TPTP Problem File: SLH0512^1.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    : Multiset_Ordering_NPC/0002_Multiset_Ordering_in_NP/prob_00591_026178__13736538_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1762 ( 645 unt; 478 typ;   0 def)
%            Number of atoms       : 3853 (2144 equ;   0 cnn)
%            Maximal formula atoms :   19 (   3 avg)
%            Number of connectives : 16763 ( 478   ~;  50   |; 445   &;14072   @)
%                                         (   0 <=>;1718  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   49 (   8 avg)
%            Number of types       :   81 (  80 usr)
%            Number of type conns  : 2543 (2543   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  401 ( 398 usr;  30 con; 0-16 aty)
%            Number of variables   : 5437 ( 325   ^;4737   !; 375   ?;5437   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:29:47.029
%------------------------------------------------------------------------------
% Could-be-implicit typings (80)
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(ty_n_t__Set__Oset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(ty_n_t__List__Olist_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(ty_n_t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(ty_n_t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(ty_n_t__List__Olist_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Nat__Onat,type,
    nat: $tType ).

% Explicit typings (398)
thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
    minus_minus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    plus_p6734580430835824111ropVar: multis2129813143740876984ropVar > multis2129813143740876984ropVar > multis2129813143740876984ropVar ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups__List_Omonoid__add_Osum__list_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_HOL_OThe_001t__Nat__Onat,type,
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thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    the_Pr7468375744313411758ropVar: ( produc7679832412844898215ropVar > $o ) > produc7679832412844898215ropVar ).

thf(sy_c_HOL_OThe_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    the_Pr3361993152900459897ropVar: ( produc2642885279816024434ropVar > $o ) > produc2642885279816024434ropVar ).

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

thf(sy_c_If_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J,type,
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thf(sy_c_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J_J,type,
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thf(sy_c_List_Oappend_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    append9216677672214865852ropVar: list_P2861839656816079527ropVar > list_P2861839656816079527ropVar > list_P2861839656816079527ropVar ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_List_Odistinct_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Odistinct_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
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thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_List_Odistinct_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Odrop_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Odrop_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Oenumerate_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Oenumerate_001t__Nat__Onat,type,
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thf(sy_c_List_Ofilter_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Ofilter_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Nat__Onat_J,type,
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    set_li6509544621384488717ropVar: list_l3415868495287753214ropVar > set_li8998331099879477720ropVar ).

thf(sy_c_List_Olist_Oset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    set_Pr1973727305987283026ropVar: list_P7811946491041149379ropVar > set_Pr5522547032670187677ropVar ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    set_Pr310918443939006630ropVar: list_P1330674724486821911ropVar > set_Pr8264647302970636017ropVar ).

thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_List_Olist__update_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
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thf(sy_c_List_Olistrel_001t__Nat__Onat_001t__Nat__Onat,type,
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    zip_Pr6094978418188712748ropVar: list_P444016040222843053ropVar > list_M6740372356978808802ropVar > list_P3821663398315094434ropVar ).

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    zip_Pr5260769330036727841ropVar: list_P8791008543742759928ropVar > list_M6740372356978808802ropVar > list_P1330674724486821911ropVar ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Nat__Onat,type,
    zip_Pr2402428817466909387ar_nat: list_P8791008543742759928ropVar > list_nat > list_P2692446888044980865ar_nat ).

thf(sy_c_List_Ozip_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    zip_Pr6829646289182870327ropVar: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P1750615084264230189ropVar ).

thf(sy_c_List__Order_Olist__order__extension_001t__Multiset____Ordering____in____NP__OPropVar,type,
    list_l6611490901775297704ropVar: ( set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar ) > ( set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar ) > $o ).

thf(sy_c_List__Order_Olist__order__extension_001t__Nat__Onat,type,
    list_l792762466888043652on_nat: ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ) > ( set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat ) > $o ).

thf(sy_c_List__Order_Olist__order__extension_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    list_l8030158259481955774ropVar: ( set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar ) > ( set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar ) > $o ).

thf(sy_c_Multiset_Oadd__mset_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset_Oadd__mset_001t__Nat__Onat,type,
    add_mset_nat: nat > multiset_nat > multiset_nat ).

thf(sy_c_Multiset_Oadd__mset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    add_ms6892841465718707280ropVar: produc2642885279816024434ropVar > multis2129813143740876984ropVar > multis2129813143740876984ropVar ).

thf(sy_c_Multiset_Ocomm__monoid__add_Osum__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    comm_m5787568287065167983et_nat: ( multiset_nat > multiset_nat > multiset_nat ) > multiset_nat > multis1201202736280713200et_nat > multiset_nat ).

thf(sy_c_Multiset_Olinorder__class_Opart_001t__Nat__Onat_001t__Nat__Onat,type,
    linord7044143797982409259at_nat: ( nat > nat ) > nat > list_nat > produc5438398690206989622st_nat ).

thf(sy_c_Multiset_Olinorder__class_Opart_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Nat__Onat,type,
    linord1115752046135255673ar_nat: ( produc2642885279816024434ropVar > nat ) > nat > list_P8791008543742759928ropVar > produc777314622398639830ropVar ).

thf(sy_c_Multiset_Olinorder__class_Osorted__list__of__multiset_001t__Nat__Onat,type,
    linord3047872887403683810et_nat: multiset_nat > list_nat ).

thf(sy_c_Multiset_Omset_001t__Multiset____Ordering____in____NP__OPropVar,type,
    mset_M6581516532424202993ropVar: list_M6740372356978808802ropVar > multis193838646080965794ropVar ).

thf(sy_c_Multiset_Omset_001t__Nat__Onat,type,
    mset_nat: list_nat > multiset_nat ).

thf(sy_c_Multiset_Omset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    mset_P5170620898817730055ropVar: list_P8791008543742759928ropVar > multis2129813143740876984ropVar ).

thf(sy_c_Multiset_Omset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    mset_P6383711406899277590at_nat: list_P6011104703257516679at_nat > multis2468970476368604999at_nat ).

thf(sy_c_Multiset_Omset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    mset_P2421302103884862786ropVar: list_P2861839656816079527ropVar > multis8516091272965721575ropVar ).

thf(sy_c_Multiset_Omset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Nat__Onat_J,type,
    mset_P6847787391757867548ar_nat: list_P2692446888044980865ar_nat > multis8346698504194622913ar_nat ).

thf(sy_c_Multiset_Omset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    mset_P7447892262639026108ropVar: list_P1750615084264230189ropVar > multis5328359229859807725ropVar ).

thf(sy_c_Multiset_Omult1_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset_Omult1_001t__Nat__Onat,type,
    mult1_nat: set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).

thf(sy_c_Multiset_Oset__mset_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
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thf(sy_c_Multiset_Oset__mset_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset_Oset__mset_001t__Nat__Onat,type,
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thf(sy_c_Multiset_Oset__mset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    set_ms882399555722365103ropVar: multis2129813143740876984ropVar > set_Pr6405597471078531282ropVar ).

thf(sy_c_Multiset_Osubseteq__mset_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset_Osubseteq__mset_001t__Nat__Onat,type,
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thf(sy_c_Multiset__Extension2_Ons__mul__ext_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis738276037714153361ropVar: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr2934421189634577799ropVar ).

thf(sy_c_Multiset__Extension2_Ons__mul__ext_001t__Nat__Onat,type,
    multis8831328596575508315xt_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr7086239977703175297et_nat ).

thf(sy_c_Multiset__Extension2_Ons__mul__ext_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    multis6478642269264501415ropVar: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr6293541671686298887ropVar ).

thf(sy_c_Multiset__Extension2_Os__mul__ext_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset__Extension2_Os__mul__ext_001t__Nat__Onat,type,
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thf(sy_c_Multiset__Extension2_Os__mul__ext_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    multis2067306628098044553ropVar: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr6293541671686298887ropVar ).

thf(sy_c_Multiset__Extension__Pair_Omultpw_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset__Extension__Pair_Omultpw_001t__Nat__Onat,type,
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thf(sy_c_Multiset__Extension__Pair_Omultpw_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxOneIJ,type,
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thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxOneJI,type,
    multis6646701651571564453xOneJI: nat > nat > multis3193088007478089820ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxZeroIJ,type,
    multis2983220944385456105ZeroIJ: nat > nat > multis3193088007478089820ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxZeroJI,type,
    multis2983220944385521703ZeroJI: nat > nat > multis3193088007478089820ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OEpsilon,type,
    multis2544335231667181926psilon: nat > multis3193088007478089820ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OGamma,type,
    multis387687052011358179_Gamma: nat > nat > multis3193088007478089820ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_Ocase__PropVar_001_Eo,type,
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thf(sy_c_Multiset__Ordering__in__NP_OPropVar_Osize__PropVar,type,
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thf(sy_c_Multiset__Ordering__in__NP_Oencoder_OencodeGammaCond_001t__Nat__Onat_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset__Ordering__in__NP_Oencoder_OencodeGammaCond_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis6911105939383758721ropVar: produc2642885279816024434ropVar > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > multis3193088007478089820ropVar > multis3193088007478089820ropVar > $o > $o > produc2642885279816024434ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__exactly__one_001t__Nat__Onat_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__exactly__one_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis8736296938027350326ropVar: produc2642885279816024434ropVar > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > list_P8791008543742759928ropVar > produc4542211886649297581ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_001t__Nat__Onat,type,
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thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula14_001t__Nat__Onat,type,
    multis778902868463715824414_nat: nat > ( multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > nat > nat ) > nat > nat > list_nat ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula14_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    multis3450153464401155806ropVar: produc2642885279816024434ropVar > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar > produc2642885279816024434ropVar ) > nat > nat > list_P8791008543742759928ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula15_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis7034272045817201159ropVar: multis3193088007478089820ropVar > multis3193088007478089820ropVar > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > list_M6740372356978808802ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula15_001t__Nat__Onat,type,
    multis902447301359355274115_nat: nat > nat > ( multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > nat > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > list_nat ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula15_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    multis5595896195939027229ropVar: produc2642885279816024434ropVar > produc2642885279816024434ropVar > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > list_P8791008543742759928ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula16_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis6448939448807493190ropVar: multis3193088007478089820ropVar > multis3193088007478089820ropVar > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > ( list_M6740372356978808802ropVar > multis3193088007478089820ropVar ) > ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > list_M6740372356978808802ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula16_001t__Nat__Onat,type,
    multis103654530569517143016_nat: nat > nat > ( multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > nat > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( list_M6740372356978808802ropVar > nat ) > ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > list_nat ).

thf(sy_c_Multiset__Ordering__in__NP_Oms__encoder_Oformula16_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    multis7741638927476898652ropVar: produc2642885279816024434ropVar > produc2642885279816024434ropVar > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > produc2642885279816024434ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > ( list_M6740372356978808802ropVar > produc2642885279816024434ropVar ) > ( nat > nat > $o ) > ( nat > nat > $o ) > nat > nat > list_P8791008543742759928ropVar ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    size_s9082923797619839828ropVar: list_l3041241048860003048ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    size_s3023201423986296836st_nat: list_list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J,type,
    size_s6779274088989432426ropVar: list_l3415868495287753214ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Multiset__Omultiset_It__Nat__Onat_J_J,type,
    size_s6386657463320973636et_nat: list_multiset_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    size_s1943289338788136526ropVar: list_M6740372356978808802ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
    size_size_list_nat: list_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    size_s8779405381146381081ropVar: list_P444016040222843053ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    size_s6752116297430514276ropVar: list_P8791008543742759928ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    size_s5460976970255530739at_nat: list_P6011104703257516679at_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J,type,
    size_s815404617008747835ropVar: list_P2861839656816079527ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    size_s836052043357872910ropVar: multis193838646080965794ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    size_s5917832649809541300et_nat: multiset_nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    size_s8378524962137626404ropVar: multis2129813143740876984ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__Multiset____Ordering____in____NP__OPropVar,type,
    size_s6253272723116879048ropVar: multis3193088007478089820ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__String__Ochar,type,
    size_size_char: char > nat ).

thf(sy_c_Order__Pair_OSN__order__pair_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    order_5896603146005108336ropVar: set_Pr3559363423829876615ropVar > set_Pr3559363423829876615ropVar > $o ).

thf(sy_c_Order__Pair_OSN__order__pair_001t__List__Olist_It__Nat__Onat_J,type,
    order_2144575240049343698st_nat: set_Pr3451248702717554689st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_Order__Pair_OSN__order__pair_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    order_7256030434733506182ropVar: set_Pr8381652856031230727ropVar > set_Pr8381652856031230727ropVar > $o ).

thf(sy_c_Order__Pair_OSN__order__pair_001t__Multiset____Ordering____in____NP__OPropVar,type,
    order_6814195989798057450ropVar: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > $o ).

thf(sy_c_Order__Pair_OSN__order__pair_001t__Nat__Onat,type,
    order_7802623214671921858ir_nat: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_Order__Pair_OSN__order__pair_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    order_9198723064055253760ropVar: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    ord_le5777773500796000884et_nat: multiset_nat > multiset_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat,type,
    ord_less_nat: nat > nat > $o ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_M_062_It__Nat__Onat_M_Eo_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc4727192421694094319st_nat: ( nat > nat > $o ) > list_nat > produc254973753779126261st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_M_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    produc3887254912111605537ropVar: ( produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o ) > list_P8791008543742759928ropVar > produc5533280449304468145ropVar ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    produc2076565587003802519ropVar: list_M6740372356978808802ropVar > list_M6740372356978808802ropVar > produc4148340297220298919ropVar ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    produc4116939308056247650ropVar: list_M6740372356978808802ropVar > list_P444016040222843053ropVar > produc3905501191202405874ropVar ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc2694037385005941721st_nat: list_nat > list_nat > produc1828647624359046049st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    produc5634919067874425663ropVar: list_nat > list_P8791008543742759928ropVar > produc8527832761551885325ropVar ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4487115339913071592st_nat: list_nat > produc1828647624359046049st_nat > produc5438398690206989622st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_001t__List__Olist_It__Nat__Onat_J,type,
    produc5924578096210284289st_nat: list_P8791008543742759928ropVar > list_nat > produc2894083670482369799st_nat ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    produc5333478016634530327ropVar: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > produc3638977808798937127ropVar ).

thf(sy_c_Product__Type_OPair_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J,type,
    produc2087848848369820614ropVar: list_P8791008543742759928ropVar > produc3638977808798937127ropVar > produc777314622398639830ropVar ).

thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J_001t__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    produc8064044144368941463ropVar: multis193838646080965794ropVar > multis193838646080965794ropVar > produc313332580105183911ropVar ).

thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Nat__Onat_J_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    produc2735455520514455641et_nat: multiset_nat > multiset_nat > produc7112813594591200289et_nat ).

thf(sy_c_Product__Type_OPair_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_001t__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    produc7134096272351878757ropVar: ( nat > nat > produc4900251693569959959ropVar ) > product_prod_nat_nat > produc4900251693569959959ropVar ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    produc5464566285301805891ropVar: ( nat > nat > produc7514035565325968913ropVar ) > product_prod_nat_nat > produc7514035565325968913ropVar ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Nat__Onat_J,type,
    produc2337209523370107711ar_nat: ( nat > nat > produc103364944588188913ar_nat ) > product_prod_nat_nat > produc103364944588188913ar_nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    produc6103969181082503513ropVar: ( nat > nat > produc7137084944585861159ropVar ) > product_prod_nat_nat > produc7137084944585861159ropVar ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    produc3167482732446468999ropVar: ( nat > produc7679832412844898215ropVar > produc2642885279816024434ropVar ) > produc5265754482477539788ropVar > produc2642885279816024434ropVar ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001_Eo,type,
    produc1692715175595045538pVar_o: ( nat > produc2642885279816024434ropVar > $o ) > produc4900251693569959959ropVar > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_001t__Multiset____Ordering____in____NP__OPropVar_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    produc5773209726166850449ropVar: ( produc7679832412844898215ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar ) > produc5178754913583666204ropVar > produc2642885279816024434ropVar ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    produc5464477786531444550ropVar: ( produc2642885279816024434ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar ) > produc4542211886649297581ropVar > list_P8791008543742759928ropVar ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Multiset____Ordering____in____NP__OPropVar_001t__Multiset____Ordering____in____NP__OPropVar,type,
    produc9119430476348611782ropVar: ( produc2642885279816024434ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar ) > produc7514035565325968913ropVar > multis3193088007478089820ropVar ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Multiset____Ordering____in____NP__OPropVar_001t__Nat__Onat,type,
    produc6539649061971739878ar_nat: ( produc2642885279816024434ropVar > multis3193088007478089820ropVar > nat ) > produc7514035565325968913ropVar > nat ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Nat__Onat_001_Eo,type,
    produc5286831101113786240_nat_o: ( produc2642885279816024434ropVar > nat > $o ) > produc103364944588188913ar_nat > $o ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001_Eo,type,
    produc6789530531619613804pVar_o: ( produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o ) > produc7137084944585861159ropVar > $o ).

thf(sy_c_Relation_OId__on_001t__Multiset____Ordering____in____NP__OPropVar,type,
    id_on_8850091095513083129ropVar: set_Mu5658373669842198716ropVar > set_Pr7799659934693882759ropVar ).

thf(sy_c_Relation_OId__on_001t__Nat__Onat,type,
    id_on_nat: set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Set_OCollect_001t__Multiset____Ordering____in____NP__OPropVar,type,
    collec8658135508005558727ropVar: ( multis3193088007478089820ropVar > $o ) > set_Mu5658373669842198716ropVar ).

thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
    collect_nat: ( nat > $o ) > set_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_Mt__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    collec7970277406990558738ropVar: ( produc4148340297220298919ropVar > $o ) > set_Pr3559363423829876615ropVar ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    collec1525075934208178781ropVar: ( produc3905501191202405874ropVar > $o ) > set_Pr1137703025709547602ropVar ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    collec1570431334306492044st_nat: ( produc1828647624359046049st_nat > $o ) > set_Pr3451248702717554689st_nat ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J,type,
    collec851525693690771858ropVar: ( produc3638977808798937127ropVar > $o ) > set_Pr8381652856031230727ropVar ).

thf(sy_c_String_Ochar_Osize__char,type,
    size_char: char > nat ).

thf(sy_c_fChoice_001t__Multiset____Ordering____in____NP__OPropVar,type,
    fChoic539809714188224432ropVar: ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar ).

thf(sy_c_fChoice_001t__Nat__Onat,type,
    fChoice_nat: ( nat > $o ) > nat ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    fChoic157639270322531067ropVar: ( produc7679832412844898215ropVar > $o ) > produc7679832412844898215ropVar ).

thf(sy_c_fChoice_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    fChoic3237874067385186502ropVar: ( produc2642885279816024434ropVar > $o ) > produc2642885279816024434ropVar ).

thf(sy_c_member_001t__List__Olist_It__Nat__Onat_J,type,
    member_list_nat: list_nat > set_list_nat > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    member6236339133593825953ropVar: list_P8791008543742759928ropVar > set_li8998331099879477720ropVar > $o ).

thf(sy_c_member_001t__Multiset__Omultiset_It__Nat__Onat_J,type,
    member_multiset_nat: multiset_nat > set_multiset_nat > $o ).

thf(sy_c_member_001t__Multiset____Ordering____in____NP__OPropVar,type,
    member8421201611525043589ropVar: multis3193088007478089820ropVar > set_Mu5658373669842198716ropVar > $o ).

thf(sy_c_member_001t__Nat__Onat,type,
    member_nat: nat > set_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_Mt__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    member7927977717557115344ropVar: produc4148340297220298919ropVar > set_Pr3559363423829876615ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    member7926529425716556827ropVar: produc3905501191202405874ropVar > set_Pr1137703025709547602ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member7340969449405702474st_nat: produc1828647624359046049st_nat > set_Pr3451248702717554689st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J,type,
    member6598729439514207908ropVar: produc8527832761551885325ropVar > set_Pr5890891477630704451ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    member964980348444692382st_nat: produc2894083670482369799st_nat > set_Pr5984347726151952189st_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J,type,
    member6468787482982540624ropVar: produc3638977808798937127ropVar > set_Pr8381652856031230727ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J_Mt__Multiset__Omultiset_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    member5226782437669593040ropVar: produc313332580105183911ropVar > set_Pr2934421189634577799ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Nat__Onat_J_Mt__Multiset__Omultiset_It__Nat__Onat_J_J,type,
    member2326506860204807626et_nat: produc7112813594591200289et_nat > set_Pr7086239977703175297et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J,type,
    member1455471568137992016ropVar: produc4518724579456149031ropVar > set_Pr6293541671686298887ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    member2619728555115997648ropVar: produc7679832412844898215ropVar > set_Pr7799659934693882759ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Nat__Onat_J,type,
    member8816771811004535154ar_nat: produc7433502520361543131ar_nat > set_Pr3981321659643448081ar_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    member9210975342120487067ropVar: produc2642885279816024434ropVar > set_Pr6405597471078531282ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    member1707986370202145254ropVar: produc9176844427374245309ropVar > set_Pr5522547032670187677ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    member1824086966109417880ropVar: produc440817675466425857ropVar > set_Pr8672156015886137399ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    member918664796156078510ropVar: produc4900251693569959959ropVar > set_Pr7708077615150402893ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    member45177508153868858ropVar: produc7514035565325968913ropVar > set_Pr8264647302970636017ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Nat__Onat_J,type,
    member5345150084029083272ar_nat: produc103364944588188913ar_nat > set_Pr7538684846379304231ar_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    member4092153873655836240ropVar: produc7137084944585861159ropVar > set_Pr3201966908590682119ropVar > $o ).

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

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

thf(sy_v_i____,type,
    i: nat ).

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

thf(sy_v_j____,type,
    j: nat ).

thf(sy_v_m,type,
    m: nat ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_on____,type,
    on: multis3193088007478089820ropVar ).

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

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

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

thf(sy_v_xs1____,type,
    xs1: list_nat ).

thf(sy_v_xs2____,type,
    xs2: list_nat ).

thf(sy_v_y____,type,
    y: multis3193088007478089820ropVar ).

thf(sy_v_ys1____,type,
    ys1: list_nat ).

thf(sy_v_ys2____,type,
    ys2: list_nat ).

thf(sy_v_ze____,type,
    ze: multis3193088007478089820ropVar ).

% Relevant facts (1268)
thf(fact_0_PropVar_Oinject_I5_J,axiom,
    ! [X51: nat,X52: nat,Y51: nat,Y52: nat] :
      ( ( ( multis2983220944385456105ZeroIJ @ X51 @ X52 )
        = ( multis2983220944385456105ZeroIJ @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% PropVar.inject(5)
thf(fact_1_PropVar_Oinject_I6_J,axiom,
    ! [X61: nat,X62: nat,Y61: nat,Y62: nat] :
      ( ( ( multis6646701651571498855xOneIJ @ X61 @ X62 )
        = ( multis6646701651571498855xOneIJ @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% PropVar.inject(6)
thf(fact_2_i,axiom,
    ord_less_nat @ i @ n ).

% i
thf(fact_3_PropVar_Odistinct_I29_J,axiom,
    ! [X51: nat,X52: nat,X61: nat,X62: nat] :
      ( ( multis2983220944385456105ZeroIJ @ X51 @ X52 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(29)
thf(fact_4_PropVar_Oinject_I1_J,axiom,
    ! [X11: nat,X12: nat,Y11: nat,Y12: nat] :
      ( ( ( multis387687052011358179_Gamma @ X11 @ X12 )
        = ( multis387687052011358179_Gamma @ Y11 @ Y12 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 ) ) ) ).

% PropVar.inject(1)
thf(fact_5_PropVar_Odistinct_I7_J,axiom,
    ! [X11: nat,X12: nat,X51: nat,X52: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(7)
thf(fact_6__C1_C_I2_J,axiom,
    ( ( nth_Pr227365692153740281ropVar
      @ ( map_na2145702811007361372ropVar
        @ ^ [J: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ i @ J ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385456105ZeroIJ @ i @ J ) @ ( multis6646701651571498855xOneIJ @ i @ J ) ) )
        @ ( upt @ zero_zero_nat @ m ) )
      @ j )
    = ( produc6194868611276206818ropVar @ y @ ( produc6936306168532358295ropVar @ ze @ on ) ) ) ).

% "1"(2)
thf(fact_7_PropVar_Osimps_I41_J,axiom,
    ! [F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,X51: nat,X52: nat] :
      ( ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) )
      = ( F5 @ X51 @ X52 ) ) ).

% PropVar.simps(41)
thf(fact_8_PropVar_Odistinct_I25_J,axiom,
    ! [X41: nat,X42: nat,X51: nat,X52: nat] :
      ( ( multis6646701651571564453xOneJI @ X41 @ X42 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(25)
thf(fact_9_PropVar_Odistinct_I21_J,axiom,
    ! [X31: nat,X32: nat,X51: nat,X52: nat] :
      ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(21)
thf(fact_10_PropVar_Osize__gen_I5_J,axiom,
    ! [X51: nat,X52: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(5)
thf(fact_11__C1_C_I1_J,axiom,
    ( ord_less_nat @ j
    @ ( size_s6752116297430514276ropVar
      @ ( map_na2145702811007361372ropVar
        @ ^ [J: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ i @ J ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385456105ZeroIJ @ i @ J ) @ ( multis6646701651571498855xOneIJ @ i @ J ) ) )
        @ ( upt @ zero_zero_nat @ m ) ) ) ) ).

% "1"(1)
thf(fact_12_PropVar_Oinject_I3_J,axiom,
    ! [X31: nat,X32: nat,Y31: nat,Y32: nat] :
      ( ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
        = ( multis2983220944385521703ZeroJI @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% PropVar.inject(3)
thf(fact_13_PropVar_Oinject_I4_J,axiom,
    ! [X41: nat,X42: nat,Y41: nat,Y42: nat] :
      ( ( ( multis6646701651571564453xOneJI @ X41 @ X42 )
        = ( multis6646701651571564453xOneJI @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% PropVar.inject(4)
thf(fact_14__C2_C_I1_J,axiom,
    member_nat @ i @ ( set_nat2 @ ( upt @ zero_zero_nat @ n ) ) ).

% "2"(1)
thf(fact_15_PropVar_Osimps_I42_J,axiom,
    ! [F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,X61: nat,X62: nat] :
      ( ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( multis6646701651571498855xOneIJ @ X61 @ X62 ) )
      = ( F6 @ X61 @ X62 ) ) ).

% PropVar.simps(42)
thf(fact_16_PropVar_Osimps_I40_J,axiom,
    ! [F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,X41: nat,X42: nat] :
      ( ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( multis6646701651571564453xOneJI @ X41 @ X42 ) )
      = ( F4 @ X41 @ X42 ) ) ).

% PropVar.simps(40)
thf(fact_17_PropVar_Osimps_I39_J,axiom,
    ! [F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,X31: nat,X32: nat] :
      ( ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( multis2983220944385521703ZeroJI @ X31 @ X32 ) )
      = ( F3 @ X31 @ X32 ) ) ).

% PropVar.simps(39)
thf(fact_18_PropVar_Osimps_I37_J,axiom,
    ! [F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,X11: nat,X12: nat] :
      ( ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = ( F1 @ X11 @ X12 ) ) ).

% PropVar.simps(37)
thf(fact_19_PropVar_Odistinct_I27_J,axiom,
    ! [X41: nat,X42: nat,X61: nat,X62: nat] :
      ( ( multis6646701651571564453xOneJI @ X41 @ X42 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(27)
thf(fact_20_PropVar_Odistinct_I23_J,axiom,
    ! [X31: nat,X32: nat,X61: nat,X62: nat] :
      ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(23)
thf(fact_21_PropVar_Odistinct_I19_J,axiom,
    ! [X31: nat,X32: nat,X41: nat,X42: nat] :
      ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
     != ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).

% PropVar.distinct(19)
thf(fact_22_PropVar_Odistinct_I5_J,axiom,
    ! [X11: nat,X12: nat,X41: nat,X42: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).

% PropVar.distinct(5)
thf(fact_23_PropVar_Odistinct_I3_J,axiom,
    ! [X11: nat,X12: nat,X31: nat,X32: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis2983220944385521703ZeroJI @ X31 @ X32 ) ) ).

% PropVar.distinct(3)
thf(fact_24_PropVar_Osize__gen_I6_J,axiom,
    ! [X61: nat,X62: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis6646701651571498855xOneIJ @ X61 @ X62 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(6)
thf(fact_25_PropVar_Osize__gen_I4_J,axiom,
    ! [X41: nat,X42: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis6646701651571564453xOneJI @ X41 @ X42 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(4)
thf(fact_26_PropVar_Osize__gen_I3_J,axiom,
    ! [X31: nat,X32: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis2983220944385521703ZeroJI @ X31 @ X32 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(3)
thf(fact_27_PropVar_Osize__gen_I1_J,axiom,
    ! [X11: nat,X12: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(1)
thf(fact_28_PropVar_Ocase__distrib,axiom,
    ! [H: $o > $o,F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,PropVar: multis3193088007478089820ropVar] :
      ( ( H @ ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ PropVar ) )
      = ( multis8641292176345589959pVar_o
        @ ^ [X1: nat,X2: nat] : ( H @ ( F1 @ X1 @ X2 ) )
        @ ^ [X: nat] : ( H @ ( F2 @ X ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F3 @ X1 @ X2 ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F4 @ X1 @ X2 ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F5 @ X1 @ X2 ) )
        @ ^ [X1: nat,X2: nat] : ( H @ ( F6 @ X1 @ X2 ) )
        @ PropVar ) ) ).

% PropVar.case_distrib
thf(fact_29_PropVar_Odistinct_I9_J,axiom,
    ! [X11: nat,X12: nat,X61: nat,X62: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(9)
thf(fact_30_nth__map,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( nth_Pr227365692153740281ropVar @ ( map_Pr3565048146863404228ropVar @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr227365692153740281ropVar @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_31_nth__map,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > nat] :
      ( ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( nth_nat @ ( map_Pr7810381211709419518ar_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_Pr227365692153740281ropVar @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_32_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr227365692153740281ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_33_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Mu4537670848183934179ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_34_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_35_map__nth,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( map_na2466558649495876678ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s1943289338788136526ropVar @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_36_map__nth,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( map_na2145702811007361372ropVar @ ( nth_Pr227365692153740281ropVar @ Xs ) @ ( upt @ zero_zero_nat @ ( size_s6752116297430514276ropVar @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_37_map__nth,axiom,
    ! [Xs: list_nat] :
      ( ( map_nat_nat @ ( nth_nat @ Xs ) @ ( upt @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) )
      = Xs ) ).

% map_nth
thf(fact_38_length__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( size_s1943289338788136526ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_39_length__map,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( size_s6752116297430514276ropVar @ ( map_Pr3565048146863404228ropVar @ F @ Xs ) )
      = ( size_s6752116297430514276ropVar @ Xs ) ) ).

% length_map
thf(fact_40_length__map,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( size_s6752116297430514276ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_41_length__map,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar] :
      ( ( size_size_list_nat @ ( map_Pr7810381211709419518ar_nat @ F @ Xs ) )
      = ( size_s6752116297430514276ropVar @ Xs ) ) ).

% length_map
thf(fact_42_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_43_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_44_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_45_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_46_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_47_map__equality__iff,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,G: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Ys ) )
      = ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Pr227365692153740281ropVar @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_48_map__equality__iff,axiom,
    ! [F: produc2642885279816024434ropVar > multis3193088007478089820ropVar,Xs: list_P8791008543742759928ropVar,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_Pr3335389669906741166ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Pr227365692153740281ropVar @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_49_map__equality__iff,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Pr227365692153740281ropVar @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_50_map__equality__iff,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_Pr3565048146863404228ropVar @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_Pr227365692153740281ropVar @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_51_map__equality__iff,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: produc2642885279816024434ropVar > multis3193088007478089820ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_Pr3335389669906741166ropVar @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_Pr227365692153740281ropVar @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_52_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: produc2642885279816024434ropVar > nat,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Pr7810381211709419518ar_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_Pr227365692153740281ropVar @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_53_map__equality__iff,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_54_map__equality__iff,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_55_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I ) )
              = ( G @ ( nth_nat @ Ys @ I ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_56_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs2: list_nat] : Xs2 ) ) ).

% map_ident
thf(fact_57_nth__equalityI,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Xs ) )
           => ( ( nth_Pr227365692153740281ropVar @ Xs @ I2 )
              = ( nth_Pr227365692153740281ropVar @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_58_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_59_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
    ! [A: multis3193088007478089820ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( member8421201611525043589ropVar @ A @ ( collec8658135508005558727ropVar @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_61_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_62_Collect__mem__eq,axiom,
    ! [A2: set_Mu5658373669842198716ropVar] :
      ( ( collec8658135508005558727ropVar
        @ ^ [X: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_63_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > produc2642885279816024434ropVar > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X3: produc2642885279816024434ropVar] : ( P @ I @ X3 ) ) )
      = ( ? [Xs2: list_P8791008543742759928ropVar] :
            ( ( ( size_s6752116297430514276ropVar @ Xs2 )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_Pr227365692153740281ropVar @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_64_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ K )
           => ? [X3: nat] : ( P @ I @ X3 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K )
               => ( P @ I @ ( nth_nat @ Xs2 @ I ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_65_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_P8791008543742759928ropVar,Z: list_P8791008543742759928ropVar] : ( Y = Z ) )
    = ( ^ [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
          ( ( ( size_s6752116297430514276ropVar @ Xs2 )
            = ( size_s6752116297430514276ropVar @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Xs2 ) )
             => ( ( nth_Pr227365692153740281ropVar @ Xs2 @ I )
                = ( nth_Pr227365692153740281ropVar @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_66_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I )
                = ( nth_nat @ Ys2 @ I ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_67_map__eq__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > produc2642885279816024434ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_68_map__eq__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_69_map__eq__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_70_ex__map__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_na2145702811007361372ropVar @ F @ Xs2 ) ) )
      = ( ! [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Ys ) )
           => ? [Y2: nat] :
                ( X
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_71_ex__map__conv,axiom,
    ! [Ys: list_nat,F: nat > nat] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_nat_nat @ F @ Xs2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ? [Y2: nat] :
                ( X
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_72_ex__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_na2466558649495876678ropVar @ F @ Xs2 ) ) )
      = ( ! [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Ys ) )
           => ? [Y2: nat] :
                ( X
                = ( F @ Y2 ) ) ) ) ) ).

% ex_map_conv
thf(fact_73_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_na2145702811007361372ropVar @ F @ Xs )
          = ( map_na2145702811007361372ropVar @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_74_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ Xs )
          = ( map_na2466558649495876678ropVar @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_75_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) )
       => ( ( map_nat_nat @ F @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_76_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_77_map__idI,axiom,
    ! [Xs: list_M6740372356978808802ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ( ( F @ X4 )
            = X4 ) )
     => ( ( map_Mu2265489581466636612ropVar @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_78_map__ext,axiom,
    ! [Xs: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Xs ) ) ) ).

% map_ext
thf(fact_79_map__ext,axiom,
    ! [Xs: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Xs ) ) ) ).

% map_ext
thf(fact_80_map__ext,axiom,
    ! [Xs: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X4 )
            = ( G @ X4 ) ) )
     => ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_81_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_nat_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_82_list_Omap__ident__strong,axiom,
    ! [T: list_M6740372356978808802ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ! [Z2: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ Z2 @ ( set_Mu146256361055586289ropVar @ T ) )
         => ( ( F @ Z2 )
            = Z2 ) )
     => ( ( map_Mu2265489581466636612ropVar @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_83_list_Oinj__map__strong,axiom,
    ! [X5: list_nat,Xa: list_nat,F: nat > produc2642885279816024434ropVar,Fa: nat > produc2642885279816024434ropVar] :
      ( ! [Z2: nat,Za: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X5 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_na2145702811007361372ropVar @ F @ X5 )
          = ( map_na2145702811007361372ropVar @ Fa @ Xa ) )
       => ( X5 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_84_list_Oinj__map__strong,axiom,
    ! [X5: list_nat,Xa: list_nat,F: nat > multis3193088007478089820ropVar,Fa: nat > multis3193088007478089820ropVar] :
      ( ! [Z2: nat,Za: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X5 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_na2466558649495876678ropVar @ F @ X5 )
          = ( map_na2466558649495876678ropVar @ Fa @ Xa ) )
       => ( X5 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_85_list_Oinj__map__strong,axiom,
    ! [X5: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
      ( ! [Z2: nat,Za: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X5 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z2 )
                = ( Fa @ Za ) )
             => ( Z2 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F @ X5 )
          = ( map_nat_nat @ Fa @ Xa ) )
       => ( X5 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_86_list_Omap__cong0,axiom,
    ! [X5: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X5 ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_na2145702811007361372ropVar @ F @ X5 )
        = ( map_na2145702811007361372ropVar @ G @ X5 ) ) ) ).

% list.map_cong0
thf(fact_87_list_Omap__cong0,axiom,
    ! [X5: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X5 ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_na2466558649495876678ropVar @ F @ X5 )
        = ( map_na2466558649495876678ropVar @ G @ X5 ) ) ) ).

% list.map_cong0
thf(fact_88_list_Omap__cong0,axiom,
    ! [X5: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [Z2: nat] :
          ( ( member_nat @ Z2 @ ( set_nat2 @ X5 ) )
         => ( ( F @ Z2 )
            = ( G @ Z2 ) ) )
     => ( ( map_nat_nat @ F @ X5 )
        = ( map_nat_nat @ G @ X5 ) ) ) ).

% list.map_cong0
thf(fact_89_list_Omap__cong,axiom,
    ! [X5: list_nat,Ya: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ( X5 = Ya )
     => ( ! [Z2: nat] :
            ( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_na2145702811007361372ropVar @ F @ X5 )
          = ( map_na2145702811007361372ropVar @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_90_list_Omap__cong,axiom,
    ! [X5: list_nat,Ya: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ( X5 = Ya )
     => ( ! [Z2: nat] :
            ( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ X5 )
          = ( map_na2466558649495876678ropVar @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_91_list_Omap__cong,axiom,
    ! [X5: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
      ( ( X5 = Ya )
     => ( ! [Z2: nat] :
            ( ( member_nat @ Z2 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z2 )
              = ( G @ Z2 ) ) )
       => ( ( map_nat_nat @ F @ X5 )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_92_length__pos__if__in__set,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s1943289338788136526ropVar @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_93_length__pos__if__in__set,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6752116297430514276ropVar @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_94_length__pos__if__in__set,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_95_all__set__conv__all__nth,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1943289338788136526ropVar @ Xs ) )
           => ( P @ ( nth_Mu4537670848183934179ropVar @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_96_all__set__conv__all__nth,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ! [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Xs ) )
           => ( P @ ( nth_Pr227365692153740281ropVar @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_97_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_98_all__nth__imp__all__set,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o,X5: multis3193088007478089820ropVar] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Xs ) )
         => ( P @ ( nth_Mu4537670848183934179ropVar @ Xs @ I2 ) ) )
     => ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ( P @ X5 ) ) ) ).

% all_nth_imp_all_set
thf(fact_99_all__nth__imp__all__set,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o,X5: produc2642885279816024434ropVar] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Xs ) )
         => ( P @ ( nth_Pr227365692153740281ropVar @ Xs @ I2 ) ) )
     => ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
       => ( P @ X5 ) ) ) ).

% all_nth_imp_all_set
thf(fact_100_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X5: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
       => ( P @ X5 ) ) ) ).

% all_nth_imp_all_set
thf(fact_101_in__set__conv__nth,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s1943289338788136526ropVar @ Xs ) )
            & ( ( nth_Mu4537670848183934179ropVar @ Xs @ I )
              = X5 ) ) ) ) ).

% in_set_conv_nth
thf(fact_102_in__set__conv__nth,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Xs ) )
            & ( ( nth_Pr227365692153740281ropVar @ Xs @ I )
              = X5 ) ) ) ) ).

% in_set_conv_nth
thf(fact_103_in__set__conv__nth,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
      = ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I )
              = X5 ) ) ) ) ).

% in_set_conv_nth
thf(fact_104_list__ball__nth,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ord_less_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( ! [X4: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_Mu4537670848183934179ropVar @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_105_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ! [X4: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_Pr227365692153740281ropVar @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_106_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_107_nth__mem,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( ord_less_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( member8421201611525043589ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ N ) @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% nth_mem
thf(fact_108_nth__mem,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar] :
      ( ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( member9210975342120487067ropVar @ ( nth_Pr227365692153740281ropVar @ Xs @ N ) @ ( set_Pr3946994337432900359ropVar @ Xs ) ) ) ).

% nth_mem
thf(fact_109_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_110_zero__reorient,axiom,
    ! [X5: nat] :
      ( ( zero_zero_nat = X5 )
      = ( X5 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_111_linorder__neqE__nat,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( X5 != Y3 )
     => ( ~ ( ord_less_nat @ X5 @ Y3 )
       => ( ord_less_nat @ Y3 @ X5 ) ) ) ).

% linorder_neqE_nat
thf(fact_112_infinite__descent,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ~ ( P @ N2 )
         => ? [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
              & ~ ( P @ M ) ) )
     => ( P @ N ) ) ).

% infinite_descent
thf(fact_113_nat__less__induct,axiom,
    ! [P: nat > $o,N: nat] :
      ( ! [N2: nat] :
          ( ! [M: nat] :
              ( ( ord_less_nat @ M @ N2 )
             => ( P @ M ) )
         => ( P @ N2 ) )
     => ( P @ N ) ) ).

% nat_less_induct
thf(fact_114_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_115_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_116_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_117_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_118_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_119_size__neq__size__imp__neq,axiom,
    ! [X5: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ X5 )
       != ( size_s6752116297430514276ropVar @ Y3 ) )
     => ( X5 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_120_size__neq__size__imp__neq,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( ( size_s6253272723116879048ropVar @ X5 )
       != ( size_s6253272723116879048ropVar @ Y3 ) )
     => ( X5 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_121_size__neq__size__imp__neq,axiom,
    ! [X5: char,Y3: char] :
      ( ( ( size_size_char @ X5 )
       != ( size_size_char @ Y3 ) )
     => ( X5 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_122_size__neq__size__imp__neq,axiom,
    ! [X5: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X5 )
       != ( size_size_list_nat @ Y3 ) )
     => ( X5 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_123_neq__if__length__neq,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
       != ( size_s6752116297430514276ropVar @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_124_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_125_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_P8791008543742759928ropVar] :
      ( ( size_s6752116297430514276ropVar @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_126_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_127_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_128_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_129_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_130_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_131_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_132_infinite__descent0,axiom,
    ! [P: nat > $o,N: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N2: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( ~ ( P @ N2 )
             => ? [M: nat] :
                  ( ( ord_less_nat @ M @ N2 )
                  & ~ ( P @ M ) ) ) )
       => ( P @ N ) ) ) ).

% infinite_descent0
thf(fact_133_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_134_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_135_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_136_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_137_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_138_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_139_length__induct,axiom,
    ! [P: list_P8791008543742759928ropVar > $o,Xs: list_P8791008543742759928ropVar] :
      ( ! [Xs3: list_P8791008543742759928ropVar] :
          ( ! [Ys3: list_P8791008543742759928ropVar] :
              ( ( ord_less_nat @ ( size_s6752116297430514276ropVar @ Ys3 ) @ ( size_s6752116297430514276ropVar @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_140_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys3 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_141_map__eq__imp__length__eq,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,G: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Ys ) )
     => ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_142_map__eq__imp__length__eq,axiom,
    ! [F: produc2642885279816024434ropVar > multis3193088007478089820ropVar,Xs: list_P8791008543742759928ropVar,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_Pr3335389669906741166ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
     => ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_143_map__eq__imp__length__eq,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_144_map__eq__imp__length__eq,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_Pr3565048146863404228ropVar @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_145_map__eq__imp__length__eq,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: produc2642885279816024434ropVar > multis3193088007478089820ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_Pr3335389669906741166ropVar @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_146_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: produc2642885279816024434ropVar > nat,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Pr7810381211709419518ar_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_147_map__eq__imp__length__eq,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_148_map__eq__imp__length__eq,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_149_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_150_PropVar_Oexhaust,axiom,
    ! [Y3: multis3193088007478089820ropVar] :
      ( ! [X112: nat,X122: nat] :
          ( Y3
         != ( multis387687052011358179_Gamma @ X112 @ X122 ) )
     => ( ! [X22: nat] :
            ( Y3
           != ( multis2544335231667181926psilon @ X22 ) )
       => ( ! [X312: nat,X322: nat] :
              ( Y3
             != ( multis2983220944385521703ZeroJI @ X312 @ X322 ) )
         => ( ! [X412: nat,X422: nat] :
                ( Y3
               != ( multis6646701651571564453xOneJI @ X412 @ X422 ) )
           => ( ! [X512: nat,X522: nat] :
                  ( Y3
                 != ( multis2983220944385456105ZeroIJ @ X512 @ X522 ) )
             => ~ ! [X612: nat,X622: nat] :
                    ( Y3
                   != ( multis6646701651571498855xOneIJ @ X612 @ X622 ) ) ) ) ) ) ) ).

% PropVar.exhaust
thf(fact_151_old_Oprod_Oinject,axiom,
    ! [A: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar,A3: multis3193088007478089820ropVar,B2: produc7679832412844898215ropVar] :
      ( ( ( produc6194868611276206818ropVar @ A @ B )
        = ( produc6194868611276206818ropVar @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_152_old_Oprod_Oinject,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A3: multis3193088007478089820ropVar,B2: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ A @ B )
        = ( produc6936306168532358295ropVar @ A3 @ B2 ) )
      = ( ( A = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_153_prod_Oinject,axiom,
    ! [X13: multis3193088007478089820ropVar,X23: produc7679832412844898215ropVar,Y1: multis3193088007478089820ropVar,Y22: produc7679832412844898215ropVar] :
      ( ( ( produc6194868611276206818ropVar @ X13 @ X23 )
        = ( produc6194868611276206818ropVar @ Y1 @ Y22 ) )
      = ( ( X13 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
thf(fact_154_prod_Oinject,axiom,
    ! [X13: multis3193088007478089820ropVar,X23: multis3193088007478089820ropVar,Y1: multis3193088007478089820ropVar,Y22: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ X13 @ X23 )
        = ( produc6936306168532358295ropVar @ Y1 @ Y22 ) )
      = ( ( X13 = Y1 )
        & ( X23 = Y22 ) ) ) ).

% prod.inject
thf(fact_155_prod__induct3,axiom,
    ! [P: produc2642885279816024434ropVar > $o,X5: produc2642885279816024434ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] : ( P @ ( produc6194868611276206818ropVar @ A4 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) )
     => ( P @ X5 ) ) ).

% prod_induct3
thf(fact_156_PropVar_Oinject_I2_J,axiom,
    ! [X23: nat,Y22: nat] :
      ( ( ( multis2544335231667181926psilon @ X23 )
        = ( multis2544335231667181926psilon @ Y22 ) )
      = ( X23 = Y22 ) ) ).

% PropVar.inject(2)
thf(fact_157_PropVar_Osize_I8_J,axiom,
    ! [X23: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis2544335231667181926psilon @ X23 ) )
      = zero_zero_nat ) ).

% PropVar.size(8)
thf(fact_158_PropVar_Odistinct_I1_J,axiom,
    ! [X11: nat,X12: nat,X23: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis2544335231667181926psilon @ X23 ) ) ).

% PropVar.distinct(1)
thf(fact_159_PropVar_Odistinct_I17_J,axiom,
    ! [X23: nat,X61: nat,X62: nat] :
      ( ( multis2544335231667181926psilon @ X23 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(17)
thf(fact_160_PropVar_Odistinct_I15_J,axiom,
    ! [X23: nat,X51: nat,X52: nat] :
      ( ( multis2544335231667181926psilon @ X23 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(15)
thf(fact_161_PropVar_Odistinct_I13_J,axiom,
    ! [X23: nat,X41: nat,X42: nat] :
      ( ( multis2544335231667181926psilon @ X23 )
     != ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).

% PropVar.distinct(13)
thf(fact_162_PropVar_Odistinct_I11_J,axiom,
    ! [X23: nat,X31: nat,X32: nat] :
      ( ( multis2544335231667181926psilon @ X23 )
     != ( multis2983220944385521703ZeroJI @ X31 @ X32 ) ) ).

% PropVar.distinct(11)
thf(fact_163_PropVar_Osimps_I38_J,axiom,
    ! [F1: nat > nat > $o,F2: nat > $o,F3: nat > nat > $o,F4: nat > nat > $o,F5: nat > nat > $o,F6: nat > nat > $o,X23: nat] :
      ( ( multis8641292176345589959pVar_o @ F1 @ F2 @ F3 @ F4 @ F5 @ F6 @ ( multis2544335231667181926psilon @ X23 ) )
      = ( F2 @ X23 ) ) ).

% PropVar.simps(38)
thf(fact_164_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2642885279816024434ropVar] :
      ~ ! [A4: multis3193088007478089820ropVar,B3: produc7679832412844898215ropVar] :
          ( Y3
         != ( produc6194868611276206818ropVar @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_165_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc7679832412844898215ropVar] :
      ~ ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( Y3
         != ( produc6936306168532358295ropVar @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_166_surj__pair,axiom,
    ! [P2: produc2642885279816024434ropVar] :
    ? [X4: multis3193088007478089820ropVar,Y4: produc7679832412844898215ropVar] :
      ( P2
      = ( produc6194868611276206818ropVar @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_167_surj__pair,axiom,
    ! [P2: produc7679832412844898215ropVar] :
    ? [X4: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
      ( P2
      = ( produc6936306168532358295ropVar @ X4 @ Y4 ) ) ).

% surj_pair
thf(fact_168_prod__cases,axiom,
    ! [P: produc2642885279816024434ropVar > $o,P2: produc2642885279816024434ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: produc7679832412844898215ropVar] : ( P @ ( produc6194868611276206818ropVar @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_169_prod__cases,axiom,
    ! [P: produc7679832412844898215ropVar > $o,P2: produc7679832412844898215ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] : ( P @ ( produc6936306168532358295ropVar @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_170_Pair__inject,axiom,
    ! [A: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar,A3: multis3193088007478089820ropVar,B2: produc7679832412844898215ropVar] :
      ( ( ( produc6194868611276206818ropVar @ A @ B )
        = ( produc6194868611276206818ropVar @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_171_Pair__inject,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A3: multis3193088007478089820ropVar,B2: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ A @ B )
        = ( produc6936306168532358295ropVar @ A3 @ B2 ) )
     => ~ ( ( A = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_172_PropVar_Osize__gen_I2_J,axiom,
    ! [X23: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis2544335231667181926psilon @ X23 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(2)
thf(fact_173_PropVar_Osize_I7_J,axiom,
    ! [X11: nat,X12: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = zero_zero_nat ) ).

% PropVar.size(7)
thf(fact_174_PropVar_Osize_I12_J,axiom,
    ! [X61: nat,X62: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis6646701651571498855xOneIJ @ X61 @ X62 ) )
      = zero_zero_nat ) ).

% PropVar.size(12)
thf(fact_175_PropVar_Osize_I11_J,axiom,
    ! [X51: nat,X52: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) )
      = zero_zero_nat ) ).

% PropVar.size(11)
thf(fact_176_PropVar_Osize_I10_J,axiom,
    ! [X41: nat,X42: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis6646701651571564453xOneJI @ X41 @ X42 ) )
      = zero_zero_nat ) ).

% PropVar.size(10)
thf(fact_177_PropVar_Osize_I9_J,axiom,
    ! [X31: nat,X32: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis2983220944385521703ZeroJI @ X31 @ X32 ) )
      = zero_zero_nat ) ).

% PropVar.size(9)
thf(fact_178_prod__cases3,axiom,
    ! [Y3: produc2642885279816024434ropVar] :
      ~ ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] :
          ( Y3
         != ( produc6194868611276206818ropVar @ A4 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_179_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C2: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_180_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_181_nth__zip,axiom,
    ! [I3: nat,Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s8779405381146381081ropVar @ Ys ) )
       => ( ( nth_Pr227365692153740281ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) @ I3 )
          = ( produc6194868611276206818ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ I3 ) @ ( nth_Pr6144466203599902510ropVar @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_182_nth__zip,axiom,
    ! [I3: nat,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Ys ) )
       => ( ( nth_Pr6144466203599902510ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) @ I3 )
          = ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ I3 ) @ ( nth_Mu4537670848183934179ropVar @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_183_nth__zip,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Ys ) )
       => ( ( nth_Pr4218712991885816750ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs @ Ys ) @ I3 )
          = ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs @ I3 ) @ ( nth_Pr227365692153740281ropVar @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_184_nth__zip,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,Ys: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr6781381391575616938ar_nat @ ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys ) @ I3 )
          = ( produc5526329313106567659ar_nat @ ( nth_Pr227365692153740281ropVar @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_185_nth__zip,axiom,
    ! [I3: nat,Xs: list_nat,Ys: list_P8791008543742759928ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Ys ) )
       => ( ( nth_Pr2354896103702612176ropVar @ ( zip_na5961122453619627049ropVar @ Xs @ Ys ) @ I3 )
          = ( produc9085022949259285321ropVar @ ( nth_nat @ Xs @ I3 ) @ ( nth_Pr227365692153740281ropVar @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_186_nth__zip,axiom,
    ! [I3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I3 )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_187_length__code,axiom,
    ( size_s6752116297430514276ropVar
    = ( gen_le3870443462522697605ropVar @ zero_zero_nat ) ) ).

% length_code
thf(fact_188_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_189_pos_I3_J,axiom,
    ! [Xs: list_nat,X5: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X4 )
              = X5 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y5 )
                    = X5 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% pos(3)
thf(fact_190_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_191_distinct__upt,axiom,
    ! [I3: nat,J2: nat] : ( distinct_nat @ ( upt @ I3 @ J2 ) ) ).

% distinct_upt
thf(fact_192_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_193_zip__same,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ B ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Xs ) ) )
      = ( ( member8421201611525043589ropVar @ A @ ( set_Mu146256361055586289ropVar @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_194_in__set__zipE,axiom,
    ! [X5: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_195_in__set__zipE,axiom,
    ! [X5: nat,Y3: multis3193088007478089820ropVar,Xs: list_nat,Ys: list_M6740372356978808802ropVar] :
      ( ( member1824086966109417880ropVar @ ( produc1720817013404193843ropVar @ X5 @ Y3 ) @ ( set_Pr1649027773232293932ropVar @ ( zip_na7122055390739322131ropVar @ Xs @ Ys ) ) )
     => ~ ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ~ ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_196_in__set__zipE,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: nat,Xs: list_M6740372356978808802ropVar,Ys: list_nat] :
      ( ( member8816771811004535154ar_nat @ ( produc1224776155581759701ar_nat @ X5 @ Y3 ) @ ( set_Pr8641712618127411206ar_nat @ ( zip_Mu6626014532916887989ar_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_197_in__set__zipE,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: produc7679832412844898215ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X5 @ Y3 ) @ ( set_Pr3946994337432900359ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) ) )
     => ~ ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( member2619728555115997648ropVar @ Y3 @ ( set_Pr4075413451802982460ropVar @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_198_in__set__zipE,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) )
     => ~ ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_199_set__zip__leftD,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: produc7679832412844898215ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X5 @ Y3 ) @ ( set_Pr3946994337432900359ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) ) )
     => ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_200_set__zip__leftD,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) )
     => ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_201_set__zip__rightD,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: produc7679832412844898215ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X5 @ Y3 ) @ ( set_Pr3946994337432900359ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) ) )
     => ( member2619728555115997648ropVar @ Y3 @ ( set_Pr4075413451802982460ropVar @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_202_set__zip__rightD,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) )
     => ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_203_zip__same__conv__map,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( zip_Mu2706791923038522295ropVar @ Xs @ Xs )
      = ( map_Mu7320851716970821775ropVar
        @ ^ [X: multis3193088007478089820ropVar] : ( produc6936306168532358295ropVar @ X @ X )
        @ Xs ) ) ).

% zip_same_conv_map
thf(fact_204_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar,X5: multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_s8779405381146381081ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ~ ! [Y4: produc7679832412844898215ropVar] :
              ~ ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X5 @ Y4 ) @ ( set_Pr3946994337432900359ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_205_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,X5: multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_s1943289338788136526ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ~ ! [Y4: multis3193088007478089820ropVar] :
              ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y4 ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_206_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P8791008543742759928ropVar,X5: multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ~ ! [Y4: produc2642885279816024434ropVar] :
              ~ ( member1707986370202145254ropVar @ ( produc4268072148187102125ropVar @ X5 @ Y4 ) @ ( set_Pr1973727305987283026ropVar @ ( zip_Mu7420031413393175245ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_207_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_nat,X5: multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ~ ! [Y4: nat] :
              ~ ( member8816771811004535154ar_nat @ ( produc1224776155581759701ar_nat @ X5 @ Y4 ) @ ( set_Pr8641712618127411206ar_nat @ ( zip_Mu6626014532916887989ar_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_208_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
       => ~ ! [Y4: produc2642885279816024434ropVar] :
              ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y4 ) @ ( set_Pr1712790080554429628ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_209_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,X5: produc2642885279816024434ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
       => ~ ! [Y4: nat] :
              ~ ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X5 @ Y4 ) @ ( set_Pr7230637098968509724ar_nat @ ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_210_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,X5: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y4: produc2642885279816024434ropVar] :
              ~ ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ X5 @ Y4 ) @ ( set_Pr2804151811095504962ropVar @ ( zip_na5961122453619627049ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_211_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_nat,X5: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y4: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_212_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar,Y3: produc7679832412844898215ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_s8779405381146381081ropVar @ Ys ) )
     => ( ( member2619728555115997648ropVar @ Y3 @ ( set_Pr4075413451802982460ropVar @ Ys ) )
       => ~ ! [X4: multis3193088007478089820ropVar] :
              ~ ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X4 @ Y3 ) @ ( set_Pr3946994337432900359ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_213_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_s1943289338788136526ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Ys ) )
       => ~ ! [X4: multis3193088007478089820ropVar] :
              ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_214_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s1943289338788136526ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Ys ) )
       => ~ ! [X4: produc2642885279816024434ropVar] :
              ~ ( member45177508153868858ropVar @ ( produc2108810064830654721ropVar @ X4 @ Y3 ) @ ( set_Pr310918443939006630ropVar @ ( zip_Pr5260769330036727841ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_215_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ Ys ) )
       => ~ ! [X4: produc2642885279816024434ropVar] :
              ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ ( set_Pr1712790080554429628ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_216_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Y3: nat] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: produc2642885279816024434ropVar] :
              ~ ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X4 @ Y3 ) @ ( set_Pr7230637098968509724ar_nat @ ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_217_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1943289338788136526ropVar @ Ys ) )
     => ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member1824086966109417880ropVar @ ( produc1720817013404193843ropVar @ X4 @ Y3 ) @ ( set_Pr1649027773232293932ropVar @ ( zip_na7122055390739322131ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_218_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ X4 @ Y3 ) @ ( set_Pr2804151811095504962ropVar @ ( zip_na5961122453619627049ropVar @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_219_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Y3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_220_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_P8791008543742759928ropVar,I3: nat,J2: nat] :
      ( ( distin2267423591311547883ropVar @ Xs )
     => ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
       => ( ( ord_less_nat @ J2 @ ( size_s6752116297430514276ropVar @ Xs ) )
         => ( ( ( nth_Pr227365692153740281ropVar @ Xs @ I3 )
              = ( nth_Pr227365692153740281ropVar @ Xs @ J2 ) )
            = ( I3 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_221_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I3: nat,J2: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Xs @ J2 ) )
            = ( I3 = J2 ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_222_distinct__conv__nth,axiom,
    ( distin2267423591311547883ropVar
    = ( ^ [Xs2: list_P8791008543742759928ropVar] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Xs2 ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_s6752116297430514276ropVar @ Xs2 ) )
             => ( ( I != J )
               => ( ( nth_Pr227365692153740281ropVar @ Xs2 @ I )
                 != ( nth_Pr227365692153740281ropVar @ Xs2 @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_223_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs2: list_nat] :
        ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs2 ) )
         => ! [J: nat] :
              ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs2 ) )
             => ( ( I != J )
               => ( ( nth_nat @ Xs2 @ I )
                 != ( nth_nat @ Xs2 @ J ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_224_distinct__Ex1,axiom,
    ! [Xs: list_M6740372356978808802ropVar,X5: multis3193088007478089820ropVar] :
      ( ( distin2675581014522817237ropVar @ Xs )
     => ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s1943289338788136526ropVar @ Xs ) )
            & ( ( nth_Mu4537670848183934179ropVar @ Xs @ X4 )
              = X5 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s1943289338788136526ropVar @ Xs ) )
                  & ( ( nth_Mu4537670848183934179ropVar @ Xs @ Y5 )
                    = X5 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_225_distinct__Ex1,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( distin2267423591311547883ropVar @ Xs )
     => ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_s6752116297430514276ropVar @ Xs ) )
            & ( ( nth_Pr227365692153740281ropVar @ Xs @ X4 )
              = X5 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s6752116297430514276ropVar @ Xs ) )
                  & ( ( nth_Pr227365692153740281ropVar @ Xs @ Y5 )
                    = X5 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_226_distinct__Ex1,axiom,
    ! [Xs: list_nat,X5: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
       => ? [X4: nat] :
            ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X4 )
              = X5 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y5 )
                    = X5 ) )
               => ( Y5 = X4 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_227_pos_I1_J,axiom,
    ! [Xs: list_nat,X5: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
       => ( ord_less_nat @ ( pos_of @ Xs @ X5 ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% pos(1)
thf(fact_228_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C2: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_229_pos_I2_J,axiom,
    ! [Xs: list_nat,X5: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
       => ( ( nth_nat @ Xs @ ( pos_of @ Xs @ X5 ) )
          = X5 ) ) ) ).

% pos(2)
thf(fact_230_p__xs_I3_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ xs1 ) )
     => ? [X4: nat] :
          ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ xs1 ) )
          & ( ( nth_nat @ xs1 @ X4 )
            = X5 )
          & ! [Y5: nat] :
              ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ xs1 ) )
                & ( ( nth_nat @ xs1 @ Y5 )
                  = X5 ) )
             => ( Y5 = X4 ) ) ) ) ).

% p_xs(3)
thf(fact_231_p__ys_I3_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ ys1 ) )
     => ? [X4: nat] :
          ( ( ord_less_nat @ X4 @ ( size_size_list_nat @ ys1 ) )
          & ( ( nth_nat @ ys1 @ X4 )
            = X5 )
          & ! [Y5: nat] :
              ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ ys1 ) )
                & ( ( nth_nat @ ys1 @ Y5 )
                  = X5 ) )
             => ( Y5 = X4 ) ) ) ) ).

% p_ys(3)
thf(fact_232_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_233__092_060open_062pos__of_A_092_060equiv_062_A_092_060lambda_062xs_Ai_O_ATHE_Ap_O_Ap_A_060_Alength_Axs_A_092_060and_062_Axs_A_B_Ap_A_061_Ai_092_060close_062,axiom,
    ( pos_of
    = ( ^ [Xs2: list_nat,I: nat] :
          ( the_nat
          @ ^ [P3: nat] :
              ( ( ord_less_nat @ P3 @ ( size_size_list_nat @ Xs2 ) )
              & ( ( nth_nat @ Xs2 @ P3 )
                = I ) ) ) ) ) ).

% \<open>pos_of \<equiv> \<lambda>xs i. THE p. p < length xs \<and> xs ! p = i\<close>
thf(fact_234_distinct__swap,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,J2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_s6752116297430514276ropVar @ Xs ) )
       => ( ( distin2267423591311547883ropVar @ ( list_u8100953110031273952ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ ( nth_Pr227365692153740281ropVar @ Xs @ J2 ) ) @ J2 @ ( nth_Pr227365692153740281ropVar @ Xs @ I3 ) ) )
          = ( distin2267423591311547883ropVar @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_235_distinct__swap,axiom,
    ! [I3: nat,Xs: list_nat,J2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_236__092_060open_062distinct_Axs1_092_060close_062,axiom,
    distinct_nat @ xs1 ).

% \<open>distinct xs1\<close>
thf(fact_237__092_060open_062distinct_Ays1_092_060close_062,axiom,
    distinct_nat @ ys1 ).

% \<open>distinct ys1\<close>
thf(fact_238_p__ys_I2_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ ys1 ) )
     => ( ( nth_nat @ ys1 @ ( pos_of @ ys1 @ X5 ) )
        = X5 ) ) ).

% p_ys(2)
thf(fact_239_p__xs_I2_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ xs1 ) )
     => ( ( nth_nat @ xs1 @ ( pos_of @ xs1 @ X5 ) )
        = X5 ) ) ).

% p_xs(2)
thf(fact_240_p__ys_I1_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ ys1 ) )
     => ( ord_less_nat @ ( pos_of @ ys1 @ X5 ) @ ( size_size_list_nat @ ys1 ) ) ) ).

% p_ys(1)
thf(fact_241_p__xs_I1_J,axiom,
    ! [X5: nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ xs1 ) )
     => ( ord_less_nat @ ( pos_of @ xs1 @ X5 ) @ ( size_size_list_nat @ xs1 ) ) ) ).

% p_xs(1)
thf(fact_242_pos__of__def,axiom,
    ! [Xs: list_nat,I3: nat] :
      ( ( pos_of @ Xs @ I3 )
      = ( the_nat
        @ ^ [P3: nat] :
            ( ( ord_less_nat @ P3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ P3 )
              = I3 ) ) ) ) ).

% pos_of_def
thf(fact_243_j__ys1_I1_J,axiom,
    ! [J2: nat] :
      ( ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
     => ( member_nat @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) @ ( set_nat2 @ xs1 ) ) ) ).

% j_ys1(1)
thf(fact_244_length__list__update,axiom,
    ! [Xs: list_P8791008543742759928ropVar,I3: nat,X5: produc2642885279816024434ropVar] :
      ( ( size_s6752116297430514276ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ X5 ) )
      = ( size_s6752116297430514276ropVar @ Xs ) ) ).

% length_list_update
thf(fact_245_length__list__update,axiom,
    ! [Xs: list_nat,I3: nat,X5: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I3 @ X5 ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_246_list__update__id,axiom,
    ! [Xs: list_P8791008543742759928ropVar,I3: nat] :
      ( ( list_u8100953110031273952ropVar @ Xs @ I3 @ ( nth_Pr227365692153740281ropVar @ Xs @ I3 ) )
      = Xs ) ).

% list_update_id
thf(fact_247_list__update__id,axiom,
    ! [Xs: list_nat,I3: nat] :
      ( ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ I3 ) )
      = Xs ) ).

% list_update_id
thf(fact_248_nth__list__update__neq,axiom,
    ! [I3: nat,J2: nat,Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( I3 != J2 )
     => ( ( nth_Pr227365692153740281ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ X5 ) @ J2 )
        = ( nth_Pr227365692153740281ropVar @ Xs @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_249_nth__list__update__neq,axiom,
    ! [I3: nat,J2: nat,Xs: list_nat,X5: nat] :
      ( ( I3 != J2 )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X5 ) @ J2 )
        = ( nth_nat @ Xs @ J2 ) ) ) ).

% nth_list_update_neq
thf(fact_250_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( nth_Pr227365692153740281ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ X5 ) @ I3 )
        = X5 ) ) ).

% nth_list_update_eq
thf(fact_251_nth__list__update__eq,axiom,
    ! [I3: nat,Xs: list_nat,X5: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X5 ) @ I3 )
        = X5 ) ) ).

% nth_list_update_eq
thf(fact_252_set__swap,axiom,
    ! [I3: nat,Xs: list_M6740372356978808802ropVar,J2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_s1943289338788136526ropVar @ Xs ) )
       => ( ( set_Mu146256361055586289ropVar @ ( list_u5207168704986282698ropVar @ ( list_u5207168704986282698ropVar @ Xs @ I3 @ ( nth_Mu4537670848183934179ropVar @ Xs @ J2 ) ) @ J2 @ ( nth_Mu4537670848183934179ropVar @ Xs @ I3 ) ) )
          = ( set_Mu146256361055586289ropVar @ Xs ) ) ) ) ).

% set_swap
thf(fact_253_set__swap,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,J2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_s6752116297430514276ropVar @ Xs ) )
       => ( ( set_Pr3946994337432900359ropVar @ ( list_u8100953110031273952ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ ( nth_Pr227365692153740281ropVar @ Xs @ J2 ) ) @ J2 @ ( nth_Pr227365692153740281ropVar @ Xs @ I3 ) ) )
          = ( set_Pr3946994337432900359ropVar @ Xs ) ) ) ) ).

% set_swap
thf(fact_254_set__swap,axiom,
    ! [I3: nat,Xs: list_nat,J2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I3 @ ( nth_nat @ Xs @ J2 ) ) @ J2 @ ( nth_nat @ Xs @ I3 ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_255_j__ys1_I3_J,axiom,
    ! [J2: nat] :
      ( ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
     => ( v2 @ ( multis387687052011358179_Gamma @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) @ J2 ) ) ) ).

% j_ys1(3)
thf(fact_256_zip__update,axiom,
    ! [Xs: list_M6740372356978808802ropVar,I3: nat,X5: multis3193088007478089820ropVar,Ys: list_P444016040222843053ropVar,Y3: produc7679832412844898215ropVar] :
      ( ( zip_Mu6605103971575351810ropVar @ ( list_u5207168704986282698ropVar @ Xs @ I3 @ X5 ) @ ( list_u4256948785802686485ropVar @ Ys @ I3 @ Y3 ) )
      = ( list_u8100953110031273952ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) @ I3 @ ( produc6194868611276206818ropVar @ X5 @ Y3 ) ) ) ).

% zip_update
thf(fact_257_zip__update,axiom,
    ! [Xs: list_M6740372356978808802ropVar,I3: nat,X5: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( zip_Mu2706791923038522295ropVar @ ( list_u5207168704986282698ropVar @ Xs @ I3 @ X5 ) @ ( list_u5207168704986282698ropVar @ Ys @ I3 @ Y3 ) )
      = ( list_u4256948785802686485ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) @ I3 @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) ) ) ).

% zip_update
thf(fact_258_map__update,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,K: nat,Y3: nat] :
      ( ( map_na2145702811007361372ropVar @ F @ ( list_update_nat @ Xs @ K @ Y3 ) )
      = ( list_u8100953110031273952ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_259_map__update,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,K: nat,Y3: nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( list_update_nat @ Xs @ K @ Y3 ) )
      = ( list_u5207168704986282698ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_260_map__update,axiom,
    ! [F: nat > nat,Xs: list_nat,K: nat,Y3: nat] :
      ( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y3 ) )
      = ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y3 ) ) ) ).

% map_update
thf(fact_261_set__update__memI,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,X5: multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) )
     => ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ ( list_u5207168704986282698ropVar @ Xs @ N @ X5 ) ) ) ) ).

% set_update_memI
thf(fact_262_set__update__memI,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ ( list_u8100953110031273952ropVar @ Xs @ N @ X5 ) ) ) ) ).

% set_update_memI
thf(fact_263_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X5: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X5 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X5 ) ) ) ) ).

% set_update_memI
thf(fact_264_nth__list__update,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,J2: nat,X5: produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( ( I3 = J2 )
         => ( ( nth_Pr227365692153740281ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ X5 ) @ J2 )
            = X5 ) )
        & ( ( I3 != J2 )
         => ( ( nth_Pr227365692153740281ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ X5 ) @ J2 )
            = ( nth_Pr227365692153740281ropVar @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_265_nth__list__update,axiom,
    ! [I3: nat,Xs: list_nat,J2: nat,X5: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I3 = J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X5 ) @ J2 )
            = X5 ) )
        & ( ( I3 != J2 )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I3 @ X5 ) @ J2 )
            = ( nth_nat @ Xs @ J2 ) ) ) ) ) ).

% nth_list_update
thf(fact_266_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( ( list_u8100953110031273952ropVar @ Xs @ I3 @ X5 )
          = Xs )
        = ( ( nth_Pr227365692153740281ropVar @ Xs @ I3 )
          = X5 ) ) ) ).

% list_update_same_conv
thf(fact_267_list__update__same__conv,axiom,
    ! [I3: nat,Xs: list_nat,X5: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I3 @ X5 )
          = Xs )
        = ( ( nth_nat @ Xs @ I3 )
          = X5 ) ) ) ).

% list_update_same_conv
thf(fact_268_j__ys1_I2_J,axiom,
    ! [J2: nat] :
      ( ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
     => ( cns @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) @ J2 ) ) ).

% j_ys1(2)
thf(fact_269_the__sym__eq__trivial,axiom,
    ! [X5: nat] :
      ( ( the_nat
        @ ( ^ [Y: nat,Z: nat] : ( Y = Z )
          @ X5 ) )
      = X5 ) ).

% the_sym_eq_trivial
thf(fact_270_the__eq__trivial,axiom,
    ! [A: nat] :
      ( ( the_nat
        @ ^ [X: nat] : ( X = A ) )
      = A ) ).

% the_eq_trivial
thf(fact_271_the__equality,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( P @ A )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A ) )
       => ( ( the_nat @ P )
          = A ) ) ) ).

% the_equality
thf(fact_272_Id__on__set,axiom,
    ! [Xs: list_nat] :
      ( ( id_on_nat @ ( set_nat2 @ Xs ) )
      = ( set_Pr5648618587558075414at_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_273_Id__on__set,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( id_on_8850091095513083129ropVar @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( set_Pr4075413451802982460ropVar
        @ ( map_Mu7320851716970821775ropVar
          @ ^ [X: multis3193088007478089820ropVar] : ( produc6936306168532358295ropVar @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_274_listrel1__iff__update,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R ) )
      = ( ? [Y2: multis3193088007478089820ropVar,N3: nat] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ N3 ) @ Y2 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s1943289338788136526ropVar @ Xs ) )
            & ( Ys
              = ( list_u5207168704986282698ropVar @ Xs @ N3 @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_275_listrel1__iff__update,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R ) )
      = ( ? [Y2: produc2642885279816024434ropVar,N3: nat] :
            ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs @ N3 ) @ Y2 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s6752116297430514276ropVar @ Xs ) )
            & ( Ys
              = ( list_u8100953110031273952ropVar @ Xs @ N3 @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_276_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
      = ( ? [Y2: nat,N3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ Y2 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N3 @ Y2 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_277_the1__equality,axiom,
    ! [P: nat > $o,A: nat] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P @ A )
       => ( ( the_nat @ P )
          = A ) ) ) ).

% the1_equality
thf(fact_278_listrel1__eq__len,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R ) )
     => ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_279_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_280_theI,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( P @ A )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A ) )
       => ( P @ ( the_nat @ P ) ) ) ) ).

% theI
thf(fact_281_theI_H,axiom,
    ! [P: nat > $o] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( P @ ( the_nat @ P ) ) ) ).

% theI'
thf(fact_282_theI2,axiom,
    ! [P: nat > $o,A: nat,Q: nat > $o] :
      ( ( P @ A )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A ) )
       => ( ! [X4: nat] :
              ( ( P @ X4 )
             => ( Q @ X4 ) )
         => ( Q @ ( the_nat @ P ) ) ) ) ) ).

% theI2
thf(fact_283_If__def,axiom,
    ( if_nat
    = ( ^ [P4: $o,X: nat,Y2: nat] :
          ( the_nat
          @ ^ [Z3: nat] :
              ( ( P4
               => ( Z3 = X ) )
              & ( ~ P4
               => ( Z3 = Y2 ) ) ) ) ) ) ).

% If_def
thf(fact_284_the1I2,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( the_nat @ P ) ) ) ) ).

% the1I2
thf(fact_285_Id__onI,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ ( id_on_nat @ A2 ) ) ) ).

% Id_onI
thf(fact_286_Id__onI,axiom,
    ! [A: multis3193088007478089820ropVar,A2: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ A @ A2 )
     => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ A ) @ ( id_on_8850091095513083129ropVar @ A2 ) ) ) ).

% Id_onI
thf(fact_287_Id__on__iff,axiom,
    ! [X5: nat,Y3: nat,A2: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ ( id_on_nat @ A2 ) )
      = ( ( X5 = Y3 )
        & ( member_nat @ X5 @ A2 ) ) ) ).

% Id_on_iff
thf(fact_288_Id__on__iff,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,A2: set_Mu5658373669842198716ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ ( id_on_8850091095513083129ropVar @ A2 ) )
      = ( ( X5 = Y3 )
        & ( member8421201611525043589ropVar @ X5 @ A2 ) ) ) ).

% Id_on_iff
thf(fact_289_Id__on__eqI,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( A = B )
     => ( ( member_nat @ A @ A2 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( id_on_nat @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_290_Id__on__eqI,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A2: set_Mu5658373669842198716ropVar] :
      ( ( A = B )
     => ( ( member8421201611525043589ropVar @ A @ A2 )
       => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ B ) @ ( id_on_8850091095513083129ropVar @ A2 ) ) ) ) ).

% Id_on_eqI
thf(fact_291_Id__onE,axiom,
    ! [C3: product_prod_nat_nat,A2: set_nat] :
      ( ( member8440522571783428010at_nat @ C3 @ ( id_on_nat @ A2 ) )
     => ~ ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( C3
             != ( product_Pair_nat_nat @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_292_Id__onE,axiom,
    ! [C3: produc7679832412844898215ropVar,A2: set_Mu5658373669842198716ropVar] :
      ( ( member2619728555115997648ropVar @ C3 @ ( id_on_8850091095513083129ropVar @ A2 ) )
     => ~ ! [X4: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X4 @ A2 )
           => ( C3
             != ( produc6936306168532358295ropVar @ X4 @ X4 ) ) ) ) ).

% Id_onE
thf(fact_293_listrel__iff__nth,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar,R: set_Pr6405597471078531282ropVar] :
      ( ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Xs @ Ys ) @ ( listre2663080307769418044ropVar @ R ) )
      = ( ( ( size_s1943289338788136526ropVar @ Xs )
          = ( size_s8779405381146381081ropVar @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s1943289338788136526ropVar @ Xs ) )
           => ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ N3 ) @ ( nth_Pr6144466203599902510ropVar @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_294_listrel__iff__nth,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre4495854419763884273ropVar @ R ) )
      = ( ( ( size_s1943289338788136526ropVar @ Xs )
          = ( size_s1943289338788136526ropVar @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s1943289338788136526ropVar @ Xs ) )
           => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ N3 ) @ ( nth_Mu4537670848183934179ropVar @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_295_listrel__iff__nth,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre2294694963173455985ropVar @ R ) )
      = ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s6752116297430514276ropVar @ Xs ) )
           => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs @ N3 ) @ ( nth_Pr227365692153740281ropVar @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_296_listrel__iff__nth,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs @ Ys ) @ ( listre5688682141216627473ar_nat @ R ) )
      = ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s6752116297430514276ropVar @ Xs ) )
           => ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ ( nth_Pr227365692153740281ropVar @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_297_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs @ Ys ) @ ( listre24003740514569327ropVar @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ ( nth_nat @ Xs @ N3 ) @ ( nth_Pr227365692153740281ropVar @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_298_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_299_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,Ns_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,As: list_M6740372356978808802ropVar,Bs: list_M6740372356978808802ropVar,NS: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( list_l6611490901775297704ropVar @ S_list @ Ns_list )
     => ( ( ( size_s1943289338788136526ropVar @ As )
          = ( size_s1943289338788136526ropVar @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Bs ) )
             => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ As @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Bs @ I2 ) ) @ NS ) )
         => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_300_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar,Ns_list: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar,As: list_P8791008543742759928ropVar,Bs: list_P8791008543742759928ropVar,NS: set_Pr3201966908590682119ropVar,S2: set_Pr3201966908590682119ropVar] :
      ( ( list_l8030158259481955774ropVar @ S_list @ Ns_list )
     => ( ( ( size_s6752116297430514276ropVar @ As )
          = ( size_s6752116297430514276ropVar @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Bs ) )
             => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ As @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Bs @ I2 ) ) @ NS ) )
         => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_301_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ( ( size_size_list_nat @ As )
          = ( size_size_list_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_302_j__ys2_I3_J,axiom,
    ! [J2: nat] :
      ( ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
     => ( v2 @ ( multis387687052011358179_Gamma @ ( i_of_j2 @ J2 ) @ J2 ) ) ) ).

% j_ys2(3)
thf(fact_303_distinct__product,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( ( distinct_nat @ Ys )
       => ( distin6923225563576452346at_nat @ ( product_nat_nat @ Xs @ Ys ) ) ) ) ).

% distinct_product
thf(fact_304_listrel__eq__len,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre2294694963173455985ropVar @ R ) )
     => ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_305_listrel__eq__len,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs @ Ys ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_306_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs @ Ys ) @ ( listre24003740514569327ropVar @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_307_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_308_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,S2: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ! [A4: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ S2 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
       => ( ! [A4: nat,B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ NS )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( S_list @ S2 @ NS ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) ) @ ( S_list @ S2 @ NS ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_309_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,Ns_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,S2: set_Pr7799659934693882759ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS: set_Pr7799659934693882759ropVar,As: list_M6740372356978808802ropVar,Bs: list_M6740372356978808802ropVar] :
      ( ( list_l6611490901775297704ropVar @ S_list @ Ns_list )
     => ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ S2 )
           => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
       => ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
              ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ NS )
             => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
         => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As @ Bs ) @ ( S_list @ S2 @ NS ) )
           => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F @ As ) @ ( map_Mu2265489581466636612ropVar @ F @ Bs ) ) @ ( S_list @ S2 @ NS ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_310_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,S2: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ! [A4: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ S2 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
       => ( ! [A4: nat,B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ NS )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_311_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,Ns_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,S2: set_Pr7799659934693882759ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS: set_Pr7799659934693882759ropVar,As: list_M6740372356978808802ropVar,Bs: list_M6740372356978808802ropVar] :
      ( ( list_l6611490901775297704ropVar @ S_list @ Ns_list )
     => ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ S2 )
           => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
       => ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
              ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ NS )
             => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
         => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As @ Bs ) @ ( Ns_list @ S2 @ NS ) )
           => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F @ As ) @ ( map_Mu2265489581466636612ropVar @ F @ Bs ) ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_312_pred__equals__eq2,axiom,
    ! [R2: set_Pr6405597471078531282ropVar,S2: set_Pr6405597471078531282ropVar] :
      ( ( ( ^ [X: multis3193088007478089820ropVar,Y2: produc7679832412844898215ropVar] : ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X @ Y2 ) @ R2 ) )
        = ( ^ [X: multis3193088007478089820ropVar,Y2: produc7679832412844898215ropVar] : ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X @ Y2 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_313_pred__equals__eq2,axiom,
    ! [R2: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( ( ^ [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y2 ) @ R2 ) )
        = ( ^ [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y2 ) @ S2 ) ) )
      = ( R2 = S2 ) ) ).

% pred_equals_eq2
thf(fact_314_j__ys2_I2_J,axiom,
    ! [J2: nat] :
      ( ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
     => ( cs @ ( i_of_j2 @ J2 ) @ J2 ) ) ).

% j_ys2(2)
thf(fact_315_j__ys2_I1_J,axiom,
    ! [J2: nat] :
      ( ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
     => ( member_nat @ ( i_of_j2 @ J2 ) @ ( set_nat2 @ xs2 ) ) ) ).

% j_ys2(1)
thf(fact_316_dist__ys,axiom,
    distinct_nat @ ( append_nat @ ys1 @ ys2 ) ).

% dist_ys
thf(fact_317_ys,axiom,
    ( ( mset_nat @ ( upt @ zero_zero_nat @ m ) )
    = ( plus_p6334493942879108393et_nat @ ( mset_nat @ ys1 ) @ ( mset_nat @ ys2 ) ) ) ).

% ys
thf(fact_318__092_060open_062v_H_A_092_060equiv_062_A_092_060lambda_062x_O_Acase_Ax_Aof_AGamma_Ai_Aj_A_092_060Rightarrow_062_Ai_A_092_060in_062_Aset_Axs1_A_092_060and_062_Aj_A_092_060in_062_Aset_Ays1_A_092_060and_062_Ai_A_061_Axs1_A_B_Apos__of_Ays1_Aj_A_092_060or_062_Ai_A_092_060in_062_Aset_Axs2_A_092_060and_062_Aj_A_092_060in_062_Aset_Ays2_A_092_060and_062_Ai_A_061_Ai__of__j2_Aj_A_124_AEpsilon_Ai_A_092_060Rightarrow_062_Ai_A_092_060in_062_Aset_Axs1_092_060close_062,axiom,
    ( v
    = ( multis8641292176345589959pVar_o
      @ ^ [I: nat,J: nat] :
          ( ( ( member_nat @ I @ ( set_nat2 @ xs1 ) )
            & ( member_nat @ J @ ( set_nat2 @ ys1 ) )
            & ( I
              = ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) ) )
          | ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
            & ( member_nat @ J @ ( set_nat2 @ ys2 ) )
            & ( I
              = ( i_of_j2 @ J ) ) ) )
      @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs1 ) )
      @ ^ [A5: nat,B4: nat] : undefined_o
      @ ^ [A5: nat,B4: nat] : undefined_o
      @ ^ [A5: nat,B4: nat] : undefined_o
      @ ^ [A5: nat,B4: nat] : undefined_o ) ) ).

% \<open>v' \<equiv> \<lambda>x. case x of Gamma i j \<Rightarrow> i \<in> set xs1 \<and> j \<in> set ys1 \<and> i = xs1 ! pos_of ys1 j \<or> i \<in> set xs2 \<and> j \<in> set ys2 \<and> i = i_of_j2 j | Epsilon i \<Rightarrow> i \<in> set xs1\<close>
thf(fact_319_list__order__extension_Ointro,axiom,
    ! [S_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,Ns_list: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar] :
      ( ! [S3: set_Pr7799659934693882759ropVar,NS2: set_Pr7799659934693882759ropVar] :
          ( ( order_6814195989798057450ropVar @ S3 @ NS2 )
         => ( order_5896603146005108336ropVar @ ( S_list @ S3 @ NS2 ) @ ( Ns_list @ S3 @ NS2 ) ) )
     => ( ! [S3: set_Pr7799659934693882759ropVar,F7: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS2: set_Pr7799659934693882759ropVar,As2: list_M6740372356978808802ropVar,Bs2: list_M6740372356978808802ropVar] :
            ( ! [A6: multis3193088007478089820ropVar,B5: multis3193088007478089820ropVar] :
                ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B5 ) @ S3 )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ S3 ) )
           => ( ! [A6: multis3193088007478089820ropVar,B5: multis3193088007478089820ropVar] :
                  ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B5 ) @ NS2 )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ NS2 ) )
             => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As2 @ Bs2 ) @ ( S_list @ S3 @ NS2 ) )
               => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F7 @ As2 ) @ ( map_Mu2265489581466636612ropVar @ F7 @ Bs2 ) ) @ ( S_list @ S3 @ NS2 ) ) ) ) )
       => ( ! [S3: set_Pr7799659934693882759ropVar,F7: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS2: set_Pr7799659934693882759ropVar,As2: list_M6740372356978808802ropVar,Bs2: list_M6740372356978808802ropVar] :
              ( ! [A6: multis3193088007478089820ropVar,B5: multis3193088007478089820ropVar] :
                  ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B5 ) @ S3 )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ S3 ) )
             => ( ! [A6: multis3193088007478089820ropVar,B5: multis3193088007478089820ropVar] :
                    ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B5 ) @ NS2 )
                   => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ NS2 ) )
               => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As2 @ Bs2 ) @ ( Ns_list @ S3 @ NS2 ) )
                 => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F7 @ As2 ) @ ( map_Mu2265489581466636612ropVar @ F7 @ Bs2 ) ) @ ( Ns_list @ S3 @ NS2 ) ) ) ) )
         => ( ! [As2: list_M6740372356978808802ropVar,Bs2: list_M6740372356978808802ropVar,NS2: set_Pr7799659934693882759ropVar,S3: set_Pr7799659934693882759ropVar] :
                ( ( ( size_s1943289338788136526ropVar @ As2 )
                  = ( size_s1943289338788136526ropVar @ Bs2 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s1943289338788136526ropVar @ Bs2 ) )
                     => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ As2 @ I4 ) @ ( nth_Mu4537670848183934179ropVar @ Bs2 @ I4 ) ) @ NS2 ) )
                 => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As2 @ Bs2 ) @ ( Ns_list @ S3 @ NS2 ) ) ) )
           => ( list_l6611490901775297704ropVar @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_320_list__order__extension_Ointro,axiom,
    ! [S_list: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar,Ns_list: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar] :
      ( ! [S3: set_Pr3201966908590682119ropVar,NS2: set_Pr3201966908590682119ropVar] :
          ( ( order_9198723064055253760ropVar @ S3 @ NS2 )
         => ( order_7256030434733506182ropVar @ ( S_list @ S3 @ NS2 ) @ ( Ns_list @ S3 @ NS2 ) ) )
     => ( ! [S3: set_Pr3201966908590682119ropVar,F7: produc2642885279816024434ropVar > produc2642885279816024434ropVar,NS2: set_Pr3201966908590682119ropVar,As2: list_P8791008543742759928ropVar,Bs2: list_P8791008543742759928ropVar] :
            ( ! [A6: produc2642885279816024434ropVar,B5: produc2642885279816024434ropVar] :
                ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A6 @ B5 ) @ S3 )
               => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ S3 ) )
           => ( ! [A6: produc2642885279816024434ropVar,B5: produc2642885279816024434ropVar] :
                  ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A6 @ B5 ) @ NS2 )
                 => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ NS2 ) )
             => ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As2 @ Bs2 ) @ ( S_list @ S3 @ NS2 ) )
               => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( map_Pr3565048146863404228ropVar @ F7 @ As2 ) @ ( map_Pr3565048146863404228ropVar @ F7 @ Bs2 ) ) @ ( S_list @ S3 @ NS2 ) ) ) ) )
       => ( ! [S3: set_Pr3201966908590682119ropVar,F7: produc2642885279816024434ropVar > produc2642885279816024434ropVar,NS2: set_Pr3201966908590682119ropVar,As2: list_P8791008543742759928ropVar,Bs2: list_P8791008543742759928ropVar] :
              ( ! [A6: produc2642885279816024434ropVar,B5: produc2642885279816024434ropVar] :
                  ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A6 @ B5 ) @ S3 )
                 => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ S3 ) )
             => ( ! [A6: produc2642885279816024434ropVar,B5: produc2642885279816024434ropVar] :
                    ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A6 @ B5 ) @ NS2 )
                   => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ NS2 ) )
               => ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As2 @ Bs2 ) @ ( Ns_list @ S3 @ NS2 ) )
                 => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( map_Pr3565048146863404228ropVar @ F7 @ As2 ) @ ( map_Pr3565048146863404228ropVar @ F7 @ Bs2 ) ) @ ( Ns_list @ S3 @ NS2 ) ) ) ) )
         => ( ! [As2: list_P8791008543742759928ropVar,Bs2: list_P8791008543742759928ropVar,NS2: set_Pr3201966908590682119ropVar,S3: set_Pr3201966908590682119ropVar] :
                ( ( ( size_s6752116297430514276ropVar @ As2 )
                  = ( size_s6752116297430514276ropVar @ Bs2 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s6752116297430514276ropVar @ Bs2 ) )
                     => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ As2 @ I4 ) @ ( nth_Pr227365692153740281ropVar @ Bs2 @ I4 ) ) @ NS2 ) )
                 => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As2 @ Bs2 ) @ ( Ns_list @ S3 @ NS2 ) ) ) )
           => ( list_l8030158259481955774ropVar @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_321_list__order__extension_Ointro,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
      ( ! [S3: set_Pr1261947904930325089at_nat,NS2: set_Pr1261947904930325089at_nat] :
          ( ( order_7802623214671921858ir_nat @ S3 @ NS2 )
         => ( order_2144575240049343698st_nat @ ( S_list @ S3 @ NS2 ) @ ( Ns_list @ S3 @ NS2 ) ) )
     => ( ! [S3: set_Pr1261947904930325089at_nat,F7: nat > nat,NS2: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
            ( ! [A6: nat,B5: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S3 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ S3 ) )
           => ( ! [A6: nat,B5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ NS2 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ NS2 ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( S_list @ S3 @ NS2 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F7 @ As2 ) @ ( map_nat_nat @ F7 @ Bs2 ) ) @ ( S_list @ S3 @ NS2 ) ) ) ) )
       => ( ! [S3: set_Pr1261947904930325089at_nat,F7: nat > nat,NS2: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
              ( ! [A6: nat,B5: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ S3 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ S3 ) )
             => ( ! [A6: nat,B5: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B5 ) @ NS2 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F7 @ A6 ) @ ( F7 @ B5 ) ) @ NS2 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list @ S3 @ NS2 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F7 @ As2 ) @ ( map_nat_nat @ F7 @ Bs2 ) ) @ ( Ns_list @ S3 @ NS2 ) ) ) ) )
         => ( ! [As2: list_nat,Bs2: list_nat,NS2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
                ( ( ( size_size_list_nat @ As2 )
                  = ( size_size_list_nat @ Bs2 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Bs2 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As2 @ I4 ) @ ( nth_nat @ Bs2 @ I4 ) ) @ NS2 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list @ S3 @ NS2 ) ) ) )
           => ( list_l792762466888043652on_nat @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_322_list__order__extension__def,axiom,
    ( list_l6611490901775297704ropVar
    = ( ^ [S_list2: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar,Ns_list2: set_Pr7799659934693882759ropVar > set_Pr7799659934693882759ropVar > set_Pr3559363423829876615ropVar] :
          ( ! [S4: set_Pr7799659934693882759ropVar,NS3: set_Pr7799659934693882759ropVar] :
              ( ( order_6814195989798057450ropVar @ S4 @ NS3 )
             => ( order_5896603146005108336ropVar @ ( S_list2 @ S4 @ NS3 ) @ ( Ns_list2 @ S4 @ NS3 ) ) )
          & ! [S4: set_Pr7799659934693882759ropVar,F8: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS3: set_Pr7799659934693882759ropVar,As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar] :
              ( ! [A5: multis3193088007478089820ropVar,B4: multis3193088007478089820ropVar] :
                  ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A5 @ B4 ) @ S4 )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ S4 ) )
             => ( ! [A5: multis3193088007478089820ropVar,B4: multis3193088007478089820ropVar] :
                    ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A5 @ B4 ) @ NS3 )
                   => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ NS3 ) )
               => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As3 @ Bs3 ) @ ( S_list2 @ S4 @ NS3 ) )
                 => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F8 @ As3 ) @ ( map_Mu2265489581466636612ropVar @ F8 @ Bs3 ) ) @ ( S_list2 @ S4 @ NS3 ) ) ) ) )
          & ! [S4: set_Pr7799659934693882759ropVar,F8: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS3: set_Pr7799659934693882759ropVar,As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar] :
              ( ! [A5: multis3193088007478089820ropVar,B4: multis3193088007478089820ropVar] :
                  ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A5 @ B4 ) @ S4 )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ S4 ) )
             => ( ! [A5: multis3193088007478089820ropVar,B4: multis3193088007478089820ropVar] :
                    ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A5 @ B4 ) @ NS3 )
                   => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ NS3 ) )
               => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As3 @ Bs3 ) @ ( Ns_list2 @ S4 @ NS3 ) )
                 => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F8 @ As3 ) @ ( map_Mu2265489581466636612ropVar @ F8 @ Bs3 ) ) @ ( Ns_list2 @ S4 @ NS3 ) ) ) ) )
          & ! [As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar,NS3: set_Pr7799659934693882759ropVar,S4: set_Pr7799659934693882759ropVar] :
              ( ( ( size_s1943289338788136526ropVar @ As3 )
                = ( size_s1943289338788136526ropVar @ Bs3 ) )
             => ( ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( size_s1943289338788136526ropVar @ Bs3 ) )
                   => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ As3 @ I ) @ ( nth_Mu4537670848183934179ropVar @ Bs3 @ I ) ) @ NS3 ) )
               => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As3 @ Bs3 ) @ ( Ns_list2 @ S4 @ NS3 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_323_list__order__extension__def,axiom,
    ( list_l8030158259481955774ropVar
    = ( ^ [S_list2: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar,Ns_list2: set_Pr3201966908590682119ropVar > set_Pr3201966908590682119ropVar > set_Pr8381652856031230727ropVar] :
          ( ! [S4: set_Pr3201966908590682119ropVar,NS3: set_Pr3201966908590682119ropVar] :
              ( ( order_9198723064055253760ropVar @ S4 @ NS3 )
             => ( order_7256030434733506182ropVar @ ( S_list2 @ S4 @ NS3 ) @ ( Ns_list2 @ S4 @ NS3 ) ) )
          & ! [S4: set_Pr3201966908590682119ropVar,F8: produc2642885279816024434ropVar > produc2642885279816024434ropVar,NS3: set_Pr3201966908590682119ropVar,As3: list_P8791008543742759928ropVar,Bs3: list_P8791008543742759928ropVar] :
              ( ! [A5: produc2642885279816024434ropVar,B4: produc2642885279816024434ropVar] :
                  ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A5 @ B4 ) @ S4 )
                 => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ S4 ) )
             => ( ! [A5: produc2642885279816024434ropVar,B4: produc2642885279816024434ropVar] :
                    ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A5 @ B4 ) @ NS3 )
                   => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ NS3 ) )
               => ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As3 @ Bs3 ) @ ( S_list2 @ S4 @ NS3 ) )
                 => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( map_Pr3565048146863404228ropVar @ F8 @ As3 ) @ ( map_Pr3565048146863404228ropVar @ F8 @ Bs3 ) ) @ ( S_list2 @ S4 @ NS3 ) ) ) ) )
          & ! [S4: set_Pr3201966908590682119ropVar,F8: produc2642885279816024434ropVar > produc2642885279816024434ropVar,NS3: set_Pr3201966908590682119ropVar,As3: list_P8791008543742759928ropVar,Bs3: list_P8791008543742759928ropVar] :
              ( ! [A5: produc2642885279816024434ropVar,B4: produc2642885279816024434ropVar] :
                  ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A5 @ B4 ) @ S4 )
                 => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ S4 ) )
             => ( ! [A5: produc2642885279816024434ropVar,B4: produc2642885279816024434ropVar] :
                    ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A5 @ B4 ) @ NS3 )
                   => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ NS3 ) )
               => ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As3 @ Bs3 ) @ ( Ns_list2 @ S4 @ NS3 ) )
                 => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( map_Pr3565048146863404228ropVar @ F8 @ As3 ) @ ( map_Pr3565048146863404228ropVar @ F8 @ Bs3 ) ) @ ( Ns_list2 @ S4 @ NS3 ) ) ) ) )
          & ! [As3: list_P8791008543742759928ropVar,Bs3: list_P8791008543742759928ropVar,NS3: set_Pr3201966908590682119ropVar,S4: set_Pr3201966908590682119ropVar] :
              ( ( ( size_s6752116297430514276ropVar @ As3 )
                = ( size_s6752116297430514276ropVar @ Bs3 ) )
             => ( ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Bs3 ) )
                   => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ As3 @ I ) @ ( nth_Pr227365692153740281ropVar @ Bs3 @ I ) ) @ NS3 ) )
               => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As3 @ Bs3 ) @ ( Ns_list2 @ S4 @ NS3 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_324_list__order__extension__def,axiom,
    ( list_l792762466888043652on_nat
    = ( ^ [S_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
          ( ! [S4: set_Pr1261947904930325089at_nat,NS3: set_Pr1261947904930325089at_nat] :
              ( ( order_7802623214671921858ir_nat @ S4 @ NS3 )
             => ( order_2144575240049343698st_nat @ ( S_list2 @ S4 @ NS3 ) @ ( Ns_list2 @ S4 @ NS3 ) ) )
          & ! [S4: set_Pr1261947904930325089at_nat,F8: nat > nat,NS3: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
              ( ! [A5: nat,B4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ S4 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ S4 ) )
             => ( ! [A5: nat,B4: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ NS3 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ NS3 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( S_list2 @ S4 @ NS3 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F8 @ As3 ) @ ( map_nat_nat @ F8 @ Bs3 ) ) @ ( S_list2 @ S4 @ NS3 ) ) ) ) )
          & ! [S4: set_Pr1261947904930325089at_nat,F8: nat > nat,NS3: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
              ( ! [A5: nat,B4: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ S4 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ S4 ) )
             => ( ! [A5: nat,B4: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A5 @ B4 ) @ NS3 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F8 @ A5 ) @ ( F8 @ B4 ) ) @ NS3 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list2 @ S4 @ NS3 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F8 @ As3 ) @ ( map_nat_nat @ F8 @ Bs3 ) ) @ ( Ns_list2 @ S4 @ NS3 ) ) ) ) )
          & ! [As3: list_nat,Bs3: list_nat,NS3: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
              ( ( ( size_size_list_nat @ As3 )
                = ( size_size_list_nat @ Bs3 ) )
             => ( ! [I: nat] :
                    ( ( ord_less_nat @ I @ ( size_size_list_nat @ Bs3 ) )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As3 @ I ) @ ( nth_nat @ Bs3 @ I ) ) @ NS3 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list2 @ S4 @ NS3 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_325_v_H__def,axiom,
    ! [X5: multis3193088007478089820ropVar] :
      ( ( v @ X5 )
      = ( multis8641292176345589959pVar_o
        @ ^ [I: nat,J: nat] :
            ( ( ( member_nat @ I @ ( set_nat2 @ xs1 ) )
              & ( member_nat @ J @ ( set_nat2 @ ys1 ) )
              & ( I
                = ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) ) )
            | ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
              & ( member_nat @ J @ ( set_nat2 @ ys2 ) )
              & ( I
                = ( i_of_j2 @ J ) ) ) )
        @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs1 ) )
        @ ^ [A5: nat,B4: nat] : undefined_o
        @ ^ [A5: nat,B4: nat] : undefined_o
        @ ^ [A5: nat,B4: nat] : undefined_o
        @ ^ [A5: nat,B4: nat] : undefined_o
        @ X5 ) ) ).

% v'_def
thf(fact_326_add__left__cancel,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A @ B )
        = ( plus_p6334493942879108393et_nat @ A @ C3 ) )
      = ( B = C3 ) ) ).

% add_left_cancel
thf(fact_327_add__left__cancel,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C3 ) )
      = ( B = C3 ) ) ).

% add_left_cancel
thf(fact_328_add__right__cancel,axiom,
    ! [B: multiset_nat,A: multiset_nat,C3: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ B @ A )
        = ( plus_p6334493942879108393et_nat @ C3 @ A ) )
      = ( B = C3 ) ) ).

% add_right_cancel
thf(fact_329_add__right__cancel,axiom,
    ! [B: nat,A: nat,C3: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C3 @ A ) )
      = ( B = C3 ) ) ).

% add_right_cancel
thf(fact_330_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C3: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C3 )
      = ( append_nat @ A @ ( append_nat @ B @ C3 ) ) ) ).

% append.assoc
thf(fact_331_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_332_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_333_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_334_xs,axiom,
    ( ( mset_nat @ ( upt @ zero_zero_nat @ n ) )
    = ( plus_p6334493942879108393et_nat @ ( mset_nat @ xs1 ) @ ( mset_nat @ xs2 ) ) ) ).

% xs
thf(fact_335_dist__xs,axiom,
    distinct_nat @ ( append_nat @ xs1 @ xs2 ) ).

% dist_xs
thf(fact_336_add__0,axiom,
    ! [A: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A )
      = A ) ).

% add_0
thf(fact_337_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_338_zero__eq__add__iff__both__eq__0,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X5 @ Y3 ) )
      = ( ( X5 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_339_add__eq__0__iff__both__eq__0,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X5 @ Y3 )
        = zero_zero_nat )
      = ( ( X5 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_340_add__cancel__right__right,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( A
        = ( plus_p6334493942879108393et_nat @ A @ B ) )
      = ( B = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_right_right
thf(fact_341_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_342_add__cancel__right__left,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( A
        = ( plus_p6334493942879108393et_nat @ B @ A ) )
      = ( B = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_right_left
thf(fact_343_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_344_add__cancel__left__right,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A @ B )
        = A )
      = ( B = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_left_right
thf(fact_345_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_346_add__cancel__left__left,axiom,
    ! [B: multiset_nat,A: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ B @ A )
        = A )
      = ( B = zero_z7348594199698428585et_nat ) ) ).

% add_cancel_left_left
thf(fact_347_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_348_add_Oright__neutral,axiom,
    ! [A: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
      = A ) ).

% add.right_neutral
thf(fact_349_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_350_add__less__cancel__right,axiom,
    ! [A: nat,C3: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_351_add__less__cancel__left,axiom,
    ! [C3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_352_append__eq__append__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Us: list_P8791008543742759928ropVar,Vs: list_P8791008543742759928ropVar] :
      ( ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        | ( ( size_s6752116297430514276ropVar @ Us )
          = ( size_s6752116297430514276ropVar @ Vs ) ) )
     => ( ( ( append4352292717583037453ropVar @ Xs @ Us )
          = ( append4352292717583037453ropVar @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_353_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_354_union__eq__empty,axiom,
    ! [M3: multiset_nat,N4: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ M3 @ N4 )
        = zero_z7348594199698428585et_nat )
      = ( ( M3 = zero_z7348594199698428585et_nat )
        & ( N4 = zero_z7348594199698428585et_nat ) ) ) ).

% union_eq_empty
thf(fact_355_empty__eq__union,axiom,
    ! [M3: multiset_nat,N4: multiset_nat] :
      ( ( zero_z7348594199698428585et_nat
        = ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
      = ( ( M3 = zero_z7348594199698428585et_nat )
        & ( N4 = zero_z7348594199698428585et_nat ) ) ) ).

% empty_eq_union
thf(fact_356_subset__mset_Ozero__eq__add__iff__both__eq__0,axiom,
    ! [X5: multiset_nat,Y3: multiset_nat] :
      ( ( zero_z7348594199698428585et_nat
        = ( plus_p6334493942879108393et_nat @ X5 @ Y3 ) )
      = ( ( X5 = zero_z7348594199698428585et_nat )
        & ( Y3 = zero_z7348594199698428585et_nat ) ) ) ).

% subset_mset.zero_eq_add_iff_both_eq_0
thf(fact_357_subset__mset_Oadd__eq__0__iff__both__eq__0,axiom,
    ! [X5: multiset_nat,Y3: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ X5 @ Y3 )
        = zero_z7348594199698428585et_nat )
      = ( ( X5 = zero_z7348594199698428585et_nat )
        & ( Y3 = zero_z7348594199698428585et_nat ) ) ) ).

% subset_mset.add_eq_0_iff_both_eq_0
thf(fact_358_map__append,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Ys: list_nat] :
      ( ( map_na2145702811007361372ropVar @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append4352292717583037453ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ ( map_na2145702811007361372ropVar @ F @ Ys ) ) ) ).

% map_append
thf(fact_359_map__append,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append2684835082924936439ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ F @ Ys ) ) ) ).

% map_append
thf(fact_360_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_361_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_362_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_363_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_364_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_365_mset__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( mset_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ ( mset_nat @ Ys ) ) ) ).

% mset_append
thf(fact_366_zip__append,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Us: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Vs: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Us ) )
     => ( ( zip_Pr6829646289182870327ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ ( append4352292717583037453ropVar @ Us @ Vs ) )
        = ( append7972353524314683330ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs @ Us ) @ ( zip_Pr6829646289182870327ropVar @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_367_zip__append,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Us: list_nat,Ys: list_P8791008543742759928ropVar,Vs: list_nat] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Pr2402428817466909387ar_nat @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append4419790923233094806ar_nat @ ( zip_Pr2402428817466909387ar_nat @ Xs @ Us ) @ ( zip_Pr2402428817466909387ar_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_368_zip__append,axiom,
    ! [Xs: list_nat,Us: list_P8791008543742759928ropVar,Ys: list_nat,Vs: list_P8791008543742759928ropVar] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Us ) )
     => ( ( zip_na5961122453619627049ropVar @ ( append_nat @ Xs @ Ys ) @ ( append4352292717583037453ropVar @ Us @ Vs ) )
        = ( append9216677672214865852ropVar @ ( zip_na5961122453619627049ropVar @ Xs @ Us ) @ ( zip_na5961122453619627049ropVar @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_369_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_370_size__mset,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( size_s8378524962137626404ropVar @ ( mset_P5170620898817730055ropVar @ Xs ) )
      = ( size_s6752116297430514276ropVar @ Xs ) ) ).

% size_mset
thf(fact_371_size__mset,axiom,
    ! [Xs: list_nat] :
      ( ( size_s5917832649809541300et_nat @ ( mset_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% size_mset
thf(fact_372__092_060open_062i__of__j2_A_092_060equiv_062_A_092_060lambda_062j_O_ASOME_Ai_O_Ai_A_092_060in_062_Aset_Axs2_A_092_060and_062_Acs_Ai_Aj_092_060close_062,axiom,
    ( i_of_j2
    = ( ^ [J: nat] :
          ( fChoice_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
              & ( cs @ I @ J ) ) ) ) ) ).

% \<open>i_of_j2 \<equiv> \<lambda>j. SOME i. i \<in> set xs2 \<and> cs i j\<close>
thf(fact_373_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ C3 )
      = ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C3 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_374_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C3 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C3 ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_375_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: multiset_nat,J2: multiset_nat,K: multiset_nat,L: multiset_nat] :
      ( ( ( I3 = J2 )
        & ( K = L ) )
     => ( ( plus_p6334493942879108393et_nat @ I3 @ K )
        = ( plus_p6334493942879108393et_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_376_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I3: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I3 = J2 )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I3 @ K )
        = ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_377_group__cancel_Oadd1,axiom,
    ! [A2: multiset_nat,K: multiset_nat,A: multiset_nat,B: multiset_nat] :
      ( ( A2
        = ( plus_p6334493942879108393et_nat @ K @ A ) )
     => ( ( plus_p6334493942879108393et_nat @ A2 @ B )
        = ( plus_p6334493942879108393et_nat @ K @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_378_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_379_group__cancel_Oadd2,axiom,
    ! [B6: multiset_nat,K: multiset_nat,B: multiset_nat,A: multiset_nat] :
      ( ( B6
        = ( plus_p6334493942879108393et_nat @ K @ B ) )
     => ( ( plus_p6334493942879108393et_nat @ A @ B6 )
        = ( plus_p6334493942879108393et_nat @ K @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_380_group__cancel_Oadd2,axiom,
    ! [B6: nat,K: nat,B: nat,A: nat] :
      ( ( B6
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B6 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_381_add_Oassoc,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ C3 )
      = ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C3 ) ) ) ).

% add.assoc
thf(fact_382_add_Oassoc,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C3 )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C3 ) ) ) ).

% add.assoc
thf(fact_383_add_Ocommute,axiom,
    ( plus_p6334493942879108393et_nat
    = ( ^ [A5: multiset_nat,B4: multiset_nat] : ( plus_p6334493942879108393et_nat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_384_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A5: nat,B4: nat] : ( plus_plus_nat @ B4 @ A5 ) ) ) ).

% add.commute
thf(fact_385_add_Oleft__commute,axiom,
    ! [B: multiset_nat,A: multiset_nat,C3: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ B @ ( plus_p6334493942879108393et_nat @ A @ C3 ) )
      = ( plus_p6334493942879108393et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C3 ) ) ) ).

% add.left_commute
thf(fact_386_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C3: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C3 ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C3 ) ) ) ).

% add.left_commute
thf(fact_387_add__left__imp__eq,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A @ B )
        = ( plus_p6334493942879108393et_nat @ A @ C3 ) )
     => ( B = C3 ) ) ).

% add_left_imp_eq
thf(fact_388_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C3 ) )
     => ( B = C3 ) ) ).

% add_left_imp_eq
thf(fact_389_add__right__imp__eq,axiom,
    ! [B: multiset_nat,A: multiset_nat,C3: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ B @ A )
        = ( plus_p6334493942879108393et_nat @ C3 @ A ) )
     => ( B = C3 ) ) ).

% add_right_imp_eq
thf(fact_390_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C3: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C3 @ A ) )
     => ( B = C3 ) ) ).

% add_right_imp_eq
thf(fact_391_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_392_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_393_union__le__mono1,axiom,
    ! [B6: multiset_nat,D: multiset_nat,C4: multiset_nat] :
      ( ( ord_le5777773500796000884et_nat @ B6 @ D )
     => ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ B6 @ C4 ) @ ( plus_p6334493942879108393et_nat @ D @ C4 ) ) ) ).

% union_le_mono1
thf(fact_394_union__le__mono2,axiom,
    ! [B6: multiset_nat,D: multiset_nat,C4: multiset_nat] :
      ( ( ord_le5777773500796000884et_nat @ B6 @ D )
     => ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ C4 @ B6 ) @ ( plus_p6334493942879108393et_nat @ C4 @ D ) ) ) ).

% union_le_mono2
thf(fact_395_union__less__mono,axiom,
    ! [A2: multiset_nat,C4: multiset_nat,B6: multiset_nat,D: multiset_nat] :
      ( ( ord_le5777773500796000884et_nat @ A2 @ C4 )
     => ( ( ord_le5777773500796000884et_nat @ B6 @ D )
       => ( ord_le5777773500796000884et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) @ ( plus_p6334493942879108393et_nat @ C4 @ D ) ) ) ) ).

% union_less_mono
thf(fact_396_ex__mset,axiom,
    ! [X7: multiset_nat] :
    ? [Xs3: list_nat] :
      ( ( mset_nat @ Xs3 )
      = X7 ) ).

% ex_mset
thf(fact_397_union__assoc,axiom,
    ! [M3: multiset_nat,N4: multiset_nat,K2: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) @ K2 )
      = ( plus_p6334493942879108393et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ K2 ) ) ) ).

% union_assoc
thf(fact_398_union__lcomm,axiom,
    ! [M3: multiset_nat,N4: multiset_nat,K2: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ K2 ) )
      = ( plus_p6334493942879108393et_nat @ N4 @ ( plus_p6334493942879108393et_nat @ M3 @ K2 ) ) ) ).

% union_lcomm
thf(fact_399_union__commute,axiom,
    ( plus_p6334493942879108393et_nat
    = ( ^ [M4: multiset_nat,N5: multiset_nat] : ( plus_p6334493942879108393et_nat @ N5 @ M4 ) ) ) ).

% union_commute
thf(fact_400_union__left__cancel,axiom,
    ! [K2: multiset_nat,M3: multiset_nat,N4: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ K2 @ M3 )
        = ( plus_p6334493942879108393et_nat @ K2 @ N4 ) )
      = ( M3 = N4 ) ) ).

% union_left_cancel
thf(fact_401_union__right__cancel,axiom,
    ! [M3: multiset_nat,K2: multiset_nat,N4: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ M3 @ K2 )
        = ( plus_p6334493942879108393et_nat @ N4 @ K2 ) )
      = ( M3 = N4 ) ) ).

% union_right_cancel
thf(fact_402_multi__union__self__other__eq,axiom,
    ! [A2: multiset_nat,X7: multiset_nat,Y6: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ A2 @ X7 )
        = ( plus_p6334493942879108393et_nat @ A2 @ Y6 ) )
     => ( X7 = Y6 ) ) ).

% multi_union_self_other_eq
thf(fact_403_ex__mset__zip__left,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Xs4: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( mset_P5170620898817730055ropVar @ Xs4 )
          = ( mset_P5170620898817730055ropVar @ Xs ) )
       => ? [Ys4: list_P8791008543742759928ropVar] :
            ( ( ( size_s6752116297430514276ropVar @ Ys4 )
              = ( size_s6752116297430514276ropVar @ Xs4 ) )
            & ( ( mset_P7447892262639026108ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs4 @ Ys4 ) )
              = ( mset_P7447892262639026108ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_404_ex__mset__zip__left,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Xs4: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( mset_P5170620898817730055ropVar @ Xs4 )
          = ( mset_P5170620898817730055ropVar @ Xs ) )
       => ? [Ys4: list_nat] :
            ( ( ( size_size_list_nat @ Ys4 )
              = ( size_s6752116297430514276ropVar @ Xs4 ) )
            & ( ( mset_P6847787391757867548ar_nat @ ( zip_Pr2402428817466909387ar_nat @ Xs4 @ Ys4 ) )
              = ( mset_P6847787391757867548ar_nat @ ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_405_ex__mset__zip__left,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Xs4: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( mset_nat @ Xs4 )
          = ( mset_nat @ Xs ) )
       => ? [Ys4: list_P8791008543742759928ropVar] :
            ( ( ( size_s6752116297430514276ropVar @ Ys4 )
              = ( size_size_list_nat @ Xs4 ) )
            & ( ( mset_P2421302103884862786ropVar @ ( zip_na5961122453619627049ropVar @ Xs4 @ Ys4 ) )
              = ( mset_P2421302103884862786ropVar @ ( zip_na5961122453619627049ropVar @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_406_ex__mset__zip__left,axiom,
    ! [Xs: list_nat,Ys: list_nat,Xs4: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( mset_nat @ Xs4 )
          = ( mset_nat @ Xs ) )
       => ? [Ys4: list_nat] :
            ( ( ( size_size_list_nat @ Ys4 )
              = ( size_size_list_nat @ Xs4 ) )
            & ( ( mset_P6383711406899277590at_nat @ ( zip_nat_nat @ Xs4 @ Ys4 ) )
              = ( mset_P6383711406899277590at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_407_mset__eq__setD,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( mset_M6581516532424202993ropVar @ Xs )
        = ( mset_M6581516532424202993ropVar @ Ys ) )
     => ( ( set_Mu146256361055586289ropVar @ Xs )
        = ( set_Mu146256361055586289ropVar @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_408_mset__eq__setD,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( set_nat2 @ Xs )
        = ( set_nat2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_409_mset__eq__length,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( mset_P5170620898817730055ropVar @ Xs )
        = ( mset_P5170620898817730055ropVar @ Ys ) )
     => ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% mset_eq_length
thf(fact_410_mset__eq__length,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% mset_eq_length
thf(fact_411_add_Ocomm__neutral,axiom,
    ! [A: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
      = A ) ).

% add.comm_neutral
thf(fact_412_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_413_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_414_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_415_add__less__imp__less__right,axiom,
    ! [A: nat,C3: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_416_add__less__imp__less__left,axiom,
    ! [C3: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_417_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) ) ) ).

% add_strict_right_mono
thf(fact_418_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_419_add__strict__mono,axiom,
    ! [A: nat,B: nat,C3: nat,D2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C3 @ D2 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ D2 ) ) ) ) ).

% add_strict_mono
thf(fact_420_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I3: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I3 @ J2 )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_421_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I3: nat,J2: nat,K: nat,L: nat] :
      ( ( ( I3 = J2 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_422_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I3: nat,J2: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I3 @ J2 )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_423_map__eq__append__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Ys: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( append4352292717583037453ropVar @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na2145702811007361372ropVar @ F @ Us2 ) )
            & ( Zs
              = ( map_na2145702811007361372ropVar @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_424_map__eq__append__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( append2684835082924936439ropVar @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na2466558649495876678ropVar @ F @ Us2 ) )
            & ( Zs
              = ( map_na2466558649495876678ropVar @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_425_map__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_426_append__eq__map__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( ( append4352292717583037453ropVar @ Ys @ Zs )
        = ( map_na2145702811007361372ropVar @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na2145702811007361372ropVar @ F @ Us2 ) )
            & ( Zs
              = ( map_na2145702811007361372ropVar @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_427_append__eq__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( ( append2684835082924936439ropVar @ Ys @ Zs )
        = ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na2466558649495876678ropVar @ F @ Us2 ) )
            & ( Zs
              = ( map_na2466558649495876678ropVar @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_428_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_429_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( distinct_nat @ Xs )
        = ( distinct_nat @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_430_empty__neutral_I1_J,axiom,
    ! [X5: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ X5 )
      = X5 ) ).

% empty_neutral(1)
thf(fact_431_empty__neutral_I2_J,axiom,
    ! [X5: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ X5 @ zero_z7348594199698428585et_nat )
      = X5 ) ).

% empty_neutral(2)
thf(fact_432_set__eq__iff__mset__eq__distinct,axiom,
    ! [X5: list_M6740372356978808802ropVar,Y3: list_M6740372356978808802ropVar] :
      ( ( distin2675581014522817237ropVar @ X5 )
     => ( ( distin2675581014522817237ropVar @ Y3 )
       => ( ( ( set_Mu146256361055586289ropVar @ X5 )
            = ( set_Mu146256361055586289ropVar @ Y3 ) )
          = ( ( mset_M6581516532424202993ropVar @ X5 )
            = ( mset_M6581516532424202993ropVar @ Y3 ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_433_set__eq__iff__mset__eq__distinct,axiom,
    ! [X5: list_nat,Y3: list_nat] :
      ( ( distinct_nat @ X5 )
     => ( ( distinct_nat @ Y3 )
       => ( ( ( set_nat2 @ X5 )
            = ( set_nat2 @ Y3 ) )
          = ( ( mset_nat @ X5 )
            = ( mset_nat @ Y3 ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_434_pos__add__strict,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C3 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C3 ) ) ) ) ).

% pos_add_strict
thf(fact_435_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_436_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_437_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_438_append__listrel1I,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_439_list__update__append1,axiom,
    ! [I3: nat,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( list_u8100953110031273952ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ I3 @ X5 )
        = ( append4352292717583037453ropVar @ ( list_u8100953110031273952ropVar @ Xs @ I3 @ X5 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_440_list__update__append1,axiom,
    ! [I3: nat,Xs: list_nat,Ys: list_nat,X5: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I3 @ X5 )
        = ( append_nat @ ( list_update_nat @ Xs @ I3 @ X5 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_441_mset__swap,axiom,
    ! [I3: nat,Ls: list_P8791008543742759928ropVar,J2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Ls ) )
     => ( ( ord_less_nat @ J2 @ ( size_s6752116297430514276ropVar @ Ls ) )
       => ( ( mset_P5170620898817730055ropVar @ ( list_u8100953110031273952ropVar @ ( list_u8100953110031273952ropVar @ Ls @ J2 @ ( nth_Pr227365692153740281ropVar @ Ls @ I3 ) ) @ I3 @ ( nth_Pr227365692153740281ropVar @ Ls @ J2 ) ) )
          = ( mset_P5170620898817730055ropVar @ Ls ) ) ) ) ).

% mset_swap
thf(fact_442_mset__swap,axiom,
    ! [I3: nat,Ls: list_nat,J2: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ls ) )
     => ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Ls ) )
       => ( ( mset_nat @ ( list_update_nat @ ( list_update_nat @ Ls @ J2 @ ( nth_nat @ Ls @ I3 ) ) @ I3 @ ( nth_nat @ Ls @ J2 ) ) )
          = ( mset_nat @ Ls ) ) ) ) ).

% mset_swap
thf(fact_443_i__of__j2__def,axiom,
    ! [J2: nat] :
      ( ( i_of_j2 @ J2 )
      = ( fChoice_nat
        @ ^ [I: nat] :
            ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
            & ( cs @ I @ J2 ) ) ) ) ).

% i_of_j2_def
thf(fact_444_mset__map__split,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Ys1: list_P8791008543742759928ropVar,Ys22: list_P8791008543742759928ropVar] :
      ( ( ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) )
        = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Ys1 ) @ ( mset_P5170620898817730055ropVar @ Ys22 ) ) )
     => ? [Zs1: list_nat,Zs2: list_nat] :
          ( ( ( mset_nat @ Xs )
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
          & ( Ys1
            = ( map_na2145702811007361372ropVar @ F @ Zs1 ) )
          & ( Ys22
            = ( map_na2145702811007361372ropVar @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_445_mset__map__split,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys1: list_M6740372356978808802ropVar,Ys22: list_M6740372356978808802ropVar] :
      ( ( ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys1 ) @ ( mset_M6581516532424202993ropVar @ Ys22 ) ) )
     => ? [Zs1: list_nat,Zs2: list_nat] :
          ( ( ( mset_nat @ Xs )
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
          & ( Ys1
            = ( map_na2466558649495876678ropVar @ F @ Zs1 ) )
          & ( Ys22
            = ( map_na2466558649495876678ropVar @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_446_mset__map__split,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys1: list_nat,Ys22: list_nat] :
      ( ( ( mset_nat @ ( map_nat_nat @ F @ Xs ) )
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys22 ) ) )
     => ? [Zs1: list_nat,Zs2: list_nat] :
          ( ( ( mset_nat @ Xs )
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Zs1 ) @ ( mset_nat @ Zs2 ) ) )
          & ( Ys1
            = ( map_nat_nat @ F @ Zs1 ) )
          & ( Ys22
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% mset_map_split
thf(fact_447_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_448_verit__sum__simplify,axiom,
    ! [A: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ A @ zero_z7348594199698428585et_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_449_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_450__092_060open_062v_A_092_060equiv_062_A_092_060lambda_062x_O_Acase_Ax_Aof_AAuxZeroJI_Ai_Aj_A_092_060Rightarrow_062_A_092_060not_062_ABex_A_Iset_A_Idrop_Ai_A_Imap_A_I_092_060lambda_062i_O_AGamma_Ai_Aj_J_A_0910_O_O_060n_093_J_J_J_Av_H_A_124_AAuxOneJI_Ai_Aj_A_092_060Rightarrow_062_A_092_060exists_062_By_O_Ay_A_092_060in_062_Aset_A_Idrop_Ai_A_Imap_A_I_092_060lambda_062i_O_AGamma_Ai_Aj_J_A_0910_O_O_060n_093_J_J_A_092_060and_062_Av_H_Ay_A_124_AAuxZeroIJ_Ai_Aj_A_092_060Rightarrow_062_A_092_060not_062_ABex_A_Iset_A_Idrop_Aj_A_Imap_A_IGamma_Ai_J_A_0910_O_O_060m_093_J_J_J_Av_H_A_124_AAuxOneIJ_Ai_Aj_A_092_060Rightarrow_062_A_092_060exists_062_By_O_Ay_A_092_060in_062_Aset_A_Idrop_Aj_A_Imap_A_IGamma_Ai_J_A_0910_O_O_060m_093_J_J_A_092_060and_062_Av_H_Ay_A_124_A___A_092_060Rightarrow_062_Av_H_Ax_092_060close_062,axiom,
    ( v2
    = ( ^ [X: multis3193088007478089820ropVar] :
          ( multis8641292176345589959pVar_o
          @ ^ [Nat1: nat,Nat2: nat] : ( v @ X )
          @ ^ [Nat: nat] : ( v @ X )
          @ ^ [I: nat,J: nat] :
              ~ ? [Y2: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y2
                    @ ( set_Mu146256361055586289ropVar
                      @ ( drop_M2293212096379261346ropVar @ I
                        @ ( map_na2466558649495876678ropVar
                          @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J )
                          @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                  & ( v @ Y2 ) )
          @ ^ [I: nat,J: nat] :
            ? [Y2: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Y2
                @ ( set_Mu146256361055586289ropVar
                  @ ( drop_M2293212096379261346ropVar @ I
                    @ ( map_na2466558649495876678ropVar
                      @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J )
                      @ ( upt @ zero_zero_nat @ n ) ) ) ) )
              & ( v @ Y2 )
              & ! [Z3: multis3193088007478089820ropVar] :
                  ( ( ( member8421201611525043589ropVar @ Z3
                      @ ( set_Mu146256361055586289ropVar
                        @ ( drop_M2293212096379261346ropVar @ I
                          @ ( map_na2466558649495876678ropVar
                            @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J )
                            @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                    & ( v @ Z3 ) )
                 => ( Z3 = Y2 ) ) )
          @ ^ [I: nat,J: nat] :
              ~ ? [Y2: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                  & ( v @ Y2 ) )
          @ ^ [I: nat,J: nat] :
            ? [Y2: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
              & ( v @ Y2 )
              & ! [Z3: multis3193088007478089820ropVar] :
                  ( ( ( member8421201611525043589ropVar @ Z3 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                    & ( v @ Z3 ) )
                 => ( Z3 = Y2 ) ) )
          @ X ) ) ) ).

% \<open>v \<equiv> \<lambda>x. case x of AuxZeroJI i j \<Rightarrow> \<not> Bex (set (drop i (map (\<lambda>i. Gamma i j) [0..<n]))) v' | AuxOneJI i j \<Rightarrow> \<exists>!y. y \<in> set (drop i (map (\<lambda>i. Gamma i j) [0..<n])) \<and> v' y | AuxZeroIJ i j \<Rightarrow> \<not> Bex (set (drop j (map (Gamma i) [0..<m]))) v' | AuxOneIJ i j \<Rightarrow> \<exists>!y. y \<in> set (drop j (map (Gamma i) [0..<m])) \<and> v' y | _ \<Rightarrow> v' x\<close>
thf(fact_451_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_452_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_453_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_454_drop__drop,axiom,
    ! [N: nat,M2: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( drop_M2293212096379261346ropVar @ M2 @ Xs ) )
      = ( drop_M2293212096379261346ropVar @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ).

% drop_drop
thf(fact_455_drop__drop,axiom,
    ! [N: nat,M2: nat,Xs: list_nat] :
      ( ( drop_nat @ N @ ( drop_nat @ M2 @ Xs ) )
      = ( drop_nat @ ( plus_plus_nat @ N @ M2 ) @ Xs ) ) ).

% drop_drop
thf(fact_456_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_457_length__append,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( size_s6752116297430514276ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s6752116297430514276ropVar @ Xs ) @ ( size_s6752116297430514276ropVar @ Ys ) ) ) ).

% length_append
thf(fact_458_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_459_drop0,axiom,
    ( ( drop_M2293212096379261346ropVar @ zero_zero_nat )
    = ( ^ [X: list_M6740372356978808802ropVar] : X ) ) ).

% drop0
thf(fact_460_drop0,axiom,
    ( ( drop_nat @ zero_zero_nat )
    = ( ^ [X: list_nat] : X ) ) ).

% drop0
thf(fact_461_size__union,axiom,
    ! [M3: multiset_nat,N4: multiset_nat] :
      ( ( size_s5917832649809541300et_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
      = ( plus_plus_nat @ ( size_s5917832649809541300et_nat @ M3 ) @ ( size_s5917832649809541300et_nat @ N4 ) ) ) ).

% size_union
thf(fact_462_nth__append__length__plus,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,N: nat] :
      ( ( nth_Pr227365692153740281ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s6752116297430514276ropVar @ Xs ) @ N ) )
      = ( nth_Pr227365692153740281ropVar @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_463_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_464_drop__update__cancel,axiom,
    ! [N: nat,M2: nat,Xs: list_M6740372356978808802ropVar,X5: multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( drop_M2293212096379261346ropVar @ M2 @ ( list_u5207168704986282698ropVar @ Xs @ N @ X5 ) )
        = ( drop_M2293212096379261346ropVar @ M2 @ Xs ) ) ) ).

% drop_update_cancel
thf(fact_465_drop__update__cancel,axiom,
    ! [N: nat,M2: nat,Xs: list_nat,X5: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( drop_nat @ M2 @ ( list_update_nat @ Xs @ N @ X5 ) )
        = ( drop_nat @ M2 @ Xs ) ) ) ).

% drop_update_cancel
thf(fact_466_nth__upt,axiom,
    ! [I3: nat,K: nat,J2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 )
     => ( ( nth_nat @ ( upt @ I3 @ J2 ) @ K )
        = ( plus_plus_nat @ I3 @ K ) ) ) ).

% nth_upt
thf(fact_467_v__def,axiom,
    ! [X5: multis3193088007478089820ropVar] :
      ( ( v2 @ X5 )
      = ( multis8641292176345589959pVar_o
        @ ^ [Nat1: nat,Nat2: nat] : ( v @ X5 )
        @ ^ [Nat: nat] : ( v @ X5 )
        @ ^ [I: nat,J: nat] :
            ~ ? [X: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X
                  @ ( set_Mu146256361055586289ropVar
                    @ ( drop_M2293212096379261346ropVar @ I
                      @ ( map_na2466558649495876678ropVar
                        @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J )
                        @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                & ( v @ X ) )
        @ ^ [I: nat,J: nat] :
          ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X
              @ ( set_Mu146256361055586289ropVar
                @ ( drop_M2293212096379261346ropVar @ I
                  @ ( map_na2466558649495876678ropVar
                    @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J )
                    @ ( upt @ zero_zero_nat @ n ) ) ) ) )
            & ( v @ X )
            & ! [Y2: multis3193088007478089820ropVar] :
                ( ( ( member8421201611525043589ropVar @ Y2
                    @ ( set_Mu146256361055586289ropVar
                      @ ( drop_M2293212096379261346ropVar @ I
                        @ ( map_na2466558649495876678ropVar
                          @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J )
                          @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                  & ( v @ Y2 ) )
               => ( Y2 = X ) ) )
        @ ^ [I: nat,J: nat] :
            ~ ? [X: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                & ( v @ X ) )
        @ ^ [I: nat,J: nat] :
          ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
            & ( v @ X )
            & ! [Y2: multis3193088007478089820ropVar] :
                ( ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                  & ( v @ Y2 ) )
               => ( Y2 = X ) ) )
        @ X5 ) ) ).

% v_def
thf(fact_468_drop__zip,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( drop_P7722043279991717613ropVar @ N @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) )
      = ( zip_Mu2706791923038522295ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) @ ( drop_M2293212096379261346ropVar @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_469_drop__zip,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,Ys: list_nat] :
      ( ( drop_P2520701842345426197ar_nat @ N @ ( zip_Mu6626014532916887989ar_nat @ Xs @ Ys ) )
      = ( zip_Mu6626014532916887989ar_nat @ ( drop_M2293212096379261346ropVar @ N @ Xs ) @ ( drop_nat @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_470_drop__zip,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_M6740372356978808802ropVar] :
      ( ( drop_P4751389034305084731ropVar @ N @ ( zip_na7122055390739322131ropVar @ Xs @ Ys ) )
      = ( zip_na7122055390739322131ropVar @ ( drop_nat @ N @ Xs ) @ ( drop_M2293212096379261346ropVar @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_471_drop__zip,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_P8868858903918902087at_nat @ N @ ( zip_nat_nat @ Xs @ Ys ) )
      = ( zip_nat_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ N @ Ys ) ) ) ).

% drop_zip
thf(fact_472_verit__sko__forall__indirect2,axiom,
    ! [X5: nat,P: nat > $o,P5: nat > $o] :
      ( ( X5
        = ( fChoice_nat
          @ ^ [X: nat] :
              ~ ( P @ X ) ) )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
            = ( P5 @ X4 ) )
       => ( ( ! [X3: nat] : ( P5 @ X3 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_473_verit__sko__forall__indirect,axiom,
    ! [X5: nat,P: nat > $o] :
      ( ( X5
        = ( fChoice_nat
          @ ^ [X: nat] :
              ~ ( P @ X ) ) )
     => ( ( ! [X3: nat] : ( P @ X3 ) )
        = ( P @ X5 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_474_verit__sko__ex__indirect2,axiom,
    ! [X5: nat,P: nat > $o,P5: nat > $o] :
      ( ( X5
        = ( fChoice_nat @ P ) )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
            = ( P5 @ X4 ) )
       => ( ( ? [X3: nat] : ( P5 @ X3 ) )
          = ( P @ X5 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_475_verit__sko__ex__indirect,axiom,
    ! [X5: nat,P: nat > $o] :
      ( ( X5
        = ( fChoice_nat @ P ) )
     => ( ( ? [X3: nat] : ( P @ X3 ) )
        = ( P @ X5 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_476_verit__sko__forall_H_H,axiom,
    ! [B6: nat,A2: nat,P: nat > $o] :
      ( ( B6 = A2 )
     => ( ( ( fChoice_nat @ P )
          = A2 )
        = ( ( fChoice_nat @ P )
          = B6 ) ) ) ).

% verit_sko_forall''
thf(fact_477_verit__sko__forall_H,axiom,
    ! [P: nat > $o,A2: $o] :
      ( ( ( P
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = A2 )
     => ( ( ! [X3: nat] : ( P @ X3 ) )
        = A2 ) ) ).

% verit_sko_forall'
thf(fact_478_verit__sko__forall,axiom,
    ( ( ^ [P6: nat > $o] :
        ! [X8: nat] : ( P6 @ X8 ) )
    = ( ^ [P4: nat > $o] :
          ( P4
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ~ ( P4 @ X ) ) ) ) ) ).

% verit_sko_forall
thf(fact_479_verit__sko__ex_H,axiom,
    ! [P: nat > $o,A2: $o] :
      ( ( ( P @ ( fChoice_nat @ P ) )
        = A2 )
     => ( ( ? [X3: nat] : ( P @ X3 ) )
        = A2 ) ) ).

% verit_sko_ex'
thf(fact_480_drop__0,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ zero_zero_nat @ Xs )
      = Xs ) ).

% drop_0
thf(fact_481_drop__0,axiom,
    ! [Xs: list_nat] :
      ( ( drop_nat @ zero_zero_nat @ Xs )
      = Xs ) ).

% drop_0
thf(fact_482_in__set__dropD,axiom,
    ! [X5: nat,N: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ ( drop_nat @ N @ Xs ) ) )
     => ( member_nat @ X5 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_dropD
thf(fact_483_in__set__dropD,axiom,
    ! [X5: multis3193088007478089820ropVar,N: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) ) )
     => ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% in_set_dropD
thf(fact_484_drop__map,axiom,
    ! [N: nat,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( map_Mu2265489581466636612ropVar @ F @ Xs ) )
      = ( map_Mu2265489581466636612ropVar @ F @ ( drop_M2293212096379261346ropVar @ N @ Xs ) ) ) ).

% drop_map
thf(fact_485_drop__map,axiom,
    ! [N: nat,F: multis3193088007478089820ropVar > nat,Xs: list_M6740372356978808802ropVar] :
      ( ( drop_nat @ N @ ( map_Mu1970517791673442536ar_nat @ F @ Xs ) )
      = ( map_Mu1970517791673442536ar_nat @ F @ ( drop_M2293212096379261346ropVar @ N @ Xs ) ) ) ).

% drop_map
thf(fact_486_drop__map,axiom,
    ! [N: nat,F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( drop_P4829783966340880056ropVar @ N @ ( map_na2145702811007361372ropVar @ F @ Xs ) )
      = ( map_na2145702811007361372ropVar @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_487_drop__map,axiom,
    ! [N: nat,F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( map_na2466558649495876678ropVar @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_488_drop__map,axiom,
    ! [N: nat,F: nat > nat,Xs: list_nat] :
      ( ( drop_nat @ N @ ( map_nat_nat @ F @ Xs ) )
      = ( map_nat_nat @ F @ ( drop_nat @ N @ Xs ) ) ) ).

% drop_map
thf(fact_489_distinct__drop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,I3: nat] :
      ( ( distin2675581014522817237ropVar @ Xs )
     => ( distin2675581014522817237ropVar @ ( drop_M2293212096379261346ropVar @ I3 @ Xs ) ) ) ).

% distinct_drop
thf(fact_490_distinct__drop,axiom,
    ! [Xs: list_nat,I3: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( drop_nat @ I3 @ Xs ) ) ) ).

% distinct_drop
thf(fact_491_plus__nat_Oadd__0,axiom,
    ! [N: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N )
      = N ) ).

% plus_nat.add_0
thf(fact_492_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_493_add__lessD1,axiom,
    ! [I3: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ K )
     => ( ord_less_nat @ I3 @ K ) ) ).

% add_lessD1
thf(fact_494_add__less__mono,axiom,
    ! [I3: nat,J2: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I3 @ J2 )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ L ) ) ) ) ).

% add_less_mono
thf(fact_495_not__add__less1,axiom,
    ! [I3: nat,J2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I3 @ J2 ) @ I3 ) ).

% not_add_less1
thf(fact_496_not__add__less2,axiom,
    ! [J2: nat,I3: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J2 @ I3 ) @ I3 ) ).

% not_add_less2
thf(fact_497_add__less__mono1,axiom,
    ! [I3: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ J2 )
     => ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% add_less_mono1
thf(fact_498_trans__less__add1,axiom,
    ! [I3: nat,J2: nat,M2: nat] :
      ( ( ord_less_nat @ I3 @ J2 )
     => ( ord_less_nat @ I3 @ ( plus_plus_nat @ J2 @ M2 ) ) ) ).

% trans_less_add1
thf(fact_499_trans__less__add2,axiom,
    ! [I3: nat,J2: nat,M2: nat] :
      ( ( ord_less_nat @ I3 @ J2 )
     => ( ord_less_nat @ I3 @ ( plus_plus_nat @ M2 @ J2 ) ) ) ).

% trans_less_add2
thf(fact_500_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_501_less__imp__add__positive,axiom,
    ! [I3: nat,J2: nat] :
      ( ( ord_less_nat @ I3 @ J2 )
     => ? [K4: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K4 )
          & ( ( plus_plus_nat @ I3 @ K4 )
            = J2 ) ) ) ).

% less_imp_add_positive
thf(fact_502_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_503_gen__length__def,axiom,
    ( gen_le3870443462522697605ropVar
    = ( ^ [N3: nat,Xs2: list_P8791008543742759928ropVar] : ( plus_plus_nat @ N3 @ ( size_s6752116297430514276ropVar @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_504_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N3: nat,Xs2: list_nat] : ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_505_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_506_some__sym__eq__trivial,axiom,
    ! [X5: nat] :
      ( ( fChoice_nat
        @ ( ^ [Y: nat,Z: nat] : ( Y = Z )
          @ X5 ) )
      = X5 ) ).

% some_sym_eq_trivial
thf(fact_507_some__eq__trivial,axiom,
    ! [X5: nat] :
      ( ( fChoice_nat
        @ ^ [Y2: nat] : ( Y2 = X5 ) )
      = X5 ) ).

% some_eq_trivial
thf(fact_508_some__equality,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( P @ A )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( X4 = A ) )
       => ( ( fChoice_nat @ P )
          = A ) ) ) ).

% some_equality
thf(fact_509_ns__mul__ext__elim,axiom,
    ! [Xs: multis193838646080965794ropVar,Ys: multis193838646080965794ropVar,NS: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis738276037714153361ropVar @ NS @ S2 ) )
     => ? [Xs12: list_M6740372356978808802ropVar,Xs22: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs12 ) @ ( mset_M6581516532424202993ropVar @ Xs22 ) ) )
          & ? [Ys12: list_M6740372356978808802ropVar,Ys23: list_M6740372356978808802ropVar] :
              ( ( Ys
                = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys12 ) @ ( mset_M6581516532424202993ropVar @ Ys23 ) ) )
              & ( ( size_s1943289338788136526ropVar @ Xs12 )
                = ( size_s1943289338788136526ropVar @ Ys12 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s1943289338788136526ropVar @ Ys12 ) )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs12 @ I4 ) @ ( nth_Mu4537670848183934179ropVar @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Ys23 ) )
                 => ? [Xa2: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Xs22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_510_ns__mul__ext__elim,axiom,
    ! [Xs: multis2129813143740876984ropVar,Ys: multis2129813143740876984ropVar,NS: set_Pr3201966908590682119ropVar,S2: set_Pr3201966908590682119ropVar] :
      ( ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ Xs @ Ys ) @ ( multis6478642269264501415ropVar @ NS @ S2 ) )
     => ? [Xs12: list_P8791008543742759928ropVar,Xs22: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Xs12 ) @ ( mset_P5170620898817730055ropVar @ Xs22 ) ) )
          & ? [Ys12: list_P8791008543742759928ropVar,Ys23: list_P8791008543742759928ropVar] :
              ( ( Ys
                = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Ys12 ) @ ( mset_P5170620898817730055ropVar @ Ys23 ) ) )
              & ( ( size_s6752116297430514276ropVar @ Xs12 )
                = ( size_s6752116297430514276ropVar @ Ys12 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s6752116297430514276ropVar @ Ys12 ) )
                 => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs12 @ I4 ) @ ( nth_Pr227365692153740281ropVar @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: produc2642885279816024434ropVar] :
                  ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Ys23 ) )
                 => ? [Xa2: produc2642885279816024434ropVar] :
                      ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Xs22 ) )
                      & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_511_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) )
     => ? [Xs12: list_nat,Xs22: list_nat] :
          ( ( Xs
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs22 ) ) )
          & ? [Ys12: list_nat,Ys23: list_nat] :
              ( ( Ys
                = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys23 ) ) )
              & ( ( size_size_list_nat @ Xs12 )
                = ( size_size_list_nat @ Ys12 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys12 ) )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs12 @ I4 ) @ ( nth_nat @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: nat] :
                  ( ( member_nat @ X6 @ ( set_nat2 @ Ys23 ) )
                 => ? [Xa2: nat] :
                      ( ( member_nat @ Xa2 @ ( set_nat2 @ Xs22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_512_ns__mul__ext__intro,axiom,
    ! [Xs: multis193838646080965794ropVar,Xs1: list_M6740372356978808802ropVar,Xs23: list_M6740372356978808802ropVar,Ys: multis193838646080965794ropVar,Ys1: list_M6740372356978808802ropVar,Ys22: list_M6740372356978808802ropVar,NS: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( Xs
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs1 ) @ ( mset_M6581516532424202993ropVar @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys1 ) @ ( mset_M6581516532424202993ropVar @ Ys22 ) ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs1 )
            = ( size_s1943289338788136526ropVar @ Ys1 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Ys1 ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs1 @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Ys1 @ I2 ) ) @ NS ) )
           => ( ! [Y4: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y4 @ ( set_Mu146256361055586289ropVar @ Ys22 ) )
                 => ? [X6: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs23 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X6 @ Y4 ) @ S2 ) ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis738276037714153361ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_513_ns__mul__ext__intro,axiom,
    ! [Xs: multis2129813143740876984ropVar,Xs1: list_P8791008543742759928ropVar,Xs23: list_P8791008543742759928ropVar,Ys: multis2129813143740876984ropVar,Ys1: list_P8791008543742759928ropVar,Ys22: list_P8791008543742759928ropVar,NS: set_Pr3201966908590682119ropVar,S2: set_Pr3201966908590682119ropVar] :
      ( ( Xs
        = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Xs1 ) @ ( mset_P5170620898817730055ropVar @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Ys1 ) @ ( mset_P5170620898817730055ropVar @ Ys22 ) ) )
       => ( ( ( size_s6752116297430514276ropVar @ Xs1 )
            = ( size_s6752116297430514276ropVar @ Ys1 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Ys1 ) )
               => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs1 @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Ys1 @ I2 ) ) @ NS ) )
           => ( ! [Y4: produc2642885279816024434ropVar] :
                  ( ( member9210975342120487067ropVar @ Y4 @ ( set_Pr3946994337432900359ropVar @ Ys22 ) )
                 => ? [X6: produc2642885279816024434ropVar] :
                      ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs23 ) )
                      & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X6 @ Y4 ) @ S2 ) ) )
             => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ Xs @ Ys ) @ ( multis6478642269264501415ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_514_ns__mul__ext__intro,axiom,
    ! [Xs: multiset_nat,Xs1: list_nat,Xs23: list_nat,Ys: multiset_nat,Ys1: list_nat,Ys22: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( Xs
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs1 ) @ ( mset_nat @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys22 ) ) )
       => ( ( ( size_size_list_nat @ Xs1 )
            = ( size_size_list_nat @ Ys1 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys1 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I2 ) @ ( nth_nat @ Ys1 @ I2 ) ) @ NS ) )
           => ( ! [Y4: nat] :
                  ( ( member_nat @ Y4 @ ( set_nat2 @ Ys22 ) )
                 => ? [X6: nat] :
                      ( ( member_nat @ X6 @ ( set_nat2 @ Xs23 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y4 ) @ S2 ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_515_listrelp__listrel__eq,axiom,
    ! [R: set_Pr6405597471078531282ropVar] :
      ( ( listre1108540755941494666ropVar
        @ ^ [X: multis3193088007478089820ropVar,Y2: produc7679832412844898215ropVar] : ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X @ Y2 ) @ R ) )
      = ( ^ [X: list_M6740372356978808802ropVar,Y2: list_P444016040222843053ropVar] : ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ X @ Y2 ) @ ( listre2663080307769418044ropVar @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_516_listrelp__listrel__eq,axiom,
    ! [R: set_Pr7799659934693882759ropVar] :
      ( ( listre514068680659036479ropVar
        @ ^ [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y2 ) @ R ) )
      = ( ^ [X: list_M6740372356978808802ropVar,Y2: list_M6740372356978808802ropVar] : ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ X @ Y2 ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_517_drop__upt,axiom,
    ! [M2: nat,I3: nat,J2: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I3 @ J2 ) )
      = ( upt @ ( plus_plus_nat @ I3 @ M2 ) @ J2 ) ) ).

% drop_upt
thf(fact_518_someI2,axiom,
    ! [P: nat > $o,A: nat,Q: nat > $o] :
      ( ( P @ A )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( fChoice_nat @ P ) ) ) ) ).

% someI2
thf(fact_519_someI__ex,axiom,
    ! [P: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( P @ ( fChoice_nat @ P ) ) ) ).

% someI_ex
thf(fact_520_someI2__ex,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( Q @ X4 ) )
       => ( Q @ ( fChoice_nat @ P ) ) ) ) ).

% someI2_ex
thf(fact_521_someI2__bex,axiom,
    ! [A2: set_Mu5658373669842198716ropVar,P: multis3193088007478089820ropVar > $o,Q: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X4: multis3193088007478089820ropVar] :
            ( ( ( member8421201611525043589ropVar @ X4 @ A2 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoic539809714188224432ropVar
            @ ^ [X: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_522_someI2__bex,axiom,
    ! [A2: set_nat,P: nat > $o,Q: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ A2 )
          & ( P @ X6 ) )
     => ( ! [X4: nat] :
            ( ( ( member_nat @ X4 @ A2 )
              & ( P @ X4 ) )
           => ( Q @ X4 ) )
       => ( Q
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_523_some__eq__ex,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X3: nat] : ( P @ X3 ) ) ) ).

% some_eq_ex
thf(fact_524_some1__equality,axiom,
    ! [P: nat > $o,A: nat] :
      ( ? [X6: nat] :
          ( ( P @ X6 )
          & ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( Y4 = X6 ) ) )
     => ( ( P @ A )
       => ( ( fChoice_nat @ P )
          = A ) ) ) ).

% some1_equality
thf(fact_525_all__ns__ns__mul__ext,axiom,
    ! [As: list_M6740372356978808802ropVar,Bs: list_M6740372356978808802ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ As )
        = ( size_s1943289338788136526ropVar @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Bs ) )
           => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ As @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Bs @ I2 ) ) @ Ns ) )
       => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ As ) @ ( mset_M6581516532424202993ropVar @ Bs ) ) @ ( multis738276037714153361ropVar @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_526_all__ns__ns__mul__ext,axiom,
    ! [As: list_P8791008543742759928ropVar,Bs: list_P8791008543742759928ropVar,Ns: set_Pr3201966908590682119ropVar,S: set_Pr3201966908590682119ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ As )
        = ( size_s6752116297430514276ropVar @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Bs ) )
           => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ As @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Bs @ I2 ) ) @ Ns ) )
       => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ ( mset_P5170620898817730055ropVar @ As ) @ ( mset_P5170620898817730055ropVar @ Bs ) ) @ ( multis6478642269264501415ropVar @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_527_all__ns__ns__mul__ext,axiom,
    ! [As: list_nat,Bs: list_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ Ns ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As ) @ ( mset_nat @ Bs ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_528_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( P @ A4 @ B3 )
          = ( P @ B3 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B3: nat] :
              ( ( P @ A4 @ B3 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_529_map__upt__eqI,axiom,
    ! [Xs: list_M6740372356978808802ropVar,N: nat,M2: nat,F: nat > multis3193088007478089820ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( minus_minus_nat @ N @ M2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Xs ) )
           => ( ( nth_Mu4537670848183934179ropVar @ Xs @ I2 )
              = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ ( upt @ M2 @ N ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_530_map__upt__eqI,axiom,
    ! [Xs: list_P8791008543742759928ropVar,N: nat,M2: nat,F: nat > produc2642885279816024434ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( minus_minus_nat @ N @ M2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Xs ) )
           => ( ( nth_Pr227365692153740281ropVar @ Xs @ I2 )
              = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) )
       => ( ( map_na2145702811007361372ropVar @ F @ ( upt @ M2 @ N ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_531_map__upt__eqI,axiom,
    ! [Xs: list_nat,N: nat,M2: nat,F: nat > nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( minus_minus_nat @ N @ M2 ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) )
       => ( ( map_nat_nat @ F @ ( upt @ M2 @ N ) )
          = Xs ) ) ) ).

% map_upt_eqI
thf(fact_532_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_533_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_534_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_535_add__diff__cancel__right_H,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_536_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_537_add__diff__cancel__right,axiom,
    ! [A: multiset_nat,C3: multiset_nat,B: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ C3 ) @ ( plus_p6334493942879108393et_nat @ B @ C3 ) )
      = ( minus_8522176038001411705et_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_538_add__diff__cancel__right,axiom,
    ! [A: nat,C3: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C3 ) @ ( plus_plus_nat @ B @ C3 ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_539_add__diff__cancel__left_H,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_540_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_541_add__diff__cancel__left,axiom,
    ! [C3: multiset_nat,A: multiset_nat,B: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C3 @ A ) @ ( plus_p6334493942879108393et_nat @ C3 @ B ) )
      = ( minus_8522176038001411705et_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_542_add__diff__cancel__left,axiom,
    ! [C3: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C3 @ A ) @ ( plus_plus_nat @ C3 @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_543_diff__0__eq__0,axiom,
    ! [N: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_544_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_545_diff__diff__left,axiom,
    ! [I3: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
      = ( minus_minus_nat @ I3 @ ( plus_plus_nat @ J2 @ K ) ) ) ).

% diff_diff_left
thf(fact_546_diff__add__zero,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ A @ ( plus_p6334493942879108393et_nat @ A @ B ) )
      = zero_z7348594199698428585et_nat ) ).

% diff_add_zero
thf(fact_547_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_548_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_549_length__drop,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar] :
      ( ( size_s1943289338788136526ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) )
      = ( minus_minus_nat @ ( size_s1943289338788136526ropVar @ Xs ) @ N ) ) ).

% length_drop
thf(fact_550_length__drop,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar] :
      ( ( size_s6752116297430514276ropVar @ ( drop_P4829783966340880056ropVar @ N @ Xs ) )
      = ( minus_minus_nat @ ( size_s6752116297430514276ropVar @ Xs ) @ N ) ) ).

% length_drop
thf(fact_551_length__drop,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( drop_nat @ N @ Xs ) )
      = ( minus_minus_nat @ ( size_size_list_nat @ Xs ) @ N ) ) ).

% length_drop
thf(fact_552_length__upt,axiom,
    ! [I3: nat,J2: nat] :
      ( ( size_size_list_nat @ ( upt @ I3 @ J2 ) )
      = ( minus_minus_nat @ J2 @ I3 ) ) ).

% length_upt
thf(fact_553_drop__append,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( drop_M2293212096379261346ropVar @ N @ ( append2684835082924936439ropVar @ Xs @ Ys ) )
      = ( append2684835082924936439ropVar @ ( drop_M2293212096379261346ropVar @ N @ Xs ) @ ( drop_M2293212096379261346ropVar @ ( minus_minus_nat @ N @ ( size_s1943289338788136526ropVar @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_554_drop__append,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( drop_P4829783966340880056ropVar @ N @ ( append4352292717583037453ropVar @ Xs @ Ys ) )
      = ( append4352292717583037453ropVar @ ( drop_P4829783966340880056ropVar @ N @ Xs ) @ ( drop_P4829783966340880056ropVar @ ( minus_minus_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_555_drop__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( drop_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( drop_nat @ N @ Xs ) @ ( drop_nat @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_556_diff__commute,axiom,
    ! [I3: nat,J2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J2 ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I3 @ K ) @ J2 ) ) ).

% diff_commute
thf(fact_557_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C3: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C3 ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C3 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_558_diff__diff__eq,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ A @ B ) @ C3 )
      = ( minus_8522176038001411705et_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ C3 ) ) ) ).

% diff_diff_eq
thf(fact_559_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C3: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C3 )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C3 ) ) ) ).

% diff_diff_eq
thf(fact_560_add__implies__diff,axiom,
    ! [C3: multiset_nat,B: multiset_nat,A: multiset_nat] :
      ( ( ( plus_p6334493942879108393et_nat @ C3 @ B )
        = A )
     => ( C3
        = ( minus_8522176038001411705et_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_561_add__implies__diff,axiom,
    ! [C3: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C3 @ B )
        = A )
     => ( C3
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_562_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_563_diffs0__imp__equal,axiom,
    ! [M2: nat,N: nat] :
      ( ( ( minus_minus_nat @ M2 @ N )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N @ M2 )
          = zero_zero_nat )
       => ( M2 = N ) ) ) ).

% diffs0_imp_equal
thf(fact_564_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_565_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_566_diff__add__inverse2,axiom,
    ! [M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N ) @ N )
      = M2 ) ).

% diff_add_inverse2
thf(fact_567_diff__add__inverse,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N @ M2 ) @ N )
      = M2 ) ).

% diff_add_inverse
thf(fact_568_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N @ K ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% diff_cancel2
thf(fact_569_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( minus_minus_nat @ M2 @ N ) ) ).

% Nat.diff_cancel
thf(fact_570_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_571_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_572_diff__add__0,axiom,
    ! [N: nat,M2: nat] :
      ( ( minus_minus_nat @ N @ ( plus_plus_nat @ N @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_573_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_574_less__diff__conv,axiom,
    ! [I3: nat,J2: nat,K: nat] :
      ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ J2 @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I3 @ K ) @ J2 ) ) ).

% less_diff_conv
thf(fact_575_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D3: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D3 ) )
           => ( P @ D3 ) ) ) ) ).

% nat_diff_split
thf(fact_576_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D3: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D3 ) )
                & ~ ( P @ D3 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_577_nth__append,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
       => ( ( nth_Pr227365692153740281ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ N )
          = ( nth_Pr227365692153740281ropVar @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
       => ( ( nth_Pr227365692153740281ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ N )
          = ( nth_Pr227365692153740281ropVar @ Ys @ ( minus_minus_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_578_nth__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ N )
          = ( nth_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_579_list__update__append,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
       => ( ( list_u8100953110031273952ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ N @ X5 )
          = ( append4352292717583037453ropVar @ ( list_u8100953110031273952ropVar @ Xs @ N @ X5 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) )
       => ( ( list_u8100953110031273952ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) @ N @ X5 )
          = ( append4352292717583037453ropVar @ Xs @ ( list_u8100953110031273952ropVar @ Ys @ ( minus_minus_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) ) @ X5 ) ) ) ) ) ).

% list_update_append
thf(fact_580_list__update__append,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat,X5: nat] :
      ( ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X5 )
          = ( append_nat @ ( list_update_nat @ Xs @ N @ X5 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
       => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ N @ X5 )
          = ( append_nat @ Xs @ ( list_update_nat @ Ys @ ( minus_minus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ X5 ) ) ) ) ) ).

% list_update_append
thf(fact_581_nth__map__upt,axiom,
    ! [I3: nat,N: nat,M2: nat,F: nat > produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ N @ M2 ) )
     => ( ( nth_Pr227365692153740281ropVar @ ( map_na2145702811007361372ropVar @ F @ ( upt @ M2 @ N ) ) @ I3 )
        = ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) ) ).

% nth_map_upt
thf(fact_582_nth__map__upt,axiom,
    ! [I3: nat,N: nat,M2: nat,F: nat > multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ N @ M2 ) )
     => ( ( nth_Mu4537670848183934179ropVar @ ( map_na2466558649495876678ropVar @ F @ ( upt @ M2 @ N ) ) @ I3 )
        = ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) ) ).

% nth_map_upt
thf(fact_583_nth__map__upt,axiom,
    ! [I3: nat,N: nat,M2: nat,F: nat > nat] :
      ( ( ord_less_nat @ I3 @ ( minus_minus_nat @ N @ M2 ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M2 @ N ) ) @ I3 )
        = ( F @ ( plus_plus_nat @ M2 @ I3 ) ) ) ) ).

% nth_map_upt
thf(fact_584_ns__mul__extI__old,axiom,
    ! [A2: multis193838646080965794ropVar,Xs: list_M6740372356978808802ropVar,A22: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,Ys: list_M6740372356978808802ropVar,B22: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( A2
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) @ A22 ) )
     => ( ( B6
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys ) @ B22 ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs )
            = ( size_s1943289338788136526ropVar @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Ys ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Ys @ I2 ) ) @ Ns ) )
           => ( ! [B3: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ B3 @ ( set_ms1147286257884310937ropVar @ B22 ) )
                 => ? [A6: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B3 ) @ S ) ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis738276037714153361ropVar @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_585_ns__mul__extI__old,axiom,
    ! [A2: multis2129813143740876984ropVar,Xs: list_P8791008543742759928ropVar,A22: multis2129813143740876984ropVar,B6: multis2129813143740876984ropVar,Ys: list_P8791008543742759928ropVar,B22: multis2129813143740876984ropVar,Ns: set_Pr3201966908590682119ropVar,S: set_Pr3201966908590682119ropVar] :
      ( ( A2
        = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Xs ) @ A22 ) )
     => ( ( B6
          = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Ys ) @ B22 ) )
       => ( ( ( size_s6752116297430514276ropVar @ Xs )
            = ( size_s6752116297430514276ropVar @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Ys ) )
               => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Ys @ I2 ) ) @ Ns ) )
           => ( ! [B3: produc2642885279816024434ropVar] :
                  ( ( member9210975342120487067ropVar @ B3 @ ( set_ms882399555722365103ropVar @ B22 ) )
                 => ? [A6: produc2642885279816024434ropVar] :
                      ( ( member9210975342120487067ropVar @ A6 @ ( set_ms882399555722365103ropVar @ A22 ) )
                      & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A6 @ B3 ) @ S ) ) )
             => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ A2 @ B6 ) @ ( multis6478642269264501415ropVar @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_586_ns__mul__extI__old,axiom,
    ! [A2: multiset_nat,Xs: list_nat,A22: multiset_nat,B6: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( A2
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
     => ( ( B6
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ Ns ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( set_mset_nat @ B22 ) )
                 => ? [A6: nat] :
                      ( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ S ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_587_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_M6740372356978808802ropVar,Bs: list_M6740372356978808802ropVar,NS: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ As )
        = ( size_s1943289338788136526ropVar @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Bs ) )
           => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ As @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Bs @ I2 ) ) @ NS ) )
       => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As @ Bs )
          @ ( collec7970277406990558738ropVar
            @ ( produc6401158782714314988pVar_o
              @ ^ [As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar] : ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ As3 ) @ ( mset_M6581516532424202993ropVar @ Bs3 ) ) @ ( multis738276037714153361ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_588_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_P8791008543742759928ropVar,Bs: list_P8791008543742759928ropVar,NS: set_Pr3201966908590682119ropVar,S2: set_Pr3201966908590682119ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ As )
        = ( size_s6752116297430514276ropVar @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Bs ) )
           => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ As @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Bs @ I2 ) ) @ NS ) )
       => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ As @ Bs )
          @ ( collec851525693690771858ropVar
            @ ( produc8620918753924295276pVar_o
              @ ^ [As3: list_P8791008543742759928ropVar,Bs3: list_P8791008543742759928ropVar] : ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ ( mset_P5170620898817730055ropVar @ As3 ) @ ( mset_P5170620898817730055ropVar @ Bs3 ) ) @ ( multis6478642269264501415ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_589_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ NS ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs )
          @ ( collec1570431334306492044st_nat
            @ ( produc6382697403170970154_nat_o
              @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_590_mul__ext__list_Ons__map,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ S2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ NS )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_591_mul__ext__list_Ons__map,axiom,
    ! [S2: set_Pr7799659934693882759ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS: set_Pr7799659934693882759ropVar,As: list_M6740372356978808802ropVar,Bs: list_M6740372356978808802ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ S2 )
         => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
     => ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ NS )
           => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
       => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As @ Bs )
            @ ( collec7970277406990558738ropVar
              @ ( produc6401158782714314988pVar_o
                @ ^ [As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar] : ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ As3 ) @ ( mset_M6581516532424202993ropVar @ Bs3 ) ) @ ( multis738276037714153361ropVar @ NS @ S2 ) ) ) ) )
         => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F @ As ) @ ( map_Mu2265489581466636612ropVar @ F @ Bs ) )
            @ ( collec7970277406990558738ropVar
              @ ( produc6401158782714314988pVar_o
                @ ^ [As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar] : ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ As3 ) @ ( mset_M6581516532424202993ropVar @ Bs3 ) ) @ ( multis738276037714153361ropVar @ NS @ S2 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_592_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_P8791008543742759928ropVar,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_s6752116297430514276ropVar @ Xs ) )
     => ( ( nth_Pr2354896103702612176ropVar @ ( enumer8415853101840577367ropVar @ N @ Xs ) @ M2 )
        = ( produc9085022949259285321ropVar @ ( plus_plus_nat @ N @ M2 ) @ ( nth_Pr227365692153740281ropVar @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_593_nth__enumerate__eq,axiom,
    ! [M2: nat,Xs: list_nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N @ Xs ) @ M2 )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N @ M2 ) @ ( nth_nat @ Xs @ M2 ) ) ) ) ).

% nth_enumerate_eq
thf(fact_594_multpw__listE,axiom,
    ! [X7: multis193838646080965794ropVar,Y6: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ X7 @ Y6 ) @ ( multis9091339148773653984ropVar @ Ns ) )
     => ~ ! [Xs3: list_M6740372356978808802ropVar,Ys5: list_M6740372356978808802ropVar] :
            ( ( ( size_s1943289338788136526ropVar @ Xs3 )
              = ( size_s1943289338788136526ropVar @ Ys5 ) )
           => ( ( X7
                = ( mset_M6581516532424202993ropVar @ Xs3 ) )
             => ( ( Y6
                  = ( mset_M6581516532424202993ropVar @ Ys5 ) )
               => ~ ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s1943289338788136526ropVar @ Ys5 ) )
                     => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs3 @ I4 ) @ ( nth_Mu4537670848183934179ropVar @ Ys5 @ I4 ) ) @ Ns ) ) ) ) ) ) ).

% multpw_listE
thf(fact_595_multpw__listE,axiom,
    ! [X7: multis2129813143740876984ropVar,Y6: multis2129813143740876984ropVar,Ns: set_Pr3201966908590682119ropVar] :
      ( ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ X7 @ Y6 ) @ ( multis1527098867523604214ropVar @ Ns ) )
     => ~ ! [Xs3: list_P8791008543742759928ropVar,Ys5: list_P8791008543742759928ropVar] :
            ( ( ( size_s6752116297430514276ropVar @ Xs3 )
              = ( size_s6752116297430514276ropVar @ Ys5 ) )
           => ( ( X7
                = ( mset_P5170620898817730055ropVar @ Xs3 ) )
             => ( ( Y6
                  = ( mset_P5170620898817730055ropVar @ Ys5 ) )
               => ~ ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s6752116297430514276ropVar @ Ys5 ) )
                     => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs3 @ I4 ) @ ( nth_Pr227365692153740281ropVar @ Ys5 @ I4 ) ) @ Ns ) ) ) ) ) ) ).

% multpw_listE
thf(fact_596_multpw__listE,axiom,
    ! [X7: multiset_nat,Y6: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X7 @ Y6 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
     => ~ ! [Xs3: list_nat,Ys5: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = ( size_size_list_nat @ Ys5 ) )
           => ( ( X7
                = ( mset_nat @ Xs3 ) )
             => ( ( Y6
                  = ( mset_nat @ Ys5 ) )
               => ~ ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys5 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs3 @ I4 ) @ ( nth_nat @ Ys5 @ I4 ) ) @ Ns ) ) ) ) ) ) ).

% multpw_listE
thf(fact_597_diff__diff__add__mset,axiom,
    ! [M3: multiset_nat,N4: multiset_nat,P: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M3 @ N4 ) @ P )
      = ( minus_8522176038001411705et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ P ) ) ) ).

% diff_diff_add_mset
thf(fact_598_length__enumerate,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar] :
      ( ( size_s815404617008747835ropVar @ ( enumer8415853101840577367ropVar @ N @ Xs ) )
      = ( size_s6752116297430514276ropVar @ Xs ) ) ).

% length_enumerate
thf(fact_599_length__enumerate,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_600_Eps__case__prod__eq,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: produc7679832412844898215ropVar] :
      ( ( fChoic3237874067385186502ropVar
        @ ( produc9122939242341821665pVar_o
          @ ^ [X9: multis3193088007478089820ropVar,Y7: produc7679832412844898215ropVar] :
              ( ( X5 = X9 )
              & ( Y3 = Y7 ) ) ) )
      = ( produc6194868611276206818ropVar @ X5 @ Y3 ) ) ).

% Eps_case_prod_eq
thf(fact_601_Eps__case__prod__eq,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( fChoic157639270322531067ropVar
        @ ( produc1988339464994610156pVar_o
          @ ^ [X9: multis3193088007478089820ropVar,Y7: multis3193088007478089820ropVar] :
              ( ( X5 = X9 )
              & ( Y3 = Y7 ) ) ) )
      = ( produc6936306168532358295ropVar @ X5 @ Y3 ) ) ).

% Eps_case_prod_eq
thf(fact_602_The__split__eq,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: produc7679832412844898215ropVar] :
      ( ( the_Pr3361993152900459897ropVar
        @ ( produc9122939242341821665pVar_o
          @ ^ [X9: multis3193088007478089820ropVar,Y7: produc7679832412844898215ropVar] :
              ( ( X5 = X9 )
              & ( Y3 = Y7 ) ) ) )
      = ( produc6194868611276206818ropVar @ X5 @ Y3 ) ) ).

% The_split_eq
thf(fact_603_The__split__eq,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( the_Pr7468375744313411758ropVar
        @ ( produc1988339464994610156pVar_o
          @ ^ [X9: multis3193088007478089820ropVar,Y7: multis3193088007478089820ropVar] :
              ( ( X5 = X9 )
              & ( Y3 = Y7 ) ) ) )
      = ( produc6936306168532358295ropVar @ X5 @ Y3 ) ) ).

% The_split_eq
thf(fact_604_case__prodI2,axiom,
    ! [P2: produc2642885279816024434ropVar,C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > $o] :
      ( ! [A4: multis3193088007478089820ropVar,B3: produc7679832412844898215ropVar] :
          ( ( P2
            = ( produc6194868611276206818ropVar @ A4 @ B3 ) )
         => ( C3 @ A4 @ B3 ) )
     => ( produc9122939242341821665pVar_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_605_case__prodI2,axiom,
    ! [P2: produc7679832412844898215ropVar,C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > $o] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( ( P2
            = ( produc6936306168532358295ropVar @ A4 @ B3 ) )
         => ( C3 @ A4 @ B3 ) )
     => ( produc1988339464994610156pVar_o @ C3 @ P2 ) ) ).

% case_prodI2
thf(fact_606_case__prodI,axiom,
    ! [F: multis3193088007478089820ropVar > produc7679832412844898215ropVar > $o,A: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar] :
      ( ( F @ A @ B )
     => ( produc9122939242341821665pVar_o @ F @ ( produc6194868611276206818ropVar @ A @ B ) ) ) ).

% case_prodI
thf(fact_607_case__prodI,axiom,
    ! [F: multis3193088007478089820ropVar > multis3193088007478089820ropVar > $o,A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar] :
      ( ( F @ A @ B )
     => ( produc1988339464994610156pVar_o @ F @ ( produc6936306168532358295ropVar @ A @ B ) ) ) ).

% case_prodI
thf(fact_608_set__mset__mset,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( set_ms1147286257884310937ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) )
      = ( set_Mu146256361055586289ropVar @ Xs ) ) ).

% set_mset_mset
thf(fact_609_set__mset__mset,axiom,
    ! [Xs: list_nat] :
      ( ( set_mset_nat @ ( mset_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_mset_mset
thf(fact_610_in__diffD,axiom,
    ! [A: nat,M3: multiset_nat,N4: multiset_nat] :
      ( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M3 @ N4 ) ) )
     => ( member_nat @ A @ ( set_mset_nat @ M3 ) ) ) ).

% in_diffD
thf(fact_611_in__diffD,axiom,
    ! [A: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar,N4: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ ( minus_287902380581138057ropVar @ M3 @ N4 ) ) )
     => ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ M3 ) ) ) ).

% in_diffD
thf(fact_612_multiset__nonemptyE,axiom,
    ! [A2: multiset_nat] :
      ( ( A2 != zero_z7348594199698428585et_nat )
     => ~ ! [X4: nat] :
            ~ ( member_nat @ X4 @ ( set_mset_nat @ A2 ) ) ) ).

% multiset_nonemptyE
thf(fact_613_multiset__nonemptyE,axiom,
    ! [A2: multis193838646080965794ropVar] :
      ( ( A2 != zero_z6226653793019358809ropVar )
     => ~ ! [X4: multis3193088007478089820ropVar] :
            ~ ( member8421201611525043589ropVar @ X4 @ ( set_ms1147286257884310937ropVar @ A2 ) ) ) ).

% multiset_nonemptyE
thf(fact_614_union__iff,axiom,
    ! [A: multis3193088007478089820ropVar,A2: multis193838646080965794ropVar,B6: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ ( plus_p1339091108561003481ropVar @ A2 @ B6 ) ) )
      = ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ A2 ) )
        | ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ B6 ) ) ) ) ).

% union_iff
thf(fact_615_union__iff,axiom,
    ! [A: nat,A2: multiset_nat,B6: multiset_nat] :
      ( ( member_nat @ A @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) ) )
      = ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
        | ( member_nat @ A @ ( set_mset_nat @ B6 ) ) ) ) ).

% union_iff
thf(fact_616_Multiset_Odiff__add,axiom,
    ! [M3: multiset_nat,N4: multiset_nat,Q: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ M3 @ ( plus_p6334493942879108393et_nat @ N4 @ Q ) )
      = ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M3 @ N4 ) @ Q ) ) ).

% Multiset.diff_add
thf(fact_617_diff__union__cancelL,axiom,
    ! [N4: multiset_nat,M3: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ N4 @ M3 ) @ N4 )
      = M3 ) ).

% diff_union_cancelL
thf(fact_618_diff__union__cancelR,axiom,
    ! [M3: multiset_nat,N4: multiset_nat] :
      ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ M3 @ N4 ) @ N4 )
      = M3 ) ).

% diff_union_cancelR
thf(fact_619_split__paired__Eps,axiom,
    ( fChoic3237874067385186502ropVar
    = ( ^ [P4: produc2642885279816024434ropVar > $o] :
          ( fChoic3237874067385186502ropVar
          @ ( produc9122939242341821665pVar_o
            @ ^ [A5: multis3193088007478089820ropVar,B4: produc7679832412844898215ropVar] : ( P4 @ ( produc6194868611276206818ropVar @ A5 @ B4 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_620_split__paired__Eps,axiom,
    ( fChoic157639270322531067ropVar
    = ( ^ [P4: produc7679832412844898215ropVar > $o] :
          ( fChoic157639270322531067ropVar
          @ ( produc1988339464994610156pVar_o
            @ ^ [A5: multis3193088007478089820ropVar,B4: multis3193088007478089820ropVar] : ( P4 @ ( produc6936306168532358295ropVar @ A5 @ B4 ) ) ) ) ) ) ).

% split_paired_Eps
thf(fact_621_case__prodE,axiom,
    ! [C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > $o,P2: produc2642885279816024434ropVar] :
      ( ( produc9122939242341821665pVar_o @ C3 @ P2 )
     => ~ ! [X4: multis3193088007478089820ropVar,Y4: produc7679832412844898215ropVar] :
            ( ( P2
              = ( produc6194868611276206818ropVar @ X4 @ Y4 ) )
           => ~ ( C3 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_622_case__prodE,axiom,
    ! [C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > $o,P2: produc7679832412844898215ropVar] :
      ( ( produc1988339464994610156pVar_o @ C3 @ P2 )
     => ~ ! [X4: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
            ( ( P2
              = ( produc6936306168532358295ropVar @ X4 @ Y4 ) )
           => ~ ( C3 @ X4 @ Y4 ) ) ) ).

% case_prodE
thf(fact_623_case__prodD,axiom,
    ! [F: multis3193088007478089820ropVar > produc7679832412844898215ropVar > $o,A: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar] :
      ( ( produc9122939242341821665pVar_o @ F @ ( produc6194868611276206818ropVar @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_624_case__prodD,axiom,
    ! [F: multis3193088007478089820ropVar > multis3193088007478089820ropVar > $o,A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar] :
      ( ( produc1988339464994610156pVar_o @ F @ ( produc6936306168532358295ropVar @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_625_ns__mul__extE,axiom,
    ! [A2: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis738276037714153361ropVar @ Ns @ S ) )
     => ~ ! [A1: multis193838646080965794ropVar,A23: multis193838646080965794ropVar] :
            ( ( A2
              = ( plus_p1339091108561003481ropVar @ A1 @ A23 ) )
           => ! [B1: multis193838646080965794ropVar,B23: multis193838646080965794ropVar] :
                ( ( B6
                  = ( plus_p1339091108561003481ropVar @ B1 @ B23 ) )
               => ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A1 @ B1 ) @ ( multis9091339148773653984ropVar @ Ns ) )
                 => ~ ! [B5: multis3193088007478089820ropVar] :
                        ( ( member8421201611525043589ropVar @ B5 @ ( set_ms1147286257884310937ropVar @ B23 ) )
                       => ? [A4: multis3193088007478089820ropVar] :
                            ( ( member8421201611525043589ropVar @ A4 @ ( set_ms1147286257884310937ropVar @ A23 ) )
                            & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ).

% ns_mul_extE
thf(fact_626_ns__mul__extE,axiom,
    ! [A2: multiset_nat,B6: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
     => ~ ! [A1: multiset_nat,A23: multiset_nat] :
            ( ( A2
              = ( plus_p6334493942879108393et_nat @ A1 @ A23 ) )
           => ! [B1: multiset_nat,B23: multiset_nat] :
                ( ( B6
                  = ( plus_p6334493942879108393et_nat @ B1 @ B23 ) )
               => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A1 @ B1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
                 => ~ ! [B5: nat] :
                        ( ( member_nat @ B5 @ ( set_mset_nat @ B23 ) )
                       => ? [A4: nat] :
                            ( ( member_nat @ A4 @ ( set_mset_nat @ A23 ) )
                            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ).

% ns_mul_extE
thf(fact_627_ns__mul__extI,axiom,
    ! [A2: multis193838646080965794ropVar,A12: multis193838646080965794ropVar,A22: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,B12: multis193838646080965794ropVar,B22: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( A2
        = ( plus_p1339091108561003481ropVar @ A12 @ A22 ) )
     => ( ( B6
          = ( plus_p1339091108561003481ropVar @ B12 @ B22 ) )
       => ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A12 @ B12 ) @ ( multis9091339148773653984ropVar @ Ns ) )
         => ( ! [B3: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ B3 @ ( set_ms1147286257884310937ropVar @ B22 ) )
               => ? [A6: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A22 ) )
                    & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B3 ) @ S ) ) )
           => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis738276037714153361ropVar @ Ns @ S ) ) ) ) ) ) ).

% ns_mul_extI
thf(fact_628_ns__mul__extI,axiom,
    ! [A2: multiset_nat,A12: multiset_nat,A22: multiset_nat,B6: multiset_nat,B12: multiset_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( A2
        = ( plus_p6334493942879108393et_nat @ A12 @ A22 ) )
     => ( ( B6
          = ( plus_p6334493942879108393et_nat @ B12 @ B22 ) )
       => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A12 @ B12 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
         => ( ! [B3: nat] :
                ( ( member_nat @ B3 @ ( set_mset_nat @ B22 ) )
               => ? [A6: nat] :
                    ( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
                    & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ S ) ) )
           => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ) ) ).

% ns_mul_extI
thf(fact_629_in__multiset__in__set,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) ) )
      = ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_630_in__multiset__in__set,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
      = ( member_nat @ X5 @ ( set_nat2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_631_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr6810726163584182713ropVar @ ( produc1416661995297908400ropVar @ H ) @ ( zip_Mu2706791923038522295ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_632_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr6582868597391307571ar_nat @ ( produc7431473831949406332ar_nat @ H ) @ ( zip_Mu2706791923038522295ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_633_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > nat > multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr193826012314142715ropVar @ ( produc7927514689771840474ropVar @ H ) @ ( zip_Mu6626014532916887989ar_nat @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_634_map2__map__map,axiom,
    ! [H: multis3193088007478089820ropVar > nat > nat,F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr1531560619341599857at_nat @ ( produc2601766963269458130at_nat @ H ) @ ( zip_Mu6626014532916887989ar_nat @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_635_map2__map__map,axiom,
    ! [H: nat > multis3193088007478089820ropVar > multis3193088007478089820ropVar,F: nat > nat,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr1062706098098207061ropVar @ ( produc222539320033202172ropVar @ H ) @ ( zip_na7122055390739322131ropVar @ ( map_nat_nat @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_636_map2__map__map,axiom,
    ! [H: nat > multis3193088007478089820ropVar > nat,F: nat > nat,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr3446757514997570711ar_nat @ ( produc518888235098666032ar_nat @ H ) @ ( zip_na7122055390739322131ropVar @ ( map_nat_nat @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_637_map2__map__map,axiom,
    ! [H: nat > nat > multis3193088007478089820ropVar,F: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr6281086966775181663ropVar @ ( produc1014929092921100174ropVar @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_638_map2__map__map,axiom,
    ! [H: nat > nat > nat,F: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( map_Pr3938374229010428429at_nat @ ( produc6842872674320459806at_nat @ H ) @ ( zip_nat_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_639_map2__map__map,axiom,
    ! [H: produc2642885279816024434ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr1871985665542393807ropVar @ ( produc9119430476348611782ropVar @ H ) @ ( zip_Pr5260769330036727841ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_na2466558649495876678ropVar
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_640_map2__map__map,axiom,
    ! [H: produc2642885279816024434ropVar > multis3193088007478089820ropVar > nat,F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( map_Pr7962067919199238045ar_nat @ ( produc6539649061971739878ar_nat @ H ) @ ( zip_Pr5260769330036727841ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Xs ) ) )
      = ( map_nat_nat
        @ ^ [X: nat] : ( H @ ( F @ X ) @ ( G @ X ) )
        @ Xs ) ) ).

% map2_map_map
thf(fact_641_listrel__def,axiom,
    ( listre2663080307769418044ropVar
    = ( ^ [R3: set_Pr6405597471078531282ropVar] :
          ( collec1525075934208178781ropVar
          @ ( produc6787239627795551841pVar_o
            @ ( listre1108540755941494666ropVar
              @ ^ [X: multis3193088007478089820ropVar,Y2: produc7679832412844898215ropVar] : ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X @ Y2 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_642_listrel__def,axiom,
    ( listre4495854419763884273ropVar
    = ( ^ [R3: set_Pr7799659934693882759ropVar] :
          ( collec7970277406990558738ropVar
          @ ( produc6401158782714314988pVar_o
            @ ( listre514068680659036479ropVar
              @ ^ [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y2 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_643_enumerate__map__upt,axiom,
    ! [N: nat,F: nat > produc2642885279816024434ropVar,M2: nat] :
      ( ( enumer8415853101840577367ropVar @ N @ ( map_na2145702811007361372ropVar @ F @ ( upt @ N @ M2 ) ) )
      = ( map_na8440032292881660845ropVar
        @ ^ [K3: nat] : ( produc9085022949259285321ropVar @ K3 @ ( F @ K3 ) )
        @ ( upt @ N @ M2 ) ) ) ).

% enumerate_map_upt
thf(fact_644_enumerate__map__upt,axiom,
    ! [N: nat,F: nat > multis3193088007478089820ropVar,M2: nat] :
      ( ( enumer7962022052585343041ropVar @ N @ ( map_na2466558649495876678ropVar @ F @ ( upt @ N @ M2 ) ) )
      = ( map_na8756600662017154455ropVar
        @ ^ [K3: nat] : ( produc1720817013404193843ropVar @ K3 @ ( F @ K3 ) )
        @ ( upt @ N @ M2 ) ) ) ).

% enumerate_map_upt
thf(fact_645_enumerate__map__upt,axiom,
    ! [N: nat,F: nat > nat,M2: nat] :
      ( ( enumerate_nat @ N @ ( map_nat_nat @ F @ ( upt @ N @ M2 ) ) )
      = ( map_na7298421622053143531at_nat
        @ ^ [K3: nat] : ( product_Pair_nat_nat @ K3 @ ( F @ K3 ) )
        @ ( upt @ N @ M2 ) ) ) ).

% enumerate_map_upt
thf(fact_646_nth__mem__mset,axiom,
    ! [I3: nat,Ls: list_M6740372356978808802ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s1943289338788136526ropVar @ Ls ) )
     => ( member8421201611525043589ropVar @ ( nth_Mu4537670848183934179ropVar @ Ls @ I3 ) @ ( set_ms1147286257884310937ropVar @ ( mset_M6581516532424202993ropVar @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_647_nth__mem__mset,axiom,
    ! [I3: nat,Ls: list_P8791008543742759928ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Ls ) )
     => ( member9210975342120487067ropVar @ ( nth_Pr227365692153740281ropVar @ Ls @ I3 ) @ ( set_ms882399555722365103ropVar @ ( mset_P5170620898817730055ropVar @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_648_nth__mem__mset,axiom,
    ! [I3: nat,Ls: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ls ) )
     => ( member_nat @ ( nth_nat @ Ls @ I3 ) @ ( set_mset_nat @ ( mset_nat @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_649_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( enumer8415853101840577367ropVar @ N @ ( append4352292717583037453ropVar @ Xs @ Ys ) )
      = ( append9216677672214865852ropVar @ ( enumer8415853101840577367ropVar @ N @ Xs ) @ ( enumer8415853101840577367ropVar @ ( plus_plus_nat @ N @ ( size_s6752116297430514276ropVar @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_650_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_nat,Ys: list_nat] :
      ( ( enumerate_nat @ N @ ( append_nat @ Xs @ Ys ) )
      = ( append985823374593552924at_nat @ ( enumerate_nat @ N @ Xs ) @ ( enumerate_nat @ ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_651_enumerate__eq__zip,axiom,
    ( enumer8415853101840577367ropVar
    = ( ^ [N3: nat,Xs2: list_P8791008543742759928ropVar] : ( zip_na5961122453619627049ropVar @ ( upt @ N3 @ ( plus_plus_nat @ N3 @ ( size_s6752116297430514276ropVar @ Xs2 ) ) ) @ Xs2 ) ) ) ).

% enumerate_eq_zip
thf(fact_652_enumerate__eq__zip,axiom,
    ( enumerate_nat
    = ( ^ [N3: nat,Xs2: list_nat] : ( zip_nat_nat @ ( upt @ N3 @ ( plus_plus_nat @ N3 @ ( size_size_list_nat @ Xs2 ) ) ) @ Xs2 ) ) ) ).

% enumerate_eq_zip
thf(fact_653_multpw__listI,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,X7: multis193838646080965794ropVar,Y6: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( ( size_s1943289338788136526ropVar @ Xs )
        = ( size_s1943289338788136526ropVar @ Ys ) )
     => ( ( X7
          = ( mset_M6581516532424202993ropVar @ Xs ) )
       => ( ( Y6
            = ( mset_M6581516532424202993ropVar @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Ys ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Ys @ I2 ) ) @ Ns ) )
           => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ X7 @ Y6 ) @ ( multis9091339148773653984ropVar @ Ns ) ) ) ) ) ) ).

% multpw_listI
thf(fact_654_multpw__listI,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,X7: multis2129813143740876984ropVar,Y6: multis2129813143740876984ropVar,Ns: set_Pr3201966908590682119ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( X7
          = ( mset_P5170620898817730055ropVar @ Xs ) )
       => ( ( Y6
            = ( mset_P5170620898817730055ropVar @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Ys ) )
               => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Ys @ I2 ) ) @ Ns ) )
           => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ X7 @ Y6 ) @ ( multis1527098867523604214ropVar @ Ns ) ) ) ) ) ) ).

% multpw_listI
thf(fact_655_multpw__listI,axiom,
    ! [Xs: list_nat,Ys: list_nat,X7: multiset_nat,Y6: multiset_nat,Ns: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( X7
          = ( mset_nat @ Xs ) )
       => ( ( Y6
            = ( mset_nat @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ Ns ) )
           => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ X7 @ Y6 ) @ ( multis5152147736693828940pw_nat @ Ns ) ) ) ) ) ) ).

% multpw_listI
thf(fact_656_s__mul__extI__old,axiom,
    ! [A2: multis193838646080965794ropVar,Xs: list_M6740372356978808802ropVar,A22: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,Ys: list_M6740372356978808802ropVar,B22: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( A2
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs ) @ A22 ) )
     => ( ( B6
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys ) @ B22 ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs )
            = ( size_s1943289338788136526ropVar @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Ys ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Ys @ I2 ) ) @ Ns ) )
           => ( ( A22 != zero_z6226653793019358809ropVar )
             => ( ! [B3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ B3 @ ( set_ms1147286257884310937ropVar @ B22 ) )
                   => ? [A6: multis3193088007478089820ropVar] :
                        ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A22 ) )
                        & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B3 ) @ S ) ) )
               => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis4691071231837107059ropVar @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_657_s__mul__extI__old,axiom,
    ! [A2: multis2129813143740876984ropVar,Xs: list_P8791008543742759928ropVar,A22: multis2129813143740876984ropVar,B6: multis2129813143740876984ropVar,Ys: list_P8791008543742759928ropVar,B22: multis2129813143740876984ropVar,Ns: set_Pr3201966908590682119ropVar,S: set_Pr3201966908590682119ropVar] :
      ( ( A2
        = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Xs ) @ A22 ) )
     => ( ( B6
          = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Ys ) @ B22 ) )
       => ( ( ( size_s6752116297430514276ropVar @ Xs )
            = ( size_s6752116297430514276ropVar @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Ys ) )
               => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Ys @ I2 ) ) @ Ns ) )
           => ( ( A22 != zero_z6774573742726938735ropVar )
             => ( ! [B3: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ B3 @ ( set_ms882399555722365103ropVar @ B22 ) )
                   => ? [A6: produc2642885279816024434ropVar] :
                        ( ( member9210975342120487067ropVar @ A6 @ ( set_ms882399555722365103ropVar @ A22 ) )
                        & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A6 @ B3 ) @ S ) ) )
               => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ A2 @ B6 ) @ ( multis2067306628098044553ropVar @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_658_s__mul__extI__old,axiom,
    ! [A2: multiset_nat,Xs: list_nat,A22: multiset_nat,B6: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( A2
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
     => ( ( B6
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I2 ) @ ( nth_nat @ Ys @ I2 ) ) @ Ns ) )
           => ( ( A22 != zero_z7348594199698428585et_nat )
             => ( ! [B3: nat] :
                    ( ( member_nat @ B3 @ ( set_mset_nat @ B22 ) )
                   => ? [A6: nat] :
                        ( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ S ) ) )
               => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_659_mem__case__prodI,axiom,
    ! [Z4: nat,C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > set_nat,A: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar] :
      ( ( member_nat @ Z4 @ ( C3 @ A @ B ) )
     => ( member_nat @ Z4 @ ( produc6886536763941271165et_nat @ C3 @ ( produc6194868611276206818ropVar @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_660_mem__case__prodI,axiom,
    ! [Z4: multis3193088007478089820ropVar,C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > set_Mu5658373669842198716ropVar,A: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( C3 @ A @ B ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc7209848115417554821ropVar @ C3 @ ( produc6194868611276206818ropVar @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_661_mem__case__prodI,axiom,
    ! [Z4: nat,C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > set_nat,A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar] :
      ( ( member_nat @ Z4 @ ( C3 @ A @ B ) )
     => ( member_nat @ Z4 @ ( produc5513644333188418610et_nat @ C3 @ ( produc6936306168532358295ropVar @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_662_mem__case__prodI,axiom,
    ! [Z4: multis3193088007478089820ropVar,C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > set_Mu5658373669842198716ropVar,A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( C3 @ A @ B ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc1375130672541687568ropVar @ C3 @ ( produc6936306168532358295ropVar @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_663_mem__case__prodI2,axiom,
    ! [P2: produc2642885279816024434ropVar,Z4: nat,C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > set_nat] :
      ( ! [A4: multis3193088007478089820ropVar,B3: produc7679832412844898215ropVar] :
          ( ( P2
            = ( produc6194868611276206818ropVar @ A4 @ B3 ) )
         => ( member_nat @ Z4 @ ( C3 @ A4 @ B3 ) ) )
     => ( member_nat @ Z4 @ ( produc6886536763941271165et_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_664_mem__case__prodI2,axiom,
    ! [P2: produc2642885279816024434ropVar,Z4: multis3193088007478089820ropVar,C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > set_Mu5658373669842198716ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: produc7679832412844898215ropVar] :
          ( ( P2
            = ( produc6194868611276206818ropVar @ A4 @ B3 ) )
         => ( member8421201611525043589ropVar @ Z4 @ ( C3 @ A4 @ B3 ) ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc7209848115417554821ropVar @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_665_mem__case__prodI2,axiom,
    ! [P2: produc7679832412844898215ropVar,Z4: nat,C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > set_nat] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( ( P2
            = ( produc6936306168532358295ropVar @ A4 @ B3 ) )
         => ( member_nat @ Z4 @ ( C3 @ A4 @ B3 ) ) )
     => ( member_nat @ Z4 @ ( produc5513644333188418610et_nat @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_666_mem__case__prodI2,axiom,
    ! [P2: produc7679832412844898215ropVar,Z4: multis3193088007478089820ropVar,C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > set_Mu5658373669842198716ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( ( P2
            = ( produc6936306168532358295ropVar @ A4 @ B3 ) )
         => ( member8421201611525043589ropVar @ Z4 @ ( C3 @ A4 @ B3 ) ) )
     => ( member8421201611525043589ropVar @ Z4 @ ( produc1375130672541687568ropVar @ C3 @ P2 ) ) ) ).

% mem_case_prodI2
thf(fact_667_mem__case__prodE,axiom,
    ! [Z4: nat,C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > set_nat,P2: produc2642885279816024434ropVar] :
      ( ( member_nat @ Z4 @ ( produc6886536763941271165et_nat @ C3 @ P2 ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Y4: produc7679832412844898215ropVar] :
            ( ( P2
              = ( produc6194868611276206818ropVar @ X4 @ Y4 ) )
           => ~ ( member_nat @ Z4 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_668_mem__case__prodE,axiom,
    ! [Z4: multis3193088007478089820ropVar,C3: multis3193088007478089820ropVar > produc7679832412844898215ropVar > set_Mu5658373669842198716ropVar,P2: produc2642885279816024434ropVar] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( produc7209848115417554821ropVar @ C3 @ P2 ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Y4: produc7679832412844898215ropVar] :
            ( ( P2
              = ( produc6194868611276206818ropVar @ X4 @ Y4 ) )
           => ~ ( member8421201611525043589ropVar @ Z4 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_669_mem__case__prodE,axiom,
    ! [Z4: nat,C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > set_nat,P2: produc7679832412844898215ropVar] :
      ( ( member_nat @ Z4 @ ( produc5513644333188418610et_nat @ C3 @ P2 ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
            ( ( P2
              = ( produc6936306168532358295ropVar @ X4 @ Y4 ) )
           => ~ ( member_nat @ Z4 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_670_mem__case__prodE,axiom,
    ! [Z4: multis3193088007478089820ropVar,C3: multis3193088007478089820ropVar > multis3193088007478089820ropVar > set_Mu5658373669842198716ropVar,P2: produc7679832412844898215ropVar] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( produc1375130672541687568ropVar @ C3 @ P2 ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
            ( ( P2
              = ( produc6936306168532358295ropVar @ X4 @ Y4 ) )
           => ~ ( member8421201611525043589ropVar @ Z4 @ ( C3 @ X4 @ Y4 ) ) ) ) ).

% mem_case_prodE
thf(fact_671_zip__left__commute,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
      ( ( zip_Mu6605103971575351810ropVar @ Xs @ ( zip_Mu2706791923038522295ropVar @ Ys @ Zs ) )
      = ( map_Pr3565048146863404228ropVar
        @ ( produc6440250867464900155ropVar
          @ ^ [Y2: multis3193088007478089820ropVar] :
              ( produc6542419149356799942ropVar
              @ ^ [X: multis3193088007478089820ropVar,Z3: multis3193088007478089820ropVar] : ( produc6194868611276206818ropVar @ X @ ( produc6936306168532358295ropVar @ Y2 @ Z3 ) ) ) )
        @ ( zip_Mu6605103971575351810ropVar @ Ys @ ( zip_Mu2706791923038522295ropVar @ Xs @ Zs ) ) ) ) ).

% zip_left_commute
thf(fact_672_zip__assoc,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
      ( ( zip_Mu6605103971575351810ropVar @ Xs @ ( zip_Mu2706791923038522295ropVar @ Ys @ Zs ) )
      = ( map_Pr5238634059687723162ropVar
        @ ( produc5773209726166850449ropVar
          @ ( produc2679423467814395815ropVar
            @ ^ [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Z3: multis3193088007478089820ropVar] : ( produc6194868611276206818ropVar @ X @ ( produc6936306168532358295ropVar @ Y2 @ Z3 ) ) ) )
        @ ( zip_Pr6094978418188712748ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) @ Zs ) ) ) ).

% zip_assoc
thf(fact_673_zip__commute,axiom,
    ( zip_Mu6605103971575351810ropVar
    = ( ^ [Xs2: list_M6740372356978808802ropVar,Ys2: list_P444016040222843053ropVar] :
          ( map_Pr5238634059687723162ropVar
          @ ( produc5773209726166850449ropVar
            @ ^ [X: produc7679832412844898215ropVar,Y2: multis3193088007478089820ropVar] : ( produc6194868611276206818ropVar @ Y2 @ X ) )
          @ ( zip_Pr6094978418188712748ropVar @ Ys2 @ Xs2 ) ) ) ) ).

% zip_commute
thf(fact_674_zip__commute,axiom,
    ( zip_Mu2706791923038522295ropVar
    = ( ^ [Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
          ( map_Pr788171021273317892ropVar
          @ ( produc5035923070952770043ropVar
            @ ^ [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( produc6936306168532358295ropVar @ Y2 @ X ) )
          @ ( zip_Mu2706791923038522295ropVar @ Ys2 @ Xs2 ) ) ) ) ).

% zip_commute
thf(fact_675_zip__map1,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_P444016040222843053ropVar] :
      ( ( zip_Mu6605103971575351810ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ Ys )
      = ( map_Pr5017561720213080160ropVar
        @ ( produc3167482732446468999ropVar
          @ ^ [X: nat] : ( produc6194868611276206818ropVar @ ( F @ X ) ) )
        @ ( zip_na5672676981451767646ropVar @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_676_zip__map1,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_M6740372356978808802ropVar] :
      ( ( zip_Mu2706791923038522295ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ Ys )
      = ( map_Pr7683091257510837664ropVar
        @ ( produc8169964424091799367ropVar
          @ ^ [X: nat] : ( produc6936306168532358295ropVar @ ( F @ X ) ) )
        @ ( zip_na7122055390739322131ropVar @ Xs @ Ys ) ) ) ).

% zip_map1
thf(fact_677_zip__map2,axiom,
    ! [Xs: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( zip_Mu2706791923038522295ropVar @ Xs @ ( map_na2466558649495876678ropVar @ F @ Ys ) )
      = ( map_Pr50427201417919046ropVar
        @ ( produc3672616437829085477ropVar
          @ ^ [X: multis3193088007478089820ropVar,Y2: nat] : ( produc6936306168532358295ropVar @ X @ ( F @ Y2 ) ) )
        @ ( zip_Mu6626014532916887989ar_nat @ Xs @ Ys ) ) ) ).

% zip_map2
thf(fact_678_zip__map__map,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( zip_Pr6829646289182870327ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ ( map_na2145702811007361372ropVar @ G @ Ys ) )
      = ( map_Pr8369245140016153642ropVar
        @ ( produc6103969181082503513ropVar
          @ ^ [X: nat,Y2: nat] : ( produc6842008660352832535ropVar @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_679_zip__map__map,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( zip_Pr5260769330036727841ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( map_Pr4235916416374289428ropVar
        @ ( produc5464566285301805891ropVar
          @ ^ [X: nat,Y2: nat] : ( produc2108810064830654721ropVar @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_680_zip__map__map,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( zip_Pr2402428817466909387ar_nat @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr8718630781927822894ar_nat
        @ ( produc2337209523370107711ar_nat
          @ ^ [X: nat,Y2: nat] : ( produc5526329313106567659ar_nat @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_681_zip__map__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( zip_Mu7420031413393175245ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2145702811007361372ropVar @ G @ Ys ) )
      = ( map_Pr5898725278422565824ropVar
        @ ( produc7127375147350082287ropVar
          @ ^ [X: nat,Y2: nat] : ( produc4268072148187102125ropVar @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_682_zip__map__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( zip_Mu6626014532916887989ar_nat @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr1488799624602950936ar_nat
        @ ( produc1654110814936199209ar_nat
          @ ^ [X: nat,Y2: nat] : ( produc1224776155581759701ar_nat @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_683_zip__map__map,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( zip_na5961122453619627049ropVar @ ( map_nat_nat @ F @ Xs ) @ ( map_na2145702811007361372ropVar @ G @ Ys ) )
      = ( map_Pr4292145494054818132ropVar
        @ ( produc7134096272351878757ropVar
          @ ^ [X: nat,Y2: nat] : ( produc9085022949259285321ropVar @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_684_zip__map__map,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( zip_na7122055390739322131ropVar @ ( map_nat_nat @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( map_Pr3719486816562609470ropVar
        @ ( produc3884798006895857743ropVar
          @ ^ [X: nat,Y2: nat] : ( produc1720817013404193843ropVar @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_685_zip__map__map,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ G @ Ys ) )
      = ( map_Pr8058819605623181956at_nat
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y2: nat] : ( product_Pair_nat_nat @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_686_zip__map__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( zip_Mu2706791923038522295ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ G @ Ys ) )
      = ( map_Pr8688549323584812970ropVar
        @ ( produc2843008504863863001ropVar
          @ ^ [X: nat,Y2: nat] : ( produc6936306168532358295ropVar @ ( F @ X ) @ ( G @ Y2 ) ) )
        @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_map_map
thf(fact_687_all__s__s__mul__ext,axiom,
    ! [A2: multiset_nat,B6: multiset_nat,S: set_Pr1261947904930325089at_nat,Ns: set_Pr1261947904930325089at_nat] :
      ( ( A2 != zero_z7348594199698428585et_nat )
     => ( ! [B3: nat] :
            ( ( member_nat @ B3 @ ( set_mset_nat @ B6 ) )
           => ? [A6: nat] :
                ( ( member_nat @ A6 @ ( set_mset_nat @ A2 ) )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ S ) ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_688_all__s__s__mul__ext,axiom,
    ! [A2: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,S: set_Pr7799659934693882759ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( A2 != zero_z6226653793019358809ropVar )
     => ( ! [B3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ B3 @ ( set_ms1147286257884310937ropVar @ B6 ) )
           => ? [A6: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A2 ) )
                & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B3 ) @ S ) ) )
       => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis4691071231837107059ropVar @ Ns @ S ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_689_s__mul__ext__ne__extend__left,axiom,
    ! [A2: multiset_nat,B6: multiset_nat,C4: multiset_nat,W: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( A2 != zero_z7348594199698428585et_nat )
     => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ B6 @ C4 ) @ ( multis8831328596575508315xt_nat @ W @ S2 ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) @ C4 ) @ ( multis4215214144601311993xt_nat @ W @ S2 ) ) ) ) ).

% s_mul_ext_ne_extend_left
thf(fact_690_s__mul__extE,axiom,
    ! [A2: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis4691071231837107059ropVar @ Ns @ S ) )
     => ~ ! [A1: multis193838646080965794ropVar,A23: multis193838646080965794ropVar] :
            ( ( A2
              = ( plus_p1339091108561003481ropVar @ A1 @ A23 ) )
           => ! [B1: multis193838646080965794ropVar,B23: multis193838646080965794ropVar] :
                ( ( B6
                  = ( plus_p1339091108561003481ropVar @ B1 @ B23 ) )
               => ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A1 @ B1 ) @ ( multis9091339148773653984ropVar @ Ns ) )
                 => ( ( A23 != zero_z6226653793019358809ropVar )
                   => ~ ! [B5: multis3193088007478089820ropVar] :
                          ( ( member8421201611525043589ropVar @ B5 @ ( set_ms1147286257884310937ropVar @ B23 ) )
                         => ? [A4: multis3193088007478089820ropVar] :
                              ( ( member8421201611525043589ropVar @ A4 @ ( set_ms1147286257884310937ropVar @ A23 ) )
                              & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ) ).

% s_mul_extE
thf(fact_691_s__mul__extE,axiom,
    ! [A2: multiset_nat,B6: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) )
     => ~ ! [A1: multiset_nat,A23: multiset_nat] :
            ( ( A2
              = ( plus_p6334493942879108393et_nat @ A1 @ A23 ) )
           => ! [B1: multiset_nat,B23: multiset_nat] :
                ( ( B6
                  = ( plus_p6334493942879108393et_nat @ B1 @ B23 ) )
               => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A1 @ B1 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
                 => ( ( A23 != zero_z7348594199698428585et_nat )
                   => ~ ! [B5: nat] :
                          ( ( member_nat @ B5 @ ( set_mset_nat @ B23 ) )
                         => ? [A4: nat] :
                              ( ( member_nat @ A4 @ ( set_mset_nat @ A23 ) )
                              & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ) ).

% s_mul_extE
thf(fact_692_s__mul__extI,axiom,
    ! [A2: multis193838646080965794ropVar,A12: multis193838646080965794ropVar,A22: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,B12: multis193838646080965794ropVar,B22: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( A2
        = ( plus_p1339091108561003481ropVar @ A12 @ A22 ) )
     => ( ( B6
          = ( plus_p1339091108561003481ropVar @ B12 @ B22 ) )
       => ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A12 @ B12 ) @ ( multis9091339148773653984ropVar @ Ns ) )
         => ( ( A22 != zero_z6226653793019358809ropVar )
           => ( ! [B3: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ B3 @ ( set_ms1147286257884310937ropVar @ B22 ) )
                 => ? [A6: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ A6 @ ( set_ms1147286257884310937ropVar @ A22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A6 @ B3 ) @ S ) ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis4691071231837107059ropVar @ Ns @ S ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_693_s__mul__extI,axiom,
    ! [A2: multiset_nat,A12: multiset_nat,A22: multiset_nat,B6: multiset_nat,B12: multiset_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( A2
        = ( plus_p6334493942879108393et_nat @ A12 @ A22 ) )
     => ( ( B6
          = ( plus_p6334493942879108393et_nat @ B12 @ B22 ) )
       => ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A12 @ B12 ) @ ( multis5152147736693828940pw_nat @ Ns ) )
         => ( ( A22 != zero_z7348594199698428585et_nat )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( set_mset_nat @ B22 ) )
                 => ? [A6: nat] :
                      ( ( member_nat @ A6 @ ( set_mset_nat @ A22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B3 ) @ S ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_694_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A2: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar,C4: multis193838646080965794ropVar,D: multis193838646080965794ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A2 @ B6 ) @ ( multis738276037714153361ropVar @ Ns @ S ) )
     => ( ( C4 != zero_z6226653793019358809ropVar )
       => ( ! [D4: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ D4 @ ( set_ms1147286257884310937ropVar @ D ) )
             => ? [C5: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ C5 @ ( set_ms1147286257884310937ropVar @ C4 ) )
                  & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ C5 @ D4 ) @ S ) ) )
         => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( plus_p1339091108561003481ropVar @ A2 @ C4 ) @ ( plus_p1339091108561003481ropVar @ B6 @ D ) ) @ ( multis4691071231837107059ropVar @ Ns @ S ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_695_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A2: multiset_nat,B6: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat,C4: multiset_nat,D: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A2 @ B6 ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) )
     => ( ( C4 != zero_z7348594199698428585et_nat )
       => ( ! [D4: nat] :
              ( ( member_nat @ D4 @ ( set_mset_nat @ D ) )
             => ? [C5: nat] :
                  ( ( member_nat @ C5 @ ( set_mset_nat @ C4 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C5 @ D4 ) @ S ) ) )
         => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C4 ) @ ( plus_p6334493942879108393et_nat @ B6 @ D ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_696_mul__ext__list_Olist__order__extension__axioms,axiom,
    ( list_l792762466888043652on_nat
    @ ^ [S5: set_Pr1261947904930325089at_nat,Ns2: set_Pr1261947904930325089at_nat] :
        ( collec1570431334306492044st_nat
        @ ( produc6382697403170970154_nat_o
          @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis4215214144601311993xt_nat @ Ns2 @ S5 ) ) ) )
    @ ^ [S5: set_Pr1261947904930325089at_nat,Ns2: set_Pr1261947904930325089at_nat] :
        ( collec1570431334306492044st_nat
        @ ( produc6382697403170970154_nat_o
          @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis8831328596575508315xt_nat @ Ns2 @ S5 ) ) ) ) ) ).

% mul_ext_list.list_order_extension_axioms
thf(fact_697_mul__ext__list_Os__map,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ S2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
     => ( ! [A4: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ B3 ) @ NS )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) ) ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_698_mul__ext__list_Os__map,axiom,
    ! [S2: set_Pr7799659934693882759ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,NS: set_Pr7799659934693882759ropVar,As: list_M6740372356978808802ropVar,Bs: list_M6740372356978808802ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ S2 )
         => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ S2 ) )
     => ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A4 @ B3 ) @ NS )
           => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( F @ A4 ) @ ( F @ B3 ) ) @ NS ) )
       => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ As @ Bs )
            @ ( collec7970277406990558738ropVar
              @ ( produc6401158782714314988pVar_o
                @ ^ [As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar] : ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ As3 ) @ ( mset_M6581516532424202993ropVar @ Bs3 ) ) @ ( multis4691071231837107059ropVar @ NS @ S2 ) ) ) ) )
         => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( map_Mu2265489581466636612ropVar @ F @ As ) @ ( map_Mu2265489581466636612ropVar @ F @ Bs ) )
            @ ( collec7970277406990558738ropVar
              @ ( produc6401158782714314988pVar_o
                @ ^ [As3: list_M6740372356978808802ropVar,Bs3: list_M6740372356978808802ropVar] : ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ As3 ) @ ( mset_M6581516532424202993ropVar @ Bs3 ) ) @ ( multis4691071231837107059ropVar @ NS @ S2 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_699_mul__ext__list_Oextension,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,NS: set_Pr1261947904930325089at_nat] :
      ( ( order_7802623214671921858ir_nat @ S2 @ NS )
     => ( order_2144575240049343698st_nat
        @ ( collec1570431334306492044st_nat
          @ ( produc6382697403170970154_nat_o
            @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) ) ) )
        @ ( collec1570431334306492044st_nat
          @ ( produc6382697403170970154_nat_o
            @ ^ [As3: list_nat,Bs3: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As3 ) @ ( mset_nat @ Bs3 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) ) ) ).

% mul_ext_list.extension
thf(fact_700_s__mul__ext__elim,axiom,
    ! [Xs: multis193838646080965794ropVar,Ys: multis193838646080965794ropVar,NS: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis4691071231837107059ropVar @ NS @ S2 ) )
     => ? [Xs12: list_M6740372356978808802ropVar,Xs22: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs12 ) @ ( mset_M6581516532424202993ropVar @ Xs22 ) ) )
          & ? [Ys12: list_M6740372356978808802ropVar,Ys23: list_M6740372356978808802ropVar] :
              ( ( Ys
                = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys12 ) @ ( mset_M6581516532424202993ropVar @ Ys23 ) ) )
              & ( ( size_s1943289338788136526ropVar @ Xs12 )
                = ( size_s1943289338788136526ropVar @ Ys12 ) )
              & ( Xs22 != nil_Mu4132399263956683874ropVar )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s1943289338788136526ropVar @ Ys12 ) )
                 => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs12 @ I4 ) @ ( nth_Mu4537670848183934179ropVar @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Ys23 ) )
                 => ? [Xa2: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Xs22 ) )
                      & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_701_s__mul__ext__elim,axiom,
    ! [Xs: multis2129813143740876984ropVar,Ys: multis2129813143740876984ropVar,NS: set_Pr3201966908590682119ropVar,S2: set_Pr3201966908590682119ropVar] :
      ( ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ Xs @ Ys ) @ ( multis2067306628098044553ropVar @ NS @ S2 ) )
     => ? [Xs12: list_P8791008543742759928ropVar,Xs22: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Xs12 ) @ ( mset_P5170620898817730055ropVar @ Xs22 ) ) )
          & ? [Ys12: list_P8791008543742759928ropVar,Ys23: list_P8791008543742759928ropVar] :
              ( ( Ys
                = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Ys12 ) @ ( mset_P5170620898817730055ropVar @ Ys23 ) ) )
              & ( ( size_s6752116297430514276ropVar @ Xs12 )
                = ( size_s6752116297430514276ropVar @ Ys12 ) )
              & ( Xs22 != nil_Pr6119456312066593144ropVar )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s6752116297430514276ropVar @ Ys12 ) )
                 => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs12 @ I4 ) @ ( nth_Pr227365692153740281ropVar @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: produc2642885279816024434ropVar] :
                  ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Ys23 ) )
                 => ? [Xa2: produc2642885279816024434ropVar] :
                      ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Xs22 ) )
                      & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_702_s__mul__ext__elim,axiom,
    ! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) )
     => ? [Xs12: list_nat,Xs22: list_nat] :
          ( ( Xs
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs22 ) ) )
          & ? [Ys12: list_nat,Ys23: list_nat] :
              ( ( Ys
                = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys23 ) ) )
              & ( ( size_size_list_nat @ Xs12 )
                = ( size_size_list_nat @ Ys12 ) )
              & ( Xs22 != nil_nat )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys12 ) )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs12 @ I4 ) @ ( nth_nat @ Ys12 @ I4 ) ) @ NS ) )
              & ! [X6: nat] :
                  ( ( member_nat @ X6 @ ( set_nat2 @ Ys23 ) )
                 => ? [Xa2: nat] :
                      ( ( member_nat @ Xa2 @ ( set_nat2 @ Xs22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_703_s__mul__ext__intro,axiom,
    ! [Xs: multis193838646080965794ropVar,Xs1: list_M6740372356978808802ropVar,Xs23: list_M6740372356978808802ropVar,Ys: multis193838646080965794ropVar,Ys1: list_M6740372356978808802ropVar,Ys22: list_M6740372356978808802ropVar,NS: set_Pr7799659934693882759ropVar,S2: set_Pr7799659934693882759ropVar] :
      ( ( Xs
        = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Xs1 ) @ ( mset_M6581516532424202993ropVar @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p1339091108561003481ropVar @ ( mset_M6581516532424202993ropVar @ Ys1 ) @ ( mset_M6581516532424202993ropVar @ Ys22 ) ) )
       => ( ( ( size_s1943289338788136526ropVar @ Xs1 )
            = ( size_s1943289338788136526ropVar @ Ys1 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1943289338788136526ropVar @ Ys1 ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( nth_Mu4537670848183934179ropVar @ Xs1 @ I2 ) @ ( nth_Mu4537670848183934179ropVar @ Ys1 @ I2 ) ) @ NS ) )
           => ( ( Xs23 != nil_Mu4132399263956683874ropVar )
             => ( ! [Y4: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ Y4 @ ( set_Mu146256361055586289ropVar @ Ys22 ) )
                   => ? [X6: multis3193088007478089820ropVar] :
                        ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs23 ) )
                        & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X6 @ Y4 ) @ S2 ) ) )
               => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Xs @ Ys ) @ ( multis4691071231837107059ropVar @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_704_s__mul__ext__intro,axiom,
    ! [Xs: multis2129813143740876984ropVar,Xs1: list_P8791008543742759928ropVar,Xs23: list_P8791008543742759928ropVar,Ys: multis2129813143740876984ropVar,Ys1: list_P8791008543742759928ropVar,Ys22: list_P8791008543742759928ropVar,NS: set_Pr3201966908590682119ropVar,S2: set_Pr3201966908590682119ropVar] :
      ( ( Xs
        = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Xs1 ) @ ( mset_P5170620898817730055ropVar @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p6734580430835824111ropVar @ ( mset_P5170620898817730055ropVar @ Ys1 ) @ ( mset_P5170620898817730055ropVar @ Ys22 ) ) )
       => ( ( ( size_s6752116297430514276ropVar @ Xs1 )
            = ( size_s6752116297430514276ropVar @ Ys1 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s6752116297430514276ropVar @ Ys1 ) )
               => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( nth_Pr227365692153740281ropVar @ Xs1 @ I2 ) @ ( nth_Pr227365692153740281ropVar @ Ys1 @ I2 ) ) @ NS ) )
           => ( ( Xs23 != nil_Pr6119456312066593144ropVar )
             => ( ! [Y4: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ Y4 @ ( set_Pr3946994337432900359ropVar @ Ys22 ) )
                   => ? [X6: produc2642885279816024434ropVar] :
                        ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs23 ) )
                        & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X6 @ Y4 ) @ S2 ) ) )
               => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ Xs @ Ys ) @ ( multis2067306628098044553ropVar @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_705_s__mul__ext__intro,axiom,
    ! [Xs: multiset_nat,Xs1: list_nat,Xs23: list_nat,Ys: multiset_nat,Ys1: list_nat,Ys22: list_nat,NS: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( Xs
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs1 ) @ ( mset_nat @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys22 ) ) )
       => ( ( ( size_size_list_nat @ Xs1 )
            = ( size_size_list_nat @ Ys1 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys1 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I2 ) @ ( nth_nat @ Ys1 @ I2 ) ) @ NS ) )
           => ( ( Xs23 != nil_nat )
             => ( ! [Y4: nat] :
                    ( ( member_nat @ Y4 @ ( set_nat2 @ Ys22 ) )
                   => ? [X6: nat] :
                        ( ( member_nat @ X6 @ ( set_nat2 @ Xs23 ) )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X6 @ Y4 ) @ S2 ) ) )
               => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_706_subset__mset_Osum__mset__0__iff,axiom,
    ! [M3: multis1201202736280713200et_nat] :
      ( ( ( comm_m5787568287065167983et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ M3 )
        = zero_z7348594199698428585et_nat )
      = ( ! [X: multiset_nat] :
            ( ( member_multiset_nat @ X @ ( set_ms4188662328148412963et_nat @ M3 ) )
           => ( X = zero_z7348594199698428585et_nat ) ) ) ) ).

% subset_mset.sum_mset_0_iff
thf(fact_707_listrel__iff__zip,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar,R: set_Pr6405597471078531282ropVar] :
      ( ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Xs @ Ys ) @ ( listre2663080307769418044ropVar @ R ) )
      = ( ( ( size_s1943289338788136526ropVar @ Xs )
          = ( size_s8779405381146381081ropVar @ Ys ) )
        & ! [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) ) )
           => ( produc9122939242341821665pVar_o
              @ ^ [Y2: multis3193088007478089820ropVar,Z3: produc7679832412844898215ropVar] : ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ Y2 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_708_listrel__iff__zip,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre4495854419763884273ropVar @ R ) )
      = ( ( ( size_s1943289338788136526ropVar @ Xs )
          = ( size_s1943289338788136526ropVar @ Ys ) )
        & ! [X: produc7679832412844898215ropVar] :
            ( ( member2619728555115997648ropVar @ X @ ( set_Pr4075413451802982460ropVar @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) )
           => ( produc1988339464994610156pVar_o
              @ ^ [Y2: multis3193088007478089820ropVar,Z3: multis3193088007478089820ropVar] : ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Y2 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_709_listrel__iff__zip,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre2294694963173455985ropVar @ R ) )
      = ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        & ! [X: produc7137084944585861159ropVar] :
            ( ( member4092153873655836240ropVar @ X @ ( set_Pr1712790080554429628ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs @ Ys ) ) )
           => ( produc6789530531619613804pVar_o
              @ ^ [Y2: produc2642885279816024434ropVar,Z3: produc2642885279816024434ropVar] : ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ Y2 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_710_listrel__iff__zip,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs @ Ys ) @ ( listre5688682141216627473ar_nat @ R ) )
      = ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [X: produc103364944588188913ar_nat] :
            ( ( member5345150084029083272ar_nat @ X @ ( set_Pr7230637098968509724ar_nat @ ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys ) ) )
           => ( produc5286831101113786240_nat_o
              @ ^ [Y2: produc2642885279816024434ropVar,Z3: nat] : ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ Y2 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_711_listrel__iff__zip,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs @ Ys ) @ ( listre24003740514569327ropVar @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
        & ! [X: produc4900251693569959959ropVar] :
            ( ( member918664796156078510ropVar @ X @ ( set_Pr2804151811095504962ropVar @ ( zip_na5961122453619627049ropVar @ Xs @ Ys ) ) )
           => ( produc1692715175595045538pVar_o
              @ ^ [Y2: nat,Z3: produc2642885279816024434ropVar] : ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ Y2 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_712_listrel__iff__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
           => ( produc6081775807080527818_nat_o
              @ ^ [Y2: nat,Z3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y2 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_713_map__is__Nil__conv,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% map_is_Nil_conv
thf(fact_714_map__is__Nil__conv,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ Xs )
        = nil_Pr6119456312066593144ropVar )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% map_is_Nil_conv
thf(fact_715_map__is__Nil__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = nil_Pr6119456312066593144ropVar )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_716_map__is__Nil__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = nil_Mu4132399263956683874ropVar )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_717_map__is__Nil__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_718_Nil__is__map__conv,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar] :
      ( ( nil_nat
        = ( map_Pr7810381211709419518ar_nat @ F @ Xs ) )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% Nil_is_map_conv
thf(fact_719_Nil__is__map__conv,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( map_Pr3565048146863404228ropVar @ F @ Xs ) )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% Nil_is_map_conv
thf(fact_720_Nil__is__map__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( map_na2145702811007361372ropVar @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_721_Nil__is__map__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( nil_Mu4132399263956683874ropVar
        = ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_722_Nil__is__map__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_723_list_Omap__disc__iff,axiom,
    ! [F: produc2642885279816024434ropVar > nat,A: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ A )
        = nil_nat )
      = ( A = nil_Pr6119456312066593144ropVar ) ) ).

% list.map_disc_iff
thf(fact_724_list_Omap__disc__iff,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,A: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ A )
        = nil_Pr6119456312066593144ropVar )
      = ( A = nil_Pr6119456312066593144ropVar ) ) ).

% list.map_disc_iff
thf(fact_725_list_Omap__disc__iff,axiom,
    ! [F: nat > produc2642885279816024434ropVar,A: list_nat] :
      ( ( ( map_na2145702811007361372ropVar @ F @ A )
        = nil_Pr6119456312066593144ropVar )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_726_list_Omap__disc__iff,axiom,
    ! [F: nat > multis3193088007478089820ropVar,A: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ A )
        = nil_Mu4132399263956683874ropVar )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_727_list_Omap__disc__iff,axiom,
    ! [F: nat > nat,A: list_nat] :
      ( ( ( map_nat_nat @ F @ A )
        = nil_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_728_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_729_append_Oright__neutral,axiom,
    ! [A: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ A @ nil_Pr6119456312066593144ropVar )
      = A ) ).

% append.right_neutral
thf(fact_730_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_731_append__Nil2,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ Xs @ nil_Pr6119456312066593144ropVar )
      = Xs ) ).

% append_Nil2
thf(fact_732_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_733_append__self__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr6119456312066593144ropVar ) ) ).

% append_self_conv
thf(fact_734_self__append__conv,axiom,
    ! [Y3: list_nat,Ys: list_nat] :
      ( ( Y3
        = ( append_nat @ Y3 @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_735_self__append__conv,axiom,
    ! [Y3: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( Y3
        = ( append4352292717583037453ropVar @ Y3 @ Ys ) )
      = ( Ys = nil_Pr6119456312066593144ropVar ) ) ).

% self_append_conv
thf(fact_736_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_737_append__self__conv2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% append_self_conv2
thf(fact_738_self__append__conv2,axiom,
    ! [Y3: list_nat,Xs: list_nat] :
      ( ( Y3
        = ( append_nat @ Xs @ Y3 ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_739_self__append__conv2,axiom,
    ! [Y3: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( Y3
        = ( append4352292717583037453ropVar @ Xs @ Y3 ) )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% self_append_conv2
thf(fact_740_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_741_Nil__is__append__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( append4352292717583037453ropVar @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        & ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% Nil_is_append_conv
thf(fact_742_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_743_append__is__Nil__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ Ys )
        = nil_Pr6119456312066593144ropVar )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        & ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% append_is_Nil_conv
thf(fact_744_zip__Nil,axiom,
    ! [Ys: list_P444016040222843053ropVar] :
      ( ( zip_Mu6605103971575351810ropVar @ nil_Mu4132399263956683874ropVar @ Ys )
      = nil_Pr6119456312066593144ropVar ) ).

% zip_Nil
thf(fact_745_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_746_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar] :
      ( ( nil_Pr2504600128863385745ropVar
        = ( zip_na5961122453619627049ropVar @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% Nil_eq_zip_iff
thf(fact_747_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat] :
      ( ( nil_Pr6931085416736390507ar_nat
        = ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_748_Nil__eq__zip__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( nil_Pr6755641420451177005ropVar
        = ( zip_Pr6829646289182870327ropVar @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        | ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% Nil_eq_zip_iff
thf(fact_749_Nil__eq__zip__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) )
      = ( ( Xs = nil_Mu4132399263956683874ropVar )
        | ( Ys = nil_Pr4408211540142832045ropVar ) ) ) ).

% Nil_eq_zip_iff
thf(fact_750_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_751_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar] :
      ( ( ( zip_na5961122453619627049ropVar @ Xs @ Ys )
        = nil_Pr2504600128863385745ropVar )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% zip_eq_Nil_iff
thf(fact_752_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat] :
      ( ( ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys )
        = nil_Pr6931085416736390507ar_nat )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_753_zip__eq__Nil__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( zip_Pr6829646289182870327ropVar @ Xs @ Ys )
        = nil_Pr6755641420451177005ropVar )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        | ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% zip_eq_Nil_iff
thf(fact_754_zip__eq__Nil__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( ( zip_Mu6605103971575351810ropVar @ Xs @ Ys )
        = nil_Pr6119456312066593144ropVar )
      = ( ( Xs = nil_Mu4132399263956683874ropVar )
        | ( Ys = nil_Pr4408211540142832045ropVar ) ) ) ).

% zip_eq_Nil_iff
thf(fact_755_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X5: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X5 )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_756_list__update__nonempty,axiom,
    ! [Xs: list_P8791008543742759928ropVar,K: nat,X5: produc2642885279816024434ropVar] :
      ( ( ( list_u8100953110031273952ropVar @ Xs @ K @ X5 )
        = nil_Pr6119456312066593144ropVar )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% list_update_nonempty
thf(fact_757_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumerate_nat @ N @ nil_nat )
      = nil_Pr5478986624290739719at_nat ) ).

% enumerate_simps(1)
thf(fact_758_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumer8415853101840577367ropVar @ N @ nil_Pr6119456312066593144ropVar )
      = nil_Pr2504600128863385745ropVar ) ).

% enumerate_simps(1)
thf(fact_759_length__0__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% length_0_conv
thf(fact_760_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_761_mset__zero__iff,axiom,
    ! [X5: list_P8791008543742759928ropVar] :
      ( ( ( mset_P5170620898817730055ropVar @ X5 )
        = zero_z6774573742726938735ropVar )
      = ( X5 = nil_Pr6119456312066593144ropVar ) ) ).

% mset_zero_iff
thf(fact_762_mset__zero__iff,axiom,
    ! [X5: list_nat] :
      ( ( ( mset_nat @ X5 )
        = zero_z7348594199698428585et_nat )
      = ( X5 = nil_nat ) ) ).

% mset_zero_iff
thf(fact_763_mset__zero__iff__right,axiom,
    ! [X5: list_P8791008543742759928ropVar] :
      ( ( zero_z6774573742726938735ropVar
        = ( mset_P5170620898817730055ropVar @ X5 ) )
      = ( X5 = nil_Pr6119456312066593144ropVar ) ) ).

% mset_zero_iff_right
thf(fact_764_mset__zero__iff__right,axiom,
    ! [X5: list_nat] :
      ( ( zero_z7348594199698428585et_nat
        = ( mset_nat @ X5 ) )
      = ( X5 = nil_nat ) ) ).

% mset_zero_iff_right
thf(fact_765_length__greater__0__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s6752116297430514276ropVar @ Xs ) )
      = ( Xs != nil_Pr6119456312066593144ropVar ) ) ).

% length_greater_0_conv
thf(fact_766_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_767_list_Osimps_I8_J,axiom,
    ! [F: produc2642885279816024434ropVar > nat] :
      ( ( map_Pr7810381211709419518ar_nat @ F @ nil_Pr6119456312066593144ropVar )
      = nil_nat ) ).

% list.simps(8)
thf(fact_768_list_Osimps_I8_J,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar] :
      ( ( map_Pr3565048146863404228ropVar @ F @ nil_Pr6119456312066593144ropVar )
      = nil_Pr6119456312066593144ropVar ) ).

% list.simps(8)
thf(fact_769_list_Osimps_I8_J,axiom,
    ! [F: nat > produc2642885279816024434ropVar] :
      ( ( map_na2145702811007361372ropVar @ F @ nil_nat )
      = nil_Pr6119456312066593144ropVar ) ).

% list.simps(8)
thf(fact_770_list_Osimps_I8_J,axiom,
    ! [F: nat > multis3193088007478089820ropVar] :
      ( ( map_na2466558649495876678ropVar @ F @ nil_nat )
      = nil_Mu4132399263956683874ropVar ) ).

% list.simps(8)
thf(fact_771_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_772_zip_Osimps_I1_J,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( zip_Mu6605103971575351810ropVar @ Xs @ nil_Pr4408211540142832045ropVar )
      = nil_Pr6119456312066593144ropVar ) ).

% zip.simps(1)
thf(fact_773_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_774_eq__Nil__appendI,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_775_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_776_append_Oleft__neutral,axiom,
    ! [A: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ A )
      = A ) ).

% append.left_neutral
thf(fact_777_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_778_append__Nil,axiom,
    ! [Ys: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_779_distinct_Osimps_I1_J,axiom,
    distinct_nat @ nil_nat ).

% distinct.simps(1)
thf(fact_780_distinct_Osimps_I1_J,axiom,
    distin2267423591311547883ropVar @ nil_Pr6119456312066593144ropVar ).

% distinct.simps(1)
thf(fact_781_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_M2293212096379261346ropVar @ N @ nil_Mu4132399263956683874ropVar )
      = nil_Mu4132399263956683874ropVar ) ).

% drop_Nil
thf(fact_782_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_nat @ N @ nil_nat )
      = nil_nat ) ).

% drop_Nil
thf(fact_783_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_P4829783966340880056ropVar @ N @ nil_Pr6119456312066593144ropVar )
      = nil_Pr6119456312066593144ropVar ) ).

% drop_Nil
thf(fact_784_list__update_Osimps_I1_J,axiom,
    ! [I3: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I3 @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_785_list__update_Osimps_I1_J,axiom,
    ! [I3: nat,V: produc2642885279816024434ropVar] :
      ( ( list_u8100953110031273952ropVar @ nil_Pr6119456312066593144ropVar @ I3 @ V )
      = nil_Pr6119456312066593144ropVar ) ).

% list_update.simps(1)
thf(fact_786_list__update__code_I1_J,axiom,
    ! [I3: nat,Y3: nat] :
      ( ( list_update_nat @ nil_nat @ I3 @ Y3 )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_787_list__update__code_I1_J,axiom,
    ! [I3: nat,Y3: produc2642885279816024434ropVar] :
      ( ( list_u8100953110031273952ropVar @ nil_Pr6119456312066593144ropVar @ I3 @ Y3 )
      = nil_Pr6119456312066593144ropVar ) ).

% list_update_code(1)
thf(fact_788_product_Osimps_I1_J,axiom,
    ! [Uu: list_P444016040222843053ropVar] :
      ( ( produc4762506738712241808ropVar @ nil_Mu4132399263956683874ropVar @ Uu )
      = nil_Pr6119456312066593144ropVar ) ).

% product.simps(1)
thf(fact_789_listrelp_ONil,axiom,
    ! [R: nat > nat > $o] : ( listrelp_nat_nat @ R @ nil_nat @ nil_nat ) ).

% listrelp.Nil
thf(fact_790_listrelp_ONil,axiom,
    ! [R: nat > produc2642885279816024434ropVar > $o] : ( listre2196087975860241569ropVar @ R @ nil_nat @ nil_Pr6119456312066593144ropVar ) ).

% listrelp.Nil
thf(fact_791_listrelp_ONil,axiom,
    ! [R: produc2642885279816024434ropVar > nat > $o] : ( listre7860766376562299715ar_nat @ R @ nil_Pr6119456312066593144ropVar @ nil_nat ) ).

% listrelp.Nil
thf(fact_792_listrelp_ONil,axiom,
    ! [R: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] : ( listre4042120741191764159ropVar @ R @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar ) ).

% listrelp.Nil
thf(fact_793_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_length_nat @ N @ nil_nat )
      = N ) ).

% gen_length_code(1)
thf(fact_794_gen__length__code_I1_J,axiom,
    ! [N: nat] :
      ( ( gen_le3870443462522697605ropVar @ N @ nil_Pr6119456312066593144ropVar )
      = N ) ).

% gen_length_code(1)
thf(fact_795_list_Osize_I3_J,axiom,
    ( ( size_s6752116297430514276ropVar @ nil_Pr6119456312066593144ropVar )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_796_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_797_upt__0,axiom,
    ! [I3: nat] :
      ( ( upt @ I3 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_798_mset_Osimps_I1_J,axiom,
    ( ( mset_P5170620898817730055ropVar @ nil_Pr6119456312066593144ropVar )
    = zero_z6774573742726938735ropVar ) ).

% mset.simps(1)
thf(fact_799_mset_Osimps_I1_J,axiom,
    ( ( mset_nat @ nil_nat )
    = zero_z7348594199698428585et_nat ) ).

% mset.simps(1)
thf(fact_800_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_801_not__Nil__listrel1,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Xs ) @ ( listre6655839188755619153ropVar @ R ) ) ).

% not_Nil_listrel1
thf(fact_802_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_803_not__listrel1__Nil,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ nil_Pr6119456312066593144ropVar ) @ ( listre6655839188755619153ropVar @ R ) ) ).

% not_listrel1_Nil
thf(fact_804_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_805_listrel__Nil2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs @ nil_nat ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% listrel_Nil2
thf(fact_806_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs @ nil_Pr6119456312066593144ropVar ) @ ( listre24003740514569327ropVar @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_807_listrel__Nil2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ nil_Pr6119456312066593144ropVar ) @ ( listre2294694963173455985ropVar @ R ) )
     => ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% listrel_Nil2
thf(fact_808_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_809_listrel__Nil1,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ nil_nat @ Xs ) @ ( listre24003740514569327ropVar @ R ) )
     => ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% listrel_Nil1
thf(fact_810_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ nil_Pr6119456312066593144ropVar @ Xs ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_811_listrel__Nil1,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Xs ) @ ( listre2294694963173455985ropVar @ R ) )
     => ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% listrel_Nil1
thf(fact_812_listrel_ONil,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).

% listrel.Nil
thf(fact_813_listrel_ONil,axiom,
    ! [R: set_Pr7708077615150402893ropVar] : ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ nil_nat @ nil_Pr6119456312066593144ropVar ) @ ( listre24003740514569327ropVar @ R ) ) ).

% listrel.Nil
thf(fact_814_listrel_ONil,axiom,
    ! [R: set_Pr7538684846379304231ar_nat] : ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ nil_Pr6119456312066593144ropVar @ nil_nat ) @ ( listre5688682141216627473ar_nat @ R ) ) ).

% listrel.Nil
thf(fact_815_listrel_ONil,axiom,
    ! [R: set_Pr3201966908590682119ropVar] : ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar ) @ ( listre2294694963173455985ropVar @ R ) ) ).

% listrel.Nil
thf(fact_816_list__eq__iff__zip__eq,axiom,
    ( ( ^ [Y: list_P8791008543742759928ropVar,Z: list_P8791008543742759928ropVar] : ( Y = Z ) )
    = ( ^ [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
          ( ( ( size_s6752116297430514276ropVar @ Xs2 )
            = ( size_s6752116297430514276ropVar @ Ys2 ) )
          & ! [X: produc7137084944585861159ropVar] :
              ( ( member4092153873655836240ropVar @ X @ ( set_Pr1712790080554429628ropVar @ ( zip_Pr6829646289182870327ropVar @ Xs2 @ Ys2 ) ) )
             => ( produc6789530531619613804pVar_o
                @ ^ [Y: produc2642885279816024434ropVar,Z: produc2642885279816024434ropVar] : ( Y = Z )
                @ X ) ) ) ) ) ).

% list_eq_iff_zip_eq
thf(fact_817_list__eq__iff__zip__eq,axiom,
    ( ( ^ [Y: list_nat,Z: list_nat] : ( Y = Z ) )
    = ( ^ [Xs2: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs2 @ Ys2 ) ) )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y: nat,Z: nat] : ( Y = Z )
                @ X ) ) ) ) ) ).

% list_eq_iff_zip_eq
thf(fact_818_case__prod__Pair__iden,axiom,
    ! [P2: produc2642885279816024434ropVar] :
      ( ( produc6440250867464900155ropVar @ produc6194868611276206818ropVar @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_819_case__prod__Pair__iden,axiom,
    ! [P2: produc7679832412844898215ropVar] :
      ( ( produc5035923070952770043ropVar @ produc6936306168532358295ropVar @ P2 )
      = P2 ) ).

% case_prod_Pair_iden
thf(fact_820_ms__encoder_Oencoding__sound_I1_J,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,S: nat > produc2642885279816024434ropVar,T: nat > produc2642885279816024434ropVar,S2: set_Pr3201966908590682119ropVar,Cns: nat > nat > $o,NS: set_Pr3201966908590682119ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( set_nat2 @ ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis6478642269264501415ropVar @ NS @ S2 ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(1)
thf(fact_821_ms__encoder_Oencoding__sound_I1_J,axiom,
    ! [Enc_False: multis3193088007478089820ropVar,Enc_True: multis3193088007478089820ropVar,Enc_pos: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_neg: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_all: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Enc_conj: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o,Cs: nat > nat > $o,S: nat > produc2642885279816024434ropVar,T: nat > produc2642885279816024434ropVar,S2: set_Pr3201966908590682119ropVar,Cns: nat > nat > $o,NS: set_Pr3201966908590682119ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis6617357583593421813ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis7034272045817201159ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis6478642269264501415ropVar @ NS @ S2 ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(1)
thf(fact_822_ms__encoder_Oencoding__sound_I1_J,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,S: nat > nat,T: nat > nat,S2: set_Pr1261947904930325089at_nat,Cns: nat > nat > $o,NS: set_Pr1261947904930325089at_nat,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( set_nat2 @ ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ ( map_nat_nat @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_nat @ ( map_nat_nat @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(1)
thf(fact_823_ms__encoder_Oencoding__sound_I1_J,axiom,
    ! [Enc_False: multis3193088007478089820ropVar,Enc_True: multis3193088007478089820ropVar,Enc_pos: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_neg: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_all: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Enc_conj: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o,Cs: nat > nat > $o,S: nat > nat,T: nat > nat,S2: set_Pr1261947904930325089at_nat,Cns: nat > nat > $o,NS: set_Pr1261947904930325089at_nat,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis6617357583593421813ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis7034272045817201159ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ ( map_nat_nat @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_nat @ ( map_nat_nat @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis8831328596575508315xt_nat @ NS @ S2 ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(1)
thf(fact_824_ms__encoder_Oencoding__sound_I1_J,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,S: nat > multis3193088007478089820ropVar,T: nat > multis3193088007478089820ropVar,S2: set_Pr7799659934693882759ropVar,Cns: nat > nat > $o,NS: set_Pr7799659934693882759ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( set_nat2 @ ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis738276037714153361ropVar @ NS @ S2 ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(1)
thf(fact_825_ms__encoder_Oencoding__sound_I1_J,axiom,
    ! [Enc_False: multis3193088007478089820ropVar,Enc_True: multis3193088007478089820ropVar,Enc_pos: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_neg: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_all: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Enc_conj: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o,Cs: nat > nat > $o,S: nat > multis3193088007478089820ropVar,T: nat > multis3193088007478089820ropVar,S2: set_Pr7799659934693882759ropVar,Cns: nat > nat > $o,NS: set_Pr7799659934693882759ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis6617357583593421813ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis7034272045817201159ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis738276037714153361ropVar @ NS @ S2 ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(1)
thf(fact_826_transpose__rectangle,axiom,
    ! [Xs: list_l3041241048860003048ropVar,N: nat] :
      ( ( ( Xs = nil_li2787573280958550248ropVar )
       => ( N = zero_zero_nat ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s9082923797619839828ropVar @ Xs ) )
           => ( ( size_s1943289338788136526ropVar @ ( nth_li1789610779111697641ropVar @ Xs @ I2 ) )
              = N ) )
       => ( ( transp2038539320897558508ropVar @ Xs )
          = ( map_na5381558106008680140ropVar
            @ ^ [I: nat] :
                ( map_na2466558649495876678ropVar
                @ ^ [J: nat] : ( nth_Mu4537670848183934179ropVar @ ( nth_li1789610779111697641ropVar @ Xs @ J ) @ I )
                @ ( upt @ zero_zero_nat @ ( size_s9082923797619839828ropVar @ Xs ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% transpose_rectangle
thf(fact_827_transpose__rectangle,axiom,
    ! [Xs: list_l3415868495287753214ropVar,N: nat] :
      ( ( ( Xs = nil_li2679230897895549182ropVar )
       => ( N = zero_zero_nat ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6779274088989432426ropVar @ Xs ) )
           => ( ( size_s6752116297430514276ropVar @ ( nth_li6497732868662353663ropVar @ Xs @ I2 ) )
              = N ) )
       => ( ( transp9078802294292144386ropVar @ Xs )
          = ( map_na2676018863266313954ropVar
            @ ^ [I: nat] :
                ( map_na2145702811007361372ropVar
                @ ^ [J: nat] : ( nth_Pr227365692153740281ropVar @ ( nth_li6497732868662353663ropVar @ Xs @ J ) @ I )
                @ ( upt @ zero_zero_nat @ ( size_s6779274088989432426ropVar @ Xs ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% transpose_rectangle
thf(fact_828_transpose__rectangle,axiom,
    ! [Xs: list_list_nat,N: nat] :
      ( ( ( Xs = nil_list_nat )
       => ( N = zero_zero_nat ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs ) )
           => ( ( size_size_list_nat @ ( nth_list_nat @ Xs @ I2 ) )
              = N ) )
       => ( ( transpose_nat @ Xs )
          = ( map_nat_list_nat
            @ ^ [I: nat] :
                ( map_nat_nat
                @ ^ [J: nat] : ( nth_nat @ ( nth_list_nat @ Xs @ J ) @ I )
                @ ( upt @ zero_zero_nat @ ( size_s3023201423986296836st_nat @ Xs ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% transpose_rectangle
thf(fact_829_transpose__empty,axiom,
    ! [Xs: list_list_nat] :
      ( ( ( transpose_nat @ Xs )
        = nil_list_nat )
      = ( ! [X: list_nat] :
            ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
           => ( X = nil_nat ) ) ) ) ).

% transpose_empty
thf(fact_830_transpose__empty,axiom,
    ! [Xs: list_l3415868495287753214ropVar] :
      ( ( ( transp9078802294292144386ropVar @ Xs )
        = nil_li2679230897895549182ropVar )
      = ( ! [X: list_P8791008543742759928ropVar] :
            ( ( member6236339133593825953ropVar @ X @ ( set_li6509544621384488717ropVar @ Xs ) )
           => ( X = nil_Pr6119456312066593144ropVar ) ) ) ) ).

% transpose_empty
thf(fact_831_transpose__map__map,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_list_nat] :
      ( ( transp9078802294292144386ropVar @ ( map_li4979107708160704082ropVar @ ( map_na2145702811007361372ropVar @ F ) @ Xs ) )
      = ( map_li4979107708160704082ropVar @ ( map_na2145702811007361372ropVar @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_832_transpose__map__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_list_nat] :
      ( ( transp2038539320897558508ropVar @ ( map_li5946103070670478908ropVar @ ( map_na2466558649495876678ropVar @ F ) @ Xs ) )
      = ( map_li5946103070670478908ropVar @ ( map_na2466558649495876678ropVar @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_833_transpose__map__map,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) )
      = ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_834_ms__encoder_Oencoding__sound_I2_J,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,S: nat > produc2642885279816024434ropVar,T: nat > produc2642885279816024434ropVar,S2: set_Pr3201966908590682119ropVar,Cns: nat > nat > $o,NS: set_Pr3201966908590682119ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( set_nat2 @ ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ ( set_nat2 @ ( multis103654530569517143016_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 ) ) )
                 => ( Eval @ V @ X4 ) )
             => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis2067306628098044553ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(2)
thf(fact_835_ms__encoder_Oencoding__sound_I2_J,axiom,
    ! [Enc_False: multis3193088007478089820ropVar,Enc_True: multis3193088007478089820ropVar,Enc_pos: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_neg: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_all: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Enc_conj: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o,Cs: nat > nat > $o,S: nat > produc2642885279816024434ropVar,T: nat > produc2642885279816024434ropVar,S2: set_Pr3201966908590682119ropVar,Cns: nat > nat > $o,NS: set_Pr3201966908590682119ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis6617357583593421813ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis7034272045817201159ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( ! [X4: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis6448939448807493190ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 ) ) )
                 => ( Eval @ V @ X4 ) )
             => ( member1455471568137992016ropVar @ ( produc3978907599504985111ropVar @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_P5170620898817730055ropVar @ ( map_na2145702811007361372ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis2067306628098044553ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(2)
thf(fact_836_ms__encoder_Oencoding__sound_I2_J,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,S: nat > nat,T: nat > nat,S2: set_Pr1261947904930325089at_nat,Cns: nat > nat > $o,NS: set_Pr1261947904930325089at_nat,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( set_nat2 @ ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ ( set_nat2 @ ( multis103654530569517143016_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 ) ) )
                 => ( Eval @ V @ X4 ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ ( map_nat_nat @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_nat @ ( map_nat_nat @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(2)
thf(fact_837_ms__encoder_Oencoding__sound_I2_J,axiom,
    ! [Enc_False: multis3193088007478089820ropVar,Enc_True: multis3193088007478089820ropVar,Enc_pos: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_neg: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_all: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Enc_conj: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o,Cs: nat > nat > $o,S: nat > nat,T: nat > nat,S2: set_Pr1261947904930325089at_nat,Cns: nat > nat > $o,NS: set_Pr1261947904930325089at_nat,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis6617357583593421813ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis7034272045817201159ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( ! [X4: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis6448939448807493190ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 ) ) )
                 => ( Eval @ V @ X4 ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ ( map_nat_nat @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_nat @ ( map_nat_nat @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis4215214144601311993xt_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(2)
thf(fact_838_ms__encoder_Oencoding__sound_I2_J,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,S: nat > multis3193088007478089820ropVar,T: nat > multis3193088007478089820ropVar,S2: set_Pr7799659934693882759ropVar,Cns: nat > nat > $o,NS: set_Pr7799659934693882759ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ ( set_nat2 @ ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ ( set_nat2 @ ( multis103654530569517143016_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 ) ) )
                 => ( Eval @ V @ X4 ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis4691071231837107059ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(2)
thf(fact_839_ms__encoder_Oencoding__sound_I2_J,axiom,
    ! [Enc_False: multis3193088007478089820ropVar,Enc_True: multis3193088007478089820ropVar,Enc_pos: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_neg: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar,Enc_not_all: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Enc_conj: list_M6740372356978808802ropVar > multis3193088007478089820ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > multis3193088007478089820ropVar > $o,Cs: nat > nat > $o,S: nat > multis3193088007478089820ropVar,T: nat > multis3193088007478089820ropVar,S2: set_Pr7799659934693882759ropVar,Cns: nat > nat > $o,NS: set_Pr7799659934693882759ropVar,N: nat,M2: nat,V: multis3193088007478089820ropVar > $o] :
      ( ( multis6617357583593421813ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ! [I2: nat,J3: nat] :
            ( ( Cs @ I2 @ J3 )
            = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ S2 ) )
       => ( ! [I2: nat,J3: nat] :
              ( ( Cns @ I2 @ J3 )
              = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I2 ) @ ( T @ J3 ) ) @ NS ) )
         => ( ! [X4: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis7034272045817201159ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) )
               => ( Eval @ V @ X4 ) )
           => ( ! [X4: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( multis6448939448807493190ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 ) ) )
                 => ( Eval @ V @ X4 ) )
             => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ S @ ( upt @ zero_zero_nat @ N ) ) ) @ ( mset_M6581516532424202993ropVar @ ( map_na2466558649495876678ropVar @ T @ ( upt @ zero_zero_nat @ M2 ) ) ) ) @ ( multis4691071231837107059ropVar @ NS @ S2 ) ) ) ) ) ) ) ).

% ms_encoder.encoding_sound(2)
thf(fact_840_subset__mset_Osum__list__update,axiom,
    ! [K: nat,Xs: list_multiset_nat,X5: multiset_nat] :
      ( ( ord_less_nat @ K @ ( size_s6386657463320973636et_nat @ Xs ) )
     => ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ ( list_u3438943574295160626et_nat @ Xs @ K @ X5 ) )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) @ X5 ) @ ( nth_multiset_nat @ Xs @ K ) ) ) ) ).

% subset_mset.sum_list_update
thf(fact_841_mset__update,axiom,
    ! [I3: nat,Ls: list_P8791008543742759928ropVar,V: produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Ls ) )
     => ( ( mset_P5170620898817730055ropVar @ ( list_u8100953110031273952ropVar @ Ls @ I3 @ V ) )
        = ( add_ms6892841465718707280ropVar @ V @ ( minus_2570738613443076255ropVar @ ( mset_P5170620898817730055ropVar @ Ls ) @ ( add_ms6892841465718707280ropVar @ ( nth_Pr227365692153740281ropVar @ Ls @ I3 ) @ zero_z6774573742726938735ropVar ) ) ) ) ) ).

% mset_update
thf(fact_842_mset__update,axiom,
    ! [I3: nat,Ls: list_nat,V: nat] :
      ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ls ) )
     => ( ( mset_nat @ ( list_update_nat @ Ls @ I3 @ V ) )
        = ( add_mset_nat @ V @ ( minus_8522176038001411705et_nat @ ( mset_nat @ Ls ) @ ( add_mset_nat @ ( nth_nat @ Ls @ I3 ) @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% mset_update
thf(fact_843_encode__sum__0__1__main_Ocases,axiom,
    ! [X5: list_P8791008543742759928ropVar] :
      ( ! [X4: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar] :
          ( X5
         != ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ nil_Pr6119456312066593144ropVar ) )
     => ( ! [X4: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar,V2: produc2642885279816024434ropVar,Va: list_P8791008543742759928ropVar] :
            ( X5
           != ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ ( cons_P1504208554336013608ropVar @ V2 @ Va ) ) )
       => ( X5 = nil_Pr6119456312066593144ropVar ) ) ) ).

% encode_sum_0_1_main.cases
thf(fact_844_list_Oinject,axiom,
    ! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar,Y21: produc2642885279816024434ropVar,Y222: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X21 @ X222 )
        = ( cons_P1504208554336013608ropVar @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_845_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_846_union__mset__add__mset__right,axiom,
    ! [A2: multiset_nat,A: nat,B6: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ A2 @ ( add_mset_nat @ A @ B6 ) )
      = ( add_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) ) ) ).

% union_mset_add_mset_right
thf(fact_847_union__mset__add__mset__left,axiom,
    ! [A: nat,A2: multiset_nat,B6: multiset_nat] :
      ( ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ A @ A2 ) @ B6 )
      = ( add_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) ) ) ).

% union_mset_add_mset_left
thf(fact_848_append1__eq__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X5 @ nil_Pr6119456312066593144ropVar ) )
        = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ Y3 @ nil_Pr6119456312066593144ropVar ) ) )
      = ( ( Xs = Ys )
        & ( X5 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_849_append1__eq__conv,axiom,
    ! [Xs: list_nat,X5: nat,Ys: list_nat,Y3: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X5 @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X5 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_850_nth__Cons__0,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ zero_zero_nat )
      = X5 ) ).

% nth_Cons_0
thf(fact_851_nth__Cons__0,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X5 @ Xs ) @ zero_zero_nat )
      = X5 ) ).

% nth_Cons_0
thf(fact_852_diff__add__mset__swap,axiom,
    ! [B: nat,A2: multiset_nat,M3: multiset_nat] :
      ( ~ ( member_nat @ B @ ( set_mset_nat @ A2 ) )
     => ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ M3 ) @ A2 )
        = ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ M3 @ A2 ) ) ) ) ).

% diff_add_mset_swap
thf(fact_853_diff__add__mset__swap,axiom,
    ! [B: multis3193088007478089820ropVar,A2: multis193838646080965794ropVar,M3: multis193838646080965794ropVar] :
      ( ~ ( member8421201611525043589ropVar @ B @ ( set_ms1147286257884310937ropVar @ A2 ) )
     => ( ( minus_287902380581138057ropVar @ ( add_ms1730626103851491130ropVar @ B @ M3 ) @ A2 )
        = ( add_ms1730626103851491130ropVar @ B @ ( minus_287902380581138057ropVar @ M3 @ A2 ) ) ) ) ).

% diff_add_mset_swap
thf(fact_854_subset__mset_Osum__list__eq__0__iff,axiom,
    ! [Ns: list_multiset_nat] :
      ( ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Ns )
        = zero_z7348594199698428585et_nat )
      = ( ! [X: multiset_nat] :
            ( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Ns ) )
           => ( X = zero_z7348594199698428585et_nat ) ) ) ) ).

% subset_mset.sum_list_eq_0_iff
thf(fact_855_nth__append__length,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( nth_Pr227365692153740281ropVar @ ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X5 @ Ys ) ) @ ( size_s6752116297430514276ropVar @ Xs ) )
      = X5 ) ).

% nth_append_length
thf(fact_856_nth__append__length,axiom,
    ! [Xs: list_nat,X5: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X5 @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X5 ) ).

% nth_append_length
thf(fact_857_list__update__length,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ( list_u8100953110031273952ropVar @ ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X5 @ Ys ) ) @ ( size_s6752116297430514276ropVar @ Xs ) @ Y3 )
      = ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) ) ).

% list_update_length
thf(fact_858_list__update__length,axiom,
    ! [Xs: list_nat,X5: nat,Ys: list_nat,Y3: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X5 @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y3 )
      = ( append_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) ) ).

% list_update_length
thf(fact_859_diff__union__swap2,axiom,
    ! [Y3: nat,M3: multiset_nat,X5: nat] :
      ( ( member_nat @ Y3 @ ( set_mset_nat @ M3 ) )
     => ( ( minus_8522176038001411705et_nat @ ( add_mset_nat @ X5 @ M3 ) @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) )
        = ( add_mset_nat @ X5 @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% diff_union_swap2
thf(fact_860_diff__union__swap2,axiom,
    ! [Y3: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar,X5: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ Y3 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ( ( minus_287902380581138057ropVar @ ( add_ms1730626103851491130ropVar @ X5 @ M3 ) @ ( add_ms1730626103851491130ropVar @ Y3 @ zero_z6226653793019358809ropVar ) )
        = ( add_ms1730626103851491130ropVar @ X5 @ ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ Y3 @ zero_z6226653793019358809ropVar ) ) ) ) ) ).

% diff_union_swap2
thf(fact_861_insert__DiffM,axiom,
    ! [X5: nat,M3: multiset_nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ M3 ) )
     => ( ( add_mset_nat @ X5 @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) ) )
        = M3 ) ) ).

% insert_DiffM
thf(fact_862_insert__DiffM,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ( ( add_ms1730626103851491130ropVar @ X5 @ ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) ) )
        = M3 ) ) ).

% insert_DiffM
thf(fact_863_Cons__listrel1__Cons,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre6655839188755619153ropVar @ R ) )
      = ( ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y3 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X5 = Y3 )
          & ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_864_Cons__listrel1__Cons,axiom,
    ! [X5: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X5 = Y3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_865_Cons__listrel1__Cons,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) ) @ ( listre3736066117146447419ropVar @ R ) )
      = ( ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ R )
          & ( Xs = Ys ) )
        | ( ( X5 = Y3 )
          & ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_866_zip__Cons__Cons,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( zip_Pr6829646289182870327ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
      = ( cons_P3372013117981404637ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y3 ) @ ( zip_Pr6829646289182870327ropVar @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_867_zip__Cons__Cons,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: nat,Ys: list_nat] :
      ( ( zip_Pr2402428817466909387ar_nat @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
      = ( cons_P501864517277197755ar_nat @ ( produc5526329313106567659ar_nat @ X5 @ Y3 ) @ ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_868_zip__Cons__Cons,axiom,
    ! [X5: nat,Xs: list_nat,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( zip_na5961122453619627049ropVar @ ( cons_nat @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
      = ( cons_P5298751266258968801ropVar @ ( produc9085022949259285321ropVar @ X5 @ Y3 ) @ ( zip_na5961122453619627049ropVar @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_869_zip__Cons__Cons,axiom,
    ! [X5: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( zip_nat_nat @ ( cons_nat @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) )
      = ( cons_P6512896166579812791at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ ( zip_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_870_zip__Cons__Cons,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Y3: produc7679832412844898215ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( zip_Mu6605103971575351810ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ ( cons_P6664127749089838429ropVar @ Y3 @ Ys ) )
      = ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X5 @ Y3 ) @ ( zip_Mu6605103971575351810ropVar @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_871_zip__Cons__Cons,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( zip_Mu2706791923038522295ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
      = ( cons_P6664127749089838429ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ ( zip_Mu2706791923038522295ropVar @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_872_mset__single__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar] :
      ( ( ( mset_P5170620898817730055ropVar @ Xs )
        = ( add_ms6892841465718707280ropVar @ X5 @ zero_z6774573742726938735ropVar ) )
      = ( Xs
        = ( cons_P1504208554336013608ropVar @ X5 @ nil_Pr6119456312066593144ropVar ) ) ) ).

% mset_single_iff
thf(fact_873_mset__single__iff,axiom,
    ! [Xs: list_nat,X5: nat] :
      ( ( ( mset_nat @ Xs )
        = ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) )
      = ( Xs
        = ( cons_nat @ X5 @ nil_nat ) ) ) ).

% mset_single_iff
thf(fact_874_mset__single__iff__right,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ( add_ms6892841465718707280ropVar @ X5 @ zero_z6774573742726938735ropVar )
        = ( mset_P5170620898817730055ropVar @ Xs ) )
      = ( Xs
        = ( cons_P1504208554336013608ropVar @ X5 @ nil_Pr6119456312066593144ropVar ) ) ) ).

% mset_single_iff_right
thf(fact_875_mset__single__iff__right,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat )
        = ( mset_nat @ Xs ) )
      = ( Xs
        = ( cons_nat @ X5 @ nil_nat ) ) ) ).

% mset_single_iff_right
thf(fact_876_ns__mul__ext__singleton,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,Ns: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ B ) @ Ns )
     => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) @ ( add_ms1730626103851491130ropVar @ B @ zero_z6226653793019358809ropVar ) ) @ ( multis738276037714153361ropVar @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton
thf(fact_877_s__mul__ext__singleton,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,S: set_Pr7799659934693882759ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ B ) @ S )
     => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) @ ( add_ms1730626103851491130ropVar @ B @ zero_z6226653793019358809ropVar ) ) @ ( multis4691071231837107059ropVar @ Ns @ S ) ) ) ).

% s_mul_ext_singleton
thf(fact_878_map__eq__Cons__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
      = ( ? [Z3: nat,Zs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs3 ) )
            & ( ( F @ Z3 )
              = Y3 )
            & ( ( map_na2466558649495876678ropVar @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_879_map__eq__Cons__conv,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
      = ( ? [Z3: produc2642885279816024434ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ Z3 @ Zs3 ) )
            & ( ( F @ Z3 )
              = Y3 )
            & ( ( map_Pr3565048146863404228ropVar @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_880_map__eq__Cons__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
      = ( ? [Z3: nat,Zs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs3 ) )
            & ( ( F @ Z3 )
              = Y3 )
            & ( ( map_na2145702811007361372ropVar @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_881_map__eq__Cons__conv,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar,Y3: nat,Ys: list_nat] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
      = ( ? [Z3: produc2642885279816024434ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ Z3 @ Zs3 ) )
            & ( ( F @ Z3 )
              = Y3 )
            & ( ( map_Pr7810381211709419518ar_nat @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_882_map__eq__Cons__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
      = ( ? [Z3: nat,Zs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z3 @ Zs3 ) )
            & ( ( F @ Z3 )
              = Y3 )
            & ( ( map_nat_nat @ F @ Zs3 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_883_Cons__eq__map__conv,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( cons_M4277272395645347858ropVar @ X5 @ Xs )
        = ( map_na2466558649495876678ropVar @ F @ Ys ) )
      = ( ? [Z3: nat,Zs3: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs3 ) )
            & ( X5
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_na2466558649495876678ropVar @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_884_Cons__eq__map__conv,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
        = ( map_Pr3565048146863404228ropVar @ F @ Ys ) )
      = ( ? [Z3: produc2642885279816024434ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Ys
              = ( cons_P1504208554336013608ropVar @ Z3 @ Zs3 ) )
            & ( X5
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_Pr3565048146863404228ropVar @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_885_Cons__eq__map__conv,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
        = ( map_na2145702811007361372ropVar @ F @ Ys ) )
      = ( ? [Z3: nat,Zs3: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs3 ) )
            & ( X5
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_na2145702811007361372ropVar @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_886_Cons__eq__map__conv,axiom,
    ! [X5: nat,Xs: list_nat,F: produc2642885279816024434ropVar > nat,Ys: list_P8791008543742759928ropVar] :
      ( ( ( cons_nat @ X5 @ Xs )
        = ( map_Pr7810381211709419518ar_nat @ F @ Ys ) )
      = ( ? [Z3: produc2642885279816024434ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Ys
              = ( cons_P1504208554336013608ropVar @ Z3 @ Zs3 ) )
            & ( X5
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_Pr7810381211709419518ar_nat @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_887_Cons__eq__map__conv,axiom,
    ! [X5: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X5 @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
      = ( ? [Z3: nat,Zs3: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z3 @ Zs3 ) )
            & ( X5
              = ( F @ Z3 ) )
            & ( Xs
              = ( map_nat_nat @ F @ Zs3 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_888_map__eq__Cons__D,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
     => ? [Z2: nat,Zs4: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs4 ) )
          & ( ( F @ Z2 )
            = Y3 )
          & ( ( map_na2466558649495876678ropVar @ F @ Zs4 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_889_map__eq__Cons__D,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
     => ? [Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) )
          & ( ( F @ Z2 )
            = Y3 )
          & ( ( map_Pr3565048146863404228ropVar @ F @ Zs4 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_890_map__eq__Cons__D,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
     => ? [Z2: nat,Zs4: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs4 ) )
          & ( ( F @ Z2 )
            = Y3 )
          & ( ( map_na2145702811007361372ropVar @ F @ Zs4 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_891_map__eq__Cons__D,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar,Y3: nat,Ys: list_nat] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
     => ? [Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) )
          & ( ( F @ Z2 )
            = Y3 )
          & ( ( map_Pr7810381211709419518ar_nat @ F @ Zs4 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_892_map__eq__Cons__D,axiom,
    ! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
     => ? [Z2: nat,Zs4: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z2 @ Zs4 ) )
          & ( ( F @ Z2 )
            = Y3 )
          & ( ( map_nat_nat @ F @ Zs4 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_893_Cons__eq__map__D,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( cons_M4277272395645347858ropVar @ X5 @ Xs )
        = ( map_na2466558649495876678ropVar @ F @ Ys ) )
     => ? [Z2: nat,Zs4: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z2 @ Zs4 ) )
          & ( X5
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_na2466558649495876678ropVar @ F @ Zs4 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_894_Cons__eq__map__D,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
        = ( map_Pr3565048146863404228ropVar @ F @ Ys ) )
     => ? [Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( ( Ys
            = ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) )
          & ( X5
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_Pr3565048146863404228ropVar @ F @ Zs4 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_895_Cons__eq__map__D,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
        = ( map_na2145702811007361372ropVar @ F @ Ys ) )
     => ? [Z2: nat,Zs4: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z2 @ Zs4 ) )
          & ( X5
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_na2145702811007361372ropVar @ F @ Zs4 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_896_Cons__eq__map__D,axiom,
    ! [X5: nat,Xs: list_nat,F: produc2642885279816024434ropVar > nat,Ys: list_P8791008543742759928ropVar] :
      ( ( ( cons_nat @ X5 @ Xs )
        = ( map_Pr7810381211709419518ar_nat @ F @ Ys ) )
     => ? [Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( ( Ys
            = ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) )
          & ( X5
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_Pr7810381211709419518ar_nat @ F @ Zs4 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_897_Cons__eq__map__D,axiom,
    ! [X5: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X5 @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
     => ? [Z2: nat,Zs4: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z2 @ Zs4 ) )
          & ( X5
            = ( F @ Z2 ) )
          & ( Xs
            = ( map_nat_nat @ F @ Zs4 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_898_list_Osimps_I9_J,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( ( map_Pr3565048146863404228ropVar @ F @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) )
      = ( cons_P1504208554336013608ropVar @ ( F @ X21 ) @ ( map_Pr3565048146863404228ropVar @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_899_list_Osimps_I9_J,axiom,
    ! [F: produc2642885279816024434ropVar > nat,X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( ( map_Pr7810381211709419518ar_nat @ F @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_Pr7810381211709419518ar_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_900_list_Osimps_I9_J,axiom,
    ! [F: nat > multis3193088007478089820ropVar,X21: nat,X222: list_nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_M4277272395645347858ropVar @ ( F @ X21 ) @ ( map_na2466558649495876678ropVar @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_901_list_Osimps_I9_J,axiom,
    ! [F: nat > produc2642885279816024434ropVar,X21: nat,X222: list_nat] :
      ( ( map_na2145702811007361372ropVar @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_P1504208554336013608ropVar @ ( F @ X21 ) @ ( map_na2145702811007361372ropVar @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_902_list_Osimps_I9_J,axiom,
    ! [F: nat > nat,X21: nat,X222: list_nat] :
      ( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_903_list_Odistinct_I1_J,axiom,
    ! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( nil_Pr6119456312066593144ropVar
     != ( cons_P1504208554336013608ropVar @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_904_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_905_list_OdiscI,axiom,
    ! [List: list_P8791008543742759928ropVar,X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( ( List
        = ( cons_P1504208554336013608ropVar @ X21 @ X222 ) )
     => ( List != nil_Pr6119456312066593144ropVar ) ) ).

% list.discI
thf(fact_906_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_907_list_Oexhaust,axiom,
    ! [Y3: list_P8791008543742759928ropVar] :
      ( ( Y3 != nil_Pr6119456312066593144ropVar )
     => ~ ! [X212: produc2642885279816024434ropVar,X223: list_P8791008543742759928ropVar] :
            ( Y3
           != ( cons_P1504208554336013608ropVar @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_908_list_Oexhaust,axiom,
    ! [Y3: list_nat] :
      ( ( Y3 != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y3
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_909_min__list_Ocases,axiom,
    ! [X5: list_nat] :
      ( ! [X4: nat,Xs3: list_nat] :
          ( X5
         != ( cons_nat @ X4 @ Xs3 ) )
     => ( X5 = nil_nat ) ) ).

% min_list.cases
thf(fact_910_transpose_Ocases,axiom,
    ! [X5: list_l3415868495287753214ropVar] :
      ( ( X5 != nil_li2679230897895549182ropVar )
     => ( ! [Xss: list_l3415868495287753214ropVar] :
            ( X5
           != ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss ) )
       => ~ ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Xss: list_l3415868495287753214ropVar] :
              ( X5
             != ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_911_transpose_Ocases,axiom,
    ! [X5: list_list_nat] :
      ( ( X5 != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X5
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X4: nat,Xs3: list_nat,Xss: list_list_nat] :
              ( X5
             != ( cons_list_nat @ ( cons_nat @ X4 @ Xs3 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_912_remdups__adj_Ocases,axiom,
    ! [X5: list_P8791008543742759928ropVar] :
      ( ( X5 != nil_Pr6119456312066593144ropVar )
     => ( ! [X4: produc2642885279816024434ropVar] :
            ( X5
           != ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X4: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
              ( X5
             != ( cons_P1504208554336013608ropVar @ X4 @ ( cons_P1504208554336013608ropVar @ Y4 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_913_remdups__adj_Ocases,axiom,
    ! [X5: list_nat] :
      ( ( X5 != nil_nat )
     => ( ! [X4: nat] :
            ( X5
           != ( cons_nat @ X4 @ nil_nat ) )
       => ~ ! [X4: nat,Y4: nat,Xs3: list_nat] :
              ( X5
             != ( cons_nat @ X4 @ ( cons_nat @ Y4 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_914_neq__Nil__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
      = ( ? [Y2: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar] :
            ( Xs
            = ( cons_P1504208554336013608ropVar @ Y2 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_915_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y2: nat,Ys2: list_nat] :
            ( Xs
            = ( cons_nat @ Y2 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_916_list__induct2_H,axiom,
    ! [P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > $o,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar )
     => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ nil_Pr6119456312066593144ropVar )
       => ( ! [Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] : ( P @ nil_Pr6119456312066593144ropVar @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
         => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
                ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_917_list__induct2_H,axiom,
    ! [P: list_P8791008543742759928ropVar > list_nat > $o,Xs: list_P8791008543742759928ropVar,Ys: list_nat] :
      ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat )
     => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ nil_nat )
       => ( ! [Y4: nat,Ys5: list_nat] : ( P @ nil_Pr6119456312066593144ropVar @ ( cons_nat @ Y4 @ Ys5 ) )
         => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: nat,Ys5: list_nat] :
                ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_918_list__induct2_H,axiom,
    ! [P: list_nat > list_P8791008543742759928ropVar > $o,Xs: list_nat,Ys: list_P8791008543742759928ropVar] :
      ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar )
     => ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_Pr6119456312066593144ropVar )
       => ( ! [Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] : ( P @ nil_nat @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
         => ( ! [X4: nat,Xs3: list_nat,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
                ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_919_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X4: nat,Xs3: list_nat] : ( P @ ( cons_nat @ X4 @ Xs3 ) @ nil_nat )
       => ( ! [Y4: nat,Ys5: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys5 ) )
         => ( ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat] :
                ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_920_list__nonempty__induct,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > $o] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ( ! [X4: produc2642885279816024434ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) )
       => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
              ( ( Xs3 != nil_Pr6119456312066593144ropVar )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_921_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
       => ( ! [X4: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_922_successively_Ocases,axiom,
    ! [X5: produc5533280449304468145ropVar] :
      ( ! [P7: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
          ( X5
         != ( produc3887254912111605537ropVar @ P7 @ nil_Pr6119456312066593144ropVar ) )
     => ( ! [P7: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X4: produc2642885279816024434ropVar] :
            ( X5
           != ( produc3887254912111605537ropVar @ P7 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) )
       => ~ ! [P7: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X4: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
              ( X5
             != ( produc3887254912111605537ropVar @ P7 @ ( cons_P1504208554336013608ropVar @ X4 @ ( cons_P1504208554336013608ropVar @ Y4 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_923_successively_Ocases,axiom,
    ! [X5: produc254973753779126261st_nat] :
      ( ! [P7: nat > nat > $o] :
          ( X5
         != ( produc4727192421694094319st_nat @ P7 @ nil_nat ) )
     => ( ! [P7: nat > nat > $o,X4: nat] :
            ( X5
           != ( produc4727192421694094319st_nat @ P7 @ ( cons_nat @ X4 @ nil_nat ) ) )
       => ~ ! [P7: nat > nat > $o,X4: nat,Y4: nat,Xs3: list_nat] :
              ( X5
             != ( produc4727192421694094319st_nat @ P7 @ ( cons_nat @ X4 @ ( cons_nat @ Y4 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_924_sorted__wrt_Ocases,axiom,
    ! [X5: produc5533280449304468145ropVar] :
      ( ! [P7: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
          ( X5
         != ( produc3887254912111605537ropVar @ P7 @ nil_Pr6119456312066593144ropVar ) )
     => ~ ! [P7: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
            ( X5
           != ( produc3887254912111605537ropVar @ P7 @ ( cons_P1504208554336013608ropVar @ X4 @ Ys5 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_925_sorted__wrt_Ocases,axiom,
    ! [X5: produc254973753779126261st_nat] :
      ( ! [P7: nat > nat > $o] :
          ( X5
         != ( produc4727192421694094319st_nat @ P7 @ nil_nat ) )
     => ~ ! [P7: nat > nat > $o,X4: nat,Ys5: list_nat] :
            ( X5
           != ( produc4727192421694094319st_nat @ P7 @ ( cons_nat @ X4 @ Ys5 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_926_listrelp_OCons,axiom,
    ! [R: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X5: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( R @ X5 @ Y3 )
     => ( ( listre4042120741191764159ropVar @ R @ Xs @ Ys )
       => ( listre4042120741191764159ropVar @ R @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_927_listrelp_OCons,axiom,
    ! [R: produc2642885279816024434ropVar > nat > $o,X5: produc2642885279816024434ropVar,Y3: nat,Xs: list_P8791008543742759928ropVar,Ys: list_nat] :
      ( ( R @ X5 @ Y3 )
     => ( ( listre7860766376562299715ar_nat @ R @ Xs @ Ys )
       => ( listre7860766376562299715ar_nat @ R @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_928_listrelp_OCons,axiom,
    ! [R: nat > produc2642885279816024434ropVar > $o,X5: nat,Y3: produc2642885279816024434ropVar,Xs: list_nat,Ys: list_P8791008543742759928ropVar] :
      ( ( R @ X5 @ Y3 )
     => ( ( listre2196087975860241569ropVar @ R @ Xs @ Ys )
       => ( listre2196087975860241569ropVar @ R @ ( cons_nat @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_929_listrelp_OCons,axiom,
    ! [R: nat > nat > $o,X5: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( R @ X5 @ Y3 )
     => ( ( listrelp_nat_nat @ R @ Xs @ Ys )
       => ( listrelp_nat_nat @ R @ ( cons_nat @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) ) ) ).

% listrelp.Cons
thf(fact_930_append__Cons,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ Ys )
      = ( cons_P1504208554336013608ropVar @ X5 @ ( append4352292717583037453ropVar @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_931_append__Cons,axiom,
    ! [X5: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X5 @ Xs ) @ Ys )
      = ( cons_nat @ X5 @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_932_Cons__eq__appendI,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs1: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X5 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append4352292717583037453ropVar @ Xs1 @ Zs ) )
       => ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
          = ( append4352292717583037453ropVar @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_933_Cons__eq__appendI,axiom,
    ! [X5: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X5 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X5 @ Xs )
          = ( append_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_934_distinct__length__2__or__more,axiom,
    ! [A: produc2642885279816024434ropVar,B: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( distin2267423591311547883ropVar @ ( cons_P1504208554336013608ropVar @ A @ ( cons_P1504208554336013608ropVar @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distin2267423591311547883ropVar @ ( cons_P1504208554336013608ropVar @ A @ Xs ) )
        & ( distin2267423591311547883ropVar @ ( cons_P1504208554336013608ropVar @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_935_distinct__length__2__or__more,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ A @ ( cons_nat @ B @ Xs ) ) )
      = ( ( A != B )
        & ( distinct_nat @ ( cons_nat @ A @ Xs ) )
        & ( distinct_nat @ ( cons_nat @ B @ Xs ) ) ) ) ).

% distinct_length_2_or_more
thf(fact_936_mset_Osimps_I2_J,axiom,
    ! [A: produc2642885279816024434ropVar,X5: list_P8791008543742759928ropVar] :
      ( ( mset_P5170620898817730055ropVar @ ( cons_P1504208554336013608ropVar @ A @ X5 ) )
      = ( add_ms6892841465718707280ropVar @ A @ ( mset_P5170620898817730055ropVar @ X5 ) ) ) ).

% mset.simps(2)
thf(fact_937_mset_Osimps_I2_J,axiom,
    ! [A: nat,X5: list_nat] :
      ( ( mset_nat @ ( cons_nat @ A @ X5 ) )
      = ( add_mset_nat @ A @ ( mset_nat @ X5 ) ) ) ).

% mset.simps(2)
thf(fact_938_not__Cons__self2,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_939_not__Cons__self2,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( cons_nat @ X5 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_940_list_Oset__intros_I2_J,axiom,
    ! [Y3: multis3193088007478089820ropVar,X222: list_M6740372356978808802ropVar,X21: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ X222 ) )
     => ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_941_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar,X21: produc2642885279816024434ropVar] :
      ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ X222 ) )
     => ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_942_list_Oset__intros_I2_J,axiom,
    ! [Y3: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_943_list_Oset__intros_I1_J,axiom,
    ! [X21: multis3193088007478089820ropVar,X222: list_M6740372356978808802ropVar] : ( member8421201611525043589ropVar @ X21 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_944_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] : ( member9210975342120487067ropVar @ X21 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_945_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_946_list_Oset__cases,axiom,
    ! [E: multis3193088007478089820ropVar,A: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ E @ ( set_Mu146256361055586289ropVar @ A ) )
     => ( ! [Z22: list_M6740372356978808802ropVar] :
            ( A
           != ( cons_M4277272395645347858ropVar @ E @ Z22 ) )
       => ~ ! [Z1: multis3193088007478089820ropVar,Z22: list_M6740372356978808802ropVar] :
              ( ( A
                = ( cons_M4277272395645347858ropVar @ Z1 @ Z22 ) )
             => ~ ( member8421201611525043589ropVar @ E @ ( set_Mu146256361055586289ropVar @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_947_list_Oset__cases,axiom,
    ! [E: produc2642885279816024434ropVar,A: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ E @ ( set_Pr3946994337432900359ropVar @ A ) )
     => ( ! [Z22: list_P8791008543742759928ropVar] :
            ( A
           != ( cons_P1504208554336013608ropVar @ E @ Z22 ) )
       => ~ ! [Z1: produc2642885279816024434ropVar,Z22: list_P8791008543742759928ropVar] :
              ( ( A
                = ( cons_P1504208554336013608ropVar @ Z1 @ Z22 ) )
             => ~ ( member9210975342120487067ropVar @ E @ ( set_Pr3946994337432900359ropVar @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_948_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_949_set__ConsD,axiom,
    ! [Y3: multis3193088007478089820ropVar,X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) ) )
     => ( ( Y3 = X5 )
        | ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_950_set__ConsD,axiom,
    ! [Y3: produc2642885279816024434ropVar,X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) ) )
     => ( ( Y3 = X5 )
        | ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_951_set__ConsD,axiom,
    ! [Y3: nat,X5: nat,Xs: list_nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X5 @ Xs ) ) )
     => ( ( Y3 = X5 )
        | ( member_nat @ Y3 @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_952_mset__add,axiom,
    ! [A: nat,A2: multiset_nat] :
      ( ( member_nat @ A @ ( set_mset_nat @ A2 ) )
     => ~ ! [B7: multiset_nat] :
            ( A2
           != ( add_mset_nat @ A @ B7 ) ) ) ).

% mset_add
thf(fact_953_mset__add,axiom,
    ! [A: multis3193088007478089820ropVar,A2: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ A2 ) )
     => ~ ! [B7: multis193838646080965794ropVar] :
            ( A2
           != ( add_ms1730626103851491130ropVar @ A @ B7 ) ) ) ).

% mset_add
thf(fact_954_multi__member__split,axiom,
    ! [X5: nat,M3: multiset_nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ M3 ) )
     => ? [A7: multiset_nat] :
          ( M3
          = ( add_mset_nat @ X5 @ A7 ) ) ) ).

% multi_member_split
thf(fact_955_multi__member__split,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ? [A7: multis193838646080965794ropVar] :
          ( M3
          = ( add_ms1730626103851491130ropVar @ X5 @ A7 ) ) ) ).

% multi_member_split
thf(fact_956_insert__noteq__member,axiom,
    ! [B: nat,B6: multiset_nat,C3: nat,C4: multiset_nat] :
      ( ( ( add_mset_nat @ B @ B6 )
        = ( add_mset_nat @ C3 @ C4 ) )
     => ( ( B != C3 )
       => ( member_nat @ C3 @ ( set_mset_nat @ B6 ) ) ) ) ).

% insert_noteq_member
thf(fact_957_insert__noteq__member,axiom,
    ! [B: multis3193088007478089820ropVar,B6: multis193838646080965794ropVar,C3: multis3193088007478089820ropVar,C4: multis193838646080965794ropVar] :
      ( ( ( add_ms1730626103851491130ropVar @ B @ B6 )
        = ( add_ms1730626103851491130ropVar @ C3 @ C4 ) )
     => ( ( B != C3 )
       => ( member8421201611525043589ropVar @ C3 @ ( set_ms1147286257884310937ropVar @ B6 ) ) ) ) ).

% insert_noteq_member
thf(fact_958_union__single__eq__member,axiom,
    ! [X5: nat,M3: multiset_nat,N4: multiset_nat] :
      ( ( ( add_mset_nat @ X5 @ M3 )
        = N4 )
     => ( member_nat @ X5 @ ( set_mset_nat @ N4 ) ) ) ).

% union_single_eq_member
thf(fact_959_union__single__eq__member,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar,N4: multis193838646080965794ropVar] :
      ( ( ( add_ms1730626103851491130ropVar @ X5 @ M3 )
        = N4 )
     => ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ N4 ) ) ) ).

% union_single_eq_member
thf(fact_960_zip__eq__ConsE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Xy: produc7137084944585861159ropVar,Xys: list_P1750615084264230189ropVar] :
      ( ( ( zip_Pr6829646289182870327ropVar @ Xs @ Ys )
        = ( cons_P3372013117981404637ropVar @ Xy @ Xys ) )
     => ~ ! [X4: produc2642885279816024434ropVar,Xs5: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ X4 @ Xs5 ) )
           => ! [Y4: produc2642885279816024434ropVar,Ys4: list_P8791008543742759928ropVar] :
                ( ( Ys
                  = ( cons_P1504208554336013608ropVar @ Y4 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6842008660352832535ropVar @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_Pr6829646289182870327ropVar @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_961_zip__eq__ConsE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Xy: produc103364944588188913ar_nat,Xys: list_P2692446888044980865ar_nat] :
      ( ( ( zip_Pr2402428817466909387ar_nat @ Xs @ Ys )
        = ( cons_P501864517277197755ar_nat @ Xy @ Xys ) )
     => ~ ! [X4: produc2642885279816024434ropVar,Xs5: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ X4 @ Xs5 ) )
           => ! [Y4: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y4 @ Ys4 ) )
               => ( ( Xy
                    = ( produc5526329313106567659ar_nat @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_Pr2402428817466909387ar_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_962_zip__eq__ConsE,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Xy: produc4900251693569959959ropVar,Xys: list_P2861839656816079527ropVar] :
      ( ( ( zip_na5961122453619627049ropVar @ Xs @ Ys )
        = ( cons_P5298751266258968801ropVar @ Xy @ Xys ) )
     => ~ ! [X4: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs5 ) )
           => ! [Y4: produc2642885279816024434ropVar,Ys4: list_P8791008543742759928ropVar] :
                ( ( Ys
                  = ( cons_P1504208554336013608ropVar @ Y4 @ Ys4 ) )
               => ( ( Xy
                    = ( produc9085022949259285321ropVar @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_na5961122453619627049ropVar @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_963_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 ) )
     => ~ ! [X4: nat,Xs5: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs5 ) )
           => ! [Y4: nat,Ys4: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y4 @ Ys4 ) )
               => ( ( Xy
                    = ( product_Pair_nat_nat @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_nat_nat @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_964_zip__eq__ConsE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar,Xy: produc2642885279816024434ropVar,Xys: list_P8791008543742759928ropVar] :
      ( ( ( zip_Mu6605103971575351810ropVar @ Xs @ Ys )
        = ( cons_P1504208554336013608ropVar @ Xy @ Xys ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Xs5: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( cons_M4277272395645347858ropVar @ X4 @ Xs5 ) )
           => ! [Y4: produc7679832412844898215ropVar,Ys4: list_P444016040222843053ropVar] :
                ( ( Ys
                  = ( cons_P6664127749089838429ropVar @ Y4 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6194868611276206818ropVar @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_Mu6605103971575351810ropVar @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_965_zip__eq__ConsE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,Xy: produc7679832412844898215ropVar,Xys: list_P444016040222843053ropVar] :
      ( ( ( zip_Mu2706791923038522295ropVar @ Xs @ Ys )
        = ( cons_P6664127749089838429ropVar @ Xy @ Xys ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Xs5: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( cons_M4277272395645347858ropVar @ X4 @ Xs5 ) )
           => ! [Y4: multis3193088007478089820ropVar,Ys4: list_M6740372356978808802ropVar] :
                ( ( Ys
                  = ( cons_M4277272395645347858ropVar @ Y4 @ Ys4 ) )
               => ( ( Xy
                    = ( produc6936306168532358295ropVar @ X4 @ Y4 ) )
                 => ( Xys
                   != ( zip_Mu2706791923038522295ropVar @ Xs5 @ Ys4 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_966_single__is__union,axiom,
    ! [A: nat,M3: multiset_nat,N4: multiset_nat] :
      ( ( ( add_mset_nat @ A @ zero_z7348594199698428585et_nat )
        = ( plus_p6334493942879108393et_nat @ M3 @ N4 ) )
      = ( ( ( ( add_mset_nat @ A @ zero_z7348594199698428585et_nat )
            = M3 )
          & ( N4 = zero_z7348594199698428585et_nat ) )
        | ( ( M3 = zero_z7348594199698428585et_nat )
          & ( ( add_mset_nat @ A @ zero_z7348594199698428585et_nat )
            = N4 ) ) ) ) ).

% single_is_union
thf(fact_967_union__is__single,axiom,
    ! [M3: multiset_nat,N4: multiset_nat,A: nat] :
      ( ( ( plus_p6334493942879108393et_nat @ M3 @ N4 )
        = ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
      = ( ( ( M3
            = ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
          & ( N4 = zero_z7348594199698428585et_nat ) )
        | ( ( M3 = zero_z7348594199698428585et_nat )
          & ( N4
            = ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% union_is_single
thf(fact_968_add__mset__add__single,axiom,
    ( add_mset_nat
    = ( ^ [A5: nat,A8: multiset_nat] : ( plus_p6334493942879108393et_nat @ A8 @ ( add_mset_nat @ A5 @ zero_z7348594199698428585et_nat ) ) ) ) ).

% add_mset_add_single
thf(fact_969_multi__member__last,axiom,
    ! [X5: nat] : ( member_nat @ X5 @ ( set_mset_nat @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) ) ) ).

% multi_member_last
thf(fact_970_multi__member__last,axiom,
    ! [X5: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) ) ) ).

% multi_member_last
thf(fact_971_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: 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 ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat,Z2: nat,Zs4: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_nat @ Zs4 ) )
                   => ( ( ( size_size_list_nat @ Zs4 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_972_list__induct4,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Zs: list_nat,Ws: list_nat,P: list_P8791008543742759928ropVar > list_nat > list_nat > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ 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 ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_nat @ nil_nat )
           => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: nat,Ys5: list_nat,Z2: nat,Zs4: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_nat @ Zs4 ) )
                   => ( ( ( size_size_list_nat @ Zs4 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_973_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_nat,P: list_nat > list_P8791008543742759928ropVar > list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: nat,Zs4: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s6752116297430514276ropVar @ Ys5 ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                      = ( size_size_list_nat @ Zs4 ) )
                   => ( ( ( size_size_list_nat @ Zs4 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_974_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_P8791008543742759928ropVar,Ws: list_nat,P: list_nat > list_nat > list_P8791008543742759928ropVar > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_Pr6119456312066593144ropVar @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat,Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_s6752116297430514276ropVar @ Zs4 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs4 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_975_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ws: list_P8791008543742759928ropVar,P: list_nat > list_nat > list_nat > list_P8791008543742759928ropVar > $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_s6752116297430514276ropVar @ Ws ) )
         => ( ( P @ nil_nat @ nil_nat @ nil_nat @ nil_Pr6119456312066593144ropVar )
           => ( ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat,Z2: nat,Zs4: list_nat,W2: produc2642885279816024434ropVar,Ws2: list_P8791008543742759928ropVar] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_nat @ Zs4 ) )
                   => ( ( ( size_size_list_nat @ Zs4 )
                        = ( size_s6752116297430514276ropVar @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) @ ( cons_P1504208554336013608ropVar @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_976_list__induct4,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_nat,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_nat )
           => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: nat,Zs4: list_nat,W2: nat,Ws2: list_nat] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                    = ( size_s6752116297430514276ropVar @ Ys5 ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                      = ( size_size_list_nat @ Zs4 ) )
                   => ( ( ( size_size_list_nat @ Zs4 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_977_list__induct4,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Zs: list_P8791008543742759928ropVar,Ws: list_nat,P: list_P8791008543742759928ropVar > list_nat > list_P8791008543742759928ropVar > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_Pr6119456312066593144ropVar @ nil_nat )
           => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: nat,Ys5: list_nat,Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar,W2: nat,Ws2: list_nat] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_s6752116297430514276ropVar @ Zs4 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs4 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_978_list__induct4,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Zs: list_nat,Ws: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > list_nat > list_nat > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s6752116297430514276ropVar @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_nat @ nil_Pr6119456312066593144ropVar )
           => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: nat,Ys5: list_nat,Z2: nat,Zs4: list_nat,W2: produc2642885279816024434ropVar,Ws2: list_P8791008543742759928ropVar] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                    = ( size_size_list_nat @ Ys5 ) )
                 => ( ( ( size_size_list_nat @ Ys5 )
                      = ( size_size_list_nat @ Zs4 ) )
                   => ( ( ( size_size_list_nat @ Zs4 )
                        = ( size_s6752116297430514276ropVar @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) @ ( cons_P1504208554336013608ropVar @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_979_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,Ws: list_nat,P: list_nat > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat )
           => ( ! [X4: nat,Xs3: list_nat,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar,W2: nat,Ws2: list_nat] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s6752116297430514276ropVar @ Ys5 ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                      = ( size_s6752116297430514276ropVar @ Zs4 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs4 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) @ ( cons_nat @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_980_list__induct4,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_P8791008543742759928ropVar,P: list_nat > list_P8791008543742759928ropVar > list_nat > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s6752116297430514276ropVar @ Ws ) )
         => ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_Pr6119456312066593144ropVar )
           => ( ! [X4: nat,Xs3: list_nat,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: nat,Zs4: list_nat,W2: produc2642885279816024434ropVar,Ws2: list_P8791008543742759928ropVar] :
                  ( ( ( size_size_list_nat @ Xs3 )
                    = ( size_s6752116297430514276ropVar @ Ys5 ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                      = ( size_size_list_nat @ Zs4 ) )
                   => ( ( ( size_size_list_nat @ Zs4 )
                        = ( size_s6752116297430514276ropVar @ Ws2 ) )
                     => ( ( P @ Xs3 @ Ys5 @ Zs4 @ Ws2 )
                       => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) @ ( cons_P1504208554336013608ropVar @ W2 @ Ws2 ) ) ) ) ) )
             => ( P @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_981_list__induct3,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar )
         => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
                ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                  = ( size_s6752116297430514276ropVar @ Ys5 ) )
               => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                    = ( size_s6752116297430514276ropVar @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_982_list__induct3,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Zs: list_nat,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat )
         => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: nat,Zs4: list_nat] :
                ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                  = ( size_s6752116297430514276ropVar @ Ys5 ) )
               => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                    = ( size_size_list_nat @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_983_list__induct3,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Zs: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > list_nat > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_Pr6119456312066593144ropVar )
         => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: nat,Ys5: list_nat,Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
                ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_s6752116297430514276ropVar @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_984_list__induct3,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,Zs: list_nat,P: list_P8791008543742759928ropVar > list_nat > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_nat )
         => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: nat,Ys5: list_nat,Z2: nat,Zs4: list_nat] :
                ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_size_list_nat @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_985_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,P: list_nat > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar )
         => ( ! [X4: nat,Xs3: list_nat,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_s6752116297430514276ropVar @ Ys5 ) )
               => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                    = ( size_s6752116297430514276ropVar @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_986_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,Zs: list_nat,P: list_nat > list_P8791008543742759928ropVar > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys )
          = ( size_size_list_nat @ Zs ) )
       => ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar @ nil_nat )
         => ( ! [X4: nat,Xs3: list_nat,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar,Z2: nat,Zs4: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_s6752116297430514276ropVar @ Ys5 ) )
               => ( ( ( size_s6752116297430514276ropVar @ Ys5 )
                    = ( size_size_list_nat @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_987_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_P8791008543742759928ropVar,P: list_nat > list_nat > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ( size_size_list_nat @ Ys )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_Pr6119456312066593144ropVar )
         => ( ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat,Z2: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_s6752116297430514276ropVar @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_P1504208554336013608ropVar @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_988_list__induct3,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,P: 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 ) )
       => ( ( P @ nil_nat @ nil_nat @ nil_nat )
         => ( ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat,Z2: nat,Zs4: list_nat] :
                ( ( ( size_size_list_nat @ Xs3 )
                  = ( size_size_list_nat @ Ys5 ) )
               => ( ( ( size_size_list_nat @ Ys5 )
                    = ( size_size_list_nat @ Zs4 ) )
                 => ( ( P @ Xs3 @ Ys5 @ Zs4 )
                   => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) @ ( cons_nat @ Z2 @ Zs4 ) ) ) ) )
           => ( P @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_989_list__induct2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar )
       => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
              ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                = ( size_s6752116297430514276ropVar @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_990_list__induct2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_nat,P: list_P8791008543742759928ropVar > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat )
       => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: nat,Ys5: list_nat] :
              ( ( ( size_s6752116297430514276ropVar @ Xs3 )
                = ( size_size_list_nat @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_991_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_P8791008543742759928ropVar,P: list_nat > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s6752116297430514276ropVar @ Ys ) )
     => ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar )
       => ( ! [X4: nat,Xs3: list_nat,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_s6752116297430514276ropVar @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_992_list__induct2,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: list_nat > list_nat > $o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( P @ nil_nat @ nil_nat )
       => ( ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat] :
              ( ( ( size_size_list_nat @ Xs3 )
                = ( size_size_list_nat @ Ys5 ) )
             => ( ( P @ Xs3 @ Ys5 )
               => ( P @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) ) ) )
         => ( P @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_993_rev__nonempty__induct,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > $o] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ( ! [X4: produc2642885279816024434ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) )
       => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
              ( ( Xs3 != nil_Pr6119456312066593144ropVar )
             => ( ( P @ Xs3 )
               => ( P @ ( append4352292717583037453ropVar @ Xs3 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_994_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X4: nat] : ( P @ ( cons_nat @ X4 @ nil_nat ) )
       => ( ! [X4: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P @ Xs3 )
               => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X4 @ nil_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_995_append__eq__Cons__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Ys @ Zs )
        = ( cons_P1504208554336013608ropVar @ X5 @ Xs ) )
      = ( ( ( Ys = nil_Pr6119456312066593144ropVar )
          & ( Zs
            = ( cons_P1504208554336013608ropVar @ X5 @ Xs ) ) )
        | ? [Ys6: list_P8791008543742759928ropVar] :
            ( ( Ys
              = ( cons_P1504208554336013608ropVar @ X5 @ Ys6 ) )
            & ( ( append4352292717583037453ropVar @ Ys6 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_996_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,X5: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( cons_nat @ X5 @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs
            = ( cons_nat @ X5 @ Xs ) ) )
        | ? [Ys6: list_nat] :
            ( ( Ys
              = ( cons_nat @ X5 @ Ys6 ) )
            & ( ( append_nat @ Ys6 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_997_Cons__eq__append__conv,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
        = ( append4352292717583037453ropVar @ Ys @ Zs ) )
      = ( ( ( Ys = nil_Pr6119456312066593144ropVar )
          & ( ( cons_P1504208554336013608ropVar @ X5 @ Xs )
            = Zs ) )
        | ? [Ys6: list_P8791008543742759928ropVar] :
            ( ( ( cons_P1504208554336013608ropVar @ X5 @ Ys6 )
              = Ys )
            & ( Xs
              = ( append4352292717583037453ropVar @ Ys6 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_998_Cons__eq__append__conv,axiom,
    ! [X5: nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X5 @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X5 @ Xs )
            = Zs ) )
        | ? [Ys6: list_nat] :
            ( ( ( cons_nat @ X5 @ Ys6 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys6 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_999_rev__exhaust,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ~ ! [Ys5: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar] :
            ( Xs
           != ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ Y4 @ nil_Pr6119456312066593144ropVar ) ) ) ) ).

% rev_exhaust
thf(fact_1000_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys5: list_nat,Y4: nat] :
            ( Xs
           != ( append_nat @ Ys5 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_1001_rev__induct,axiom,
    ! [P: list_P8791008543742759928ropVar > $o,Xs: list_P8791008543742759928ropVar] :
      ( ( P @ nil_Pr6119456312066593144ropVar )
     => ( ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
            ( ( P @ Xs3 )
           => ( P @ ( append4352292717583037453ropVar @ Xs3 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1002_rev__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X4: nat,Xs3: list_nat] :
            ( ( P @ Xs3 )
           => ( P @ ( append_nat @ Xs3 @ ( cons_nat @ X4 @ nil_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_1003_split__list__first__prop__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,X: multis3193088007478089820ropVar] :
            ( ? [Zs3: list_M6740372356978808802ropVar] :
                ( Xs
                = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y2: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Ys2 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1004_split__list__first__prop__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ? [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,X: produc2642885279816024434ropVar] :
            ( ? [Zs3: list_P8791008543742759928ropVar] :
                ( Xs
                = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y2: produc2642885279816024434ropVar] :
                ( ( member9210975342120487067ropVar @ Y2 @ ( set_Pr3946994337432900359ropVar @ Ys2 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1005_split__list__first__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_nat,X: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs
                = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y2: nat] :
                ( ( member_nat @ Y2 @ ( set_nat2 @ Ys2 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_1006_split__list__last__prop__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,X: multis3193088007478089820ropVar,Zs3: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y2: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1007_split__list__last__prop__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ? [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,X: produc2642885279816024434ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y2: produc2642885279816024434ropVar] :
                ( ( member9210975342120487067ropVar @ Y2 @ ( set_Pr3946994337432900359ropVar @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1008_split__list__last__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_nat,X: nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs3 ) ) )
            & ( P @ X )
            & ! [Y2: nat] :
                ( ( member_nat @ Y2 @ ( set_nat2 @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_1009_in__set__conv__decomp__first,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X5 @ Zs3 ) ) )
            & ~ ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1010_in__set__conv__decomp__first,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X5 @ Zs3 ) ) )
            & ~ ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1011_in__set__conv__decomp__first,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X5 @ Zs3 ) ) )
            & ~ ( member_nat @ X5 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_1012_in__set__conv__decomp__last,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X5 @ Zs3 ) ) )
            & ~ ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1013_in__set__conv__decomp__last,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X5 @ Zs3 ) ) )
            & ~ ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1014_in__set__conv__decomp__last,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X5 @ Zs3 ) ) )
            & ~ ( member_nat @ X5 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_1015_split__list__first__propE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_M6740372356978808802ropVar,X4: multis3193088007478089820ropVar] :
            ( ? [Zs4: list_M6740372356978808802ropVar] :
                ( Xs
                = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs4 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ Xa3 @ ( set_Mu146256361055586289ropVar @ Ys5 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1016_split__list__first__propE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X6: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
            ( ? [Zs4: list_P8791008543742759928ropVar] :
                ( Xs
                = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs4 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa3: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ Xa3 @ ( set_Pr3946994337432900359ropVar @ Ys5 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1017_split__list__first__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_nat,X4: nat] :
            ( ? [Zs4: list_nat] :
                ( Xs
                = ( append_nat @ Ys5 @ ( cons_nat @ X4 @ Zs4 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ ( set_nat2 @ Ys5 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_1018_split__list__last__propE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_M6740372356978808802ropVar,X4: multis3193088007478089820ropVar,Zs4: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs4 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ Xa3 @ ( set_Mu146256361055586289ropVar @ Zs4 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1019_split__list__last__propE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X6: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs4 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa3: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ Xa3 @ ( set_Pr3946994337432900359ropVar @ Zs4 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1020_split__list__last__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_nat,X4: nat,Zs4: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys5 @ ( cons_nat @ X4 @ Zs4 ) ) )
           => ( ( P @ X4 )
             => ~ ! [Xa3: nat] :
                    ( ( member_nat @ Xa3 @ ( set_nat2 @ Zs4 ) )
                   => ~ ( P @ Xa3 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_1021_split__list__first__prop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_M6740372356978808802ropVar,X4: multis3193088007478089820ropVar] :
          ( ? [Zs4: list_M6740372356978808802ropVar] :
              ( Xs
              = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs4 ) ) )
          & ( P @ X4 )
          & ! [Xa3: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Xa3 @ ( set_Mu146256361055586289ropVar @ Ys5 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_1022_split__list__first__prop,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X6: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
          ( ? [Zs4: list_P8791008543742759928ropVar] :
              ( Xs
              = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs4 ) ) )
          & ( P @ X4 )
          & ! [Xa3: produc2642885279816024434ropVar] :
              ( ( member9210975342120487067ropVar @ Xa3 @ ( set_Pr3946994337432900359ropVar @ Ys5 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_1023_split__list__first__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_nat,X4: nat] :
          ( ? [Zs4: list_nat] :
              ( Xs
              = ( append_nat @ Ys5 @ ( cons_nat @ X4 @ Zs4 ) ) )
          & ( P @ X4 )
          & ! [Xa3: nat] :
              ( ( member_nat @ Xa3 @ ( set_nat2 @ Ys5 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_first_prop
thf(fact_1024_split__list__last__prop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_M6740372356978808802ropVar,X4: multis3193088007478089820ropVar,Zs4: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs4 ) ) )
          & ( P @ X4 )
          & ! [Xa3: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Xa3 @ ( set_Mu146256361055586289ropVar @ Zs4 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_1025_split__list__last__prop,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X6: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs4 ) ) )
          & ( P @ X4 )
          & ! [Xa3: produc2642885279816024434ropVar] :
              ( ( member9210975342120487067ropVar @ Xa3 @ ( set_Pr3946994337432900359ropVar @ Zs4 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_1026_split__list__last__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_nat,X4: nat,Zs4: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys5 @ ( cons_nat @ X4 @ Zs4 ) ) )
          & ( P @ X4 )
          & ! [Xa3: nat] :
              ( ( member_nat @ Xa3 @ ( set_nat2 @ Zs4 ) )
             => ~ ( P @ Xa3 ) ) ) ) ).

% split_list_last_prop
thf(fact_1027_in__set__conv__decomp,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
            ( Xs
            = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X5 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1028_in__set__conv__decomp,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( Xs
            = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X5 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1029_in__set__conv__decomp,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs3: list_nat] :
            ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X5 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_1030_append__Cons__eq__iff,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,Xs4: list_M6740372356978808802ropVar,Ys7: list_M6740372356978808802ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ( ~ ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Ys ) )
       => ( ( ( append2684835082924936439ropVar @ Xs @ ( cons_M4277272395645347858ropVar @ X5 @ Ys ) )
            = ( append2684835082924936439ropVar @ Xs4 @ ( cons_M4277272395645347858ropVar @ X5 @ Ys7 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1031_append__Cons__eq__iff,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Xs4: list_P8791008543742759928ropVar,Ys7: list_P8791008543742759928ropVar] :
      ( ~ ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ( ~ ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Ys ) )
       => ( ( ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X5 @ Ys ) )
            = ( append4352292717583037453ropVar @ Xs4 @ ( cons_P1504208554336013608ropVar @ X5 @ Ys7 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1032_append__Cons__eq__iff,axiom,
    ! [X5: nat,Xs: list_nat,Ys: list_nat,Xs4: list_nat,Ys7: list_nat] :
      ( ~ ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat @ X5 @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X5 @ Ys ) )
            = ( append_nat @ Xs4 @ ( cons_nat @ X5 @ Ys7 ) ) )
          = ( ( Xs = Xs4 )
            & ( Ys = Ys7 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_1033_split__list__propE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_M6740372356978808802ropVar,X4: multis3193088007478089820ropVar] :
            ( ? [Zs4: list_M6740372356978808802ropVar] :
                ( Xs
                = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs4 ) ) )
           => ~ ( P @ X4 ) ) ) ).

% split_list_propE
thf(fact_1034_split__list__propE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X6: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
            ( ? [Zs4: list_P8791008543742759928ropVar] :
                ( Xs
                = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs4 ) ) )
           => ~ ( P @ X4 ) ) ) ).

% split_list_propE
thf(fact_1035_split__list__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ~ ! [Ys5: list_nat,X4: nat] :
            ( ? [Zs4: list_nat] :
                ( Xs
                = ( append_nat @ Ys5 @ ( cons_nat @ X4 @ Zs4 ) ) )
           => ~ ( P @ X4 ) ) ) ).

% split_list_propE
thf(fact_1036_split__list__first,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ? [Ys5: list_M6740372356978808802ropVar,Zs4: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X5 @ Zs4 ) ) )
          & ~ ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Ys5 ) ) ) ) ).

% split_list_first
thf(fact_1037_split__list__first,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ? [Ys5: list_P8791008543742759928ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X5 @ Zs4 ) ) )
          & ~ ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Ys5 ) ) ) ) ).

% split_list_first
thf(fact_1038_split__list__first,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
     => ? [Ys5: list_nat,Zs4: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys5 @ ( cons_nat @ X5 @ Zs4 ) ) )
          & ~ ( member_nat @ X5 @ ( set_nat2 @ Ys5 ) ) ) ) ).

% split_list_first
thf(fact_1039_split__list__prop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X6: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X6 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_M6740372356978808802ropVar,X4: multis3193088007478089820ropVar] :
          ( ? [Zs4: list_M6740372356978808802ropVar] :
              ( Xs
              = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs4 ) ) )
          & ( P @ X4 ) ) ) ).

% split_list_prop
thf(fact_1040_split__list__prop,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X6: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X6 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
          ( ? [Zs4: list_P8791008543742759928ropVar] :
              ( Xs
              = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs4 ) ) )
          & ( P @ X4 ) ) ) ).

% split_list_prop
thf(fact_1041_split__list__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X6: nat] :
          ( ( member_nat @ X6 @ ( set_nat2 @ Xs ) )
          & ( P @ X6 ) )
     => ? [Ys5: list_nat,X4: nat] :
          ( ? [Zs4: list_nat] :
              ( Xs
              = ( append_nat @ Ys5 @ ( cons_nat @ X4 @ Zs4 ) ) )
          & ( P @ X4 ) ) ) ).

% split_list_prop
thf(fact_1042_split__list__last,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ? [Ys5: list_M6740372356978808802ropVar,Zs4: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X5 @ Zs4 ) ) )
          & ~ ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Zs4 ) ) ) ) ).

% split_list_last
thf(fact_1043_split__list__last,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ? [Ys5: list_P8791008543742759928ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X5 @ Zs4 ) ) )
          & ~ ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Zs4 ) ) ) ) ).

% split_list_last
thf(fact_1044_split__list__last,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
     => ? [Ys5: list_nat,Zs4: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys5 @ ( cons_nat @ X5 @ Zs4 ) ) )
          & ~ ( member_nat @ X5 @ ( set_nat2 @ Zs4 ) ) ) ) ).

% split_list_last
thf(fact_1045_split__list,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ? [Ys5: list_M6740372356978808802ropVar,Zs4: list_M6740372356978808802ropVar] :
          ( Xs
          = ( append2684835082924936439ropVar @ Ys5 @ ( cons_M4277272395645347858ropVar @ X5 @ Zs4 ) ) ) ) ).

% split_list
thf(fact_1046_split__list,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ? [Ys5: list_P8791008543742759928ropVar,Zs4: list_P8791008543742759928ropVar] :
          ( Xs
          = ( append4352292717583037453ropVar @ Ys5 @ ( cons_P1504208554336013608ropVar @ X5 @ Zs4 ) ) ) ) ).

% split_list
thf(fact_1047_split__list,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
     => ? [Ys5: list_nat,Zs4: list_nat] :
          ( Xs
          = ( append_nat @ Ys5 @ ( cons_nat @ X5 @ Zs4 ) ) ) ) ).

% split_list
thf(fact_1048_subset__eq__mset__impl_Ocases,axiom,
    ! [X5: produc3638977808798937127ropVar] :
      ( ! [Ys5: list_P8791008543742759928ropVar] :
          ( X5
         != ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys5 ) )
     => ~ ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Ys5: list_P8791008543742759928ropVar] :
            ( X5
           != ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ Ys5 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_1049_subset__eq__mset__impl_Ocases,axiom,
    ! [X5: produc1828647624359046049st_nat] :
      ( ! [Ys5: list_nat] :
          ( X5
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys5 ) )
     => ~ ! [X4: nat,Xs3: list_nat,Ys5: list_nat] :
            ( X5
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs3 ) @ Ys5 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_1050_shuffles_Ocases,axiom,
    ! [X5: produc3638977808798937127ropVar] :
      ( ! [Ys5: list_P8791008543742759928ropVar] :
          ( X5
         != ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys5 ) )
     => ( ! [Xs3: list_P8791008543742759928ropVar] :
            ( X5
           != ( produc5333478016634530327ropVar @ Xs3 @ nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
              ( X5
             != ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) ) ) ) ) ).

% shuffles.cases
thf(fact_1051_shuffles_Ocases,axiom,
    ! [X5: produc1828647624359046049st_nat] :
      ( ! [Ys5: list_nat] :
          ( X5
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys5 ) )
     => ( ! [Xs3: list_nat] :
            ( X5
           != ( produc2694037385005941721st_nat @ Xs3 @ nil_nat ) )
       => ~ ! [X4: nat,Xs3: list_nat,Y4: nat,Ys5: list_nat] :
              ( X5
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs3 ) @ ( cons_nat @ Y4 @ Ys5 ) ) ) ) ) ).

% shuffles.cases
thf(fact_1052_distinct__singleton,axiom,
    ! [X5: produc2642885279816024434ropVar] : ( distin2267423591311547883ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ nil_Pr6119456312066593144ropVar ) ) ).

% distinct_singleton
thf(fact_1053_distinct__singleton,axiom,
    ! [X5: nat] : ( distinct_nat @ ( cons_nat @ X5 @ nil_nat ) ) ).

% distinct_singleton
thf(fact_1054_distinct_Osimps_I2_J,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( distin2675581014522817237ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) )
      = ( ~ ( member8421201611525043589ropVar @ X5 @ ( set_Mu146256361055586289ropVar @ Xs ) )
        & ( distin2675581014522817237ropVar @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_1055_distinct_Osimps_I2_J,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( distin2267423591311547883ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) )
      = ( ~ ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
        & ( distin2267423591311547883ropVar @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_1056_distinct_Osimps_I2_J,axiom,
    ! [X5: nat,Xs: list_nat] :
      ( ( distinct_nat @ ( cons_nat @ X5 @ Xs ) )
      = ( ~ ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
        & ( distinct_nat @ Xs ) ) ) ).

% distinct.simps(2)
thf(fact_1057_list__update__code_I2_J,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ( list_u8100953110031273952ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ zero_zero_nat @ Y3 )
      = ( cons_P1504208554336013608ropVar @ Y3 @ Xs ) ) ).

% list_update_code(2)
thf(fact_1058_list__update__code_I2_J,axiom,
    ! [X5: nat,Xs: list_nat,Y3: nat] :
      ( ( list_update_nat @ ( cons_nat @ X5 @ Xs ) @ zero_zero_nat @ Y3 )
      = ( cons_nat @ Y3 @ Xs ) ) ).

% list_update_code(2)
thf(fact_1059_nth__via__drop,axiom,
    ! [N: nat,Xs: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( drop_M2293212096379261346ropVar @ N @ Xs )
        = ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
     => ( ( nth_Mu4537670848183934179ropVar @ Xs @ N )
        = Y3 ) ) ).

% nth_via_drop
thf(fact_1060_nth__via__drop,axiom,
    ! [N: nat,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( drop_P4829783966340880056ropVar @ N @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
     => ( ( nth_Pr227365692153740281ropVar @ Xs @ N )
        = Y3 ) ) ).

% nth_via_drop
thf(fact_1061_nth__via__drop,axiom,
    ! [N: nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( ( drop_nat @ N @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
     => ( ( nth_nat @ Xs @ N )
        = Y3 ) ) ).

% nth_via_drop
thf(fact_1062_listrel1I2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar,X5: produc2642885279816024434ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R ) )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ X5 @ Ys ) ) @ ( listre6655839188755619153ropVar @ R ) ) ) ).

% listrel1I2
thf(fact_1063_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,X5: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs ) @ ( cons_nat @ X5 @ Ys ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I2
thf(fact_1064_listrelp_Osimps,axiom,
    ( listre4042120741191764159ropVar
    = ( ^ [R3: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,A13: list_P8791008543742759928ropVar,A24: list_P8791008543742759928ropVar] :
          ( ( ( A13 = nil_Pr6119456312066593144ropVar )
            & ( A24 = nil_Pr6119456312066593144ropVar ) )
          | ? [X: produc2642885279816024434ropVar,Y2: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
              ( ( A13
                = ( cons_P1504208554336013608ropVar @ X @ Xs2 ) )
              & ( A24
                = ( cons_P1504208554336013608ropVar @ Y2 @ Ys2 ) )
              & ( R3 @ X @ Y2 )
              & ( listre4042120741191764159ropVar @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ).

% listrelp.simps
thf(fact_1065_listrelp_Osimps,axiom,
    ( listre7860766376562299715ar_nat
    = ( ^ [R3: produc2642885279816024434ropVar > nat > $o,A13: list_P8791008543742759928ropVar,A24: list_nat] :
          ( ( ( A13 = nil_Pr6119456312066593144ropVar )
            & ( A24 = nil_nat ) )
          | ? [X: produc2642885279816024434ropVar,Y2: nat,Xs2: list_P8791008543742759928ropVar,Ys2: list_nat] :
              ( ( A13
                = ( cons_P1504208554336013608ropVar @ X @ Xs2 ) )
              & ( A24
                = ( cons_nat @ Y2 @ Ys2 ) )
              & ( R3 @ X @ Y2 )
              & ( listre7860766376562299715ar_nat @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ).

% listrelp.simps
thf(fact_1066_listrelp_Osimps,axiom,
    ( listre2196087975860241569ropVar
    = ( ^ [R3: nat > produc2642885279816024434ropVar > $o,A13: list_nat,A24: list_P8791008543742759928ropVar] :
          ( ( ( A13 = nil_nat )
            & ( A24 = nil_Pr6119456312066593144ropVar ) )
          | ? [X: nat,Y2: produc2642885279816024434ropVar,Xs2: list_nat,Ys2: list_P8791008543742759928ropVar] :
              ( ( A13
                = ( cons_nat @ X @ Xs2 ) )
              & ( A24
                = ( cons_P1504208554336013608ropVar @ Y2 @ Ys2 ) )
              & ( R3 @ X @ Y2 )
              & ( listre2196087975860241569ropVar @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ).

% listrelp.simps
thf(fact_1067_listrelp_Osimps,axiom,
    ( listrelp_nat_nat
    = ( ^ [R3: nat > nat > $o,A13: list_nat,A24: list_nat] :
          ( ( ( A13 = nil_nat )
            & ( A24 = nil_nat ) )
          | ? [X: nat,Y2: nat,Xs2: list_nat,Ys2: list_nat] :
              ( ( A13
                = ( cons_nat @ X @ Xs2 ) )
              & ( A24
                = ( cons_nat @ Y2 @ Ys2 ) )
              & ( R3 @ X @ Y2 )
              & ( listrelp_nat_nat @ R3 @ Xs2 @ Ys2 ) ) ) ) ) ).

% listrelp.simps
thf(fact_1068_listrelp_Ocases,axiom,
    ! [R: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,A14: list_P8791008543742759928ropVar,A25: list_P8791008543742759928ropVar] :
      ( ( listre4042120741191764159ropVar @ R @ A14 @ A25 )
     => ( ( ( A14 = nil_Pr6119456312066593144ropVar )
         => ( A25 != nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X4: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
              ( ( A14
                = ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) )
             => ! [Ys5: list_P8791008543742759928ropVar] :
                  ( ( A25
                    = ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
                 => ( ( R @ X4 @ Y4 )
                   => ~ ( listre4042120741191764159ropVar @ R @ Xs3 @ Ys5 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_1069_listrelp_Ocases,axiom,
    ! [R: produc2642885279816024434ropVar > nat > $o,A14: list_P8791008543742759928ropVar,A25: list_nat] :
      ( ( listre7860766376562299715ar_nat @ R @ A14 @ A25 )
     => ( ( ( A14 = nil_Pr6119456312066593144ropVar )
         => ( A25 != nil_nat ) )
       => ~ ! [X4: produc2642885279816024434ropVar,Y4: nat,Xs3: list_P8791008543742759928ropVar] :
              ( ( A14
                = ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) )
             => ! [Ys5: list_nat] :
                  ( ( A25
                    = ( cons_nat @ Y4 @ Ys5 ) )
                 => ( ( R @ X4 @ Y4 )
                   => ~ ( listre7860766376562299715ar_nat @ R @ Xs3 @ Ys5 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_1070_listrelp_Ocases,axiom,
    ! [R: nat > produc2642885279816024434ropVar > $o,A14: list_nat,A25: list_P8791008543742759928ropVar] :
      ( ( listre2196087975860241569ropVar @ R @ A14 @ A25 )
     => ( ( ( A14 = nil_nat )
         => ( A25 != nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X4: nat,Y4: produc2642885279816024434ropVar,Xs3: list_nat] :
              ( ( A14
                = ( cons_nat @ X4 @ Xs3 ) )
             => ! [Ys5: list_P8791008543742759928ropVar] :
                  ( ( A25
                    = ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
                 => ( ( R @ X4 @ Y4 )
                   => ~ ( listre2196087975860241569ropVar @ R @ Xs3 @ Ys5 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_1071_listrelp_Ocases,axiom,
    ! [R: nat > nat > $o,A14: list_nat,A25: list_nat] :
      ( ( listrelp_nat_nat @ R @ A14 @ A25 )
     => ( ( ( A14 = nil_nat )
         => ( A25 != nil_nat ) )
       => ~ ! [X4: nat,Y4: nat,Xs3: list_nat] :
              ( ( A14
                = ( cons_nat @ X4 @ Xs3 ) )
             => ! [Ys5: list_nat] :
                  ( ( A25
                    = ( cons_nat @ Y4 @ Ys5 ) )
                 => ( ( R @ X4 @ Y4 )
                   => ~ ( listrelp_nat_nat @ R @ Xs3 @ Ys5 ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_1072_multi__member__this,axiom,
    ! [X5: multis3193088007478089820ropVar,XS: multis193838646080965794ropVar] : ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ ( plus_p1339091108561003481ropVar @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) @ XS ) ) ) ).

% multi_member_this
thf(fact_1073_multi__member__this,axiom,
    ! [X5: nat,XS: multiset_nat] : ( member_nat @ X5 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ).

% multi_member_this
thf(fact_1074_multi__member__skip,axiom,
    ! [X5: multis3193088007478089820ropVar,XS: multis193838646080965794ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ XS ) )
     => ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ ( plus_p1339091108561003481ropVar @ ( add_ms1730626103851491130ropVar @ Y3 @ zero_z6226653793019358809ropVar ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_1075_multi__member__skip,axiom,
    ! [X5: nat,XS: multiset_nat,Y3: nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ XS ) )
     => ( member_nat @ X5 @ ( set_mset_nat @ ( plus_p6334493942879108393et_nat @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_1076_more__than__one__mset__mset__diff,axiom,
    ! [A: nat,M3: multiset_nat] :
      ( ( member_nat @ A @ ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
     => ( ( set_mset_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
        = ( set_mset_nat @ M3 ) ) ) ).

% more_than_one_mset_mset_diff
thf(fact_1077_more__than__one__mset__mset__diff,axiom,
    ! [A: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) ) ) )
     => ( ( set_ms1147286257884310937ropVar @ ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) ) )
        = ( set_ms1147286257884310937ropVar @ M3 ) ) ) ).

% more_than_one_mset_mset_diff
thf(fact_1078_multiset__add__sub__el__shuffle,axiom,
    ! [C3: nat,B6: multiset_nat,B: nat] :
      ( ( member_nat @ C3 @ ( set_mset_nat @ B6 ) )
     => ( ( B != C3 )
       => ( ( add_mset_nat @ B @ ( minus_8522176038001411705et_nat @ B6 @ ( add_mset_nat @ C3 @ zero_z7348594199698428585et_nat ) ) )
          = ( minus_8522176038001411705et_nat @ ( add_mset_nat @ B @ B6 ) @ ( add_mset_nat @ C3 @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% multiset_add_sub_el_shuffle
thf(fact_1079_multiset__add__sub__el__shuffle,axiom,
    ! [C3: multis3193088007478089820ropVar,B6: multis193838646080965794ropVar,B: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ C3 @ ( set_ms1147286257884310937ropVar @ B6 ) )
     => ( ( B != C3 )
       => ( ( add_ms1730626103851491130ropVar @ B @ ( minus_287902380581138057ropVar @ B6 @ ( add_ms1730626103851491130ropVar @ C3 @ zero_z6226653793019358809ropVar ) ) )
          = ( minus_287902380581138057ropVar @ ( add_ms1730626103851491130ropVar @ B @ B6 ) @ ( add_ms1730626103851491130ropVar @ C3 @ zero_z6226653793019358809ropVar ) ) ) ) ) ).

% multiset_add_sub_el_shuffle
thf(fact_1080_add__mset__remove__trivial__eq,axiom,
    ! [N4: multiset_nat,A: nat] :
      ( ( N4
        = ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) )
      = ( member_nat @ A @ ( set_mset_nat @ N4 ) ) ) ).

% add_mset_remove_trivial_eq
thf(fact_1081_add__mset__remove__trivial__eq,axiom,
    ! [N4: multis193838646080965794ropVar,A: multis3193088007478089820ropVar] :
      ( ( N4
        = ( add_ms1730626103851491130ropVar @ A @ ( minus_287902380581138057ropVar @ N4 @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) ) ) )
      = ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ N4 ) ) ) ).

% add_mset_remove_trivial_eq
thf(fact_1082_add__mset__remove__trivial__If,axiom,
    ! [A: nat,N4: multiset_nat] :
      ( ( ( member_nat @ A @ ( set_mset_nat @ N4 ) )
       => ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
          = N4 ) )
      & ( ~ ( member_nat @ A @ ( set_mset_nat @ N4 ) )
       => ( ( add_mset_nat @ A @ ( minus_8522176038001411705et_nat @ N4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) )
          = ( add_mset_nat @ A @ N4 ) ) ) ) ).

% add_mset_remove_trivial_If
thf(fact_1083_add__mset__remove__trivial__If,axiom,
    ! [A: multis3193088007478089820ropVar,N4: multis193838646080965794ropVar] :
      ( ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ N4 ) )
       => ( ( add_ms1730626103851491130ropVar @ A @ ( minus_287902380581138057ropVar @ N4 @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) ) )
          = N4 ) )
      & ( ~ ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ N4 ) )
       => ( ( add_ms1730626103851491130ropVar @ A @ ( minus_287902380581138057ropVar @ N4 @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) ) )
          = ( add_ms1730626103851491130ropVar @ A @ N4 ) ) ) ) ).

% add_mset_remove_trivial_If
thf(fact_1084_multi__drop__mem__not__eq,axiom,
    ! [C3: nat,B6: multiset_nat] :
      ( ( member_nat @ C3 @ ( set_mset_nat @ B6 ) )
     => ( ( minus_8522176038001411705et_nat @ B6 @ ( add_mset_nat @ C3 @ zero_z7348594199698428585et_nat ) )
       != B6 ) ) ).

% multi_drop_mem_not_eq
thf(fact_1085_multi__drop__mem__not__eq,axiom,
    ! [C3: multis3193088007478089820ropVar,B6: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ C3 @ ( set_ms1147286257884310937ropVar @ B6 ) )
     => ( ( minus_287902380581138057ropVar @ B6 @ ( add_ms1730626103851491130ropVar @ C3 @ zero_z6226653793019358809ropVar ) )
       != B6 ) ) ).

% multi_drop_mem_not_eq
thf(fact_1086_diff__single__eq__union,axiom,
    ! [X5: nat,M3: multiset_nat,N4: multiset_nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ M3 ) )
     => ( ( ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) )
          = N4 )
        = ( M3
          = ( add_mset_nat @ X5 @ N4 ) ) ) ) ).

% diff_single_eq_union
thf(fact_1087_diff__single__eq__union,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar,N4: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ( ( ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) )
          = N4 )
        = ( M3
          = ( add_ms1730626103851491130ropVar @ X5 @ N4 ) ) ) ) ).

% diff_single_eq_union
thf(fact_1088_diff__single__trivial,axiom,
    ! [X5: nat,M3: multiset_nat] :
      ( ~ ( member_nat @ X5 @ ( set_mset_nat @ M3 ) )
     => ( ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) )
        = M3 ) ) ).

% diff_single_trivial
thf(fact_1089_diff__single__trivial,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ( ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) )
        = M3 ) ) ).

% diff_single_trivial
thf(fact_1090_multpw__split1R,axiom,
    ! [X5: multis3193088007478089820ropVar,X7: multis193838646080965794ropVar,Y6: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( add_ms1730626103851491130ropVar @ X5 @ X7 ) @ Y6 ) @ ( multis9091339148773653984ropVar @ Ns ) )
     => ~ ! [Z2: multis3193088007478089820ropVar,Z5: multis193838646080965794ropVar] :
            ( ( Y6
              = ( add_ms1730626103851491130ropVar @ Z2 @ Z5 ) )
           => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Z2 ) @ Ns )
             => ~ ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ X7 @ Z5 ) @ ( multis9091339148773653984ropVar @ Ns ) ) ) ) ) ).

% multpw_split1R
thf(fact_1091_multpw__split1L,axiom,
    ! [X7: multis193838646080965794ropVar,Y3: multis3193088007478089820ropVar,Y6: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ X7 @ ( add_ms1730626103851491130ropVar @ Y3 @ Y6 ) ) @ ( multis9091339148773653984ropVar @ Ns ) )
     => ~ ! [Z2: multis3193088007478089820ropVar,Z5: multis193838646080965794ropVar] :
            ( ( X7
              = ( add_ms1730626103851491130ropVar @ Z2 @ Z5 ) )
           => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Z2 @ Y3 ) @ Ns )
             => ~ ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ Z5 @ Y6 ) @ ( multis9091339148773653984ropVar @ Ns ) ) ) ) ) ).

% multpw_split1L
thf(fact_1092_multpw_Oadd,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,Ns: set_Pr7799659934693882759ropVar,X7: multis193838646080965794ropVar,Y6: multis193838646080965794ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ Ns )
     => ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ X7 @ Y6 ) @ ( multis9091339148773653984ropVar @ Ns ) )
       => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( add_ms1730626103851491130ropVar @ X5 @ X7 ) @ ( add_ms1730626103851491130ropVar @ Y3 @ Y6 ) ) @ ( multis9091339148773653984ropVar @ Ns ) ) ) ) ).

% multpw.add
thf(fact_1093_same__length__different,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( Xs != Ys )
     => ( ( ( size_s6752116297430514276ropVar @ Xs )
          = ( size_s6752116297430514276ropVar @ Ys ) )
       => ? [Pre: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar,Xs5: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys4: list_P8791008543742759928ropVar] :
            ( ( X4 != Y4 )
            & ( Xs
              = ( append4352292717583037453ropVar @ Pre @ ( append4352292717583037453ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) @ Xs5 ) ) )
            & ( Ys
              = ( append4352292717583037453ropVar @ Pre @ ( append4352292717583037453ropVar @ ( cons_P1504208554336013608ropVar @ Y4 @ nil_Pr6119456312066593144ropVar ) @ Ys4 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1094_same__length__different,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
       => ? [Pre: list_nat,X4: nat,Xs5: list_nat,Y4: nat,Ys4: list_nat] :
            ( ( X4 != Y4 )
            & ( Xs
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ X4 @ nil_nat ) @ Xs5 ) ) )
            & ( Ys
              = ( append_nat @ Pre @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Ys4 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1095_not__distinct__decomp,axiom,
    ! [Ws: list_P8791008543742759928ropVar] :
      ( ~ ( distin2267423591311547883ropVar @ Ws )
     => ? [Xs3: list_P8791008543742759928ropVar,Ys5: list_P8791008543742759928ropVar,Zs4: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar] :
          ( Ws
          = ( append4352292717583037453ropVar @ Xs3 @ ( append4352292717583037453ropVar @ ( cons_P1504208554336013608ropVar @ Y4 @ nil_Pr6119456312066593144ropVar ) @ ( append4352292717583037453ropVar @ Ys5 @ ( append4352292717583037453ropVar @ ( cons_P1504208554336013608ropVar @ Y4 @ nil_Pr6119456312066593144ropVar ) @ Zs4 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_1096_not__distinct__decomp,axiom,
    ! [Ws: list_nat] :
      ( ~ ( distinct_nat @ Ws )
     => ? [Xs3: list_nat,Ys5: list_nat,Zs4: list_nat,Y4: nat] :
          ( Ws
          = ( append_nat @ Xs3 @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ ( append_nat @ Ys5 @ ( append_nat @ ( cons_nat @ Y4 @ nil_nat ) @ Zs4 ) ) ) ) ) ) ).

% not_distinct_decomp
thf(fact_1097_not__distinct__conv__prefix,axiom,
    ! [As: list_M6740372356978808802ropVar] :
      ( ( ~ ( distin2675581014522817237ropVar @ As ) )
      = ( ? [Xs2: list_M6740372356978808802ropVar,Y2: multis3193088007478089820ropVar,Ys2: list_M6740372356978808802ropVar] :
            ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
            & ( distin2675581014522817237ropVar @ Xs2 )
            & ( As
              = ( append2684835082924936439ropVar @ Xs2 @ ( cons_M4277272395645347858ropVar @ Y2 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1098_not__distinct__conv__prefix,axiom,
    ! [As: list_P8791008543742759928ropVar] :
      ( ( ~ ( distin2267423591311547883ropVar @ As ) )
      = ( ? [Xs2: list_P8791008543742759928ropVar,Y2: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar] :
            ( ( member9210975342120487067ropVar @ Y2 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
            & ( distin2267423591311547883ropVar @ Xs2 )
            & ( As
              = ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ Y2 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1099_not__distinct__conv__prefix,axiom,
    ! [As: list_nat] :
      ( ( ~ ( distinct_nat @ As ) )
      = ( ? [Xs2: list_nat,Y2: nat,Ys2: list_nat] :
            ( ( member_nat @ Y2 @ ( set_nat2 @ Xs2 ) )
            & ( distinct_nat @ Xs2 )
            & ( As
              = ( append_nat @ Xs2 @ ( cons_nat @ Y2 @ Ys2 ) ) ) ) ) ) ).

% not_distinct_conv_prefix
thf(fact_1100_Cons__listrel1E2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre6655839188755619153ropVar @ R ) )
     => ( ! [X4: produc2642885279816024434ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ X4 @ Ys ) )
           => ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ R ) )
       => ~ ! [Zs4: list_P8791008543742759928ropVar] :
              ( ( Xs
                = ( cons_P1504208554336013608ropVar @ Y3 @ Zs4 ) )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Zs4 @ Ys ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_1101_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R ) )
     => ( ! [X4: nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R ) )
       => ~ ! [Zs4: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y3 @ Zs4 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs4 @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_1102_Cons__listrel1E2,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) ) @ ( listre3736066117146447419ropVar @ R ) )
     => ( ! [X4: multis3193088007478089820ropVar] :
            ( ( Xs
              = ( cons_M4277272395645347858ropVar @ X4 @ Ys ) )
           => ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R ) )
       => ~ ! [Zs4: list_M6740372356978808802ropVar] :
              ( ( Xs
                = ( cons_M4277272395645347858ropVar @ Y3 @ Zs4 ) )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Zs4 @ Ys ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_1103_Cons__listrel1E1,axiom,
    ! [X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ Ys ) @ ( listre6655839188755619153ropVar @ R ) )
     => ( ! [Y4: produc2642885279816024434ropVar] :
            ( ( Ys
              = ( cons_P1504208554336013608ropVar @ Y4 @ Xs ) )
           => ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y4 ) @ R ) )
       => ~ ! [Zs4: list_P8791008543742759928ropVar] :
              ( ( Ys
                = ( cons_P1504208554336013608ropVar @ X5 @ Zs4 ) )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Zs4 ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_1104_Cons__listrel1E1,axiom,
    ! [X5: nat,Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs ) @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ! [Y4: nat] :
            ( ( Ys
              = ( cons_nat @ Y4 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y4 ) @ R ) )
       => ~ ! [Zs4: list_nat] :
              ( ( Ys
                = ( cons_nat @ X5 @ Zs4 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs4 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_1105_Cons__listrel1E1,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ Ys ) @ ( listre3736066117146447419ropVar @ R ) )
     => ( ! [Y4: multis3193088007478089820ropVar] :
            ( ( Ys
              = ( cons_M4277272395645347858ropVar @ Y4 @ Xs ) )
           => ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y4 ) @ R ) )
       => ~ ! [Zs4: list_M6740372356978808802ropVar] :
              ( ( Ys
                = ( cons_M4277272395645347858ropVar @ X5 @ Zs4 ) )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Zs4 ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_1106_listrel1I1,axiom,
    ! [X5: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y3 ) @ R )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Xs ) ) @ ( listre6655839188755619153ropVar @ R ) ) ) ).

% listrel1I1
thf(fact_1107_listrel1I1,axiom,
    ! [X5: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Xs ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_1108_listrel1I1,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ R )
     => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Xs ) ) @ ( listre3736066117146447419ropVar @ R ) ) ) ).

% listrel1I1
thf(fact_1109_listrel__Cons2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre2294694963173455985ropVar @ R ) )
     => ~ ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) )
           => ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ R )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs3 @ Ys ) @ ( listre2294694963173455985ropVar @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1110_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre24003740514569327ropVar @ R ) )
     => ~ ! [X4: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs3 ) )
           => ( ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ X4 @ Y3 ) @ R )
             => ~ ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs3 @ Ys ) @ ( listre24003740514569327ropVar @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1111_listrel__Cons2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Y3: nat,Ys: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ~ ! [X4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) )
           => ( ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X4 @ Y3 ) @ R )
             => ~ ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs3 @ Ys ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1112_listrel__Cons2,axiom,
    ! [Xs: list_nat,Y3: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [X4: nat,Xs3: list_nat] :
            ( ( Xs
              = ( cons_nat @ X4 @ Xs3 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1113_listrel__Cons2,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Y3: produc7679832412844898215ropVar,Ys: list_P444016040222843053ropVar,R: set_Pr6405597471078531282ropVar] :
      ( ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Xs @ ( cons_P6664127749089838429ropVar @ Y3 @ Ys ) ) @ ( listre2663080307769418044ropVar @ R ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Xs3: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( cons_M4277272395645347858ropVar @ X4 @ Xs3 ) )
           => ( ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X4 @ Y3 ) @ R )
             => ~ ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Xs3 @ Ys ) @ ( listre2663080307769418044ropVar @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1114_listrel__Cons2,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) ) @ ( listre4495854419763884273ropVar @ R ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Xs3: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( cons_M4277272395645347858ropVar @ X4 @ Xs3 ) )
           => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs3 @ Ys ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1115_listrel__Cons1,axiom,
    ! [Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) @ Xs ) @ ( listre2294694963173455985ropVar @ R ) )
     => ~ ! [Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
           => ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ Y3 @ Y4 ) @ R )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Ys @ Ys5 ) @ ( listre2294694963173455985ropVar @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1116_listrel__Cons1,axiom,
    ! [Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Xs: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) @ Xs ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ~ ! [Y4: nat,Ys5: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys5 ) )
           => ( ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ Y3 @ Y4 ) @ R )
             => ~ ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Ys @ Ys5 ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1117_listrel__Cons1,axiom,
    ! [Y3: nat,Ys: list_nat,Xs: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listre24003740514569327ropVar @ R ) )
     => ~ ! [Y4: produc2642885279816024434ropVar,Ys5: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
           => ( ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ Y3 @ Y4 ) @ R )
             => ~ ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Ys @ Ys5 ) @ ( listre24003740514569327ropVar @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1118_listrel__Cons1,axiom,
    ! [Y3: nat,Ys: list_nat,Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y3 @ Ys ) @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [Y4: nat,Ys5: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y4 @ Ys5 ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Y4 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Ys5 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1119_listrel__Cons1,axiom,
    ! [Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,Xs: list_P444016040222843053ropVar,R: set_Pr6405597471078531282ropVar] :
      ( ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) @ Xs ) @ ( listre2663080307769418044ropVar @ R ) )
     => ~ ! [Y4: produc7679832412844898215ropVar,Ys5: list_P444016040222843053ropVar] :
            ( ( Xs
              = ( cons_P6664127749089838429ropVar @ Y4 @ Ys5 ) )
           => ( ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ Y3 @ Y4 ) @ R )
             => ~ ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Ys @ Ys5 ) @ ( listre2663080307769418044ropVar @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1120_listrel__Cons1,axiom,
    ! [Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,Xs: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) @ Xs ) @ ( listre4495854419763884273ropVar @ R ) )
     => ~ ! [Y4: multis3193088007478089820ropVar,Ys5: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( cons_M4277272395645347858ropVar @ Y4 @ Ys5 ) )
           => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Y3 @ Y4 ) @ R )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Ys @ Ys5 ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1121_listrel_OCons,axiom,
    ! [X5: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y3 ) @ R )
     => ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre2294694963173455985ropVar @ R ) )
       => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre2294694963173455985ropVar @ R ) ) ) ) ).

% listrel.Cons
thf(fact_1122_listrel_OCons,axiom,
    ! [X5: produc2642885279816024434ropVar,Y3: nat,R: set_Pr7538684846379304231ar_nat,Xs: list_P8791008543742759928ropVar,Ys: list_nat] :
      ( ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X5 @ Y3 ) @ R )
     => ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs @ Ys ) @ ( listre5688682141216627473ar_nat @ R ) )
       => ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_1123_listrel_OCons,axiom,
    ! [X5: nat,Y3: produc2642885279816024434ropVar,R: set_Pr7708077615150402893ropVar,Xs: list_nat,Ys: list_P8791008543742759928ropVar] :
      ( ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ X5 @ Y3 ) @ R )
     => ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs @ Ys ) @ ( listre24003740514569327ropVar @ R ) )
       => ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ ( cons_nat @ X5 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre24003740514569327ropVar @ R ) ) ) ) ).

% listrel.Cons
thf(fact_1124_listrel_OCons,axiom,
    ! [X5: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_1125_listrel_OCons,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: produc7679832412844898215ropVar,R: set_Pr6405597471078531282ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X5 @ Y3 ) @ R )
     => ( ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Xs @ Ys ) @ ( listre2663080307769418044ropVar @ R ) )
       => ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ ( cons_P6664127749089838429ropVar @ Y3 @ Ys ) ) @ ( listre2663080307769418044ropVar @ R ) ) ) ) ).

% listrel.Cons
thf(fact_1126_listrel_OCons,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ R )
     => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre4495854419763884273ropVar @ R ) )
       => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ).

% listrel.Cons
thf(fact_1127_multpw__single,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ Ns )
     => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) @ ( add_ms1730626103851491130ropVar @ Y3 @ zero_z6226653793019358809ropVar ) ) @ ( multis9091339148773653984ropVar @ Ns ) ) ) ).

% multpw_single
thf(fact_1128_multpw_Osimps,axiom,
    ! [A14: multis193838646080965794ropVar,A25: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A14 @ A25 ) @ ( multis9091339148773653984ropVar @ Ns ) )
      = ( ( ( A14 = zero_z6226653793019358809ropVar )
          & ( A25 = zero_z6226653793019358809ropVar ) )
        | ? [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,X3: multis193838646080965794ropVar,Y8: multis193838646080965794ropVar] :
            ( ( A14
              = ( add_ms1730626103851491130ropVar @ X @ X3 ) )
            & ( A25
              = ( add_ms1730626103851491130ropVar @ Y2 @ Y8 ) )
            & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y2 ) @ Ns )
            & ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ X3 @ Y8 ) @ ( multis9091339148773653984ropVar @ Ns ) ) ) ) ) ).

% multpw.simps
thf(fact_1129_multpw_Ocases,axiom,
    ! [A14: multis193838646080965794ropVar,A25: multis193838646080965794ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ A14 @ A25 ) @ ( multis9091339148773653984ropVar @ Ns ) )
     => ( ( ( A14 = zero_z6226653793019358809ropVar )
         => ( A25 != zero_z6226653793019358809ropVar ) )
       => ~ ! [X4: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar,X10: multis193838646080965794ropVar] :
              ( ( A14
                = ( add_ms1730626103851491130ropVar @ X4 @ X10 ) )
             => ! [Y9: multis193838646080965794ropVar] :
                  ( ( A25
                    = ( add_ms1730626103851491130ropVar @ Y4 @ Y9 ) )
                 => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y4 ) @ Ns )
                   => ~ ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ X10 @ Y9 ) @ ( multis9091339148773653984ropVar @ Ns ) ) ) ) ) ) ) ).

% multpw.cases
thf(fact_1130_insert__DiffM2,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ( ( plus_p1339091108561003481ropVar @ ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) ) @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) )
        = M3 ) ) ).

% insert_DiffM2
thf(fact_1131_insert__DiffM2,axiom,
    ! [X5: nat,M3: multiset_nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ M3 ) )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) )
        = M3 ) ) ).

% insert_DiffM2
thf(fact_1132_diff__union__single__conv,axiom,
    ! [A: multis3193088007478089820ropVar,J4: multis193838646080965794ropVar,I5: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ J4 ) )
     => ( ( minus_287902380581138057ropVar @ ( plus_p1339091108561003481ropVar @ I5 @ J4 ) @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) )
        = ( plus_p1339091108561003481ropVar @ I5 @ ( minus_287902380581138057ropVar @ J4 @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) ) ) ) ) ).

% diff_union_single_conv
thf(fact_1133_diff__union__single__conv,axiom,
    ! [A: nat,J4: multiset_nat,I5: multiset_nat] :
      ( ( member_nat @ A @ ( set_mset_nat @ J4 ) )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ I5 @ J4 ) @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) )
        = ( plus_p6334493942879108393et_nat @ I5 @ ( minus_8522176038001411705et_nat @ J4 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% diff_union_single_conv
thf(fact_1134_ns__mul__ext__singleton2,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,S: set_Pr7799659934693882759ropVar,Ns: set_Pr7799659934693882759ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ B ) @ S )
     => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) @ ( add_ms1730626103851491130ropVar @ B @ zero_z6226653793019358809ropVar ) ) @ ( multis738276037714153361ropVar @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton2
thf(fact_1135_listrel_Osimps,axiom,
    ! [A14: list_P8791008543742759928ropVar,A25: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ A14 @ A25 ) @ ( listre2294694963173455985ropVar @ R ) )
      = ( ( ( A14 = nil_Pr6119456312066593144ropVar )
          & ( A25 = nil_Pr6119456312066593144ropVar ) )
        | ? [X: produc2642885279816024434ropVar,Y2: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
            ( ( A14
              = ( cons_P1504208554336013608ropVar @ X @ Xs2 ) )
            & ( A25
              = ( cons_P1504208554336013608ropVar @ Y2 @ Ys2 ) )
            & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X @ Y2 ) @ R )
            & ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( listre2294694963173455985ropVar @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_1136_listrel_Osimps,axiom,
    ! [A14: list_P8791008543742759928ropVar,A25: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ A14 @ A25 ) @ ( listre5688682141216627473ar_nat @ R ) )
      = ( ( ( A14 = nil_Pr6119456312066593144ropVar )
          & ( A25 = nil_nat ) )
        | ? [X: produc2642885279816024434ropVar,Y2: nat,Xs2: list_P8791008543742759928ropVar,Ys2: list_nat] :
            ( ( A14
              = ( cons_P1504208554336013608ropVar @ X @ Xs2 ) )
            & ( A25
              = ( cons_nat @ Y2 @ Ys2 ) )
            & ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X @ Y2 ) @ R )
            & ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs2 @ Ys2 ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_1137_listrel_Osimps,axiom,
    ! [A14: list_nat,A25: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ A14 @ A25 ) @ ( listre24003740514569327ropVar @ R ) )
      = ( ( ( A14 = nil_nat )
          & ( A25 = nil_Pr6119456312066593144ropVar ) )
        | ? [X: nat,Y2: produc2642885279816024434ropVar,Xs2: list_nat,Ys2: list_P8791008543742759928ropVar] :
            ( ( A14
              = ( cons_nat @ X @ Xs2 ) )
            & ( A25
              = ( cons_P1504208554336013608ropVar @ Y2 @ Ys2 ) )
            & ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ X @ Y2 ) @ R )
            & ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs2 @ Ys2 ) @ ( listre24003740514569327ropVar @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_1138_listrel_Osimps,axiom,
    ! [A14: list_nat,A25: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A14 @ A25 ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( A14 = nil_nat )
          & ( A25 = nil_nat ) )
        | ? [X: nat,Y2: nat,Xs2: list_nat,Ys2: list_nat] :
            ( ( A14
              = ( cons_nat @ X @ Xs2 ) )
            & ( A25
              = ( cons_nat @ Y2 @ Ys2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y2 ) @ R )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_1139_listrel_Osimps,axiom,
    ! [A14: list_M6740372356978808802ropVar,A25: list_P444016040222843053ropVar,R: set_Pr6405597471078531282ropVar] :
      ( ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ A14 @ A25 ) @ ( listre2663080307769418044ropVar @ R ) )
      = ( ( ( A14 = nil_Mu4132399263956683874ropVar )
          & ( A25 = nil_Pr4408211540142832045ropVar ) )
        | ? [X: multis3193088007478089820ropVar,Y2: produc7679832412844898215ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_P444016040222843053ropVar] :
            ( ( A14
              = ( cons_M4277272395645347858ropVar @ X @ Xs2 ) )
            & ( A25
              = ( cons_P6664127749089838429ropVar @ Y2 @ Ys2 ) )
            & ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X @ Y2 ) @ R )
            & ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Xs2 @ Ys2 ) @ ( listre2663080307769418044ropVar @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_1140_listrel_Osimps,axiom,
    ! [A14: list_M6740372356978808802ropVar,A25: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ A14 @ A25 ) @ ( listre4495854419763884273ropVar @ R ) )
      = ( ( ( A14 = nil_Mu4132399263956683874ropVar )
          & ( A25 = nil_Mu4132399263956683874ropVar ) )
        | ? [X: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
            ( ( A14
              = ( cons_M4277272395645347858ropVar @ X @ Xs2 ) )
            & ( A25
              = ( cons_M4277272395645347858ropVar @ Y2 @ Ys2 ) )
            & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y2 ) @ R )
            & ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_1141_listrel_Ocases,axiom,
    ! [A14: list_P8791008543742759928ropVar,A25: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ A14 @ A25 ) @ ( listre2294694963173455985ropVar @ R ) )
     => ( ( ( A14 = nil_Pr6119456312066593144ropVar )
         => ( A25 != nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X4: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] :
              ( ( A14
                = ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) )
             => ! [Ys5: list_P8791008543742759928ropVar] :
                  ( ( A25
                    = ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
                 => ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y4 ) @ R )
                   => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs3 @ Ys5 ) @ ( listre2294694963173455985ropVar @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1142_listrel_Ocases,axiom,
    ! [A14: list_P8791008543742759928ropVar,A25: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ A14 @ A25 ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ( ( ( A14 = nil_Pr6119456312066593144ropVar )
         => ( A25 != nil_nat ) )
       => ~ ! [X4: produc2642885279816024434ropVar,Y4: nat,Xs3: list_P8791008543742759928ropVar] :
              ( ( A14
                = ( cons_P1504208554336013608ropVar @ X4 @ Xs3 ) )
             => ! [Ys5: list_nat] :
                  ( ( A25
                    = ( cons_nat @ Y4 @ Ys5 ) )
                 => ( ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X4 @ Y4 ) @ R )
                   => ~ ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs3 @ Ys5 ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1143_listrel_Ocases,axiom,
    ! [A14: list_nat,A25: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ A14 @ A25 ) @ ( listre24003740514569327ropVar @ R ) )
     => ( ( ( A14 = nil_nat )
         => ( A25 != nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X4: nat,Y4: produc2642885279816024434ropVar,Xs3: list_nat] :
              ( ( A14
                = ( cons_nat @ X4 @ Xs3 ) )
             => ! [Ys5: list_P8791008543742759928ropVar] :
                  ( ( A25
                    = ( cons_P1504208554336013608ropVar @ Y4 @ Ys5 ) )
                 => ( ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ X4 @ Y4 ) @ R )
                   => ~ ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs3 @ Ys5 ) @ ( listre24003740514569327ropVar @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1144_listrel_Ocases,axiom,
    ! [A14: list_nat,A25: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A14 @ A25 ) @ ( listrel_nat_nat @ R ) )
     => ( ( ( A14 = nil_nat )
         => ( A25 != nil_nat ) )
       => ~ ! [X4: nat,Y4: nat,Xs3: list_nat] :
              ( ( A14
                = ( cons_nat @ X4 @ Xs3 ) )
             => ! [Ys5: list_nat] :
                  ( ( A25
                    = ( cons_nat @ Y4 @ Ys5 ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys5 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1145_listrel_Ocases,axiom,
    ! [A14: list_M6740372356978808802ropVar,A25: list_P444016040222843053ropVar,R: set_Pr6405597471078531282ropVar] :
      ( ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ A14 @ A25 ) @ ( listre2663080307769418044ropVar @ R ) )
     => ( ( ( A14 = nil_Mu4132399263956683874ropVar )
         => ( A25 != nil_Pr4408211540142832045ropVar ) )
       => ~ ! [X4: multis3193088007478089820ropVar,Y4: produc7679832412844898215ropVar,Xs3: list_M6740372356978808802ropVar] :
              ( ( A14
                = ( cons_M4277272395645347858ropVar @ X4 @ Xs3 ) )
             => ! [Ys5: list_P444016040222843053ropVar] :
                  ( ( A25
                    = ( cons_P6664127749089838429ropVar @ Y4 @ Ys5 ) )
                 => ( ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X4 @ Y4 ) @ R )
                   => ~ ( member7926529425716556827ropVar @ ( produc4116939308056247650ropVar @ Xs3 @ Ys5 ) @ ( listre2663080307769418044ropVar @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1146_listrel_Ocases,axiom,
    ! [A14: list_M6740372356978808802ropVar,A25: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ A14 @ A25 ) @ ( listre4495854419763884273ropVar @ R ) )
     => ( ( ( A14 = nil_Mu4132399263956683874ropVar )
         => ( A25 != nil_Mu4132399263956683874ropVar ) )
       => ~ ! [X4: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar,Xs3: list_M6740372356978808802ropVar] :
              ( ( A14
                = ( cons_M4277272395645347858ropVar @ X4 @ Xs3 ) )
             => ! [Ys5: list_M6740372356978808802ropVar] :
                  ( ( A25
                    = ( cons_M4277272395645347858ropVar @ Y4 @ Ys5 ) )
                 => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y4 ) @ R )
                   => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs3 @ Ys5 ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1147_listrel1E,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R ) )
     => ~ ! [X4: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar] :
            ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y4 ) @ R )
           => ! [Us3: list_P8791008543742759928ropVar,Vs3: list_P8791008543742759928ropVar] :
                ( ( Xs
                  = ( append4352292717583037453ropVar @ Us3 @ ( cons_P1504208554336013608ropVar @ X4 @ Vs3 ) ) )
               => ( Ys
                 != ( append4352292717583037453ropVar @ Us3 @ ( cons_P1504208554336013608ropVar @ Y4 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1148_listrel1E,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ~ ! [X4: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R )
           => ! [Us3: list_nat,Vs3: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us3 @ ( cons_nat @ X4 @ Vs3 ) ) )
               => ( Ys
                 != ( append_nat @ Us3 @ ( cons_nat @ Y4 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1149_listrel1E,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R ) )
     => ~ ! [X4: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y4 ) @ R )
           => ! [Us3: list_M6740372356978808802ropVar,Vs3: list_M6740372356978808802ropVar] :
                ( ( Xs
                  = ( append2684835082924936439ropVar @ Us3 @ ( cons_M4277272395645347858ropVar @ X4 @ Vs3 ) ) )
               => ( Ys
                 != ( append2684835082924936439ropVar @ Us3 @ ( cons_M4277272395645347858ropVar @ Y4 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1150_listrel1I,axiom,
    ! [X5: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar,Xs: list_P8791008543742759928ropVar,Us: list_P8791008543742759928ropVar,Vs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y3 ) @ R )
     => ( ( Xs
          = ( append4352292717583037453ropVar @ Us @ ( cons_P1504208554336013608ropVar @ X5 @ Vs ) ) )
       => ( ( Ys
            = ( append4352292717583037453ropVar @ Us @ ( cons_P1504208554336013608ropVar @ Y3 @ Vs ) ) )
         => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% listrel1I
thf(fact_1151_listrel1I,axiom,
    ! [X5: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R )
     => ( ( Xs
          = ( append_nat @ Us @ ( cons_nat @ X5 @ Vs ) ) )
       => ( ( Ys
            = ( append_nat @ Us @ ( cons_nat @ Y3 @ Vs ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_1152_listrel1I,axiom,
    ! [X5: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar,Xs: list_M6740372356978808802ropVar,Us: list_M6740372356978808802ropVar,Vs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ R )
     => ( ( Xs
          = ( append2684835082924936439ropVar @ Us @ ( cons_M4277272395645347858ropVar @ X5 @ Vs ) ) )
       => ( ( Ys
            = ( append2684835082924936439ropVar @ Us @ ( cons_M4277272395645347858ropVar @ Y3 @ Vs ) ) )
         => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% listrel1I
thf(fact_1153_size__Diff1__less,axiom,
    ! [X5: nat,M3: multiset_nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ M3 ) )
     => ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M3 ) ) ) ).

% size_Diff1_less
thf(fact_1154_size__Diff1__less,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ( ord_less_nat @ ( size_s836052043357872910ropVar @ ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) ) ) @ ( size_s836052043357872910ropVar @ M3 ) ) ) ).

% size_Diff1_less
thf(fact_1155_size__Diff2__less,axiom,
    ! [X5: nat,M3: multiset_nat,Y3: nat] :
      ( ( member_nat @ X5 @ ( set_mset_nat @ M3 ) )
     => ( ( member_nat @ Y3 @ ( set_mset_nat @ M3 ) )
       => ( ord_less_nat @ ( size_s5917832649809541300et_nat @ ( minus_8522176038001411705et_nat @ ( minus_8522176038001411705et_nat @ M3 @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) ) @ ( add_mset_nat @ Y3 @ zero_z7348594199698428585et_nat ) ) ) @ ( size_s5917832649809541300et_nat @ M3 ) ) ) ) ).

% size_Diff2_less
thf(fact_1156_size__Diff2__less,axiom,
    ! [X5: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ M3 ) )
     => ( ( member8421201611525043589ropVar @ Y3 @ ( set_ms1147286257884310937ropVar @ M3 ) )
       => ( ord_less_nat @ ( size_s836052043357872910ropVar @ ( minus_287902380581138057ropVar @ ( minus_287902380581138057ropVar @ M3 @ ( add_ms1730626103851491130ropVar @ X5 @ zero_z6226653793019358809ropVar ) ) @ ( add_ms1730626103851491130ropVar @ Y3 @ zero_z6226653793019358809ropVar ) ) ) @ ( size_s836052043357872910ropVar @ M3 ) ) ) ) ).

% size_Diff2_less
thf(fact_1157_product_Osimps_I2_J,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( produc4762506738712241808ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ Ys )
      = ( append4352292717583037453ropVar @ ( map_Pr6673589034486973647ropVar @ ( produc6194868611276206818ropVar @ X5 ) @ Ys ) @ ( produc4762506738712241808ropVar @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_1158_product_Osimps_I2_J,axiom,
    ! [X5: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( produc3968825954260087365ropVar @ ( cons_M4277272395645347858ropVar @ X5 @ Xs ) @ Ys )
      = ( append5590948776005684034ropVar @ ( map_Mu7320851716970821775ropVar @ ( produc6936306168532358295ropVar @ X5 ) @ Ys ) @ ( produc3968825954260087365ropVar @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_1159_ms__encoder_Oformula16__def,axiom,
    ! [Enc_False: produc2642885279816024434ropVar,Enc_True: produc2642885279816024434ropVar,Enc_pos: multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_neg: multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_impl: multis3193088007478089820ropVar > produc2642885279816024434ropVar > produc2642885279816024434ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_not_all: list_M6740372356978808802ropVar > produc2642885279816024434ropVar,Enc_conj: list_P8791008543742759928ropVar > produc2642885279816024434ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > produc2642885279816024434ropVar > $o,Cs: nat > nat > $o,Cns: nat > nat > $o,N: nat,M2: nat] :
      ( ( multis8181102237825478411ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ( multis7741638927476898652ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 )
        = ( cons_P1504208554336013608ropVar @ ( Enc_not_all @ ( map_na2466558649495876678ropVar @ multis2544335231667181926psilon @ ( upt @ zero_zero_nat @ N ) ) ) @ ( multis5595896195939027229ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) ) ) ).

% ms_encoder.formula16_def
thf(fact_1160_ms__encoder_Oformula16__def,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,Cns: nat > nat > $o,N: nat,M2: nat] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ( multis103654530569517143016_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Cs @ Cns @ N @ M2 )
        = ( cons_nat @ ( Enc_not_all @ ( map_na2466558649495876678ropVar @ multis2544335231667181926psilon @ ( upt @ zero_zero_nat @ N ) ) ) @ ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 ) ) ) ) ).

% ms_encoder.formula16_def
thf(fact_1161_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X5: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X5 @ nil_Pr6119456312066593144ropVar ) ) @ ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ Y3 @ nil_Pr6119456312066593144ropVar ) ) ) @ ( listre6655839188755619153ropVar @ R ) )
      = ( ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R ) )
          & ( X5 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X5 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_1162_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X5: nat,Ys: list_nat,Y3: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X5 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( X5 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_1163_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,X5: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ Xs @ ( cons_M4277272395645347858ropVar @ X5 @ nil_Mu4132399263956683874ropVar ) ) @ ( append2684835082924936439ropVar @ Ys @ ( cons_M4277272395645347858ropVar @ Y3 @ nil_Mu4132399263956683874ropVar ) ) ) @ ( listre3736066117146447419ropVar @ R ) )
      = ( ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R ) )
          & ( X5 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X5 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_1164_ms__encoder_Oformula14__def,axiom,
    ! [Enc_False: produc2642885279816024434ropVar,Enc_True: produc2642885279816024434ropVar,Enc_pos: multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_neg: multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_impl: multis3193088007478089820ropVar > produc2642885279816024434ropVar > produc2642885279816024434ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_not_all: list_M6740372356978808802ropVar > produc2642885279816024434ropVar,Enc_conj: list_P8791008543742759928ropVar > produc2642885279816024434ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > produc2642885279816024434ropVar > $o,N: nat,M2: nat] :
      ( ( multis8181102237825478411ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ( multis3450153464401155806ropVar @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ N @ M2 )
        = ( append4352292717583037453ropVar
          @ ( maps_n6440817060062774047ropVar
            @ ^ [J: nat] :
                ( produc5464477786531444550ropVar @ cons_P1504208554336013608ropVar
                @ ( multis8736296938027350326ropVar @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite
                  @ ( map_na2145702811007361372ropVar
                    @ ^ [I: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ I @ J ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385521703ZeroJI @ I @ J ) @ ( multis6646701651571564453xOneJI @ I @ J ) ) )
                    @ ( upt @ zero_zero_nat @ N ) ) ) )
            @ ( upt @ zero_zero_nat @ M2 ) )
          @ ( maps_n6440817060062774047ropVar
            @ ^ [I: nat] :
                ( produc5464477786531444550ropVar
                @ ^ [One2: produc2642885279816024434ropVar] : ( cons_P1504208554336013608ropVar @ ( Enc_impl @ ( multis2544335231667181926psilon @ I ) @ One2 ) )
                @ ( multis8736296938027350326ropVar @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite
                  @ ( map_na2145702811007361372ropVar
                    @ ^ [J: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ I @ J ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385456105ZeroIJ @ I @ J ) @ ( multis6646701651571498855xOneIJ @ I @ J ) ) )
                    @ ( upt @ zero_zero_nat @ M2 ) ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% ms_encoder.formula14_def
thf(fact_1165_ms__encoder_Oformula14__def,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,N: nat,M2: nat] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ( multis778902868463715824414_nat @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ N @ M2 )
        = ( append_nat
          @ ( maps_nat_nat
            @ ^ [J: nat] :
                ( produc6860903796091618494st_nat @ cons_nat
                @ ( multis5159292385415766206ropVar @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite
                  @ ( map_na2145702811007361372ropVar
                    @ ^ [I: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ I @ J ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385521703ZeroJI @ I @ J ) @ ( multis6646701651571564453xOneJI @ I @ J ) ) )
                    @ ( upt @ zero_zero_nat @ N ) ) ) )
            @ ( upt @ zero_zero_nat @ M2 ) )
          @ ( maps_nat_nat
            @ ^ [I: nat] :
                ( produc6860903796091618494st_nat
                @ ^ [One2: nat] : ( cons_nat @ ( Enc_impl @ ( multis2544335231667181926psilon @ I ) @ One2 ) )
                @ ( multis5159292385415766206ropVar @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite
                  @ ( map_na2145702811007361372ropVar
                    @ ^ [J: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ I @ J ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385456105ZeroIJ @ I @ J ) @ ( multis6646701651571498855xOneIJ @ I @ J ) ) )
                    @ ( upt @ zero_zero_nat @ M2 ) ) ) )
            @ ( upt @ zero_zero_nat @ N ) ) ) ) ) ).

% ms_encoder.formula14_def
thf(fact_1166_sorted__list__of__multiset__singleton,axiom,
    ! [X5: nat] :
      ( ( linord3047872887403683810et_nat @ ( add_mset_nat @ X5 @ zero_z7348594199698428585et_nat ) )
      = ( cons_nat @ X5 @ nil_nat ) ) ).

% sorted_list_of_multiset_singleton
thf(fact_1167_subset__mset_Oelem__le__sum__list,axiom,
    ! [K: nat,Ns: list_multiset_nat] :
      ( ( ord_less_nat @ K @ ( size_s6386657463320973636et_nat @ Ns ) )
     => ( subseteq_mset_nat @ ( nth_multiset_nat @ Ns @ K ) @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Ns ) ) ) ).

% subset_mset.elem_le_sum_list
thf(fact_1168_part__code_I2_J,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Pivot: nat,X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( linord1115752046135255673ar_nat @ F @ Pivot @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) )
      = ( produc2088294038180475963ropVar
        @ ^ [Lts: list_P8791008543742759928ropVar] :
            ( produc4669712208314498474ropVar
            @ ^ [Eqs: list_P8791008543742759928ropVar,Gts: list_P8791008543742759928ropVar] : ( if_Pro6661055909275588252ropVar @ ( ord_less_nat @ ( F @ X5 ) @ Pivot ) @ ( produc2087848848369820614ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Lts ) @ ( produc5333478016634530327ropVar @ Eqs @ Gts ) ) @ ( if_Pro6661055909275588252ropVar @ ( ord_less_nat @ Pivot @ ( F @ X5 ) ) @ ( produc2087848848369820614ropVar @ Lts @ ( produc5333478016634530327ropVar @ Eqs @ ( cons_P1504208554336013608ropVar @ X5 @ Gts ) ) ) @ ( produc2087848848369820614ropVar @ Lts @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X5 @ Eqs ) @ Gts ) ) ) ) )
        @ ( linord1115752046135255673ar_nat @ F @ Pivot @ Xs ) ) ) ).

% part_code(2)
thf(fact_1169_part__code_I2_J,axiom,
    ! [F: nat > nat,Pivot: nat,X5: nat,Xs: list_nat] :
      ( ( linord7044143797982409259at_nat @ F @ Pivot @ ( cons_nat @ X5 @ Xs ) )
      = ( produc759784954885767755st_nat
        @ ^ [Lts: list_nat] :
            ( produc952337756555705636st_nat
            @ ^ [Eqs: list_nat,Gts: list_nat] : ( if_Pro7837850173255959792st_nat @ ( ord_less_nat @ ( F @ X5 ) @ Pivot ) @ ( produc4487115339913071592st_nat @ ( cons_nat @ X5 @ Lts ) @ ( produc2694037385005941721st_nat @ Eqs @ Gts ) ) @ ( if_Pro7837850173255959792st_nat @ ( ord_less_nat @ Pivot @ ( F @ X5 ) ) @ ( produc4487115339913071592st_nat @ Lts @ ( produc2694037385005941721st_nat @ Eqs @ ( cons_nat @ X5 @ Gts ) ) ) @ ( produc4487115339913071592st_nat @ Lts @ ( produc2694037385005941721st_nat @ ( cons_nat @ X5 @ Eqs ) @ Gts ) ) ) ) )
        @ ( linord7044143797982409259at_nat @ F @ Pivot @ Xs ) ) ) ).

% part_code(2)
thf(fact_1170_subset__mset_Oadd__le__cancel__left,axiom,
    ! [C3: multiset_nat,A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C3 @ A ) @ ( plus_p6334493942879108393et_nat @ C3 @ B ) )
      = ( subseteq_mset_nat @ A @ B ) ) ).

% subset_mset.add_le_cancel_left
thf(fact_1171_subset__mset_Oadd__le__cancel__right,axiom,
    ! [A: multiset_nat,C3: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C3 ) @ ( plus_p6334493942879108393et_nat @ B @ C3 ) )
      = ( subseteq_mset_nat @ A @ B ) ) ).

% subset_mset.add_le_cancel_right
thf(fact_1172_mset__subset__eq__mono__add__left__cancel,axiom,
    ! [C4: multiset_nat,A2: multiset_nat,B6: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C4 @ A2 ) @ ( plus_p6334493942879108393et_nat @ C4 @ B6 ) )
      = ( subseteq_mset_nat @ A2 @ B6 ) ) ).

% mset_subset_eq_mono_add_left_cancel
thf(fact_1173_mset__subset__eq__mono__add__right__cancel,axiom,
    ! [A2: multiset_nat,C4: multiset_nat,B6: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C4 ) @ ( plus_p6334493942879108393et_nat @ B6 @ C4 ) )
      = ( subseteq_mset_nat @ A2 @ B6 ) ) ).

% mset_subset_eq_mono_add_right_cancel
thf(fact_1174_mset__sorted__list__of__multiset,axiom,
    ! [M3: multiset_nat] :
      ( ( mset_nat @ ( linord3047872887403683810et_nat @ M3 ) )
      = M3 ) ).

% mset_sorted_list_of_multiset
thf(fact_1175_subset__mset_Ole__add__same__cancel2,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ B @ A ) )
      = ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B ) ) ).

% subset_mset.le_add_same_cancel2
thf(fact_1176_subset__mset_Ole__add__same__cancel1,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ ( plus_p6334493942879108393et_nat @ A @ B ) )
      = ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B ) ) ).

% subset_mset.le_add_same_cancel1
thf(fact_1177_subset__mset_Oadd__le__same__cancel2,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ B )
      = ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.add_le_same_cancel2
thf(fact_1178_subset__mset_Oadd__le__same__cancel1,axiom,
    ! [B: multiset_nat,A: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ B @ A ) @ B )
      = ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.add_le_same_cancel1
thf(fact_1179_mset__subset__eq__multiset__union__diff__commute,axiom,
    ! [B6: multiset_nat,A2: multiset_nat,C4: multiset_nat] :
      ( ( subseteq_mset_nat @ B6 @ A2 )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ A2 @ B6 ) @ C4 )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C4 ) @ B6 ) ) ) ).

% mset_subset_eq_multiset_union_diff_commute
thf(fact_1180_subset__mset_Oadd__diff__assoc2,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B @ A ) @ C3 )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B @ C3 ) @ A ) ) ) ).

% subset_mset.add_diff_assoc2
thf(fact_1181_subset__mset_Oadd__diff__assoc,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( plus_p6334493942879108393et_nat @ C3 @ ( minus_8522176038001411705et_nat @ B @ A ) )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C3 @ B ) @ A ) ) ) ).

% subset_mset.add_diff_assoc
thf(fact_1182_sorted__list__of__multiset__empty,axiom,
    ( ( linord3047872887403683810et_nat @ zero_z7348594199698428585et_nat )
    = nil_nat ) ).

% sorted_list_of_multiset_empty
thf(fact_1183_set__sorted__list__of__multiset,axiom,
    ! [M3: multiset_nat] :
      ( ( set_nat2 @ ( linord3047872887403683810et_nat @ M3 ) )
      = ( set_mset_nat @ M3 ) ) ).

% set_sorted_list_of_multiset
thf(fact_1184_single__subset__iff,axiom,
    ! [A: nat,M3: multiset_nat] :
      ( ( subseteq_mset_nat @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) @ M3 )
      = ( member_nat @ A @ ( set_mset_nat @ M3 ) ) ) ).

% single_subset_iff
thf(fact_1185_single__subset__iff,axiom,
    ! [A: multis3193088007478089820ropVar,M3: multis193838646080965794ropVar] :
      ( ( subset828532044049478595ropVar @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) @ M3 )
      = ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ M3 ) ) ) ).

% single_subset_iff
thf(fact_1186_mset__subset__eqD,axiom,
    ! [A2: multiset_nat,B6: multiset_nat,X5: nat] :
      ( ( subseteq_mset_nat @ A2 @ B6 )
     => ( ( member_nat @ X5 @ ( set_mset_nat @ A2 ) )
       => ( member_nat @ X5 @ ( set_mset_nat @ B6 ) ) ) ) ).

% mset_subset_eqD
thf(fact_1187_mset__subset__eqD,axiom,
    ! [A2: multis193838646080965794ropVar,B6: multis193838646080965794ropVar,X5: multis3193088007478089820ropVar] :
      ( ( subset828532044049478595ropVar @ A2 @ B6 )
     => ( ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ A2 ) )
       => ( member8421201611525043589ropVar @ X5 @ ( set_ms1147286257884310937ropVar @ B6 ) ) ) ) ).

% mset_subset_eqD
thf(fact_1188_subset__mset_Oadd__mono,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat,D2: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( subseteq_mset_nat @ C3 @ D2 )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C3 ) @ ( plus_p6334493942879108393et_nat @ B @ D2 ) ) ) ) ).

% subset_mset.add_mono
thf(fact_1189_subset__mset_Oless__eqE,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ~ ! [C: multiset_nat] :
            ( B
           != ( plus_p6334493942879108393et_nat @ A @ C ) ) ) ).

% subset_mset.less_eqE
thf(fact_1190_subset__mset_Ole__iff__add,axiom,
    ( subseteq_mset_nat
    = ( ^ [A5: multiset_nat,B4: multiset_nat] :
        ? [C2: multiset_nat] :
          ( B4
          = ( plus_p6334493942879108393et_nat @ A5 @ C2 ) ) ) ) ).

% subset_mset.le_iff_add
thf(fact_1191_subset__mset_Oadd__left__mono,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C3 @ A ) @ ( plus_p6334493942879108393et_nat @ C3 @ B ) ) ) ).

% subset_mset.add_left_mono
thf(fact_1192_subset__mset_Oadd__right__mono,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C3 ) @ ( plus_p6334493942879108393et_nat @ B @ C3 ) ) ) ).

% subset_mset.add_right_mono
thf(fact_1193_subset__mset_Oadd__le__imp__le__left,axiom,
    ! [C3: multiset_nat,A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C3 @ A ) @ ( plus_p6334493942879108393et_nat @ C3 @ B ) )
     => ( subseteq_mset_nat @ A @ B ) ) ).

% subset_mset.add_le_imp_le_left
thf(fact_1194_subset__mset_Oadd__le__imp__le__right,axiom,
    ! [A: multiset_nat,C3: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C3 ) @ ( plus_p6334493942879108393et_nat @ B @ C3 ) )
     => ( subseteq_mset_nat @ A @ B ) ) ).

% subset_mset.add_le_imp_le_right
thf(fact_1195_mset__subset__eq__add__left,axiom,
    ! [A2: multiset_nat,B6: multiset_nat] : ( subseteq_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) ) ).

% mset_subset_eq_add_left
thf(fact_1196_mset__subset__eq__mono__add,axiom,
    ! [A2: multiset_nat,B6: multiset_nat,C4: multiset_nat,D: multiset_nat] :
      ( ( subseteq_mset_nat @ A2 @ B6 )
     => ( ( subseteq_mset_nat @ C4 @ D )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A2 @ C4 ) @ ( plus_p6334493942879108393et_nat @ B6 @ D ) ) ) ) ).

% mset_subset_eq_mono_add
thf(fact_1197_mset__subset__eq__add__right,axiom,
    ! [B6: multiset_nat,A2: multiset_nat] : ( subseteq_mset_nat @ B6 @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) ) ).

% mset_subset_eq_add_right
thf(fact_1198_mset__subset__eq__exists__conv,axiom,
    ( subseteq_mset_nat
    = ( ^ [A8: multiset_nat,B8: multiset_nat] :
        ? [C6: multiset_nat] :
          ( B8
          = ( plus_p6334493942879108393et_nat @ A8 @ C6 ) ) ) ) ).

% mset_subset_eq_exists_conv
thf(fact_1199_subset__mset_Oadd__nonpos__eq__0__iff,axiom,
    ! [X5: multiset_nat,Y3: multiset_nat] :
      ( ( subseteq_mset_nat @ X5 @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ Y3 @ zero_z7348594199698428585et_nat )
       => ( ( ( plus_p6334493942879108393et_nat @ X5 @ Y3 )
            = zero_z7348594199698428585et_nat )
          = ( ( X5 = zero_z7348594199698428585et_nat )
            & ( Y3 = zero_z7348594199698428585et_nat ) ) ) ) ) ).

% subset_mset.add_nonpos_eq_0_iff
thf(fact_1200_subset__mset_Oadd__nonneg__eq__0__iff,axiom,
    ! [X5: multiset_nat,Y3: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X5 )
     => ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ Y3 )
       => ( ( ( plus_p6334493942879108393et_nat @ X5 @ Y3 )
            = zero_z7348594199698428585et_nat )
          = ( ( X5 = zero_z7348594199698428585et_nat )
            & ( Y3 = zero_z7348594199698428585et_nat ) ) ) ) ) ).

% subset_mset.add_nonneg_eq_0_iff
thf(fact_1201_subset__mset_Oadd__nonpos__nonpos,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ B @ zero_z7348594199698428585et_nat )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) @ zero_z7348594199698428585et_nat ) ) ) ).

% subset_mset.add_nonpos_nonpos
thf(fact_1202_subset__mset_Oadd__nonneg__nonneg,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A )
     => ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ B )
       => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( plus_p6334493942879108393et_nat @ A @ B ) ) ) ) ).

% subset_mset.add_nonneg_nonneg
thf(fact_1203_subset__mset_Oadd__increasing2,axiom,
    ! [C3: multiset_nat,B: multiset_nat,A: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ C3 )
     => ( ( subseteq_mset_nat @ B @ A )
       => ( subseteq_mset_nat @ B @ ( plus_p6334493942879108393et_nat @ A @ C3 ) ) ) ) ).

% subset_mset.add_increasing2
thf(fact_1204_subset__mset_Oadd__decreasing2,axiom,
    ! [C3: multiset_nat,A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ C3 @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ A @ B )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C3 ) @ B ) ) ) ).

% subset_mset.add_decreasing2
thf(fact_1205_subset__mset_Oadd__increasing,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ A )
     => ( ( subseteq_mset_nat @ B @ C3 )
       => ( subseteq_mset_nat @ B @ ( plus_p6334493942879108393et_nat @ A @ C3 ) ) ) ) ).

% subset_mset.add_increasing
thf(fact_1206_subset__mset_Oadd__decreasing,axiom,
    ! [A: multiset_nat,C3: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ zero_z7348594199698428585et_nat )
     => ( ( subseteq_mset_nat @ C3 @ B )
       => ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ A @ C3 ) @ B ) ) ) ).

% subset_mset.add_decreasing
thf(fact_1207_subset__mset_Odiff__add,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B @ A ) @ A )
        = B ) ) ).

% subset_mset.diff_add
thf(fact_1208_subset__mset_Ole__add__diff,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( subseteq_mset_nat @ C3 @ ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B @ C3 ) @ A ) ) ) ).

% subset_mset.le_add_diff
thf(fact_1209_subset__mset_Ole__diff__conv2,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( subseteq_mset_nat @ C3 @ ( minus_8522176038001411705et_nat @ B @ A ) )
        = ( subseteq_mset_nat @ ( plus_p6334493942879108393et_nat @ C3 @ A ) @ B ) ) ) ).

% subset_mset.le_diff_conv2
thf(fact_1210_subset__mset_Odiff__add__assoc,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C3 @ B ) @ A )
        = ( plus_p6334493942879108393et_nat @ C3 @ ( minus_8522176038001411705et_nat @ B @ A ) ) ) ) ).

% subset_mset.diff_add_assoc
thf(fact_1211_subset__mset_Odiff__add__assoc2,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ B @ C3 ) @ A )
        = ( plus_p6334493942879108393et_nat @ ( minus_8522176038001411705et_nat @ B @ A ) @ C3 ) ) ) ).

% subset_mset.diff_add_assoc2
thf(fact_1212_subset__mset_Odiff__diff__right,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( minus_8522176038001411705et_nat @ C3 @ ( minus_8522176038001411705et_nat @ B @ A ) )
        = ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ C3 @ A ) @ B ) ) ) ).

% subset_mset.diff_diff_right
thf(fact_1213_subset__mset_Oadd__diff__inverse,axiom,
    ! [A: multiset_nat,B: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( plus_p6334493942879108393et_nat @ A @ ( minus_8522176038001411705et_nat @ B @ A ) )
        = B ) ) ).

% subset_mset.add_diff_inverse
thf(fact_1214_subset__mset_Ole__imp__diff__is__add,axiom,
    ! [A: multiset_nat,B: multiset_nat,C3: multiset_nat] :
      ( ( subseteq_mset_nat @ A @ B )
     => ( ( subseteq_mset_nat @ A @ B )
       => ( ( ( minus_8522176038001411705et_nat @ B @ A )
            = C3 )
          = ( B
            = ( plus_p6334493942879108393et_nat @ C3 @ A ) ) ) ) ) ).

% subset_mset.le_imp_diff_is_add
thf(fact_1215_subset__eq__diff__conv,axiom,
    ! [A2: multiset_nat,C4: multiset_nat,B6: multiset_nat] :
      ( ( subseteq_mset_nat @ ( minus_8522176038001411705et_nat @ A2 @ C4 ) @ B6 )
      = ( subseteq_mset_nat @ A2 @ ( plus_p6334493942879108393et_nat @ B6 @ C4 ) ) ) ).

% subset_eq_diff_conv
thf(fact_1216_multiset__diff__union__assoc,axiom,
    ! [C4: multiset_nat,B6: multiset_nat,A2: multiset_nat] :
      ( ( subseteq_mset_nat @ C4 @ B6 )
     => ( ( minus_8522176038001411705et_nat @ ( plus_p6334493942879108393et_nat @ A2 @ B6 ) @ C4 )
        = ( plus_p6334493942879108393et_nat @ A2 @ ( minus_8522176038001411705et_nat @ B6 @ C4 ) ) ) ) ).

% multiset_diff_union_assoc
thf(fact_1217_transpose_Osimps_I2_J,axiom,
    ! [Xss2: list_list_nat] :
      ( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
      = ( transpose_nat @ Xss2 ) ) ).

% transpose.simps(2)
thf(fact_1218_transpose_Osimps_I2_J,axiom,
    ! [Xss2: list_l3415868495287753214ropVar] :
      ( ( transp9078802294292144386ropVar @ ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss2 ) )
      = ( transp9078802294292144386ropVar @ Xss2 ) ) ).

% transpose.simps(2)
thf(fact_1219_maps__simps_I2_J,axiom,
    ! [F: nat > list_nat] :
      ( ( maps_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1220_maps__simps_I2_J,axiom,
    ! [F: nat > list_P8791008543742759928ropVar] :
      ( ( maps_n6440817060062774047ropVar @ F @ nil_nat )
      = nil_Pr6119456312066593144ropVar ) ).

% maps_simps(2)
thf(fact_1221_maps__simps_I2_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_nat] :
      ( ( maps_P2882123423910056385ar_nat @ F @ nil_Pr6119456312066593144ropVar )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1222_maps__simps_I2_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_P8791008543742759928ropVar] :
      ( ( maps_P2685787764321406401ropVar @ F @ nil_Pr6119456312066593144ropVar )
      = nil_Pr6119456312066593144ropVar ) ).

% maps_simps(2)
thf(fact_1223_mset__subset__eq__single,axiom,
    ! [A: nat,B6: multiset_nat] :
      ( ( member_nat @ A @ ( set_mset_nat @ B6 ) )
     => ( subseteq_mset_nat @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) @ B6 ) ) ).

% mset_subset_eq_single
thf(fact_1224_mset__subset__eq__single,axiom,
    ! [A: multis3193088007478089820ropVar,B6: multis193838646080965794ropVar] :
      ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ B6 ) )
     => ( subset828532044049478595ropVar @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) @ B6 ) ) ).

% mset_subset_eq_single
thf(fact_1225_multi__subset__induct,axiom,
    ! [F9: multiset_nat,A2: multiset_nat,P: multiset_nat > $o] :
      ( ( subseteq_mset_nat @ F9 @ A2 )
     => ( ( P @ zero_z7348594199698428585et_nat )
       => ( ! [A4: nat,F10: multiset_nat] :
              ( ( member_nat @ A4 @ ( set_mset_nat @ A2 ) )
             => ( ( P @ F10 )
               => ( P @ ( add_mset_nat @ A4 @ F10 ) ) ) )
         => ( P @ F9 ) ) ) ) ).

% multi_subset_induct
thf(fact_1226_multi__subset__induct,axiom,
    ! [F9: multis193838646080965794ropVar,A2: multis193838646080965794ropVar,P: multis193838646080965794ropVar > $o] :
      ( ( subset828532044049478595ropVar @ F9 @ A2 )
     => ( ( P @ zero_z6226653793019358809ropVar )
       => ( ! [A4: multis3193088007478089820ropVar,F10: multis193838646080965794ropVar] :
              ( ( member8421201611525043589ropVar @ A4 @ ( set_ms1147286257884310937ropVar @ A2 ) )
             => ( ( P @ F10 )
               => ( P @ ( add_ms1730626103851491130ropVar @ A4 @ F10 ) ) ) )
         => ( P @ F9 ) ) ) ) ).

% multi_subset_induct
thf(fact_1227_subset__mset_Osum__list__nonneg,axiom,
    ! [Xs: list_multiset_nat] :
      ( ! [X4: multiset_nat] :
          ( ( member_multiset_nat @ X4 @ ( set_multiset_nat2 @ Xs ) )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X4 ) )
     => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) ) ) ).

% subset_mset.sum_list_nonneg
thf(fact_1228_subset__mset_Osum__list__nonpos,axiom,
    ! [Xs: list_multiset_nat] :
      ( ! [X4: multiset_nat] :
          ( ( member_multiset_nat @ X4 @ ( set_multiset_nat2 @ Xs ) )
         => ( subseteq_mset_nat @ X4 @ zero_z7348594199698428585et_nat ) )
     => ( subseteq_mset_nat @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) @ zero_z7348594199698428585et_nat ) ) ).

% subset_mset.sum_list_nonpos
thf(fact_1229_subset__mset_Omember__le__sum__list,axiom,
    ! [X5: multiset_nat,Xs: list_multiset_nat] :
      ( ( member_multiset_nat @ X5 @ ( set_multiset_nat2 @ Xs ) )
     => ( subseteq_mset_nat @ X5 @ ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs ) ) ) ).

% subset_mset.member_le_sum_list
thf(fact_1230_subset__mset_Osum__list__nonneg__eq__0__iff,axiom,
    ! [Xs: list_multiset_nat] :
      ( ! [X4: multiset_nat] :
          ( ( member_multiset_nat @ X4 @ ( set_multiset_nat2 @ Xs ) )
         => ( subseteq_mset_nat @ zero_z7348594199698428585et_nat @ X4 ) )
     => ( ( ( groups2887787882517827221et_nat @ plus_p6334493942879108393et_nat @ zero_z7348594199698428585et_nat @ Xs )
          = zero_z7348594199698428585et_nat )
        = ( ! [X: multiset_nat] :
              ( ( member_multiset_nat @ X @ ( set_multiset_nat2 @ Xs ) )
             => ( X = zero_z7348594199698428585et_nat ) ) ) ) ) ).

% subset_mset.sum_list_nonneg_eq_0_iff
thf(fact_1231_insert__subset__eq__iff,axiom,
    ! [A: nat,A2: multiset_nat,B6: multiset_nat] :
      ( ( subseteq_mset_nat @ ( add_mset_nat @ A @ A2 ) @ B6 )
      = ( ( member_nat @ A @ ( set_mset_nat @ B6 ) )
        & ( subseteq_mset_nat @ A2 @ ( minus_8522176038001411705et_nat @ B6 @ ( add_mset_nat @ A @ zero_z7348594199698428585et_nat ) ) ) ) ) ).

% insert_subset_eq_iff
thf(fact_1232_insert__subset__eq__iff,axiom,
    ! [A: multis3193088007478089820ropVar,A2: multis193838646080965794ropVar,B6: multis193838646080965794ropVar] :
      ( ( subset828532044049478595ropVar @ ( add_ms1730626103851491130ropVar @ A @ A2 ) @ B6 )
      = ( ( member8421201611525043589ropVar @ A @ ( set_ms1147286257884310937ropVar @ B6 ) )
        & ( subset828532044049478595ropVar @ A2 @ ( minus_287902380581138057ropVar @ B6 @ ( add_ms1730626103851491130ropVar @ A @ zero_z6226653793019358809ropVar ) ) ) ) ) ).

% insert_subset_eq_iff
thf(fact_1233_maps__simps_I1_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_nat,X5: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( maps_P2882123423910056385ar_nat @ F @ ( cons_P1504208554336013608ropVar @ X5 @ Xs ) )
      = ( append_nat @ ( F @ X5 ) @ ( maps_P2882123423910056385ar_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1234_maps__simps_I1_J,axiom,
    ! [F: nat > list_nat,X5: nat,Xs: list_nat] :
      ( ( maps_nat_nat @ F @ ( cons_nat @ X5 @ Xs ) )
      = ( append_nat @ ( F @ X5 ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_1235_ms__encoder_Oformula15__def,axiom,
    ! [Enc_False: produc2642885279816024434ropVar,Enc_True: produc2642885279816024434ropVar,Enc_pos: multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_neg: multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_impl: multis3193088007478089820ropVar > produc2642885279816024434ropVar > produc2642885279816024434ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc2642885279816024434ropVar,Enc_not_all: list_M6740372356978808802ropVar > produc2642885279816024434ropVar,Enc_conj: list_P8791008543742759928ropVar > produc2642885279816024434ropVar,Eval: ( multis3193088007478089820ropVar > $o ) > produc2642885279816024434ropVar > $o,Cs: nat > nat > $o,Cns: nat > nat > $o,N: nat,M2: nat] :
      ( ( multis8181102237825478411ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ( multis5595896195939027229ropVar @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 )
        = ( append4352292717583037453ropVar
          @ ( maps_n6440817060062774047ropVar
            @ ^ [I: nat] :
                ( maps_n6440817060062774047ropVar
                @ ^ [J: nat] :
                    ( if_lis1975434623467068862ropVar
                    @ ( ( Cs @ I @ J )
                      & ( Cns @ I @ J ) )
                    @ nil_Pr6119456312066593144ropVar
                    @ ( cons_P1504208554336013608ropVar @ ( multis6911105939383758721ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ ( multis387687052011358179_Gamma @ I @ J ) @ ( multis2544335231667181926psilon @ I ) @ ( Cs @ I @ J ) @ ( Cns @ I @ J ) ) @ nil_Pr6119456312066593144ropVar ) )
                @ ( upt @ zero_zero_nat @ M2 ) )
            @ ( upt @ zero_zero_nat @ N ) )
          @ ( multis3450153464401155806ropVar @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ N @ M2 ) ) ) ) ).

% ms_encoder.formula15_def
thf(fact_1236_ms__encoder_Oformula15__def,axiom,
    ! [Enc_False: nat,Enc_True: nat,Enc_pos: multis3193088007478089820ropVar > nat,Enc_neg: multis3193088007478089820ropVar > nat,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_impl: multis3193088007478089820ropVar > nat > nat,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat,Enc_not_all: list_M6740372356978808802ropVar > nat,Enc_conj: list_nat > nat,Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,Cs: nat > nat > $o,Cns: nat > nat > $o,N: nat,M2: nat] :
      ( ( multis4476563104974320119er_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Enc_not_all @ Enc_conj @ Eval )
     => ( ( multis902447301359355274115_nat @ Enc_False @ Enc_True @ Enc_pos @ Enc_neg @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ Enc_var_impl @ Enc_not_and @ Cs @ Cns @ N @ M2 )
        = ( append_nat
          @ ( maps_nat_nat
            @ ^ [I: nat] :
                ( maps_nat_nat
                @ ^ [J: nat] :
                    ( if_list_nat
                    @ ( ( Cs @ I @ J )
                      & ( Cns @ I @ J ) )
                    @ nil_nat
                    @ ( cons_nat @ ( multis8432056474937814451ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ ( multis387687052011358179_Gamma @ I @ J ) @ ( multis2544335231667181926psilon @ I ) @ ( Cs @ I @ J ) @ ( Cns @ I @ J ) ) @ nil_nat ) )
                @ ( upt @ zero_zero_nat @ M2 ) )
            @ ( upt @ zero_zero_nat @ N ) )
          @ ( multis778902868463715824414_nat @ Enc_False @ Enc_pos @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ Enc_ite @ Enc_impl @ N @ M2 ) ) ) ) ).

% ms_encoder.formula15_def
thf(fact_1237_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_lists_nat @ N @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_1238_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list6927042657299049856ropVar @ N @ nil_Pr6119456312066593144ropVar )
          = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list6927042657299049856ropVar @ N @ nil_Pr6119456312066593144ropVar )
          = nil_li2679230897895549182ropVar ) ) ) ).

% n_lists_Nil
thf(fact_1239_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_1240_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( n_list6927042657299049856ropVar @ zero_zero_nat @ Xs )
      = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) ).

% n_lists.simps(1)
thf(fact_1241_length__n__lists__elem,axiom,
    ! [Ys: list_P8791008543742759928ropVar,N: nat,Xs: list_P8791008543742759928ropVar] :
      ( ( member6236339133593825953ropVar @ Ys @ ( set_li6509544621384488717ropVar @ ( n_list6927042657299049856ropVar @ N @ Xs ) ) )
     => ( ( size_s6752116297430514276ropVar @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1242_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_1243_distinct__n__lists,axiom,
    ! [Xs: list_nat,N: nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_list_nat @ ( n_lists_nat @ N @ Xs ) ) ) ).

% distinct_n_lists
thf(fact_1244_mult1E,axiom,
    ! [N4: multis193838646080965794ropVar,M3: multis193838646080965794ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ N4 @ M3 ) @ ( mult1_1560173407137554895ropVar @ R ) )
     => ~ ! [A4: multis3193088007478089820ropVar,M0: multis193838646080965794ropVar] :
            ( ( M3
              = ( add_ms1730626103851491130ropVar @ A4 @ M0 ) )
           => ! [K5: multis193838646080965794ropVar] :
                ( ( N4
                  = ( plus_p1339091108561003481ropVar @ M0 @ K5 ) )
               => ~ ! [B5: multis3193088007478089820ropVar] :
                      ( ( member8421201611525043589ropVar @ B5 @ ( set_ms1147286257884310937ropVar @ K5 ) )
                     => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ B5 @ A4 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_1245_mult1E,axiom,
    ! [N4: multiset_nat,M3: multiset_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ N4 @ M3 ) @ ( mult1_nat @ R ) )
     => ~ ! [A4: nat,M0: multiset_nat] :
            ( ( M3
              = ( add_mset_nat @ A4 @ M0 ) )
           => ! [K5: multiset_nat] :
                ( ( N4
                  = ( plus_p6334493942879108393et_nat @ M0 @ K5 ) )
               => ~ ! [B5: nat] :
                      ( ( member_nat @ B5 @ ( set_mset_nat @ K5 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A4 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_1246_mult1__union,axiom,
    ! [B6: multiset_nat,D: multiset_nat,R: set_Pr1261947904930325089at_nat,C4: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ B6 @ D ) @ ( mult1_nat @ R ) )
     => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ C4 @ B6 ) @ ( plus_p6334493942879108393et_nat @ C4 @ D ) ) @ ( mult1_nat @ R ) ) ) ).

% mult1_union
thf(fact_1247_less__add,axiom,
    ! [N4: multis193838646080965794ropVar,A: multis3193088007478089820ropVar,M02: multis193838646080965794ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ N4 @ ( add_ms1730626103851491130ropVar @ A @ M02 ) ) @ ( mult1_1560173407137554895ropVar @ R ) )
     => ( ? [M5: multis193838646080965794ropVar] :
            ( ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ M5 @ M02 ) @ ( mult1_1560173407137554895ropVar @ R ) )
            & ( N4
              = ( add_ms1730626103851491130ropVar @ A @ M5 ) ) )
        | ? [K5: multis193838646080965794ropVar] :
            ( ! [B5: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ B5 @ ( set_ms1147286257884310937ropVar @ K5 ) )
               => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ B5 @ A ) @ R ) )
            & ( N4
              = ( plus_p1339091108561003481ropVar @ M02 @ K5 ) ) ) ) ) ).

% less_add
thf(fact_1248_less__add,axiom,
    ! [N4: multiset_nat,A: nat,M02: multiset_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ N4 @ ( add_mset_nat @ A @ M02 ) ) @ ( mult1_nat @ R ) )
     => ( ? [M5: multiset_nat] :
            ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ M5 @ M02 ) @ ( mult1_nat @ R ) )
            & ( N4
              = ( add_mset_nat @ A @ M5 ) ) )
        | ? [K5: multiset_nat] :
            ( ! [B5: nat] :
                ( ( member_nat @ B5 @ ( set_mset_nat @ K5 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B5 @ A ) @ R ) )
            & ( N4
              = ( plus_p6334493942879108393et_nat @ M02 @ K5 ) ) ) ) ) ).

% less_add
thf(fact_1249_mult1I,axiom,
    ! [M3: multis193838646080965794ropVar,A: multis3193088007478089820ropVar,M02: multis193838646080965794ropVar,N4: multis193838646080965794ropVar,K2: multis193838646080965794ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( M3
        = ( add_ms1730626103851491130ropVar @ A @ M02 ) )
     => ( ( N4
          = ( plus_p1339091108561003481ropVar @ M02 @ K2 ) )
       => ( ! [B3: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ B3 @ ( set_ms1147286257884310937ropVar @ K2 ) )
             => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ B3 @ A ) @ R ) )
         => ( member5226782437669593040ropVar @ ( produc8064044144368941463ropVar @ N4 @ M3 ) @ ( mult1_1560173407137554895ropVar @ R ) ) ) ) ) ).

% mult1I
thf(fact_1250_mult1I,axiom,
    ! [M3: multiset_nat,A: nat,M02: multiset_nat,N4: multiset_nat,K2: multiset_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( M3
        = ( add_mset_nat @ A @ M02 ) )
     => ( ( N4
          = ( plus_p6334493942879108393et_nat @ M02 @ K2 ) )
       => ( ! [B3: nat] :
              ( ( member_nat @ B3 @ ( set_mset_nat @ K2 ) )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ B3 @ A ) @ R ) )
         => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ N4 @ M3 ) @ ( mult1_nat @ R ) ) ) ) ) ).

% mult1I
thf(fact_1251_distinct__product__lists,axiom,
    ! [Xss2: list_list_nat] :
      ( ! [X4: list_nat] :
          ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss2 ) )
         => ( distinct_nat @ X4 ) )
     => ( distinct_list_nat @ ( product_lists_nat @ Xss2 ) ) ) ).

% distinct_product_lists
thf(fact_1252_nth__transpose,axiom,
    ! [I3: nat,Xs: list_l3415868495287753214ropVar] :
      ( ( ord_less_nat @ I3 @ ( size_s6779274088989432426ropVar @ ( transp9078802294292144386ropVar @ Xs ) ) )
     => ( ( nth_li6497732868662353663ropVar @ ( transp9078802294292144386ropVar @ Xs ) @ I3 )
        = ( map_li1623070805142362430ropVar
          @ ^ [Xs2: list_P8791008543742759928ropVar] : ( nth_Pr227365692153740281ropVar @ Xs2 @ I3 )
          @ ( filter4529039315287521013ropVar
            @ ^ [Ys2: list_P8791008543742759928ropVar] : ( ord_less_nat @ I3 @ ( size_s6752116297430514276ropVar @ Ys2 ) )
            @ Xs ) ) ) ) ).

% nth_transpose
thf(fact_1253_nth__transpose,axiom,
    ! [I3: nat,Xs: list_list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ ( transpose_nat @ Xs ) ) )
     => ( ( nth_list_nat @ ( transpose_nat @ Xs ) @ I3 )
        = ( map_list_nat_nat
          @ ^ [Xs2: list_nat] : ( nth_nat @ Xs2 @ I3 )
          @ ( filter_list_nat
            @ ^ [Ys2: list_nat] : ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys2 ) )
            @ Xs ) ) ) ) ).

% nth_transpose
thf(fact_1254_filter__True,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( filter_nat @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_1255_filter__True,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ( P @ X4 ) )
     => ( ( filter8895658007216384985ropVar @ P @ Xs )
        = Xs ) ) ).

% filter_True
thf(fact_1256_filter__append,axiom,
    ! [P: nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( filter_nat @ P @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( filter_nat @ P @ Xs ) @ ( filter_nat @ P @ Ys ) ) ) ).

% filter_append
thf(fact_1257_set__filter,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( set_nat2 @ ( filter_nat @ P @ Xs ) )
      = ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
            & ( P @ X ) ) ) ) ).

% set_filter
thf(fact_1258_set__filter,axiom,
    ! [P: multis3193088007478089820ropVar > $o,Xs: list_M6740372356978808802ropVar] :
      ( ( set_Mu146256361055586289ropVar @ ( filter8895658007216384985ropVar @ P @ Xs ) )
      = ( collec8658135508005558727ropVar
        @ ^ [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
            & ( P @ X ) ) ) ) ).

% set_filter
thf(fact_1259_filter__False,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ! [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
         => ~ ( P @ X4 ) )
     => ( ( filter9064576222288807663ropVar @ P @ Xs )
        = nil_Pr6119456312066593144ropVar ) ) ).

% filter_False
thf(fact_1260_filter__False,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ~ ( P @ X4 ) )
     => ( ( filter_nat @ P @ Xs )
        = nil_nat ) ) ).

% filter_False
thf(fact_1261_filter__False,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ! [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ~ ( P @ X4 ) )
     => ( ( filter8895658007216384985ropVar @ P @ Xs )
        = nil_Mu4132399263956683874ropVar ) ) ).

% filter_False
thf(fact_1262_distinct__filter,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ ( filter_nat @ P @ Xs ) ) ) ).

% distinct_filter
thf(fact_1263_distinct__map__filter,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,P: nat > $o] :
      ( ( distin2675581014522817237ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
     => ( distin2675581014522817237ropVar @ ( map_na2466558649495876678ropVar @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filter
thf(fact_1264_distinct__map__filter,axiom,
    ! [F: nat > nat,Xs: list_nat,P: nat > $o] :
      ( ( distinct_nat @ ( map_nat_nat @ F @ Xs ) )
     => ( distinct_nat @ ( map_nat_nat @ F @ ( filter_nat @ P @ Xs ) ) ) ) ).

% distinct_map_filter
thf(fact_1265_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1266_upt__eq__Cons__conv,axiom,
    ! [I3: nat,J2: nat,X5: nat,Xs: list_nat] :
      ( ( ( upt @ I3 @ J2 )
        = ( cons_nat @ X5 @ Xs ) )
      = ( ( ord_less_nat @ I3 @ J2 )
        & ( I3 = X5 )
        & ( ( upt @ ( plus_plus_nat @ I3 @ one_one_nat ) @ J2 )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_1267_hd__upt,axiom,
    ! [I3: nat,J2: nat] :
      ( ( ord_less_nat @ I3 @ J2 )
     => ( ( hd_nat @ ( upt @ I3 @ J2 ) )
        = I3 ) ) ).

% hd_upt

% Helper facts (15)
thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( if_nat @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X5: nat,Y3: nat] :
      ( ( if_nat @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X3: nat] : ( P @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X5: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X5: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__Multiset____Ordering____in____NP__OPropVar_T,axiom,
    ! [P: multis3193088007478089820ropVar > $o] :
      ( ( P @ ( fChoic539809714188224432ropVar @ P ) )
      = ( ? [X3: multis3193088007478089820ropVar] : ( P @ X3 ) ) ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_T,axiom,
    ! [P: produc7679832412844898215ropVar > $o] :
      ( ( P @ ( fChoic157639270322531067ropVar @ P ) )
      = ( ? [X3: produc7679832412844898215ropVar] : ( P @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
    ! [X5: produc5438398690206989622st_nat,Y3: produc5438398690206989622st_nat] :
      ( ( if_Pro7837850173255959792st_nat @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J_J_T,axiom,
    ! [X5: produc5438398690206989622st_nat,Y3: produc5438398690206989622st_nat] :
      ( ( if_Pro7837850173255959792st_nat @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_fChoice_1_1_fChoice_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_T,axiom,
    ! [P: produc2642885279816024434ropVar > $o] :
      ( ( P @ ( fChoic3237874067385186502ropVar @ P ) )
      = ( ? [X3: produc2642885279816024434ropVar] : ( P @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_T,axiom,
    ! [X5: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( if_lis1975434623467068862ropVar @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_T,axiom,
    ! [X5: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( if_lis1975434623467068862ropVar @ $true @ X5 @ Y3 )
      = X5 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J_J_T,axiom,
    ! [X5: produc777314622398639830ropVar,Y3: produc777314622398639830ropVar] :
      ( ( if_Pro6661055909275588252ropVar @ $false @ X5 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_J_J_T,axiom,
    ! [X5: produc777314622398639830ropVar,Y3: produc777314622398639830ropVar] :
      ( ( if_Pro6661055909275588252ropVar @ $true @ X5 @ Y3 )
      = X5 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ze
    = ( multis2983220944385456105ZeroIJ @ i @ j ) ) ).

%------------------------------------------------------------------------------