447184179um_a_b,X21: produc4970237071031658413um_a_b] :
      ( ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ X222 ) )
     => ( member6480126262776431190um_a_b @ Y @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_640_list_Oset__intros_I2_J,axiom,
    ! [Y: sum_sum_a_b,X222: list_Sum_sum_a_b,X21: sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ X222 ) )
     => ( member_Sum_sum_a_b @ Y @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_641_list_Oset__intros_I2_J,axiom,
    ! [Y: produc6336967135452217047_b_nat,X222: list_P667245297671117661_b_nat,X21: produc6336967135452217047_b_nat] :
      ( ( member8290135809484590976_b_nat @ Y @ ( set_Pr176305205332118508_b_nat @ X222 ) )
     => ( member8290135809484590976_b_nat @ Y @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_642_list_Oset__intros_I2_J,axiom,
    ! [Y: produc7779921308951573458um_a_b,X222: list_P5598968436384720482um_a_b,X21: produc7779921308951573458um_a_b] :
      ( ( member4052926750399300073um_a_b @ Y @ ( set_Pr7399620718385288829um_a_b @ X222 ) )
     => ( member4052926750399300073um_a_b @ Y @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_643_list_Oset__intros_I2_J,axiom,
    ! [Y: produc5919649888572472791um_a_b,X222: list_P6497659457863146077um_a_b,X21: produc5919649888572472791um_a_b] :
      ( ( member7872818562604846720um_a_b @ Y @ ( set_Pr8982359995307150060um_a_b @ X222 ) )
     => ( member7872818562604846720um_a_b @ Y @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_644_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_nat_nat,X222: list_P6011104703257516679at_nat,X21: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ X222 ) )
     => ( member8440522571783428010at_nat @ Y @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_645_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_646_impossible__Cons,axiom,
    ! [Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) )
     => ( Xs
       != ( cons_nat @ X @ Ys ) ) ) ).

% impossible_Cons
thf(fact_647_set__subset__Cons,axiom,
    ! [Xs: list_P911570089447184179um_a_b,X: produc4970237071031658413um_a_b] : ( ord_le1431929631403617965um_a_b @ ( set_Pr2436422074441463490um_a_b @ Xs ) @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_648_set__subset__Cons,axiom,
    ! [Xs: list_Sum_sum_a_b,X: sum_sum_a_b] : ( ord_le9019793522827316924um_a_b @ ( set_Sum_sum_a_b2 @ Xs ) @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_649_set__subset__Cons,axiom,
    ! [Xs: list_P667245297671117661_b_nat,X: produc6336967135452217047_b_nat] : ( ord_le8626435567010533079_b_nat @ ( set_Pr176305205332118508_b_nat @ Xs ) @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_650_set__subset__Cons,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,X: produc7779921308951573458um_a_b] : ( ord_le5806309049788436072um_a_b @ ( set_Pr7399620718385288829um_a_b @ Xs ) @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_651_set__subset__Cons,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,X: produc5919649888572472791um_a_b] : ( ord_le5233477690347785687um_a_b @ ( set_Pr8982359995307150060um_a_b @ Xs ) @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_652_set__subset__Cons,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_653_set__subset__Cons,axiom,
    ! [Xs: list_nat,X: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_654_set__subset__Cons,axiom,
    ! [Xs: list_a,X: a] : ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ ( cons_a @ X @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_655_distinct_Osimps_I2_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( distin3760490539933216678um_a_b @ ( cons_P4211151491411527779um_a_b @ X @ Xs ) )
      = ( ~ ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
        & ( distin3760490539933216678um_a_b @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_656_distinct_Osimps_I2_J,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( distinct_Sum_sum_a_b @ ( cons_Sum_sum_a_b @ X @ Xs ) )
      = ( ~ ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
        & ( distinct_Sum_sum_a_b @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_657_distinct_Osimps_I2_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( distin7255498516875091152_b_nat @ ( cons_P4197526505774637965_b_nat @ X @ Xs ) )
      = ( ~ ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Xs ) )
        & ( distin7255498516875091152_b_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_658_distinct_Osimps_I2_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( distin2733616231799517337um_a_b @ ( cons_P7543864023423915932um_a_b @ X @ Xs ) )
      = ( ~ ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
        & ( distin2733616231799517337um_a_b @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_659_distinct_Osimps_I2_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( distin6838181269995346896um_a_b @ ( cons_P3780209258894893709um_a_b @ X @ Xs ) )
      = ( ~ ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
        & ( distin6838181269995346896um_a_b @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_660_distinct_Osimps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( distin6923225563576452346at_nat @ ( cons_P6512896166579812791at_nat @ X @ Xs ) )
      = ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( distin6923225563576452346at_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_661_distinct_Osimps_I2_J,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ X @ Xs ) )
      = ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
        & ( distinct_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_662_subset__subseqs,axiom,
    ! [X2: set_Pr7446880501794416397um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( ord_le1431929631403617965um_a_b @ X2 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
     => ( member5262719511116389814um_a_b @ X2 @ ( image_6206480448382822303um_a_b @ set_Pr2436422074441463490um_a_b @ ( set_li6530523165260044232um_a_b @ ( subseq770281342586140802um_a_b @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_663_subset__subseqs,axiom,
    ! [X2: set_Sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( ord_le9019793522827316924um_a_b @ X2 @ ( set_Sum_sum_a_b2 @ Xs ) )
     => ( member4060935254435997939um_a_b @ X2 @ ( image_3885185050486581529um_a_b @ set_Sum_sum_a_b2 @ ( set_list_Sum_sum_a_b2 @ ( subseqs_Sum_sum_a_b @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_664_subset__subseqs,axiom,
    ! [X2: set_Pr7634312471471294263_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( ord_le8626435567010533079_b_nat @ X2 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
     => ( member4516920322559128544_b_nat @ X2 @ ( image_8279935827611976479_b_nat @ set_Pr176305205332118508_b_nat @ ( set_li4576262994822683122_b_nat @ ( subseq5583020958151495084_b_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_665_subset__subseqs,axiom,
    ! [X2: set_Pr5956926064383532552um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( ord_le5806309049788436072um_a_b @ X2 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
     => ( member8475297673577648159um_a_b @ X2 @ ( image_3456916899849611505um_a_b @ set_Pr7399620718385288829um_a_b @ ( set_li7179527725634549645um_a_b @ ( subseq353916544268657853um_a_b @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_666_subset__subseqs,axiom,
    ! [X2: set_Pr4241354594808546871um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( ord_le5233477690347785687um_a_b @ X2 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
     => ( member1123962445896381152um_a_b @ X2 @ ( image_5678317050538148639um_a_b @ set_Pr8982359995307150060um_a_b @ ( set_li1183305118159935730um_a_b @ ( subseq5165703711271750828um_a_b @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_667_subset__subseqs,axiom,
    ! [X2: set_Pr1261947904930325089at_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_le3146513528884898305at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( member2643936169264416010at_nat @ X2 @ ( image_2829140658542241439at_nat @ set_Pr5648618587558075414at_nat @ ( set_li2962362208618583836at_nat @ ( subseq4535541509918465494at_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_668_subset__subseqs,axiom,
    ! [X2: set_nat,Xs: list_nat] :
      ( ( ord_less_eq_set_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( member_set_nat @ X2 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_669_subset__subseqs,axiom,
    ! [X2: set_a,Xs: list_a] :
      ( ( ord_less_eq_set_a @ X2 @ ( set_a2 @ Xs ) )
     => ( member_set_a @ X2 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_670_in__set__simps_I2_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: produc4970237071031658413um_a_b] :
      ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ Y @ nil_Pr8520116601857385651um_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_671_in__set__simps_I2_J,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ Y @ nil_Sum_sum_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_672_in__set__simps_I2_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Y: produc6336967135452217047_b_nat] :
      ( ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ Y @ nil_Pr3750691640972353501_b_nat ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_673_in__set__simps_I2_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Y: produc7779921308951573458um_a_b] :
      ( ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ Y @ nil_Pr1090988463983433548um_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_674_in__set__simps_I2_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Y: produc5919649888572472791um_a_b] :
      ( ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ Y @ nil_Pr3333374394092609245um_a_b ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_675_in__set__simps_I2_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ Y @ nil_Pr5478986624290739719at_nat ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_676_in__set__simps_I2_J,axiom,
    ! [X: nat,Y: nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( X = Y ) ) ).

% in_set_simps(2)
thf(fact_677_in__set__simps_I3_J,axiom,
    ! [X: produc4970237071031658413um_a_b] :
      ~ ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ nil_Pr8520116601857385651um_a_b ) ) ).

% in_set_simps(3)
thf(fact_678_in__set__simps_I3_J,axiom,
    ! [X: sum_sum_a_b] :
      ~ ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ nil_Sum_sum_a_b ) ) ).

% in_set_simps(3)
thf(fact_679_in__set__simps_I3_J,axiom,
    ! [X: produc6336967135452217047_b_nat] :
      ~ ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ nil_Pr3750691640972353501_b_nat ) ) ).

% in_set_simps(3)
thf(fact_680_in__set__simps_I3_J,axiom,
    ! [X: produc7779921308951573458um_a_b] :
      ~ ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ nil_Pr1090988463983433548um_a_b ) ) ).

% in_set_simps(3)
thf(fact_681_in__set__simps_I3_J,axiom,
    ! [X: produc5919649888572472791um_a_b] :
      ~ ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ nil_Pr3333374394092609245um_a_b ) ) ).

% in_set_simps(3)
thf(fact_682_in__set__simps_I3_J,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ nil_Pr5478986624290739719at_nat ) ) ).

% in_set_simps(3)
thf(fact_683_in__set__simps_I3_J,axiom,
    ! [X: nat] :
      ~ ( member_nat @ X @ ( set_nat2 @ nil_nat ) ) ).

% in_set_simps(3)
thf(fact_684_in__set__simps_I1_J,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: produc4970237071031658413um_a_b,Z2: produc4970237071031658413um_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ Y @ ( cons_P4211151491411527779um_a_b @ Z2 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ Z2 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_685_in__set__simps_I1_J,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b,Z2: sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ Y @ ( cons_Sum_sum_a_b @ Z2 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ Z2 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_686_in__set__simps_I1_J,axiom,
    ! [X: produc6336967135452217047_b_nat,Y: produc6336967135452217047_b_nat,Z2: produc6336967135452217047_b_nat,Ys: list_P667245297671117661_b_nat] :
      ( ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ Y @ ( cons_P4197526505774637965_b_nat @ Z2 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ Z2 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_687_in__set__simps_I1_J,axiom,
    ! [X: produc7779921308951573458um_a_b,Y: produc7779921308951573458um_a_b,Z2: produc7779921308951573458um_a_b,Ys: list_P5598968436384720482um_a_b] :
      ( ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ Y @ ( cons_P7543864023423915932um_a_b @ Z2 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ Z2 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_688_in__set__simps_I1_J,axiom,
    ! [X: produc5919649888572472791um_a_b,Y: produc5919649888572472791um_a_b,Z2: produc5919649888572472791um_a_b,Ys: list_P6497659457863146077um_a_b] :
      ( ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ Y @ ( cons_P3780209258894893709um_a_b @ Z2 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ Z2 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_689_in__set__simps_I1_J,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,Z2: product_prod_nat_nat,Ys: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ Y @ ( cons_P6512896166579812791at_nat @ Z2 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ Z2 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_690_in__set__simps_I1_J,axiom,
    ! [X: nat,Y: nat,Z2: nat,Ys: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Y @ ( cons_nat @ Z2 @ Ys ) ) ) )
      = ( ( X = Y )
        | ( member_nat @ X @ ( set_nat2 @ ( cons_nat @ Z2 @ Ys ) ) ) ) ) ).

% in_set_simps(1)
thf(fact_691_Cons__in__lex,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b,Y: sum_sum_a_b,Ys: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b] :
      ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ ( cons_Sum_sum_a_b @ X @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) @ ( lex_Sum_sum_a_b @ R3 ) )
      = ( ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ R3 )
          & ( ( size_s5949480688510517322um_a_b @ Xs )
            = ( size_s5949480688510517322um_a_b @ Ys ) ) )
        | ( ( X = Y )
          & ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs @ Ys ) @ ( lex_Sum_sum_a_b @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_692_Cons__in__lex,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( lex_nat @ R3 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
          & ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lex_nat @ R3 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_693_the__elem__set,axiom,
    ! [X: produc4970237071031658413um_a_b] :
      ( ( the_el2633929365587109666um_a_b @ ( set_Pr2436422074441463490um_a_b @ ( cons_P4211151491411527779um_a_b @ X @ nil_Pr8520116601857385651um_a_b ) ) )
      = X ) ).

% the_elem_set
thf(fact_694_the__elem__set,axiom,
    ! [X: sum_sum_a_b] :
      ( ( the_elem_Sum_sum_a_b @ ( set_Sum_sum_a_b2 @ ( cons_Sum_sum_a_b @ X @ nil_Sum_sum_a_b ) ) )
      = X ) ).

% the_elem_set
thf(fact_695_the__elem__set,axiom,
    ! [X: produc6336967135452217047_b_nat] :
      ( ( the_el40913752972000332_b_nat @ ( set_Pr176305205332118508_b_nat @ ( cons_P4197526505774637965_b_nat @ X @ nil_Pr3750691640972353501_b_nat ) ) )
      = X ) ).

% the_elem_set
thf(fact_696_the__elem__set,axiom,
    ! [X: produc7779921308951573458um_a_b] :
      ( ( the_el7404150159202758173um_a_b @ ( set_Pr7399620718385288829um_a_b @ ( cons_P7543864023423915932um_a_b @ X @ nil_Pr1090988463983433548um_a_b ) ) )
      = X ) ).

% the_elem_set
thf(fact_697_the__elem__set,axiom,
    ! [X: produc5919649888572472791um_a_b] :
      ( ( the_el8846968542947031884um_a_b @ ( set_Pr8982359995307150060um_a_b @ ( cons_P3780209258894893709um_a_b @ X @ nil_Pr3333374394092609245um_a_b ) ) )
      = X ) ).

% the_elem_set
thf(fact_698_the__elem__set,axiom,
    ! [X: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( set_Pr5648618587558075414at_nat @ ( cons_P6512896166579812791at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) )
      = X ) ).

% the_elem_set
thf(fact_699_the__elem__set,axiom,
    ! [X: nat] :
      ( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X @ nil_nat ) ) )
      = X ) ).

% the_elem_set
thf(fact_700_not__in__set__insert,axiom,
    ! [X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ~ ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
     => ( ( insert4007205078261549769um_a_b @ X @ Xs )
        = ( cons_P4211151491411527779um_a_b @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_701_not__in__set__insert,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ~ ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
     => ( ( insert_Sum_sum_a_b @ X @ Xs )
        = ( cons_Sum_sum_a_b @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_702_not__in__set__insert,axiom,
    ! [X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ~ ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Xs ) )
     => ( ( insert2368410386661493235_b_nat @ X @ Xs )
        = ( cons_P4197526505774637965_b_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_703_not__in__set__insert,axiom,
    ! [X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ~ ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
     => ( ( insert6442401040785751478um_a_b @ X @ Xs )
        = ( cons_P7543864023423915932um_a_b @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_704_not__in__set__insert,axiom,
    ! [X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ~ ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
     => ( ( insert1951093139781748979um_a_b @ X @ Xs )
        = ( cons_P3780209258894893709um_a_b @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_705_not__in__set__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( insert8944034826898310173at_nat @ X @ Xs )
        = ( cons_P6512896166579812791at_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_706_not__in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = ( cons_nat @ X @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_707_in__set__insert,axiom,
    ! [X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
     => ( ( insert4007205078261549769um_a_b @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_708_in__set__insert,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
     => ( ( insert_Sum_sum_a_b @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_709_in__set__insert,axiom,
    ! [X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Xs ) )
     => ( ( insert2368410386661493235_b_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_710_in__set__insert,axiom,
    ! [X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
     => ( ( insert6442401040785751478um_a_b @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_711_in__set__insert,axiom,
    ! [X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
     => ( ( insert1951093139781748979um_a_b @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_712_in__set__insert,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( insert8944034826898310173at_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_713_in__set__insert,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_714_GreatestI__nat,axiom,
    ! [P2: nat > $o,K: nat,B3: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_nat
thf(fact_715_Greatest__le__nat,axiom,
    ! [P2: nat > $o,K: nat,B3: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% Greatest_le_nat
thf(fact_716_GreatestI__ex__nat,axiom,
    ! [P2: nat > $o,B3: nat] :
      ( ? [X_1: nat] : ( P2 @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ( P2 @ ( order_Greatest_nat @ P2 ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_717_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K: nat,B3: nat] :
      ( ( P2 @ K )
     => ( ! [Y3: nat] :
            ( ( P2 @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B3 ) )
       => ? [X5: nat] :
            ( ( P2 @ X5 )
            & ! [Y7: nat] :
                ( ( P2 @ Y7 )
               => ( ord_less_eq_nat @ Y7 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_718_nat__le__linear,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
      | ( ord_less_eq_nat @ N @ M2 ) ) ).

% nat_le_linear
thf(fact_719_le__antisym,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_eq_nat @ N @ M2 )
       => ( M2 = N ) ) ) ).

% le_antisym
thf(fact_720_eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 = N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% eq_imp_le
thf(fact_721_le__trans,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ J2 @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_722_le__refl,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ N @ N ) ).

% le_refl
thf(fact_723_List_Oinsert__def,axiom,
    ( insert4007205078261549769um_a_b
    = ( ^ [X4: produc4970237071031658413um_a_b,Xs2: list_P911570089447184179um_a_b] : ( if_lis2231744601426488825um_a_b @ ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ Xs2 ) ) @ Xs2 @ ( cons_P4211151491411527779um_a_b @ X4 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_724_List_Oinsert__def,axiom,
    ( insert_Sum_sum_a_b
    = ( ^ [X4: sum_sum_a_b,Xs2: list_Sum_sum_a_b] : ( if_list_Sum_sum_a_b @ ( member_Sum_sum_a_b @ X4 @ ( set_Sum_sum_a_b2 @ Xs2 ) ) @ Xs2 @ ( cons_Sum_sum_a_b @ X4 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_725_List_Oinsert__def,axiom,
    ( insert2368410386661493235_b_nat
    = ( ^ [X4: produc6336967135452217047_b_nat,Xs2: list_P667245297671117661_b_nat] : ( if_lis3161153797521331747_b_nat @ ( member8290135809484590976_b_nat @ X4 @ ( set_Pr176305205332118508_b_nat @ Xs2 ) ) @ Xs2 @ ( cons_P4197526505774637965_b_nat @ X4 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_726_List_Oinsert__def,axiom,
    ( insert6442401040785751478um_a_b
    = ( ^ [X4: produc7779921308951573458um_a_b,Xs2: list_P5598968436384720482um_a_b] : ( if_lis4596758700692144156um_a_b @ ( member4052926750399300073um_a_b @ X4 @ ( set_Pr7399620718385288829um_a_b @ Xs2 ) ) @ Xs2 @ ( cons_P7543864023423915932um_a_b @ X4 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_727_List_Oinsert__def,axiom,
    ( insert1951093139781748979um_a_b
    = ( ^ [X4: produc5919649888572472791um_a_b,Xs2: list_P6497659457863146077um_a_b] : ( if_lis8991567957713360163um_a_b @ ( member7872818562604846720um_a_b @ X4 @ ( set_Pr8982359995307150060um_a_b @ Xs2 ) ) @ Xs2 @ ( cons_P3780209258894893709um_a_b @ X4 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_728_List_Oinsert__def,axiom,
    ( insert8944034826898310173at_nat
    = ( ^ [X4: product_prod_nat_nat,Xs2: list_P6011104703257516679at_nat] : ( if_lis9186351972506106189at_nat @ ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) ) @ Xs2 @ ( cons_P6512896166579812791at_nat @ X4 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_729_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X4: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X4 @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_730_comp__sunion__with_Ocases,axiom,
    ! [X: produc7737026988361515175um_a_b] :
      ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K2: sum_sum_a_b,V2: sum_sum_a_b,As2: list_P911570089447184179um_a_b,K3: sum_sum_a_b,V3: sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
          ( X
         != ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
            ( X
           != ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ nil_Pr8520116601857385651um_a_b @ Bs ) ) )
       => ~ ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,As2: list_P911570089447184179um_a_b] :
              ( X
             != ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ As2 @ nil_Pr8520116601857385651um_a_b ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_731_comp__sunion__with_Ocases,axiom,
    ! [X: produc4052817018830251980um_a_b] :
      ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K2: sum_sum_a_b,V2: produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b,K3: sum_sum_a_b,V3: produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
          ( X
         != ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
            ( X
           != ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ nil_Pr7649374833807421942um_a_b @ Bs ) ) )
       => ~ ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b] :
              ( X
             != ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ As2 @ nil_Pr7649374833807421942um_a_b ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_732_comp__sunion__with_Ocases,axiom,
    ! [X: produc8464943063266627625_b_nat] :
      ( ! [F3: sum_sum_a_b > nat > nat > nat,K2: sum_sum_a_b,V2: nat,As2: list_P667245297671117661_b_nat,K3: sum_sum_a_b,V3: nat,Bs: list_P667245297671117661_b_nat] :
          ( X
         != ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: sum_sum_a_b > nat > nat > nat,Bs: list_P667245297671117661_b_nat] :
            ( X
           != ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ nil_Pr3750691640972353501_b_nat @ Bs ) ) )
       => ~ ! [F3: sum_sum_a_b > nat > nat > nat,As2: list_P667245297671117661_b_nat] :
              ( X
             != ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ As2 @ nil_Pr3750691640972353501_b_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_733_comp__sunion__with_Ocases,axiom,
    ! [X: produc5120744747776269730um_a_b] :
      ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b,K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
          ( X
         != ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
            ( X
           != ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ nil_Pr1090988463983433548um_a_b @ Bs ) ) )
       => ~ ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b] :
              ( X
             != ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ As2 @ nil_Pr1090988463983433548um_a_b ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_734_comp__sunion__with_Ocases,axiom,
    ! [X: produc1341479252616905341um_a_b] :
      ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b,K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
          ( X
         != ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
            ( X
           != ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ nil_Pr3333374394092609245um_a_b @ Bs ) ) )
       => ~ ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,As2: list_P6497659457863146077um_a_b] :
              ( X
             != ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ As2 @ nil_Pr3333374394092609245um_a_b ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_735_comp__sunion__with_Ocases,axiom,
    ! [X: produc4961808859980711423at_nat] :
      ( ! [F3: nat > nat > nat > nat,K2: nat,V2: nat,As2: list_P6011104703257516679at_nat,K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
          ( X
         != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
            ( X
           != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
       => ~ ! [F3: nat > nat > nat > nat,As2: list_P6011104703257516679at_nat] :
              ( X
             != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ As2 @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% comp_sunion_with.cases
thf(fact_736_sunion__with_Ocases,axiom,
    ! [X: produc5120744747776269730um_a_b] :
      ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b,K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
          ( X
         != ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
            ( X
           != ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ nil_Pr1090988463983433548um_a_b @ Bs ) ) )
       => ~ ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b] :
              ( X
             != ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ As2 @ nil_Pr1090988463983433548um_a_b ) ) ) ) ) ).

% sunion_with.cases
thf(fact_737_sunion__with_Ocases,axiom,
    ! [X: produc1341479252616905341um_a_b] :
      ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b,K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
          ( X
         != ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
            ( X
           != ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ nil_Pr3333374394092609245um_a_b @ Bs ) ) )
       => ~ ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,As2: list_P6497659457863146077um_a_b] :
              ( X
             != ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ As2 @ nil_Pr3333374394092609245um_a_b ) ) ) ) ) ).

% sunion_with.cases
thf(fact_738_sunion__with_Ocases,axiom,
    ! [X: produc4961808859980711423at_nat] :
      ( ! [F3: nat > nat > nat > nat,K2: nat,V2: nat,As2: list_P6011104703257516679at_nat,K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
          ( X
         != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
     => ( ! [F3: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
            ( X
           != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
       => ~ ! [F3: nat > nat > nat > nat,As2: list_P6011104703257516679at_nat] :
              ( X
             != ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ As2 @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% sunion_with.cases
thf(fact_739_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc6661094113385059586um_a_b] :
      ( ! [K2: sum_sum_a_b] :
          ( X
         != ( produc6895264061556658612um_a_b @ K2 @ nil_Pr8520116601857385651um_a_b ) )
     => ~ ! [K2: sum_sum_a_b,K3: sum_sum_a_b,V2: sum_sum_a_b,Xs3: list_P911570089447184179um_a_b] :
            ( X
           != ( produc6895264061556658612um_a_b @ K2 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_740_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc8659213103137179411um_a_b] :
      ( ! [K2: sum_sum_a_b] :
          ( X
         != ( produc7465751454420652107um_a_b @ K2 @ nil_Pr7649374833807421942um_a_b ) )
     => ~ ! [K2: sum_sum_a_b,K3: sum_sum_a_b,V2: produc4970237071031658413um_a_b,Xs3: list_P6716435105863944716um_a_b] :
            ( X
           != ( produc7465751454420652107um_a_b @ K2 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_741_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc8787139405198277676_b_nat] :
      ( ! [K2: sum_sum_a_b] :
          ( X
         != ( produc4871069856000220126_b_nat @ K2 @ nil_Pr3750691640972353501_b_nat ) )
     => ~ ! [K2: sum_sum_a_b,K3: sum_sum_a_b,V2: nat,Xs3: list_P667245297671117661_b_nat] :
            ( X
           != ( produc4871069856000220126_b_nat @ K2 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V2 ) @ Xs3 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_742_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc7925846991739328147um_a_b] :
      ( ! [K2: nat] :
          ( X
         != ( produc7902279242088003275um_a_b @ K2 @ nil_Pr1090988463983433548um_a_b ) )
     => ~ ! [K2: nat,K3: nat,V2: produc4970237071031658413um_a_b,Xs3: list_P5598968436384720482um_a_b] :
            ( X
           != ( produc7902279242088003275um_a_b @ K2 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_743_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc8393995592613569922um_a_b] :
      ( ! [K2: nat] :
          ( X
         != ( produc1183721963040099124um_a_b @ K2 @ nil_Pr3333374394092609245um_a_b ) )
     => ~ ! [K2: nat,K3: nat,V2: sum_sum_a_b,Xs3: list_P6497659457863146077um_a_b] :
            ( X
           != ( produc1183721963040099124um_a_b @ K2 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_744_AssocList_Odelete__aux_Ocases,axiom,
    ! [X: produc8472197452120411308at_nat] :
      ( ! [K2: nat] :
          ( X
         != ( produc6109913384486294878at_nat @ K2 @ nil_Pr5478986624290739719at_nat ) )
     => ~ ! [K2: nat,K3: nat,V2: nat,Xs3: list_P6011104703257516679at_nat] :
            ( X
           != ( produc6109913384486294878at_nat @ K2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V2 ) @ Xs3 ) ) ) ) ).

% AssocList.delete_aux.cases
thf(fact_745_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_746_ord_Osunion__with_Opinduct,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,A0: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,A1: list_P911570089447184179um_a_b,A22: list_P911570089447184179um_a_b,P2: ( sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b ) > list_P911570089447184179um_a_b > list_P911570089447184179um_a_b > $o] :
      ( ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ A0 @ ( produc2669586549142030935um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K2: sum_sum_a_b,V2: sum_sum_a_b,As2: list_P911570089447184179um_a_b,K3: sum_sum_a_b,V3: sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
            ( ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
              ( ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ nil_Pr8520116601857385651um_a_b @ Bs ) ) )
             => ( P2 @ F3 @ nil_Pr8520116601857385651um_a_b @ Bs ) )
         => ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,As2: list_P911570089447184179um_a_b] :
                ( ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ As2 @ nil_Pr8520116601857385651um_a_b ) ) )
               => ( P2 @ F3 @ As2 @ nil_Pr8520116601857385651um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_747_ord_Osunion__with_Opinduct,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,A0: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,A1: list_P6716435105863944716um_a_b,A22: list_P6716435105863944716um_a_b,P2: ( sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b ) > list_P6716435105863944716um_a_b > list_P6716435105863944716um_a_b > $o] :
      ( ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ A0 @ ( produc4082290306366126769um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K2: sum_sum_a_b,V2: produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b,K3: sum_sum_a_b,V3: produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
            ( ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
              ( ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ nil_Pr7649374833807421942um_a_b @ Bs ) ) )
             => ( P2 @ F3 @ nil_Pr7649374833807421942um_a_b @ Bs ) )
         => ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b] :
                ( ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ As2 @ nil_Pr7649374833807421942um_a_b ) ) )
               => ( P2 @ F3 @ As2 @ nil_Pr7649374833807421942um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_748_ord_Osunion__with_Opinduct,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,A0: sum_sum_a_b > nat > nat > nat,A1: list_P667245297671117661_b_nat,A22: list_P667245297671117661_b_nat,P2: ( sum_sum_a_b > nat > nat > nat ) > list_P667245297671117661_b_nat > list_P667245297671117661_b_nat > $o] :
      ( ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ A0 @ ( produc8028711988652938967_b_nat @ A1 @ A22 ) ) )
     => ( ! [F3: sum_sum_a_b > nat > nat > nat,K2: sum_sum_a_b,V2: nat,As2: list_P667245297671117661_b_nat,K3: sum_sum_a_b,V3: nat,Bs: list_P667245297671117661_b_nat] :
            ( ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: sum_sum_a_b > nat > nat > nat,Bs: list_P667245297671117661_b_nat] :
              ( ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ nil_Pr3750691640972353501_b_nat @ Bs ) ) )
             => ( P2 @ F3 @ nil_Pr3750691640972353501_b_nat @ Bs ) )
         => ( ! [F3: sum_sum_a_b > nat > nat > nat,As2: list_P667245297671117661_b_nat] :
                ( ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ As2 @ nil_Pr3750691640972353501_b_nat ) ) )
               => ( P2 @ F3 @ As2 @ nil_Pr3750691640972353501_b_nat ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_749_ord_Osunion__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,A1: list_P5598968436384720482um_a_b,A22: list_P5598968436384720482um_a_b,P2: ( nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b ) > list_P5598968436384720482um_a_b > list_P5598968436384720482um_a_b > $o] :
      ( ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ A0 @ ( produc8507494088392525149um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b,K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
            ( ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
              ( ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ nil_Pr1090988463983433548um_a_b @ Bs ) ) )
             => ( P2 @ F3 @ nil_Pr1090988463983433548um_a_b @ Bs ) )
         => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b] :
                ( ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ As2 @ nil_Pr1090988463983433548um_a_b ) ) )
               => ( P2 @ F3 @ As2 @ nil_Pr1090988463983433548um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_750_ord_Osunion__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,A1: list_P6497659457863146077um_a_b,A22: list_P6497659457863146077um_a_b,P2: ( nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b ) > list_P6497659457863146077um_a_b > list_P6497659457863146077um_a_b > $o] :
      ( ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ A0 @ ( produc8821755967955457751um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b,K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
            ( ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
              ( ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ nil_Pr3333374394092609245um_a_b @ Bs ) ) )
             => ( P2 @ F3 @ nil_Pr3333374394092609245um_a_b @ Bs ) )
         => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,As2: list_P6497659457863146077um_a_b] :
                ( ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ As2 @ nil_Pr3333374394092609245um_a_b ) ) )
               => ( P2 @ F3 @ As2 @ nil_Pr3333374394092609245um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_751_ord_Osunion__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P2: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > nat > nat > nat,K2: nat,V2: nat,As2: list_P6011104703257516679at_nat,K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > nat > nat > nat,Bs: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Bs ) ) )
             => ( P2 @ F3 @ nil_Pr5478986624290739719at_nat @ Bs ) )
         => ( ! [F3: nat > nat > nat > nat,As2: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ As2 @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P2 @ F3 @ As2 @ nil_Pr5478986624290739719at_nat ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sunion_with.pinduct
thf(fact_752_ord_Osinter__with_Opinduct,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,A0: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,A1: list_P911570089447184179um_a_b,A22: list_P911570089447184179um_a_b,P2: ( sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b ) > list_P911570089447184179um_a_b > list_P911570089447184179um_a_b > $o] :
      ( ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ A0 @ ( produc2669586549142030935um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K2: sum_sum_a_b,V2: sum_sum_a_b,As2: list_P911570089447184179um_a_b,K3: sum_sum_a_b,V3: sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
            ( ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Uu: list_P911570089447184179um_a_b] :
              ( ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ nil_Pr8520116601857385651um_a_b @ Uu ) ) )
             => ( P2 @ F3 @ nil_Pr8520116601857385651um_a_b @ Uu ) )
         => ( ! [F3: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Uv: list_P911570089447184179um_a_b] :
                ( ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F3 @ ( produc2669586549142030935um_a_b @ Uv @ nil_Pr8520116601857385651um_a_b ) ) )
               => ( P2 @ F3 @ Uv @ nil_Pr8520116601857385651um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_753_ord_Osinter__with_Opinduct,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,A0: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,A1: list_P6716435105863944716um_a_b,A22: list_P6716435105863944716um_a_b,P2: ( sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b ) > list_P6716435105863944716um_a_b > list_P6716435105863944716um_a_b > $o] :
      ( ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ A0 @ ( produc4082290306366126769um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K2: sum_sum_a_b,V2: produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b,K3: sum_sum_a_b,V3: produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
            ( ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Uu: list_P6716435105863944716um_a_b] :
              ( ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ nil_Pr7649374833807421942um_a_b @ Uu ) ) )
             => ( P2 @ F3 @ nil_Pr7649374833807421942um_a_b @ Uu ) )
         => ( ! [F3: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Uv: list_P6716435105863944716um_a_b] :
                ( ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F3 @ ( produc4082290306366126769um_a_b @ Uv @ nil_Pr7649374833807421942um_a_b ) ) )
               => ( P2 @ F3 @ Uv @ nil_Pr7649374833807421942um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_754_ord_Osinter__with_Opinduct,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,A0: sum_sum_a_b > nat > nat > nat,A1: list_P667245297671117661_b_nat,A22: list_P667245297671117661_b_nat,P2: ( sum_sum_a_b > nat > nat > nat ) > list_P667245297671117661_b_nat > list_P667245297671117661_b_nat > $o] :
      ( ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ A0 @ ( produc8028711988652938967_b_nat @ A1 @ A22 ) ) )
     => ( ! [F3: sum_sum_a_b > nat > nat > nat,K2: sum_sum_a_b,V2: nat,As2: list_P667245297671117661_b_nat,K3: sum_sum_a_b,V3: nat,Bs: list_P667245297671117661_b_nat] :
            ( ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: sum_sum_a_b > nat > nat > nat,Uu: list_P667245297671117661_b_nat] :
              ( ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ nil_Pr3750691640972353501_b_nat @ Uu ) ) )
             => ( P2 @ F3 @ nil_Pr3750691640972353501_b_nat @ Uu ) )
         => ( ! [F3: sum_sum_a_b > nat > nat > nat,Uv: list_P667245297671117661_b_nat] :
                ( ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F3 @ ( produc8028711988652938967_b_nat @ Uv @ nil_Pr3750691640972353501_b_nat ) ) )
               => ( P2 @ F3 @ Uv @ nil_Pr3750691640972353501_b_nat ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_755_ord_Osinter__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,A1: list_P5598968436384720482um_a_b,A22: list_P5598968436384720482um_a_b,P2: ( nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b ) > list_P5598968436384720482um_a_b > list_P5598968436384720482um_a_b > $o] :
      ( ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ A0 @ ( produc8507494088392525149um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b,K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
            ( ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Uu: list_P5598968436384720482um_a_b] :
              ( ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ nil_Pr1090988463983433548um_a_b @ Uu ) ) )
             => ( P2 @ F3 @ nil_Pr1090988463983433548um_a_b @ Uu ) )
         => ( ! [F3: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Uv: list_P5598968436384720482um_a_b] :
                ( ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F3 @ ( produc8507494088392525149um_a_b @ Uv @ nil_Pr1090988463983433548um_a_b ) ) )
               => ( P2 @ F3 @ Uv @ nil_Pr1090988463983433548um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_756_ord_Osinter__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,A1: list_P6497659457863146077um_a_b,A22: list_P6497659457863146077um_a_b,P2: ( nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b ) > list_P6497659457863146077um_a_b > list_P6497659457863146077um_a_b > $o] :
      ( ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ A0 @ ( produc8821755967955457751um_a_b @ A1 @ A22 ) ) )
     => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b,K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
            ( ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Uu: list_P6497659457863146077um_a_b] :
              ( ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ nil_Pr3333374394092609245um_a_b @ Uu ) ) )
             => ( P2 @ F3 @ nil_Pr3333374394092609245um_a_b @ Uu ) )
         => ( ! [F3: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Uv: list_P6497659457863146077um_a_b] :
                ( ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F3 @ ( produc8821755967955457751um_a_b @ Uv @ nil_Pr3333374394092609245um_a_b ) ) )
               => ( P2 @ F3 @ Uv @ nil_Pr3333374394092609245um_a_b ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_757_ord_Osinter__with_Opinduct,axiom,
    ! [Less: nat > nat > $o,A0: nat > nat > nat > nat,A1: list_P6011104703257516679at_nat,A22: list_P6011104703257516679at_nat,P2: ( nat > nat > nat > nat ) > list_P6011104703257516679at_nat > list_P6011104703257516679at_nat > $o] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ A0 @ ( produc5943733680697469783at_nat @ A1 @ A22 ) ) )
     => ( ! [F3: nat > nat > nat > nat,K2: nat,V2: nat,As2: list_P6011104703257516679at_nat,K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
            ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
           => ( ( ( Less @ K3 @ K2 )
               => ( P2 @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ Bs ) )
             => ( ( ~ ( Less @ K3 @ K2 )
                 => ( ( Less @ K2 @ K3 )
                   => ( P2 @ F3 @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
               => ( ( ~ ( Less @ K3 @ K2 )
                   => ( ~ ( Less @ K2 @ K3 )
                     => ( P2 @ F3 @ As2 @ Bs ) ) )
                 => ( P2 @ F3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) )
       => ( ! [F3: nat > nat > nat > nat,Uu: list_P6011104703257516679at_nat] :
              ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Uu ) ) )
             => ( P2 @ F3 @ nil_Pr5478986624290739719at_nat @ Uu ) )
         => ( ! [F3: nat > nat > nat > nat,Uv: list_P6011104703257516679at_nat] :
                ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F3 @ ( produc5943733680697469783at_nat @ Uv @ nil_Pr5478986624290739719at_nat ) ) )
               => ( P2 @ F3 @ Uv @ nil_Pr5478986624290739719at_nat ) )
           => ( P2 @ A0 @ A1 @ A22 ) ) ) ) ) ).

% ord.sinter_with.pinduct
thf(fact_758_AssocList_Oupdate__with__aux_Osimps_I1_J,axiom,
    ! [V: sum_sum_a_b,K: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b] :
      ( ( update7109155850483007580um_a_b @ V @ K @ F @ nil_Pr8520116601857385651um_a_b )
      = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ ( F @ V ) ) @ nil_Pr8520116601857385651um_a_b ) ) ).

% AssocList.update_with_aux.simps(1)
thf(fact_759_AssocList_Oupdate__with__aux_Osimps_I1_J,axiom,
    ! [V: produc4970237071031658413um_a_b,K: sum_sum_a_b,F: produc4970237071031658413um_a_b > produc4970237071031658413um_a_b] :
      ( ( update2011012084792080581um_a_b @ V @ K @ F @ nil_Pr7649374833807421942um_a_b )
      = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ ( F @ V ) ) @ nil_Pr7649374833807421942um_a_b ) ) ).

% AssocList.update_with_aux.simps(1)
thf(fact_760_AssocList_Oupdate__with__aux_Osimps_I1_J,axiom,
    ! [V: nat,K: sum_sum_a_b,F: nat > nat] :
      ( ( update3120058914886408838um_a_b @ V @ K @ F @ nil_Pr3750691640972353501_b_nat )
      = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ ( F @ V ) ) @ nil_Pr3750691640972353501_b_nat ) ) ).

% AssocList.update_with_aux.simps(1)
thf(fact_761_AssocList_Oupdate__with__aux_Osimps_I1_J,axiom,
    ! [V: produc4970237071031658413um_a_b,K: nat,F: produc4970237071031658413um_a_b > produc4970237071031658413um_a_b] :
      ( ( update1964132014378800879_b_nat @ V @ K @ F @ nil_Pr1090988463983433548um_a_b )
      = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ ( F @ V ) ) @ nil_Pr1090988463983433548um_a_b ) ) ).

% AssocList.update_with_aux.simps(1)
thf(fact_762_AssocList_Oupdate__with__aux_Osimps_I1_J,axiom,
    ! [V: sum_sum_a_b,K: nat,F: sum_sum_a_b > sum_sum_a_b] :
      ( ( update8114415933154093190_b_nat @ V @ K @ F @ nil_Pr3333374394092609245um_a_b )
      = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ ( F @ V ) ) @ nil_Pr3333374394092609245um_a_b ) ) ).

% AssocList.update_with_aux.simps(1)
thf(fact_763_AssocList_Oupdate__with__aux_Osimps_I1_J,axiom,
    ! [V: nat,K: nat,F: nat > nat] :
      ( ( update8447963936283049648at_nat @ V @ K @ F @ nil_Pr5478986624290739719at_nat )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ V ) ) @ nil_Pr5478986624290739719at_nat ) ) ).

% AssocList.update_with_aux.simps(1)
thf(fact_764_listrel_Ocases,axiom,
    ! [A1: list_Sum_sum_a_b,A22: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b] :
      ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ A1 @ A22 ) @ ( listre1816753519460753727um_a_b @ R3 ) )
     => ( ( ( A1 = nil_Sum_sum_a_b )
         => ( A22 != nil_Sum_sum_a_b ) )
       => ~ ! [X5: sum_sum_a_b,Y3: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
              ( ( A1
                = ( cons_Sum_sum_a_b @ X5 @ Xs3 ) )
             => ! [Ys3: list_Sum_sum_a_b] :
                  ( ( A22
                    = ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
                 => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ Y3 ) @ R3 )
                   => ~ ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs3 @ Ys3 ) @ ( listre1816753519460753727um_a_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_765_listrel_Ocases,axiom,
    ! [A1: list_Sum_sum_a_b,A22: list_P911570089447184179um_a_b,R3: set_Pr4696577323406728626um_a_b] :
      ( ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ A1 @ A22 ) @ ( listre2533333633648478420um_a_b @ R3 ) )
     => ( ( ( A1 = nil_Sum_sum_a_b )
         => ( A22 != nil_Pr8520116601857385651um_a_b ) )
       => ~ ! [X5: sum_sum_a_b,Y3: produc4970237071031658413um_a_b,Xs3: list_Sum_sum_a_b] :
              ( ( A1
                = ( cons_Sum_sum_a_b @ X5 @ Xs3 ) )
             => ! [Ys3: list_P911570089447184179um_a_b] :
                  ( ( A22
                    = ( cons_P4211151491411527779um_a_b @ Y3 @ Ys3 ) )
                 => ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X5 @ Y3 ) @ R3 )
                   => ~ ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ Xs3 @ Ys3 ) @ ( listre2533333633648478420um_a_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_766_listrel_Ocases,axiom,
    ! [A1: list_Sum_sum_a_b,A22: list_nat,R3: set_Pr7634312471471294263_b_nat] :
      ( ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ A1 @ A22 ) @ ( listre3679324330735508329_b_nat @ R3 ) )
     => ( ( ( A1 = nil_Sum_sum_a_b )
         => ( A22 != nil_nat ) )
       => ~ ! [X5: sum_sum_a_b,Y3: nat,Xs3: list_Sum_sum_a_b] :
              ( ( A1
                = ( cons_Sum_sum_a_b @ X5 @ Xs3 ) )
             => ! [Ys3: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys3 ) )
                 => ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X5 @ Y3 ) @ R3 )
                   => ~ ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ Xs3 @ Ys3 ) @ ( listre3679324330735508329_b_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_767_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_P911570089447184179um_a_b,R3: set_Pr5956926064383532552um_a_b] :
      ( ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ A1 @ A22 ) @ ( listre3466312577575861290um_a_b @ R3 ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_Pr8520116601857385651um_a_b ) )
       => ~ ! [X5: nat,Y3: produc4970237071031658413um_a_b,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X5 @ Xs3 ) )
             => ! [Ys3: list_P911570089447184179um_a_b] :
                  ( ( A22
                    = ( cons_P4211151491411527779um_a_b @ Y3 @ Ys3 ) )
                 => ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X5 @ Y3 ) @ R3 )
                   => ~ ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ Xs3 @ Ys3 ) @ ( listre3466312577575861290um_a_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_768_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_Sum_sum_a_b,R3: set_Pr4241354594808546871um_a_b] :
      ( ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ A1 @ A22 ) @ ( listre7908339349322599785um_a_b @ R3 ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_Sum_sum_a_b ) )
       => ~ ! [X5: nat,Y3: sum_sum_a_b,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X5 @ Xs3 ) )
             => ! [Ys3: list_Sum_sum_a_b] :
                  ( ( A22
                    = ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
                 => ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X5 @ Y3 ) @ R3 )
                   => ~ ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ Xs3 @ Ys3 ) @ ( listre7908339349322599785um_a_b @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_769_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R3 ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X5: nat,Y3: nat,Xs3: list_nat] :
              ( ( A1
                = ( cons_nat @ X5 @ Xs3 ) )
             => ! [Ys3: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y3 @ Ys3 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R3 )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_770_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R3 ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_771_listrel_OCons,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b,Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ R3 )
     => ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs @ Ys ) @ ( listre1816753519460753727um_a_b @ R3 ) )
       => ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ ( cons_Sum_sum_a_b @ X @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) @ ( listre1816753519460753727um_a_b @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_772_listrel_OCons,axiom,
    ! [X: sum_sum_a_b,Y: produc4970237071031658413um_a_b,R3: set_Pr4696577323406728626um_a_b,Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X @ Y ) @ R3 )
     => ( ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ Xs @ Ys ) @ ( listre2533333633648478420um_a_b @ R3 ) )
       => ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ ( cons_Sum_sum_a_b @ X @ Xs ) @ ( cons_P4211151491411527779um_a_b @ Y @ Ys ) ) @ ( listre2533333633648478420um_a_b @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_773_listrel_OCons,axiom,
    ! [X: sum_sum_a_b,Y: nat,R3: set_Pr7634312471471294263_b_nat,Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X @ Y ) @ R3 )
     => ( ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ Xs @ Ys ) @ ( listre3679324330735508329_b_nat @ R3 ) )
       => ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ ( cons_Sum_sum_a_b @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listre3679324330735508329_b_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_774_listrel_OCons,axiom,
    ! [X: nat,Y: produc4970237071031658413um_a_b,R3: set_Pr5956926064383532552um_a_b,Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X @ Y ) @ R3 )
     => ( ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ Xs @ Ys ) @ ( listre3466312577575861290um_a_b @ R3 ) )
       => ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ ( cons_nat @ X @ Xs ) @ ( cons_P4211151491411527779um_a_b @ Y @ Ys ) ) @ ( listre3466312577575861290um_a_b @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_775_listrel_OCons,axiom,
    ! [X: nat,Y: sum_sum_a_b,R3: set_Pr4241354594808546871um_a_b,Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X @ Y ) @ R3 )
     => ( ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ Xs @ Ys ) @ ( listre7908339349322599785um_a_b @ R3 ) )
       => ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ ( cons_nat @ X @ Xs ) @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) @ ( listre7908339349322599785um_a_b @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_776_listrel_OCons,axiom,
    ! [X: nat,Y: nat,R3: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R3 )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R3 ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) ) ) ) ).

% listrel.Cons
thf(fact_777_listrel__Cons1,axiom,
    ! [Y: sum_sum_a_b,Ys: list_Sum_sum_a_b,Xs: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b] :
      ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ ( cons_Sum_sum_a_b @ Y @ Ys ) @ Xs ) @ ( listre1816753519460753727um_a_b @ R3 ) )
     => ~ ! [Y3: sum_sum_a_b,Ys3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
           => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ Y @ Y3 ) @ R3 )
             => ~ ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Ys @ Ys3 ) @ ( listre1816753519460753727um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_778_listrel__Cons1,axiom,
    ! [Y: sum_sum_a_b,Ys: list_Sum_sum_a_b,Xs: list_P911570089447184179um_a_b,R3: set_Pr4696577323406728626um_a_b] :
      ( ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ ( cons_Sum_sum_a_b @ Y @ Ys ) @ Xs ) @ ( listre2533333633648478420um_a_b @ R3 ) )
     => ~ ! [Y3: produc4970237071031658413um_a_b,Ys3: list_P911570089447184179um_a_b] :
            ( ( Xs
              = ( cons_P4211151491411527779um_a_b @ Y3 @ Ys3 ) )
           => ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ Y @ Y3 ) @ R3 )
             => ~ ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ Ys @ Ys3 ) @ ( listre2533333633648478420um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_779_listrel__Cons1,axiom,
    ! [Y: sum_sum_a_b,Ys: list_Sum_sum_a_b,Xs: list_nat,R3: set_Pr7634312471471294263_b_nat] :
      ( ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ ( cons_Sum_sum_a_b @ Y @ Ys ) @ Xs ) @ ( listre3679324330735508329_b_nat @ R3 ) )
     => ~ ! [Y3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys3 ) )
           => ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ Y @ Y3 ) @ R3 )
             => ~ ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ Ys @ Ys3 ) @ ( listre3679324330735508329_b_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_780_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_P911570089447184179um_a_b,R3: set_Pr5956926064383532552um_a_b] :
      ( ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listre3466312577575861290um_a_b @ R3 ) )
     => ~ ! [Y3: produc4970237071031658413um_a_b,Ys3: list_P911570089447184179um_a_b] :
            ( ( Xs
              = ( cons_P4211151491411527779um_a_b @ Y3 @ Ys3 ) )
           => ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ Y @ Y3 ) @ R3 )
             => ~ ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ Ys @ Ys3 ) @ ( listre3466312577575861290um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_781_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_Sum_sum_a_b,R3: set_Pr4241354594808546871um_a_b] :
      ( ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listre7908339349322599785um_a_b @ R3 ) )
     => ~ ! [Y3: sum_sum_a_b,Ys3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ Y3 @ Ys3 ) )
           => ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ Y @ Y3 ) @ R3 )
             => ~ ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ Ys @ Ys3 ) @ ( listre7908339349322599785um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_782_listrel__Cons1,axiom,
    ! [Y: nat,Ys: list_nat,Xs: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [Y3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y @ Y3 ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys3 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_783_listrel__Cons2,axiom,
    ! [Xs: list_Sum_sum_a_b,Y: sum_sum_a_b,Ys: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b] :
      ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) @ ( listre1816753519460753727um_a_b @ R3 ) )
     => ~ ! [X5: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ X5 @ Xs3 ) )
           => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ Y ) @ R3 )
             => ~ ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs3 @ Ys ) @ ( listre1816753519460753727um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_784_listrel__Cons2,axiom,
    ! [Xs: list_Sum_sum_a_b,Y: produc4970237071031658413um_a_b,Ys: list_P911570089447184179um_a_b,R3: set_Pr4696577323406728626um_a_b] :
      ( ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ Xs @ ( cons_P4211151491411527779um_a_b @ Y @ Ys ) ) @ ( listre2533333633648478420um_a_b @ R3 ) )
     => ~ ! [X5: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ X5 @ Xs3 ) )
           => ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X5 @ Y ) @ R3 )
             => ~ ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ Xs3 @ Ys ) @ ( listre2533333633648478420um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_785_listrel__Cons2,axiom,
    ! [Xs: list_Sum_sum_a_b,Y: nat,Ys: list_nat,R3: set_Pr7634312471471294263_b_nat] :
      ( ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listre3679324330735508329_b_nat @ R3 ) )
     => ~ ! [X5: sum_sum_a_b,Xs3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( cons_Sum_sum_a_b @ X5 @ Xs3 ) )
           => ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X5 @ Y ) @ R3 )
             => ~ ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ Xs3 @ Ys ) @ ( listre3679324330735508329_b_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_786_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: produc4970237071031658413um_a_b,Ys: list_P911570089447184179um_a_b,R3: set_Pr5956926064383532552um_a_b] :
      ( ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ Xs @ ( cons_P4211151491411527779um_a_b @ Y @ Ys ) ) @ ( listre3466312577575861290um_a_b @ R3 ) )
     => ~ ! [X5: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X5 @ Xs3 ) )
           => ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X5 @ Y ) @ R3 )
             => ~ ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ Xs3 @ Ys ) @ ( listre3466312577575861290um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_787_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: sum_sum_a_b,Ys: list_Sum_sum_a_b,R3: set_Pr4241354594808546871um_a_b] :
      ( ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ Xs @ ( cons_Sum_sum_a_b @ Y @ Ys ) ) @ ( listre7908339349322599785um_a_b @ R3 ) )
     => ~ ! [X5: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X5 @ Xs3 ) )
           => ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X5 @ Y ) @ R3 )
             => ~ ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ Xs3 @ Ys ) @ ( listre7908339349322599785um_a_b @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_788_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y: nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat @ R3 ) )
     => ~ ! [X5: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X5 @ Xs3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y ) @ R3 )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_789_listrel_Osimps,axiom,
    ! [A1: list_Sum_sum_a_b,A22: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b] :
      ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ A1 @ A22 ) @ ( listre1816753519460753727um_a_b @ R3 ) )
      = ( ( ( A1 = nil_Sum_sum_a_b )
          & ( A22 = nil_Sum_sum_a_b ) )
        | ? [X4: sum_sum_a_b,Y5: sum_sum_a_b,Xs2: list_Sum_sum_a_b,Ys2: list_Sum_sum_a_b] :
            ( ( A1
              = ( cons_Sum_sum_a_b @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) )
            & ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X4 @ Y5 ) @ R3 )
            & ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs2 @ Ys2 ) @ ( listre1816753519460753727um_a_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_790_listrel_Osimps,axiom,
    ! [A1: list_Sum_sum_a_b,A22: list_P911570089447184179um_a_b,R3: set_Pr4696577323406728626um_a_b] :
      ( ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ A1 @ A22 ) @ ( listre2533333633648478420um_a_b @ R3 ) )
      = ( ( ( A1 = nil_Sum_sum_a_b )
          & ( A22 = nil_Pr8520116601857385651um_a_b ) )
        | ? [X4: sum_sum_a_b,Y5: produc4970237071031658413um_a_b,Xs2: list_Sum_sum_a_b,Ys2: list_P911570089447184179um_a_b] :
            ( ( A1
              = ( cons_Sum_sum_a_b @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_P4211151491411527779um_a_b @ Y5 @ Ys2 ) )
            & ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X4 @ Y5 ) @ R3 )
            & ( member8081343546124835977um_a_b @ ( produc1837043907632934820um_a_b @ Xs2 @ Ys2 ) @ ( listre2533333633648478420um_a_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_791_listrel_Osimps,axiom,
    ! [A1: list_Sum_sum_a_b,A22: list_nat,R3: set_Pr7634312471471294263_b_nat] :
      ( ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ A1 @ A22 ) @ ( listre3679324330735508329_b_nat @ R3 ) )
      = ( ( ( A1 = nil_Sum_sum_a_b )
          & ( A22 = nil_nat ) )
        | ? [X4: sum_sum_a_b,Y5: nat,Xs2: list_Sum_sum_a_b,Ys2: list_nat] :
            ( ( A1
              = ( cons_Sum_sum_a_b @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y5 @ Ys2 ) )
            & ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X4 @ Y5 ) @ R3 )
            & ( member3308341281527368992st_nat @ ( produc8652313775564450735st_nat @ Xs2 @ Ys2 ) @ ( listre3679324330735508329_b_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_792_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_P911570089447184179um_a_b,R3: set_Pr5956926064383532552um_a_b] :
      ( ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ A1 @ A22 ) @ ( listre3466312577575861290um_a_b @ R3 ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_Pr8520116601857385651um_a_b ) )
        | ? [X4: nat,Y5: produc4970237071031658413um_a_b,Xs2: list_nat,Ys2: list_P911570089447184179um_a_b] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_P4211151491411527779um_a_b @ Y5 @ Ys2 ) )
            & ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X4 @ Y5 ) @ R3 )
            & ( member6609540720157959135um_a_b @ ( produc9007428411043754746um_a_b @ Xs2 @ Ys2 ) @ ( listre3466312577575861290um_a_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_793_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_Sum_sum_a_b,R3: set_Pr4241354594808546871um_a_b] :
      ( ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ A1 @ A22 ) @ ( listre7908339349322599785um_a_b @ R3 ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_Sum_sum_a_b ) )
        | ? [X4: nat,Y5: sum_sum_a_b,Xs2: list_nat,Ys2: list_Sum_sum_a_b] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_Sum_sum_a_b @ Y5 @ Ys2 ) )
            & ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X4 @ Y5 ) @ R3 )
            & ( member7969790780479983392um_a_b @ ( produc5768412803108495535um_a_b @ Xs2 @ Ys2 ) @ ( listre7908339349322599785um_a_b @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_794_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R3 ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X4: nat,Y5: nat,Xs2: list_nat,Ys2: list_nat] :
            ( ( A1
              = ( cons_nat @ X4 @ Xs2 ) )
            & ( A22
              = ( cons_nat @ Y5 @ Ys2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R3 )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R3 ) ) ) ) ) ).

% listrel.simps
thf(fact_795_ord_Osunion__with_Opelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P911570089447184179um_a_b,Xb: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( ( rBT_su4305455488528285193um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: sum_sum_a_b,V2: sum_sum_a_b,As2: list_P911570089447184179um_a_b] :
              ( ( Xa
                = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: sum_sum_a_b,V3: sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
                  ( ( Xb
                    = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ ( rBT_su4305455488528285193um_a_b @ Less @ X @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ ( rBT_su4305455488528285193um_a_b @ Less @ X @ As2 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su4305455488528285193um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr8520116601857385651um_a_b )
             => ( ( Y = Xb )
               => ~ ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ nil_Pr8520116601857385651um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr8520116601857385651um_a_b )
               => ( ( Y = Xa )
                 => ~ ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ Xa @ nil_Pr8520116601857385651um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_796_ord_Osunion__with_Opelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P6716435105863944716um_a_b,Xb: list_P6716435105863944716um_a_b,Y: list_P6716435105863944716um_a_b] :
      ( ( ( rBT_su544253560460886922um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: sum_sum_a_b,V2: produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b] :
              ( ( Xa
                = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: sum_sum_a_b,V3: produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
                  ( ( Xb
                    = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ ( rBT_su544253560460886922um_a_b @ Less @ X @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ ( rBT_su544253560460886922um_a_b @ Less @ X @ As2 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su544253560460886922um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr7649374833807421942um_a_b )
             => ( ( Y = Xb )
               => ~ ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ nil_Pr7649374833807421942um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr7649374833807421942um_a_b )
               => ( ( Y = Xa )
                 => ~ ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ Xa @ nil_Pr7649374833807421942um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_797_ord_Osunion__with_Opelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > nat > nat > nat,Xa: list_P667245297671117661_b_nat,Xb: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( ( rBT_su1686119633526928435_b_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ Xa @ Xb ) ) )
       => ( ! [K2: sum_sum_a_b,V2: nat,As2: list_P667245297671117661_b_nat] :
              ( ( Xa
                = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) )
             => ! [K3: sum_sum_a_b,V3: nat,Bs: list_P667245297671117661_b_nat] :
                  ( ( Xb
                    = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ ( rBT_su1686119633526928435_b_nat @ Less @ X @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ ( rBT_su1686119633526928435_b_nat @ Less @ X @ As2 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su1686119633526928435_b_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr3750691640972353501_b_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ nil_Pr3750691640972353501_b_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr3750691640972353501_b_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ Xa @ nil_Pr3750691640972353501_b_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_798_ord_Osunion__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P5598968436384720482um_a_b,Xb: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( ( rBT_su7166556703742098144um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b] :
              ( ( Xa
                = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
                  ( ( Xb
                    = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ ( rBT_su7166556703742098144um_a_b @ Less @ X @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ ( rBT_su7166556703742098144um_a_b @ Less @ X @ As2 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su7166556703742098144um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr1090988463983433548um_a_b )
             => ( ( Y = Xb )
               => ~ ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ nil_Pr1090988463983433548um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr1090988463983433548um_a_b )
               => ( ( Y = Xa )
                 => ~ ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ Xa @ nil_Pr1090988463983433548um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_799_ord_Osunion__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P6497659457863146077um_a_b,Xb: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( ( rBT_su5915134652114019891um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b] :
              ( ( Xa
                = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
                  ( ( Xb
                    = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ ( rBT_su5915134652114019891um_a_b @ Less @ X @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ ( rBT_su5915134652114019891um_a_b @ Less @ X @ As2 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su5915134652114019891um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr3333374394092609245um_a_b )
             => ( ( Y = Xb )
               => ~ ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ nil_Pr3333374394092609245um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr3333374394092609245um_a_b )
               => ( ( Y = Xa )
                 => ~ ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ Xa @ nil_Pr3333374394092609245um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_800_ord_Osunion__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_su5597997906817964125at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V2: nat,As2: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) )
             => ! [K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
             => ( ( Y = Xb )
               => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = Xa )
                 => ~ ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sunion_with.pelims
thf(fact_801_ord_Osinter__with_Opelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P911570089447184179um_a_b,Xb: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( ( rBT_si7236827711767104150um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: sum_sum_a_b,V2: sum_sum_a_b,As2: list_P911570089447184179um_a_b] :
              ( ( Xa
                = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: sum_sum_a_b,V3: sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
                  ( ( Xb
                    = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( rBT_si7236827711767104150um_a_b @ Less @ X @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( rBT_si7236827711767104150um_a_b @ Less @ X @ As2 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si7236827711767104150um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr8520116601857385651um_a_b )
             => ( ( Y = nil_Pr8520116601857385651um_a_b )
               => ~ ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ nil_Pr8520116601857385651um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr8520116601857385651um_a_b )
               => ( ( Y = nil_Pr8520116601857385651um_a_b )
                 => ~ ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ X @ ( produc2669586549142030935um_a_b @ Xa @ nil_Pr8520116601857385651um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_802_ord_Osinter__with_Opelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P6716435105863944716um_a_b,Xb: list_P6716435105863944716um_a_b,Y: list_P6716435105863944716um_a_b] :
      ( ( ( rBT_si936693504573915325um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: sum_sum_a_b,V2: produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b] :
              ( ( Xa
                = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: sum_sum_a_b,V3: produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
                  ( ( Xb
                    = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( rBT_si936693504573915325um_a_b @ Less @ X @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( rBT_si936693504573915325um_a_b @ Less @ X @ As2 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si936693504573915325um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr7649374833807421942um_a_b )
             => ( ( Y = nil_Pr7649374833807421942um_a_b )
               => ~ ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ nil_Pr7649374833807421942um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr7649374833807421942um_a_b )
               => ( ( Y = nil_Pr7649374833807421942um_a_b )
                 => ~ ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ X @ ( produc4082290306366126769um_a_b @ Xa @ nil_Pr7649374833807421942um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_803_ord_Osinter__with_Opelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > nat > nat > nat,Xa: list_P667245297671117661_b_nat,Xb: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( ( rBT_si3280174512405941952_b_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ Xa @ Xb ) ) )
       => ( ! [K2: sum_sum_a_b,V2: nat,As2: list_P667245297671117661_b_nat] :
              ( ( Xa
                = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) )
             => ! [K3: sum_sum_a_b,V3: nat,Bs: list_P667245297671117661_b_nat] :
                  ( ( Xb
                    = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( rBT_si3280174512405941952_b_nat @ Less @ X @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( rBT_si3280174512405941952_b_nat @ Less @ X @ As2 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si3280174512405941952_b_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr3750691640972353501_b_nat )
             => ( ( Y = nil_Pr3750691640972353501_b_nat )
               => ~ ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ nil_Pr3750691640972353501_b_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr3750691640972353501_b_nat )
               => ( ( Y = nil_Pr3750691640972353501_b_nat )
                 => ~ ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ X @ ( produc8028711988652938967_b_nat @ Xa @ nil_Pr3750691640972353501_b_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_804_ord_Osinter__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P5598968436384720482um_a_b,Xb: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( ( rBT_si3716287490892065299um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b] :
              ( ( Xa
                = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
                  ( ( Xb
                    = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( rBT_si3716287490892065299um_a_b @ Less @ X @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( rBT_si3716287490892065299um_a_b @ Less @ X @ As2 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si3716287490892065299um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr1090988463983433548um_a_b )
             => ( ( Y = nil_Pr1090988463983433548um_a_b )
               => ~ ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ nil_Pr1090988463983433548um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr1090988463983433548um_a_b )
               => ( ( Y = nil_Pr1090988463983433548um_a_b )
                 => ~ ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ X @ ( produc8507494088392525149um_a_b @ Xa @ nil_Pr1090988463983433548um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_805_ord_Osinter__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P6497659457863146077um_a_b,Xb: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( ( rBT_si7509189530993033408um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b] :
              ( ( Xa
                = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) )
             => ! [K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
                  ( ( Xb
                    = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( rBT_si7509189530993033408um_a_b @ Less @ X @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( rBT_si7509189530993033408um_a_b @ Less @ X @ As2 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si7509189530993033408um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr3333374394092609245um_a_b )
             => ( ( Y = nil_Pr3333374394092609245um_a_b )
               => ~ ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ nil_Pr3333374394092609245um_a_b @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr3333374394092609245um_a_b )
               => ( ( Y = nil_Pr3333374394092609245um_a_b )
                 => ~ ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ X @ ( produc8821755967955457751um_a_b @ Xa @ nil_Pr3333374394092609245um_a_b ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_806_ord_Osinter__with_Opelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_si2101226886118949098at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ Xb ) ) )
       => ( ! [K2: nat,V2: nat,As2: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) )
             => ! [K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
                  ( ( Xb
                    = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) )
                 => ( ( ( ( Less @ K3 @ K2 )
                       => ( Y
                          = ( rBT_si2101226886118949098at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                      & ( ~ ( Less @ K3 @ K2 )
                       => ( ( ( Less @ K2 @ K3 )
                           => ( Y
                              = ( rBT_si2101226886118949098at_nat @ Less @ X @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
                          & ( ~ ( Less @ K2 @ K3 )
                           => ( Y
                              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) )
                   => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) ) ) ) )
         => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
             => ( ( Y = nil_Pr5478986624290739719at_nat )
               => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ nil_Pr5478986624290739719at_nat @ Xb ) ) ) ) )
           => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
               => ( ( Y = nil_Pr5478986624290739719at_nat )
                 => ~ ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ X @ ( produc5943733680697469783at_nat @ Xa @ nil_Pr5478986624290739719at_nat ) ) ) ) ) ) ) ) ) ).

% ord.sinter_with.pelims
thf(fact_807_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K: sum_sum_a_b,V: sum_sum_a_b,As3: list_P911570089447184179um_a_b,K4: sum_sum_a_b,V4: sum_sum_a_b,Bs2: list_P911570089447184179um_a_b] :
      ( ( accp_P2761727076427072734um_a_b @ ( rBT_su8940943015884164556um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F @ ( produc2669586549142030935um_a_b @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ ( rBT_su4305455488528285193um_a_b @ Less @ F @ As3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su4305455488528285193um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_808_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K: sum_sum_a_b,V: produc4970237071031658413um_a_b,As3: list_P6716435105863944716um_a_b,K4: sum_sum_a_b,V4: produc4970237071031658413um_a_b,Bs2: list_P6716435105863944716um_a_b] :
      ( ( accp_P3837254257538658773um_a_b @ ( rBT_su4708641767375808007um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F @ ( produc4082290306366126769um_a_b @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ ( rBT_su544253560460886922um_a_b @ Less @ F @ As3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su544253560460886922um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_809_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > nat > nat > nat,K: sum_sum_a_b,V: nat,As3: list_P667245297671117661_b_nat,K4: sum_sum_a_b,V4: nat,Bs2: list_P667245297671117661_b_nat] :
      ( ( accp_P4841378303068227168_b_nat @ ( rBT_su1841271182767887862_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F @ ( produc8028711988652938967_b_nat @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
            = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ ( rBT_su1686119633526928435_b_nat @ Less @ F @ As3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su1686119633526928435_b_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_810_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K: nat,V: produc4970237071031658413um_a_b,As3: list_P5598968436384720482um_a_b,K4: nat,V4: produc4970237071031658413um_a_b,Bs2: list_P5598968436384720482um_a_b] :
      ( ( accp_P7243937427905268139um_a_b @ ( rBT_su5785667328470027101um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ ( rBT_su7166556703742098144um_a_b @ Less @ F @ As3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su7166556703742098144um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_811_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K: nat,V: sum_sum_a_b,As3: list_P6497659457863146077um_a_b,K4: nat,V4: sum_sum_a_b,Bs2: list_P6497659457863146077um_a_b] :
      ( ( accp_P3495745604126287028um_a_b @ ( rBT_su6070286201354979318um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ ( rBT_su5915134652114019891um_a_b @ Less @ F @ As3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su5915134652114019891um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_812_ord_Osunion__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > nat > nat > nat,K: nat,V: nat,As3: list_P6011104703257516679at_nat,K4: nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_su3626503011957378080at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sunion_with.psimps(1)
thf(fact_813_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K: sum_sum_a_b,V: sum_sum_a_b,As3: list_P911570089447184179um_a_b,K4: sum_sum_a_b,V4: sum_sum_a_b,Bs2: list_P911570089447184179um_a_b] :
      ( ( accp_P2761727076427072734um_a_b @ ( rBT_si931034614487634303um_a_b @ Less ) @ ( produc4128843237555951321um_a_b @ F @ ( produc2669586549142030935um_a_b @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( rBT_si7236827711767104150um_a_b @ Less @ F @ As3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si7236827711767104150um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_814_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K: sum_sum_a_b,V: produc4970237071031658413um_a_b,As3: list_P6716435105863944716um_a_b,K4: sum_sum_a_b,V4: produc4970237071031658413um_a_b,Bs2: list_P6716435105863944716um_a_b] :
      ( ( accp_P3837254257538658773um_a_b @ ( rBT_si231391089102964884um_a_b @ Less ) @ ( produc1452291671118998076um_a_b @ F @ ( produc4082290306366126769um_a_b @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( rBT_si936693504573915325um_a_b @ Less @ F @ As3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si936693504573915325um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_815_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,F: sum_sum_a_b > nat > nat > nat,K: sum_sum_a_b,V: nat,As3: list_P667245297671117661_b_nat,K4: sum_sum_a_b,V4: nat,Bs2: list_P667245297671117661_b_nat] :
      ( ( accp_P4841378303068227168_b_nat @ ( rBT_si3311849766672747433_b_nat @ Less ) @ ( produc3073544346786878555_b_nat @ F @ ( produc8028711988652938967_b_nat @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
            = ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( rBT_si3280174512405941952_b_nat @ Less @ F @ As3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si3280174512405941952_b_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_816_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,K: nat,V: produc4970237071031658413um_a_b,As3: list_P5598968436384720482um_a_b,K4: nat,V4: produc4970237071031658413um_a_b,Bs2: list_P5598968436384720482um_a_b] :
      ( ( accp_P7243937427905268139um_a_b @ ( rBT_si1793801883848063978um_a_b @ Less ) @ ( produc4266320775040431122um_a_b @ F @ ( produc8507494088392525149um_a_b @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( rBT_si3716287490892065299um_a_b @ Less @ F @ As3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si3716287490892065299um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_817_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,K: nat,V: sum_sum_a_b,As3: list_P6497659457863146077um_a_b,K4: nat,V4: sum_sum_a_b,Bs2: list_P6497659457863146077um_a_b] :
      ( ( accp_P3495745604126287028um_a_b @ ( rBT_si7540864785259838889um_a_b @ Less ) @ ( produc4236031883622431407um_a_b @ F @ ( produc8821755967955457751um_a_b @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
            = ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( rBT_si7509189530993033408um_a_b @ Less @ F @ As3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si7509189530993033408um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_818_ord_Osinter__with_Opsimps_I1_J,axiom,
    ! [Less: nat > nat > $o,F: nat > nat > nat > nat,K: nat,V: nat,As3: list_P6011104703257516679at_nat,K4: nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( accp_P8135403078564911670at_nat @ ( rBT_si7171397677145600979at_nat @ Less ) @ ( produc8375354190041410609at_nat @ F @ ( produc5943733680697469783at_nat @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
     => ( ( ( Less @ K4 @ K )
         => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
            = ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ Bs2 ) ) )
        & ( ~ ( Less @ K4 @ K )
         => ( ( ( Less @ K @ K4 )
             => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( rBT_si2101226886118949098at_nat @ Less @ F @ As3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
            & ( ~ ( Less @ K @ K4 )
             => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ) ).

% ord.sinter_with.psimps(1)
thf(fact_819_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,K4: sum_sum_a_b,K: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,V: sum_sum_a_b,As3: list_P911570089447184179um_a_b,V4: sum_sum_a_b,Bs2: list_P911570089447184179um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ ( rBT_su4305455488528285193um_a_b @ Less @ F @ As3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su4305455488528285193um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su4305455488528285193um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_820_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,K4: sum_sum_a_b,K: sum_sum_a_b,F: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,V: produc4970237071031658413um_a_b,As3: list_P6716435105863944716um_a_b,V4: produc4970237071031658413um_a_b,Bs2: list_P6716435105863944716um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ ( rBT_su544253560460886922um_a_b @ Less @ F @ As3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su544253560460886922um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su544253560460886922um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_821_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,K4: sum_sum_a_b,K: sum_sum_a_b,F: sum_sum_a_b > nat > nat > nat,V: nat,As3: list_P667245297671117661_b_nat,V4: nat,Bs2: list_P667245297671117661_b_nat] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
          = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ ( rBT_su1686119633526928435_b_nat @ Less @ F @ As3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su1686119633526928435_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su1686119633526928435_b_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_822_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K: nat,F: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,V: produc4970237071031658413um_a_b,As3: list_P5598968436384720482um_a_b,V4: produc4970237071031658413um_a_b,Bs2: list_P5598968436384720482um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ ( rBT_su7166556703742098144um_a_b @ Less @ F @ As3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su7166556703742098144um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su7166556703742098144um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_823_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K: nat,F: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,V: sum_sum_a_b,As3: list_P6497659457863146077um_a_b,V4: sum_sum_a_b,Bs2: list_P6497659457863146077um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ ( rBT_su5915134652114019891um_a_b @ Less @ F @ As3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su5915134652114019891um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su5915134652114019891um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_824_ord_Osunion__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K: nat,F: nat > nat > nat > nat,V: nat,As3: list_P6011104703257516679at_nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ Bs2 ) ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_su5597997906817964125at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sunion_with.simps(1)
thf(fact_825_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,K4: sum_sum_a_b,K: sum_sum_a_b,F: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,V: sum_sum_a_b,As3: list_P911570089447184179um_a_b,V4: sum_sum_a_b,Bs2: list_P911570089447184179um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( rBT_si7236827711767104150um_a_b @ Less @ F @ As3 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si7236827711767104150um_a_b @ Less @ F @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V ) @ As3 ) @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si7236827711767104150um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_826_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,K4: sum_sum_a_b,K: sum_sum_a_b,F: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,V: produc4970237071031658413um_a_b,As3: list_P6716435105863944716um_a_b,V4: produc4970237071031658413um_a_b,Bs2: list_P6716435105863944716um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( rBT_si936693504573915325um_a_b @ Less @ F @ As3 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si936693504573915325um_a_b @ Less @ F @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V ) @ As3 ) @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si936693504573915325um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_827_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,K4: sum_sum_a_b,K: sum_sum_a_b,F: sum_sum_a_b > nat > nat > nat,V: nat,As3: list_P667245297671117661_b_nat,V4: nat,Bs2: list_P667245297671117661_b_nat] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
          = ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( rBT_si3280174512405941952_b_nat @ Less @ F @ As3 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si3280174512405941952_b_nat @ Less @ F @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V ) @ As3 ) @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si3280174512405941952_b_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_828_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K: nat,F: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,V: produc4970237071031658413um_a_b,As3: list_P5598968436384720482um_a_b,V4: produc4970237071031658413um_a_b,Bs2: list_P5598968436384720482um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( rBT_si3716287490892065299um_a_b @ Less @ F @ As3 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si3716287490892065299um_a_b @ Less @ F @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V ) @ As3 ) @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si3716287490892065299um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_829_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K: nat,F: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,V: sum_sum_a_b,As3: list_P6497659457863146077um_a_b,V4: sum_sum_a_b,Bs2: list_P6497659457863146077um_a_b] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
          = ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( rBT_si7509189530993033408um_a_b @ Less @ F @ As3 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si7509189530993033408um_a_b @ Less @ F @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V ) @ As3 ) @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si7509189530993033408um_a_b @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_830_ord_Osinter__with_Osimps_I1_J,axiom,
    ! [Less: nat > nat > $o,K4: nat,K: nat,F: nat > nat > nat > nat,V: nat,As3: list_P6011104703257516679at_nat,V4: nat,Bs2: list_P6011104703257516679at_nat] :
      ( ( ( Less @ K4 @ K )
       => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
          = ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ Bs2 ) ) )
      & ( ~ ( Less @ K4 @ K )
       => ( ( ( Less @ K @ K4 )
           => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( rBT_si2101226886118949098at_nat @ Less @ F @ As3 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) ) ) )
          & ( ~ ( Less @ K @ K4 )
           => ( ( rBT_si2101226886118949098at_nat @ Less @ F @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V ) @ As3 ) @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V4 ) @ Bs2 ) )
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ ( F @ K @ V @ V4 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ F @ As3 @ Bs2 ) ) ) ) ) ) ) ).

% ord.sinter_with.simps(1)
thf(fact_831_ord_Osunion__with_Oelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P911570089447184179um_a_b,Xb: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( ( rBT_su4305455488528285193um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: sum_sum_a_b,V2: sum_sum_a_b,As2: list_P911570089447184179um_a_b] :
            ( ( Xa
              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: sum_sum_a_b,V3: sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
                ( ( Xb
                  = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ ( rBT_su4305455488528285193um_a_b @ Less @ X @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ ( rBT_su4305455488528285193um_a_b @ Less @ X @ As2 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su4305455488528285193um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr8520116601857385651um_a_b )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr8520116601857385651um_a_b )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_832_ord_Osunion__with_Oelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P6716435105863944716um_a_b,Xb: list_P6716435105863944716um_a_b,Y: list_P6716435105863944716um_a_b] :
      ( ( ( rBT_su544253560460886922um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: sum_sum_a_b,V2: produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b] :
            ( ( Xa
              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: sum_sum_a_b,V3: produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
                ( ( Xb
                  = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ ( rBT_su544253560460886922um_a_b @ Less @ X @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ ( rBT_su544253560460886922um_a_b @ Less @ X @ As2 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su544253560460886922um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr7649374833807421942um_a_b )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr7649374833807421942um_a_b )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_833_ord_Osunion__with_Oelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > nat > nat > nat,Xa: list_P667245297671117661_b_nat,Xb: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( ( rBT_su1686119633526928435_b_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: sum_sum_a_b,V2: nat,As2: list_P667245297671117661_b_nat] :
            ( ( Xa
              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) )
           => ! [K3: sum_sum_a_b,V3: nat,Bs: list_P667245297671117661_b_nat] :
                ( ( Xb
                  = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ ( rBT_su1686119633526928435_b_nat @ Less @ X @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ ( rBT_su1686119633526928435_b_nat @ Less @ X @ As2 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su1686119633526928435_b_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr3750691640972353501_b_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr3750691640972353501_b_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_834_ord_Osunion__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P5598968436384720482um_a_b,Xb: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( ( rBT_su7166556703742098144um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b] :
            ( ( Xa
              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
                ( ( Xb
                  = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ ( rBT_su7166556703742098144um_a_b @ Less @ X @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ ( rBT_su7166556703742098144um_a_b @ Less @ X @ As2 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su7166556703742098144um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr1090988463983433548um_a_b )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr1090988463983433548um_a_b )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_835_ord_Osunion__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P6497659457863146077um_a_b,Xb: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( ( rBT_su5915134652114019891um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b] :
            ( ( Xa
              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
                ( ( Xb
                  = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ ( rBT_su5915134652114019891um_a_b @ Less @ X @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ ( rBT_su5915134652114019891um_a_b @ Less @ X @ As2 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su5915134652114019891um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr3333374394092609245um_a_b )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr3333374394092609245um_a_b )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_836_ord_Osunion__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_su5597997906817964125at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V2: nat,As2: list_P6011104703257516679at_nat] :
            ( ( Xa
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) )
           => ! [K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_su5597997906817964125at_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
           => ( Y != Xb ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != Xa ) ) ) ) ) ).

% ord.sunion_with.elims
thf(fact_837_ord_Osinter__with_Oelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P911570089447184179um_a_b,Xb: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( ( rBT_si7236827711767104150um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: sum_sum_a_b,V2: sum_sum_a_b,As2: list_P911570089447184179um_a_b] :
            ( ( Xa
              = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: sum_sum_a_b,V3: sum_sum_a_b,Bs: list_P911570089447184179um_a_b] :
                ( ( Xb
                  = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si7236827711767104150um_a_b @ Less @ X @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si7236827711767104150um_a_b @ Less @ X @ As2 @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si7236827711767104150um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr8520116601857385651um_a_b )
           => ( Y != nil_Pr8520116601857385651um_a_b ) )
         => ~ ( ( Xb = nil_Pr8520116601857385651um_a_b )
             => ( Y != nil_Pr8520116601857385651um_a_b ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_838_ord_Osinter__with_Oelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P6716435105863944716um_a_b,Xb: list_P6716435105863944716um_a_b,Y: list_P6716435105863944716um_a_b] :
      ( ( ( rBT_si936693504573915325um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: sum_sum_a_b,V2: produc4970237071031658413um_a_b,As2: list_P6716435105863944716um_a_b] :
            ( ( Xa
              = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: sum_sum_a_b,V3: produc4970237071031658413um_a_b,Bs: list_P6716435105863944716um_a_b] :
                ( ( Xb
                  = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si936693504573915325um_a_b @ Less @ X @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si936693504573915325um_a_b @ Less @ X @ As2 @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si936693504573915325um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr7649374833807421942um_a_b )
           => ( Y != nil_Pr7649374833807421942um_a_b ) )
         => ~ ( ( Xb = nil_Pr7649374833807421942um_a_b )
             => ( Y != nil_Pr7649374833807421942um_a_b ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_839_ord_Osinter__with_Oelims,axiom,
    ! [Less: sum_sum_a_b > sum_sum_a_b > $o,X: sum_sum_a_b > nat > nat > nat,Xa: list_P667245297671117661_b_nat,Xb: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( ( rBT_si3280174512405941952_b_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: sum_sum_a_b,V2: nat,As2: list_P667245297671117661_b_nat] :
            ( ( Xa
              = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) )
           => ! [K3: sum_sum_a_b,V3: nat,Bs: list_P667245297671117661_b_nat] :
                ( ( Xb
                  = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si3280174512405941952_b_nat @ Less @ X @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si3280174512405941952_b_nat @ Less @ X @ As2 @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V3 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si3280174512405941952_b_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr3750691640972353501_b_nat )
           => ( Y != nil_Pr3750691640972353501_b_nat ) )
         => ~ ( ( Xb = nil_Pr3750691640972353501_b_nat )
             => ( Y != nil_Pr3750691640972353501_b_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_840_ord_Osinter__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b > produc4970237071031658413um_a_b,Xa: list_P5598968436384720482um_a_b,Xb: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( ( rBT_si3716287490892065299um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V2: produc4970237071031658413um_a_b,As2: list_P5598968436384720482um_a_b] :
            ( ( Xa
              = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: nat,V3: produc4970237071031658413um_a_b,Bs: list_P5598968436384720482um_a_b] :
                ( ( Xb
                  = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si3716287490892065299um_a_b @ Less @ X @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si3716287490892065299um_a_b @ Less @ X @ As2 @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si3716287490892065299um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr1090988463983433548um_a_b )
           => ( Y != nil_Pr1090988463983433548um_a_b ) )
         => ~ ( ( Xb = nil_Pr1090988463983433548um_a_b )
             => ( Y != nil_Pr1090988463983433548um_a_b ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_841_ord_Osinter__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > sum_sum_a_b > sum_sum_a_b > sum_sum_a_b,Xa: list_P6497659457863146077um_a_b,Xb: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( ( rBT_si7509189530993033408um_a_b @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V2: sum_sum_a_b,As2: list_P6497659457863146077um_a_b] :
            ( ( Xa
              = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) )
           => ! [K3: nat,V3: sum_sum_a_b,Bs: list_P6497659457863146077um_a_b] :
                ( ( Xb
                  = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si7509189530993033408um_a_b @ Less @ X @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si7509189530993033408um_a_b @ Less @ X @ As2 @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V3 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si7509189530993033408um_a_b @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr3333374394092609245um_a_b )
           => ( Y != nil_Pr3333374394092609245um_a_b ) )
         => ~ ( ( Xb = nil_Pr3333374394092609245um_a_b )
             => ( Y != nil_Pr3333374394092609245um_a_b ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_842_ord_Osinter__with_Oelims,axiom,
    ! [Less: nat > nat > $o,X: nat > nat > nat > nat,Xa: list_P6011104703257516679at_nat,Xb: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( rBT_si2101226886118949098at_nat @ Less @ X @ Xa @ Xb )
        = Y )
     => ( ! [K2: nat,V2: nat,As2: list_P6011104703257516679at_nat] :
            ( ( Xa
              = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) )
           => ! [K3: nat,V3: nat,Bs: list_P6011104703257516679at_nat] :
                ( ( Xb
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) )
               => ~ ( ( ( Less @ K3 @ K2 )
                     => ( Y
                        = ( rBT_si2101226886118949098at_nat @ Less @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ V2 ) @ As2 ) @ Bs ) ) )
                    & ( ~ ( Less @ K3 @ K2 )
                     => ( ( ( Less @ K2 @ K3 )
                         => ( Y
                            = ( rBT_si2101226886118949098at_nat @ Less @ X @ As2 @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V3 ) @ Bs ) ) ) )
                        & ( ~ ( Less @ K2 @ K3 )
                         => ( Y
                            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K2 @ ( X @ K2 @ V2 @ V3 ) ) @ ( rBT_si2101226886118949098at_nat @ Less @ X @ As2 @ Bs ) ) ) ) ) ) ) ) )
       => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
           => ( Y != nil_Pr5478986624290739719at_nat ) )
         => ~ ( ( Xb = nil_Pr5478986624290739719at_nat )
             => ( Y != nil_Pr5478986624290739719at_nat ) ) ) ) ) ).

% ord.sinter_with.elims
thf(fact_843_subrelI,axiom,
    ! [R3: set_Pr7446880501794416397um_a_b,S3: set_Pr7446880501794416397um_a_b] :
      ( ! [X5: sum_sum_a_b,Y3: sum_sum_a_b] :
          ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ Y3 ) @ R3 )
         => ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ Y3 ) @ S3 ) )
     => ( ord_le1431929631403617965um_a_b @ R3 @ S3 ) ) ).

% subrelI
thf(fact_844_subrelI,axiom,
    ! [R3: set_Pr4696577323406728626um_a_b,S3: set_Pr4696577323406728626um_a_b] :
      ( ! [X5: sum_sum_a_b,Y3: produc4970237071031658413um_a_b] :
          ( ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X5 @ Y3 ) @ R3 )
         => ( member152508769716563603um_a_b @ ( produc7989310782275393198um_a_b @ X5 @ Y3 ) @ S3 ) )
     => ( ord_le3617630362065423890um_a_b @ R3 @ S3 ) ) ).

% subrelI
thf(fact_845_subrelI,axiom,
    ! [R3: set_Pr7634312471471294263_b_nat,S3: set_Pr7634312471471294263_b_nat] :
      ( ! [X5: sum_sum_a_b,Y3: nat] :
          ( ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X5 @ Y3 ) @ R3 )
         => ( member8290135809484590976_b_nat @ ( produc9094079831559073295_b_nat @ X5 @ Y3 ) @ S3 ) )
     => ( ord_le8626435567010533079_b_nat @ R3 @ S3 ) ) ).

% subrelI
thf(fact_846_subrelI,axiom,
    ! [R3: set_Pr5956926064383532552um_a_b,S3: set_Pr5956926064383532552um_a_b] :
      ( ! [X5: nat,Y3: produc4970237071031658413um_a_b] :
          ( ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X5 @ Y3 ) @ R3 )
         => ( member4052926750399300073um_a_b @ ( produc6016411306719553028um_a_b @ X5 @ Y3 ) @ S3 ) )
     => ( ord_le5806309049788436072um_a_b @ R3 @ S3 ) ) ).

% subrelI
thf(fact_847_subrelI,axiom,
    ! [R3: set_Pr4241354594808546871um_a_b,S3: set_Pr4241354594808546871um_a_b] :
      ( ! [X5: nat,Y3: sum_sum_a_b] :
          ( ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X5 @ Y3 ) @ R3 )
         => ( member7872818562604846720um_a_b @ ( produc4099722813291388943um_a_b @ X5 @ Y3 ) @ S3 ) )
     => ( ord_le5233477690347785687um_a_b @ R3 @ S3 ) ) ).

% subrelI
thf(fact_848_subrelI,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ! [X5: nat,Y3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ S3 ) )
     => ( ord_le3146513528884898305at_nat @ R3 @ S3 ) ) ).

% subrelI
thf(fact_849_mlex__leq,axiom,
    ! [F: sum_sum_a_b > nat,X: sum_sum_a_b,Y: sum_sum_a_b,R: set_Pr7446880501794416397um_a_b] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ R )
       => ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X @ Y ) @ ( mlex_p8502794516353305431um_a_b @ F @ R ) ) ) ) ).

% mlex_leq
thf(fact_850_mlex__leq,axiom,
    ! [F: nat > nat,X: nat,Y: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ ( mlex_prod_nat @ F @ R ) ) ) ) ).

% mlex_leq
thf(fact_851_AssocList_Odelete__aux__eq__Nil__conv,axiom,
    ! [K: sum_sum_a_b,Ts: list_P911570089447184179um_a_b] :
      ( ( ( delete6941486676214758144um_a_b @ K @ Ts )
        = nil_Pr8520116601857385651um_a_b )
      = ( ( Ts = nil_Pr8520116601857385651um_a_b )
        | ? [V5: sum_sum_a_b] :
            ( Ts
            = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K @ V5 ) @ nil_Pr8520116601857385651um_a_b ) ) ) ) ).

% AssocList.delete_aux_eq_Nil_conv
thf(fact_852_AssocList_Odelete__aux__eq__Nil__conv,axiom,
    ! [K: sum_sum_a_b,Ts: list_P6716435105863944716um_a_b] :
      ( ( ( delete1161692237630136275um_a_b @ K @ Ts )
        = nil_Pr7649374833807421942um_a_b )
      = ( ( Ts = nil_Pr7649374833807421942um_a_b )
        | ? [V5: produc4970237071031658413um_a_b] :
            ( Ts
            = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K @ V5 ) @ nil_Pr7649374833807421942um_a_b ) ) ) ) ).

% AssocList.delete_aux_eq_Nil_conv
thf(fact_853_AssocList_Odelete__aux__eq__Nil__conv,axiom,
    ! [K: sum_sum_a_b,Ts: list_P667245297671117661_b_nat] :
      ( ( ( delete1083089001929472810_b_nat @ K @ Ts )
        = nil_Pr3750691640972353501_b_nat )
      = ( ( Ts = nil_Pr3750691640972353501_b_nat )
        | ? [V5: nat] :
            ( Ts
            = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K @ V5 ) @ nil_Pr3750691640972353501_b_nat ) ) ) ) ).

% AssocList.delete_aux_eq_Nil_conv
thf(fact_854_AssocList_Odelete__aux__eq__Nil__conv,axiom,
    ! [K: nat,Ts: list_P5598968436384720482um_a_b] :
      ( ( ( delete3348065510246518057um_a_b @ K @ Ts )
        = nil_Pr1090988463983433548um_a_b )
      = ( ( Ts = nil_Pr1090988463983433548um_a_b )
        | ? [V5: produc4970237071031658413um_a_b] :
            ( Ts
            = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K @ V5 ) @ nil_Pr1090988463983433548um_a_b ) ) ) ) ).

% AssocList.delete_aux_eq_Nil_conv
thf(fact_855_AssocList_Odelete__aux__eq__Nil__conv,axiom,
    ! [K: nat,Ts: list_P6497659457863146077um_a_b] :
      ( ( ( delete5312104020516564266um_a_b @ K @ Ts )
        = nil_Pr3333374394092609245um_a_b )
      = ( ( Ts = nil_Pr3333374394092609245um_a_b )
        | ? [V5: sum_sum_a_b] :
            ( Ts
            = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K @ V5 ) @ nil_Pr3333374394092609245um_a_b ) ) ) ) ).

% AssocList.delete_aux_eq_Nil_conv
thf(fact_856_AssocList_Odelete__aux__eq__Nil__conv,axiom,
    ! [K: nat,Ts: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ K @ Ts )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Ts = nil_Pr5478986624290739719at_nat )
        | ? [V5: nat] :
            ( Ts
            = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K @ V5 ) @ nil_Pr5478986624290739719at_nat ) ) ) ) ).

% AssocList.delete_aux_eq_Nil_conv
thf(fact_857_AssocList_Odelete__aux_Oelims,axiom,
    ! [X: sum_sum_a_b,Xa: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( ( delete6941486676214758144um_a_b @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr8520116601857385651um_a_b )
         => ( Y != nil_Pr8520116601857385651um_a_b ) )
       => ~ ! [K3: sum_sum_a_b,V2: sum_sum_a_b,Xs3: list_P911570089447184179um_a_b] :
              ( ( Xa
                = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V2 ) @ Xs3 ) )
             => ~ ( ( ( X = K3 )
                   => ( Y = Xs3 ) )
                  & ( ( X != K3 )
                   => ( Y
                      = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V2 ) @ ( delete6941486676214758144um_a_b @ X @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.elims
thf(fact_858_AssocList_Odelete__aux_Oelims,axiom,
    ! [X: sum_sum_a_b,Xa: list_P6716435105863944716um_a_b,Y: list_P6716435105863944716um_a_b] :
      ( ( ( delete1161692237630136275um_a_b @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr7649374833807421942um_a_b )
         => ( Y != nil_Pr7649374833807421942um_a_b ) )
       => ~ ! [K3: sum_sum_a_b,V2: produc4970237071031658413um_a_b,Xs3: list_P6716435105863944716um_a_b] :
              ( ( Xa
                = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V2 ) @ Xs3 ) )
             => ~ ( ( ( X = K3 )
                   => ( Y = Xs3 ) )
                  & ( ( X != K3 )
                   => ( Y
                      = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V2 ) @ ( delete1161692237630136275um_a_b @ X @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.elims
thf(fact_859_AssocList_Odelete__aux_Oelims,axiom,
    ! [X: sum_sum_a_b,Xa: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( ( delete1083089001929472810_b_nat @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr3750691640972353501_b_nat )
         => ( Y != nil_Pr3750691640972353501_b_nat ) )
       => ~ ! [K3: sum_sum_a_b,V2: nat,Xs3: list_P667245297671117661_b_nat] :
              ( ( Xa
                = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V2 ) @ Xs3 ) )
             => ~ ( ( ( X = K3 )
                   => ( Y = Xs3 ) )
                  & ( ( X != K3 )
                   => ( Y
                      = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V2 ) @ ( delete1083089001929472810_b_nat @ X @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.elims
thf(fact_860_AssocList_Odelete__aux_Oelims,axiom,
    ! [X: nat,Xa: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( ( delete3348065510246518057um_a_b @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr1090988463983433548um_a_b )
         => ( Y != nil_Pr1090988463983433548um_a_b ) )
       => ~ ! [K3: nat,V2: produc4970237071031658413um_a_b,Xs3: list_P5598968436384720482um_a_b] :
              ( ( Xa
                = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V2 ) @ Xs3 ) )
             => ~ ( ( ( X = K3 )
                   => ( Y = Xs3 ) )
                  & ( ( X != K3 )
                   => ( Y
                      = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V2 ) @ ( delete3348065510246518057um_a_b @ X @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.elims
thf(fact_861_AssocList_Odelete__aux_Oelims,axiom,
    ! [X: nat,Xa: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( ( delete5312104020516564266um_a_b @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr3333374394092609245um_a_b )
         => ( Y != nil_Pr3333374394092609245um_a_b ) )
       => ~ ! [K3: nat,V2: sum_sum_a_b,Xs3: list_P6497659457863146077um_a_b] :
              ( ( Xa
                = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V2 ) @ Xs3 ) )
             => ~ ( ( ( X = K3 )
                   => ( Y = Xs3 ) )
                  & ( ( X != K3 )
                   => ( Y
                      = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V2 ) @ ( delete5312104020516564266um_a_b @ X @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.elims
thf(fact_862_AssocList_Odelete__aux_Oelims,axiom,
    ! [X: nat,Xa: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ X @ Xa )
        = Y )
     => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
         => ( Y != nil_Pr5478986624290739719at_nat ) )
       => ~ ! [K3: nat,V2: nat,Xs3: list_P6011104703257516679at_nat] :
              ( ( Xa
                = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V2 ) @ Xs3 ) )
             => ~ ( ( ( X = K3 )
                   => ( Y = Xs3 ) )
                  & ( ( X != K3 )
                   => ( Y
                      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V2 ) @ ( delete_aux_nat_nat @ X @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.elims
thf(fact_863_list__ex1__iff,axiom,
    ( list_e9161289806097395830um_a_b
    = ( ^ [P3: produc4970237071031658413um_a_b > $o,Xs2: list_P911570089447184179um_a_b] :
        ? [X4: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ Xs2 ) )
          & ( P3 @ X4 )
          & ! [Y5: produc4970237071031658413um_a_b] :
              ( ( ( member6480126262776431190um_a_b @ Y5 @ ( set_Pr2436422074441463490um_a_b @ Xs2 ) )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_864_list__ex1__iff,axiom,
    ( list_ex1_Sum_sum_a_b
    = ( ^ [P3: sum_sum_a_b > $o,Xs2: list_Sum_sum_a_b] :
        ? [X4: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X4 @ ( set_Sum_sum_a_b2 @ Xs2 ) )
          & ( P3 @ X4 )
          & ! [Y5: sum_sum_a_b] :
              ( ( ( member_Sum_sum_a_b @ Y5 @ ( set_Sum_sum_a_b2 @ Xs2 ) )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_865_list__ex1__iff,axiom,
    ( list_e8509259699837697440_b_nat
    = ( ^ [P3: produc6336967135452217047_b_nat > $o,Xs2: list_P667245297671117661_b_nat] :
        ? [X4: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X4 @ ( set_Pr176305205332118508_b_nat @ Xs2 ) )
          & ( P3 @ X4 )
          & ! [Y5: produc6336967135452217047_b_nat] :
              ( ( ( member8290135809484590976_b_nat @ Y5 @ ( set_Pr176305205332118508_b_nat @ Xs2 ) )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_866_list__ex1__iff,axiom,
    ( list_e4826986601217043401um_a_b
    = ( ^ [P3: produc7779921308951573458um_a_b > $o,Xs2: list_P5598968436384720482um_a_b] :
        ? [X4: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X4 @ ( set_Pr7399620718385288829um_a_b @ Xs2 ) )
          & ( P3 @ X4 )
          & ! [Y5: produc7779921308951573458um_a_b] :
              ( ( ( member4052926750399300073um_a_b @ Y5 @ ( set_Pr7399620718385288829um_a_b @ Xs2 ) )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_867_list__ex1__iff,axiom,
    ( list_e8091942452957953184um_a_b
    = ( ^ [P3: produc5919649888572472791um_a_b > $o,Xs2: list_P6497659457863146077um_a_b] :
        ? [X4: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X4 @ ( set_Pr8982359995307150060um_a_b @ Xs2 ) )
          & ( P3 @ X4 )
          & ! [Y5: produc5919649888572472791um_a_b] :
              ( ( ( member7872818562604846720um_a_b @ Y5 @ ( set_Pr8982359995307150060um_a_b @ Xs2 ) )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_868_list__ex1__iff,axiom,
    ( list_e8644085759156585930at_nat
    = ( ^ [P3: product_prod_nat_nat > $o,Xs2: list_P6011104703257516679at_nat] :
        ? [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
          & ( P3 @ X4 )
          & ! [Y5: product_prod_nat_nat] :
              ( ( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Xs2 ) )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_869_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P3: nat > $o,Xs2: list_nat] :
        ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
          & ( P3 @ X4 )
          & ! [Y5: nat] :
              ( ( ( member_nat @ Y5 @ ( set_nat2 @ Xs2 ) )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_870_AssocList_Odelete__aux_Osimps_I2_J,axiom,
    ! [K: sum_sum_a_b,K4: sum_sum_a_b,V: sum_sum_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( ( K = K4 )
       => ( ( delete6941486676214758144um_a_b @ K @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K != K4 )
       => ( ( delete6941486676214758144um_a_b @ K @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V ) @ Xs ) )
          = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K4 @ V ) @ ( delete6941486676214758144um_a_b @ K @ Xs ) ) ) ) ) ).

% AssocList.delete_aux.simps(2)
thf(fact_871_AssocList_Odelete__aux_Osimps_I2_J,axiom,
    ! [K: sum_sum_a_b,K4: sum_sum_a_b,V: produc4970237071031658413um_a_b,Xs: list_P6716435105863944716um_a_b] :
      ( ( ( K = K4 )
       => ( ( delete1161692237630136275um_a_b @ K @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K != K4 )
       => ( ( delete1161692237630136275um_a_b @ K @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V ) @ Xs ) )
          = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K4 @ V ) @ ( delete1161692237630136275um_a_b @ K @ Xs ) ) ) ) ) ).

% AssocList.delete_aux.simps(2)
thf(fact_872_AssocList_Odelete__aux_Osimps_I2_J,axiom,
    ! [K: sum_sum_a_b,K4: sum_sum_a_b,V: nat,Xs: list_P667245297671117661_b_nat] :
      ( ( ( K = K4 )
       => ( ( delete1083089001929472810_b_nat @ K @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K != K4 )
       => ( ( delete1083089001929472810_b_nat @ K @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V ) @ Xs ) )
          = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K4 @ V ) @ ( delete1083089001929472810_b_nat @ K @ Xs ) ) ) ) ) ).

% AssocList.delete_aux.simps(2)
thf(fact_873_AssocList_Odelete__aux_Osimps_I2_J,axiom,
    ! [K: nat,K4: nat,V: produc4970237071031658413um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( ( K = K4 )
       => ( ( delete3348065510246518057um_a_b @ K @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K != K4 )
       => ( ( delete3348065510246518057um_a_b @ K @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V ) @ Xs ) )
          = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K4 @ V ) @ ( delete3348065510246518057um_a_b @ K @ Xs ) ) ) ) ) ).

% AssocList.delete_aux.simps(2)
thf(fact_874_AssocList_Odelete__aux_Osimps_I2_J,axiom,
    ! [K: nat,K4: nat,V: sum_sum_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( ( K = K4 )
       => ( ( delete5312104020516564266um_a_b @ K @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K != K4 )
       => ( ( delete5312104020516564266um_a_b @ K @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V ) @ Xs ) )
          = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K4 @ V ) @ ( delete5312104020516564266um_a_b @ K @ Xs ) ) ) ) ) ).

% AssocList.delete_aux.simps(2)
thf(fact_875_AssocList_Odelete__aux_Osimps_I2_J,axiom,
    ! [K: nat,K4: nat,V: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ( K = K4 )
       => ( ( delete_aux_nat_nat @ K @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K != K4 )
       => ( ( delete_aux_nat_nat @ K @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ Xs ) )
          = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K4 @ V ) @ ( delete_aux_nat_nat @ K @ Xs ) ) ) ) ) ).

% AssocList.delete_aux.simps(2)
thf(fact_876_AssocList_Odelete__aux_Opelims,axiom,
    ! [X: sum_sum_a_b,Xa: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( ( delete6941486676214758144um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P6816953145850428601um_a_b @ delete2169469174080775573um_a_b @ ( produc6895264061556658612um_a_b @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr8520116601857385651um_a_b )
           => ( ( Y = nil_Pr8520116601857385651um_a_b )
             => ~ ( accp_P6816953145850428601um_a_b @ delete2169469174080775573um_a_b @ ( produc6895264061556658612um_a_b @ X @ nil_Pr8520116601857385651um_a_b ) ) ) )
         => ~ ! [K3: sum_sum_a_b,V2: sum_sum_a_b,Xs3: list_P911570089447184179um_a_b] :
                ( ( Xa
                  = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V2 ) @ Xs3 ) )
               => ( ( ( ( X = K3 )
                     => ( Y = Xs3 ) )
                    & ( ( X != K3 )
                     => ( Y
                        = ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V2 ) @ ( delete6941486676214758144um_a_b @ X @ Xs3 ) ) ) ) )
                 => ~ ( accp_P6816953145850428601um_a_b @ delete2169469174080775573um_a_b @ ( produc6895264061556658612um_a_b @ X @ ( cons_P4211151491411527779um_a_b @ ( produc8374529264883863525um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.pelims
thf(fact_877_AssocList_Odelete__aux_Opelims,axiom,
    ! [X: sum_sum_a_b,Xa: list_P6716435105863944716um_a_b,Y: list_P6716435105863944716um_a_b] :
      ( ( ( delete1161692237630136275um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P8839113985496565916um_a_b @ delete532881961126511870um_a_b @ ( produc7465751454420652107um_a_b @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr7649374833807421942um_a_b )
           => ( ( Y = nil_Pr7649374833807421942um_a_b )
             => ~ ( accp_P8839113985496565916um_a_b @ delete532881961126511870um_a_b @ ( produc7465751454420652107um_a_b @ X @ nil_Pr7649374833807421942um_a_b ) ) ) )
         => ~ ! [K3: sum_sum_a_b,V2: produc4970237071031658413um_a_b,Xs3: list_P6716435105863944716um_a_b] :
                ( ( Xa
                  = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V2 ) @ Xs3 ) )
               => ( ( ( ( X = K3 )
                     => ( Y = Xs3 ) )
                    & ( ( X != K3 )
                     => ( Y
                        = ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V2 ) @ ( delete1161692237630136275um_a_b @ X @ Xs3 ) ) ) ) )
                 => ~ ( accp_P8839113985496565916um_a_b @ delete532881961126511870um_a_b @ ( produc7465751454420652107um_a_b @ X @ ( cons_P2376077528734995014um_a_b @ ( produc7989310782275393198um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.pelims
thf(fact_878_AssocList_Odelete__aux_Opelims,axiom,
    ! [X: sum_sum_a_b,Xa: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( ( delete1083089001929472810_b_nat @ X @ Xa )
        = Y )
     => ( ( accp_P2415167917280603619_b_nat @ delete3623115347726443455_b_nat @ ( produc4871069856000220126_b_nat @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr3750691640972353501_b_nat )
           => ( ( Y = nil_Pr3750691640972353501_b_nat )
             => ~ ( accp_P2415167917280603619_b_nat @ delete3623115347726443455_b_nat @ ( produc4871069856000220126_b_nat @ X @ nil_Pr3750691640972353501_b_nat ) ) ) )
         => ~ ! [K3: sum_sum_a_b,V2: nat,Xs3: list_P667245297671117661_b_nat] :
                ( ( Xa
                  = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V2 ) @ Xs3 ) )
               => ( ( ( ( X = K3 )
                     => ( Y = Xs3 ) )
                    & ( ( X != K3 )
                     => ( Y
                        = ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V2 ) @ ( delete1083089001929472810_b_nat @ X @ Xs3 ) ) ) ) )
                 => ~ ( accp_P2415167917280603619_b_nat @ delete3623115347726443455_b_nat @ ( produc4871069856000220126_b_nat @ X @ ( cons_P4197526505774637965_b_nat @ ( produc9094079831559073295_b_nat @ K3 @ V2 ) @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.pelims
thf(fact_879_AssocList_Odelete__aux_Opelims,axiom,
    ! [X: nat,Xa: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( ( delete3348065510246518057um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P7968901905569023004um_a_b @ delete8508285360664163412um_a_b @ ( produc7902279242088003275um_a_b @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr1090988463983433548um_a_b )
           => ( ( Y = nil_Pr1090988463983433548um_a_b )
             => ~ ( accp_P7968901905569023004um_a_b @ delete8508285360664163412um_a_b @ ( produc7902279242088003275um_a_b @ X @ nil_Pr1090988463983433548um_a_b ) ) ) )
         => ~ ! [K3: nat,V2: produc4970237071031658413um_a_b,Xs3: list_P5598968436384720482um_a_b] :
                ( ( Xa
                  = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V2 ) @ Xs3 ) )
               => ( ( ( ( X = K3 )
                     => ( Y = Xs3 ) )
                    & ( ( X != K3 )
                     => ( Y
                        = ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V2 ) @ ( delete3348065510246518057um_a_b @ X @ Xs3 ) ) ) ) )
                 => ~ ( accp_P7968901905569023004um_a_b @ delete8508285360664163412um_a_b @ ( produc7902279242088003275um_a_b @ X @ ( cons_P7543864023423915932um_a_b @ ( produc6016411306719553028um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.pelims
thf(fact_880_AssocList_Odelete__aux_Opelims,axiom,
    ! [X: nat,Xa: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( ( delete5312104020516564266um_a_b @ X @ Xa )
        = Y )
     => ( ( accp_P1964116808751817017um_a_b @ delete7852130366313534911um_a_b @ ( produc1183721963040099124um_a_b @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr3333374394092609245um_a_b )
           => ( ( Y = nil_Pr3333374394092609245um_a_b )
             => ~ ( accp_P1964116808751817017um_a_b @ delete7852130366313534911um_a_b @ ( produc1183721963040099124um_a_b @ X @ nil_Pr3333374394092609245um_a_b ) ) ) )
         => ~ ! [K3: nat,V2: sum_sum_a_b,Xs3: list_P6497659457863146077um_a_b] :
                ( ( Xa
                  = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V2 ) @ Xs3 ) )
               => ( ( ( ( X = K3 )
                     => ( Y = Xs3 ) )
                    & ( ( X != K3 )
                     => ( Y
                        = ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V2 ) @ ( delete5312104020516564266um_a_b @ X @ Xs3 ) ) ) ) )
                 => ~ ( accp_P1964116808751817017um_a_b @ delete7852130366313534911um_a_b @ ( produc1183721963040099124um_a_b @ X @ ( cons_P3780209258894893709um_a_b @ ( produc4099722813291388943um_a_b @ K3 @ V2 ) @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.pelims
thf(fact_881_AssocList_Odelete__aux_Opelims,axiom,
    ! [X: nat,Xa: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( ( delete_aux_nat_nat @ X @ Xa )
        = Y )
     => ( ( accp_P1391465874720445027at_nat @ delete4291352608092917737at_nat @ ( produc6109913384486294878at_nat @ X @ Xa ) )
       => ( ( ( Xa = nil_Pr5478986624290739719at_nat )
           => ( ( Y = nil_Pr5478986624290739719at_nat )
             => ~ ( accp_P1391465874720445027at_nat @ delete4291352608092917737at_nat @ ( produc6109913384486294878at_nat @ X @ nil_Pr5478986624290739719at_nat ) ) ) )
         => ~ ! [K3: nat,V2: nat,Xs3: list_P6011104703257516679at_nat] :
                ( ( Xa
                  = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V2 ) @ Xs3 ) )
               => ( ( ( ( X = K3 )
                     => ( Y = Xs3 ) )
                    & ( ( X != K3 )
                     => ( Y
                        = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V2 ) @ ( delete_aux_nat_nat @ X @ Xs3 ) ) ) ) )
                 => ~ ( accp_P1391465874720445027at_nat @ delete4291352608092917737at_nat @ ( produc6109913384486294878at_nat @ X @ ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ K3 @ V2 ) @ Xs3 ) ) ) ) ) ) ) ) ).

% AssocList.delete_aux.pelims
thf(fact_882_can__select__set__list__ex1,axiom,
    ! [P2: produc4970237071031658413um_a_b > $o,A5: list_P911570089447184179um_a_b] :
      ( ( can_se2965781163881960362um_a_b @ P2 @ ( set_Pr2436422074441463490um_a_b @ A5 ) )
      = ( list_e9161289806097395830um_a_b @ P2 @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_883_can__select__set__list__ex1,axiom,
    ! [P2: sum_sum_a_b > $o,A5: list_Sum_sum_a_b] :
      ( ( can_se4659341492078941673um_a_b @ P2 @ ( set_Sum_sum_a_b2 @ A5 ) )
      = ( list_ex1_Sum_sum_a_b @ P2 @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_884_can__select__set__list__ex1,axiom,
    ! [P2: produc6336967135452217047_b_nat > $o,A5: list_P667245297671117661_b_nat] :
      ( ( can_se6158366165627607252_b_nat @ P2 @ ( set_Pr176305205332118508_b_nat @ A5 ) )
      = ( list_e8509259699837697440_b_nat @ P2 @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_885_can__select__set__list__ex1,axiom,
    ! [P2: produc7779921308951573458um_a_b > $o,A5: list_P5598968436384720482um_a_b] :
      ( ( can_se7268299864462161813um_a_b @ P2 @ ( set_Pr7399620718385288829um_a_b @ A5 ) )
      = ( list_e4826986601217043401um_a_b @ P2 @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_886_can__select__set__list__ex1,axiom,
    ! [P2: produc5919649888572472791um_a_b > $o,A5: list_P6497659457863146077um_a_b] :
      ( ( can_se5741048918747862996um_a_b @ P2 @ ( set_Pr8982359995307150060um_a_b @ A5 ) )
      = ( list_e8091942452957953184um_a_b @ P2 @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_887_can__select__set__list__ex1,axiom,
    ! [P2: product_prod_nat_nat > $o,A5: list_P6011104703257516679at_nat] :
      ( ( can_se4754832747099445502at_nat @ P2 @ ( set_Pr5648618587558075414at_nat @ A5 ) )
      = ( list_e8644085759156585930at_nat @ P2 @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_888_can__select__set__list__ex1,axiom,
    ! [P2: nat > $o,A5: list_nat] :
      ( ( can_select_nat @ P2 @ ( set_nat2 @ A5 ) )
      = ( list_ex1_nat @ P2 @ A5 ) ) ).

% can_select_set_list_ex1
thf(fact_889_lenlex__length,axiom,
    ! [Ms: list_nat,Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R3 ) )
     => ( ord_less_eq_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_890_lexord__cons__cons,axiom,
    ! [A3: sum_sum_a_b,X: list_Sum_sum_a_b,B3: sum_sum_a_b,Y: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b] :
      ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ ( cons_Sum_sum_a_b @ A3 @ X ) @ ( cons_Sum_sum_a_b @ B3 @ Y ) ) @ ( lexord_Sum_sum_a_b @ R3 ) )
      = ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ A3 @ B3 ) @ R3 )
        | ( ( A3 = B3 )
          & ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ X @ Y ) @ ( lexord_Sum_sum_a_b @ R3 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_891_lexord__cons__cons,axiom,
    ! [A3: nat,X: list_nat,B3: nat,Y: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ X ) @ ( cons_nat @ B3 @ Y ) ) @ ( lexord_nat @ R3 ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ R3 )
        | ( ( A3 = B3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R3 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_892_set__rev,axiom,
    ! [Xs: list_P911570089447184179um_a_b] :
      ( ( set_Pr2436422074441463490um_a_b @ ( rev_Pr3063422751929003795um_a_b @ Xs ) )
      = ( set_Pr2436422074441463490um_a_b @ Xs ) ) ).

% set_rev
thf(fact_893_set__rev,axiom,
    ! [Xs: list_Sum_sum_a_b] :
      ( ( set_Sum_sum_a_b2 @ ( rev_Sum_sum_a_b @ Xs ) )
      = ( set_Sum_sum_a_b2 @ Xs ) ) ).

% set_rev
thf(fact_894_set__rev,axiom,
    ! [Xs: list_P667245297671117661_b_nat] :
      ( ( set_Pr176305205332118508_b_nat @ ( rev_Pr2989361455954602557_b_nat @ Xs ) )
      = ( set_Pr176305205332118508_b_nat @ Xs ) ) ).

% set_rev
thf(fact_895_set__rev,axiom,
    ! [Xs: list_P5598968436384720482um_a_b] :
      ( ( set_Pr7399620718385288829um_a_b @ ( rev_Pr3113145760173511916um_a_b @ Xs ) )
      = ( set_Pr7399620718385288829um_a_b @ Xs ) ) ).

% set_rev
thf(fact_896_set__rev,axiom,
    ! [Xs: list_P6497659457863146077um_a_b] :
      ( ( set_Pr8982359995307150060um_a_b @ ( rev_Pr2572044209074858301um_a_b @ Xs ) )
      = ( set_Pr8982359995307150060um_a_b @ Xs ) ) ).

% set_rev
thf(fact_897_set__rev,axiom,
    ! [Xs: list_P6011104703257516679at_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( rev_Pr6102188148953555047at_nat @ Xs ) )
      = ( set_Pr5648618587558075414at_nat @ Xs ) ) ).

% set_rev
thf(fact_898_set__rev,axiom,
    ! [Xs: list_nat] :
      ( ( set_nat2 @ ( rev_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_rev
thf(fact_899_length__rev,axiom,
    ! [Xs: list_nat] :
      ( ( size_size_list_nat @ ( rev_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_rev
thf(fact_900_zip__rev,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s5949480688510517322um_a_b @ Ys ) )
     => ( ( zip_Su8953288692215371525um_a_b @ ( rev_Sum_sum_a_b @ Xs ) @ ( rev_Sum_sum_a_b @ Ys ) )
        = ( rev_Pr3063422751929003795um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_901_zip__rev,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_P911570089447184179um_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s8239069388891432863um_a_b @ Ys ) )
     => ( ( zip_Su8837167750138406798um_a_b @ ( rev_Sum_sum_a_b @ Xs ) @ ( rev_Pr3063422751929003795um_a_b @ Ys ) )
        = ( rev_Pr807437698542170006um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_902_zip__rev,axiom,
    ! [Xs: list_Sum_sum_a_b,Ys: list_nat] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_Sum_sum_a_b_nat @ ( rev_Sum_sum_a_b @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr2989361455954602557_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_903_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_P911570089447184179um_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8239069388891432863um_a_b @ Ys ) )
     => ( ( zip_na2369391571721746148um_a_b @ ( rev_nat @ Xs ) @ ( rev_Pr3063422751929003795um_a_b @ Ys ) )
        = ( rev_Pr3113145760173511916um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_904_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_Sum_sum_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5949480688510517322um_a_b @ Ys ) )
     => ( ( zip_nat_Sum_sum_a_b @ ( rev_nat @ Xs ) @ ( rev_Sum_sum_a_b @ Ys ) )
        = ( rev_Pr2572044209074858301um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_905_zip__rev,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( zip_nat_nat @ ( rev_nat @ Xs ) @ ( rev_nat @ Ys ) )
        = ( rev_Pr6102188148953555047at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ).

% zip_rev
thf(fact_906_can__select__def,axiom,
    ( can_se2965781163881960362um_a_b
    = ( ^ [P3: produc4970237071031658413um_a_b > $o,A7: set_Pr7446880501794416397um_a_b] :
        ? [X4: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X4 @ A7 )
          & ( P3 @ X4 )
          & ! [Y5: produc4970237071031658413um_a_b] :
              ( ( ( member6480126262776431190um_a_b @ Y5 @ A7 )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_907_can__select__def,axiom,
    ( can_select_nat
    = ( ^ [P3: nat > $o,A7: set_nat] :
        ? [X4: nat] :
          ( ( member_nat @ X4 @ A7 )
          & ( P3 @ X4 )
          & ! [Y5: nat] :
              ( ( ( member_nat @ Y5 @ A7 )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_908_can__select__def,axiom,
    ( can_se6158366165627607252_b_nat
    = ( ^ [P3: produc6336967135452217047_b_nat > $o,A7: set_Pr7634312471471294263_b_nat] :
        ? [X4: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X4 @ A7 )
          & ( P3 @ X4 )
          & ! [Y5: produc6336967135452217047_b_nat] :
              ( ( ( member8290135809484590976_b_nat @ Y5 @ A7 )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_909_can__select__def,axiom,
    ( can_se7268299864462161813um_a_b
    = ( ^ [P3: produc7779921308951573458um_a_b > $o,A7: set_Pr5956926064383532552um_a_b] :
        ? [X4: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X4 @ A7 )
          & ( P3 @ X4 )
          & ! [Y5: produc7779921308951573458um_a_b] :
              ( ( ( member4052926750399300073um_a_b @ Y5 @ A7 )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_910_can__select__def,axiom,
    ( can_se5741048918747862996um_a_b
    = ( ^ [P3: produc5919649888572472791um_a_b > $o,A7: set_Pr4241354594808546871um_a_b] :
        ? [X4: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X4 @ A7 )
          & ( P3 @ X4 )
          & ! [Y5: produc5919649888572472791um_a_b] :
              ( ( ( member7872818562604846720um_a_b @ Y5 @ A7 )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_911_can__select__def,axiom,
    ( can_se4754832747099445502at_nat
    = ( ^ [P3: product_prod_nat_nat > $o,A7: set_Pr1261947904930325089at_nat] :
        ? [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A7 )
          & ( P3 @ X4 )
          & ! [Y5: product_prod_nat_nat] :
              ( ( ( member8440522571783428010at_nat @ Y5 @ A7 )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_912_can__select__def,axiom,
    ( can_se4659341492078941673um_a_b
    = ( ^ [P3: sum_sum_a_b > $o,A7: set_Sum_sum_a_b] :
        ? [X4: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X4 @ A7 )
          & ( P3 @ X4 )
          & ! [Y5: sum_sum_a_b] :
              ( ( ( member_Sum_sum_a_b @ Y5 @ A7 )
                & ( P3 @ Y5 ) )
             => ( Y5 = X4 ) ) ) ) ) ).

% can_select_def
thf(fact_913_lexord__irreflexive,axiom,
    ! [R3: set_Pr7446880501794416397um_a_b,Xs: list_Sum_sum_a_b] :
      ( ! [X5: sum_sum_a_b] :
          ~ ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ X5 ) @ R3 )
     => ~ ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs @ Xs ) @ ( lexord_Sum_sum_a_b @ R3 ) ) ) ).

% lexord_irreflexive
thf(fact_914_lexord__irreflexive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X5: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R3 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lexord_nat @ R3 ) ) ) ).

% lexord_irreflexive
thf(fact_915_lexord__linear,axiom,
    ! [R3: set_Pr7446880501794416397um_a_b,X: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
      ( ! [A: sum_sum_a_b,B: sum_sum_a_b] :
          ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ A @ B ) @ R3 )
          | ( A = B )
          | ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ B @ A ) @ R3 ) )
     => ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ X @ Y ) @ ( lexord_Sum_sum_a_b @ R3 ) )
        | ( X = Y )
        | ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Y @ X ) @ ( lexord_Sum_sum_a_b @ R3 ) ) ) ) ).

% lexord_linear
thf(fact_916_lexord__linear,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,X: list_nat,Y: list_nat] :
      ( ! [A: nat,B: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R3 )
          | ( A = B )
          | ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B @ A ) @ R3 ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R3 ) )
        | ( X = Y )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Y @ X ) @ ( lexord_nat @ R3 ) ) ) ) ).

% lexord_linear
thf(fact_917_lenlex__irreflexive,axiom,
    ! [R3: set_Pr7446880501794416397um_a_b,Xs: list_Sum_sum_a_b] :
      ( ! [X5: sum_sum_a_b] :
          ~ ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ X5 ) @ R3 )
     => ~ ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs @ Xs ) @ ( lenlex_Sum_sum_a_b @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_918_lenlex__irreflexive,axiom,
    ! [R3: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ! [X5: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R3 )
     => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Xs ) @ ( lenlex_nat @ R3 ) ) ) ).

% lenlex_irreflexive
thf(fact_919_lexord__partial__trans,axiom,
    ! [Xs: list_P911570089447184179um_a_b,R3: set_Pr8399320699097828935um_a_b,Ys: list_P911570089447184179um_a_b,Zs: list_P911570089447184179um_a_b] :
      ( ! [X5: produc4970237071031658413um_a_b,Y3: produc4970237071031658413um_a_b,Z3: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X5 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
         => ( ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ X5 @ Y3 ) @ R3 )
           => ( ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ Y3 @ Z3 ) @ R3 )
             => ( member1616855488514259088um_a_b @ ( produc1529531236301088727um_a_b @ X5 @ Z3 ) @ R3 ) ) ) )
     => ( ( member6189382312003533840um_a_b @ ( produc2669586549142030935um_a_b @ Xs @ Ys ) @ ( lexord3869047401015908384um_a_b @ R3 ) )
       => ( ( member6189382312003533840um_a_b @ ( produc2669586549142030935um_a_b @ Ys @ Zs ) @ ( lexord3869047401015908384um_a_b @ R3 ) )
         => ( member6189382312003533840um_a_b @ ( produc2669586549142030935um_a_b @ Xs @ Zs ) @ ( lexord3869047401015908384um_a_b @ R3 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_920_lexord__partial__trans,axiom,
    ! [Xs: list_P667245297671117661_b_nat,R3: set_Pr1916421491760925895_b_nat,Ys: list_P667245297671117661_b_nat,Zs: list_P667245297671117661_b_nat] :
      ( ! [X5: produc6336967135452217047_b_nat,Y3: produc6336967135452217047_b_nat,Z3: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X5 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
         => ( ( member681441685064050448_b_nat @ ( produc490312431154969431_b_nat @ X5 @ Y3 ) @ R3 )
           => ( ( member681441685064050448_b_nat @ ( produc490312431154969431_b_nat @ Y3 @ Z3 ) @ R3 )
             => ( member681441685064050448_b_nat @ ( produc490312431154969431_b_nat @ X5 @ Z3 ) @ R3 ) ) ) )
     => ( ( member2390596324953219984_b_nat @ ( produc8028711988652938967_b_nat @ Xs @ Ys ) @ ( lexord6550050937452159306_b_nat @ R3 ) )
       => ( ( member2390596324953219984_b_nat @ ( produc8028711988652938967_b_nat @ Ys @ Zs ) @ ( lexord6550050937452159306_b_nat @ R3 ) )
         => ( member2390596324953219984_b_nat @ ( produc8028711988652938967_b_nat @ Xs @ Zs ) @ ( lexord6550050937452159306_b_nat @ R3 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_921_lexord__partial__trans,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,R3: set_Pr3875401747812976357um_a_b,Ys: list_P5598968436384720482um_a_b,Zs: list_P5598968436384720482um_a_b] :
      ( ! [X5: produc7779921308951573458um_a_b,Y3: produc7779921308951573458um_a_b,Z3: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X5 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
         => ( ( member885238345412010542um_a_b @ ( produc8504789778075864509um_a_b @ X5 @ Y3 ) @ R3 )
           => ( ( member885238345412010542um_a_b @ ( produc8504789778075864509um_a_b @ Y3 @ Z3 ) @ R3 )
             => ( member885238345412010542um_a_b @ ( produc8504789778075864509um_a_b @ X5 @ Z3 ) @ R3 ) ) ) )
     => ( ( member7269071352107997646um_a_b @ ( produc8507494088392525149um_a_b @ Xs @ Ys ) @ ( lexord6315165880653629343um_a_b @ R3 ) )
       => ( ( member7269071352107997646um_a_b @ ( produc8507494088392525149um_a_b @ Ys @ Zs ) @ ( lexord6315165880653629343um_a_b @ R3 ) )
         => ( member7269071352107997646um_a_b @ ( produc8507494088392525149um_a_b @ Xs @ Zs ) @ ( lexord6315165880653629343um_a_b @ R3 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_922_lexord__partial__trans,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,R3: set_Pr6889002383187201223um_a_b,Ys: list_P6497659457863146077um_a_b,Zs: list_P6497659457863146077um_a_b] :
      ( ! [X5: produc5919649888572472791um_a_b,Y3: produc5919649888572472791um_a_b,Z3: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X5 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
         => ( ( member6178717281470555920um_a_b @ ( produc8963228657344478039um_a_b @ X5 @ Y3 ) @ R3 )
           => ( ( member6178717281470555920um_a_b @ ( produc8963228657344478039um_a_b @ Y3 @ Z3 ) @ R3 )
             => ( member6178717281470555920um_a_b @ ( produc8963228657344478039um_a_b @ X5 @ Z3 ) @ R3 ) ) ) )
     => ( ( member5972188106405192592um_a_b @ ( produc8821755967955457751um_a_b @ Xs @ Ys ) @ ( lexord6132733690572415050um_a_b @ R3 ) )
       => ( ( member5972188106405192592um_a_b @ ( produc8821755967955457751um_a_b @ Ys @ Zs ) @ ( lexord6132733690572415050um_a_b @ R3 ) )
         => ( member5972188106405192592um_a_b @ ( produc8821755967955457751um_a_b @ Xs @ Zs ) @ ( lexord6132733690572415050um_a_b @ R3 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_923_lexord__partial__trans,axiom,
    ! [Xs: list_P6011104703257516679at_nat,R3: set_Pr8693737435421807431at_nat,Ys: list_P6011104703257516679at_nat,Zs: list_P6011104703257516679at_nat] :
      ( ! [X5: product_prod_nat_nat,Y3: product_prod_nat_nat,Z3: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Y3 ) @ R3 )
           => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ Y3 @ Z3 ) @ R3 )
             => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X5 @ Z3 ) @ R3 ) ) ) )
     => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Ys ) @ ( lexord2841853652668343668at_nat @ R3 ) )
       => ( ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Ys @ Zs ) @ ( lexord2841853652668343668at_nat @ R3 ) )
         => ( member6693912407220327184at_nat @ ( produc5943733680697469783at_nat @ Xs @ Zs ) @ ( lexord2841853652668343668at_nat @ R3 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_924_lexord__partial__trans,axiom,
    ! [Xs: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b,Ys: list_Sum_sum_a_b,Zs: list_Sum_sum_a_b] :
      ( ! [X5: sum_sum_a_b,Y3: sum_sum_a_b,Z3: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X5 @ ( set_Sum_sum_a_b2 @ Xs ) )
         => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ Y3 ) @ R3 )
           => ( ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ Y3 @ Z3 ) @ R3 )
             => ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ X5 @ Z3 ) @ R3 ) ) ) )
     => ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs @ Ys ) @ ( lexord_Sum_sum_a_b @ R3 ) )
       => ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Ys @ Zs ) @ ( lexord_Sum_sum_a_b @ R3 ) )
         => ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Xs @ Zs ) @ ( lexord_Sum_sum_a_b @ R3 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_925_lexord__partial__trans,axiom,
    ! [Xs: list_nat,R3: set_Pr1261947904930325089at_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X5: nat,Y3: nat,Z3: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R3 )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Z3 ) @ R3 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Z3 ) @ R3 ) ) ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexord_nat @ R3 ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R3 ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs ) @ ( lexord_nat @ R3 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_926_lexord__lex,axiom,
    ! [X: list_nat,Y: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lex_nat @ R3 ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X @ Y ) @ ( lexord_nat @ R3 ) )
        & ( ( size_size_list_nat @ X )
          = ( size_size_list_nat @ Y ) ) ) ) ).

% lexord_lex
thf(fact_927_Cons__lenlex__iff,axiom,
    ! [M2: sum_sum_a_b,Ms: list_Sum_sum_a_b,N: sum_sum_a_b,Ns: list_Sum_sum_a_b,R3: set_Pr7446880501794416397um_a_b] :
      ( ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ ( cons_Sum_sum_a_b @ M2 @ Ms ) @ ( cons_Sum_sum_a_b @ N @ Ns ) ) @ ( lenlex_Sum_sum_a_b @ R3 ) )
      = ( ( ord_less_nat @ ( size_s5949480688510517322um_a_b @ Ms ) @ ( size_s5949480688510517322um_a_b @ Ns ) )
        | ( ( ( size_s5949480688510517322um_a_b @ Ms )
            = ( size_s5949480688510517322um_a_b @ Ns ) )
          & ( member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member4891539924951922422um_a_b @ ( produc5431536226049688709um_a_b @ Ms @ Ns ) @ ( lenlex_Sum_sum_a_b @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_928_Cons__lenlex__iff,axiom,
    ! [M2: nat,Ms: list_nat,N: nat,Ns: list_nat,R3: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ M2 @ Ms ) @ ( cons_nat @ N @ Ns ) ) @ ( lenlex_nat @ R3 ) )
      = ( ( ord_less_nat @ ( size_size_list_nat @ Ms ) @ ( size_size_list_nat @ Ns ) )
        | ( ( ( size_size_list_nat @ Ms )
            = ( size_size_list_nat @ Ns ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M2 @ N ) @ R3 ) )
        | ( ( M2 = N )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ms @ Ns ) @ ( lenlex_nat @ R3 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_929_lexn__length,axiom,
    ! [Xs: list_nat,Ys: list_nat,R3: set_Pr1261947904930325089at_nat,N: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( lexn_nat @ R3 @ N ) )
     => ( ( ( size_size_list_nat @ Xs )
          = N )
        & ( ( size_size_list_nat @ Ys )
          = N ) ) ) ).

% lexn_length
thf(fact_930_lexord__sufI,axiom,
    ! [U: list_nat,W2: list_nat,R3: set_Pr1261947904930325089at_nat,V: list_nat,Z2: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ W2 ) @ ( lexord_nat @ R3 ) )
     => ( ( ord_less_eq_nat @ ( size_size_list_nat @ W2 ) @ ( size_size_list_nat @ U ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ V ) @ ( append_nat @ W2 @ Z2 ) ) @ ( lexord_nat @ R3 ) ) ) ) ).

% lexord_sufI
thf(fact_931_Set_Omember__remove,axiom,
    ! [X: produc4970237071031658413um_a_b,Y: produc4970237071031658413um_a_b,A5: set_Pr7446880501794416397um_a_b] :
      ( ( member6480126262776431190um_a_b @ X @ ( remove1888975756299800594um_a_b @ Y @ A5 ) )
      = ( ( member6480126262776431190um_a_b @ X @ A5 )
        & ( X != Y ) ) ) ).

% Set.member_remove
thf(fact_932_Set_Omember__remove,axiom,
    ! [X: nat,Y: nat,A5: set_nat] :
      ( ( member_nat @ X @ ( remove_nat @ Y @ A5 ) )
      = ( ( member_nat @ X @ A5 )
        & ( X != Y ) ) ) ).

% Set.member_remove
thf(fact_933_Set_Omember__remove,axiom,
    ! [X: produc6336967135452217047_b_nat,Y: produc6336967135452217047_b_nat,A5: set_Pr7634312471471294263_b_nat] :
      ( ( member8290135809484590976_b_nat @ X @ ( remove7448331093040006460_b_nat @ Y @ A5 ) )
      = ( ( member8290135809484590976_b_nat @ X @ A5 )
        & ( X != Y ) ) ) ).

% Set.member_remove
thf(fact_934_Set_Omember__remove,axiom,
    ! [X: produc7779921308951573458um_a_b,Y: produc7779921308951573458um_a_b,A5: set_Pr5956926064383532552um_a_b] :
      ( ( member4052926750399300073um_a_b @ X @ ( remove3659418628615625005um_a_b @ Y @ A5 ) )
      = ( ( member4052926750399300073um_a_b @ X @ A5 )
        & ( X != Y ) ) ) ).

% Set.member_remove
thf(fact_935_Set_Omember__remove,axiom,
    ! [X: produc5919649888572472791um_a_b,Y: produc5919649888572472791um_a_b,A5: set_Pr4241354594808546871um_a_b] :
      ( ( member7872818562604846720um_a_b @ X @ ( remove7031013846160262204um_a_b @ Y @ A5 ) )
      = ( ( member7872818562604846720um_a_b @ X @ A5 )
        & ( X != Y ) ) ) ).

% Set.member_remove
thf(fact_936_Set_Omember__remove,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat,A5: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( remove6466555014256735590at_nat @ Y @ A5 ) )
      = ( ( member8440522571783428010at_nat @ X @ A5 )
        & ( X != Y ) ) ) ).

% Set.member_remove
thf(fact_937_Set_Omember__remove,axiom,
    ! [X: sum_sum_a_b,Y: sum_sum_a_b,A5: set_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ X @ ( remove_Sum_sum_a_b @ Y @ A5 ) )
      = ( ( member_Sum_sum_a_b @ X @ A5 )
        & ( X != Y ) ) ) ).

% Set.member_remove
thf(fact_938_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_939_zip__append,axiom,
    ! [Xs: list_Sum_sum_a_b,Us: list_Sum_sum_a_b,Ys: list_Sum_sum_a_b,Vs: list_Sum_sum_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s5949480688510517322um_a_b @ Us ) )
     => ( ( zip_Su8953288692215371525um_a_b @ ( append_Sum_sum_a_b @ Xs @ Ys ) @ ( append_Sum_sum_a_b @ Us @ Vs ) )
        = ( append482101194363978952um_a_b @ ( zip_Su8953288692215371525um_a_b @ Xs @ Us ) @ ( zip_Su8953288692215371525um_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_940_zip__append,axiom,
    ! [Xs: list_Sum_sum_a_b,Us: list_P911570089447184179um_a_b,Ys: list_Sum_sum_a_b,Vs: list_P911570089447184179um_a_b] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_s8239069388891432863um_a_b @ Us ) )
     => ( ( zip_Su8837167750138406798um_a_b @ ( append_Sum_sum_a_b @ Xs @ Ys ) @ ( append482101194363978952um_a_b @ Us @ Vs ) )
        = ( append2550848805994021281um_a_b @ ( zip_Su8837167750138406798um_a_b @ Xs @ Us ) @ ( zip_Su8837167750138406798um_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_941_zip__append,axiom,
    ! [Xs: list_Sum_sum_a_b,Us: list_nat,Ys: list_Sum_sum_a_b,Vs: list_nat] :
      ( ( ( size_s5949480688510517322um_a_b @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Sum_sum_a_b_nat @ ( append_Sum_sum_a_b @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append7648079923110641138_b_nat @ ( zip_Sum_sum_a_b_nat @ Xs @ Us ) @ ( zip_Sum_sum_a_b_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_942_zip__append,axiom,
    ! [Xs: list_nat,Us: list_P911570089447184179um_a_b,Ys: list_nat,Vs: list_P911570089447184179um_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s8239069388891432863um_a_b @ Us ) )
     => ( ( zip_na2369391571721746148um_a_b @ ( append_nat @ Xs @ Ys ) @ ( append482101194363978952um_a_b @ Us @ Vs ) )
        = ( append629602400917629175um_a_b @ ( zip_na2369391571721746148um_a_b @ Xs @ Us ) @ ( zip_na2369391571721746148um_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_943_zip__append,axiom,
    ! [Xs: list_nat,Us: list_Sum_sum_a_b,Ys: list_nat,Vs: list_Sum_sum_a_b] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5949480688510517322um_a_b @ Us ) )
     => ( ( zip_nat_Sum_sum_a_b @ ( append_nat @ Xs @ Ys ) @ ( append_Sum_sum_a_b @ Us @ Vs ) )
        = ( append7230762676230896882um_a_b @ ( zip_nat_Sum_sum_a_b @ Xs @ Us ) @ ( zip_nat_Sum_sum_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_944_zip__append,axiom,
    ! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_945_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_946_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_947_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_948_less__not__refl3,axiom,
    ! [S3: nat,T2: nat] :
      ( ( ord_less_nat @ S3 @ T2 )
     => ( S3 != T2 ) ) ).

% less_not_refl3
thf(fact_949_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_950_nat__less__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ( P2 @ M3 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N ) ) ).

% nat_less_induct
thf(fact_951_infinite__descent,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ~ ( P2 @ N3 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
              & ~ ( P2 @ M3 ) ) )
     => ( P2 @ N ) ) ).

% infinite_descent
thf(fact_952_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_953_gt__ex,axiom,
    ! [X: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X @ X_12 ) ).

% gt_ex
thf(fact_954_less__imp__neq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% less_imp_neq
thf(fact_955_order_Oasym,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A3 ) ) ).

% order.asym
thf(fact_956_ord__eq__less__trans,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( A3 = B3 )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ord_less_nat @ A3 @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_957_ord__less__eq__trans,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( B3 = C2 )
       => ( ord_less_nat @ A3 @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_958_less__induct,axiom,
    ! [P2: nat > $o,A3: nat] :
      ( ! [X5: nat] :
          ( ! [Y7: nat] :
              ( ( ord_less_nat @ Y7 @ X5 )
             => ( P2 @ Y7 ) )
         => ( P2 @ X5 ) )
     => ( P2 @ A3 ) ) ).

% less_induct
thf(fact_959_antisym__conv3,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_nat @ Y @ X )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv3
thf(fact_960_linorder__cases,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( X != Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_cases
thf(fact_961_dual__order_Oasym,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ B3 @ A3 )
     => ~ ( ord_less_nat @ A3 @ B3 ) ) ).

% dual_order.asym
thf(fact_962_dual__order_Oirrefl,axiom,
    ! [A3: nat] :
      ~ ( ord_less_nat @ A3 @ A3 ) ).

% dual_order.irrefl
thf(fact_963_exists__least__iff,axiom,
    ( ( ^ [P4: nat > $o] :
        ? [X8: nat] : ( P4 @ X8 ) )
    = ( ^ [P3: nat > $o] :
        ? [N4: nat] :
          ( ( P3 @ N4 )
          & ! [M4: nat] :
              ( ( ord_less_nat @ M4 @ N4 )
             => ~ ( P3 @ M4 ) ) ) ) ) ).

% exists_least_iff
thf(fact_964_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A3: nat,B3: nat] :
      ( ! [A: nat,B: nat] :
          ( ( ord_less_nat @ A @ B )
         => ( P2 @ A @ B ) )
     => ( ! [A: nat] : ( P2 @ A @ A )
       => ( ! [A: nat,B: nat] :
              ( ( P2 @ B @ A )
             => ( P2 @ A @ B ) )
         => ( P2 @ A3 @ B3 ) ) ) ) ).

% linorder_less_wlog
thf(fact_965_order_Ostrict__trans,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ord_less_nat @ A3 @ C2 ) ) ) ).

% order.strict_trans
thf(fact_966_not__less__iff__gr__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ( ord_less_nat @ Y @ X )
        | ( X = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_967_dual__order_Ostrict__trans,axiom,
    ! [B3: nat,A3: nat,C2: nat] :
      ( ( ord_less_nat @ B3 @ A3 )
     => ( ( ord_less_nat @ C2 @ B3 )
       => ( ord_less_nat @ C2 @ A3 ) ) ) ).

% dual_order.strict_trans
thf(fact_968_order_Ostrict__implies__not__eq,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( A3 != B3 ) ) ).

% order.strict_implies_not_eq
thf(fact_969_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ B3 @ A3 )
     => ( A3 != B3 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_970_linorder__neqE,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE
thf(fact_971_order__less__asym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_asym
thf(fact_972_linorder__neq__iff,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
      = ( ( ord_less_nat @ X @ Y )
        | ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neq_iff
thf(fact_973_order__less__asym_H,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ~ ( ord_less_nat @ B3 @ A3 ) ) ).

% order_less_asym'
thf(fact_974_order__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_trans
thf(fact_975_ord__eq__less__subst,axiom,
    ! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( A3
        = ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_976_ord__less__eq__subst,axiom,
    ! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ( F @ B3 )
          = C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_977_order__less__irrefl,axiom,
    ! [X: nat] :
      ~ ( ord_less_nat @ X @ X ) ).

% order_less_irrefl
thf(fact_978_order__less__subst1,axiom,
    ! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_979_order__less__subst2,axiom,
    ! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_980_order__less__not__sym,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_not_sym
thf(fact_981_order__less__imp__triv,axiom,
    ! [X: nat,Y: nat,P2: $o] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ X )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_982_linorder__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
      | ( X = Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_less_linear
thf(fact_983_order__less__imp__not__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( X != Y ) ) ).

% order_less_imp_not_eq
thf(fact_984_order__less__imp__not__eq2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( Y != X ) ) ).

% order_less_imp_not_eq2
thf(fact_985_order__less__imp__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ~ ( ord_less_nat @ Y @ X ) ) ).

% order_less_imp_not_less
thf(fact_986_order__le__imp__less__or__eq,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_set_a @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_987_order__le__imp__less__or__eq,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ X @ Y )
        | ( X = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_988_linorder__le__less__linear,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
      | ( ord_less_nat @ Y @ X ) ) ).

% linorder_le_less_linear
thf(fact_989_order__less__le__subst2,axiom,
    ! [A3: nat,B3: nat,F: nat > set_a,C2: set_a] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_set_a @ ( F @ B3 ) @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_990_order__less__le__subst2,axiom,
    ! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_991_order__less__le__subst1,axiom,
    ! [A3: set_a,F: set_a > set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_set_a @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_992_order__less__le__subst1,axiom,
    ! [A3: nat,F: set_a > nat,B3: set_a,C2: set_a] :
      ( ( ord_less_nat @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_993_order__less__le__subst1,axiom,
    ! [A3: set_a,F: nat > set_a,B3: nat,C2: nat] :
      ( ( ord_less_set_a @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_994_order__less__le__subst1,axiom,
    ! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_995_order__le__less__subst2,axiom,
    ! [A3: set_a,B3: set_a,F: set_a > set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ord_less_set_a @ ( F @ B3 ) @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_996_order__le__less__subst2,axiom,
    ! [A3: set_a,B3: set_a,F: set_a > nat,C2: nat] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X5: set_a,Y3: set_a] :
              ( ( ord_less_eq_set_a @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_997_order__le__less__subst2,axiom,
    ! [A3: nat,B3: nat,F: nat > set_a,C2: set_a] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_set_a @ ( F @ B3 ) @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_a @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_998_order__le__less__subst2,axiom,
    ! [A3: nat,B3: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ ( F @ B3 ) @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A3 ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_999_order__le__less__subst1,axiom,
    ! [A3: set_a,F: nat > set_a,B3: nat,C2: nat] :
      ( ( ord_less_eq_set_a @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_set_a @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_set_a @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1000_order__le__less__subst1,axiom,
    ! [A3: nat,F: nat > nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ ( F @ B3 ) )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A3 @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1001_order__less__le__trans,axiom,
    ! [X: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ Y @ Z2 )
       => ( ord_less_set_a @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1002_order__less__le__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1003_order__le__less__trans,axiom,
    ! [X: set_a,Y: set_a,Z2: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ord_less_set_a @ Y @ Z2 )
       => ( ord_less_set_a @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1004_order__le__less__trans,axiom,
    ! [X: nat,Y: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ord_less_nat @ Y @ Z2 )
       => ( ord_less_nat @ X @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1005_order__neq__le__trans,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( A3 != B3 )
     => ( ( ord_less_eq_set_a @ A3 @ B3 )
       => ( ord_less_set_a @ A3 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_1006_order__neq__le__trans,axiom,
    ! [A3: nat,B3: nat] :
      ( ( A3 != B3 )
     => ( ( ord_less_eq_nat @ A3 @ B3 )
       => ( ord_less_nat @ A3 @ B3 ) ) ) ).

% order_neq_le_trans
thf(fact_1007_order__le__neq__trans,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( A3 != B3 )
       => ( ord_less_set_a @ A3 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_1008_order__le__neq__trans,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( A3 != B3 )
       => ( ord_less_nat @ A3 @ B3 ) ) ) ).

% order_le_neq_trans
thf(fact_1009_order__less__imp__le,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_set_a @ X @ Y )
     => ( ord_less_eq_set_a @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1010_order__less__imp__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ X @ Y ) ) ).

% order_less_imp_le
thf(fact_1011_linorder__not__less,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X @ Y ) )
      = ( ord_less_eq_nat @ Y @ X ) ) ).

% linorder_not_less
thf(fact_1012_linorder__not__le,axiom,
    ! [X: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X @ Y ) )
      = ( ord_less_nat @ Y @ X ) ) ).

% linorder_not_le
thf(fact_1013_order__less__le,axiom,
    ( ord_less_set_a
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1014_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ( X4 != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1015_order__le__less,axiom,
    ( ord_less_eq_set_a
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( ord_less_set_a @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1016_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_nat @ X4 @ Y5 )
          | ( X4 = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1017_dual__order_Ostrict__implies__order,axiom,
    ! [B3: set_a,A3: set_a] :
      ( ( ord_less_set_a @ B3 @ A3 )
     => ( ord_less_eq_set_a @ B3 @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1018_dual__order_Ostrict__implies__order,axiom,
    ! [B3: nat,A3: nat] :
      ( ( ord_less_nat @ B3 @ A3 )
     => ( ord_less_eq_nat @ B3 @ A3 ) ) ).

% dual_order.strict_implies_order
thf(fact_1019_order_Ostrict__implies__order,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ord_less_set_a @ A3 @ B3 )
     => ( ord_less_eq_set_a @ A3 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_1020_order_Ostrict__implies__order,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ord_less_eq_nat @ A3 @ B3 ) ) ).

% order.strict_implies_order
thf(fact_1021_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_a
    = ( ^ [B6: set_a,A6: set_a] :
          ( ( ord_less_eq_set_a @ B6 @ A6 )
          & ~ ( ord_less_eq_set_a @ A6 @ B6 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1022_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ~ ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1023_dual__order_Ostrict__trans2,axiom,
    ! [B3: set_a,A3: set_a,C2: set_a] :
      ( ( ord_less_set_a @ B3 @ A3 )
     => ( ( ord_less_eq_set_a @ C2 @ B3 )
       => ( ord_less_set_a @ C2 @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1024_dual__order_Ostrict__trans2,axiom,
    ! [B3: nat,A3: nat,C2: nat] :
      ( ( ord_less_nat @ B3 @ A3 )
     => ( ( ord_less_eq_nat @ C2 @ B3 )
       => ( ord_less_nat @ C2 @ A3 ) ) ) ).

% dual_order.strict_trans2
thf(fact_1025_dual__order_Ostrict__trans1,axiom,
    ! [B3: set_a,A3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ B3 @ A3 )
     => ( ( ord_less_set_a @ C2 @ B3 )
       => ( ord_less_set_a @ C2 @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1026_dual__order_Ostrict__trans1,axiom,
    ! [B3: nat,A3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B3 @ A3 )
     => ( ( ord_less_nat @ C2 @ B3 )
       => ( ord_less_nat @ C2 @ A3 ) ) ) ).

% dual_order.strict_trans1
thf(fact_1027_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_a
    = ( ^ [B6: set_a,A6: set_a] :
          ( ( ord_less_eq_set_a @ B6 @ A6 )
          & ( A6 != B6 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1028_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ( A6 != B6 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1029_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_a
    = ( ^ [B6: set_a,A6: set_a] :
          ( ( ord_less_set_a @ B6 @ A6 )
          | ( A6 = B6 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1030_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( ord_less_nat @ B6 @ A6 )
          | ( A6 = B6 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1031_order_Ostrict__iff__not,axiom,
    ( ord_less_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( ( ord_less_eq_set_a @ A6 @ B6 )
          & ~ ( ord_less_eq_set_a @ B6 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1032_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ~ ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1033_order_Ostrict__trans2,axiom,
    ! [A3: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_set_a @ A3 @ B3 )
     => ( ( ord_less_eq_set_a @ B3 @ C2 )
       => ( ord_less_set_a @ A3 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1034_order_Ostrict__trans2,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ord_less_nat @ A3 @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_1035_order_Ostrict__trans1,axiom,
    ! [A3: set_a,B3: set_a,C2: set_a] :
      ( ( ord_less_eq_set_a @ A3 @ B3 )
     => ( ( ord_less_set_a @ B3 @ C2 )
       => ( ord_less_set_a @ A3 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1036_order_Ostrict__trans1,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A3 @ B3 )
     => ( ( ord_less_nat @ B3 @ C2 )
       => ( ord_less_nat @ A3 @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_1037_order_Ostrict__iff__order,axiom,
    ( ord_less_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( ( ord_less_eq_set_a @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1038_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1039_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A6: set_a,B6: set_a] :
          ( ( ord_less_set_a @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1040_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_nat @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1041_not__le__imp__less,axiom,
    ! [Y: nat,X: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X )
     => ( ord_less_nat @ X @ Y ) ) ).

% not_le_imp_less
thf(fact_1042_less__le__not__le,axiom,
    ( ord_less_set_a
    = ( ^ [X4: set_a,Y5: set_a] :
          ( ( ord_less_eq_set_a @ X4 @ Y5 )
          & ~ ( ord_less_eq_set_a @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1043_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X4: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X4 @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X4 ) ) ) ) ).

% less_le_not_le
thf(fact_1044_antisym__conv2,axiom,
    ! [X: set_a,Y: set_a] :
      ( ( ord_less_eq_set_a @ X @ Y )
     => ( ( ~ ( ord_less_set_a @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1045_antisym__conv2,axiom,
    ! [X: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X @ Y )
     => ( ( ~ ( ord_less_nat @ X @ Y ) )
        = ( X = Y ) ) ) ).

% antisym_conv2
thf(fact_1046_antisym__conv1,axiom,
    ! [X: set_a,Y: set_a] :
      ( ~ ( ord_less_set_a @ X @ Y )
     => ( ( ord_less_eq_set_a @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1047_antisym__conv1,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ( ord_less_eq_nat @ X @ Y )
        = ( X = Y ) ) ) ).

% antisym_conv1
thf(fact_1048_nless__le,axiom,
    ! [A3: set_a,B3: set_a] :
      ( ( ~ ( ord_less_set_a @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_set_a @ A3 @ B3 )
        | ( A3 = B3 ) ) ) ).

% nless_le
thf(fact_1049_nless__le,axiom,
    ! [A3: nat,B3: nat] :
      ( ( ~ ( ord_less_nat @ A3 @ B3 ) )
      = ( ~ ( ord_less_eq_nat @ A3 @ B3 )
        | ( A3 = B3 ) ) ) ).

% nless_le
thf(fact_1050_leI,axiom,
    ! [X: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X @ Y )
     => ( ord_less_eq_nat @ Y @ X ) ) ).

% leI
thf(fact_1051_leD,axiom,
    ! [Y: set_a,X: set_a] :
      ( ( ord_less_eq_set_a @ Y @ X )
     => ~ ( ord_less_set_a @ X @ Y ) ) ).

% leD
thf(fact_1052_leD,axiom,
    ! [Y: nat,X: nat] :
      ( ( ord_less_eq_nat @ Y @ X )
     => ~ ( ord_less_nat @ X @ Y ) ) ).

% leD
thf(fact_1053_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J2: nat] :
      ( ! [I3: nat,J3: nat] :
          ( ( ord_less_nat @ I3 @ J3 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J2 ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1054_le__neq__implies__less,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( M2 != N )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1055_less__or__eq__imp__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1056_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N4: nat] :
          ( ( ord_less_nat @ M4 @ N4 )
          | ( M4 = N4 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1057_less__imp__le__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_imp_le_nat
thf(fact_1058_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N4: nat] :
          ( ( ord_less_eq_nat @ M4 @ N4 )
          & ( M4 != N4 ) ) ) ) ).

% nat_less_le
thf(fact_1059_length__induct,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys5: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys5 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P2 @ Ys5 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_1060_split__list__first__prop__iff,axiom,
    ! [Xs: list_P911570089447184179um_a_b,P2: produc4970237071031658413um_a_b > $o] :
      ( ( ? [X4: produc4970237071031658413um_a_b] :
            ( ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P911570089447184179um_a_b,X4: produc4970237071031658413um_a_b] :
            ( ? [Zs3: list_P911570089447184179um_a_b] :
                ( Xs
                = ( append482101194363978952um_a_b @ Ys2 @ ( cons_P4211151491411527779um_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc4970237071031658413um_a_b] :
                ( ( member6480126262776431190um_a_b @ Y5 @ ( set_Pr2436422074441463490um_a_b @ Ys2 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1061_split__list__first__prop__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,P2: sum_sum_a_b > $o] :
      ( ( ? [X4: sum_sum_a_b] :
            ( ( member_Sum_sum_a_b @ X4 @ ( set_Sum_sum_a_b2 @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_Sum_sum_a_b,X4: sum_sum_a_b] :
            ( ? [Zs3: list_Sum_sum_a_b] :
                ( Xs
                = ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: sum_sum_a_b] :
                ( ( member_Sum_sum_a_b @ Y5 @ ( set_Sum_sum_a_b2 @ Ys2 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1062_split__list__first__prop__iff,axiom,
    ! [Xs: list_P667245297671117661_b_nat,P2: produc6336967135452217047_b_nat > $o] :
      ( ( ? [X4: produc6336967135452217047_b_nat] :
            ( ( member8290135809484590976_b_nat @ X4 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P667245297671117661_b_nat,X4: produc6336967135452217047_b_nat] :
            ( ? [Zs3: list_P667245297671117661_b_nat] :
                ( Xs
                = ( append7648079923110641138_b_nat @ Ys2 @ ( cons_P4197526505774637965_b_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc6336967135452217047_b_nat] :
                ( ( member8290135809484590976_b_nat @ Y5 @ ( set_Pr176305205332118508_b_nat @ Ys2 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1063_split__list__first__prop__iff,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,P2: produc7779921308951573458um_a_b > $o] :
      ( ( ? [X4: produc7779921308951573458um_a_b] :
            ( ( member4052926750399300073um_a_b @ X4 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P5598968436384720482um_a_b,X4: produc7779921308951573458um_a_b] :
            ( ? [Zs3: list_P5598968436384720482um_a_b] :
                ( Xs
                = ( append629602400917629175um_a_b @ Ys2 @ ( cons_P7543864023423915932um_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc7779921308951573458um_a_b] :
                ( ( member4052926750399300073um_a_b @ Y5 @ ( set_Pr7399620718385288829um_a_b @ Ys2 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1064_split__list__first__prop__iff,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,P2: produc5919649888572472791um_a_b > $o] :
      ( ( ? [X4: produc5919649888572472791um_a_b] :
            ( ( member7872818562604846720um_a_b @ X4 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P6497659457863146077um_a_b,X4: produc5919649888572472791um_a_b] :
            ( ? [Zs3: list_P6497659457863146077um_a_b] :
                ( Xs
                = ( append7230762676230896882um_a_b @ Ys2 @ ( cons_P3780209258894893709um_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc5919649888572472791um_a_b] :
                ( ( member7872818562604846720um_a_b @ Y5 @ ( set_Pr8982359995307150060um_a_b @ Ys2 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1065_split__list__first__prop__iff,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o] :
      ( ( ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,X4: product_prod_nat_nat] :
            ( ? [Zs3: list_P6011104703257516679at_nat] :
                ( Xs
                = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Ys2 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1066_split__list__first__prop__iff,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_nat,X4: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs
                = ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( set_nat2 @ Ys2 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1067_split__list__last__prop__iff,axiom,
    ! [Xs: list_P911570089447184179um_a_b,P2: produc4970237071031658413um_a_b > $o] :
      ( ( ? [X4: produc4970237071031658413um_a_b] :
            ( ( member6480126262776431190um_a_b @ X4 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P911570089447184179um_a_b,X4: produc4970237071031658413um_a_b,Zs3: list_P911570089447184179um_a_b] :
            ( ( Xs
              = ( append482101194363978952um_a_b @ Ys2 @ ( cons_P4211151491411527779um_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc4970237071031658413um_a_b] :
                ( ( member6480126262776431190um_a_b @ Y5 @ ( set_Pr2436422074441463490um_a_b @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1068_split__list__last__prop__iff,axiom,
    ! [Xs: list_Sum_sum_a_b,P2: sum_sum_a_b > $o] :
      ( ( ? [X4: sum_sum_a_b] :
            ( ( member_Sum_sum_a_b @ X4 @ ( set_Sum_sum_a_b2 @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_Sum_sum_a_b,X4: sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: sum_sum_a_b] :
                ( ( member_Sum_sum_a_b @ Y5 @ ( set_Sum_sum_a_b2 @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1069_split__list__last__prop__iff,axiom,
    ! [Xs: list_P667245297671117661_b_nat,P2: produc6336967135452217047_b_nat > $o] :
      ( ( ? [X4: produc6336967135452217047_b_nat] :
            ( ( member8290135809484590976_b_nat @ X4 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P667245297671117661_b_nat,X4: produc6336967135452217047_b_nat,Zs3: list_P667245297671117661_b_nat] :
            ( ( Xs
              = ( append7648079923110641138_b_nat @ Ys2 @ ( cons_P4197526505774637965_b_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc6336967135452217047_b_nat] :
                ( ( member8290135809484590976_b_nat @ Y5 @ ( set_Pr176305205332118508_b_nat @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1070_split__list__last__prop__iff,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,P2: produc7779921308951573458um_a_b > $o] :
      ( ( ? [X4: produc7779921308951573458um_a_b] :
            ( ( member4052926750399300073um_a_b @ X4 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P5598968436384720482um_a_b,X4: produc7779921308951573458um_a_b,Zs3: list_P5598968436384720482um_a_b] :
            ( ( Xs
              = ( append629602400917629175um_a_b @ Ys2 @ ( cons_P7543864023423915932um_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc7779921308951573458um_a_b] :
                ( ( member4052926750399300073um_a_b @ Y5 @ ( set_Pr7399620718385288829um_a_b @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1071_split__list__last__prop__iff,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,P2: produc5919649888572472791um_a_b > $o] :
      ( ( ? [X4: produc5919649888572472791um_a_b] :
            ( ( member7872818562604846720um_a_b @ X4 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P6497659457863146077um_a_b,X4: produc5919649888572472791um_a_b,Zs3: list_P6497659457863146077um_a_b] :
            ( ( Xs
              = ( append7230762676230896882um_a_b @ Ys2 @ ( cons_P3780209258894893709um_a_b @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: produc5919649888572472791um_a_b] :
                ( ( member7872818562604846720um_a_b @ Y5 @ ( set_Pr8982359995307150060um_a_b @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1072_split__list__last__prop__iff,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o] :
      ( ( ? [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,X4: product_prod_nat_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1073_split__list__last__prop__iff,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_nat,X4: nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ( P2 @ X4 )
            & ! [Y5: nat] :
                ( ( member_nat @ Y5 @ ( set_nat2 @ Zs3 ) )
               => ~ ( P2 @ Y5 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1074_in__set__conv__decomp__first,axiom,
    ! [X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
      = ( ? [Ys2: list_P911570089447184179um_a_b,Zs3: list_P911570089447184179um_a_b] :
            ( ( Xs
              = ( append482101194363978952um_a_b @ Ys2 @ ( cons_P4211151491411527779um_a_b @ X @ Zs3 ) ) )
            & ~ ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1075_in__set__conv__decomp__first,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
      = ( ? [Ys2: list_Sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X @ Zs3 ) ) )
            & ~ ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1076_in__set__conv__decomp__first,axiom,
    ! [X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Xs ) )
      = ( ? [Ys2: list_P667245297671117661_b_nat,Zs3: list_P667245297671117661_b_nat] :
            ( ( Xs
              = ( append7648079923110641138_b_nat @ Ys2 @ ( cons_P4197526505774637965_b_nat @ X @ Zs3 ) ) )
            & ~ ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1077_in__set__conv__decomp__first,axiom,
    ! [X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
      = ( ? [Ys2: list_P5598968436384720482um_a_b,Zs3: list_P5598968436384720482um_a_b] :
            ( ( Xs
              = ( append629602400917629175um_a_b @ Ys2 @ ( cons_P7543864023423915932um_a_b @ X @ Zs3 ) ) )
            & ~ ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1078_in__set__conv__decomp__first,axiom,
    ! [X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
      = ( ? [Ys2: list_P6497659457863146077um_a_b,Zs3: list_P6497659457863146077um_a_b] :
            ( ( Xs
              = ( append7230762676230896882um_a_b @ Ys2 @ ( cons_P3780209258894893709um_a_b @ X @ Zs3 ) ) )
            & ~ ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1079_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
            & ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1080_in__set__conv__decomp__first,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1081_in__set__conv__decomp__last,axiom,
    ! [X: produc4970237071031658413um_a_b,Xs: list_P911570089447184179um_a_b] :
      ( ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
      = ( ? [Ys2: list_P911570089447184179um_a_b,Zs3: list_P911570089447184179um_a_b] :
            ( ( Xs
              = ( append482101194363978952um_a_b @ Ys2 @ ( cons_P4211151491411527779um_a_b @ X @ Zs3 ) ) )
            & ~ ( member6480126262776431190um_a_b @ X @ ( set_Pr2436422074441463490um_a_b @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1082_in__set__conv__decomp__last,axiom,
    ! [X: sum_sum_a_b,Xs: list_Sum_sum_a_b] :
      ( ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Xs ) )
      = ( ? [Ys2: list_Sum_sum_a_b,Zs3: list_Sum_sum_a_b] :
            ( ( Xs
              = ( append_Sum_sum_a_b @ Ys2 @ ( cons_Sum_sum_a_b @ X @ Zs3 ) ) )
            & ~ ( member_Sum_sum_a_b @ X @ ( set_Sum_sum_a_b2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1083_in__set__conv__decomp__last,axiom,
    ! [X: produc6336967135452217047_b_nat,Xs: list_P667245297671117661_b_nat] :
      ( ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Xs ) )
      = ( ? [Ys2: list_P667245297671117661_b_nat,Zs3: list_P667245297671117661_b_nat] :
            ( ( Xs
              = ( append7648079923110641138_b_nat @ Ys2 @ ( cons_P4197526505774637965_b_nat @ X @ Zs3 ) ) )
            & ~ ( member8290135809484590976_b_nat @ X @ ( set_Pr176305205332118508_b_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1084_in__set__conv__decomp__last,axiom,
    ! [X: produc7779921308951573458um_a_b,Xs: list_P5598968436384720482um_a_b] :
      ( ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
      = ( ? [Ys2: list_P5598968436384720482um_a_b,Zs3: list_P5598968436384720482um_a_b] :
            ( ( Xs
              = ( append629602400917629175um_a_b @ Ys2 @ ( cons_P7543864023423915932um_a_b @ X @ Zs3 ) ) )
            & ~ ( member4052926750399300073um_a_b @ X @ ( set_Pr7399620718385288829um_a_b @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1085_in__set__conv__decomp__last,axiom,
    ! [X: produc5919649888572472791um_a_b,Xs: list_P6497659457863146077um_a_b] :
      ( ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
      = ( ? [Ys2: list_P6497659457863146077um_a_b,Zs3: list_P6497659457863146077um_a_b] :
            ( ( Xs
              = ( append7230762676230896882um_a_b @ Ys2 @ ( cons_P3780209258894893709um_a_b @ X @ Zs3 ) ) )
            & ~ ( member7872818562604846720um_a_b @ X @ ( set_Pr8982359995307150060um_a_b @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1086_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [Ys2: list_P6011104703257516679at_nat,Zs3: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys2 @ ( cons_P6512896166579812791at_nat @ X @ Zs3 ) ) )
            & ~ ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1087_in__set__conv__decomp__last,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
            & ~ ( member_nat @ X @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1088_split__list__first__propE,axiom,
    ! [Xs: list_P911570089447184179um_a_b,P2: produc4970237071031658413um_a_b > $o] :
      ( ? [X7: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X7 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P911570089447184179um_a_b,X5: produc4970237071031658413um_a_b] :
            ( ? [Zs2: list_P911570089447184179um_a_b] :
                ( Xs
                = ( append482101194363978952um_a_b @ Ys3 @ ( cons_P4211151491411527779um_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc4970237071031658413um_a_b] :
                    ( ( member6480126262776431190um_a_b @ Xa2 @ ( set_Pr2436422074441463490um_a_b @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1089_split__list__first__propE,axiom,
    ! [Xs: list_Sum_sum_a_b,P2: sum_sum_a_b > $o] :
      ( ? [X7: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_Sum_sum_a_b,X5: sum_sum_a_b] :
            ( ? [Zs2: list_Sum_sum_a_b] :
                ( Xs
                = ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: sum_sum_a_b] :
                    ( ( member_Sum_sum_a_b @ Xa2 @ ( set_Sum_sum_a_b2 @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1090_split__list__first__propE,axiom,
    ! [Xs: list_P667245297671117661_b_nat,P2: produc6336967135452217047_b_nat > $o] :
      ( ? [X7: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X7 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P667245297671117661_b_nat,X5: produc6336967135452217047_b_nat] :
            ( ? [Zs2: list_P667245297671117661_b_nat] :
                ( Xs
                = ( append7648079923110641138_b_nat @ Ys3 @ ( cons_P4197526505774637965_b_nat @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc6336967135452217047_b_nat] :
                    ( ( member8290135809484590976_b_nat @ Xa2 @ ( set_Pr176305205332118508_b_nat @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1091_split__list__first__propE,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,P2: produc7779921308951573458um_a_b > $o] :
      ( ? [X7: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X7 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P5598968436384720482um_a_b,X5: produc7779921308951573458um_a_b] :
            ( ? [Zs2: list_P5598968436384720482um_a_b] :
                ( Xs
                = ( append629602400917629175um_a_b @ Ys3 @ ( cons_P7543864023423915932um_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc7779921308951573458um_a_b] :
                    ( ( member4052926750399300073um_a_b @ Xa2 @ ( set_Pr7399620718385288829um_a_b @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1092_split__list__first__propE,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,P2: produc5919649888572472791um_a_b > $o] :
      ( ? [X7: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X7 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P6497659457863146077um_a_b,X5: produc5919649888572472791um_a_b] :
            ( ? [Zs2: list_P6497659457863146077um_a_b] :
                ( Xs
                = ( append7230762676230896882um_a_b @ Ys3 @ ( cons_P3780209258894893709um_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc5919649888572472791um_a_b] :
                    ( ( member7872818562604846720um_a_b @ Xa2 @ ( set_Pr8982359995307150060um_a_b @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1093_split__list__first__propE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o] :
      ( ? [X7: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X7 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P6011104703257516679at_nat,X5: product_prod_nat_nat] :
            ( ? [Zs2: list_P6011104703257516679at_nat] :
                ( Xs
                = ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa2 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1094_split__list__first__propE,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ? [X7: nat] :
          ( ( member_nat @ X7 @ ( set_nat2 @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_nat,X5: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs
                = ( append_nat @ Ys3 @ ( cons_nat @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1095_split__list__last__propE,axiom,
    ! [Xs: list_P911570089447184179um_a_b,P2: produc4970237071031658413um_a_b > $o] :
      ( ? [X7: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X7 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P911570089447184179um_a_b,X5: produc4970237071031658413um_a_b,Zs2: list_P911570089447184179um_a_b] :
            ( ( Xs
              = ( append482101194363978952um_a_b @ Ys3 @ ( cons_P4211151491411527779um_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc4970237071031658413um_a_b] :
                    ( ( member6480126262776431190um_a_b @ Xa2 @ ( set_Pr2436422074441463490um_a_b @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1096_split__list__last__propE,axiom,
    ! [Xs: list_Sum_sum_a_b,P2: sum_sum_a_b > $o] :
      ( ? [X7: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_Sum_sum_a_b,X5: sum_sum_a_b,Zs2: list_Sum_sum_a_b] :
            ( ( Xs
              = ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: sum_sum_a_b] :
                    ( ( member_Sum_sum_a_b @ Xa2 @ ( set_Sum_sum_a_b2 @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1097_split__list__last__propE,axiom,
    ! [Xs: list_P667245297671117661_b_nat,P2: produc6336967135452217047_b_nat > $o] :
      ( ? [X7: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X7 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P667245297671117661_b_nat,X5: produc6336967135452217047_b_nat,Zs2: list_P667245297671117661_b_nat] :
            ( ( Xs
              = ( append7648079923110641138_b_nat @ Ys3 @ ( cons_P4197526505774637965_b_nat @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc6336967135452217047_b_nat] :
                    ( ( member8290135809484590976_b_nat @ Xa2 @ ( set_Pr176305205332118508_b_nat @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1098_split__list__last__propE,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,P2: produc7779921308951573458um_a_b > $o] :
      ( ? [X7: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X7 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P5598968436384720482um_a_b,X5: produc7779921308951573458um_a_b,Zs2: list_P5598968436384720482um_a_b] :
            ( ( Xs
              = ( append629602400917629175um_a_b @ Ys3 @ ( cons_P7543864023423915932um_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc7779921308951573458um_a_b] :
                    ( ( member4052926750399300073um_a_b @ Xa2 @ ( set_Pr7399620718385288829um_a_b @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1099_split__list__last__propE,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,P2: produc5919649888572472791um_a_b > $o] :
      ( ? [X7: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X7 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P6497659457863146077um_a_b,X5: produc5919649888572472791um_a_b,Zs2: list_P6497659457863146077um_a_b] :
            ( ( Xs
              = ( append7230762676230896882um_a_b @ Ys3 @ ( cons_P3780209258894893709um_a_b @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: produc5919649888572472791um_a_b] :
                    ( ( member7872818562604846720um_a_b @ Xa2 @ ( set_Pr8982359995307150060um_a_b @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1100_split__list__last__propE,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o] :
      ( ? [X7: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X7 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_P6011104703257516679at_nat,X5: product_prod_nat_nat,Zs2: list_P6011104703257516679at_nat] :
            ( ( Xs
              = ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Xa2 @ ( set_Pr5648618587558075414at_nat @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1101_split__list__last__propE,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ? [X7: nat] :
          ( ( member_nat @ X7 @ ( set_nat2 @ Xs ) )
          & ( P2 @ X7 ) )
     => ~ ! [Ys3: list_nat,X5: nat,Zs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X5 @ Zs2 ) ) )
           => ( ( P2 @ X5 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
                   => ~ ( P2 @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1102_split__list__first__prop,axiom,
    ! [Xs: list_P911570089447184179um_a_b,P2: produc4970237071031658413um_a_b > $o] :
      ( ? [X7: produc4970237071031658413um_a_b] :
          ( ( member6480126262776431190um_a_b @ X7 @ ( set_Pr2436422074441463490um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ? [Ys3: list_P911570089447184179um_a_b,X5: produc4970237071031658413um_a_b] :
          ( ? [Zs2: list_P911570089447184179um_a_b] :
              ( Xs
              = ( append482101194363978952um_a_b @ Ys3 @ ( cons_P4211151491411527779um_a_b @ X5 @ Zs2 ) ) )
          & ( P2 @ X5 )
          & ! [Xa2: produc4970237071031658413um_a_b] :
              ( ( member6480126262776431190um_a_b @ Xa2 @ ( set_Pr2436422074441463490um_a_b @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1103_split__list__first__prop,axiom,
    ! [Xs: list_Sum_sum_a_b,P2: sum_sum_a_b > $o] :
      ( ? [X7: sum_sum_a_b] :
          ( ( member_Sum_sum_a_b @ X7 @ ( set_Sum_sum_a_b2 @ Xs ) )
          & ( P2 @ X7 ) )
     => ? [Ys3: list_Sum_sum_a_b,X5: sum_sum_a_b] :
          ( ? [Zs2: list_Sum_sum_a_b] :
              ( Xs
              = ( append_Sum_sum_a_b @ Ys3 @ ( cons_Sum_sum_a_b @ X5 @ Zs2 ) ) )
          & ( P2 @ X5 )
          & ! [Xa2: sum_sum_a_b] :
              ( ( member_Sum_sum_a_b @ Xa2 @ ( set_Sum_sum_a_b2 @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1104_split__list__first__prop,axiom,
    ! [Xs: list_P667245297671117661_b_nat,P2: produc6336967135452217047_b_nat > $o] :
      ( ? [X7: produc6336967135452217047_b_nat] :
          ( ( member8290135809484590976_b_nat @ X7 @ ( set_Pr176305205332118508_b_nat @ Xs ) )
          & ( P2 @ X7 ) )
     => ? [Ys3: list_P667245297671117661_b_nat,X5: produc6336967135452217047_b_nat] :
          ( ? [Zs2: list_P667245297671117661_b_nat] :
              ( Xs
              = ( append7648079923110641138_b_nat @ Ys3 @ ( cons_P4197526505774637965_b_nat @ X5 @ Zs2 ) ) )
          & ( P2 @ X5 )
          & ! [Xa2: produc6336967135452217047_b_nat] :
              ( ( member8290135809484590976_b_nat @ Xa2 @ ( set_Pr176305205332118508_b_nat @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1105_split__list__first__prop,axiom,
    ! [Xs: list_P5598968436384720482um_a_b,P2: produc7779921308951573458um_a_b > $o] :
      ( ? [X7: produc7779921308951573458um_a_b] :
          ( ( member4052926750399300073um_a_b @ X7 @ ( set_Pr7399620718385288829um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ? [Ys3: list_P5598968436384720482um_a_b,X5: produc7779921308951573458um_a_b] :
          ( ? [Zs2: list_P5598968436384720482um_a_b] :
              ( Xs
              = ( append629602400917629175um_a_b @ Ys3 @ ( cons_P7543864023423915932um_a_b @ X5 @ Zs2 ) ) )
          & ( P2 @ X5 )
          & ! [Xa2: produc7779921308951573458um_a_b] :
              ( ( member4052926750399300073um_a_b @ Xa2 @ ( set_Pr7399620718385288829um_a_b @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1106_split__list__first__prop,axiom,
    ! [Xs: list_P6497659457863146077um_a_b,P2: produc5919649888572472791um_a_b > $o] :
      ( ? [X7: produc5919649888572472791um_a_b] :
          ( ( member7872818562604846720um_a_b @ X7 @ ( set_Pr8982359995307150060um_a_b @ Xs ) )
          & ( P2 @ X7 ) )
     => ? [Ys3: list_P6497659457863146077um_a_b,X5: produc5919649888572472791um_a_b] :
          ( ? [Zs2: list_P6497659457863146077um_a_b] :
              ( Xs
              = ( append7230762676230896882um_a_b @ Ys3 @ ( cons_P3780209258894893709um_a_b @ X5 @ Zs2 ) ) )
          & ( P2 @ X5 )
          & ! [Xa2: produc5919649888572472791um_a_b] :
              ( ( member7872818562604846720um_a_b @ Xa2 @ ( set_Pr8982359995307150060um_a_b @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1107_split__list__first__prop,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P2: product_prod_nat_nat > $o] :
      ( ? [X7: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X7 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
          & ( P2 @ X7 ) )
     => ? [Ys3: list_P6011104703257516679at_nat,X5: product_prod_nat_nat] :
          ( ? [Zs2: list_P6011104703257516679at_nat] :
              ( Xs
              = ( append985823374593552924at_nat @ Ys3 @ ( cons_P6512896166579812791at_nat @ X5 @ Zs2 ) ) )
          & ( P2 @ X5 )
          & ! [Xa2: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ Xa2 @ ( set_Pr5648618587558075414at_nat @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1108_split__list__first__prop,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ? [X7: nat] :
          ( ( member_nat @ X7 @ ( set_nat2 @ Xs ) )
          & ( P2 @ X7 ) )
     => ? [Ys3: list_nat,X5: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X5 @ Zs2 ) ) )
          & ( P2 @ X5 )
          & ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
             => ~ ( P2 @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_1109_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M2: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P2 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K2 ) ) )
       => ( P2 @ M2 ) ) ) ).

% nat_descend_induct
thf(fact_1110_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_1111_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A3: nat] :
      ( ( A3 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A3 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1112_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1113_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1114_bot__nat__0_Oextremum,axiom,
    ! [A3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A3 ) ).

% bot_nat_0.extremum
thf(fact_1115_le0,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% le0
thf(fact_1116_less__eq__nat_Osimps_I1_J,axiom,
    ! [N: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N ) ).

% less_eq_nat.simps(1)
thf(fact_1117_bot__nat__0_Oextremum__unique,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
      = ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_1118_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A3: nat] :
      ( ( ord_less_eq_nat @ A3 @ zero_zero_nat )
     => ( A3 = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_1119_le__0__eq,axiom,
    ! [N: nat] :
      ( ( ord_less_eq_nat @ N @ zero_zero_nat )
      = ( N = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1120_bot__nat__0_Oextremum__strict,axiom,
    ! [A3: nat] :
      ~ ( ord_less_nat @ A3 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1121_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1122_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1123_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1124_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1125_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1126_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P2 @ N3 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N3 )
                  & ~ ( P2 @ M3 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_1127_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1128_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C5: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_1129_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_1130_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_1131_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_1132_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1133_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1134_Suc__le__mono,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ ( suc @ M2 ) )
      = ( ord_less_eq_nat @ N @ M2 ) ) ).

% Suc_le_mono
thf(fact_1135_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1136_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1137_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1138_strict__inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I3: nat] :
            ( ( J2
              = ( suc @ I3 ) )
           => ( P2 @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( P2 @ ( suc @ I3 ) )
               => ( P2 @ I3 ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_1139_less__Suc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P2 @ I3 @ J3 )
                 => ( ( P2 @ J3 @ K2 )
                   => ( P2 @ I3 @ K2 ) ) ) ) )
         => ( P2 @ I2 @ J2 ) ) ) ) ).

% less_Suc_induct
thf(fact_1140_less__trans__Suc,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ J2 @ K )
       => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1141_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_1142_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_1143_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M5: nat] :
            ( ( M2
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1144_Nat_OAll__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P2 @ I ) ) )
      = ( ( P2 @ N )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P2 @ I ) ) ) ) ).

% Nat.All_less_Suc
thf(fact_1145_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_1146_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_1147_Ex__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P2 @ I ) ) )
      = ( ( P2 @ N )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P2 @ I ) ) ) ) ).

% Ex_less_Suc
thf(fact_1148_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1149_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_1150_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1151_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I2 @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_1152_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_1153_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ K )
     => ( ( K
         != ( suc @ I2 ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I2 @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1154_n__not__Suc__n,axiom,
    ! [N: nat] :
      ( N
     != ( suc @ N ) ) ).

% n_not_Suc_n
thf(fact_1155_Suc__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( suc @ X )
        = ( suc @ Y ) )
     => ( X = Y ) ) ).

% Suc_inject
thf(fact_1156_transitive__stepwise__le,axiom,
    ! [M2: nat,N: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ! [X5: nat] : ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y3: nat,Z3: nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z3 )
               => ( R @ X5 @ Z3 ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M2 @ N ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_1157_nat__induct__at__least,axiom,
    ! [M2: nat,N: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( P2 @ M2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M2 @ N3 )
             => ( ( P2 @ N3 )
               => ( P2 @ ( suc @ N3 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_at_least
thf(fact_1158_full__nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N3 )
             => ( P2 @ M3 ) )
         => ( P2 @ N3 ) )
     => ( P2 @ N ) ) ).

% full_nat_induct
thf(fact_1159_not__less__eq__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_eq_nat @ M2 @ N ) )
      = ( ord_less_eq_nat @ ( suc @ N ) @ M2 ) ) ).

% not_less_eq_eq
thf(fact_1160_Suc__n__not__le__n,axiom,
    ! [N: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N ) @ N ) ).

% Suc_n_not_le_n
thf(fact_1161_le__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_eq_nat @ M2 @ N )
        | ( M2
          = ( suc @ N ) ) ) ) ).

% le_Suc_eq
thf(fact_1162_Suc__le__D,axiom,
    ! [N: nat,M6: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N ) @ M6 )
     => ? [M7: nat] :
          ( M6
          = ( suc @ M7 ) ) ) ).

% Suc_le_D
thf(fact_1163_le__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ M2 @ ( suc @ N ) ) ) ).

% le_SucI
thf(fact_1164_le__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_eq_nat @ M2 @ N )
       => ( M2
          = ( suc @ N ) ) ) ) ).

% le_SucE
thf(fact_1165_Suc__leD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% Suc_leD
thf(fact_1166_not0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ? [M7: nat] :
          ( N
          = ( suc @ M7 ) ) ) ).

% not0_implies_Suc
thf(fact_1167_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_1168_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_1169_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1170_zero__induct,axiom,
    ! [P2: nat > $o,K: nat] :
      ( ( P2 @ K )
     => ( ! [N3: nat] :
            ( ( P2 @ ( suc @ N3 ) )
           => ( P2 @ N3 ) )
       => ( P2 @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_1171_diff__induct,axiom,
    ! [P2: nat > nat > $o,M2: nat,N: nat] :
      ( ! [X5: nat] : ( P2 @ X5 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P2 @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X5: nat,Y3: nat] :
              ( ( P2 @ X5 @ Y3 )
             => ( P2 @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
         => ( P2 @ M2 @ N ) ) ) ) ).

% diff_induct
thf(fact_1172_nat__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P2 @ N3 )
           => ( P2 @ ( suc @ N3 ) ) )
       => ( P2 @ N ) ) ) ).

% nat_induct
thf(fact_1173_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1174_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1175_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1176_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1177_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1178_Ex__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
            & ( P2 @ I ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N )
            & ( P2 @ ( suc @ I ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1179_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M4: nat] :
            ( N
            = ( suc @ M4 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1180_All__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N ) )
           => ( P2 @ I ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N )
           => ( P2 @ ( suc @ I ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1181_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M7: nat] :
          ( N
          = ( suc @ M7 ) ) ) ).

% gr0_implies_Suc
thf(fact_1182_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J: nat] :
            ( ( M2
              = ( suc @ J ) )
            & ( ord_less_nat @ J @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1183_Suc__leI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( suc @ M2 ) @ N ) ) ).

% Suc_leI
thf(fact_1184_Suc__le__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_eq
thf(fact_1185_dec__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P2 @ I2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
             => ( ( ord_less_nat @ N3 @ J2 )
               => ( ( P2 @ N3 )
                 => ( P2 @ ( suc @ N3 ) ) ) ) )
         => ( P2 @ J2 ) ) ) ) ).

% dec_induct
thf(fact_1186_inc__induct,axiom,
    ! [I2: nat,J2: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( P2 @ J2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
             => ( ( ord_less_nat @ N3 @ J2 )
               => ( ( P2 @ ( suc @ N3 ) )
                 => ( P2 @ N3 ) ) ) )
         => ( P2 @ I2 ) ) ) ) ).

% inc_induct
thf(fact_1187_Suc__le__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_le_lessD
thf(fact_1188_le__less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% le_less_Suc_eq
thf(fact_1189_less__Suc__eq__le,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% less_Suc_eq_le
thf(fact_1190_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N4: nat] : ( ord_less_eq_nat @ ( suc @ N4 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1191_le__imp__less__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1192_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1193_min__0R,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ N @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_1194_min__0L,axiom,
    ! [N: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% min_0L
thf(fact_1195_min__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_min_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( suc @ ( ord_min_nat @ M2 @ N ) ) ) ).

% min_Suc_Suc
thf(fact_1196_zero__notin__Suc__image,axiom,
    ! [A5: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A5 ) ) ).

% zero_notin_Suc_image
thf(fact_1197_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X: nat,M: nat] :
      ( ( P2 @ X )
     => ( ! [X5: nat] :
            ( ( P2 @ X5 )
           => ( ord_less_eq_nat @ X5 @ M ) )
       => ~ ! [M7: nat] :
              ( ( P2 @ M7 )
             => ~ ! [X7: nat] :
                    ( ( P2 @ X7 )
                   => ( ord_less_eq_nat @ X7 @ M7 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1198_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_1199_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_1200_add__Suc__right,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc_right
thf(fact_1201_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1202_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_le
thf(fact_1203_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_1204_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J2 ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J2 ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_1205_add__Suc__shift,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( plus_plus_nat @ M2 @ ( suc @ N ) ) ) ).

% add_Suc_shift
thf(fact_1206_add__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N )
      = ( suc @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% add_Suc
thf(fact_1207_nat__arith_Osuc1,axiom,
    ! [A5: nat,K: nat,A3: nat] :
      ( ( A5
        = ( plus_plus_nat @ K @ A3 ) )
     => ( ( suc @ A5 )
        = ( plus_plus_nat @ K @ ( suc @ A3 ) ) ) ) ).

% nat_arith.suc1
thf(fact_1208_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_1209_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_1210_ex__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
            & ( P2 @ M4 ) ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P2 @ X4 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1211_all__nat__less__eq,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [M4: nat] :
            ( ( ord_less_nat @ M4 @ N )
           => ( P2 @ M4 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P2 @ X4 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1212_add__leE,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ~ ( ( ord_less_eq_nat @ M2 @ N )
         => ~ ( ord_less_eq_nat @ K @ N ) ) ) ).

% add_leE
thf(fact_1213_le__add1,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ N @ M2 ) ) ).

% le_add1
thf(fact_1214_le__add2,axiom,
    ! [N: nat,M2: nat] : ( ord_less_eq_nat @ N @ ( plus_plus_nat @ M2 @ N ) ) ).

% le_add2
thf(fact_1215_add__leD1,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ( ord_less_eq_nat @ M2 @ N ) ) ).

% add_leD1
thf(fact_1216_add__leD2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N )
     => ( ord_less_eq_nat @ K @ N ) ) ).

% add_leD2
thf(fact_1217_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_1218_add__le__mono,axiom,
    ! [I2: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_le_mono
thf(fact_1219_add__le__mono1,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_le_mono1
thf(fact_1220_trans__le__add1,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).

% trans_le_add1
thf(fact_1221_trans__le__add2,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).

% trans_le_add2
thf(fact_1222_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M4: nat,N4: nat] :
        ? [K5: nat] :
          ( N4
          = ( plus_plus_nat @ M4 @ K5 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1223_add__lessD1,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K )
     => ( ord_less_nat @ I2 @ K ) ) ).

% add_lessD1
thf(fact_1224_add__less__mono,axiom,
    ! [I2: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_less_mono
thf(fact_1225_not__add__less1,axiom,
    ! [I2: nat,J2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J2 ) @ I2 ) ).

% not_add_less1
thf(fact_1226_not__add__less2,axiom,
    ! [J2: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_1227_add__less__mono1,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_less_mono1
thf(fact_1228_trans__less__add1,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).

% trans_less_add1
thf(fact_1229_trans__less__add2,axiom,
    ! [I2: nat,J2: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).

% trans_less_add2
thf(fact_1230_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_1231_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M2: nat,K: nat] :
      ( ! [M7: nat,N3: nat] :
          ( ( ord_less_nat @ M7 @ N3 )
         => ( ord_less_nat @ ( F @ M7 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M2 ) @ K ) @ ( F @ ( plus_plus_nat @ M2 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1232_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q2: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q2 ) ) ) ) ).

% less_natE
thf(fact_1233_less__add__Suc1,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_1234_less__add__Suc2,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_1235_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M4: nat,N4: nat] :
        ? [K5: nat] :
          ( N4
          = ( suc @ ( plus_plus_nat @ M4 @ K5 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1236_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1237_less__imp__add__positive,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I2 @ K2 )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_1238_one__is__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1239_add__is__1,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( plus_plus_nat @ M2 @ N )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1240_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1241_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1242_diff__Suc__Suc,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_Suc_Suc
thf(fact_1243_Suc__diff__diff,axiom,
    ! [M2: nat,N: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N ) @ K ) ) ).

% Suc_diff_diff
thf(fact_1244_diff__diff__cancel,axiom,
    ! [I2: nat,N: nat] :
      ( ( ord_less_eq_nat @ I2 @ N )
     => ( ( minus_minus_nat @ N @ ( minus_minus_nat @ N @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_1245_diff__diff__left,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% diff_diff_left
thf(fact_1246_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_1247_diff__is__0__eq_H,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1248_diff__is__0__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N ) ) ).

% diff_is_0_eq
thf(fact_1249_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J2 ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1250_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J2 @ K ) @ I2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J2 @ I2 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1251_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J2 @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 ) ) ) ).

% Nat.diff_diff_right
thf(fact_1252_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1253_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) @ I2 )
        = ( minus_minus_nat @ ( suc @ J2 ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_1254_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J2 )
     => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J2 @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J2 ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_1255_min__diff,axiom,
    ! [M2: nat,I2: nat,N: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M2 @ I2 ) @ ( minus_minus_nat @ N @ I2 ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M2 @ N ) @ I2 ) ) ).

% min_diff
thf(fact_1256_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_1257_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_1258_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M2 @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1259_Suc__diff__le,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_eq_nat @ N @ M2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( suc @ ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Suc_diff_le
thf(fact_1260_diff__less__mono,axiom,
    ! [A3: nat,B3: nat,C2: nat] :
      ( ( ord_less_nat @ A3 @ B3 )
     => ( ( ord_less_eq_nat @ C2 @ A3 )
       => ( ord_less_nat @ ( minus_minus_nat @ A3 @ C2 ) @ ( minus_minus_nat @ B3 @ C2 ) ) ) ) ).

% diff_less_mono
thf(fact_1261_less__diff__iff,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( ord_less_nat @ M2 @ N ) ) ) ) ).

% less_diff_iff
thf(fact_1262_less__imp__diff__less,axiom,
    ! [J2: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J2 @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J2 @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1263_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_1264_diff__commute,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J2 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J2 ) ) ).

% diff_commute
thf(fact_1265_diff__le__mono2,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1266_le__diff__iff_H,axiom,
    ! [A3: nat,C2: nat,B3: nat] :
      ( ( ord_less_eq_nat @ A3 @ C2 )
     => ( ( ord_less_eq_nat @ B3 @ C2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C2 @ A3 ) @ ( minus_minus_nat @ C2 @ B3 ) )
          = ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% le_diff_iff'
thf(fact_1267_diff__le__self,axiom,
    ! [M2: nat,N: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ).

% diff_le_self
thf(fact_1268_diff__le__mono,axiom,
    ! [M2: nat,N: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N @ L ) ) ) ).

% diff_le_mono
thf(fact_1269_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N @ K ) )
          = ( minus_minus_nat @ M2 @ N ) ) ) ) ).

% Nat.diff_diff_eq

% Helper facts (15)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
      ( ( if_list_Sum_sum_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Sum____Type__Osum_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: list_Sum_sum_a_b,Y: list_Sum_sum_a_b] :
      ( ( if_list_Sum_sum_a_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( if_lis9186351972506106189at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_P6011104703257516679at_nat,Y: list_P6011104703257516679at_nat] :
      ( ( if_lis9186351972506106189at_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
    ! [X: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( if_lis8991567957713360163um_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
    ! [X: list_P6497659457863146077um_a_b,Y: list_P6497659457863146077um_a_b] :
      ( ( if_lis8991567957713360163um_a_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( if_lis3161153797521331747_b_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Nat__Onat_J_J_T,axiom,
    ! [X: list_P667245297671117661_b_nat,Y: list_P667245297671117661_b_nat] :
      ( ( if_lis3161153797521331747_b_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
    ! [X: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( if_lis2231744601426488825um_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_T,axiom,
    ! [X: list_P911570089447184179um_a_b,Y: list_P911570089447184179um_a_b] :
      ( ( if_lis2231744601426488825um_a_b @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_J_T,axiom,
    ! [X: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( if_lis4596758700692144156um_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Sum____Type__Osum_Itf__a_Mtf__b_J_Mt__Sum____Type__Osum_Itf__a_Mtf__b_J_J_J_J_T,axiom,
    ! [X: list_P5598968436384720482um_a_b,Y: list_P5598968436384720482um_a_b] :
      ( ( if_lis4596758700692144156um_a_b @ $true @ X @ Y )
      = X ) ).

% Conjectures (3)
thf(conj_0,hypothesis,
    member6480126262776431190um_a_b @ ( produc8374529264883863525um_a_b @ a2 @ b ) @ ( set_Pr2436422074441463490um_a_b @ ( zip_Su8953288692215371525um_a_b @ ys @ xs ) ) ).

thf(conj_1,hypothesis,
    ad_equiv_pair_a_b @ y @ ( produc8374529264883863525um_a_b @ a2 @ b ) ).

thf(conj_2,conjecture,
    ad_equiv_pair_a_b @ x @ ( produc8374529264883863525um_a_b @ a2 @ b ) ).

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