TPTP Problem File: SLH0248^1.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_00926_041467__13950844_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1923 ( 636 unt; 629 typ;   0 def)
%            Number of atoms       : 4051 (2335 equ;   0 cnn)
%            Maximal formula atoms :   24 (   3 avg)
%            Number of connectives : 16353 ( 939   ~; 133   |; 665   &;12970   @)
%                                         (   0 <=>;1646  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   36 (   8 avg)
%            Number of types       :  137 ( 136 usr)
%            Number of type conns  : 2528 (2528   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  496 ( 493 usr;  22 con; 0-14 aty)
%            Number of variables   : 5208 ( 129   ^;4438   !; 641   ?;5208   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:10.597
%------------------------------------------------------------------------------
% Could-be-implicit typings (136)
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    set_o: $tType ).

thf(ty_n_t__Num__Onum,type,
    num: $tType ).

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

% Explicit typings (493)
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    bNF_Gr5278975096840786186pVar_o: set_li5493982441917991125pVar_o > list_l985434930288970485pVar_o > set_li3954596911114931407pVar_o ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    bNF_Gr5940263876698902660pVar_o: set_li3954596911114931407pVar_o > list_l1255154591254203759pVar_o > set_li254716981354726473pVar_o ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Multiset____Ordering____in____NP__OPropVar,type,
    bNF_Gr6307721265529545847ropVar: set_li3089797549399498434ropVar > list_M6740372356978808802ropVar > set_Mu5658373669842198716ropVar ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Nat__Onat,type,
    bNF_Gr6352880689984616693cc_nat: set_list_nat > list_nat > set_nat ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    bNF_Gr7432766864382442306od_o_o: set_li6534384796922688653od_o_o > list_P4002435161011370285od_o_o > set_Product_prod_o_o ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    bNF_Gr3337489131723850622pVar_o: set_li254716981354726473pVar_o > list_P1665994083459302249pVar_o > set_Pr8633061257259478851pVar_o ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_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,
    bNF_Gr2655190671417346957ropVar: set_li8998331099879477720ropVar > list_P8791008543742759928ropVar > set_Pr6405597471078531282ropVar ).

thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    bNF_Gr6265355799493510996ropVar: set_li9140175473581139189ropVar > list_P723798329145367999ropVar > set_Pr5911590690006159205ropVar ).

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

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

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

thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
    zero_zero_nat: nat ).

thf(sy_c_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_J,type,
    if_lis1723474115809273281pVar_o: $o > list_l6341520962544925691pVar_o > list_l6341520962544925691pVar_o > list_l6341520962544925691pVar_o ).

thf(sy_c_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
    if_lis463867677060124603pVar_o: $o > list_l985434930288970485pVar_o > list_l985434930288970485pVar_o > list_l985434930288970485pVar_o ).

thf(sy_c_If_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    if_lis6453251472562810165pVar_o: $o > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o ).

thf(sy_c_If_001t__List__Olist_It__List__Olist_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    if_lis5532466218389838089ropVar: $o > list_l1644708917474426447ropVar > list_l1644708917474426447ropVar > list_l1644708917474426447ropVar ).

thf(sy_c_If_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    if_lis9215432529438213544ropVar: $o > list_M6740372356978808802ropVar > list_M6740372356978808802ropVar > list_M6740372356978808802ropVar ).

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

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    if_lis8143501808108548851od_o_o: $o > list_P4002435161011370285od_o_o > list_P4002435161011370285od_o_o > list_P4002435161011370285od_o_o ).

thf(sy_c_If_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    if_lis4159902583309516335pVar_o: $o > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o ).

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,
    if_lis1975434623467068862ropVar: $o > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar ).

thf(sy_c_List_Oappend_001_Eo,type,
    append_o: list_o > list_o > list_o ).

thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
    append9215492216909589904pVar_o: list_l6341520962544925691pVar_o > list_l6341520962544925691pVar_o > list_l6341520962544925691pVar_o ).

thf(sy_c_List_Oappend_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    append8959015459476584842pVar_o: list_l985434930288970485pVar_o > list_l985434930288970485pVar_o > list_l985434930288970485pVar_o ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Nat__Onat_J,type,
    append_list_nat: list_list_nat > list_list_nat > list_list_nat ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    append8069948826146371844pVar_o: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o ).

thf(sy_c_List_Oappend_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,
    append4316637162825710867ropVar: list_l3415868495287753214ropVar > list_l3415868495287753214ropVar > list_l3415868495287753214ropVar ).

thf(sy_c_List_Oappend_001t__List__Olist_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    append2757213247338973028ropVar: list_l1644708917474426447ropVar > list_l1644708917474426447ropVar > list_l1644708917474426447ropVar ).

thf(sy_c_List_Oappend_001t__Multiset____Ordering____in____NP__OPropVar,type,
    append2684835082924936439ropVar: list_M6740372356978808802ropVar > list_M6740372356978808802ropVar > list_M6740372356978808802ropVar ).

thf(sy_c_List_Oappend_001t__Nat__Onat,type,
    append_nat: list_nat > list_nat > list_nat ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    append2614242729457001410od_o_o: list_P4002435161011370285od_o_o > list_P4002435161011370285od_o_o > list_P4002435161011370285od_o_o ).

thf(sy_c_List_Oappend_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    append1087384436348988414pVar_o: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o ).

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,
    append4352292717583037453ropVar: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar ).

thf(sy_c_List_Oappend_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    append7708039934470209236ropVar: list_P723798329145367999ropVar > list_P723798329145367999ropVar > list_P723798329145367999ropVar ).

thf(sy_c_List_Obind_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_001t__Nat__Onat,type,
    bind_l3081483778782373880_o_nat: list_l985434930288970485pVar_o > ( list_l1255154591254203759pVar_o > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_001t__Nat__Onat,type,
    bind_l4743436673915347442_o_nat: list_l1255154591254203759pVar_o > ( list_P1665994083459302249pVar_o > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    bind_l1937305119761364945ropVar: list_l1255154591254203759pVar_o > ( list_P1665994083459302249pVar_o > list_P723798329145367999ropVar ) > list_P723798329145367999ropVar ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    bind_n4999772531149697238pVar_o: list_nat > ( nat > list_l985434930288970485pVar_o ) > list_l985434930288970485pVar_o ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    bind_n6273727003328726608pVar_o: list_nat > ( nat > list_l1255154591254203759pVar_o ) > list_l1255154591254203759pVar_o ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat,type,
    bind_nat_nat: list_nat > ( nat > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    bind_n3686386053328211786pVar_o: list_nat > ( nat > list_P1665994083459302249pVar_o ) > list_P1665994083459302249pVar_o ).

thf(sy_c_List_Obind_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,type,
    bind_n1517830492101072217ropVar: list_nat > ( nat > list_P8791008543742759928ropVar ) > list_P8791008543742759928ropVar ).

thf(sy_c_List_Obind_001t__Nat__Onat_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_001t__Nat__Onat,type,
    bind_P7202517103677919084_o_nat: list_P1665994083459302249pVar_o > ( produc8999643685545340131pVar_o > list_nat ) > list_nat ).

thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
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thf(sy_c_List_Obind_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    bind_P4196495778012640331ropVar: list_P1665994083459302249pVar_o > ( produc8999643685545340131pVar_o > list_P723798329145367999ropVar ) > list_P723798329145367999ropVar ).

thf(sy_c_List_Obind_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,
    bind_P7182508892803130363ar_nat: list_P8791008543742759928ropVar > ( produc2642885279816024434ropVar > list_nat ) > list_nat ).

thf(sy_c_List_Obind_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__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Obutlast_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
    butlas4398145439059492935pVar_o: list_l6341520962544925691pVar_o > list_l6341520962544925691pVar_o ).

thf(sy_c_List_Obutlast_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
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thf(sy_c_List_Obutlast_001t__List__Olist_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_List_Obutlast_001t__Nat__Onat,type,
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thf(sy_c_List_Ocan__select_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
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thf(sy_c_List_Ocan__select_001t__Nat__Onat,type,
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thf(sy_c_List_Oconcat_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
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thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
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thf(sy_c_List_Oconcat_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_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_Odrop_001t__Nat__Onat,type,
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    rotate3041276922869945654pVar_o: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o ).

thf(sy_c_List_Orotate1_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,
    rotate7816721141519739205ropVar: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar ).

thf(sy_c_List_Orotate1_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    rotate7455714379189909404ropVar: list_P723798329145367999ropVar > list_P723798329145367999ropVar ).

thf(sy_c_List_Osorted__wrt_001t__Nat__Onat,type,
    sorted_wrt_nat: ( nat > nat > $o ) > list_nat > $o ).

thf(sy_c_List_Osubseqs_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    subseq7541546836001800004pVar_o: list_l985434930288970485pVar_o > list_l6341520962544925691pVar_o ).

thf(sy_c_List_Osubseqs_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    subseq2663478470768632766pVar_o: list_l1255154591254203759pVar_o > list_l985434930288970485pVar_o ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

thf(sy_c_List_Osubseqs_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    subseq8245697508244618168pVar_o: list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o ).

thf(sy_c_List_Osubseqs_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,
    subseq3825174504735880391ropVar: list_P8791008543742759928ropVar > list_l3415868495287753214ropVar ).

thf(sy_c_List_Osubseqs_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    subseq8000409699144225946ropVar: list_P723798329145367999ropVar > list_l1644708917474426447ropVar ).

thf(sy_c_List_Otake_001t__Nat__Onat,type,
    take_nat: nat > list_nat > list_nat ).

thf(sy_c_List_Oupt,type,
    upt: nat > nat > list_nat ).

thf(sy_c_Multiset__Ordering__in__NP_Ocnf__encodeGammaCond,type,
    multis334970729563149064maCond: multis3193088007478089820ropVar > multis3193088007478089820ropVar > $o > $o > list_l1255154591254203759pVar_o ).

thf(sy_c_Multiset__Ordering__in__NP_Ocnf__encode__exactly__one,type,
    multis8929244183426903067ly_one: list_P8791008543742759928ropVar > produc5436468477138497005pVar_o ).

thf(sy_c_Multiset__Ordering__in__NP_Ocnf__encode__sum__0__1__main,type,
    multis70479551085940736041_main: list_P8791008543742759928ropVar > produc9206049318796031513ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001_Eo_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    multis6533897398417245987pVar_o: ( ( $o > $o ) > list_l1255154591254203759pVar_o > $o ) > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > ( $o > list_l1255154591254203759pVar_o ) > ( $o > list_l1255154591254203759pVar_o ) > ( $o > $o > list_l1255154591254203759pVar_o ) > ( $o > $o > $o > list_l1255154591254203759pVar_o ) > ( $o > $o > $o > $o > list_l1255154591254203759pVar_o ) > ( $o > $o > $o > list_l1255154591254203759pVar_o ) > ( $o > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o ) > ( $o > $o > list_l1255154591254203759pVar_o ) > ( $o > $o > list_l1255154591254203759pVar_o ) > ( list_o > list_l1255154591254203759pVar_o ) > ( list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001_Eo_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    multis8070414971136632605pVar_o: ( ( $o > $o ) > list_P1665994083459302249pVar_o > $o ) > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > ( $o > list_P1665994083459302249pVar_o ) > ( $o > list_P1665994083459302249pVar_o ) > ( $o > $o > list_P1665994083459302249pVar_o ) > ( $o > $o > $o > list_P1665994083459302249pVar_o ) > ( $o > $o > $o > $o > list_P1665994083459302249pVar_o ) > ( $o > $o > $o > list_P1665994083459302249pVar_o ) > ( $o > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o ) > ( $o > $o > list_P1665994083459302249pVar_o ) > ( $o > $o > list_P1665994083459302249pVar_o ) > ( list_o > list_P1665994083459302249pVar_o ) > ( list_l1255154591254203759pVar_o > list_P1665994083459302249pVar_o ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001_Eo_001t__Nat__Onat,type,
    multis8460249006151941916_o_nat: ( ( $o > $o ) > nat > $o ) > nat > nat > ( $o > nat ) > ( $o > nat ) > ( $o > $o > nat ) > ( $o > $o > $o > nat ) > ( $o > $o > $o > $o > nat ) > ( $o > $o > $o > nat ) > ( $o > nat > nat ) > ( $o > $o > nat ) > ( $o > $o > nat ) > ( list_o > nat ) > ( list_nat > nat ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001_Eo_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    multis42441612937804439pVar_o: ( ( $o > $o ) > produc8999643685545340131pVar_o > $o ) > produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > ( $o > produc8999643685545340131pVar_o ) > ( $o > produc8999643685545340131pVar_o ) > ( $o > $o > produc8999643685545340131pVar_o ) > ( $o > $o > $o > produc8999643685545340131pVar_o ) > ( $o > $o > $o > $o > produc8999643685545340131pVar_o ) > ( $o > $o > $o > produc8999643685545340131pVar_o ) > ( $o > produc8999643685545340131pVar_o > produc8999643685545340131pVar_o ) > ( $o > $o > produc8999643685545340131pVar_o ) > ( $o > $o > produc8999643685545340131pVar_o ) > ( list_o > produc8999643685545340131pVar_o ) > ( list_P1665994083459302249pVar_o > produc8999643685545340131pVar_o ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001_Eo_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,
    multis4555596752605892902ropVar: ( ( $o > $o ) > produc2642885279816024434ropVar > $o ) > produc2642885279816024434ropVar > produc2642885279816024434ropVar > ( $o > produc2642885279816024434ropVar ) > ( $o > produc2642885279816024434ropVar ) > ( $o > $o > produc2642885279816024434ropVar ) > ( $o > $o > $o > produc2642885279816024434ropVar ) > ( $o > $o > $o > $o > produc2642885279816024434ropVar ) > ( $o > $o > $o > produc2642885279816024434ropVar ) > ( $o > produc2642885279816024434ropVar > produc2642885279816024434ropVar ) > ( $o > $o > produc2642885279816024434ropVar ) > ( $o > $o > produc2642885279816024434ropVar ) > ( list_o > produc2642885279816024434ropVar ) > ( list_P8791008543742759928ropVar > produc2642885279816024434ropVar ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001t__Multiset____Ordering____in____NP__OPropVar_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    multis1610358401135282399pVar_o: ( ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o ) > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > ( multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o ) > ( list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001t__Multiset____Ordering____in____NP__OPropVar_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    multis4082459982090122969pVar_o: ( ( multis3193088007478089820ropVar > $o ) > list_P1665994083459302249pVar_o > $o ) > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > ( multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( list_M6740372356978808802ropVar > list_P1665994083459302249pVar_o ) > ( list_l1255154591254203759pVar_o > list_P1665994083459302249pVar_o ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001t__Multiset____Ordering____in____NP__OPropVar_001t__Nat__Onat,type,
    multis1673919852514530400ar_nat: ( ( multis3193088007478089820ropVar > $o ) > nat > $o ) > 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 ) > ( list_nat > nat ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001t__Multiset____Ordering____in____NP__OPropVar_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    multis1616071767867332691pVar_o: ( ( multis3193088007478089820ropVar > $o ) > produc8999643685545340131pVar_o > $o ) > produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > ( multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > produc8999643685545340131pVar_o > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( list_M6740372356978808802ropVar > produc8999643685545340131pVar_o ) > ( list_P1665994083459302249pVar_o > produc8999643685545340131pVar_o ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_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,
    multis885672736175375586ropVar: ( ( multis3193088007478089820ropVar > $o ) > produc2642885279816024434ropVar > $o ) > 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 ) > ( list_P8791008543742759928ropVar > produc2642885279816024434ropVar ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_001t__Multiset____Ordering____in____NP__OPropVar_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    multis8941024563284006719ropVar: ( ( multis3193088007478089820ropVar > $o ) > propos3600597766819905199ropVar > $o ) > propos3600597766819905199ropVar > propos3600597766819905199ropVar > ( multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > propos3600597766819905199ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( list_M6740372356978808802ropVar > propos3600597766819905199ropVar ) > ( list_P723798329145367999ropVar > propos3600597766819905199ropVar ) > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_OencodeGammaCond_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis6017319109170748100ropVar: list_l1255154591254203759pVar_o > ( multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > multis3193088007478089820ropVar > multis3193088007478089820ropVar > $o > $o > list_l1255154591254203759pVar_o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_OencodeGammaCond__rel_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis3344368520953930393ropVar: produc3507939890032686909od_o_o > produc3507939890032686909od_o_o > $o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__exactly__one_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_001t__Multiset____Ordering____in____NP__OPropVar,type,
    multis1909274777570284281ropVar: list_l1255154591254203759pVar_o > ( multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > list_P8791008543742759928ropVar > produc5436468477138497005pVar_o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001_Eo_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    multis4124385670053463658pVar_o: ( $o > $o > list_l1255154591254203759pVar_o ) > ( $o > $o > $o > list_l1255154591254203759pVar_o ) > ( $o > $o > $o > $o > list_l1255154591254203759pVar_o ) > list_P7815964300992299068od_o_o > produc7254072911610423961od_o_o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001_Eo_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    multis5520445341102945764pVar_o: ( $o > $o > list_P1665994083459302249pVar_o ) > ( $o > $o > $o > list_P1665994083459302249pVar_o ) > ( $o > $o > $o > $o > list_P1665994083459302249pVar_o ) > list_P7815964300992299068od_o_o > produc7892988369987590943od_o_o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001_Eo_001t__Nat__Onat,type,
    multis4961962968097414037_o_nat: ( $o > $o > nat ) > ( $o > $o > $o > nat ) > ( $o > $o > $o > $o > nat ) > list_P7815964300992299068od_o_o > produc6664232120876310204od_o_o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001_Eo_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    multis4654419120101021918pVar_o: ( $o > $o > produc8999643685545340131pVar_o ) > ( $o > $o > $o > produc8999643685545340131pVar_o ) > ( $o > $o > $o > $o > produc8999643685545340131pVar_o ) > list_P7815964300992299068od_o_o > produc7167215054540732453od_o_o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001_Eo_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,
    multis5036415435641082093ropVar: ( $o > $o > produc2642885279816024434ropVar ) > ( $o > $o > $o > produc2642885279816024434ropVar ) > ( $o > $o > $o > $o > produc2642885279816024434ropVar ) > list_P7815964300992299068od_o_o > produc4256396717503242198od_o_o ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001t__Multiset____Ordering____in____NP__OPropVar_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    multis6898674672137113254pVar_o: ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o ) > list_P8791008543742759928ropVar > produc9206049318796031513ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001t__Multiset____Ordering____in____NP__OPropVar_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    multis8843582314722954272pVar_o: ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o ) > list_P8791008543742759928ropVar > produc6485719899986225311ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001t__Multiset____Ordering____in____NP__OPropVar_001t__Nat__Onat,type,
    multis3310578786418936409ar_nat: ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > nat ) > list_P8791008543742759928ropVar > produc1716520392357129020ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001t__Multiset____Ordering____in____NP__OPropVar_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J,type,
    multis8112049843181714714pVar_o: ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o ) > list_P8791008543742759928ropVar > produc7342324450973665189ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_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,
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thf(sy_c_Multiset__Ordering__in__NP_Oencoder_Oencode__sum__0__1__main_001t__Multiset____Ordering____in____NP__OPropVar_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    multis4183284423519575736ropVar: ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > ( multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar ) > list_P8791008543742759928ropVar > produc6318262956711666589ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Opf__encode__exactly__one,type,
    multis7914470704280101590ly_one: list_P8791008543742759928ropVar > produc6657367779575603407ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_Opf__encode__sum__0__1__main,type,
    multis55434767431285815131_main: list_P8791008543742759928ropVar > produc6318262956711666589ropVar ).

thf(sy_c_Nat_OSuc,type,
    suc: nat > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_I_Eo_J,type,
    size_size_list_o: list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
    size_s6265379629157799777pVar_o: list_l985434930288970485pVar_o > 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_M_Eo_J_J_J,type,
    size_s1343900380416901083pVar_o: list_l1255154591254203759pVar_o > 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__List__Olist_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    size_s2237552474102614243ropVar: list_l1644708917474426447ropVar > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
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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__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    size_s8966483601851171411ropVar: list_P723798329145367999ropVar > nat ).

thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat,type,
    numeral_numeral_nat: num > nat ).

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat,type,
    ord_less_eq_nat: nat > nat > $o ).

thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_M_062_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
    produc8106139234514114142pVar_o: ( list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o ) > list_l985434930288970485pVar_o > produc3853458377486834414pVar_o ).

thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc7873628753273084125st_nat: ( list_l1255154591254203759pVar_o > nat ) > produc1766241633111104772st_nat > produc1879778804429160035st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_M_062_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_M_Eo_J_J_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    produc1228089817268051288pVar_o: ( list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o ) > list_l1255154591254203759pVar_o > produc3305181862331950184pVar_o ).

thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_Mt__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_J,type,
    produc1538423139514444329pVar_o: ( list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o ) > produc5436468477138497005pVar_o > produc6278577299168974967pVar_o ).

thf(sy_c_Product__Type_OPair_001_062_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc7356997260756735697st_nat: ( list_P1665994083459302249pVar_o > nat ) > produc7743431174791021054st_nat > produc7529890578683950167st_nat ).

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__Nat__Onat_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_J,type,
    produc4434012089350267529pVar_o: ( nat > list_l1255154591254203759pVar_o ) > produc2806087519671908746pVar_o > produc5051532902548563215pVar_o ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J,type,
    produc3361903918338910589pVar_o: ( nat > list_P1665994083459302249pVar_o ) > produc8682209073212595204pVar_o > produc5715669060643202051pVar_o ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc4626581765195395529st_nat: ( nat > nat ) > produc1828647624359046049st_nat > produc1616951275169580055st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    produc663790639386698097pVar_o: ( nat > produc8999643685545340131pVar_o ) > produc5599574116987515390pVar_o > produc6111255831404647415pVar_o ).

thf(sy_c_Product__Type_OPair_001_062_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_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,
    produc5722207277547347983ropVar: ( nat > produc2642885279816024434ropVar ) > produc8527832761551885325ropVar > produc5215109234026290197ropVar ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_M_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_M_Eo_J_J_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    produc3277683845840367058pVar_o: ( produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o ) > list_P1665994083459302249pVar_o > produc7264647458362488674pVar_o ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_Mt__Nat__Onat_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_Mt__List__Olist_It__Nat__Onat_J_J,type,
    produc3864453691545290181st_nat: ( produc8999643685545340131pVar_o > nat ) > produc2695100837817899640st_nat > produc3061377351390157643st_nat ).

thf(sy_c_Product__Type_OPair_001_062_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
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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_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_Mt__Nat__Onat_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__Nat__Onat_J_J,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001_Eo,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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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__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J,type,
    member4118595025335937317pVar_o: produc6450653815840180366pVar_o > set_Pr8506680132865545540pVar_o > $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_M_Eo_J_J,type,
    member6910252441317339679pVar_o: produc4726981232939080200pVar_o > set_Pr8900919299660647230pVar_o > $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_I_Eo_M_Eo_J_Mt__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    member3169104723136097040od_o_o: produc4465838814804890855od_o_o > set_Pr7011910401827753159od_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_Mt__Nat__Onat_J,type,
    member7944572178332388089_o_nat: produc5761300969954128610_o_nat > set_Pr302692805937157912_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J,type,
    member693507358219557392pVar_o: produc7972612925037412583pVar_o > set_Pr4574656050966626247pVar_o > $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_c_member_001t__Product____Type__Oprod_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_Mt__List__Olist_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
    member4962873584118313464ropVar: produc6657367779575603407ropVar > set_Pr3339951299666510767ropVar > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_Mt__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    member8566377481635068392ropVar: produc3879394860512280639ropVar > set_Pr572148484142614943ropVar > $o ).

thf(sy_c_member_001t__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    member1796676888578033094ropVar: propos3600597766819905199ropVar > set_Pr5911590690006159205ropVar > $o ).

thf(sy_v_conds,type,
    conds: list_l985434930288970485pVar_o ).

thf(sy_v_one,type,
    one: list_l1255154591254203759pVar_o ).

thf(sy_v_thesis____,type,
    thesis: $o ).

thf(sy_v_vars,type,
    vars: list_P8791008543742759928ropVar ).

% Relevant facts (1273)
thf(fact_0_False,axiom,
    vars != nil_Pr6119456312066593144ropVar ).

% False
thf(fact_1_cnf__encoder_Obex1__cong,axiom,
    ! [Cs: nat > nat > $o,S: nat > $o,T: nat > $o,S2: set_Product_prod_o_o,Cns: nat > nat > $o,NS: set_Product_prod_o_o,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,P: product_prod_o_o > $o,Q: product_prod_o_o > $o] :
      ( ! [I: nat,J: nat] :
          ( ( Cs @ I @ J )
          = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( S @ I ) @ ( T @ J ) ) @ S2 ) )
     => ( ! [I: nat,J: nat] :
            ( ( Cns @ I @ J )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( S @ I ) @ ( T @ J ) ) @ NS ) )
       => ( ( X = Y )
         => ( ! [X2: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ X2 @ Y )
               => ( ( P @ X2 )
                  = ( Q @ X2 ) ) )
           => ( ( ? [X3: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ X3 @ X )
                    & ( P @ X3 )
                    & ! [Y2: product_prod_o_o] :
                        ( ( ( member7466972457876170832od_o_o @ Y2 @ X )
                          & ( P @ Y2 ) )
                       => ( Y2 = X3 ) ) ) )
              = ( ? [X3: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ X3 @ Y )
                    & ( Q @ X3 )
                    & ! [Y2: product_prod_o_o] :
                        ( ( ( member7466972457876170832od_o_o @ Y2 @ Y )
                          & ( Q @ Y2 ) )
                       => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ).

% cnf_encoder.bex1_cong
thf(fact_2_cnf__encoder_Obex1__cong,axiom,
    ! [Cs: nat > nat > $o,S: nat > multis3193088007478089820ropVar,T: nat > multis3193088007478089820ropVar,S2: set_Pr7799659934693882759ropVar,Cns: nat > nat > $o,NS: set_Pr7799659934693882759ropVar,X: set_Mu5658373669842198716ropVar,Y: set_Mu5658373669842198716ropVar,P: multis3193088007478089820ropVar > $o,Q: multis3193088007478089820ropVar > $o] :
      ( ! [I: nat,J: nat] :
          ( ( Cs @ I @ J )
          = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I ) @ ( T @ J ) ) @ S2 ) )
     => ( ! [I: nat,J: nat] :
            ( ( Cns @ I @ J )
            = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I ) @ ( T @ J ) ) @ NS ) )
       => ( ( X = Y )
         => ( ! [X2: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X2 @ Y )
               => ( ( P @ X2 )
                  = ( Q @ X2 ) ) )
           => ( ( ? [X3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ X3 @ X )
                    & ( P @ X3 )
                    & ! [Y2: multis3193088007478089820ropVar] :
                        ( ( ( member8421201611525043589ropVar @ Y2 @ X )
                          & ( P @ Y2 ) )
                       => ( Y2 = X3 ) ) ) )
              = ( ? [X3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ X3 @ Y )
                    & ( Q @ X3 )
                    & ! [Y2: multis3193088007478089820ropVar] :
                        ( ( ( member8421201611525043589ropVar @ Y2 @ Y )
                          & ( Q @ Y2 ) )
                       => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ).

% cnf_encoder.bex1_cong
thf(fact_3_cnf__encoder_Obex1__cong,axiom,
    ! [Cs: nat > nat > $o,S: nat > multis3193088007478089820ropVar,T: nat > multis3193088007478089820ropVar,S2: set_Pr7799659934693882759ropVar,Cns: nat > nat > $o,NS: set_Pr7799659934693882759ropVar,X: set_Product_prod_o_o,Y: set_Product_prod_o_o,P: product_prod_o_o > $o,Q: product_prod_o_o > $o] :
      ( ! [I: nat,J: nat] :
          ( ( Cs @ I @ J )
          = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I ) @ ( T @ J ) ) @ S2 ) )
     => ( ! [I: nat,J: nat] :
            ( ( Cns @ I @ J )
            = ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ ( S @ I ) @ ( T @ J ) ) @ NS ) )
       => ( ( X = Y )
         => ( ! [X2: product_prod_o_o] :
                ( ( member7466972457876170832od_o_o @ X2 @ Y )
               => ( ( P @ X2 )
                  = ( Q @ X2 ) ) )
           => ( ( ? [X3: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ X3 @ X )
                    & ( P @ X3 )
                    & ! [Y2: product_prod_o_o] :
                        ( ( ( member7466972457876170832od_o_o @ Y2 @ X )
                          & ( P @ Y2 ) )
                       => ( Y2 = X3 ) ) ) )
              = ( ? [X3: product_prod_o_o] :
                    ( ( member7466972457876170832od_o_o @ X3 @ Y )
                    & ( Q @ X3 )
                    & ! [Y2: product_prod_o_o] :
                        ( ( ( member7466972457876170832od_o_o @ Y2 @ Y )
                          & ( Q @ Y2 ) )
                       => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ).

% cnf_encoder.bex1_cong
thf(fact_4_cnf__encoder_Obex1__cong,axiom,
    ! [Cs: nat > nat > $o,S: nat > $o,T: nat > $o,S2: set_Product_prod_o_o,Cns: nat > nat > $o,NS: set_Product_prod_o_o,X: set_Mu5658373669842198716ropVar,Y: set_Mu5658373669842198716ropVar,P: multis3193088007478089820ropVar > $o,Q: multis3193088007478089820ropVar > $o] :
      ( ! [I: nat,J: nat] :
          ( ( Cs @ I @ J )
          = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( S @ I ) @ ( T @ J ) ) @ S2 ) )
     => ( ! [I: nat,J: nat] :
            ( ( Cns @ I @ J )
            = ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( S @ I ) @ ( T @ J ) ) @ NS ) )
       => ( ( X = Y )
         => ( ! [X2: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X2 @ Y )
               => ( ( P @ X2 )
                  = ( Q @ X2 ) ) )
           => ( ( ? [X3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ X3 @ X )
                    & ( P @ X3 )
                    & ! [Y2: multis3193088007478089820ropVar] :
                        ( ( ( member8421201611525043589ropVar @ Y2 @ X )
                          & ( P @ Y2 ) )
                       => ( Y2 = X3 ) ) ) )
              = ( ? [X3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ X3 @ Y )
                    & ( Q @ X3 )
                    & ! [Y2: multis3193088007478089820ropVar] :
                        ( ( ( member8421201611525043589ropVar @ Y2 @ Y )
                          & ( Q @ Y2 ) )
                       => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ).

% cnf_encoder.bex1_cong
thf(fact_5_cnf__encoder_Oencode__sum__0__1__main_Ocases,axiom,
    ! [X4: list_P8791008543742759928ropVar] :
      ( ! [X2: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar] :
          ( X4
         != ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X2 @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ nil_Pr6119456312066593144ropVar ) )
     => ( ! [X2: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar,V: produc2642885279816024434ropVar,Va: list_P8791008543742759928ropVar] :
            ( X4
           != ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X2 @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ ( cons_P1504208554336013608ropVar @ V @ Va ) ) )
       => ( X4 = nil_Pr6119456312066593144ropVar ) ) ) ).

% cnf_encoder.encode_sum_0_1_main.cases
thf(fact_6_list_Oinject,axiom,
    ! [X21: produc2642885279816024434ropVar,X22: list_P8791008543742759928ropVar,Y21: produc2642885279816024434ropVar,Y22: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X21 @ X22 )
        = ( cons_P1504208554336013608ropVar @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_7_list_Oinject,axiom,
    ! [X21: produc8999643685545340131pVar_o,X22: list_P1665994083459302249pVar_o,Y21: produc8999643685545340131pVar_o,Y22: list_P1665994083459302249pVar_o] :
      ( ( ( cons_P6860203055867761049pVar_o @ X21 @ X22 )
        = ( cons_P6860203055867761049pVar_o @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_8_list_Oinject,axiom,
    ! [X21: list_P1665994083459302249pVar_o,X22: list_l1255154591254203759pVar_o,Y21: list_P1665994083459302249pVar_o,Y22: list_l1255154591254203759pVar_o] :
      ( ( ( cons_l3452452106213086495pVar_o @ X21 @ X22 )
        = ( cons_l3452452106213086495pVar_o @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_9_list_Oinject,axiom,
    ! [X21: list_l1255154591254203759pVar_o,X22: list_l985434930288970485pVar_o,Y21: list_l1255154591254203759pVar_o,Y22: list_l985434930288970485pVar_o] :
      ( ( ( cons_l7316790070876032037pVar_o @ X21 @ X22 )
        = ( cons_l7316790070876032037pVar_o @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_10_list_Oinject,axiom,
    ! [X21: nat,X22: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X22 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X22 = Y22 ) ) ) ).

% list.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: multis3193088007478089820ropVar,X23: multis3193088007478089820ropVar,Y1: multis3193088007478089820ropVar,Y23: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ X1 @ X23 )
        = ( produc6936306168532358295ropVar @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: propos3600597766819905199ropVar,X23: list_P723798329145367999ropVar,Y1: propos3600597766819905199ropVar,Y23: list_P723798329145367999ropVar] :
      ( ( ( produc5136867404843011847ropVar @ X1 @ X23 )
        = ( produc5136867404843011847ropVar @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_13_prod_Oinject,axiom,
    ! [X1: list_P723798329145367999ropVar,X23: produc7679832412844898215ropVar,Y1: list_P723798329145367999ropVar,Y23: produc7679832412844898215ropVar] :
      ( ( ( produc2463153034239240527ropVar @ X1 @ X23 )
        = ( produc2463153034239240527ropVar @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_14_prod_Oinject,axiom,
    ! [X1: list_l1255154591254203759pVar_o,X23: list_l985434930288970485pVar_o,Y1: list_l1255154591254203759pVar_o,Y23: list_l985434930288970485pVar_o] :
      ( ( ( produc4064290870397114077pVar_o @ X1 @ X23 )
        = ( produc4064290870397114077pVar_o @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_15_prod_Oinject,axiom,
    ! [X1: list_l985434930288970485pVar_o,X23: produc7679832412844898215ropVar,Y1: list_l985434930288970485pVar_o,Y23: produc7679832412844898215ropVar] :
      ( ( ( produc2520692699247754633ropVar @ X1 @ X23 )
        = ( produc2520692699247754633ropVar @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_16_prod_Oinject,axiom,
    ! [X1: $o,X23: $o,Y1: $o,Y23: $o] :
      ( ( ( product_Pair_o_o @ X1 @ X23 )
        = ( product_Pair_o_o @ Y1 @ Y23 ) )
      = ( ( X1 = Y1 )
        & ( X23 = Y23 ) ) ) ).

% prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A2: multis3193088007478089820ropVar,B2: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ A @ B )
        = ( produc6936306168532358295ropVar @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
    ! [A: propos3600597766819905199ropVar,B: list_P723798329145367999ropVar,A2: propos3600597766819905199ropVar,B2: list_P723798329145367999ropVar] :
      ( ( ( produc5136867404843011847ropVar @ A @ B )
        = ( produc5136867404843011847ropVar @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: list_P723798329145367999ropVar,B: produc7679832412844898215ropVar,A2: list_P723798329145367999ropVar,B2: produc7679832412844898215ropVar] :
      ( ( ( produc2463153034239240527ropVar @ A @ B )
        = ( produc2463153034239240527ropVar @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
    ! [A: list_l1255154591254203759pVar_o,B: list_l985434930288970485pVar_o,A2: list_l1255154591254203759pVar_o,B2: list_l985434930288970485pVar_o] :
      ( ( ( produc4064290870397114077pVar_o @ A @ B )
        = ( produc4064290870397114077pVar_o @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
    ! [A: list_l985434930288970485pVar_o,B: produc7679832412844898215ropVar,A2: list_l985434930288970485pVar_o,B2: produc7679832412844898215ropVar] :
      ( ( ( produc2520692699247754633ropVar @ A @ B )
        = ( produc2520692699247754633ropVar @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
    ! [A: $o,B: $o,A2: $o,B2: $o] :
      ( ( ( product_Pair_o_o @ A @ B )
        = ( product_Pair_o_o @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_23_prod__cases3,axiom,
    ! [Y3: produc6318262956711666589ropVar] :
      ~ ! [A3: list_P723798329145367999ropVar,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] :
          ( Y3
         != ( produc2463153034239240527ropVar @ A3 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_24_prod__cases3,axiom,
    ! [Y3: produc9206049318796031513ropVar] :
      ~ ! [A3: list_l985434930288970485pVar_o,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] :
          ( Y3
         != ( produc2520692699247754633ropVar @ A3 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_25_prod__induct3,axiom,
    ! [P: produc6318262956711666589ropVar > $o,X4: produc6318262956711666589ropVar] :
      ( ! [A3: list_P723798329145367999ropVar,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] : ( P @ ( produc2463153034239240527ropVar @ A3 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) )
     => ( P @ X4 ) ) ).

% prod_induct3
thf(fact_26_prod__induct3,axiom,
    ! [P: produc9206049318796031513ropVar > $o,X4: produc9206049318796031513ropVar] :
      ( ! [A3: list_l985434930288970485pVar_o,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] : ( P @ ( produc2520692699247754633ropVar @ A3 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) )
     => ( P @ X4 ) ) ).

% prod_induct3
thf(fact_27_assms,axiom,
    ( ( multis8929244183426903067ly_one @ vars )
    = ( produc4064290870397114077pVar_o @ one @ conds ) ) ).

% assms
thf(fact_28_map__tailrec__rev_Ocases,axiom,
    ! [X4: produc6278577299168974967pVar_o] :
      ( ! [F: list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o,Bs: list_l985434930288970485pVar_o] :
          ( X4
         != ( produc1538423139514444329pVar_o @ F @ ( produc4064290870397114077pVar_o @ nil_li1310946081133177199pVar_o @ Bs ) ) )
     => ~ ! [F: list_P1665994083459302249pVar_o > list_l1255154591254203759pVar_o,A3: list_P1665994083459302249pVar_o,As: list_l1255154591254203759pVar_o,Bs: list_l985434930288970485pVar_o] :
            ( X4
           != ( produc1538423139514444329pVar_o @ F @ ( produc4064290870397114077pVar_o @ ( cons_l3452452106213086495pVar_o @ A3 @ As ) @ Bs ) ) ) ) ).

% map_tailrec_rev.cases
thf(fact_29_successively_Ocases,axiom,
    ! [X4: produc5533280449304468145ropVar] :
      ( ! [P2: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
          ( X4
         != ( produc3887254912111605537ropVar @ P2 @ nil_Pr6119456312066593144ropVar ) )
     => ( ! [P2: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X2: produc2642885279816024434ropVar] :
            ( X4
           != ( produc3887254912111605537ropVar @ P2 @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) )
       => ~ ! [P2: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X2: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
              ( X4
             != ( produc3887254912111605537ropVar @ P2 @ ( cons_P1504208554336013608ropVar @ X2 @ ( cons_P1504208554336013608ropVar @ Y4 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_30_successively_Ocases,axiom,
    ! [X4: produc7264647458362488674pVar_o] :
      ( ! [P2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o] :
          ( X4
         != ( produc3277683845840367058pVar_o @ P2 @ nil_Pr6413368191065476585pVar_o ) )
     => ( ! [P2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X2: produc8999643685545340131pVar_o] :
            ( X4
           != ( produc3277683845840367058pVar_o @ P2 @ ( cons_P6860203055867761049pVar_o @ X2 @ nil_Pr6413368191065476585pVar_o ) ) )
       => ~ ! [P2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X2: produc8999643685545340131pVar_o,Y4: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
              ( X4
             != ( produc3277683845840367058pVar_o @ P2 @ ( cons_P6860203055867761049pVar_o @ X2 @ ( cons_P6860203055867761049pVar_o @ Y4 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_31_successively_Ocases,axiom,
    ! [X4: produc3305181862331950184pVar_o] :
      ( ! [P2: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o] :
          ( X4
         != ( produc1228089817268051288pVar_o @ P2 @ nil_li1310946081133177199pVar_o ) )
     => ( ! [P2: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X2: list_P1665994083459302249pVar_o] :
            ( X4
           != ( produc1228089817268051288pVar_o @ P2 @ ( cons_l3452452106213086495pVar_o @ X2 @ nil_li1310946081133177199pVar_o ) ) )
       => ~ ! [P2: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X2: list_P1665994083459302249pVar_o,Y4: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
              ( X4
             != ( produc1228089817268051288pVar_o @ P2 @ ( cons_l3452452106213086495pVar_o @ X2 @ ( cons_l3452452106213086495pVar_o @ Y4 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_32_successively_Ocases,axiom,
    ! [X4: produc3853458377486834414pVar_o] :
      ( ! [P2: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o] :
          ( X4
         != ( produc8106139234514114142pVar_o @ P2 @ nil_li590035233778055285pVar_o ) )
     => ( ! [P2: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X2: list_l1255154591254203759pVar_o] :
            ( X4
           != ( produc8106139234514114142pVar_o @ P2 @ ( cons_l7316790070876032037pVar_o @ X2 @ nil_li590035233778055285pVar_o ) ) )
       => ~ ! [P2: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X2: list_l1255154591254203759pVar_o,Y4: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
              ( X4
             != ( produc8106139234514114142pVar_o @ P2 @ ( cons_l7316790070876032037pVar_o @ X2 @ ( cons_l7316790070876032037pVar_o @ Y4 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_33_successively_Ocases,axiom,
    ! [X4: produc254973753779126261st_nat] :
      ( ! [P2: nat > nat > $o] :
          ( X4
         != ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
     => ( ! [P2: nat > nat > $o,X2: nat] :
            ( X4
           != ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ nil_nat ) ) )
       => ~ ! [P2: nat > nat > $o,X2: nat,Y4: nat,Xs: list_nat] :
              ( X4
             != ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ ( cons_nat @ Y4 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_34_sorted__wrt_Ocases,axiom,
    ! [X4: produc5533280449304468145ropVar] :
      ( ! [P2: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
          ( X4
         != ( produc3887254912111605537ropVar @ P2 @ nil_Pr6119456312066593144ropVar ) )
     => ~ ! [P2: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X2: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
            ( X4
           != ( produc3887254912111605537ropVar @ P2 @ ( cons_P1504208554336013608ropVar @ X2 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_35_sorted__wrt_Ocases,axiom,
    ! [X4: produc7264647458362488674pVar_o] :
      ( ! [P2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o] :
          ( X4
         != ( produc3277683845840367058pVar_o @ P2 @ nil_Pr6413368191065476585pVar_o ) )
     => ~ ! [P2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X2: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
            ( X4
           != ( produc3277683845840367058pVar_o @ P2 @ ( cons_P6860203055867761049pVar_o @ X2 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_36_sorted__wrt_Ocases,axiom,
    ! [X4: produc3305181862331950184pVar_o] :
      ( ! [P2: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o] :
          ( X4
         != ( produc1228089817268051288pVar_o @ P2 @ nil_li1310946081133177199pVar_o ) )
     => ~ ! [P2: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X2: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
            ( X4
           != ( produc1228089817268051288pVar_o @ P2 @ ( cons_l3452452106213086495pVar_o @ X2 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_37_sorted__wrt_Ocases,axiom,
    ! [X4: produc3853458377486834414pVar_o] :
      ( ! [P2: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o] :
          ( X4
         != ( produc8106139234514114142pVar_o @ P2 @ nil_li590035233778055285pVar_o ) )
     => ~ ! [P2: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X2: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] :
            ( X4
           != ( produc8106139234514114142pVar_o @ P2 @ ( cons_l7316790070876032037pVar_o @ X2 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_38_sorted__wrt_Ocases,axiom,
    ! [X4: produc254973753779126261st_nat] :
      ( ! [P2: nat > nat > $o] :
          ( X4
         != ( produc4727192421694094319st_nat @ P2 @ nil_nat ) )
     => ~ ! [P2: nat > nat > $o,X2: nat,Ys: list_nat] :
            ( X4
           != ( produc4727192421694094319st_nat @ P2 @ ( cons_nat @ X2 @ Ys ) ) ) ) ).

% sorted_wrt.cases
thf(fact_39_shuffles_Ocases,axiom,
    ! [X4: produc3638977808798937127ropVar] :
      ( ! [Ys: list_P8791008543742759928ropVar] :
          ( X4
         != ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys ) )
     => ( ! [Xs: list_P8791008543742759928ropVar] :
            ( X4
           != ( produc5333478016634530327ropVar @ Xs @ nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
              ( X4
             != ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_40_shuffles_Ocases,axiom,
    ! [X4: produc4416794199230284903pVar_o] :
      ( ! [Ys: list_P1665994083459302249pVar_o] :
          ( X4
         != ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Ys ) )
     => ( ! [Xs: list_P1665994083459302249pVar_o] :
            ( X4
           != ( produc9064796567173055959pVar_o @ Xs @ nil_Pr6413368191065476585pVar_o ) )
       => ~ ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y4: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
              ( X4
             != ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_41_shuffles_Ocases,axiom,
    ! [X4: produc2246139759411191271pVar_o] :
      ( ! [Ys: list_l1255154591254203759pVar_o] :
          ( X4
         != ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Ys ) )
     => ( ! [Xs: list_l1255154591254203759pVar_o] :
            ( X4
           != ( produc5036572322992794967pVar_o @ Xs @ nil_li1310946081133177199pVar_o ) )
       => ~ ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Y4: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
              ( X4
             != ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) @ ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_42_shuffles_Ocases,axiom,
    ! [X4: produc1241035484025441639pVar_o] :
      ( ! [Ys: list_l985434930288970485pVar_o] :
          ( X4
         != ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Ys ) )
     => ( ! [Xs: list_l985434930288970485pVar_o] :
            ( X4
           != ( produc2080502776689584855pVar_o @ Xs @ nil_li590035233778055285pVar_o ) )
       => ~ ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Y4: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] :
              ( X4
             != ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) @ ( cons_l7316790070876032037pVar_o @ Y4 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_43_shuffles_Ocases,axiom,
    ! [X4: produc1828647624359046049st_nat] :
      ( ! [Ys: list_nat] :
          ( X4
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
     => ( ! [Xs: list_nat] :
            ( X4
           != ( produc2694037385005941721st_nat @ Xs @ nil_nat ) )
       => ~ ! [X2: nat,Xs: list_nat,Y4: nat,Ys: list_nat] :
              ( X4
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y4 @ Ys ) ) ) ) ) ).

% shuffles.cases
thf(fact_44_transpose_Ocases,axiom,
    ! [X4: list_l3415868495287753214ropVar] :
      ( ( X4 != nil_li2679230897895549182ropVar )
     => ( ! [Xss: list_l3415868495287753214ropVar] :
            ( X4
           != ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss ) )
       => ~ ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Xss: list_l3415868495287753214ropVar] :
              ( X4
             != ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_45_transpose_Ocases,axiom,
    ! [X4: list_l6341520962544925691pVar_o] :
      ( ( X4 != nil_li5579170522155219707pVar_o )
     => ( ! [Xss: list_l6341520962544925691pVar_o] :
            ( X4
           != ( cons_l8096070148660426411pVar_o @ nil_li590035233778055285pVar_o @ Xss ) )
       => ~ ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Xss: list_l6341520962544925691pVar_o] :
              ( X4
             != ( cons_l8096070148660426411pVar_o @ ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_46_transpose_Ocases,axiom,
    ! [X4: list_list_nat] :
      ( ( X4 != nil_list_nat )
     => ( ! [Xss: list_list_nat] :
            ( X4
           != ( cons_list_nat @ nil_nat @ Xss ) )
       => ~ ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
              ( X4
             != ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_47_transpose_Ocases,axiom,
    ! [X4: list_l1255154591254203759pVar_o] :
      ( ( X4 != nil_li1310946081133177199pVar_o )
     => ( ! [Xss: list_l1255154591254203759pVar_o] :
            ( X4
           != ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ Xss ) )
       => ~ ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Xss: list_l1255154591254203759pVar_o] :
              ( X4
             != ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_48_transpose_Ocases,axiom,
    ! [X4: list_l985434930288970485pVar_o] :
      ( ( X4 != nil_li590035233778055285pVar_o )
     => ( ! [Xss: list_l985434930288970485pVar_o] :
            ( X4
           != ( cons_l7316790070876032037pVar_o @ nil_li1310946081133177199pVar_o @ Xss ) )
       => ~ ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Xss: list_l985434930288970485pVar_o] :
              ( X4
             != ( cons_l7316790070876032037pVar_o @ ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_49_list__nonempty__induct,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > $o] :
      ( ( Xs2 != nil_Pr6119456312066593144ropVar )
     => ( ! [X2: produc2642885279816024434ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) )
       => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
              ( ( Xs != nil_Pr6119456312066593144ropVar )
             => ( ( P @ Xs )
               => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_50_list__nonempty__induct,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
     => ( ! [X2: produc8999643685545340131pVar_o] : ( P @ ( cons_P6860203055867761049pVar_o @ X2 @ nil_Pr6413368191065476585pVar_o ) )
       => ( ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
              ( ( Xs != nil_Pr6413368191065476585pVar_o )
             => ( ( P @ Xs )
               => ( P @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_51_list__nonempty__induct,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ( Xs2 != nil_li1310946081133177199pVar_o )
     => ( ! [X2: list_P1665994083459302249pVar_o] : ( P @ ( cons_l3452452106213086495pVar_o @ X2 @ nil_li1310946081133177199pVar_o ) )
       => ( ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
              ( ( Xs != nil_li1310946081133177199pVar_o )
             => ( ( P @ Xs )
               => ( P @ ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_52_list__nonempty__induct,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ( Xs2 != nil_li590035233778055285pVar_o )
     => ( ! [X2: list_l1255154591254203759pVar_o] : ( P @ ( cons_l7316790070876032037pVar_o @ X2 @ nil_li590035233778055285pVar_o ) )
       => ( ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
              ( ( Xs != nil_li590035233778055285pVar_o )
             => ( ( P @ Xs )
               => ( P @ ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_53_list__nonempty__induct,axiom,
    ! [Xs2: list_nat,P: list_nat > $o] :
      ( ( Xs2 != nil_nat )
     => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
       => ( ! [X2: nat,Xs: list_nat] :
              ( ( Xs != nil_nat )
             => ( ( P @ Xs )
               => ( P @ ( cons_nat @ X2 @ Xs ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_54_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs2: list_nat,Ys2: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y4: nat,Ys: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y4: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_55_list__induct2_H,axiom,
    ! [P: list_P1665994083459302249pVar_o > list_nat > $o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_nat] :
      ( ( P @ nil_Pr6413368191065476585pVar_o @ nil_nat )
     => ( ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] : ( P @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y4: nat,Ys: list_nat] : ( P @ nil_Pr6413368191065476585pVar_o @ ( cons_nat @ Y4 @ Ys ) )
         => ( ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y4: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) @ ( cons_nat @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_56_list__induct2_H,axiom,
    ! [P: list_nat > list_P1665994083459302249pVar_o > $o,Xs2: list_nat,Ys2: list_P1665994083459302249pVar_o] :
      ( ( P @ nil_nat @ nil_Pr6413368191065476585pVar_o )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_Pr6413368191065476585pVar_o )
       => ( ! [Y4: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] : ( P @ nil_nat @ ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y4: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_57_list__induct2_H,axiom,
    ! [P: list_l1255154591254203759pVar_o > list_nat > $o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_nat] :
      ( ( P @ nil_li1310946081133177199pVar_o @ nil_nat )
     => ( ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] : ( P @ ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y4: nat,Ys: list_nat] : ( P @ nil_li1310946081133177199pVar_o @ ( cons_nat @ Y4 @ Ys ) )
         => ( ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Y4: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) @ ( cons_nat @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_58_list__induct2_H,axiom,
    ! [P: list_nat > list_l1255154591254203759pVar_o > $o,Xs2: list_nat,Ys2: list_l1255154591254203759pVar_o] :
      ( ( P @ nil_nat @ nil_li1310946081133177199pVar_o )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_li1310946081133177199pVar_o )
       => ( ! [Y4: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] : ( P @ nil_nat @ ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y4: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_59_list__induct2_H,axiom,
    ! [P: list_P8791008543742759928ropVar > list_nat > $o,Xs2: list_P8791008543742759928ropVar,Ys2: list_nat] :
      ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat )
     => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y4: nat,Ys: list_nat] : ( P @ nil_Pr6119456312066593144ropVar @ ( cons_nat @ Y4 @ Ys ) )
         => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_nat @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_60_list__induct2_H,axiom,
    ! [P: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( P @ nil_Pr6413368191065476585pVar_o @ nil_Pr6413368191065476585pVar_o )
     => ( ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] : ( P @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) @ nil_Pr6413368191065476585pVar_o )
       => ( ! [Y4: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] : ( P @ nil_Pr6413368191065476585pVar_o @ ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
         => ( ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Y4: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) @ ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_61_list__induct2_H,axiom,
    ! [P: list_l985434930288970485pVar_o > list_nat > $o,Xs2: list_l985434930288970485pVar_o,Ys2: list_nat] :
      ( ( P @ nil_li590035233778055285pVar_o @ nil_nat )
     => ( ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] : ( P @ ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) @ nil_nat )
       => ( ! [Y4: nat,Ys: list_nat] : ( P @ nil_li590035233778055285pVar_o @ ( cons_nat @ Y4 @ Ys ) )
         => ( ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Y4: nat,Ys: list_nat] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) @ ( cons_nat @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_62_list__induct2_H,axiom,
    ! [P: list_nat > list_P8791008543742759928ropVar > $o,Xs2: list_nat,Ys2: list_P8791008543742759928ropVar] :
      ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_Pr6119456312066593144ropVar )
       => ( ! [Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] : ( P @ nil_nat @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_63_list__induct2_H,axiom,
    ! [P: list_nat > list_l985434930288970485pVar_o > $o,Xs2: list_nat,Ys2: list_l985434930288970485pVar_o] :
      ( ( P @ nil_nat @ nil_li590035233778055285pVar_o )
     => ( ! [X2: nat,Xs: list_nat] : ( P @ ( cons_nat @ X2 @ Xs ) @ nil_li590035233778055285pVar_o )
       => ( ! [Y4: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] : ( P @ nil_nat @ ( cons_l7316790070876032037pVar_o @ Y4 @ Ys ) )
         => ( ! [X2: nat,Xs: list_nat,Y4: list_l1255154591254203759pVar_o,Ys: list_l985434930288970485pVar_o] :
                ( ( P @ Xs @ Ys )
               => ( P @ ( cons_nat @ X2 @ Xs ) @ ( cons_l7316790070876032037pVar_o @ Y4 @ Ys ) ) )
           => ( P @ Xs2 @ Ys2 ) ) ) ) ) ).

% list_induct2'
thf(fact_64_neq__Nil__conv,axiom,
    ! [Xs2: list_P8791008543742759928ropVar] :
      ( ( Xs2 != nil_Pr6119456312066593144ropVar )
      = ( ? [Y2: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] :
            ( Xs2
            = ( cons_P1504208554336013608ropVar @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_65_neq__Nil__conv,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o] :
      ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
      = ( ? [Y2: produc8999643685545340131pVar_o,Ys3: list_P1665994083459302249pVar_o] :
            ( Xs2
            = ( cons_P6860203055867761049pVar_o @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_66_neq__Nil__conv,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o] :
      ( ( Xs2 != nil_li1310946081133177199pVar_o )
      = ( ? [Y2: list_P1665994083459302249pVar_o,Ys3: list_l1255154591254203759pVar_o] :
            ( Xs2
            = ( cons_l3452452106213086495pVar_o @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_67_neq__Nil__conv,axiom,
    ! [Xs2: list_l985434930288970485pVar_o] :
      ( ( Xs2 != nil_li590035233778055285pVar_o )
      = ( ? [Y2: list_l1255154591254203759pVar_o,Ys3: list_l985434930288970485pVar_o] :
            ( Xs2
            = ( cons_l7316790070876032037pVar_o @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_68_neq__Nil__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
      = ( ? [Y2: nat,Ys3: list_nat] :
            ( Xs2
            = ( cons_nat @ Y2 @ Ys3 ) ) ) ) ).

% neq_Nil_conv
thf(fact_69_remdups__adj_Ocases,axiom,
    ! [X4: list_P8791008543742759928ropVar] :
      ( ( X4 != nil_Pr6119456312066593144ropVar )
     => ( ! [X2: produc2642885279816024434ropVar] :
            ( X4
           != ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X2: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
              ( X4
             != ( cons_P1504208554336013608ropVar @ X2 @ ( cons_P1504208554336013608ropVar @ Y4 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_70_remdups__adj_Ocases,axiom,
    ! [X4: list_P1665994083459302249pVar_o] :
      ( ( X4 != nil_Pr6413368191065476585pVar_o )
     => ( ! [X2: produc8999643685545340131pVar_o] :
            ( X4
           != ( cons_P6860203055867761049pVar_o @ X2 @ nil_Pr6413368191065476585pVar_o ) )
       => ~ ! [X2: produc8999643685545340131pVar_o,Y4: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
              ( X4
             != ( cons_P6860203055867761049pVar_o @ X2 @ ( cons_P6860203055867761049pVar_o @ Y4 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_71_remdups__adj_Ocases,axiom,
    ! [X4: list_l1255154591254203759pVar_o] :
      ( ( X4 != nil_li1310946081133177199pVar_o )
     => ( ! [X2: list_P1665994083459302249pVar_o] :
            ( X4
           != ( cons_l3452452106213086495pVar_o @ X2 @ nil_li1310946081133177199pVar_o ) )
       => ~ ! [X2: list_P1665994083459302249pVar_o,Y4: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
              ( X4
             != ( cons_l3452452106213086495pVar_o @ X2 @ ( cons_l3452452106213086495pVar_o @ Y4 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_72_remdups__adj_Ocases,axiom,
    ! [X4: list_l985434930288970485pVar_o] :
      ( ( X4 != nil_li590035233778055285pVar_o )
     => ( ! [X2: list_l1255154591254203759pVar_o] :
            ( X4
           != ( cons_l7316790070876032037pVar_o @ X2 @ nil_li590035233778055285pVar_o ) )
       => ~ ! [X2: list_l1255154591254203759pVar_o,Y4: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
              ( X4
             != ( cons_l7316790070876032037pVar_o @ X2 @ ( cons_l7316790070876032037pVar_o @ Y4 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_73_remdups__adj_Ocases,axiom,
    ! [X4: list_nat] :
      ( ( X4 != nil_nat )
     => ( ! [X2: nat] :
            ( X4
           != ( cons_nat @ X2 @ nil_nat ) )
       => ~ ! [X2: nat,Y4: nat,Xs: list_nat] :
              ( X4
             != ( cons_nat @ X2 @ ( cons_nat @ Y4 @ Xs ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_74_min__list_Ocases,axiom,
    ! [X4: list_nat] :
      ( ! [X2: nat,Xs: list_nat] :
          ( X4
         != ( cons_nat @ X2 @ Xs ) )
     => ( X4 = nil_nat ) ) ).

% min_list.cases
thf(fact_75_list_Oexhaust,axiom,
    ! [Y3: list_P8791008543742759928ropVar] :
      ( ( Y3 != nil_Pr6119456312066593144ropVar )
     => ~ ! [X212: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
            ( Y3
           != ( cons_P1504208554336013608ropVar @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_76_list_Oexhaust,axiom,
    ! [Y3: list_P1665994083459302249pVar_o] :
      ( ( Y3 != nil_Pr6413368191065476585pVar_o )
     => ~ ! [X212: produc8999643685545340131pVar_o,X222: list_P1665994083459302249pVar_o] :
            ( Y3
           != ( cons_P6860203055867761049pVar_o @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_77_list_Oexhaust,axiom,
    ! [Y3: list_l1255154591254203759pVar_o] :
      ( ( Y3 != nil_li1310946081133177199pVar_o )
     => ~ ! [X212: list_P1665994083459302249pVar_o,X222: list_l1255154591254203759pVar_o] :
            ( Y3
           != ( cons_l3452452106213086495pVar_o @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_78_list_Oexhaust,axiom,
    ! [Y3: list_l985434930288970485pVar_o] :
      ( ( Y3 != nil_li590035233778055285pVar_o )
     => ~ ! [X212: list_l1255154591254203759pVar_o,X222: list_l985434930288970485pVar_o] :
            ( Y3
           != ( cons_l7316790070876032037pVar_o @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_79_list_Oexhaust,axiom,
    ! [Y3: list_nat] :
      ( ( Y3 != nil_nat )
     => ~ ! [X212: nat,X222: list_nat] :
            ( Y3
           != ( cons_nat @ X212 @ X222 ) ) ) ).

% list.exhaust
thf(fact_80_list_OdiscI,axiom,
    ! [List: list_P8791008543742759928ropVar,X21: produc2642885279816024434ropVar,X22: list_P8791008543742759928ropVar] :
      ( ( List
        = ( cons_P1504208554336013608ropVar @ X21 @ X22 ) )
     => ( List != nil_Pr6119456312066593144ropVar ) ) ).

% list.discI
thf(fact_81_list_OdiscI,axiom,
    ! [List: list_P1665994083459302249pVar_o,X21: produc8999643685545340131pVar_o,X22: list_P1665994083459302249pVar_o] :
      ( ( List
        = ( cons_P6860203055867761049pVar_o @ X21 @ X22 ) )
     => ( List != nil_Pr6413368191065476585pVar_o ) ) ).

% list.discI
thf(fact_82_list_OdiscI,axiom,
    ! [List: list_l1255154591254203759pVar_o,X21: list_P1665994083459302249pVar_o,X22: list_l1255154591254203759pVar_o] :
      ( ( List
        = ( cons_l3452452106213086495pVar_o @ X21 @ X22 ) )
     => ( List != nil_li1310946081133177199pVar_o ) ) ).

% list.discI
thf(fact_83_list_OdiscI,axiom,
    ! [List: list_l985434930288970485pVar_o,X21: list_l1255154591254203759pVar_o,X22: list_l985434930288970485pVar_o] :
      ( ( List
        = ( cons_l7316790070876032037pVar_o @ X21 @ X22 ) )
     => ( List != nil_li590035233778055285pVar_o ) ) ).

% list.discI
thf(fact_84_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X22: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X22 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_85_list_Odistinct_I1_J,axiom,
    ! [X21: produc2642885279816024434ropVar,X22: list_P8791008543742759928ropVar] :
      ( nil_Pr6119456312066593144ropVar
     != ( cons_P1504208554336013608ropVar @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_86_list_Odistinct_I1_J,axiom,
    ! [X21: produc8999643685545340131pVar_o,X22: list_P1665994083459302249pVar_o] :
      ( nil_Pr6413368191065476585pVar_o
     != ( cons_P6860203055867761049pVar_o @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_87_list_Odistinct_I1_J,axiom,
    ! [X21: list_P1665994083459302249pVar_o,X22: list_l1255154591254203759pVar_o] :
      ( nil_li1310946081133177199pVar_o
     != ( cons_l3452452106213086495pVar_o @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_88_list_Odistinct_I1_J,axiom,
    ! [X21: list_l1255154591254203759pVar_o,X22: list_l985434930288970485pVar_o] :
      ( nil_li590035233778055285pVar_o
     != ( cons_l7316790070876032037pVar_o @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_89_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X22: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X22 ) ) ).

% list.distinct(1)
thf(fact_90_Pair__inject,axiom,
    ! [A: propos3600597766819905199ropVar,B: list_P723798329145367999ropVar,A2: propos3600597766819905199ropVar,B2: list_P723798329145367999ropVar] :
      ( ( ( produc5136867404843011847ropVar @ A @ B )
        = ( produc5136867404843011847ropVar @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_91_Pair__inject,axiom,
    ! [A: list_P723798329145367999ropVar,B: produc7679832412844898215ropVar,A2: list_P723798329145367999ropVar,B2: produc7679832412844898215ropVar] :
      ( ( ( produc2463153034239240527ropVar @ A @ B )
        = ( produc2463153034239240527ropVar @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_92_Pair__inject,axiom,
    ! [A: list_l1255154591254203759pVar_o,B: list_l985434930288970485pVar_o,A2: list_l1255154591254203759pVar_o,B2: list_l985434930288970485pVar_o] :
      ( ( ( produc4064290870397114077pVar_o @ A @ B )
        = ( produc4064290870397114077pVar_o @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_93_Pair__inject,axiom,
    ! [A: list_l985434930288970485pVar_o,B: produc7679832412844898215ropVar,A2: list_l985434930288970485pVar_o,B2: produc7679832412844898215ropVar] :
      ( ( ( produc2520692699247754633ropVar @ A @ B )
        = ( produc2520692699247754633ropVar @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_94_Pair__inject,axiom,
    ! [A: $o,B: $o,A2: $o,B2: $o] :
      ( ( ( product_Pair_o_o @ A @ B )
        = ( product_Pair_o_o @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B = ~ B2 ) ) ) ).

% Pair_inject
thf(fact_95_Pair__inject,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A2: multis3193088007478089820ropVar,B2: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ A @ B )
        = ( produc6936306168532358295ropVar @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_96_prod__cases,axiom,
    ! [P: produc6657367779575603407ropVar > $o,P3: produc6657367779575603407ropVar] :
      ( ! [A3: propos3600597766819905199ropVar,B3: list_P723798329145367999ropVar] : ( P @ ( produc5136867404843011847ropVar @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_97_prod__cases,axiom,
    ! [P: produc6318262956711666589ropVar > $o,P3: produc6318262956711666589ropVar] :
      ( ! [A3: list_P723798329145367999ropVar,B3: produc7679832412844898215ropVar] : ( P @ ( produc2463153034239240527ropVar @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_98_prod__cases,axiom,
    ! [P: produc5436468477138497005pVar_o > $o,P3: produc5436468477138497005pVar_o] :
      ( ! [A3: list_l1255154591254203759pVar_o,B3: list_l985434930288970485pVar_o] : ( P @ ( produc4064290870397114077pVar_o @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_99_prod__cases,axiom,
    ! [P: produc9206049318796031513ropVar > $o,P3: produc9206049318796031513ropVar] :
      ( ! [A3: list_l985434930288970485pVar_o,B3: produc7679832412844898215ropVar] : ( P @ ( produc2520692699247754633ropVar @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_100_prod__cases,axiom,
    ! [P: product_prod_o_o > $o,P3: product_prod_o_o] :
      ( ! [A3: $o,B3: $o] : ( P @ ( product_Pair_o_o @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_101_prod__cases,axiom,
    ! [P: produc7679832412844898215ropVar > $o,P3: produc7679832412844898215ropVar] :
      ( ! [A3: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] : ( P @ ( produc6936306168532358295ropVar @ A3 @ B3 ) )
     => ( P @ P3 ) ) ).

% prod_cases
thf(fact_102_surj__pair,axiom,
    ! [P3: produc6657367779575603407ropVar] :
    ? [X2: propos3600597766819905199ropVar,Y4: list_P723798329145367999ropVar] :
      ( P3
      = ( produc5136867404843011847ropVar @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_103_surj__pair,axiom,
    ! [P3: produc6318262956711666589ropVar] :
    ? [X2: list_P723798329145367999ropVar,Y4: produc7679832412844898215ropVar] :
      ( P3
      = ( produc2463153034239240527ropVar @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_104_surj__pair,axiom,
    ! [P3: produc5436468477138497005pVar_o] :
    ? [X2: list_l1255154591254203759pVar_o,Y4: list_l985434930288970485pVar_o] :
      ( P3
      = ( produc4064290870397114077pVar_o @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_105_surj__pair,axiom,
    ! [P3: produc9206049318796031513ropVar] :
    ? [X2: list_l985434930288970485pVar_o,Y4: produc7679832412844898215ropVar] :
      ( P3
      = ( produc2520692699247754633ropVar @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_106_surj__pair,axiom,
    ! [P3: product_prod_o_o] :
    ? [X2: $o,Y4: $o] :
      ( P3
      = ( product_Pair_o_o @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_107_surj__pair,axiom,
    ! [P3: produc7679832412844898215ropVar] :
    ? [X2: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
      ( P3
      = ( produc6936306168532358295ropVar @ X2 @ Y4 ) ) ).

% surj_pair
thf(fact_108_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc6657367779575603407ropVar] :
      ~ ! [A3: propos3600597766819905199ropVar,B3: list_P723798329145367999ropVar] :
          ( Y3
         != ( produc5136867404843011847ropVar @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_109_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc6318262956711666589ropVar] :
      ~ ! [A3: list_P723798329145367999ropVar,B3: produc7679832412844898215ropVar] :
          ( Y3
         != ( produc2463153034239240527ropVar @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_110_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc5436468477138497005pVar_o] :
      ~ ! [A3: list_l1255154591254203759pVar_o,B3: list_l985434930288970485pVar_o] :
          ( Y3
         != ( produc4064290870397114077pVar_o @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_111_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc9206049318796031513ropVar] :
      ~ ! [A3: list_l985434930288970485pVar_o,B3: produc7679832412844898215ropVar] :
          ( Y3
         != ( produc2520692699247754633ropVar @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_112_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_o_o] :
      ~ ! [A3: $o,B3: $o] :
          ( Y3
         != ( product_Pair_o_o @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_113_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc7679832412844898215ropVar] :
      ~ ! [A3: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( Y3
         != ( produc6936306168532358295ropVar @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_114_mem__Collect__eq,axiom,
    ! [A: multis3193088007478089820ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( member8421201611525043589ropVar @ A @ ( collec8658135508005558727ropVar @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_115_mem__Collect__eq,axiom,
    ! [A: product_prod_o_o,P: product_prod_o_o > $o] :
      ( ( member7466972457876170832od_o_o @ A @ ( collec3167064739498627218od_o_o @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_116_Collect__mem__eq,axiom,
    ! [A4: set_Mu5658373669842198716ropVar] :
      ( ( collec8658135508005558727ropVar
        @ ^ [X3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_117_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_o_o] :
      ( ( collec3167064739498627218od_o_o
        @ ^ [X3: product_prod_o_o] : ( member7466972457876170832od_o_o @ X3 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_118_not__Cons__self2,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( cons_P1504208554336013608ropVar @ X4 @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_119_not__Cons__self2,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_120_not__Cons__self2,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_121_not__Cons__self2,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_122_not__Cons__self2,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( cons_nat @ X4 @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_123_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc3638977808798937127ropVar] :
      ( ! [Ys: list_P8791008543742759928ropVar] :
          ( X4
         != ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys ) )
     => ~ ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
            ( X4
           != ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_124_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc4416794199230284903pVar_o] :
      ( ! [Ys: list_P1665994083459302249pVar_o] :
          ( X4
         != ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Ys ) )
     => ~ ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o,Ys: list_P1665994083459302249pVar_o] :
            ( X4
           != ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_125_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc2246139759411191271pVar_o] :
      ( ! [Ys: list_l1255154591254203759pVar_o] :
          ( X4
         != ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Ys ) )
     => ~ ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o,Ys: list_l1255154591254203759pVar_o] :
            ( X4
           != ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_126_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc1241035484025441639pVar_o] :
      ( ! [Ys: list_l985434930288970485pVar_o] :
          ( X4
         != ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Ys ) )
     => ~ ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o,Ys: list_l985434930288970485pVar_o] :
            ( X4
           != ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_127_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc1828647624359046049st_nat] :
      ( ! [Ys: list_nat] :
          ( X4
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys ) )
     => ~ ! [X2: nat,Xs: list_nat,Ys: list_nat] :
            ( X4
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_128_insert__Nil,axiom,
    ! [X4: produc2642885279816024434ropVar] :
      ( ( insert214406736284816014ropVar @ X4 @ nil_Pr6119456312066593144ropVar )
      = ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) ).

% insert_Nil
thf(fact_129_insert__Nil,axiom,
    ! [X4: produc8999643685545340131pVar_o] :
      ( ( insert5031086936754616319pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o )
      = ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) ) ).

% insert_Nil
thf(fact_130_insert__Nil,axiom,
    ! [X4: list_P1665994083459302249pVar_o] :
      ( ( insert8144517603577680005pVar_o @ X4 @ nil_li1310946081133177199pVar_o )
      = ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) ) ).

% insert_Nil
thf(fact_131_insert__Nil,axiom,
    ! [X4: list_l1255154591254203759pVar_o] :
      ( ( insert7135957767216192651pVar_o @ X4 @ nil_li590035233778055285pVar_o )
      = ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) ) ).

% insert_Nil
thf(fact_132_insert__Nil,axiom,
    ! [X4: nat] :
      ( ( insert_nat @ X4 @ nil_nat )
      = ( cons_nat @ X4 @ nil_nat ) ) ).

% insert_Nil
thf(fact_133_list__ex1__simps_I1_J,axiom,
    ! [P: produc2642885279816024434ropVar > $o] :
      ~ ( list_e1423168564539623099ropVar @ P @ nil_Pr6119456312066593144ropVar ) ).

% list_ex1_simps(1)
thf(fact_134_list__ex1__simps_I1_J,axiom,
    ! [P: produc8999643685545340131pVar_o > $o] :
      ~ ( list_e1948564213076044716pVar_o @ P @ nil_Pr6413368191065476585pVar_o ) ).

% list_ex1_simps(1)
thf(fact_135_list__ex1__simps_I1_J,axiom,
    ! [P: list_P1665994083459302249pVar_o > $o] :
      ~ ( list_e89593185773242802pVar_o @ P @ nil_li1310946081133177199pVar_o ) ).

% list_ex1_simps(1)
thf(fact_136_list__ex1__simps_I1_J,axiom,
    ! [P: list_l1255154591254203759pVar_o > $o] :
      ~ ( list_e5196591910180150072pVar_o @ P @ nil_li590035233778055285pVar_o ) ).

% list_ex1_simps(1)
thf(fact_137_list__ex1__simps_I1_J,axiom,
    ! [P: nat > $o] :
      ~ ( list_ex1_nat @ P @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_138_product_Osimps_I1_J,axiom,
    ! [Uu: list_P444016040222843053ropVar] :
      ( ( produc4762506738712241808ropVar @ nil_Mu4132399263956683874ropVar @ Uu )
      = nil_Pr6119456312066593144ropVar ) ).

% product.simps(1)
thf(fact_139_product_Osimps_I1_J,axiom,
    ! [Uu: list_o] :
      ( ( produc9083892960428848129pVar_o @ nil_Mu4132399263956683874ropVar @ Uu )
      = nil_Pr6413368191065476585pVar_o ) ).

% product.simps(1)
thf(fact_140_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( $o > $o ) > nat > $o,Enc_False: nat,Enc_True: nat,Enc_pos: $o > nat,Enc_neg: $o > nat,Enc_different: $o > $o > nat,Enc_equiv_and_not: $o > $o > $o > nat,Enc_equiv_ite: $o > $o > $o > $o > nat,Enc_ite: $o > $o > $o > nat,Enc_impl: $o > nat > nat,Enc_var_impl: $o > $o > nat,Enc_not_and: $o > $o > nat,Enc_not_all: list_o > nat,Enc_conj: list_nat > nat,X4: $o,Zero2: $o,One2: $o] :
      ( ( multis8460249006151941916_o_nat @ Eval @ 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 )
     => ( ( multis4961962968097414037_o_nat @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X4 @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
        = ( produc2647278605670320110od_o_o @ ( cons_nat @ ( Enc_different @ Zero2 @ X4 ) @ nil_nat ) @ ( product_Pair_o_o @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_141_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > nat > $o,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,X4: multis3193088007478089820ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar] :
      ( ( multis1673919852514530400ar_nat @ Eval @ 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 )
     => ( ( multis3310578786418936409ar_nat @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) @ nil_Pr6119456312066593144ropVar ) )
        = ( produc6552318697586692974ropVar @ ( cons_nat @ ( Enc_different @ Zero2 @ X4 ) @ nil_nat ) @ ( produc6936306168532358295ropVar @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_142_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > propos3600597766819905199ropVar > $o,Enc_False: propos3600597766819905199ropVar,Enc_True: propos3600597766819905199ropVar,Enc_pos: multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_neg: multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_impl: multis3193088007478089820ropVar > propos3600597766819905199ropVar > propos3600597766819905199ropVar,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > propos3600597766819905199ropVar,Enc_not_all: list_M6740372356978808802ropVar > propos3600597766819905199ropVar,Enc_conj: list_P723798329145367999ropVar > propos3600597766819905199ropVar,X4: multis3193088007478089820ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar] :
      ( ( multis8941024563284006719ropVar @ Eval @ 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 )
     => ( ( multis4183284423519575736ropVar @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) @ nil_Pr6119456312066593144ropVar ) )
        = ( produc2463153034239240527ropVar @ ( cons_P6757824865275295865ropVar @ ( Enc_different @ Zero2 @ X4 ) @ nil_Pr7606595143512524841ropVar ) @ ( produc6936306168532358295ropVar @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_143_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( $o > $o ) > produc8999643685545340131pVar_o > $o,Enc_False: produc8999643685545340131pVar_o,Enc_True: produc8999643685545340131pVar_o,Enc_pos: $o > produc8999643685545340131pVar_o,Enc_neg: $o > produc8999643685545340131pVar_o,Enc_different: $o > $o > produc8999643685545340131pVar_o,Enc_equiv_and_not: $o > $o > $o > produc8999643685545340131pVar_o,Enc_equiv_ite: $o > $o > $o > $o > produc8999643685545340131pVar_o,Enc_ite: $o > $o > $o > produc8999643685545340131pVar_o,Enc_impl: $o > produc8999643685545340131pVar_o > produc8999643685545340131pVar_o,Enc_var_impl: $o > $o > produc8999643685545340131pVar_o,Enc_not_and: $o > $o > produc8999643685545340131pVar_o,Enc_not_all: list_o > produc8999643685545340131pVar_o,Enc_conj: list_P1665994083459302249pVar_o > produc8999643685545340131pVar_o,X4: $o,Zero2: $o,One2: $o] :
      ( ( multis42441612937804439pVar_o @ Eval @ 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 )
     => ( ( multis4654419120101021918pVar_o @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X4 @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
        = ( produc67293053361515925od_o_o @ ( cons_P6860203055867761049pVar_o @ ( Enc_different @ Zero2 @ X4 ) @ nil_Pr6413368191065476585pVar_o ) @ ( product_Pair_o_o @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_144_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > produc8999643685545340131pVar_o > $o,Enc_False: produc8999643685545340131pVar_o,Enc_True: produc8999643685545340131pVar_o,Enc_pos: multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_neg: multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_impl: multis3193088007478089820ropVar > produc8999643685545340131pVar_o > produc8999643685545340131pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc8999643685545340131pVar_o,Enc_not_all: list_M6740372356978808802ropVar > produc8999643685545340131pVar_o,Enc_conj: list_P1665994083459302249pVar_o > produc8999643685545340131pVar_o,X4: multis3193088007478089820ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar] :
      ( ( multis1616071767867332691pVar_o @ Eval @ 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 )
     => ( ( multis8112049843181714714pVar_o @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) @ nil_Pr6119456312066593144ropVar ) )
        = ( produc7313651726693361173ropVar @ ( cons_P6860203055867761049pVar_o @ ( Enc_different @ Zero2 @ X4 ) @ nil_Pr6413368191065476585pVar_o ) @ ( produc6936306168532358295ropVar @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_145_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( $o > $o ) > list_P1665994083459302249pVar_o > $o,Enc_False: list_P1665994083459302249pVar_o,Enc_True: list_P1665994083459302249pVar_o,Enc_pos: $o > list_P1665994083459302249pVar_o,Enc_neg: $o > list_P1665994083459302249pVar_o,Enc_different: $o > $o > list_P1665994083459302249pVar_o,Enc_equiv_and_not: $o > $o > $o > list_P1665994083459302249pVar_o,Enc_equiv_ite: $o > $o > $o > $o > list_P1665994083459302249pVar_o,Enc_ite: $o > $o > $o > list_P1665994083459302249pVar_o,Enc_impl: $o > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o,Enc_var_impl: $o > $o > list_P1665994083459302249pVar_o,Enc_not_and: $o > $o > list_P1665994083459302249pVar_o,Enc_not_all: list_o > list_P1665994083459302249pVar_o,Enc_conj: list_l1255154591254203759pVar_o > list_P1665994083459302249pVar_o,X4: $o,Zero2: $o,One2: $o] :
      ( ( multis8070414971136632605pVar_o @ Eval @ 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 )
     => ( ( multis5520445341102945764pVar_o @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X4 @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
        = ( produc5179445976964859151od_o_o @ ( cons_l3452452106213086495pVar_o @ ( Enc_different @ Zero2 @ X4 ) @ nil_li1310946081133177199pVar_o ) @ ( product_Pair_o_o @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_146_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_P1665994083459302249pVar_o > $o,Enc_False: list_P1665994083459302249pVar_o,Enc_True: list_P1665994083459302249pVar_o,Enc_pos: multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_neg: multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_impl: multis3193088007478089820ropVar > list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_P1665994083459302249pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_P1665994083459302249pVar_o,Enc_conj: list_l1255154591254203759pVar_o > list_P1665994083459302249pVar_o,X4: multis3193088007478089820ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar] :
      ( ( multis4082459982090122969pVar_o @ Eval @ 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 )
     => ( ( multis8843582314722954272pVar_o @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) @ nil_Pr6119456312066593144ropVar ) )
        = ( produc3704348280863636367ropVar @ ( cons_l3452452106213086495pVar_o @ ( Enc_different @ Zero2 @ X4 ) @ nil_li1310946081133177199pVar_o ) @ ( produc6936306168532358295ropVar @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_147_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( $o > $o ) > produc2642885279816024434ropVar > $o,Enc_False: produc2642885279816024434ropVar,Enc_True: produc2642885279816024434ropVar,Enc_pos: $o > produc2642885279816024434ropVar,Enc_neg: $o > produc2642885279816024434ropVar,Enc_different: $o > $o > produc2642885279816024434ropVar,Enc_equiv_and_not: $o > $o > $o > produc2642885279816024434ropVar,Enc_equiv_ite: $o > $o > $o > $o > produc2642885279816024434ropVar,Enc_ite: $o > $o > $o > produc2642885279816024434ropVar,Enc_impl: $o > produc2642885279816024434ropVar > produc2642885279816024434ropVar,Enc_var_impl: $o > $o > produc2642885279816024434ropVar,Enc_not_and: $o > $o > produc2642885279816024434ropVar,Enc_not_all: list_o > produc2642885279816024434ropVar,Enc_conj: list_P8791008543742759928ropVar > produc2642885279816024434ropVar,X4: $o,Zero2: $o,One2: $o] :
      ( ( multis4555596752605892902ropVar @ Eval @ 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 )
     => ( ( multis5036415435641082093ropVar @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X4 @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
        = ( produc5768689700875911878od_o_o @ ( cons_P1504208554336013608ropVar @ ( Enc_different @ Zero2 @ X4 ) @ nil_Pr6119456312066593144ropVar ) @ ( product_Pair_o_o @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_148_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( $o > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: $o > list_l1255154591254203759pVar_o,Enc_neg: $o > list_l1255154591254203759pVar_o,Enc_different: $o > $o > list_l1255154591254203759pVar_o,Enc_equiv_and_not: $o > $o > $o > list_l1255154591254203759pVar_o,Enc_equiv_ite: $o > $o > $o > $o > list_l1255154591254203759pVar_o,Enc_ite: $o > $o > $o > list_l1255154591254203759pVar_o,Enc_impl: $o > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: $o > $o > list_l1255154591254203759pVar_o,Enc_not_and: $o > $o > list_l1255154591254203759pVar_o,Enc_not_all: list_o > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,X4: $o,Zero2: $o,One2: $o] :
      ( ( multis6533897398417245987pVar_o @ Eval @ 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 )
     => ( ( multis4124385670053463658pVar_o @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P7930012998920551916od_o_o @ ( produc2690343539598703398od_o_o @ X4 @ ( product_Pair_o_o @ Zero2 @ One2 ) ) @ nil_Pr1406083253504547388od_o_o ) )
        = ( produc6775336607957928201od_o_o @ ( cons_l7316790070876032037pVar_o @ ( Enc_different @ Zero2 @ X4 ) @ nil_li590035233778055285pVar_o ) @ ( product_Pair_o_o @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_149_encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > produc2642885279816024434ropVar > $o,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,X4: multis3193088007478089820ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar] :
      ( ( multis885672736175375586ropVar @ Eval @ 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 )
     => ( ( multis3159227008566381353ropVar @ Enc_different @ Enc_equiv_and_not @ Enc_equiv_ite @ ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) @ nil_Pr6119456312066593144ropVar ) )
        = ( produc5322602891375056454ropVar @ ( cons_P1504208554336013608ropVar @ ( Enc_different @ Zero2 @ X4 ) @ nil_Pr6119456312066593144ropVar ) @ ( produc6936306168532358295ropVar @ Zero2 @ X4 ) ) ) ) ).

% encoder.encode_sum_0_1_main.simps(1)
thf(fact_150_map__tailrec__rev_Oelims,axiom,
    ! [X4: nat > nat,Xa: list_nat,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta7164188454487880599at_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != Xb ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A3 @ As ) )
             => ( Y3
               != ( map_ta7164188454487880599at_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_151_map__tailrec__rev_Oelims,axiom,
    ! [X4: produc8999643685545340131pVar_o > nat,Xa: list_P1665994083459302249pVar_o,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta4027472834757870206_o_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_Pr6413368191065476585pVar_o )
         => ( Y3 != Xb ) )
       => ~ ! [A3: produc8999643685545340131pVar_o,As: list_P1665994083459302249pVar_o] :
              ( ( Xa
                = ( cons_P6860203055867761049pVar_o @ A3 @ As ) )
             => ( Y3
               != ( map_ta4027472834757870206_o_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_152_map__tailrec__rev_Oelims,axiom,
    ! [X4: nat > produc8999643685545340131pVar_o,Xa: list_nat,Xb: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( ( map_ta511341784408162908pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != Xb ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A3 @ As ) )
             => ( Y3
               != ( map_ta511341784408162908pVar_o @ X4 @ As @ ( cons_P6860203055867761049pVar_o @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_153_map__tailrec__rev_Oelims,axiom,
    ! [X4: list_P1665994083459302249pVar_o > nat,Xa: list_l1255154591254203759pVar_o,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta3501597723813782532_o_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_li1310946081133177199pVar_o )
         => ( Y3 != Xb ) )
       => ~ ! [A3: list_P1665994083459302249pVar_o,As: list_l1255154591254203759pVar_o] :
              ( ( Xa
                = ( cons_l3452452106213086495pVar_o @ A3 @ As ) )
             => ( Y3
               != ( map_ta3501597723813782532_o_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_154_map__tailrec__rev_Oelims,axiom,
    ! [X4: nat > list_P1665994083459302249pVar_o,Xa: list_nat,Xb: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o] :
      ( ( ( map_ta5031888053227161698pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != Xb ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A3 @ As ) )
             => ( Y3
               != ( map_ta5031888053227161698pVar_o @ X4 @ As @ ( cons_l3452452106213086495pVar_o @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_155_map__tailrec__rev_Oelims,axiom,
    ! [X4: produc2642885279816024434ropVar > nat,Xa: list_P8791008543742759928ropVar,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta4142060526428890125ar_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_Pr6119456312066593144ropVar )
         => ( Y3 != Xb ) )
       => ~ ! [A3: produc2642885279816024434ropVar,As: list_P8791008543742759928ropVar] :
              ( ( Xa
                = ( cons_P1504208554336013608ropVar @ A3 @ As ) )
             => ( Y3
               != ( map_ta4142060526428890125ar_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_156_map__tailrec__rev_Oelims,axiom,
    ! [X4: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o,Xa: list_P1665994083459302249pVar_o,Xb: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( ( map_ta8452059871828574389pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_Pr6413368191065476585pVar_o )
         => ( Y3 != Xb ) )
       => ~ ! [A3: produc8999643685545340131pVar_o,As: list_P1665994083459302249pVar_o] :
              ( ( Xa
                = ( cons_P6860203055867761049pVar_o @ A3 @ As ) )
             => ( Y3
               != ( map_ta8452059871828574389pVar_o @ X4 @ As @ ( cons_P6860203055867761049pVar_o @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_157_map__tailrec__rev_Oelims,axiom,
    ! [X4: list_l1255154591254203759pVar_o > nat,Xa: list_l985434930288970485pVar_o,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta57781623974148362_o_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_li590035233778055285pVar_o )
         => ( Y3 != Xb ) )
       => ~ ! [A3: list_l1255154591254203759pVar_o,As: list_l985434930288970485pVar_o] :
              ( ( Xa
                = ( cons_l7316790070876032037pVar_o @ A3 @ As ) )
             => ( Y3
               != ( map_ta57781623974148362_o_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_158_map__tailrec__rev_Oelims,axiom,
    ! [X4: nat > produc2642885279816024434ropVar,Xa: list_nat,Xb: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( ( map_ta7700754162581607787ropVar @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != Xb ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A3 @ As ) )
             => ( Y3
               != ( map_ta7700754162581607787ropVar @ X4 @ As @ ( cons_P1504208554336013608ropVar @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_159_map__tailrec__rev_Oelims,axiom,
    ! [X4: nat > list_l1255154591254203759pVar_o,Xa: list_nat,Xb: list_l985434930288970485pVar_o,Y3: list_l985434930288970485pVar_o] :
      ( ( ( map_ta1976070376341471720pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( ( Xa = nil_nat )
         => ( Y3 != Xb ) )
       => ~ ! [A3: nat,As: list_nat] :
              ( ( Xa
                = ( cons_nat @ A3 @ As ) )
             => ( Y3
               != ( map_ta1976070376341471720pVar_o @ X4 @ As @ ( cons_l7316790070876032037pVar_o @ ( X4 @ A3 ) @ Xb ) ) ) ) ) ) ).

% map_tailrec_rev.elims
thf(fact_160_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ~ ( lexord1893613697524570046ropVar @ Less @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ nil_Pr6119456312066593144ropVar ) ).

% ord.lexordp_eq_simps(3)
thf(fact_161_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ~ ( lexord3023072712957166895pVar_o @ Less @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ nil_Pr6413368191065476585pVar_o ) ).

% ord.lexordp_eq_simps(3)
thf(fact_162_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ~ ( lexord589138509458605493pVar_o @ Less @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ nil_li1310946081133177199pVar_o ) ).

% ord.lexordp_eq_simps(3)
thf(fact_163_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ~ ( lexord8403422104825280955pVar_o @ Less @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ nil_li590035233778055285pVar_o ) ).

% ord.lexordp_eq_simps(3)
thf(fact_164_ord_Olexordp__eq__simps_I3_J,axiom,
    ! [Less: nat > nat > $o,X4: nat,Xs2: list_nat] :
      ~ ( lexordp_eq_nat @ Less @ ( cons_nat @ X4 @ Xs2 ) @ nil_nat ) ).

% ord.lexordp_eq_simps(3)
thf(fact_165_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( lexord1893613697524570046ropVar @ Less @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) )
      = ( ( Less @ X4 @ Y3 )
        | ( ~ ( Less @ Y3 @ X4 )
          & ( lexord1893613697524570046ropVar @ Less @ Xs2 @ Ys2 ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_166_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( lexord3023072712957166895pVar_o @ Less @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) )
      = ( ( Less @ X4 @ Y3 )
        | ( ~ ( Less @ Y3 @ X4 )
          & ( lexord3023072712957166895pVar_o @ Less @ Xs2 @ Ys2 ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_167_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( lexord589138509458605493pVar_o @ Less @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) )
      = ( ( Less @ X4 @ Y3 )
        | ( ~ ( Less @ Y3 @ X4 )
          & ( lexord589138509458605493pVar_o @ Less @ Xs2 @ Ys2 ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_168_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( lexord8403422104825280955pVar_o @ Less @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) )
      = ( ( Less @ X4 @ Y3 )
        | ( ~ ( Less @ Y3 @ X4 )
          & ( lexord8403422104825280955pVar_o @ Less @ Xs2 @ Ys2 ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_169_ord_Olexordp__eq__simps_I4_J,axiom,
    ! [Less: nat > nat > $o,X4: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat] :
      ( ( lexordp_eq_nat @ Less @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) )
      = ( ( Less @ X4 @ Y3 )
        | ( ~ ( Less @ Y3 @ X4 )
          & ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 ) ) ) ) ).

% ord.lexordp_eq_simps(4)
thf(fact_170_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,Ys2: list_P8791008543742759928ropVar] : ( lexord1893613697524570046ropVar @ Less @ nil_Pr6119456312066593144ropVar @ Ys2 ) ).

% ord.lexordp_eq_simps(1)
thf(fact_171_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,Ys2: list_P1665994083459302249pVar_o] : ( lexord3023072712957166895pVar_o @ Less @ nil_Pr6413368191065476585pVar_o @ Ys2 ) ).

% ord.lexordp_eq_simps(1)
thf(fact_172_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,Ys2: list_l1255154591254203759pVar_o] : ( lexord589138509458605493pVar_o @ Less @ nil_li1310946081133177199pVar_o @ Ys2 ) ).

% ord.lexordp_eq_simps(1)
thf(fact_173_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,Ys2: list_l985434930288970485pVar_o] : ( lexord8403422104825280955pVar_o @ Less @ nil_li590035233778055285pVar_o @ Ys2 ) ).

% ord.lexordp_eq_simps(1)
thf(fact_174_ord_Olexordp__eq__simps_I1_J,axiom,
    ! [Less: nat > nat > $o,Ys2: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys2 ) ).

% ord.lexordp_eq_simps(1)
thf(fact_175_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,Xs2: list_P8791008543742759928ropVar] :
      ( ( lexord1893613697524570046ropVar @ Less @ Xs2 @ nil_Pr6119456312066593144ropVar )
      = ( Xs2 = nil_Pr6119456312066593144ropVar ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_176_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( lexord3023072712957166895pVar_o @ Less @ Xs2 @ nil_Pr6413368191065476585pVar_o )
      = ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_177_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( lexord589138509458605493pVar_o @ Less @ Xs2 @ nil_li1310946081133177199pVar_o )
      = ( Xs2 = nil_li1310946081133177199pVar_o ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_178_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,Xs2: list_l985434930288970485pVar_o] :
      ( ( lexord8403422104825280955pVar_o @ Less @ Xs2 @ nil_li590035233778055285pVar_o )
      = ( Xs2 = nil_li590035233778055285pVar_o ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_179_ord_Olexordp__eq__simps_I2_J,axiom,
    ! [Less: nat > nat > $o,Xs2: list_nat] :
      ( ( lexordp_eq_nat @ Less @ Xs2 @ nil_nat )
      = ( Xs2 = nil_nat ) ) ).

% ord.lexordp_eq_simps(2)
thf(fact_180_encoder_Oencode__sum__0__1__main_Ocong,axiom,
    multis6898674672137113254pVar_o = multis6898674672137113254pVar_o ).

% encoder.encode_sum_0_1_main.cong
thf(fact_181_ord_Olexordp__eq_OCons,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X4: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( Less @ X4 @ Y3 )
     => ( lexord1893613697524570046ropVar @ Less @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_182_ord_Olexordp__eq_OCons,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X4: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( Less @ X4 @ Y3 )
     => ( lexord3023072712957166895pVar_o @ Less @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_183_ord_Olexordp__eq_OCons,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( Less @ X4 @ Y3 )
     => ( lexord589138509458605493pVar_o @ Less @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_184_ord_Olexordp__eq_OCons,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( Less @ X4 @ Y3 )
     => ( lexord8403422104825280955pVar_o @ Less @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_185_ord_Olexordp__eq_OCons,axiom,
    ! [Less: nat > nat > $o,X4: nat,Y3: nat,Xs2: list_nat,Ys2: list_nat] :
      ( ( Less @ X4 @ Y3 )
     => ( lexordp_eq_nat @ Less @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ).

% ord.lexordp_eq.Cons
thf(fact_186_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X4: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ~ ( Less @ X4 @ Y3 )
     => ( ~ ( Less @ Y3 @ X4 )
       => ( ( lexord1893613697524570046ropVar @ Less @ Xs2 @ Ys2 )
         => ( lexord1893613697524570046ropVar @ Less @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_187_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X4: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ~ ( Less @ X4 @ Y3 )
     => ( ~ ( Less @ Y3 @ X4 )
       => ( ( lexord3023072712957166895pVar_o @ Less @ Xs2 @ Ys2 )
         => ( lexord3023072712957166895pVar_o @ Less @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_188_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ~ ( Less @ X4 @ Y3 )
     => ( ~ ( Less @ Y3 @ X4 )
       => ( ( lexord589138509458605493pVar_o @ Less @ Xs2 @ Ys2 )
         => ( lexord589138509458605493pVar_o @ Less @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_189_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ~ ( Less @ X4 @ Y3 )
     => ( ~ ( Less @ Y3 @ X4 )
       => ( ( lexord8403422104825280955pVar_o @ Less @ Xs2 @ Ys2 )
         => ( lexord8403422104825280955pVar_o @ Less @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_190_ord_Olexordp__eq_OCons__eq,axiom,
    ! [Less: nat > nat > $o,X4: nat,Y3: nat,Xs2: list_nat,Ys2: list_nat] :
      ( ~ ( Less @ X4 @ Y3 )
     => ( ~ ( Less @ Y3 @ X4 )
       => ( ( lexordp_eq_nat @ Less @ Xs2 @ Ys2 )
         => ( lexordp_eq_nat @ Less @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ) ).

% ord.lexordp_eq.Cons_eq
thf(fact_191_ord_Olexordp__eq_ONil,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,Ys2: list_P8791008543742759928ropVar] : ( lexord1893613697524570046ropVar @ Less @ nil_Pr6119456312066593144ropVar @ Ys2 ) ).

% ord.lexordp_eq.Nil
thf(fact_192_ord_Olexordp__eq_ONil,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,Ys2: list_P1665994083459302249pVar_o] : ( lexord3023072712957166895pVar_o @ Less @ nil_Pr6413368191065476585pVar_o @ Ys2 ) ).

% ord.lexordp_eq.Nil
thf(fact_193_ord_Olexordp__eq_ONil,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,Ys2: list_l1255154591254203759pVar_o] : ( lexord589138509458605493pVar_o @ Less @ nil_li1310946081133177199pVar_o @ Ys2 ) ).

% ord.lexordp_eq.Nil
thf(fact_194_ord_Olexordp__eq_ONil,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,Ys2: list_l985434930288970485pVar_o] : ( lexord8403422104825280955pVar_o @ Less @ nil_li590035233778055285pVar_o @ Ys2 ) ).

% ord.lexordp_eq.Nil
thf(fact_195_ord_Olexordp__eq_ONil,axiom,
    ! [Less: nat > nat > $o,Ys2: list_nat] : ( lexordp_eq_nat @ Less @ nil_nat @ Ys2 ) ).

% ord.lexordp_eq.Nil
thf(fact_196_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: nat > nat,A: nat,As2: list_nat,Bs2: list_nat] :
      ( ( map_ta7164188454487880599at_nat @ F2 @ ( cons_nat @ A @ As2 ) @ Bs2 )
      = ( map_ta7164188454487880599at_nat @ F2 @ As2 @ ( cons_nat @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_197_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > nat,A: produc8999643685545340131pVar_o,As2: list_P1665994083459302249pVar_o,Bs2: list_nat] :
      ( ( map_ta4027472834757870206_o_nat @ F2 @ ( cons_P6860203055867761049pVar_o @ A @ As2 ) @ Bs2 )
      = ( map_ta4027472834757870206_o_nat @ F2 @ As2 @ ( cons_nat @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_198_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: nat > produc8999643685545340131pVar_o,A: nat,As2: list_nat,Bs2: list_P1665994083459302249pVar_o] :
      ( ( map_ta511341784408162908pVar_o @ F2 @ ( cons_nat @ A @ As2 ) @ Bs2 )
      = ( map_ta511341784408162908pVar_o @ F2 @ As2 @ ( cons_P6860203055867761049pVar_o @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_199_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: list_P1665994083459302249pVar_o > nat,A: list_P1665994083459302249pVar_o,As2: list_l1255154591254203759pVar_o,Bs2: list_nat] :
      ( ( map_ta3501597723813782532_o_nat @ F2 @ ( cons_l3452452106213086495pVar_o @ A @ As2 ) @ Bs2 )
      = ( map_ta3501597723813782532_o_nat @ F2 @ As2 @ ( cons_nat @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_200_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: nat > list_P1665994083459302249pVar_o,A: nat,As2: list_nat,Bs2: list_l1255154591254203759pVar_o] :
      ( ( map_ta5031888053227161698pVar_o @ F2 @ ( cons_nat @ A @ As2 ) @ Bs2 )
      = ( map_ta5031888053227161698pVar_o @ F2 @ As2 @ ( cons_l3452452106213086495pVar_o @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_201_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: produc2642885279816024434ropVar > nat,A: produc2642885279816024434ropVar,As2: list_P8791008543742759928ropVar,Bs2: list_nat] :
      ( ( map_ta4142060526428890125ar_nat @ F2 @ ( cons_P1504208554336013608ropVar @ A @ As2 ) @ Bs2 )
      = ( map_ta4142060526428890125ar_nat @ F2 @ As2 @ ( cons_nat @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_202_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o,A: produc8999643685545340131pVar_o,As2: list_P1665994083459302249pVar_o,Bs2: list_P1665994083459302249pVar_o] :
      ( ( map_ta8452059871828574389pVar_o @ F2 @ ( cons_P6860203055867761049pVar_o @ A @ As2 ) @ Bs2 )
      = ( map_ta8452059871828574389pVar_o @ F2 @ As2 @ ( cons_P6860203055867761049pVar_o @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_203_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: list_l1255154591254203759pVar_o > nat,A: list_l1255154591254203759pVar_o,As2: list_l985434930288970485pVar_o,Bs2: list_nat] :
      ( ( map_ta57781623974148362_o_nat @ F2 @ ( cons_l7316790070876032037pVar_o @ A @ As2 ) @ Bs2 )
      = ( map_ta57781623974148362_o_nat @ F2 @ As2 @ ( cons_nat @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_204_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: nat > produc2642885279816024434ropVar,A: nat,As2: list_nat,Bs2: list_P8791008543742759928ropVar] :
      ( ( map_ta7700754162581607787ropVar @ F2 @ ( cons_nat @ A @ As2 ) @ Bs2 )
      = ( map_ta7700754162581607787ropVar @ F2 @ As2 @ ( cons_P1504208554336013608ropVar @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_205_map__tailrec__rev_Osimps_I2_J,axiom,
    ! [F2: nat > list_l1255154591254203759pVar_o,A: nat,As2: list_nat,Bs2: list_l985434930288970485pVar_o] :
      ( ( map_ta1976070376341471720pVar_o @ F2 @ ( cons_nat @ A @ As2 ) @ Bs2 )
      = ( map_ta1976070376341471720pVar_o @ F2 @ As2 @ ( cons_l7316790070876032037pVar_o @ ( F2 @ A ) @ Bs2 ) ) ) ).

% map_tailrec_rev.simps(2)
thf(fact_206_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,A1: list_P8791008543742759928ropVar,A22: list_P8791008543742759928ropVar] :
      ( ( lexord1893613697524570046ropVar @ Less @ A1 @ A22 )
     => ( ( A1 != nil_Pr6119456312066593144ropVar )
       => ( ! [X2: produc2642885279816024434ropVar] :
              ( ? [Xs: list_P8791008543742759928ropVar] :
                  ( A1
                  = ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
             => ! [Y4: produc2642885279816024434ropVar] :
                  ( ? [Ys: list_P8791008543742759928ropVar] :
                      ( A22
                      = ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) )
                 => ~ ( Less @ X2 @ Y4 ) ) )
         => ~ ! [X2: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
                ( ( A1
                  = ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
               => ! [Ys: list_P8791008543742759928ropVar] :
                    ( ( A22
                      = ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) )
                   => ( ~ ( Less @ X2 @ Y4 )
                     => ( ~ ( Less @ Y4 @ X2 )
                       => ~ ( lexord1893613697524570046ropVar @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_207_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,A1: list_P1665994083459302249pVar_o,A22: list_P1665994083459302249pVar_o] :
      ( ( lexord3023072712957166895pVar_o @ Less @ A1 @ A22 )
     => ( ( A1 != nil_Pr6413368191065476585pVar_o )
       => ( ! [X2: produc8999643685545340131pVar_o] :
              ( ? [Xs: list_P1665994083459302249pVar_o] :
                  ( A1
                  = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
             => ! [Y4: produc8999643685545340131pVar_o] :
                  ( ? [Ys: list_P1665994083459302249pVar_o] :
                      ( A22
                      = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
                 => ~ ( Less @ X2 @ Y4 ) ) )
         => ~ ! [X2: produc8999643685545340131pVar_o,Y4: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
                ( ( A1
                  = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
               => ! [Ys: list_P1665994083459302249pVar_o] :
                    ( ( A22
                      = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
                   => ( ~ ( Less @ X2 @ Y4 )
                     => ( ~ ( Less @ Y4 @ X2 )
                       => ~ ( lexord3023072712957166895pVar_o @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_208_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,A1: list_l1255154591254203759pVar_o,A22: list_l1255154591254203759pVar_o] :
      ( ( lexord589138509458605493pVar_o @ Less @ A1 @ A22 )
     => ( ( A1 != nil_li1310946081133177199pVar_o )
       => ( ! [X2: list_P1665994083459302249pVar_o] :
              ( ? [Xs: list_l1255154591254203759pVar_o] :
                  ( A1
                  = ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) )
             => ! [Y4: list_P1665994083459302249pVar_o] :
                  ( ? [Ys: list_l1255154591254203759pVar_o] :
                      ( A22
                      = ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) )
                 => ~ ( Less @ X2 @ Y4 ) ) )
         => ~ ! [X2: list_P1665994083459302249pVar_o,Y4: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
                ( ( A1
                  = ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) )
               => ! [Ys: list_l1255154591254203759pVar_o] :
                    ( ( A22
                      = ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) )
                   => ( ~ ( Less @ X2 @ Y4 )
                     => ( ~ ( Less @ Y4 @ X2 )
                       => ~ ( lexord589138509458605493pVar_o @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_209_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,A1: list_l985434930288970485pVar_o,A22: list_l985434930288970485pVar_o] :
      ( ( lexord8403422104825280955pVar_o @ Less @ A1 @ A22 )
     => ( ( A1 != nil_li590035233778055285pVar_o )
       => ( ! [X2: list_l1255154591254203759pVar_o] :
              ( ? [Xs: list_l985434930288970485pVar_o] :
                  ( A1
                  = ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) )
             => ! [Y4: list_l1255154591254203759pVar_o] :
                  ( ? [Ys: list_l985434930288970485pVar_o] :
                      ( A22
                      = ( cons_l7316790070876032037pVar_o @ Y4 @ Ys ) )
                 => ~ ( Less @ X2 @ Y4 ) ) )
         => ~ ! [X2: list_l1255154591254203759pVar_o,Y4: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
                ( ( A1
                  = ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) )
               => ! [Ys: list_l985434930288970485pVar_o] :
                    ( ( A22
                      = ( cons_l7316790070876032037pVar_o @ Y4 @ Ys ) )
                   => ( ~ ( Less @ X2 @ Y4 )
                     => ( ~ ( Less @ Y4 @ X2 )
                       => ~ ( lexord8403422104825280955pVar_o @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_210_ord_Olexordp__eq_Ocases,axiom,
    ! [Less: nat > nat > $o,A1: list_nat,A22: list_nat] :
      ( ( lexordp_eq_nat @ Less @ A1 @ A22 )
     => ( ( A1 != nil_nat )
       => ( ! [X2: nat] :
              ( ? [Xs: list_nat] :
                  ( A1
                  = ( cons_nat @ X2 @ Xs ) )
             => ! [Y4: nat] :
                  ( ? [Ys: list_nat] :
                      ( A22
                      = ( cons_nat @ Y4 @ Ys ) )
                 => ~ ( Less @ X2 @ Y4 ) ) )
         => ~ ! [X2: nat,Y4: nat,Xs: list_nat] :
                ( ( A1
                  = ( cons_nat @ X2 @ Xs ) )
               => ! [Ys: list_nat] :
                    ( ( A22
                      = ( cons_nat @ Y4 @ Ys ) )
                   => ( ~ ( Less @ X2 @ Y4 )
                     => ( ~ ( Less @ Y4 @ X2 )
                       => ~ ( lexordp_eq_nat @ Less @ Xs @ Ys ) ) ) ) ) ) ) ) ).

% ord.lexordp_eq.cases
thf(fact_211_ord_Olexordp__eq_Osimps,axiom,
    ( lexord1893613697524570046ropVar
    = ( ^ [Less2: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,A12: list_P8791008543742759928ropVar,A23: list_P8791008543742759928ropVar] :
          ( ? [Ys3: list_P8791008543742759928ropVar] :
              ( ( A12 = nil_Pr6119456312066593144ropVar )
              & ( A23 = Ys3 ) )
          | ? [X3: produc2642885279816024434ropVar,Y2: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Ys3: list_P8791008543742759928ropVar] :
              ( ( A12
                = ( cons_P1504208554336013608ropVar @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P1504208554336013608ropVar @ Y2 @ Ys3 ) )
              & ( Less2 @ X3 @ Y2 ) )
          | ? [X3: produc2642885279816024434ropVar,Y2: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar,Ys3: list_P8791008543742759928ropVar] :
              ( ( A12
                = ( cons_P1504208554336013608ropVar @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P1504208554336013608ropVar @ Y2 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y2 )
              & ~ ( Less2 @ Y2 @ X3 )
              & ( lexord1893613697524570046ropVar @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_212_ord_Olexordp__eq_Osimps,axiom,
    ( lexord3023072712957166895pVar_o
    = ( ^ [Less2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,A12: list_P1665994083459302249pVar_o,A23: list_P1665994083459302249pVar_o] :
          ( ? [Ys3: list_P1665994083459302249pVar_o] :
              ( ( A12 = nil_Pr6413368191065476585pVar_o )
              & ( A23 = Ys3 ) )
          | ? [X3: produc8999643685545340131pVar_o,Y2: produc8999643685545340131pVar_o,Xs3: list_P1665994083459302249pVar_o,Ys3: list_P1665994083459302249pVar_o] :
              ( ( A12
                = ( cons_P6860203055867761049pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P6860203055867761049pVar_o @ Y2 @ Ys3 ) )
              & ( Less2 @ X3 @ Y2 ) )
          | ? [X3: produc8999643685545340131pVar_o,Y2: produc8999643685545340131pVar_o,Xs3: list_P1665994083459302249pVar_o,Ys3: list_P1665994083459302249pVar_o] :
              ( ( A12
                = ( cons_P6860203055867761049pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P6860203055867761049pVar_o @ Y2 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y2 )
              & ~ ( Less2 @ Y2 @ X3 )
              & ( lexord3023072712957166895pVar_o @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_213_ord_Olexordp__eq_Osimps,axiom,
    ( lexord589138509458605493pVar_o
    = ( ^ [Less2: list_P1665994083459302249pVar_o > list_P1665994083459302249pVar_o > $o,A12: list_l1255154591254203759pVar_o,A23: list_l1255154591254203759pVar_o] :
          ( ? [Ys3: list_l1255154591254203759pVar_o] :
              ( ( A12 = nil_li1310946081133177199pVar_o )
              & ( A23 = Ys3 ) )
          | ? [X3: list_P1665994083459302249pVar_o,Y2: list_P1665994083459302249pVar_o,Xs3: list_l1255154591254203759pVar_o,Ys3: list_l1255154591254203759pVar_o] :
              ( ( A12
                = ( cons_l3452452106213086495pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_l3452452106213086495pVar_o @ Y2 @ Ys3 ) )
              & ( Less2 @ X3 @ Y2 ) )
          | ? [X3: list_P1665994083459302249pVar_o,Y2: list_P1665994083459302249pVar_o,Xs3: list_l1255154591254203759pVar_o,Ys3: list_l1255154591254203759pVar_o] :
              ( ( A12
                = ( cons_l3452452106213086495pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_l3452452106213086495pVar_o @ Y2 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y2 )
              & ~ ( Less2 @ Y2 @ X3 )
              & ( lexord589138509458605493pVar_o @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_214_ord_Olexordp__eq_Osimps,axiom,
    ( lexord8403422104825280955pVar_o
    = ( ^ [Less2: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,A12: list_l985434930288970485pVar_o,A23: list_l985434930288970485pVar_o] :
          ( ? [Ys3: list_l985434930288970485pVar_o] :
              ( ( A12 = nil_li590035233778055285pVar_o )
              & ( A23 = Ys3 ) )
          | ? [X3: list_l1255154591254203759pVar_o,Y2: list_l1255154591254203759pVar_o,Xs3: list_l985434930288970485pVar_o,Ys3: list_l985434930288970485pVar_o] :
              ( ( A12
                = ( cons_l7316790070876032037pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_l7316790070876032037pVar_o @ Y2 @ Ys3 ) )
              & ( Less2 @ X3 @ Y2 ) )
          | ? [X3: list_l1255154591254203759pVar_o,Y2: list_l1255154591254203759pVar_o,Xs3: list_l985434930288970485pVar_o,Ys3: list_l985434930288970485pVar_o] :
              ( ( A12
                = ( cons_l7316790070876032037pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_l7316790070876032037pVar_o @ Y2 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y2 )
              & ~ ( Less2 @ Y2 @ X3 )
              & ( lexord8403422104825280955pVar_o @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_215_ord_Olexordp__eq_Osimps,axiom,
    ( lexordp_eq_nat
    = ( ^ [Less2: nat > nat > $o,A12: list_nat,A23: list_nat] :
          ( ? [Ys3: list_nat] :
              ( ( A12 = nil_nat )
              & ( A23 = Ys3 ) )
          | ? [X3: nat,Y2: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y2 @ Ys3 ) )
              & ( Less2 @ X3 @ Y2 ) )
          | ? [X3: nat,Y2: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y2 @ Ys3 ) )
              & ~ ( Less2 @ X3 @ Y2 )
              & ~ ( Less2 @ Y2 @ X3 )
              & ( lexordp_eq_nat @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% ord.lexordp_eq.simps
thf(fact_216_cnf__encoder_Oencode__sum__0__1__main_Osimps_I1_J,axiom,
    ! [X4: multis3193088007478089820ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar] :
      ( ( multis70479551085940736041_main @ ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X4 @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) @ nil_Pr6119456312066593144ropVar ) )
      = ( produc2520692699247754633ropVar @ ( cons_l7316790070876032037pVar_o @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Zero2 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Zero2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) @ nil_li590035233778055285pVar_o ) @ ( produc6936306168532358295ropVar @ Zero2 @ X4 ) ) ) ).

% cnf_encoder.encode_sum_0_1_main.simps(1)
thf(fact_217_product__lists_Osimps_I1_J,axiom,
    ( ( produc2956654850250905953ropVar @ nil_li2679230897895549182ropVar )
    = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) ).

% product_lists.simps(1)
thf(fact_218_product__lists_Osimps_I1_J,axiom,
    ( ( produc7231796756852922590pVar_o @ nil_li5579170522155219707pVar_o )
    = ( cons_l8096070148660426411pVar_o @ nil_li590035233778055285pVar_o @ nil_li5579170522155219707pVar_o ) ) ).

% product_lists.simps(1)
thf(fact_219_product__lists_Osimps_I1_J,axiom,
    ( ( product_lists_nat @ nil_list_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% product_lists.simps(1)
thf(fact_220_product__lists_Osimps_I1_J,axiom,
    ( ( produc2846438059124571986pVar_o @ nil_li1310946081133177199pVar_o )
    = ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o ) ) ).

% product_lists.simps(1)
thf(fact_221_product__lists_Osimps_I1_J,axiom,
    ( ( produc8865489164070834776pVar_o @ nil_li590035233778055285pVar_o )
    = ( cons_l7316790070876032037pVar_o @ nil_li1310946081133177199pVar_o @ nil_li590035233778055285pVar_o ) ) ).

% product_lists.simps(1)
thf(fact_222_subseqs_Osimps_I1_J,axiom,
    ( ( subseq3825174504735880391ropVar @ nil_Pr6119456312066593144ropVar )
    = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) ).

% subseqs.simps(1)
thf(fact_223_subseqs_Osimps_I1_J,axiom,
    ( ( subseq7541546836001800004pVar_o @ nil_li590035233778055285pVar_o )
    = ( cons_l8096070148660426411pVar_o @ nil_li590035233778055285pVar_o @ nil_li5579170522155219707pVar_o ) ) ).

% subseqs.simps(1)
thf(fact_224_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_225_subseqs_Osimps_I1_J,axiom,
    ( ( subseq8245697508244618168pVar_o @ nil_Pr6413368191065476585pVar_o )
    = ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o ) ) ).

% subseqs.simps(1)
thf(fact_226_subseqs_Osimps_I1_J,axiom,
    ( ( subseq2663478470768632766pVar_o @ nil_li1310946081133177199pVar_o )
    = ( cons_l7316790070876032037pVar_o @ nil_li1310946081133177199pVar_o @ nil_li590035233778055285pVar_o ) ) ).

% subseqs.simps(1)
thf(fact_227_map__tailrec__rev_Opelims,axiom,
    ! [X4: nat > nat,Xa: list_nat,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta7164188454487880599at_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X4 @ ( produc2694037385005941721st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = Xb )
             => ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X4 @ ( produc2694037385005941721st_nat @ nil_nat @ Xb ) ) ) ) )
         => ~ ! [A3: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta7164188454487880599at_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P3924000266242761678st_nat @ map_ta8615873517111064934at_nat @ ( produc4626581765195395529st_nat @ X4 @ ( produc2694037385005941721st_nat @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_228_map__tailrec__rev_Opelims,axiom,
    ! [X4: produc8999643685545340131pVar_o > nat,Xa: list_P1665994083459302249pVar_o,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta4027472834757870206_o_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P2686006265144907906st_nat @ map_ta8734788843987761135_o_nat @ ( produc3864453691545290181st_nat @ X4 @ ( produc5426949100443613682st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Pr6413368191065476585pVar_o )
           => ( ( Y3 = Xb )
             => ~ ( accp_P2686006265144907906st_nat @ map_ta8734788843987761135_o_nat @ ( produc3864453691545290181st_nat @ X4 @ ( produc5426949100443613682st_nat @ nil_Pr6413368191065476585pVar_o @ Xb ) ) ) ) )
         => ~ ! [A3: produc8999643685545340131pVar_o,As: list_P1665994083459302249pVar_o] :
                ( ( Xa
                  = ( cons_P6860203055867761049pVar_o @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta4027472834757870206_o_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P2686006265144907906st_nat @ map_ta8734788843987761135_o_nat @ ( produc3864453691545290181st_nat @ X4 @ ( produc5426949100443613682st_nat @ ( cons_P6860203055867761049pVar_o @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_229_map__tailrec__rev_Opelims,axiom,
    ! [X4: nat > produc8999643685545340131pVar_o,Xa: list_nat,Xb: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( ( map_ta511341784408162908pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P5735884745159397678pVar_o @ map_ta5218657793638053837pVar_o @ ( produc663790639386698097pVar_o @ X4 @ ( produc697358683602405680pVar_o @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = Xb )
             => ~ ( accp_P5735884745159397678pVar_o @ map_ta5218657793638053837pVar_o @ ( produc663790639386698097pVar_o @ X4 @ ( produc697358683602405680pVar_o @ nil_nat @ Xb ) ) ) ) )
         => ~ ! [A3: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta511341784408162908pVar_o @ X4 @ As @ ( cons_P6860203055867761049pVar_o @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P5735884745159397678pVar_o @ map_ta5218657793638053837pVar_o @ ( produc663790639386698097pVar_o @ X4 @ ( produc697358683602405680pVar_o @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_230_map__tailrec__rev_Opelims,axiom,
    ! [X4: list_P1665994083459302249pVar_o > nat,Xa: list_l1255154591254203759pVar_o,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta3501597723813782532_o_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P206715725567320718st_nat @ map_ta4999941329754189045_o_nat @ ( produc7356997260756735697st_nat @ X4 @ ( produc4052226372830137592st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_li1310946081133177199pVar_o )
           => ( ( Y3 = Xb )
             => ~ ( accp_P206715725567320718st_nat @ map_ta4999941329754189045_o_nat @ ( produc7356997260756735697st_nat @ X4 @ ( produc4052226372830137592st_nat @ nil_li1310946081133177199pVar_o @ Xb ) ) ) ) )
         => ~ ! [A3: list_P1665994083459302249pVar_o,As: list_l1255154591254203759pVar_o] :
                ( ( Xa
                  = ( cons_l3452452106213086495pVar_o @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta3501597723813782532_o_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P206715725567320718st_nat @ map_ta4999941329754189045_o_nat @ ( produc7356997260756735697st_nat @ X4 @ ( produc4052226372830137592st_nat @ ( cons_l3452452106213086495pVar_o @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_231_map__tailrec__rev_Opelims,axiom,
    ! [X4: nat > list_P1665994083459302249pVar_o,Xa: list_nat,Xb: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o] :
      ( ( ( map_ta5031888053227161698pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P7615866244381348410pVar_o @ map_ta6530231659167568211pVar_o @ ( produc3361903918338910589pVar_o @ X4 @ ( produc7103779822796047286pVar_o @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = Xb )
             => ~ ( accp_P7615866244381348410pVar_o @ map_ta6530231659167568211pVar_o @ ( produc3361903918338910589pVar_o @ X4 @ ( produc7103779822796047286pVar_o @ nil_nat @ Xb ) ) ) ) )
         => ~ ! [A3: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta5031888053227161698pVar_o @ X4 @ As @ ( cons_l3452452106213086495pVar_o @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P7615866244381348410pVar_o @ map_ta6530231659167568211pVar_o @ ( produc3361903918338910589pVar_o @ X4 @ ( produc7103779822796047286pVar_o @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_232_map__tailrec__rev_Opelims,axiom,
    ! [X4: produc2642885279816024434ropVar > nat,Xa: list_P8791008543742759928ropVar,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta4142060526428890125ar_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P6679286929532158240st_nat @ map_ta8171396224712197118ar_nat @ ( produc7391651180360970723st_nat @ X4 @ ( produc5924578096210284289st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Pr6119456312066593144ropVar )
           => ( ( Y3 = Xb )
             => ~ ( accp_P6679286929532158240st_nat @ map_ta8171396224712197118ar_nat @ ( produc7391651180360970723st_nat @ X4 @ ( produc5924578096210284289st_nat @ nil_Pr6119456312066593144ropVar @ Xb ) ) ) ) )
         => ~ ! [A3: produc2642885279816024434ropVar,As: list_P8791008543742759928ropVar] :
                ( ( Xa
                  = ( cons_P1504208554336013608ropVar @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta4142060526428890125ar_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P6679286929532158240st_nat @ map_ta8171396224712197118ar_nat @ ( produc7391651180360970723st_nat @ X4 @ ( produc5924578096210284289st_nat @ ( cons_P1504208554336013608ropVar @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_233_map__tailrec__rev_Opelims,axiom,
    ! [X4: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o,Xa: list_P1665994083459302249pVar_o,Xb: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( ( map_ta8452059871828574389pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P304740604055074350pVar_o @ map_ta2726038528091471876pVar_o @ ( produc670561721131745065pVar_o @ X4 @ ( produc9064796567173055959pVar_o @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_Pr6413368191065476585pVar_o )
           => ( ( Y3 = Xb )
             => ~ ( accp_P304740604055074350pVar_o @ map_ta2726038528091471876pVar_o @ ( produc670561721131745065pVar_o @ X4 @ ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Xb ) ) ) ) )
         => ~ ! [A3: produc8999643685545340131pVar_o,As: list_P1665994083459302249pVar_o] :
                ( ( Xa
                  = ( cons_P6860203055867761049pVar_o @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta8452059871828574389pVar_o @ X4 @ As @ ( cons_P6860203055867761049pVar_o @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P304740604055074350pVar_o @ map_ta2726038528091471876pVar_o @ ( produc670561721131745065pVar_o @ X4 @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_234_map__tailrec__rev_Opelims,axiom,
    ! [X4: list_l1255154591254203759pVar_o > nat,Xa: list_l985434930288970485pVar_o,Xb: list_nat,Y3: list_nat] :
      ( ( ( map_ta57781623974148362_o_nat @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P4052322840410693018st_nat @ map_ta15663767523874171_o_nat @ ( produc7873628753273084125st_nat @ X4 @ ( produc266617407108862846st_nat @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_li590035233778055285pVar_o )
           => ( ( Y3 = Xb )
             => ~ ( accp_P4052322840410693018st_nat @ map_ta15663767523874171_o_nat @ ( produc7873628753273084125st_nat @ X4 @ ( produc266617407108862846st_nat @ nil_li590035233778055285pVar_o @ Xb ) ) ) ) )
         => ~ ! [A3: list_l1255154591254203759pVar_o,As: list_l985434930288970485pVar_o] :
                ( ( Xa
                  = ( cons_l7316790070876032037pVar_o @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta57781623974148362_o_nat @ X4 @ As @ ( cons_nat @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P4052322840410693018st_nat @ map_ta15663767523874171_o_nat @ ( produc7873628753273084125st_nat @ X4 @ ( produc266617407108862846st_nat @ ( cons_l7316790070876032037pVar_o @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_235_map__tailrec__rev_Opelims,axiom,
    ! [X4: nat > produc2642885279816024434ropVar,Xa: list_nat,Xb: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( ( map_ta7700754162581607787ropVar @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P8506009723982548044ropVar @ map_ta2506717824010138972ropVar @ ( produc5722207277547347983ropVar @ X4 @ ( produc5634919067874425663ropVar @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = Xb )
             => ~ ( accp_P8506009723982548044ropVar @ map_ta2506717824010138972ropVar @ ( produc5722207277547347983ropVar @ X4 @ ( produc5634919067874425663ropVar @ nil_nat @ Xb ) ) ) ) )
         => ~ ! [A3: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta7700754162581607787ropVar @ X4 @ As @ ( cons_P1504208554336013608ropVar @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P8506009723982548044ropVar @ map_ta2506717824010138972ropVar @ ( produc5722207277547347983ropVar @ X4 @ ( produc5634919067874425663ropVar @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_236_map__tailrec__rev_Opelims,axiom,
    ! [X4: nat > list_l1255154591254203759pVar_o,Xa: list_nat,Xb: list_l985434930288970485pVar_o,Y3: list_l985434930288970485pVar_o] :
      ( ( ( map_ta1976070376341471720pVar_o @ X4 @ Xa @ Xb )
        = Y3 )
     => ( ( accp_P7224076938530096198pVar_o @ map_ta1933952519891197529pVar_o @ ( produc4434012089350267529pVar_o @ X4 @ ( produc2181093233133072828pVar_o @ Xa @ Xb ) ) )
       => ( ( ( Xa = nil_nat )
           => ( ( Y3 = Xb )
             => ~ ( accp_P7224076938530096198pVar_o @ map_ta1933952519891197529pVar_o @ ( produc4434012089350267529pVar_o @ X4 @ ( produc2181093233133072828pVar_o @ nil_nat @ Xb ) ) ) ) )
         => ~ ! [A3: nat,As: list_nat] :
                ( ( Xa
                  = ( cons_nat @ A3 @ As ) )
               => ( ( Y3
                    = ( map_ta1976070376341471720pVar_o @ X4 @ As @ ( cons_l7316790070876032037pVar_o @ ( X4 @ A3 ) @ Xb ) ) )
                 => ~ ( accp_P7224076938530096198pVar_o @ map_ta1933952519891197529pVar_o @ ( produc4434012089350267529pVar_o @ X4 @ ( produc2181093233133072828pVar_o @ ( cons_nat @ A3 @ As ) @ Xb ) ) ) ) ) ) ) ) ).

% map_tailrec_rev.pelims
thf(fact_237_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: nat,Y4: nat,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R )
                   => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_238_listrel_Ocases,axiom,
    ! [A1: list_o,A22: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ A1 @ A22 ) @ ( listrel_o_o @ R ) )
     => ( ( ( A1 = nil_o )
         => ( A22 != nil_o ) )
       => ~ ! [X2: $o,Y4: $o,Xs: list_o] :
              ( ( A1
                = ( cons_o @ X2 @ Xs ) )
             => ! [Ys: list_o] :
                  ( ( A22
                    = ( cons_o @ Y4 @ Ys ) )
                 => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y4 ) @ R )
                   => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel_o_o @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_239_listrel_Ocases,axiom,
    ! [A1: list_M6740372356978808802ropVar,A22: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ A1 @ A22 ) @ ( listre4495854419763884273ropVar @ R ) )
     => ( ( ( A1 = nil_Mu4132399263956683874ropVar )
         => ( A22 != nil_Mu4132399263956683874ropVar ) )
       => ~ ! [X2: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
              ( ( A1
                = ( cons_M4277272395645347858ropVar @ X2 @ Xs ) )
             => ! [Ys: list_M6740372356978808802ropVar] :
                  ( ( A22
                    = ( cons_M4277272395645347858ropVar @ Y4 @ Ys ) )
                 => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y4 ) @ R )
                   => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_240_listrel_Ocases,axiom,
    ! [A1: list_P1665994083459302249pVar_o,A22: list_nat,R: set_Pr302692805937157912_o_nat] :
      ( ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ A1 @ A22 ) @ ( listre6760217120960589698_o_nat @ R ) )
     => ( ( ( A1 = nil_Pr6413368191065476585pVar_o )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: produc8999643685545340131pVar_o,Y4: nat,Xs: list_P1665994083459302249pVar_o] :
              ( ( A1
                = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( member7944572178332388089_o_nat @ ( produc7886040895744259420_o_nat @ X2 @ Y4 ) @ R )
                   => ~ ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ Xs @ Ys ) @ ( listre6760217120960589698_o_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_241_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_P1665994083459302249pVar_o,R: set_Pr8900919299660647230pVar_o] :
      ( ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ A1 @ A22 ) @ ( listre3244086070610882400pVar_o @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_Pr6413368191065476585pVar_o ) )
       => ~ ! [X2: nat,Y4: produc8999643685545340131pVar_o,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_P1665994083459302249pVar_o] :
                  ( ( A22
                    = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
                 => ( ( member6910252441317339679pVar_o @ ( produc4369909845394552122pVar_o @ X2 @ Y4 ) @ R )
                   => ~ ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ Xs @ Ys ) @ ( listre3244086070610882400pVar_o @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_242_listrel_Ocases,axiom,
    ! [A1: list_l1255154591254203759pVar_o,A22: list_nat,R: set_Pr6978669447677043102_o_nat] :
      ( ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ A1 @ A22 ) @ ( listre3060699943780466440_o_nat @ R ) )
     => ( ( ( A1 = nil_li1310946081133177199pVar_o )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: list_P1665994083459302249pVar_o,Y4: nat,Xs: list_l1255154591254203759pVar_o] :
              ( ( A1
                = ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( member8127644508941222527_o_nat @ ( produc4045138296077651938_o_nat @ X2 @ Y4 ) @ R )
                   => ~ ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ Xs @ Ys ) @ ( listre3060699943780466440_o_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_243_listrel_Ocases,axiom,
    ! [A1: list_nat,A22: list_l1255154591254203759pVar_o,R: set_Pr8506680132865545540pVar_o] :
      ( ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ A1 @ A22 ) @ ( listre4590990273193845606pVar_o @ R ) )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_li1310946081133177199pVar_o ) )
       => ~ ! [X2: nat,Y4: list_P1665994083459302249pVar_o,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_l1255154591254203759pVar_o] :
                  ( ( A22
                    = ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) )
                 => ( ( member4118595025335937317pVar_o @ ( produc5575428625491031104pVar_o @ X2 @ Y4 ) @ R )
                   => ~ ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ Xs @ Ys ) @ ( listre4590990273193845606pVar_o @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_244_listrel_Ocases,axiom,
    ! [A1: list_P723798329145367999ropVar,A22: list_l1644708917474426447ropVar,R: set_Pr3339951299666510767ropVar] :
      ( ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ A1 @ A22 ) @ ( listre1163491797370331489ropVar @ R ) )
     => ( ( ( A1 = nil_Pr7606595143512524841ropVar )
         => ( A22 != nil_li6797326493501628985ropVar ) )
       => ~ ! [X2: propos3600597766819905199ropVar,Y4: list_P723798329145367999ropVar,Xs: list_P723798329145367999ropVar] :
              ( ( A1
                = ( cons_P6757824865275295865ropVar @ X2 @ Xs ) )
             => ! [Ys: list_l1644708917474426447ropVar] :
                  ( ( A22
                    = ( cons_l2454487807130495625ropVar @ Y4 @ Ys ) )
                 => ( ( member4962873584118313464ropVar @ ( produc5136867404843011847ropVar @ X2 @ Y4 ) @ R )
                   => ~ ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ Xs @ Ys ) @ ( listre1163491797370331489ropVar @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_245_listrel_Ocases,axiom,
    ! [A1: list_P8791008543742759928ropVar,A22: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ A1 @ A22 ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ( ( ( A1 = nil_Pr6119456312066593144ropVar )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: produc2642885279816024434ropVar,Y4: nat,Xs: list_P8791008543742759928ropVar] :
              ( ( A1
                = ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X2 @ Y4 ) @ R )
                   => ~ ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs @ Ys ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_246_listrel_Ocases,axiom,
    ! [A1: list_P1665994083459302249pVar_o,A22: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ A1 @ A22 ) @ ( listre8839862472748488369pVar_o @ R ) )
     => ( ( ( A1 = nil_Pr6413368191065476585pVar_o )
         => ( A22 != nil_Pr6413368191065476585pVar_o ) )
       => ~ ! [X2: produc8999643685545340131pVar_o,Y4: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
              ( ( A1
                = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
             => ! [Ys: list_P1665994083459302249pVar_o] :
                  ( ( A22
                    = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
                 => ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X2 @ Y4 ) @ R )
                   => ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs @ Ys ) @ ( listre8839862472748488369pVar_o @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_247_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ A1 @ A22 ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_nat ) )
        | ? [X3: nat,Y2: nat,Xs3: list_nat,Ys3: list_nat] :
            ( ( A1
              = ( cons_nat @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y2 @ Ys3 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y2 ) @ R )
            & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs3 @ Ys3 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_248_listrel_Osimps,axiom,
    ! [A1: list_o,A22: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ A1 @ A22 ) @ ( listrel_o_o @ R ) )
      = ( ( ( A1 = nil_o )
          & ( A22 = nil_o ) )
        | ? [X3: $o,Y2: $o,Xs3: list_o,Ys3: list_o] :
            ( ( A1
              = ( cons_o @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_o @ Y2 @ Ys3 ) )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y2 ) @ R )
            & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs3 @ Ys3 ) @ ( listrel_o_o @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_249_listrel_Osimps,axiom,
    ! [A1: list_M6740372356978808802ropVar,A22: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ A1 @ A22 ) @ ( listre4495854419763884273ropVar @ R ) )
      = ( ( ( A1 = nil_Mu4132399263956683874ropVar )
          & ( A22 = nil_Mu4132399263956683874ropVar ) )
        | ? [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Xs3: list_M6740372356978808802ropVar,Ys3: list_M6740372356978808802ropVar] :
            ( ( A1
              = ( cons_M4277272395645347858ropVar @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_M4277272395645347858ropVar @ Y2 @ Ys3 ) )
            & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X3 @ Y2 ) @ R )
            & ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs3 @ Ys3 ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_250_listrel_Osimps,axiom,
    ! [A1: list_P1665994083459302249pVar_o,A22: list_nat,R: set_Pr302692805937157912_o_nat] :
      ( ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ A1 @ A22 ) @ ( listre6760217120960589698_o_nat @ R ) )
      = ( ( ( A1 = nil_Pr6413368191065476585pVar_o )
          & ( A22 = nil_nat ) )
        | ? [X3: produc8999643685545340131pVar_o,Y2: nat,Xs3: list_P1665994083459302249pVar_o,Ys3: list_nat] :
            ( ( A1
              = ( cons_P6860203055867761049pVar_o @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y2 @ Ys3 ) )
            & ( member7944572178332388089_o_nat @ ( produc7886040895744259420_o_nat @ X3 @ Y2 ) @ R )
            & ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ Xs3 @ Ys3 ) @ ( listre6760217120960589698_o_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_251_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_P1665994083459302249pVar_o,R: set_Pr8900919299660647230pVar_o] :
      ( ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ A1 @ A22 ) @ ( listre3244086070610882400pVar_o @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_Pr6413368191065476585pVar_o ) )
        | ? [X3: nat,Y2: produc8999643685545340131pVar_o,Xs3: list_nat,Ys3: list_P1665994083459302249pVar_o] :
            ( ( A1
              = ( cons_nat @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_P6860203055867761049pVar_o @ Y2 @ Ys3 ) )
            & ( member6910252441317339679pVar_o @ ( produc4369909845394552122pVar_o @ X3 @ Y2 ) @ R )
            & ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ Xs3 @ Ys3 ) @ ( listre3244086070610882400pVar_o @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_252_listrel_Osimps,axiom,
    ! [A1: list_l1255154591254203759pVar_o,A22: list_nat,R: set_Pr6978669447677043102_o_nat] :
      ( ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ A1 @ A22 ) @ ( listre3060699943780466440_o_nat @ R ) )
      = ( ( ( A1 = nil_li1310946081133177199pVar_o )
          & ( A22 = nil_nat ) )
        | ? [X3: list_P1665994083459302249pVar_o,Y2: nat,Xs3: list_l1255154591254203759pVar_o,Ys3: list_nat] :
            ( ( A1
              = ( cons_l3452452106213086495pVar_o @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y2 @ Ys3 ) )
            & ( member8127644508941222527_o_nat @ ( produc4045138296077651938_o_nat @ X3 @ Y2 ) @ R )
            & ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ Xs3 @ Ys3 ) @ ( listre3060699943780466440_o_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_253_listrel_Osimps,axiom,
    ! [A1: list_nat,A22: list_l1255154591254203759pVar_o,R: set_Pr8506680132865545540pVar_o] :
      ( ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ A1 @ A22 ) @ ( listre4590990273193845606pVar_o @ R ) )
      = ( ( ( A1 = nil_nat )
          & ( A22 = nil_li1310946081133177199pVar_o ) )
        | ? [X3: nat,Y2: list_P1665994083459302249pVar_o,Xs3: list_nat,Ys3: list_l1255154591254203759pVar_o] :
            ( ( A1
              = ( cons_nat @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_l3452452106213086495pVar_o @ Y2 @ Ys3 ) )
            & ( member4118595025335937317pVar_o @ ( produc5575428625491031104pVar_o @ X3 @ Y2 ) @ R )
            & ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ Xs3 @ Ys3 ) @ ( listre4590990273193845606pVar_o @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_254_listrel_Osimps,axiom,
    ! [A1: list_P723798329145367999ropVar,A22: list_l1644708917474426447ropVar,R: set_Pr3339951299666510767ropVar] :
      ( ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ A1 @ A22 ) @ ( listre1163491797370331489ropVar @ R ) )
      = ( ( ( A1 = nil_Pr7606595143512524841ropVar )
          & ( A22 = nil_li6797326493501628985ropVar ) )
        | ? [X3: propos3600597766819905199ropVar,Y2: list_P723798329145367999ropVar,Xs3: list_P723798329145367999ropVar,Ys3: list_l1644708917474426447ropVar] :
            ( ( A1
              = ( cons_P6757824865275295865ropVar @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_l2454487807130495625ropVar @ Y2 @ Ys3 ) )
            & ( member4962873584118313464ropVar @ ( produc5136867404843011847ropVar @ X3 @ Y2 ) @ R )
            & ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ Xs3 @ Ys3 ) @ ( listre1163491797370331489ropVar @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_255_listrel_Osimps,axiom,
    ! [A1: list_P8791008543742759928ropVar,A22: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ A1 @ A22 ) @ ( listre5688682141216627473ar_nat @ R ) )
      = ( ( ( A1 = nil_Pr6119456312066593144ropVar )
          & ( A22 = nil_nat ) )
        | ? [X3: produc2642885279816024434ropVar,Y2: nat,Xs3: list_P8791008543742759928ropVar,Ys3: list_nat] :
            ( ( A1
              = ( cons_P1504208554336013608ropVar @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_nat @ Y2 @ Ys3 ) )
            & ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X3 @ Y2 ) @ R )
            & ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs3 @ Ys3 ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_256_listrel_Osimps,axiom,
    ! [A1: list_P1665994083459302249pVar_o,A22: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ A1 @ A22 ) @ ( listre8839862472748488369pVar_o @ R ) )
      = ( ( ( A1 = nil_Pr6413368191065476585pVar_o )
          & ( A22 = nil_Pr6413368191065476585pVar_o ) )
        | ? [X3: produc8999643685545340131pVar_o,Y2: produc8999643685545340131pVar_o,Xs3: list_P1665994083459302249pVar_o,Ys3: list_P1665994083459302249pVar_o] :
            ( ( A1
              = ( cons_P6860203055867761049pVar_o @ X3 @ Xs3 ) )
            & ( A22
              = ( cons_P6860203055867761049pVar_o @ Y2 @ Ys3 ) )
            & ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X3 @ Y2 ) @ R )
            & ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs3 @ Ys3 ) @ ( listre8839862472748488369pVar_o @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_257_lexord__Nil__left,axiom,
    ! [Y3: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Y3 ) @ ( lexord832287969131410277ropVar @ R ) )
      = ( ? [A5: produc2642885279816024434ropVar,X3: list_P8791008543742759928ropVar] :
            ( Y3
            = ( cons_P1504208554336013608ropVar @ A5 @ X3 ) ) ) ) ).

% lexord_Nil_left
thf(fact_258_lexord__Nil__left,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Y3 ) @ ( lexord9212727487545282390pVar_o @ R ) )
      = ( ? [A5: produc8999643685545340131pVar_o,X3: list_P1665994083459302249pVar_o] :
            ( Y3
            = ( cons_P6860203055867761049pVar_o @ A5 @ X3 ) ) ) ) ).

% lexord_Nil_left
thf(fact_259_lexord__Nil__left,axiom,
    ! [Y3: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Y3 ) @ ( lexord8675461415035850844pVar_o @ R ) )
      = ( ? [A5: list_P1665994083459302249pVar_o,X3: list_l1255154591254203759pVar_o] :
            ( Y3
            = ( cons_l3452452106213086495pVar_o @ A5 @ X3 ) ) ) ) ).

% lexord_Nil_left
thf(fact_260_lexord__Nil__left,axiom,
    ! [Y3: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Y3 ) @ ( lexord4213358897963926754pVar_o @ R ) )
      = ( ? [A5: list_l1255154591254203759pVar_o,X3: list_l985434930288970485pVar_o] :
            ( Y3
            = ( cons_l7316790070876032037pVar_o @ A5 @ X3 ) ) ) ) ).

% lexord_Nil_left
thf(fact_261_lexord__Nil__left,axiom,
    ! [Y3: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Y3 ) @ ( lexord_nat @ R ) )
      = ( ? [A5: nat,X3: list_nat] :
            ( Y3
            = ( cons_nat @ A5 @ X3 ) ) ) ) ).

% lexord_Nil_left
thf(fact_262_lexord__cons__cons,axiom,
    ! [A: produc2642885279816024434ropVar,X4: list_P8791008543742759928ropVar,B: produc2642885279816024434ropVar,Y3: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ A @ X4 ) @ ( cons_P1504208554336013608ropVar @ B @ Y3 ) ) @ ( lexord832287969131410277ropVar @ R ) )
      = ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A @ B ) @ R )
        | ( ( A = B )
          & ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ X4 @ Y3 ) @ ( lexord832287969131410277ropVar @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_263_lexord__cons__cons,axiom,
    ! [A: produc8999643685545340131pVar_o,X4: list_P1665994083459302249pVar_o,B: produc8999643685545340131pVar_o,Y3: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ A @ X4 ) @ ( cons_P6860203055867761049pVar_o @ B @ Y3 ) ) @ ( lexord9212727487545282390pVar_o @ R ) )
      = ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ A @ B ) @ R )
        | ( ( A = B )
          & ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ Y3 ) @ ( lexord9212727487545282390pVar_o @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_264_lexord__cons__cons,axiom,
    ! [A: list_P1665994083459302249pVar_o,X4: list_l1255154591254203759pVar_o,B: list_P1665994083459302249pVar_o,Y3: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ A @ X4 ) @ ( cons_l3452452106213086495pVar_o @ B @ Y3 ) ) @ ( lexord8675461415035850844pVar_o @ R ) )
      = ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ A @ B ) @ R )
        | ( ( A = B )
          & ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ Y3 ) @ ( lexord8675461415035850844pVar_o @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_265_lexord__cons__cons,axiom,
    ! [A: list_l1255154591254203759pVar_o,X4: list_l985434930288970485pVar_o,B: list_l1255154591254203759pVar_o,Y3: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ A @ X4 ) @ ( cons_l7316790070876032037pVar_o @ B @ Y3 ) ) @ ( lexord4213358897963926754pVar_o @ R ) )
      = ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ A @ B ) @ R )
        | ( ( A = B )
          & ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ X4 @ Y3 ) @ ( lexord4213358897963926754pVar_o @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_266_lexord__cons__cons,axiom,
    ! [A: nat,X4: list_nat,B: nat,Y3: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ A @ X4 ) @ ( cons_nat @ B @ Y3 ) ) @ ( lexord_nat @ R ) )
      = ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ Y3 ) @ ( lexord_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_267_lexord__cons__cons,axiom,
    ! [A: $o,X4: list_o,B: $o,Y3: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ A @ X4 ) @ ( cons_o @ B @ Y3 ) ) @ ( lexord_o @ R ) )
      = ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
        | ( ( A = B )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ X4 @ Y3 ) @ ( lexord_o @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_268_lexord__cons__cons,axiom,
    ! [A: multis3193088007478089820ropVar,X4: list_M6740372356978808802ropVar,B: multis3193088007478089820ropVar,Y3: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ A @ X4 ) @ ( cons_M4277272395645347858ropVar @ B @ Y3 ) ) @ ( lexord1374882220397993039ropVar @ R ) )
      = ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ B ) @ R )
        | ( ( A = B )
          & ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ X4 @ Y3 ) @ ( lexord1374882220397993039ropVar @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_269_cnf__encoder_OencodeGammaCond_Ocases,axiom,
    ! [X4: produc3507939890032686909od_o_o] :
      ( ! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
          ( X4
         != ( produc4931073445691921453od_o_o @ Gam @ ( produc9033520932313852770od_o_o @ Eps @ ( product_Pair_o_o @ $true @ $true ) ) ) )
     => ( ! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
            ( X4
           != ( produc4931073445691921453od_o_o @ Gam @ ( produc9033520932313852770od_o_o @ Eps @ ( product_Pair_o_o @ $false @ $false ) ) ) )
       => ( ! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
              ( X4
             != ( produc4931073445691921453od_o_o @ Gam @ ( produc9033520932313852770od_o_o @ Eps @ ( product_Pair_o_o @ $false @ $true ) ) ) )
         => ~ ! [Gam: multis3193088007478089820ropVar,Eps: multis3193088007478089820ropVar] :
                ( X4
               != ( produc4931073445691921453od_o_o @ Gam @ ( produc9033520932313852770od_o_o @ Eps @ ( product_Pair_o_o @ $true @ $false ) ) ) ) ) ) ) ).

% cnf_encoder.encodeGammaCond.cases
thf(fact_270_lexord__irreflexive,axiom,
    ! [R: set_Product_prod_o_o,Xs2: list_o] :
      ( ! [X2: $o] :
          ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
     => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Xs2 ) @ ( lexord_o @ R ) ) ) ).

% lexord_irreflexive
thf(fact_271_lexord__irreflexive,axiom,
    ! [R: set_Pr7799659934693882759ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ! [X2: multis3193088007478089820ropVar] :
          ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ X2 ) @ R )
     => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Xs2 ) @ ( lexord1374882220397993039ropVar @ R ) ) ) ).

% lexord_irreflexive
thf(fact_272_lexord__linear,axiom,
    ! [R: set_Product_prod_o_o,X4: list_o,Y3: list_o] :
      ( ! [A3: $o,B3: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ B3 ) @ R )
          | ( A3 = B3 )
          | ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ B3 @ A3 ) @ R ) )
     => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ X4 @ Y3 ) @ ( lexord_o @ R ) )
        | ( X4 = Y3 )
        | ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Y3 @ X4 ) @ ( lexord_o @ R ) ) ) ) ).

% lexord_linear
thf(fact_273_lexord__linear,axiom,
    ! [R: set_Pr7799659934693882759ropVar,X4: list_M6740372356978808802ropVar,Y3: list_M6740372356978808802ropVar] :
      ( ! [A3: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A3 @ B3 ) @ R )
          | ( A3 = B3 )
          | ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ B3 @ A3 ) @ R ) )
     => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ X4 @ Y3 ) @ ( lexord1374882220397993039ropVar @ R ) )
        | ( X4 = Y3 )
        | ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Y3 @ X4 ) @ ( lexord1374882220397993039ropVar @ R ) ) ) ) ).

% lexord_linear
thf(fact_274_listrel_ONil,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).

% listrel.Nil
thf(fact_275_listrel_ONil,axiom,
    ! [R: set_Pr302692805937157912_o_nat] : ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ nil_Pr6413368191065476585pVar_o @ nil_nat ) @ ( listre6760217120960589698_o_nat @ R ) ) ).

% listrel.Nil
thf(fact_276_listrel_ONil,axiom,
    ! [R: set_Pr8900919299660647230pVar_o] : ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ nil_nat @ nil_Pr6413368191065476585pVar_o ) @ ( listre3244086070610882400pVar_o @ R ) ) ).

% listrel.Nil
thf(fact_277_listrel_ONil,axiom,
    ! [R: set_Pr6978669447677043102_o_nat] : ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ nil_li1310946081133177199pVar_o @ nil_nat ) @ ( listre3060699943780466440_o_nat @ R ) ) ).

% listrel.Nil
thf(fact_278_listrel_ONil,axiom,
    ! [R: set_Pr8506680132865545540pVar_o] : ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ nil_nat @ nil_li1310946081133177199pVar_o ) @ ( listre4590990273193845606pVar_o @ R ) ) ).

% listrel.Nil
thf(fact_279_listrel_ONil,axiom,
    ! [R: set_Pr7538684846379304231ar_nat] : ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ nil_Pr6119456312066593144ropVar @ nil_nat ) @ ( listre5688682141216627473ar_nat @ R ) ) ).

% listrel.Nil
thf(fact_280_listrel_ONil,axiom,
    ! [R: set_Pr4574656050966626247pVar_o] : ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_Pr6413368191065476585pVar_o ) @ ( listre8839862472748488369pVar_o @ R ) ) ).

% listrel.Nil
thf(fact_281_listrel_ONil,axiom,
    ! [R: set_Pr8287603889165557668_o_nat] : ( member1238775061299713819st_nat @ ( produc266617407108862846st_nat @ nil_li590035233778055285pVar_o @ nil_nat ) @ ( listre6026826826888118798_o_nat @ R ) ) ).

% listrel.Nil
thf(fact_282_listrel_ONil,axiom,
    ! [R: set_Pr7708077615150402893ropVar] : ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ nil_nat @ nil_Pr6119456312066593144ropVar ) @ ( listre24003740514569327ropVar @ R ) ) ).

% listrel.Nil
thf(fact_283_listrel_ONil,axiom,
    ! [R: set_Pr6085839080702481098pVar_o] : ( member2278620947860517793pVar_o @ ( produc2181093233133072828pVar_o @ nil_nat @ nil_li590035233778055285pVar_o ) @ ( listre7945115579255442156pVar_o @ R ) ) ).

% listrel.Nil
thf(fact_284_listrel__Nil1,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel_nat_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil1
thf(fact_285_listrel__Nil1,axiom,
    ! [Xs2: list_nat,R: set_Pr302692805937157912_o_nat] :
      ( ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ nil_Pr6413368191065476585pVar_o @ Xs2 ) @ ( listre6760217120960589698_o_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil1
thf(fact_286_listrel__Nil1,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,R: set_Pr8900919299660647230pVar_o] :
      ( ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ nil_nat @ Xs2 ) @ ( listre3244086070610882400pVar_o @ R ) )
     => ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% listrel_Nil1
thf(fact_287_listrel__Nil1,axiom,
    ! [Xs2: list_nat,R: set_Pr6978669447677043102_o_nat] :
      ( ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ nil_li1310946081133177199pVar_o @ Xs2 ) @ ( listre3060699943780466440_o_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil1
thf(fact_288_listrel__Nil1,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,R: set_Pr8506680132865545540pVar_o] :
      ( ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ nil_nat @ Xs2 ) @ ( listre4590990273193845606pVar_o @ R ) )
     => ( Xs2 = nil_li1310946081133177199pVar_o ) ) ).

% listrel_Nil1
thf(fact_289_listrel__Nil1,axiom,
    ! [Xs2: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ nil_Pr6119456312066593144ropVar @ Xs2 ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil1
thf(fact_290_listrel__Nil1,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Xs2 ) @ ( listre8839862472748488369pVar_o @ R ) )
     => ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% listrel_Nil1
thf(fact_291_listrel__Nil1,axiom,
    ! [Xs2: list_nat,R: set_Pr8287603889165557668_o_nat] :
      ( ( member1238775061299713819st_nat @ ( produc266617407108862846st_nat @ nil_li590035233778055285pVar_o @ Xs2 ) @ ( listre6026826826888118798_o_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil1
thf(fact_292_listrel__Nil1,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ nil_nat @ Xs2 ) @ ( listre24003740514569327ropVar @ R ) )
     => ( Xs2 = nil_Pr6119456312066593144ropVar ) ) ).

% listrel_Nil1
thf(fact_293_listrel__Nil1,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,R: set_Pr6085839080702481098pVar_o] :
      ( ( member2278620947860517793pVar_o @ ( produc2181093233133072828pVar_o @ nil_nat @ Xs2 ) @ ( listre7945115579255442156pVar_o @ R ) )
     => ( Xs2 = nil_li590035233778055285pVar_o ) ) ).

% listrel_Nil1
thf(fact_294_listrel__Nil2,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel_nat_nat @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil2
thf(fact_295_listrel__Nil2,axiom,
    ! [Xs2: list_nat,R: set_Pr8900919299660647230pVar_o] :
      ( ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ Xs2 @ nil_Pr6413368191065476585pVar_o ) @ ( listre3244086070610882400pVar_o @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil2
thf(fact_296_listrel__Nil2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,R: set_Pr302692805937157912_o_nat] :
      ( ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ Xs2 @ nil_nat ) @ ( listre6760217120960589698_o_nat @ R ) )
     => ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% listrel_Nil2
thf(fact_297_listrel__Nil2,axiom,
    ! [Xs2: list_nat,R: set_Pr8506680132865545540pVar_o] :
      ( ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ Xs2 @ nil_li1310946081133177199pVar_o ) @ ( listre4590990273193845606pVar_o @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil2
thf(fact_298_listrel__Nil2,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,R: set_Pr6978669447677043102_o_nat] :
      ( ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ Xs2 @ nil_nat ) @ ( listre3060699943780466440_o_nat @ R ) )
     => ( Xs2 = nil_li1310946081133177199pVar_o ) ) ).

% listrel_Nil2
thf(fact_299_listrel__Nil2,axiom,
    ! [Xs2: list_nat,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs2 @ nil_Pr6119456312066593144ropVar ) @ ( listre24003740514569327ropVar @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil2
thf(fact_300_listrel__Nil2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ nil_Pr6413368191065476585pVar_o ) @ ( listre8839862472748488369pVar_o @ R ) )
     => ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% listrel_Nil2
thf(fact_301_listrel__Nil2,axiom,
    ! [Xs2: list_nat,R: set_Pr6085839080702481098pVar_o] :
      ( ( member2278620947860517793pVar_o @ ( produc2181093233133072828pVar_o @ Xs2 @ nil_li590035233778055285pVar_o ) @ ( listre7945115579255442156pVar_o @ R ) )
     => ( Xs2 = nil_nat ) ) ).

% listrel_Nil2
thf(fact_302_listrel__Nil2,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs2 @ nil_nat ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ( Xs2 = nil_Pr6119456312066593144ropVar ) ) ).

% listrel_Nil2
thf(fact_303_listrel__Nil2,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,R: set_Pr8287603889165557668_o_nat] :
      ( ( member1238775061299713819st_nat @ ( produc266617407108862846st_nat @ Xs2 @ nil_nat ) @ ( listre6026826826888118798_o_nat @ R ) )
     => ( Xs2 = nil_li590035233778055285pVar_o ) ) ).

% listrel_Nil2
thf(fact_304_lexord__Nil__right,axiom,
    ! [X4: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) @ ( lexord832287969131410277ropVar @ R ) ) ).

% lexord_Nil_right
thf(fact_305_lexord__Nil__right,axiom,
    ! [X4: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) @ ( lexord9212727487545282390pVar_o @ R ) ) ).

% lexord_Nil_right
thf(fact_306_lexord__Nil__right,axiom,
    ! [X4: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) @ ( lexord8675461415035850844pVar_o @ R ) ) ).

% lexord_Nil_right
thf(fact_307_lexord__Nil__right,axiom,
    ! [X4: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ X4 @ nil_li590035233778055285pVar_o ) @ ( lexord4213358897963926754pVar_o @ R ) ) ).

% lexord_Nil_right
thf(fact_308_lexord__Nil__right,axiom,
    ! [X4: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ nil_nat ) @ ( lexord_nat @ R ) ) ).

% lexord_Nil_right
thf(fact_309_listrel__Cons2,axiom,
    ! [Xs2: list_nat,Y3: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [X2: nat,Xs: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X2 @ Xs ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys2 ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_310_listrel__Cons2,axiom,
    ! [Xs2: list_o,Y3: $o,Ys2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ ( cons_o @ Y3 @ Ys2 ) ) @ ( listrel_o_o @ R ) )
     => ~ ! [X2: $o,Xs: list_o] :
            ( ( Xs2
              = ( cons_o @ X2 @ Xs ) )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R )
             => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys2 ) @ ( listrel_o_o @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_311_listrel__Cons2,axiom,
    ! [Xs2: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys2 ) ) @ ( listre4495854419763884273ropVar @ R ) )
     => ~ ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
            ( ( Xs2
              = ( cons_M4277272395645347858ropVar @ X2 @ Xs ) )
           => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ R )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys2 ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_312_listrel__Cons2,axiom,
    ! [Xs2: list_nat,Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr8900919299660647230pVar_o] :
      ( ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) @ ( listre3244086070610882400pVar_o @ R ) )
     => ~ ! [X2: nat,Xs: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X2 @ Xs ) )
           => ( ( member6910252441317339679pVar_o @ ( produc4369909845394552122pVar_o @ X2 @ Y3 ) @ R )
             => ~ ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ Xs @ Ys2 ) @ ( listre3244086070610882400pVar_o @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_313_listrel__Cons2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Y3: nat,Ys2: list_nat,R: set_Pr302692805937157912_o_nat] :
      ( ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listre6760217120960589698_o_nat @ R ) )
     => ~ ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
           => ( ( member7944572178332388089_o_nat @ ( produc7886040895744259420_o_nat @ X2 @ Y3 ) @ R )
             => ~ ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ Xs @ Ys2 ) @ ( listre6760217120960589698_o_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_314_listrel__Cons2,axiom,
    ! [Xs2: list_nat,Y3: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,R: set_Pr8506680132865545540pVar_o] :
      ( ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) @ ( listre4590990273193845606pVar_o @ R ) )
     => ~ ! [X2: nat,Xs: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X2 @ Xs ) )
           => ( ( member4118595025335937317pVar_o @ ( produc5575428625491031104pVar_o @ X2 @ Y3 ) @ R )
             => ~ ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ Xs @ Ys2 ) @ ( listre4590990273193845606pVar_o @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_315_listrel__Cons2,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Y3: nat,Ys2: list_nat,R: set_Pr6978669447677043102_o_nat] :
      ( ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listre3060699943780466440_o_nat @ R ) )
     => ~ ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
            ( ( Xs2
              = ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) )
           => ( ( member8127644508941222527_o_nat @ ( produc4045138296077651938_o_nat @ X2 @ Y3 ) @ R )
             => ~ ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ Xs @ Ys2 ) @ ( listre3060699943780466440_o_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_316_listrel__Cons2,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Y3: list_P723798329145367999ropVar,Ys2: list_l1644708917474426447ropVar,R: set_Pr3339951299666510767ropVar] :
      ( ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ Xs2 @ ( cons_l2454487807130495625ropVar @ Y3 @ Ys2 ) ) @ ( listre1163491797370331489ropVar @ R ) )
     => ~ ! [X2: propos3600597766819905199ropVar,Xs: list_P723798329145367999ropVar] :
            ( ( Xs2
              = ( cons_P6757824865275295865ropVar @ X2 @ Xs ) )
           => ( ( member4962873584118313464ropVar @ ( produc5136867404843011847ropVar @ X2 @ Y3 ) @ R )
             => ~ ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ Xs @ Ys2 ) @ ( listre1163491797370331489ropVar @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_317_listrel__Cons2,axiom,
    ! [Xs2: list_nat,Y3: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,R: set_Pr7708077615150402893ropVar] :
      ( ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) ) @ ( listre24003740514569327ropVar @ R ) )
     => ~ ! [X2: nat,Xs: list_nat] :
            ( ( Xs2
              = ( cons_nat @ X2 @ Xs ) )
           => ( ( member918664796156078510ropVar @ ( produc9085022949259285321ropVar @ X2 @ Y3 ) @ R )
             => ~ ( member6598729439514207908ropVar @ ( produc5634919067874425663ropVar @ Xs @ Ys2 ) @ ( listre24003740514569327ropVar @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_318_listrel__Cons2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) @ ( listre8839862472748488369pVar_o @ R ) )
     => ~ ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
           => ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X2 @ Y3 ) @ R )
             => ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs @ Ys2 ) @ ( listre8839862472748488369pVar_o @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_319_listrel__Cons1,axiom,
    ! [Y3: nat,Ys2: list_nat,Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ Y3 @ Ys2 ) @ Xs2 ) @ ( listrel_nat_nat @ R ) )
     => ~ ! [Y4: nat,Ys: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys ) )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y3 @ Y4 ) @ R )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Ys ) @ ( listrel_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_320_listrel__Cons1,axiom,
    ! [Y3: $o,Ys2: list_o,Xs2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ Y3 @ Ys2 ) @ Xs2 ) @ ( listrel_o_o @ R ) )
     => ~ ! [Y4: $o,Ys: list_o] :
            ( ( Xs2
              = ( cons_o @ Y4 @ Ys ) )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y3 @ Y4 ) @ R )
             => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Ys ) @ ( listrel_o_o @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_321_listrel__Cons1,axiom,
    ! [Y3: multis3193088007478089820ropVar,Ys2: list_M6740372356978808802ropVar,Xs2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre4495854419763884273ropVar @ R ) )
     => ~ ! [Y4: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
            ( ( Xs2
              = ( cons_M4277272395645347858ropVar @ Y4 @ Ys ) )
           => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Y3 @ Y4 ) @ R )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Ys2 @ Ys ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_322_listrel__Cons1,axiom,
    ! [Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,Xs2: list_nat,R: set_Pr302692805937157912_o_nat] :
      ( ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre6760217120960589698_o_nat @ R ) )
     => ~ ! [Y4: nat,Ys: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys ) )
           => ( ( member7944572178332388089_o_nat @ ( produc7886040895744259420_o_nat @ Y3 @ Y4 ) @ R )
             => ~ ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ Ys2 @ Ys ) @ ( listre6760217120960589698_o_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_323_listrel__Cons1,axiom,
    ! [Y3: nat,Ys2: list_nat,Xs2: list_P1665994083459302249pVar_o,R: set_Pr8900919299660647230pVar_o] :
      ( ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ ( cons_nat @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre3244086070610882400pVar_o @ R ) )
     => ~ ! [Y4: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
           => ( ( member6910252441317339679pVar_o @ ( produc4369909845394552122pVar_o @ Y3 @ Y4 ) @ R )
             => ~ ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ Ys2 @ Ys ) @ ( listre3244086070610882400pVar_o @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_324_listrel__Cons1,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,Xs2: list_nat,R: set_Pr6978669447677043102_o_nat] :
      ( ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre3060699943780466440_o_nat @ R ) )
     => ~ ! [Y4: nat,Ys: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys ) )
           => ( ( member8127644508941222527_o_nat @ ( produc4045138296077651938_o_nat @ Y3 @ Y4 ) @ R )
             => ~ ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ Ys2 @ Ys ) @ ( listre3060699943780466440_o_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_325_listrel__Cons1,axiom,
    ! [Y3: nat,Ys2: list_nat,Xs2: list_l1255154591254203759pVar_o,R: set_Pr8506680132865545540pVar_o] :
      ( ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ ( cons_nat @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre4590990273193845606pVar_o @ R ) )
     => ~ ! [Y4: list_P1665994083459302249pVar_o,Ys: list_l1255154591254203759pVar_o] :
            ( ( Xs2
              = ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) )
           => ( ( member4118595025335937317pVar_o @ ( produc5575428625491031104pVar_o @ Y3 @ Y4 ) @ R )
             => ~ ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ Ys2 @ Ys ) @ ( listre4590990273193845606pVar_o @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_326_listrel__Cons1,axiom,
    ! [Y3: propos3600597766819905199ropVar,Ys2: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar,R: set_Pr3339951299666510767ropVar] :
      ( ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ ( cons_P6757824865275295865ropVar @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre1163491797370331489ropVar @ R ) )
     => ~ ! [Y4: list_P723798329145367999ropVar,Ys: list_l1644708917474426447ropVar] :
            ( ( Xs2
              = ( cons_l2454487807130495625ropVar @ Y4 @ Ys ) )
           => ( ( member4962873584118313464ropVar @ ( produc5136867404843011847ropVar @ Y3 @ Y4 ) @ R )
             => ~ ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ Ys2 @ Ys ) @ ( listre1163491797370331489ropVar @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_327_listrel__Cons1,axiom,
    ! [Y3: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,Xs2: list_nat,R: set_Pr7538684846379304231ar_nat] :
      ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre5688682141216627473ar_nat @ R ) )
     => ~ ! [Y4: nat,Ys: list_nat] :
            ( ( Xs2
              = ( cons_nat @ Y4 @ Ys ) )
           => ( ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ Y3 @ Y4 ) @ R )
             => ~ ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Ys2 @ Ys ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_328_listrel__Cons1,axiom,
    ! [Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) @ Xs2 ) @ ( listre8839862472748488369pVar_o @ R ) )
     => ~ ! [Y4: produc8999643685545340131pVar_o,Ys: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
           => ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ Y3 @ Y4 ) @ R )
             => ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Ys2 @ Ys ) @ ( listre8839862472748488369pVar_o @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_329_listrel_OCons,axiom,
    ! [X4: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel_nat_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listrel_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_330_listrel_OCons,axiom,
    ! [X4: $o,Y3: $o,R: set_Product_prod_o_o,Xs2: list_o,Ys2: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R )
     => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel_o_o @ R ) )
       => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X4 @ Xs2 ) @ ( cons_o @ Y3 @ Ys2 ) ) @ ( listrel_o_o @ R ) ) ) ) ).

% listrel.Cons
thf(fact_331_listrel_OCons,axiom,
    ! [X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R )
     => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( listre4495854419763884273ropVar @ R ) )
       => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X4 @ Xs2 ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys2 ) ) @ ( listre4495854419763884273ropVar @ R ) ) ) ) ).

% listrel.Cons
thf(fact_332_listrel_OCons,axiom,
    ! [X4: produc8999643685545340131pVar_o,Y3: nat,R: set_Pr302692805937157912_o_nat,Xs2: list_P1665994083459302249pVar_o,Ys2: list_nat] :
      ( ( member7944572178332388089_o_nat @ ( produc7886040895744259420_o_nat @ X4 @ Y3 ) @ R )
     => ( ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ Xs2 @ Ys2 ) @ ( listre6760217120960589698_o_nat @ R ) )
       => ( member443142782555292559st_nat @ ( produc5426949100443613682st_nat @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listre6760217120960589698_o_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_333_listrel_OCons,axiom,
    ! [X4: nat,Y3: produc8999643685545340131pVar_o,R: set_Pr8900919299660647230pVar_o,Xs2: list_nat,Ys2: list_P1665994083459302249pVar_o] :
      ( ( member6910252441317339679pVar_o @ ( produc4369909845394552122pVar_o @ X4 @ Y3 ) @ R )
     => ( ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ Xs2 @ Ys2 ) @ ( listre3244086070610882400pVar_o @ R ) )
       => ( member3347616061724908309pVar_o @ ( produc697358683602405680pVar_o @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) @ ( listre3244086070610882400pVar_o @ R ) ) ) ) ).

% listrel.Cons
thf(fact_334_listrel_OCons,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Y3: nat,R: set_Pr6978669447677043102_o_nat,Xs2: list_l1255154591254203759pVar_o,Ys2: list_nat] :
      ( ( member8127644508941222527_o_nat @ ( produc4045138296077651938_o_nat @ X4 @ Y3 ) @ R )
     => ( ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ Xs2 @ Ys2 ) @ ( listre3060699943780466440_o_nat @ R ) )
       => ( member50098248654648213st_nat @ ( produc4052226372830137592st_nat @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listre3060699943780466440_o_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_335_listrel_OCons,axiom,
    ! [X4: nat,Y3: list_P1665994083459302249pVar_o,R: set_Pr8506680132865545540pVar_o,Xs2: list_nat,Ys2: list_l1255154591254203759pVar_o] :
      ( ( member4118595025335937317pVar_o @ ( produc5575428625491031104pVar_o @ X4 @ Y3 ) @ R )
     => ( ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ Xs2 @ Ys2 ) @ ( listre4590990273193845606pVar_o @ R ) )
       => ( member988876147076222363pVar_o @ ( produc7103779822796047286pVar_o @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) @ ( listre4590990273193845606pVar_o @ R ) ) ) ) ).

% listrel.Cons
thf(fact_336_listrel_OCons,axiom,
    ! [X4: propos3600597766819905199ropVar,Y3: list_P723798329145367999ropVar,R: set_Pr3339951299666510767ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_l1644708917474426447ropVar] :
      ( ( member4962873584118313464ropVar @ ( produc5136867404843011847ropVar @ X4 @ Y3 ) @ R )
     => ( ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ Xs2 @ Ys2 ) @ ( listre1163491797370331489ropVar @ R ) )
       => ( member7824094204520356248ropVar @ ( produc7466835187682555559ropVar @ ( cons_P6757824865275295865ropVar @ X4 @ Xs2 ) @ ( cons_l2454487807130495625ropVar @ Y3 @ Ys2 ) ) @ ( listre1163491797370331489ropVar @ R ) ) ) ) ).

% listrel.Cons
thf(fact_337_listrel_OCons,axiom,
    ! [X4: produc2642885279816024434ropVar,Y3: nat,R: set_Pr7538684846379304231ar_nat,Xs2: list_P8791008543742759928ropVar,Ys2: list_nat] :
      ( ( member5345150084029083272ar_nat @ ( produc5526329313106567659ar_nat @ X4 @ Y3 ) @ R )
     => ( ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ Xs2 @ Ys2 ) @ ( listre5688682141216627473ar_nat @ R ) )
       => ( member964980348444692382st_nat @ ( produc5924578096210284289st_nat @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listre5688682141216627473ar_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_338_listrel_OCons,axiom,
    ! [X4: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,R: set_Pr4574656050966626247pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X4 @ Y3 ) @ R )
     => ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Ys2 ) @ ( listre8839862472748488369pVar_o @ R ) )
       => ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) @ ( listre8839862472748488369pVar_o @ R ) ) ) ) ).

% listrel.Cons
thf(fact_339_cnf__encoder_OencodeGammaCond_Oelims,axiom,
    ! [X4: multis3193088007478089820ropVar,Xa: multis3193088007478089820ropVar,Xb: $o,Xc: $o,Y3: list_l1255154591254203759pVar_o] :
      ( ( ( multis334970729563149064maCond @ X4 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( Xb
         => ( Xc
           => ( Y3 != nil_li1310946081133177199pVar_o ) ) )
       => ( ( ~ Xb
           => ( ~ Xc
             => ( Y3
               != ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) ) ) )
         => ( ( ~ Xb
             => ( Xc
               => ( Y3
                 != ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Xa @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ) )
           => ~ ( Xb
               => ( ~ Xc
                 => ( Y3
                   != ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Xa @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ) ) ) ) ) ) ).

% cnf_encoder.encodeGammaCond.elims
thf(fact_340_cnf__encoder_OencodeGammaCond_Osimps_I2_J,axiom,
    ! [Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis334970729563149064maCond @ Gam2 @ Eps2 @ $false @ $false )
      = ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Gam2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) ) ).

% cnf_encoder.encodeGammaCond.simps(2)
thf(fact_341_cnf__encoder_OencodeGammaCond_Osimps_I3_J,axiom,
    ! [Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis334970729563149064maCond @ Gam2 @ Eps2 @ $false @ $true )
      = ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Gam2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Eps2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ).

% cnf_encoder.encodeGammaCond.simps(3)
thf(fact_342_cnf__encoder_OencodeGammaCond_Osimps_I4_J,axiom,
    ! [Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis334970729563149064maCond @ Gam2 @ Eps2 @ $true @ $false )
      = ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Gam2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Eps2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ).

% cnf_encoder.encodeGammaCond.simps(4)
thf(fact_343_cnf__encoder_Oenc__sound_I11_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
      = ( ~ ( ( Alpha @ X4 )
            & ( Alpha @ Y3 ) ) ) ) ).

% cnf_encoder.enc_sound(11)
thf(fact_344_cnf__encoder_Oenc__sound_I10_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
      = ( ( Alpha @ X4 )
       => ( Alpha @ Y3 ) ) ) ).

% cnf_encoder.enc_sound(10)
thf(fact_345_cnf__encoder_Oenc__sound_I8_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Z: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) )
      = ( ( ( Alpha @ X4 )
         => ( Alpha @ Y3 ) )
        & ( ~ ( Alpha @ X4 )
         => ( Alpha @ Z ) ) ) ) ).

% cnf_encoder.enc_sound(8)
thf(fact_346_cnf__encoder_Oenc__sound_I7_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,U: multis3193088007478089820ropVar,Z: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) ) ) )
      = ( ( Alpha @ X4 )
        = ( ( ( Alpha @ Y3 )
           => ( Alpha @ Z ) )
          & ( ~ ( Alpha @ Y3 )
           => ( Alpha @ U ) ) ) ) ) ).

% cnf_encoder.enc_sound(7)
thf(fact_347_cnf__encoder_Oenc__sound_I6_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,Z: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) ) )
      = ( ( Alpha @ X4 )
        = ( ( Alpha @ Y3 )
          & ~ ( Alpha @ Z ) ) ) ) ).

% cnf_encoder.enc_sound(6)
thf(fact_348_cnf__encoder_Oenc__sound_I5_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y3 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) )
      = ( ( Alpha @ X4 )
        = ( ~ ( Alpha @ Y3 ) ) ) ) ).

% cnf_encoder.enc_sound(5)
thf(fact_349_cnf__encoder_Oenc__sound_I2_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o] : ( propos2885041177628046573ropVar @ Alpha @ nil_li1310946081133177199pVar_o ) ).

% cnf_encoder.enc_sound(2)
thf(fact_350_cnf__encoder_OencodeGammaCond,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar,S: $o,Ns: $o] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( multis334970729563149064maCond @ Gam2 @ Eps2 @ S @ Ns ) )
      = ( ( Alpha @ Gam2 )
       => ( ( ( Alpha @ Eps2 )
           => Ns )
          & ( ~ ( Alpha @ Eps2 )
           => S ) ) ) ) ).

% cnf_encoder.encodeGammaCond
thf(fact_351_cnf__encoder_Oenc__sound_I1_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o] :
      ~ ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o ) ) ).

% cnf_encoder.enc_sound(1)
thf(fact_352_cnf__encoder_Oenc__sound_I3_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) )
      = ( Alpha @ X4 ) ) ).

% cnf_encoder.enc_sound(3)
thf(fact_353_cnf__encoder_Oenc__sound_I4_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) )
      = ( ~ ( Alpha @ X4 ) ) ) ).

% cnf_encoder.enc_sound(4)
thf(fact_354_cnf__encoder_OencodeGammaCond_Osimps_I1_J,axiom,
    ! [Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis334970729563149064maCond @ Gam2 @ Eps2 @ $true @ $true )
      = nil_li1310946081133177199pVar_o ) ).

% cnf_encoder.encodeGammaCond.simps(1)
thf(fact_355_cnf__encoder_OencodeGammaCond_Opelims,axiom,
    ! [X4: multis3193088007478089820ropVar,Xa: multis3193088007478089820ropVar,Xb: $o,Xc: $o,Y3: list_l1255154591254203759pVar_o] :
      ( ( ( multis334970729563149064maCond @ X4 @ Xa @ Xb @ Xc )
        = Y3 )
     => ( ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ Xb @ Xc ) ) ) )
       => ( ( Xb
           => ( Xc
             => ( ( Y3 = nil_li1310946081133177199pVar_o )
               => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $true @ $true ) ) ) ) ) ) )
         => ( ( ~ Xb
             => ( ~ Xc
               => ( ( Y3
                    = ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o ) )
                 => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $false @ $false ) ) ) ) ) ) )
           => ( ( ~ Xb
               => ( Xc
                 => ( ( Y3
                      = ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Xa @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
                   => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $false @ $true ) ) ) ) ) ) )
             => ~ ( Xb
                 => ( ~ Xc
                   => ( ( Y3
                        = ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Xa @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
                     => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $true @ $false ) ) ) ) ) ) ) ) ) ) ) ) ).

% cnf_encoder.encodeGammaCond.pelims
thf(fact_356_Cons__listrel1__Cons,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) ) @ ( listre6655839188755619153ropVar @ R ) )
      = ( ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ R )
          & ( Xs2 = Ys2 ) )
        | ( ( X4 = Y3 )
          & ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_357_Cons__listrel1__Cons,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) @ ( listre2444918065581779778pVar_o @ R ) )
      = ( ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X4 @ Y3 ) @ R )
          & ( Xs2 = Ys2 ) )
        | ( ( X4 = Y3 )
          & ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Ys2 ) @ ( listre2444918065581779778pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_358_Cons__listrel1__Cons,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) @ ( listre4950778487325574728pVar_o @ R ) )
      = ( ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ Y3 ) @ R )
          & ( Xs2 = Ys2 ) )
        | ( ( X4 = Y3 )
          & ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ Ys2 ) @ ( listre4950778487325574728pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_359_Cons__listrel1__Cons,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) ) @ ( listre8274924349387117774pVar_o @ R ) )
      = ( ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ Y3 ) @ R )
          & ( Xs2 = Ys2 ) )
        | ( ( X4 = Y3 )
          & ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_360_Cons__listrel1__Cons,axiom,
    ! [X4: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
          & ( Xs2 = Ys2 ) )
        | ( ( X4 = Y3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_361_Cons__listrel1__Cons,axiom,
    ! [X4: $o,Xs2: list_o,Y3: $o,Ys2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X4 @ Xs2 ) @ ( cons_o @ Y3 @ Ys2 ) ) @ ( listrel1_o @ R ) )
      = ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R )
          & ( Xs2 = Ys2 ) )
        | ( ( X4 = Y3 )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_362_Cons__listrel1__Cons,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X4 @ Xs2 ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys2 ) ) @ ( listre3736066117146447419ropVar @ R ) )
      = ( ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R )
          & ( Xs2 = Ys2 ) )
        | ( ( X4 = Y3 )
          & ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_363_lexord__append__left__rightI,axiom,
    ! [A: propos3600597766819905199ropVar,B: propos3600597766819905199ropVar,R: set_Pr572148484142614943ropVar,U: list_P723798329145367999ropVar,X4: list_P723798329145367999ropVar,Y3: list_P723798329145367999ropVar] :
      ( ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ A @ B ) @ R )
     => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ U @ ( cons_P6757824865275295865ropVar @ A @ X4 ) ) @ ( append7708039934470209236ropVar @ U @ ( cons_P6757824865275295865ropVar @ B @ Y3 ) ) ) @ ( lexord5395303177630463356ropVar @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_364_lexord__append__left__rightI,axiom,
    ! [A: produc2642885279816024434ropVar,B: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar,U: list_P8791008543742759928ropVar,X4: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ A @ B ) @ R )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( append4352292717583037453ropVar @ U @ ( cons_P1504208554336013608ropVar @ A @ X4 ) ) @ ( append4352292717583037453ropVar @ U @ ( cons_P1504208554336013608ropVar @ B @ Y3 ) ) ) @ ( lexord832287969131410277ropVar @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_365_lexord__append__left__rightI,axiom,
    ! [A: produc8999643685545340131pVar_o,B: produc8999643685545340131pVar_o,R: set_Pr4574656050966626247pVar_o,U: list_P1665994083459302249pVar_o,X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ A @ B ) @ R )
     => ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( append1087384436348988414pVar_o @ U @ ( cons_P6860203055867761049pVar_o @ A @ X4 ) ) @ ( append1087384436348988414pVar_o @ U @ ( cons_P6860203055867761049pVar_o @ B @ Y3 ) ) ) @ ( lexord9212727487545282390pVar_o @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_366_lexord__append__left__rightI,axiom,
    ! [A: list_P1665994083459302249pVar_o,B: list_P1665994083459302249pVar_o,R: set_Pr8855275280618702407pVar_o,U: list_l1255154591254203759pVar_o,X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ A @ B ) @ R )
     => ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( append8069948826146371844pVar_o @ U @ ( cons_l3452452106213086495pVar_o @ A @ X4 ) ) @ ( append8069948826146371844pVar_o @ U @ ( cons_l3452452106213086495pVar_o @ B @ Y3 ) ) ) @ ( lexord8675461415035850844pVar_o @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_367_lexord__append__left__rightI,axiom,
    ! [A: list_l1255154591254203759pVar_o,B: list_l1255154591254203759pVar_o,R: set_Pr3139130378708280007pVar_o,U: list_l985434930288970485pVar_o,X4: list_l985434930288970485pVar_o,Y3: list_l985434930288970485pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ A @ B ) @ R )
     => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ U @ ( cons_l7316790070876032037pVar_o @ A @ X4 ) ) @ ( append8959015459476584842pVar_o @ U @ ( cons_l7316790070876032037pVar_o @ B @ Y3 ) ) ) @ ( lexord4213358897963926754pVar_o @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_368_lexord__append__left__rightI,axiom,
    ! [A: nat,B: nat,R: set_Pr1261947904930325089at_nat,U: list_nat,X4: list_nat,Y3: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ U @ ( cons_nat @ A @ X4 ) ) @ ( append_nat @ U @ ( cons_nat @ B @ Y3 ) ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_369_lexord__append__left__rightI,axiom,
    ! [A: $o,B: $o,R: set_Product_prod_o_o,U: list_o,X4: list_o,Y3: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ R )
     => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ U @ ( cons_o @ A @ X4 ) ) @ ( append_o @ U @ ( cons_o @ B @ Y3 ) ) ) @ ( lexord_o @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_370_lexord__append__left__rightI,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar,U: list_M6740372356978808802ropVar,X4: list_M6740372356978808802ropVar,Y3: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A @ B ) @ R )
     => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ U @ ( cons_M4277272395645347858ropVar @ A @ X4 ) ) @ ( append2684835082924936439ropVar @ U @ ( cons_M4277272395645347858ropVar @ B @ Y3 ) ) ) @ ( lexord1374882220397993039ropVar @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_371_Nil__lenlex__iff1,axiom,
    ! [Ns: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ns ) @ ( lenlex46774552128460413ropVar @ R ) )
      = ( Ns != nil_Pr6119456312066593144ropVar ) ) ).

% Nil_lenlex_iff1
thf(fact_372_Nil__lenlex__iff1,axiom,
    ! [Ns: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Ns ) @ ( lenlex2260575179428607598pVar_o @ R ) )
      = ( Ns != nil_Pr6413368191065476585pVar_o ) ) ).

% Nil_lenlex_iff1
thf(fact_373_Nil__lenlex__iff1,axiom,
    ! [Ns: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Ns ) @ ( lenlex4014889434604538740pVar_o @ R ) )
      = ( Ns != nil_li1310946081133177199pVar_o ) ) ).

% Nil_lenlex_iff1
thf(fact_374_Nil__lenlex__iff1,axiom,
    ! [Ns: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Ns ) @ ( lenlex3313702719058564090pVar_o @ R ) )
      = ( Ns != nil_li590035233778055285pVar_o ) ) ).

% Nil_lenlex_iff1
thf(fact_375_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_376_partition_Osimps_I1_J,axiom,
    ! [P: produc2642885279816024434ropVar > $o] :
      ( ( partit1257260948940242929ropVar @ P @ nil_Pr6119456312066593144ropVar )
      = ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar ) ) ).

% partition.simps(1)
thf(fact_377_partition_Osimps_I1_J,axiom,
    ! [P: produc8999643685545340131pVar_o > $o] :
      ( ( partit856451379440778722pVar_o @ P @ nil_Pr6413368191065476585pVar_o )
      = ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_Pr6413368191065476585pVar_o ) ) ).

% partition.simps(1)
thf(fact_378_partition_Osimps_I1_J,axiom,
    ! [P: list_P1665994083459302249pVar_o > $o] :
      ( ( partit188521249368705256pVar_o @ P @ nil_li1310946081133177199pVar_o )
      = ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ nil_li1310946081133177199pVar_o ) ) ).

% partition.simps(1)
thf(fact_379_partition_Osimps_I1_J,axiom,
    ! [P: list_l1255154591254203759pVar_o > $o] :
      ( ( partit2223932869413013358pVar_o @ P @ nil_li590035233778055285pVar_o )
      = ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ nil_li590035233778055285pVar_o ) ) ).

% partition.simps(1)
thf(fact_380_partition_Osimps_I1_J,axiom,
    ! [P: nat > $o] :
      ( ( partition_nat @ P @ nil_nat )
      = ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).

% partition.simps(1)
thf(fact_381_bind__simps_I1_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_382_bind__simps_I1_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > list_nat] :
      ( ( bind_P7202517103677919084_o_nat @ nil_Pr6413368191065476585pVar_o @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_383_bind__simps_I1_J,axiom,
    ! [F2: nat > list_P1665994083459302249pVar_o] :
      ( ( bind_n3686386053328211786pVar_o @ nil_nat @ F2 )
      = nil_Pr6413368191065476585pVar_o ) ).

% bind_simps(1)
thf(fact_384_bind__simps_I1_J,axiom,
    ! [F2: list_P1665994083459302249pVar_o > list_nat] :
      ( ( bind_l4743436673915347442_o_nat @ nil_li1310946081133177199pVar_o @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_385_bind__simps_I1_J,axiom,
    ! [F2: nat > list_l1255154591254203759pVar_o] :
      ( ( bind_n6273727003328726608pVar_o @ nil_nat @ F2 )
      = nil_li1310946081133177199pVar_o ) ).

% bind_simps(1)
thf(fact_386_bind__simps_I1_J,axiom,
    ! [F2: produc2642885279816024434ropVar > list_nat] :
      ( ( bind_P7182508892803130363ar_nat @ nil_Pr6119456312066593144ropVar @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_387_bind__simps_I1_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > list_P1665994083459302249pVar_o] :
      ( ( bind_P2079531178914768455pVar_o @ nil_Pr6413368191065476585pVar_o @ F2 )
      = nil_Pr6413368191065476585pVar_o ) ).

% bind_simps(1)
thf(fact_388_bind__simps_I1_J,axiom,
    ! [F2: list_l1255154591254203759pVar_o > list_nat] :
      ( ( bind_l3081483778782373880_o_nat @ nil_li590035233778055285pVar_o @ F2 )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_389_bind__simps_I1_J,axiom,
    ! [F2: nat > list_P8791008543742759928ropVar] :
      ( ( bind_n1517830492101072217ropVar @ nil_nat @ F2 )
      = nil_Pr6119456312066593144ropVar ) ).

% bind_simps(1)
thf(fact_390_bind__simps_I1_J,axiom,
    ! [F2: nat > list_l985434930288970485pVar_o] :
      ( ( bind_n4999772531149697238pVar_o @ nil_nat @ F2 )
      = nil_li590035233778055285pVar_o ) ).

% bind_simps(1)
thf(fact_391_listrelp_Ocases,axiom,
    ! [R: nat > nat > $o,A1: list_nat,A22: list_nat] :
      ( ( listrelp_nat_nat @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: nat,Y4: nat,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listrelp_nat_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_392_listrelp_Ocases,axiom,
    ! [R: produc8999643685545340131pVar_o > nat > $o,A1: list_P1665994083459302249pVar_o,A22: list_nat] :
      ( ( listre2270229272446257844_o_nat @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_Pr6413368191065476585pVar_o )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: produc8999643685545340131pVar_o,Y4: nat,Xs: list_P1665994083459302249pVar_o] :
              ( ( A1
                = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre2270229272446257844_o_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_393_listrelp_Ocases,axiom,
    ! [R: nat > produc8999643685545340131pVar_o > $o,A1: list_nat,A22: list_P1665994083459302249pVar_o] :
      ( ( listre7977470258951326354pVar_o @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_Pr6413368191065476585pVar_o ) )
       => ~ ! [X2: nat,Y4: produc8999643685545340131pVar_o,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_P1665994083459302249pVar_o] :
                  ( ( A22
                    = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre7977470258951326354pVar_o @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_394_listrelp_Ocases,axiom,
    ! [R: list_P1665994083459302249pVar_o > nat > $o,A1: list_l1255154591254203759pVar_o,A22: list_nat] :
      ( ( listre7159049289101154618_o_nat @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_li1310946081133177199pVar_o )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: list_P1665994083459302249pVar_o,Y4: nat,Xs: list_l1255154591254203759pVar_o] :
              ( ( A1
                = ( cons_l3452452106213086495pVar_o @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre7159049289101154618_o_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_395_listrelp_Ocases,axiom,
    ! [R: nat > list_P1665994083459302249pVar_o > $o,A1: list_nat,A22: list_l1255154591254203759pVar_o] :
      ( ( listre8689339618514533784pVar_o @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_li1310946081133177199pVar_o ) )
       => ~ ! [X2: nat,Y4: list_P1665994083459302249pVar_o,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_l1255154591254203759pVar_o] :
                  ( ( A22
                    = ( cons_l3452452106213086495pVar_o @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre8689339618514533784pVar_o @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_396_listrelp_Ocases,axiom,
    ! [R: produc2642885279816024434ropVar > nat > $o,A1: list_P8791008543742759928ropVar,A22: list_nat] :
      ( ( listre7860766376562299715ar_nat @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_Pr6119456312066593144ropVar )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: produc2642885279816024434ropVar,Y4: nat,Xs: list_P8791008543742759928ropVar] :
              ( ( A1
                = ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre7860766376562299715ar_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_397_listrelp_Ocases,axiom,
    ! [R: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,A1: list_P1665994083459302249pVar_o,A22: list_P1665994083459302249pVar_o] :
      ( ( listre612986413372629247pVar_o @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_Pr6413368191065476585pVar_o )
         => ( A22 != nil_Pr6413368191065476585pVar_o ) )
       => ~ ! [X2: produc8999643685545340131pVar_o,Y4: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
              ( ( A1
                = ( cons_P6860203055867761049pVar_o @ X2 @ Xs ) )
             => ! [Ys: list_P1665994083459302249pVar_o] :
                  ( ( A22
                    = ( cons_P6860203055867761049pVar_o @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre612986413372629247pVar_o @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_398_listrelp_Ocases,axiom,
    ! [R: list_l1255154591254203759pVar_o > nat > $o,A1: list_l985434930288970485pVar_o,A22: list_nat] :
      ( ( listre8257602174308424512_o_nat @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_li590035233778055285pVar_o )
         => ( A22 != nil_nat ) )
       => ~ ! [X2: list_l1255154591254203759pVar_o,Y4: nat,Xs: list_l985434930288970485pVar_o] :
              ( ( A1
                = ( cons_l7316790070876032037pVar_o @ X2 @ Xs ) )
             => ! [Ys: list_nat] :
                  ( ( A22
                    = ( cons_nat @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre8257602174308424512_o_nat @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_399_listrelp_Ocases,axiom,
    ! [R: nat > produc2642885279816024434ropVar > $o,A1: list_nat,A22: list_P8791008543742759928ropVar] :
      ( ( listre2196087975860241569ropVar @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X2: nat,Y4: produc2642885279816024434ropVar,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_P8791008543742759928ropVar] :
                  ( ( A22
                    = ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre2196087975860241569ropVar @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_400_listrelp_Ocases,axiom,
    ! [R: nat > list_l1255154591254203759pVar_o > $o,A1: list_nat,A22: list_l985434930288970485pVar_o] :
      ( ( listre952518889820972062pVar_o @ R @ A1 @ A22 )
     => ( ( ( A1 = nil_nat )
         => ( A22 != nil_li590035233778055285pVar_o ) )
       => ~ ! [X2: nat,Y4: list_l1255154591254203759pVar_o,Xs: list_nat] :
              ( ( A1
                = ( cons_nat @ X2 @ Xs ) )
             => ! [Ys: list_l985434930288970485pVar_o] :
                  ( ( A22
                    = ( cons_l7316790070876032037pVar_o @ Y4 @ Ys ) )
                 => ( ( R @ X2 @ Y4 )
                   => ~ ( listre952518889820972062pVar_o @ R @ Xs @ Ys ) ) ) ) ) ) ).

% listrelp.cases
thf(fact_401_listrelp_Osimps,axiom,
    ( listrelp_nat_nat
    = ( ^ [R2: nat > nat > $o,A12: list_nat,A23: list_nat] :
          ( ( ( A12 = nil_nat )
            & ( A23 = nil_nat ) )
          | ? [X3: nat,Y2: nat,Xs3: list_nat,Ys3: list_nat] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listrelp_nat_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_402_listrelp_Osimps,axiom,
    ( listre2270229272446257844_o_nat
    = ( ^ [R2: produc8999643685545340131pVar_o > nat > $o,A12: list_P1665994083459302249pVar_o,A23: list_nat] :
          ( ( ( A12 = nil_Pr6413368191065476585pVar_o )
            & ( A23 = nil_nat ) )
          | ? [X3: produc8999643685545340131pVar_o,Y2: nat,Xs3: list_P1665994083459302249pVar_o,Ys3: list_nat] :
              ( ( A12
                = ( cons_P6860203055867761049pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre2270229272446257844_o_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_403_listrelp_Osimps,axiom,
    ( listre7977470258951326354pVar_o
    = ( ^ [R2: nat > produc8999643685545340131pVar_o > $o,A12: list_nat,A23: list_P1665994083459302249pVar_o] :
          ( ( ( A12 = nil_nat )
            & ( A23 = nil_Pr6413368191065476585pVar_o ) )
          | ? [X3: nat,Y2: produc8999643685545340131pVar_o,Xs3: list_nat,Ys3: list_P1665994083459302249pVar_o] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P6860203055867761049pVar_o @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre7977470258951326354pVar_o @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_404_listrelp_Osimps,axiom,
    ( listre7159049289101154618_o_nat
    = ( ^ [R2: list_P1665994083459302249pVar_o > nat > $o,A12: list_l1255154591254203759pVar_o,A23: list_nat] :
          ( ( ( A12 = nil_li1310946081133177199pVar_o )
            & ( A23 = nil_nat ) )
          | ? [X3: list_P1665994083459302249pVar_o,Y2: nat,Xs3: list_l1255154591254203759pVar_o,Ys3: list_nat] :
              ( ( A12
                = ( cons_l3452452106213086495pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre7159049289101154618_o_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_405_listrelp_Osimps,axiom,
    ( listre8689339618514533784pVar_o
    = ( ^ [R2: nat > list_P1665994083459302249pVar_o > $o,A12: list_nat,A23: list_l1255154591254203759pVar_o] :
          ( ( ( A12 = nil_nat )
            & ( A23 = nil_li1310946081133177199pVar_o ) )
          | ? [X3: nat,Y2: list_P1665994083459302249pVar_o,Xs3: list_nat,Ys3: list_l1255154591254203759pVar_o] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_l3452452106213086495pVar_o @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre8689339618514533784pVar_o @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_406_listrelp_Osimps,axiom,
    ( listre7860766376562299715ar_nat
    = ( ^ [R2: produc2642885279816024434ropVar > nat > $o,A12: list_P8791008543742759928ropVar,A23: list_nat] :
          ( ( ( A12 = nil_Pr6119456312066593144ropVar )
            & ( A23 = nil_nat ) )
          | ? [X3: produc2642885279816024434ropVar,Y2: nat,Xs3: list_P8791008543742759928ropVar,Ys3: list_nat] :
              ( ( A12
                = ( cons_P1504208554336013608ropVar @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre7860766376562299715ar_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_407_listrelp_Osimps,axiom,
    ( listre612986413372629247pVar_o
    = ( ^ [R2: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,A12: list_P1665994083459302249pVar_o,A23: list_P1665994083459302249pVar_o] :
          ( ( ( A12 = nil_Pr6413368191065476585pVar_o )
            & ( A23 = nil_Pr6413368191065476585pVar_o ) )
          | ? [X3: produc8999643685545340131pVar_o,Y2: produc8999643685545340131pVar_o,Xs3: list_P1665994083459302249pVar_o,Ys3: list_P1665994083459302249pVar_o] :
              ( ( A12
                = ( cons_P6860203055867761049pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P6860203055867761049pVar_o @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre612986413372629247pVar_o @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_408_listrelp_Osimps,axiom,
    ( listre8257602174308424512_o_nat
    = ( ^ [R2: list_l1255154591254203759pVar_o > nat > $o,A12: list_l985434930288970485pVar_o,A23: list_nat] :
          ( ( ( A12 = nil_li590035233778055285pVar_o )
            & ( A23 = nil_nat ) )
          | ? [X3: list_l1255154591254203759pVar_o,Y2: nat,Xs3: list_l985434930288970485pVar_o,Ys3: list_nat] :
              ( ( A12
                = ( cons_l7316790070876032037pVar_o @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_nat @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre8257602174308424512_o_nat @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_409_listrelp_Osimps,axiom,
    ( listre2196087975860241569ropVar
    = ( ^ [R2: nat > produc2642885279816024434ropVar > $o,A12: list_nat,A23: list_P8791008543742759928ropVar] :
          ( ( ( A12 = nil_nat )
            & ( A23 = nil_Pr6119456312066593144ropVar ) )
          | ? [X3: nat,Y2: produc2642885279816024434ropVar,Xs3: list_nat,Ys3: list_P8791008543742759928ropVar] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_P1504208554336013608ropVar @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre2196087975860241569ropVar @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_410_listrelp_Osimps,axiom,
    ( listre952518889820972062pVar_o
    = ( ^ [R2: nat > list_l1255154591254203759pVar_o > $o,A12: list_nat,A23: list_l985434930288970485pVar_o] :
          ( ( ( A12 = nil_nat )
            & ( A23 = nil_li590035233778055285pVar_o ) )
          | ? [X3: nat,Y2: list_l1255154591254203759pVar_o,Xs3: list_nat,Ys3: list_l985434930288970485pVar_o] :
              ( ( A12
                = ( cons_nat @ X3 @ Xs3 ) )
              & ( A23
                = ( cons_l7316790070876032037pVar_o @ Y2 @ Ys3 ) )
              & ( R2 @ X3 @ Y2 )
              & ( listre952518889820972062pVar_o @ R2 @ Xs3 @ Ys3 ) ) ) ) ) ).

% listrelp.simps
thf(fact_411_same__append__eq,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ Ys2 )
        = ( append8959015459476584842pVar_o @ Xs2 @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_412_same__append__eq,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys2 )
        = ( append_nat @ Xs2 @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_413_same__append__eq,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ Ys2 )
        = ( append7708039934470209236ropVar @ Xs2 @ Zs ) )
      = ( Ys2 = Zs ) ) ).

% same_append_eq
thf(fact_414_append__same__eq,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Ys2 @ Xs2 )
        = ( append8959015459476584842pVar_o @ Zs @ Xs2 ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_415_append__same__eq,axiom,
    ! [Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys2 @ Xs2 )
        = ( append_nat @ Zs @ Xs2 ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_416_append__same__eq,axiom,
    ! [Ys2: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Ys2 @ Xs2 )
        = ( append7708039934470209236ropVar @ Zs @ Xs2 ) )
      = ( Ys2 = Zs ) ) ).

% append_same_eq
thf(fact_417_append__assoc,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ( append8959015459476584842pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) @ Zs )
      = ( append8959015459476584842pVar_o @ Xs2 @ ( append8959015459476584842pVar_o @ Ys2 @ Zs ) ) ) ).

% append_assoc
thf(fact_418_append__assoc,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs2 @ Ys2 ) @ Zs )
      = ( append_nat @ Xs2 @ ( append_nat @ Ys2 @ Zs ) ) ) ).

% append_assoc
thf(fact_419_append__assoc,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ( append7708039934470209236ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) @ Zs )
      = ( append7708039934470209236ropVar @ Xs2 @ ( append7708039934470209236ropVar @ Ys2 @ Zs ) ) ) ).

% append_assoc
thf(fact_420_append_Oassoc,axiom,
    ! [A: list_l985434930288970485pVar_o,B: list_l985434930288970485pVar_o,C2: list_l985434930288970485pVar_o] :
      ( ( append8959015459476584842pVar_o @ ( append8959015459476584842pVar_o @ A @ B ) @ C2 )
      = ( append8959015459476584842pVar_o @ A @ ( append8959015459476584842pVar_o @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_421_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
      = ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_422_append_Oassoc,axiom,
    ! [A: list_P723798329145367999ropVar,B: list_P723798329145367999ropVar,C2: list_P723798329145367999ropVar] :
      ( ( append7708039934470209236ropVar @ ( append7708039934470209236ropVar @ A @ B ) @ C2 )
      = ( append7708039934470209236ropVar @ A @ ( append7708039934470209236ropVar @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_423_append_Oright__neutral,axiom,
    ! [A: list_P723798329145367999ropVar] :
      ( ( append7708039934470209236ropVar @ A @ nil_Pr7606595143512524841ropVar )
      = A ) ).

% append.right_neutral
thf(fact_424_append_Oright__neutral,axiom,
    ! [A: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ A @ nil_Pr6119456312066593144ropVar )
      = A ) ).

% append.right_neutral
thf(fact_425_append_Oright__neutral,axiom,
    ! [A: list_P1665994083459302249pVar_o] :
      ( ( append1087384436348988414pVar_o @ A @ nil_Pr6413368191065476585pVar_o )
      = A ) ).

% append.right_neutral
thf(fact_426_append_Oright__neutral,axiom,
    ! [A: list_l1255154591254203759pVar_o] :
      ( ( append8069948826146371844pVar_o @ A @ nil_li1310946081133177199pVar_o )
      = A ) ).

% append.right_neutral
thf(fact_427_append_Oright__neutral,axiom,
    ! [A: list_l985434930288970485pVar_o] :
      ( ( append8959015459476584842pVar_o @ A @ nil_li590035233778055285pVar_o )
      = A ) ).

% append.right_neutral
thf(fact_428_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_429_append__Nil2,axiom,
    ! [Xs2: list_P723798329145367999ropVar] :
      ( ( append7708039934470209236ropVar @ Xs2 @ nil_Pr7606595143512524841ropVar )
      = Xs2 ) ).

% append_Nil2
thf(fact_430_append__Nil2,axiom,
    ! [Xs2: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ Xs2 @ nil_Pr6119456312066593144ropVar )
      = Xs2 ) ).

% append_Nil2
thf(fact_431_append__Nil2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o] :
      ( ( append1087384436348988414pVar_o @ Xs2 @ nil_Pr6413368191065476585pVar_o )
      = Xs2 ) ).

% append_Nil2
thf(fact_432_append__Nil2,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o] :
      ( ( append8069948826146371844pVar_o @ Xs2 @ nil_li1310946081133177199pVar_o )
      = Xs2 ) ).

% append_Nil2
thf(fact_433_append__Nil2,axiom,
    ! [Xs2: list_l985434930288970485pVar_o] :
      ( ( append8959015459476584842pVar_o @ Xs2 @ nil_li590035233778055285pVar_o )
      = Xs2 ) ).

% append_Nil2
thf(fact_434_append__Nil2,axiom,
    ! [Xs2: list_nat] :
      ( ( append_nat @ Xs2 @ nil_nat )
      = Xs2 ) ).

% append_Nil2
thf(fact_435_append__self__conv,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ Ys2 )
        = Xs2 )
      = ( Ys2 = nil_Pr7606595143512524841ropVar ) ) ).

% append_self_conv
thf(fact_436_append__self__conv,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs2 @ Ys2 )
        = Xs2 )
      = ( Ys2 = nil_Pr6119456312066593144ropVar ) ) ).

% append_self_conv
thf(fact_437_append__self__conv,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( ( append1087384436348988414pVar_o @ Xs2 @ Ys2 )
        = Xs2 )
      = ( Ys2 = nil_Pr6413368191065476585pVar_o ) ) ).

% append_self_conv
thf(fact_438_append__self__conv,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( ( append8069948826146371844pVar_o @ Xs2 @ Ys2 )
        = Xs2 )
      = ( Ys2 = nil_li1310946081133177199pVar_o ) ) ).

% append_self_conv
thf(fact_439_append__self__conv,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ Ys2 )
        = Xs2 )
      = ( Ys2 = nil_li590035233778055285pVar_o ) ) ).

% append_self_conv
thf(fact_440_append__self__conv,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys2 )
        = Xs2 )
      = ( Ys2 = nil_nat ) ) ).

% append_self_conv
thf(fact_441_self__append__conv,axiom,
    ! [Y3: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( Y3
        = ( append7708039934470209236ropVar @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_Pr7606595143512524841ropVar ) ) ).

% self_append_conv
thf(fact_442_self__append__conv,axiom,
    ! [Y3: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( Y3
        = ( append4352292717583037453ropVar @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_Pr6119456312066593144ropVar ) ) ).

% self_append_conv
thf(fact_443_self__append__conv,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( Y3
        = ( append1087384436348988414pVar_o @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_Pr6413368191065476585pVar_o ) ) ).

% self_append_conv
thf(fact_444_self__append__conv,axiom,
    ! [Y3: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( Y3
        = ( append8069948826146371844pVar_o @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_li1310946081133177199pVar_o ) ) ).

% self_append_conv
thf(fact_445_self__append__conv,axiom,
    ! [Y3: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( Y3
        = ( append8959015459476584842pVar_o @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_li590035233778055285pVar_o ) ) ).

% self_append_conv
thf(fact_446_self__append__conv,axiom,
    ! [Y3: list_nat,Ys2: list_nat] :
      ( ( Y3
        = ( append_nat @ Y3 @ Ys2 ) )
      = ( Ys2 = nil_nat ) ) ).

% self_append_conv
thf(fact_447_append__self__conv2,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ Ys2 )
        = Ys2 )
      = ( Xs2 = nil_Pr7606595143512524841ropVar ) ) ).

% append_self_conv2
thf(fact_448_append__self__conv2,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs2 @ Ys2 )
        = Ys2 )
      = ( Xs2 = nil_Pr6119456312066593144ropVar ) ) ).

% append_self_conv2
thf(fact_449_append__self__conv2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( ( append1087384436348988414pVar_o @ Xs2 @ Ys2 )
        = Ys2 )
      = ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% append_self_conv2
thf(fact_450_append__self__conv2,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( ( append8069948826146371844pVar_o @ Xs2 @ Ys2 )
        = Ys2 )
      = ( Xs2 = nil_li1310946081133177199pVar_o ) ) ).

% append_self_conv2
thf(fact_451_append__self__conv2,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ Ys2 )
        = Ys2 )
      = ( Xs2 = nil_li590035233778055285pVar_o ) ) ).

% append_self_conv2
thf(fact_452_append__self__conv2,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys2 )
        = Ys2 )
      = ( Xs2 = nil_nat ) ) ).

% append_self_conv2
thf(fact_453_self__append__conv2,axiom,
    ! [Y3: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( Y3
        = ( append7708039934470209236ropVar @ Xs2 @ Y3 ) )
      = ( Xs2 = nil_Pr7606595143512524841ropVar ) ) ).

% self_append_conv2
thf(fact_454_self__append__conv2,axiom,
    ! [Y3: list_P8791008543742759928ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( Y3
        = ( append4352292717583037453ropVar @ Xs2 @ Y3 ) )
      = ( Xs2 = nil_Pr6119456312066593144ropVar ) ) ).

% self_append_conv2
thf(fact_455_self__append__conv2,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( Y3
        = ( append1087384436348988414pVar_o @ Xs2 @ Y3 ) )
      = ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% self_append_conv2
thf(fact_456_self__append__conv2,axiom,
    ! [Y3: list_l1255154591254203759pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( Y3
        = ( append8069948826146371844pVar_o @ Xs2 @ Y3 ) )
      = ( Xs2 = nil_li1310946081133177199pVar_o ) ) ).

% self_append_conv2
thf(fact_457_self__append__conv2,axiom,
    ! [Y3: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( Y3
        = ( append8959015459476584842pVar_o @ Xs2 @ Y3 ) )
      = ( Xs2 = nil_li590035233778055285pVar_o ) ) ).

% self_append_conv2
thf(fact_458_self__append__conv2,axiom,
    ! [Y3: list_nat,Xs2: list_nat] :
      ( ( Y3
        = ( append_nat @ Xs2 @ Y3 ) )
      = ( Xs2 = nil_nat ) ) ).

% self_append_conv2
thf(fact_459_Nil__is__append__conv,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( nil_Pr7606595143512524841ropVar
        = ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) )
      = ( ( Xs2 = nil_Pr7606595143512524841ropVar )
        & ( Ys2 = nil_Pr7606595143512524841ropVar ) ) ) ).

% Nil_is_append_conv
thf(fact_460_Nil__is__append__conv,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) )
      = ( ( Xs2 = nil_Pr6119456312066593144ropVar )
        & ( Ys2 = nil_Pr6119456312066593144ropVar ) ) ) ).

% Nil_is_append_conv
thf(fact_461_Nil__is__append__conv,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( nil_Pr6413368191065476585pVar_o
        = ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) )
      = ( ( Xs2 = nil_Pr6413368191065476585pVar_o )
        & ( Ys2 = nil_Pr6413368191065476585pVar_o ) ) ) ).

% Nil_is_append_conv
thf(fact_462_Nil__is__append__conv,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( nil_li1310946081133177199pVar_o
        = ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) )
      = ( ( Xs2 = nil_li1310946081133177199pVar_o )
        & ( Ys2 = nil_li1310946081133177199pVar_o ) ) ) ).

% Nil_is_append_conv
thf(fact_463_Nil__is__append__conv,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( nil_li590035233778055285pVar_o
        = ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
      = ( ( Xs2 = nil_li590035233778055285pVar_o )
        & ( Ys2 = nil_li590035233778055285pVar_o ) ) ) ).

% Nil_is_append_conv
thf(fact_464_Nil__is__append__conv,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs2 @ Ys2 ) )
      = ( ( Xs2 = nil_nat )
        & ( Ys2 = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_465_append__is__Nil__conv,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ Ys2 )
        = nil_Pr7606595143512524841ropVar )
      = ( ( Xs2 = nil_Pr7606595143512524841ropVar )
        & ( Ys2 = nil_Pr7606595143512524841ropVar ) ) ) ).

% append_is_Nil_conv
thf(fact_466_append__is__Nil__conv,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs2 @ Ys2 )
        = nil_Pr6119456312066593144ropVar )
      = ( ( Xs2 = nil_Pr6119456312066593144ropVar )
        & ( Ys2 = nil_Pr6119456312066593144ropVar ) ) ) ).

% append_is_Nil_conv
thf(fact_467_append__is__Nil__conv,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( ( append1087384436348988414pVar_o @ Xs2 @ Ys2 )
        = nil_Pr6413368191065476585pVar_o )
      = ( ( Xs2 = nil_Pr6413368191065476585pVar_o )
        & ( Ys2 = nil_Pr6413368191065476585pVar_o ) ) ) ).

% append_is_Nil_conv
thf(fact_468_append__is__Nil__conv,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( ( append8069948826146371844pVar_o @ Xs2 @ Ys2 )
        = nil_li1310946081133177199pVar_o )
      = ( ( Xs2 = nil_li1310946081133177199pVar_o )
        & ( Ys2 = nil_li1310946081133177199pVar_o ) ) ) ).

% append_is_Nil_conv
thf(fact_469_append__is__Nil__conv,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ Ys2 )
        = nil_li590035233778055285pVar_o )
      = ( ( Xs2 = nil_li590035233778055285pVar_o )
        & ( Ys2 = nil_li590035233778055285pVar_o ) ) ) ).

% append_is_Nil_conv
thf(fact_470_append__is__Nil__conv,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys2 )
        = nil_nat )
      = ( ( Xs2 = nil_nat )
        & ( Ys2 = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_471_append1__eq__conv,axiom,
    ! [Xs2: list_P723798329145367999ropVar,X4: propos3600597766819905199ropVar,Ys2: list_P723798329145367999ropVar,Y3: propos3600597766819905199ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ ( cons_P6757824865275295865ropVar @ X4 @ nil_Pr7606595143512524841ropVar ) )
        = ( append7708039934470209236ropVar @ Ys2 @ ( cons_P6757824865275295865ropVar @ Y3 @ nil_Pr7606595143512524841ropVar ) ) )
      = ( ( Xs2 = Ys2 )
        & ( X4 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_472_append1__eq__conv,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) )
        = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ Y3 @ nil_Pr6119456312066593144ropVar ) ) )
      = ( ( Xs2 = Ys2 )
        & ( X4 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_473_append1__eq__conv,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o] :
      ( ( ( append1087384436348988414pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) )
        = ( append1087384436348988414pVar_o @ Ys2 @ ( cons_P6860203055867761049pVar_o @ Y3 @ nil_Pr6413368191065476585pVar_o ) ) )
      = ( ( Xs2 = Ys2 )
        & ( X4 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_474_append1__eq__conv,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( ( append8069948826146371844pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) )
        = ( append8069948826146371844pVar_o @ Ys2 @ ( cons_l3452452106213086495pVar_o @ Y3 @ nil_li1310946081133177199pVar_o ) ) )
      = ( ( Xs2 = Ys2 )
        & ( X4 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_475_append1__eq__conv,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) )
        = ( append8959015459476584842pVar_o @ Ys2 @ ( cons_l7316790070876032037pVar_o @ Y3 @ nil_li590035233778055285pVar_o ) ) )
      = ( ( Xs2 = Ys2 )
        & ( X4 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_476_append1__eq__conv,axiom,
    ! [Xs2: list_nat,X4: nat,Ys2: list_nat,Y3: nat] :
      ( ( ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) )
        = ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) )
      = ( ( Xs2 = Ys2 )
        & ( X4 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_477_bind__simps_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat,F2: nat > list_nat] :
      ( ( bind_nat_nat @ ( cons_nat @ X4 @ Xs2 ) @ F2 )
      = ( append_nat @ ( F2 @ X4 ) @ ( bind_nat_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_478_bind__simps_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat,F2: nat > list_P723798329145367999ropVar] :
      ( ( bind_n8853754276869496968ropVar @ ( cons_nat @ X4 @ Xs2 ) @ F2 )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( bind_n8853754276869496968ropVar @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_479_bind__simps_I2_J,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,F2: produc8999643685545340131pVar_o > list_nat] :
      ( ( bind_P7202517103677919084_o_nat @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ F2 )
      = ( append_nat @ ( F2 @ X4 ) @ ( bind_P7202517103677919084_o_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_480_bind__simps_I2_J,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,F2: produc8999643685545340131pVar_o > list_P723798329145367999ropVar] :
      ( ( bind_P4196495778012640331ropVar @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ F2 )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( bind_P4196495778012640331ropVar @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_481_bind__simps_I2_J,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,F2: list_P1665994083459302249pVar_o > list_nat] :
      ( ( bind_l4743436673915347442_o_nat @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ F2 )
      = ( append_nat @ ( F2 @ X4 ) @ ( bind_l4743436673915347442_o_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_482_bind__simps_I2_J,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,F2: produc2642885279816024434ropVar > list_nat] :
      ( ( bind_P7182508892803130363ar_nat @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ F2 )
      = ( append_nat @ ( F2 @ X4 ) @ ( bind_P7182508892803130363ar_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_483_bind__simps_I2_J,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,F2: list_P1665994083459302249pVar_o > list_P723798329145367999ropVar] :
      ( ( bind_l1937305119761364945ropVar @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ F2 )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( bind_l1937305119761364945ropVar @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_484_bind__simps_I2_J,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,F2: list_l1255154591254203759pVar_o > list_nat] :
      ( ( bind_l3081483778782373880_o_nat @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ F2 )
      = ( append_nat @ ( F2 @ X4 ) @ ( bind_l3081483778782373880_o_nat @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_485_bind__simps_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat,F2: nat > list_l985434930288970485pVar_o] :
      ( ( bind_n4999772531149697238pVar_o @ ( cons_nat @ X4 @ Xs2 ) @ F2 )
      = ( append8959015459476584842pVar_o @ ( F2 @ X4 ) @ ( bind_n4999772531149697238pVar_o @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_486_bind__simps_I2_J,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,F2: produc2642885279816024434ropVar > list_P723798329145367999ropVar] :
      ( ( bind_P8041276001586520922ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ F2 )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( bind_P8041276001586520922ropVar @ Xs2 @ F2 ) ) ) ).

% bind_simps(2)
thf(fact_487_append__listrel1I,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o,Us: list_l985434930288970485pVar_o,Vs: list_l985434930288970485pVar_o] :
      ( ( ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs2 = Ys2 )
          & ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Us @ Vs ) @ ( listre8274924349387117774pVar_o @ R ) ) ) )
     => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Us ) @ ( append8959015459476584842pVar_o @ Ys2 @ Vs ) ) @ ( listre8274924349387117774pVar_o @ R ) ) ) ).

% append_listrel1I
thf(fact_488_append__listrel1I,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs2 = Ys2 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Us ) @ ( append_nat @ Ys2 @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_489_append__listrel1I,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,R: set_Pr572148484142614943ropVar,Us: list_P723798329145367999ropVar,Vs: list_P723798329145367999ropVar] :
      ( ( ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Xs2 @ Ys2 ) @ ( listre5117138414868592272ropVar @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs2 = Ys2 )
          & ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Us @ Vs ) @ ( listre5117138414868592272ropVar @ R ) ) ) )
     => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Us ) @ ( append7708039934470209236ropVar @ Ys2 @ Vs ) ) @ ( listre5117138414868592272ropVar @ R ) ) ) ).

% append_listrel1I
thf(fact_490_append__eq__append__conv2,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o,Ts: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ Ys2 )
        = ( append8959015459476584842pVar_o @ Zs @ Ts ) )
      = ( ? [Us2: list_l985434930288970485pVar_o] :
            ( ( ( Xs2
                = ( append8959015459476584842pVar_o @ Zs @ Us2 ) )
              & ( ( append8959015459476584842pVar_o @ Us2 @ Ys2 )
                = Ts ) )
            | ( ( ( append8959015459476584842pVar_o @ Xs2 @ Us2 )
                = Zs )
              & ( Ys2
                = ( append8959015459476584842pVar_o @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_491_append__eq__append__conv2,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs2 @ Ys2 )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs2
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys2 )
                = Ts ) )
            | ( ( ( append_nat @ Xs2 @ Us2 )
                = Zs )
              & ( Ys2
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_492_append__eq__append__conv2,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar,Ts: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ Ys2 )
        = ( append7708039934470209236ropVar @ Zs @ Ts ) )
      = ( ? [Us2: list_P723798329145367999ropVar] :
            ( ( ( Xs2
                = ( append7708039934470209236ropVar @ Zs @ Us2 ) )
              & ( ( append7708039934470209236ropVar @ Us2 @ Ys2 )
                = Ts ) )
            | ( ( ( append7708039934470209236ropVar @ Xs2 @ Us2 )
                = Zs )
              & ( Ys2
                = ( append7708039934470209236ropVar @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_493_append__eq__appendI,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Xs1: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Us: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys2
          = ( append8959015459476584842pVar_o @ Xs1 @ Us ) )
       => ( ( append8959015459476584842pVar_o @ Xs2 @ Ys2 )
          = ( append8959015459476584842pVar_o @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_494_append__eq__appendI,axiom,
    ! [Xs2: list_nat,Xs1: list_nat,Zs: list_nat,Ys2: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys2
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs2 @ Ys2 )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_495_append__eq__appendI,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Xs1: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Us: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys2
          = ( append7708039934470209236ropVar @ Xs1 @ Us ) )
       => ( ( append7708039934470209236ropVar @ Xs2 @ Ys2 )
          = ( append7708039934470209236ropVar @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_496_Cons__eq__appendI,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs1: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ( ( cons_P6757824865275295865ropVar @ X4 @ Xs1 )
        = Ys2 )
     => ( ( Xs2
          = ( append7708039934470209236ropVar @ Xs1 @ Zs ) )
       => ( ( cons_P6757824865275295865ropVar @ X4 @ Xs2 )
          = ( append7708039934470209236ropVar @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_497_Cons__eq__appendI,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs1: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Xs2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X4 @ Xs1 )
        = Ys2 )
     => ( ( Xs2
          = ( append4352292717583037453ropVar @ Xs1 @ Zs ) )
       => ( ( cons_P1504208554336013608ropVar @ X4 @ Xs2 )
          = ( append4352292717583037453ropVar @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_498_Cons__eq__appendI,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs1: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o,Zs: list_P1665994083459302249pVar_o] :
      ( ( ( cons_P6860203055867761049pVar_o @ X4 @ Xs1 )
        = Ys2 )
     => ( ( Xs2
          = ( append1087384436348988414pVar_o @ Xs1 @ Zs ) )
       => ( ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 )
          = ( append1087384436348988414pVar_o @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_499_Cons__eq__appendI,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs1: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o,Xs2: list_l1255154591254203759pVar_o,Zs: list_l1255154591254203759pVar_o] :
      ( ( ( cons_l3452452106213086495pVar_o @ X4 @ Xs1 )
        = Ys2 )
     => ( ( Xs2
          = ( append8069948826146371844pVar_o @ Xs1 @ Zs ) )
       => ( ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 )
          = ( append8069948826146371844pVar_o @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_500_Cons__eq__appendI,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs1: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ( ( cons_l7316790070876032037pVar_o @ X4 @ Xs1 )
        = Ys2 )
     => ( ( Xs2
          = ( append8959015459476584842pVar_o @ Xs1 @ Zs ) )
       => ( ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 )
          = ( append8959015459476584842pVar_o @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_501_Cons__eq__appendI,axiom,
    ! [X4: nat,Xs1: list_nat,Ys2: list_nat,Xs2: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X4 @ Xs1 )
        = Ys2 )
     => ( ( Xs2
          = ( append_nat @ Xs1 @ Zs ) )
       => ( ( cons_nat @ X4 @ Xs2 )
          = ( append_nat @ Ys2 @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_502_append__Cons,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( append7708039934470209236ropVar @ ( cons_P6757824865275295865ropVar @ X4 @ Xs2 ) @ Ys2 )
      = ( cons_P6757824865275295865ropVar @ X4 @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_503_append__Cons,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ Ys2 )
      = ( cons_P1504208554336013608ropVar @ X4 @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_504_append__Cons,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( append1087384436348988414pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ Ys2 )
      = ( cons_P6860203055867761049pVar_o @ X4 @ ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_505_append__Cons,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( append8069948826146371844pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ Ys2 )
      = ( cons_l3452452106213086495pVar_o @ X4 @ ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_506_append__Cons,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( append8959015459476584842pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ Ys2 )
      = ( cons_l7316790070876032037pVar_o @ X4 @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_507_append__Cons,axiom,
    ! [X4: nat,Xs2: list_nat,Ys2: list_nat] :
      ( ( append_nat @ ( cons_nat @ X4 @ Xs2 ) @ Ys2 )
      = ( cons_nat @ X4 @ ( append_nat @ Xs2 @ Ys2 ) ) ) ).

% append_Cons
thf(fact_508_append__Nil,axiom,
    ! [Ys2: list_P723798329145367999ropVar] :
      ( ( append7708039934470209236ropVar @ nil_Pr7606595143512524841ropVar @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_509_append__Nil,axiom,
    ! [Ys2: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_510_append__Nil,axiom,
    ! [Ys2: list_P1665994083459302249pVar_o] :
      ( ( append1087384436348988414pVar_o @ nil_Pr6413368191065476585pVar_o @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_511_append__Nil,axiom,
    ! [Ys2: list_l1255154591254203759pVar_o] :
      ( ( append8069948826146371844pVar_o @ nil_li1310946081133177199pVar_o @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_512_append__Nil,axiom,
    ! [Ys2: list_l985434930288970485pVar_o] :
      ( ( append8959015459476584842pVar_o @ nil_li590035233778055285pVar_o @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_513_append__Nil,axiom,
    ! [Ys2: list_nat] :
      ( ( append_nat @ nil_nat @ Ys2 )
      = Ys2 ) ).

% append_Nil
thf(fact_514_append_Oleft__neutral,axiom,
    ! [A: list_P723798329145367999ropVar] :
      ( ( append7708039934470209236ropVar @ nil_Pr7606595143512524841ropVar @ A )
      = A ) ).

% append.left_neutral
thf(fact_515_append_Oleft__neutral,axiom,
    ! [A: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ A )
      = A ) ).

% append.left_neutral
thf(fact_516_append_Oleft__neutral,axiom,
    ! [A: list_P1665994083459302249pVar_o] :
      ( ( append1087384436348988414pVar_o @ nil_Pr6413368191065476585pVar_o @ A )
      = A ) ).

% append.left_neutral
thf(fact_517_append_Oleft__neutral,axiom,
    ! [A: list_l1255154591254203759pVar_o] :
      ( ( append8069948826146371844pVar_o @ nil_li1310946081133177199pVar_o @ A )
      = A ) ).

% append.left_neutral
thf(fact_518_append_Oleft__neutral,axiom,
    ! [A: list_l985434930288970485pVar_o] :
      ( ( append8959015459476584842pVar_o @ nil_li590035233778055285pVar_o @ A )
      = A ) ).

% append.left_neutral
thf(fact_519_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_520_eq__Nil__appendI,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( Xs2 = Ys2 )
     => ( Xs2
        = ( append7708039934470209236ropVar @ nil_Pr7606595143512524841ropVar @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_521_eq__Nil__appendI,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( Xs2 = Ys2 )
     => ( Xs2
        = ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_522_eq__Nil__appendI,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( Xs2 = Ys2 )
     => ( Xs2
        = ( append1087384436348988414pVar_o @ nil_Pr6413368191065476585pVar_o @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_523_eq__Nil__appendI,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( Xs2 = Ys2 )
     => ( Xs2
        = ( append8069948826146371844pVar_o @ nil_li1310946081133177199pVar_o @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_524_eq__Nil__appendI,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( Xs2 = Ys2 )
     => ( Xs2
        = ( append8959015459476584842pVar_o @ nil_li590035233778055285pVar_o @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_525_eq__Nil__appendI,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( Xs2 = Ys2 )
     => ( Xs2
        = ( append_nat @ nil_nat @ Ys2 ) ) ) ).

% eq_Nil_appendI
thf(fact_526_ord_Olexordp__eq__pref,axiom,
    ! [Less: list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o > $o,U: list_l985434930288970485pVar_o,V2: list_l985434930288970485pVar_o] : ( lexord8403422104825280955pVar_o @ Less @ U @ ( append8959015459476584842pVar_o @ U @ V2 ) ) ).

% ord.lexordp_eq_pref
thf(fact_527_ord_Olexordp__eq__pref,axiom,
    ! [Less: nat > nat > $o,U: list_nat,V2: list_nat] : ( lexordp_eq_nat @ Less @ U @ ( append_nat @ U @ V2 ) ) ).

% ord.lexordp_eq_pref
thf(fact_528_ord_Olexordp__eq__pref,axiom,
    ! [Less: propos3600597766819905199ropVar > propos3600597766819905199ropVar > $o,U: list_P723798329145367999ropVar,V2: list_P723798329145367999ropVar] : ( lexord2075174576499698531ropVar @ Less @ U @ ( append7708039934470209236ropVar @ U @ V2 ) ) ).

% ord.lexordp_eq_pref
thf(fact_529_listrel1I,axiom,
    ! [X4: propos3600597766819905199ropVar,Y3: propos3600597766819905199ropVar,R: set_Pr572148484142614943ropVar,Xs2: list_P723798329145367999ropVar,Us: list_P723798329145367999ropVar,Vs: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append7708039934470209236ropVar @ Us @ ( cons_P6757824865275295865ropVar @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append7708039934470209236ropVar @ Us @ ( cons_P6757824865275295865ropVar @ Y3 @ Vs ) ) )
         => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Xs2 @ Ys2 ) @ ( listre5117138414868592272ropVar @ R ) ) ) ) ) ).

% listrel1I
thf(fact_530_listrel1I,axiom,
    ! [X4: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar,Xs2: list_P8791008543742759928ropVar,Us: list_P8791008543742759928ropVar,Vs: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append4352292717583037453ropVar @ Us @ ( cons_P1504208554336013608ropVar @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append4352292717583037453ropVar @ Us @ ( cons_P1504208554336013608ropVar @ Y3 @ Vs ) ) )
         => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% listrel1I
thf(fact_531_listrel1I,axiom,
    ! [X4: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,R: set_Pr4574656050966626247pVar_o,Xs2: list_P1665994083459302249pVar_o,Us: list_P1665994083459302249pVar_o,Vs: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append1087384436348988414pVar_o @ Us @ ( cons_P6860203055867761049pVar_o @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append1087384436348988414pVar_o @ Us @ ( cons_P6860203055867761049pVar_o @ Y3 @ Vs ) ) )
         => ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Ys2 ) @ ( listre2444918065581779778pVar_o @ R ) ) ) ) ) ).

% listrel1I
thf(fact_532_listrel1I,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,R: set_Pr8855275280618702407pVar_o,Xs2: list_l1255154591254203759pVar_o,Us: list_l1255154591254203759pVar_o,Vs: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append8069948826146371844pVar_o @ Us @ ( cons_l3452452106213086495pVar_o @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append8069948826146371844pVar_o @ Us @ ( cons_l3452452106213086495pVar_o @ Y3 @ Vs ) ) )
         => ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ Ys2 ) @ ( listre4950778487325574728pVar_o @ R ) ) ) ) ) ).

% listrel1I
thf(fact_533_listrel1I,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o,R: set_Pr3139130378708280007pVar_o,Xs2: list_l985434930288970485pVar_o,Us: list_l985434930288970485pVar_o,Vs: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append8959015459476584842pVar_o @ Us @ ( cons_l7316790070876032037pVar_o @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append8959015459476584842pVar_o @ Us @ ( cons_l7316790070876032037pVar_o @ Y3 @ Vs ) ) )
         => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) ) ) ) ) ).

% listrel1I
thf(fact_534_listrel1I,axiom,
    ! [X4: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Us: list_nat,Vs: list_nat,Ys2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append_nat @ Us @ ( cons_nat @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append_nat @ Us @ ( cons_nat @ Y3 @ Vs ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_535_listrel1I,axiom,
    ! [X4: $o,Y3: $o,R: set_Product_prod_o_o,Xs2: list_o,Us: list_o,Vs: list_o,Ys2: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append_o @ Us @ ( cons_o @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append_o @ Us @ ( cons_o @ Y3 @ Vs ) ) )
         => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) ) ) ) ) ).

% listrel1I
thf(fact_536_listrel1I,axiom,
    ! [X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar,Xs2: list_M6740372356978808802ropVar,Us: list_M6740372356978808802ropVar,Vs: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R )
     => ( ( Xs2
          = ( append2684835082924936439ropVar @ Us @ ( cons_M4277272395645347858ropVar @ X4 @ Vs ) ) )
       => ( ( Ys2
            = ( append2684835082924936439ropVar @ Us @ ( cons_M4277272395645347858ropVar @ Y3 @ Vs ) ) )
         => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% listrel1I
thf(fact_537_listrel1E,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,R: set_Pr572148484142614943ropVar] :
      ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Xs2 @ Ys2 ) @ ( listre5117138414868592272ropVar @ R ) )
     => ~ ! [X2: propos3600597766819905199ropVar,Y4: propos3600597766819905199ropVar] :
            ( ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ X2 @ Y4 ) @ R )
           => ! [Us3: list_P723798329145367999ropVar,Vs2: list_P723798329145367999ropVar] :
                ( ( Xs2
                  = ( append7708039934470209236ropVar @ Us3 @ ( cons_P6757824865275295865ropVar @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append7708039934470209236ropVar @ Us3 @ ( cons_P6757824865275295865ropVar @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_538_listrel1E,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( listre6655839188755619153ropVar @ R ) )
     => ~ ! [X2: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar] :
            ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y4 ) @ R )
           => ! [Us3: list_P8791008543742759928ropVar,Vs2: list_P8791008543742759928ropVar] :
                ( ( Xs2
                  = ( append4352292717583037453ropVar @ Us3 @ ( cons_P1504208554336013608ropVar @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append4352292717583037453ropVar @ Us3 @ ( cons_P1504208554336013608ropVar @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_539_listrel1E,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Ys2 ) @ ( listre2444918065581779778pVar_o @ R ) )
     => ~ ! [X2: produc8999643685545340131pVar_o,Y4: produc8999643685545340131pVar_o] :
            ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X2 @ Y4 ) @ R )
           => ! [Us3: list_P1665994083459302249pVar_o,Vs2: list_P1665994083459302249pVar_o] :
                ( ( Xs2
                  = ( append1087384436348988414pVar_o @ Us3 @ ( cons_P6860203055867761049pVar_o @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append1087384436348988414pVar_o @ Us3 @ ( cons_P6860203055867761049pVar_o @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_540_listrel1E,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ Ys2 ) @ ( listre4950778487325574728pVar_o @ R ) )
     => ~ ! [X2: list_P1665994083459302249pVar_o,Y4: list_P1665994083459302249pVar_o] :
            ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X2 @ Y4 ) @ R )
           => ! [Us3: list_l1255154591254203759pVar_o,Vs2: list_l1255154591254203759pVar_o] :
                ( ( Xs2
                  = ( append8069948826146371844pVar_o @ Us3 @ ( cons_l3452452106213086495pVar_o @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append8069948826146371844pVar_o @ Us3 @ ( cons_l3452452106213086495pVar_o @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_541_listrel1E,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) )
     => ~ ! [X2: list_l1255154591254203759pVar_o,Y4: list_l1255154591254203759pVar_o] :
            ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X2 @ Y4 ) @ R )
           => ! [Us3: list_l985434930288970485pVar_o,Vs2: list_l985434930288970485pVar_o] :
                ( ( Xs2
                  = ( append8959015459476584842pVar_o @ Us3 @ ( cons_l7316790070876032037pVar_o @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append8959015459476584842pVar_o @ Us3 @ ( cons_l7316790070876032037pVar_o @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_542_listrel1E,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
     => ~ ! [X2: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R )
           => ! [Us3: list_nat,Vs2: list_nat] :
                ( ( Xs2
                  = ( append_nat @ Us3 @ ( cons_nat @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append_nat @ Us3 @ ( cons_nat @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_543_listrel1E,axiom,
    ! [Xs2: list_o,Ys2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) )
     => ~ ! [X2: $o,Y4: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y4 ) @ R )
           => ! [Us3: list_o,Vs2: list_o] :
                ( ( Xs2
                  = ( append_o @ Us3 @ ( cons_o @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append_o @ Us3 @ ( cons_o @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_544_listrel1E,axiom,
    ! [Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( listre3736066117146447419ropVar @ R ) )
     => ~ ! [X2: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y4 ) @ R )
           => ! [Us3: list_M6740372356978808802ropVar,Vs2: list_M6740372356978808802ropVar] :
                ( ( Xs2
                  = ( append2684835082924936439ropVar @ Us3 @ ( cons_M4277272395645347858ropVar @ X2 @ Vs2 ) ) )
               => ( Ys2
                 != ( append2684835082924936439ropVar @ Us3 @ ( cons_M4277272395645347858ropVar @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_545_listrelp_OCons,axiom,
    ! [R: nat > nat > $o,X4: nat,Y3: nat,Xs2: list_nat,Ys2: list_nat] :
      ( ( R @ X4 @ Y3 )
     => ( ( listrelp_nat_nat @ R @ Xs2 @ Ys2 )
       => ( listrelp_nat_nat @ R @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_546_listrelp_OCons,axiom,
    ! [R: produc8999643685545340131pVar_o > nat > $o,X4: produc8999643685545340131pVar_o,Y3: nat,Xs2: list_P1665994083459302249pVar_o,Ys2: list_nat] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre2270229272446257844_o_nat @ R @ Xs2 @ Ys2 )
       => ( listre2270229272446257844_o_nat @ R @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_547_listrelp_OCons,axiom,
    ! [R: nat > produc8999643685545340131pVar_o > $o,X4: nat,Y3: produc8999643685545340131pVar_o,Xs2: list_nat,Ys2: list_P1665994083459302249pVar_o] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre7977470258951326354pVar_o @ R @ Xs2 @ Ys2 )
       => ( listre7977470258951326354pVar_o @ R @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_548_listrelp_OCons,axiom,
    ! [R: list_P1665994083459302249pVar_o > nat > $o,X4: list_P1665994083459302249pVar_o,Y3: nat,Xs2: list_l1255154591254203759pVar_o,Ys2: list_nat] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre7159049289101154618_o_nat @ R @ Xs2 @ Ys2 )
       => ( listre7159049289101154618_o_nat @ R @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_549_listrelp_OCons,axiom,
    ! [R: nat > list_P1665994083459302249pVar_o > $o,X4: nat,Y3: list_P1665994083459302249pVar_o,Xs2: list_nat,Ys2: list_l1255154591254203759pVar_o] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre8689339618514533784pVar_o @ R @ Xs2 @ Ys2 )
       => ( listre8689339618514533784pVar_o @ R @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_550_listrelp_OCons,axiom,
    ! [R: produc2642885279816024434ropVar > nat > $o,X4: produc2642885279816024434ropVar,Y3: nat,Xs2: list_P8791008543742759928ropVar,Ys2: list_nat] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre7860766376562299715ar_nat @ R @ Xs2 @ Ys2 )
       => ( listre7860766376562299715ar_nat @ R @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_551_listrelp_OCons,axiom,
    ! [R: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o,X4: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre612986413372629247pVar_o @ R @ Xs2 @ Ys2 )
       => ( listre612986413372629247pVar_o @ R @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_552_listrelp_OCons,axiom,
    ! [R: list_l1255154591254203759pVar_o > nat > $o,X4: list_l1255154591254203759pVar_o,Y3: nat,Xs2: list_l985434930288970485pVar_o,Ys2: list_nat] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre8257602174308424512_o_nat @ R @ Xs2 @ Ys2 )
       => ( listre8257602174308424512_o_nat @ R @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_553_listrelp_OCons,axiom,
    ! [R: nat > produc2642885279816024434ropVar > $o,X4: nat,Y3: produc2642885279816024434ropVar,Xs2: list_nat,Ys2: list_P8791008543742759928ropVar] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre2196087975860241569ropVar @ R @ Xs2 @ Ys2 )
       => ( listre2196087975860241569ropVar @ R @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_554_listrelp_OCons,axiom,
    ! [R: nat > list_l1255154591254203759pVar_o > $o,X4: nat,Y3: list_l1255154591254203759pVar_o,Xs2: list_nat,Ys2: list_l985434930288970485pVar_o] :
      ( ( R @ X4 @ Y3 )
     => ( ( listre952518889820972062pVar_o @ R @ Xs2 @ Ys2 )
       => ( listre952518889820972062pVar_o @ R @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) ) ) ) ).

% listrelp.Cons
thf(fact_555_listrelp_ONil,axiom,
    ! [R: nat > nat > $o] : ( listrelp_nat_nat @ R @ nil_nat @ nil_nat ) ).

% listrelp.Nil
thf(fact_556_listrelp_ONil,axiom,
    ! [R: produc8999643685545340131pVar_o > nat > $o] : ( listre2270229272446257844_o_nat @ R @ nil_Pr6413368191065476585pVar_o @ nil_nat ) ).

% listrelp.Nil
thf(fact_557_listrelp_ONil,axiom,
    ! [R: nat > produc8999643685545340131pVar_o > $o] : ( listre7977470258951326354pVar_o @ R @ nil_nat @ nil_Pr6413368191065476585pVar_o ) ).

% listrelp.Nil
thf(fact_558_listrelp_ONil,axiom,
    ! [R: list_P1665994083459302249pVar_o > nat > $o] : ( listre7159049289101154618_o_nat @ R @ nil_li1310946081133177199pVar_o @ nil_nat ) ).

% listrelp.Nil
thf(fact_559_listrelp_ONil,axiom,
    ! [R: nat > list_P1665994083459302249pVar_o > $o] : ( listre8689339618514533784pVar_o @ R @ nil_nat @ nil_li1310946081133177199pVar_o ) ).

% listrelp.Nil
thf(fact_560_listrelp_ONil,axiom,
    ! [R: produc2642885279816024434ropVar > nat > $o] : ( listre7860766376562299715ar_nat @ R @ nil_Pr6119456312066593144ropVar @ nil_nat ) ).

% listrelp.Nil
thf(fact_561_listrelp_ONil,axiom,
    ! [R: produc8999643685545340131pVar_o > produc8999643685545340131pVar_o > $o] : ( listre612986413372629247pVar_o @ R @ nil_Pr6413368191065476585pVar_o @ nil_Pr6413368191065476585pVar_o ) ).

% listrelp.Nil
thf(fact_562_listrelp_ONil,axiom,
    ! [R: list_l1255154591254203759pVar_o > nat > $o] : ( listre8257602174308424512_o_nat @ R @ nil_li590035233778055285pVar_o @ nil_nat ) ).

% listrelp.Nil
thf(fact_563_listrelp_ONil,axiom,
    ! [R: nat > produc2642885279816024434ropVar > $o] : ( listre2196087975860241569ropVar @ R @ nil_nat @ nil_Pr6119456312066593144ropVar ) ).

% listrelp.Nil
thf(fact_564_listrelp_ONil,axiom,
    ! [R: nat > list_l1255154591254203759pVar_o > $o] : ( listre952518889820972062pVar_o @ R @ nil_nat @ nil_li590035233778055285pVar_o ) ).

% listrelp.Nil
thf(fact_565_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_P723798329145367999ropVar,X4: propos3600597766819905199ropVar,Ys2: list_P723798329145367999ropVar,Y3: propos3600597766819905199ropVar,R: set_Pr572148484142614943ropVar] :
      ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ Xs2 @ ( cons_P6757824865275295865ropVar @ X4 @ nil_Pr7606595143512524841ropVar ) ) @ ( append7708039934470209236ropVar @ Ys2 @ ( cons_P6757824865275295865ropVar @ Y3 @ nil_Pr7606595143512524841ropVar ) ) ) @ ( listre5117138414868592272ropVar @ R ) )
      = ( ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Xs2 @ Ys2 ) @ ( listre5117138414868592272ropVar @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_566_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) @ ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ Y3 @ nil_Pr6119456312066593144ropVar ) ) ) @ ( listre6655839188755619153ropVar @ R ) )
      = ( ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( listre6655839188755619153ropVar @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_567_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) ) @ ( append1087384436348988414pVar_o @ Ys2 @ ( cons_P6860203055867761049pVar_o @ Y3 @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( listre2444918065581779778pVar_o @ R ) )
      = ( ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Ys2 ) @ ( listre2444918065581779778pVar_o @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_568_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) ) @ ( append8069948826146371844pVar_o @ Ys2 @ ( cons_l3452452106213086495pVar_o @ Y3 @ nil_li1310946081133177199pVar_o ) ) ) @ ( listre4950778487325574728pVar_o @ R ) )
      = ( ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ Ys2 ) @ ( listre4950778487325574728pVar_o @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_569_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) ) @ ( append8959015459476584842pVar_o @ Ys2 @ ( cons_l7316790070876032037pVar_o @ Y3 @ nil_li590035233778055285pVar_o ) ) ) @ ( listre8274924349387117774pVar_o @ R ) )
      = ( ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_570_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_nat,X4: nat,Ys2: list_nat,Y3: nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) @ ( append_nat @ Ys2 @ ( cons_nat @ Y3 @ nil_nat ) ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_571_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_o,X4: $o,Ys2: list_o,Y3: $o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ ( cons_o @ X4 @ nil_o ) ) @ ( append_o @ Ys2 @ ( cons_o @ Y3 @ nil_o ) ) ) @ ( listrel1_o @ R ) )
      = ( ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( listrel1_o @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_572_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_M6740372356978808802ropVar,X4: multis3193088007478089820ropVar,Ys2: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ Xs2 @ ( cons_M4277272395645347858ropVar @ X4 @ nil_Mu4132399263956683874ropVar ) ) @ ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ Y3 @ nil_Mu4132399263956683874ropVar ) ) ) @ ( listre3736066117146447419ropVar @ R ) )
      = ( ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( listre3736066117146447419ropVar @ R ) )
          & ( X4 = Y3 ) )
        | ( ( Xs2 = Ys2 )
          & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_573_rev__nonempty__induct,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ( Xs2 != nil_Pr7606595143512524841ropVar )
     => ( ! [X2: propos3600597766819905199ropVar] : ( P @ ( cons_P6757824865275295865ropVar @ X2 @ nil_Pr7606595143512524841ropVar ) )
       => ( ! [X2: propos3600597766819905199ropVar,Xs: list_P723798329145367999ropVar] :
              ( ( Xs != nil_Pr7606595143512524841ropVar )
             => ( ( P @ Xs )
               => ( P @ ( append7708039934470209236ropVar @ Xs @ ( cons_P6757824865275295865ropVar @ X2 @ nil_Pr7606595143512524841ropVar ) ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_574_rev__nonempty__induct,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > $o] :
      ( ( Xs2 != nil_Pr6119456312066593144ropVar )
     => ( ! [X2: produc2642885279816024434ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) )
       => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
              ( ( Xs != nil_Pr6119456312066593144ropVar )
             => ( ( P @ Xs )
               => ( P @ ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_575_rev__nonempty__induct,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
     => ( ! [X2: produc8999643685545340131pVar_o] : ( P @ ( cons_P6860203055867761049pVar_o @ X2 @ nil_Pr6413368191065476585pVar_o ) )
       => ( ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
              ( ( Xs != nil_Pr6413368191065476585pVar_o )
             => ( ( P @ Xs )
               => ( P @ ( append1087384436348988414pVar_o @ Xs @ ( cons_P6860203055867761049pVar_o @ X2 @ nil_Pr6413368191065476585pVar_o ) ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_576_rev__nonempty__induct,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ( Xs2 != nil_li1310946081133177199pVar_o )
     => ( ! [X2: list_P1665994083459302249pVar_o] : ( P @ ( cons_l3452452106213086495pVar_o @ X2 @ nil_li1310946081133177199pVar_o ) )
       => ( ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
              ( ( Xs != nil_li1310946081133177199pVar_o )
             => ( ( P @ Xs )
               => ( P @ ( append8069948826146371844pVar_o @ Xs @ ( cons_l3452452106213086495pVar_o @ X2 @ nil_li1310946081133177199pVar_o ) ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_577_rev__nonempty__induct,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ( Xs2 != nil_li590035233778055285pVar_o )
     => ( ! [X2: list_l1255154591254203759pVar_o] : ( P @ ( cons_l7316790070876032037pVar_o @ X2 @ nil_li590035233778055285pVar_o ) )
       => ( ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
              ( ( Xs != nil_li590035233778055285pVar_o )
             => ( ( P @ Xs )
               => ( P @ ( append8959015459476584842pVar_o @ Xs @ ( cons_l7316790070876032037pVar_o @ X2 @ nil_li590035233778055285pVar_o ) ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_578_rev__nonempty__induct,axiom,
    ! [Xs2: list_nat,P: list_nat > $o] :
      ( ( Xs2 != nil_nat )
     => ( ! [X2: nat] : ( P @ ( cons_nat @ X2 @ nil_nat ) )
       => ( ! [X2: nat,Xs: list_nat] :
              ( ( Xs != nil_nat )
             => ( ( P @ Xs )
               => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) ) )
         => ( P @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_579_append__eq__Cons__conv,axiom,
    ! [Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar,X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Ys2 @ Zs )
        = ( cons_P6757824865275295865ropVar @ X4 @ Xs2 ) )
      = ( ( ( Ys2 = nil_Pr7606595143512524841ropVar )
          & ( Zs
            = ( cons_P6757824865275295865ropVar @ X4 @ Xs2 ) ) )
        | ? [Ys4: list_P723798329145367999ropVar] :
            ( ( Ys2
              = ( cons_P6757824865275295865ropVar @ X4 @ Ys4 ) )
            & ( ( append7708039934470209236ropVar @ Ys4 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_580_append__eq__Cons__conv,axiom,
    ! [Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Ys2 @ Zs )
        = ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
      = ( ( ( Ys2 = nil_Pr6119456312066593144ropVar )
          & ( Zs
            = ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) ) )
        | ? [Ys4: list_P8791008543742759928ropVar] :
            ( ( Ys2
              = ( cons_P1504208554336013608ropVar @ X4 @ Ys4 ) )
            & ( ( append4352292717583037453ropVar @ Ys4 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_581_append__eq__Cons__conv,axiom,
    ! [Ys2: list_P1665994083459302249pVar_o,Zs: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( ( append1087384436348988414pVar_o @ Ys2 @ Zs )
        = ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
      = ( ( ( Ys2 = nil_Pr6413368191065476585pVar_o )
          & ( Zs
            = ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) ) )
        | ? [Ys4: list_P1665994083459302249pVar_o] :
            ( ( Ys2
              = ( cons_P6860203055867761049pVar_o @ X4 @ Ys4 ) )
            & ( ( append1087384436348988414pVar_o @ Ys4 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_582_append__eq__Cons__conv,axiom,
    ! [Ys2: list_l1255154591254203759pVar_o,Zs: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( ( append8069948826146371844pVar_o @ Ys2 @ Zs )
        = ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
      = ( ( ( Ys2 = nil_li1310946081133177199pVar_o )
          & ( Zs
            = ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) ) )
        | ? [Ys4: list_l1255154591254203759pVar_o] :
            ( ( Ys2
              = ( cons_l3452452106213086495pVar_o @ X4 @ Ys4 ) )
            & ( ( append8069948826146371844pVar_o @ Ys4 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_583_append__eq__Cons__conv,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Ys2 @ Zs )
        = ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
      = ( ( ( Ys2 = nil_li590035233778055285pVar_o )
          & ( Zs
            = ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) ) )
        | ? [Ys4: list_l985434930288970485pVar_o] :
            ( ( Ys2
              = ( cons_l7316790070876032037pVar_o @ X4 @ Ys4 ) )
            & ( ( append8959015459476584842pVar_o @ Ys4 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_584_append__eq__Cons__conv,axiom,
    ! [Ys2: list_nat,Zs: list_nat,X4: nat,Xs2: list_nat] :
      ( ( ( append_nat @ Ys2 @ Zs )
        = ( cons_nat @ X4 @ Xs2 ) )
      = ( ( ( Ys2 = nil_nat )
          & ( Zs
            = ( cons_nat @ X4 @ Xs2 ) ) )
        | ? [Ys4: list_nat] :
            ( ( Ys2
              = ( cons_nat @ X4 @ Ys4 ) )
            & ( ( append_nat @ Ys4 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_585_Cons__eq__append__conv,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ( ( cons_P6757824865275295865ropVar @ X4 @ Xs2 )
        = ( append7708039934470209236ropVar @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_Pr7606595143512524841ropVar )
          & ( ( cons_P6757824865275295865ropVar @ X4 @ Xs2 )
            = Zs ) )
        | ? [Ys4: list_P723798329145367999ropVar] :
            ( ( ( cons_P6757824865275295865ropVar @ X4 @ Ys4 )
              = Ys2 )
            & ( Xs2
              = ( append7708039934470209236ropVar @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_586_Cons__eq__append__conv,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X4 @ Xs2 )
        = ( append4352292717583037453ropVar @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_Pr6119456312066593144ropVar )
          & ( ( cons_P1504208554336013608ropVar @ X4 @ Xs2 )
            = Zs ) )
        | ? [Ys4: list_P8791008543742759928ropVar] :
            ( ( ( cons_P1504208554336013608ropVar @ X4 @ Ys4 )
              = Ys2 )
            & ( Xs2
              = ( append4352292717583037453ropVar @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_587_Cons__eq__append__conv,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o,Zs: list_P1665994083459302249pVar_o] :
      ( ( ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 )
        = ( append1087384436348988414pVar_o @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_Pr6413368191065476585pVar_o )
          & ( ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 )
            = Zs ) )
        | ? [Ys4: list_P1665994083459302249pVar_o] :
            ( ( ( cons_P6860203055867761049pVar_o @ X4 @ Ys4 )
              = Ys2 )
            & ( Xs2
              = ( append1087384436348988414pVar_o @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_588_Cons__eq__append__conv,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o,Zs: list_l1255154591254203759pVar_o] :
      ( ( ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 )
        = ( append8069948826146371844pVar_o @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_li1310946081133177199pVar_o )
          & ( ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 )
            = Zs ) )
        | ? [Ys4: list_l1255154591254203759pVar_o] :
            ( ( ( cons_l3452452106213086495pVar_o @ X4 @ Ys4 )
              = Ys2 )
            & ( Xs2
              = ( append8069948826146371844pVar_o @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_589_Cons__eq__append__conv,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ( ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 )
        = ( append8959015459476584842pVar_o @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_li590035233778055285pVar_o )
          & ( ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 )
            = Zs ) )
        | ? [Ys4: list_l985434930288970485pVar_o] :
            ( ( ( cons_l7316790070876032037pVar_o @ X4 @ Ys4 )
              = Ys2 )
            & ( Xs2
              = ( append8959015459476584842pVar_o @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_590_Cons__eq__append__conv,axiom,
    ! [X4: nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( cons_nat @ X4 @ Xs2 )
        = ( append_nat @ Ys2 @ Zs ) )
      = ( ( ( Ys2 = nil_nat )
          & ( ( cons_nat @ X4 @ Xs2 )
            = Zs ) )
        | ? [Ys4: list_nat] :
            ( ( ( cons_nat @ X4 @ Ys4 )
              = Ys2 )
            & ( Xs2
              = ( append_nat @ Ys4 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_591_rev__exhaust,axiom,
    ! [Xs2: list_P723798329145367999ropVar] :
      ( ( Xs2 != nil_Pr7606595143512524841ropVar )
     => ~ ! [Ys: list_P723798329145367999ropVar,Y4: propos3600597766819905199ropVar] :
            ( Xs2
           != ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ Y4 @ nil_Pr7606595143512524841ropVar ) ) ) ) ).

% rev_exhaust
thf(fact_592_rev__exhaust,axiom,
    ! [Xs2: list_P8791008543742759928ropVar] :
      ( ( Xs2 != nil_Pr6119456312066593144ropVar )
     => ~ ! [Ys: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar] :
            ( Xs2
           != ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ Y4 @ nil_Pr6119456312066593144ropVar ) ) ) ) ).

% rev_exhaust
thf(fact_593_rev__exhaust,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o] :
      ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
     => ~ ! [Ys: list_P1665994083459302249pVar_o,Y4: produc8999643685545340131pVar_o] :
            ( Xs2
           != ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ Y4 @ nil_Pr6413368191065476585pVar_o ) ) ) ) ).

% rev_exhaust
thf(fact_594_rev__exhaust,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o] :
      ( ( Xs2 != nil_li1310946081133177199pVar_o )
     => ~ ! [Ys: list_l1255154591254203759pVar_o,Y4: list_P1665994083459302249pVar_o] :
            ( Xs2
           != ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ Y4 @ nil_li1310946081133177199pVar_o ) ) ) ) ).

% rev_exhaust
thf(fact_595_rev__exhaust,axiom,
    ! [Xs2: list_l985434930288970485pVar_o] :
      ( ( Xs2 != nil_li590035233778055285pVar_o )
     => ~ ! [Ys: list_l985434930288970485pVar_o,Y4: list_l1255154591254203759pVar_o] :
            ( Xs2
           != ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ Y4 @ nil_li590035233778055285pVar_o ) ) ) ) ).

% rev_exhaust
thf(fact_596_rev__exhaust,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ~ ! [Ys: list_nat,Y4: nat] :
            ( Xs2
           != ( append_nat @ Ys @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_597_rev__induct,axiom,
    ! [P: list_P723798329145367999ropVar > $o,Xs2: list_P723798329145367999ropVar] :
      ( ( P @ nil_Pr7606595143512524841ropVar )
     => ( ! [X2: propos3600597766819905199ropVar,Xs: list_P723798329145367999ropVar] :
            ( ( P @ Xs )
           => ( P @ ( append7708039934470209236ropVar @ Xs @ ( cons_P6757824865275295865ropVar @ X2 @ nil_Pr7606595143512524841ropVar ) ) ) )
       => ( P @ Xs2 ) ) ) ).

% rev_induct
thf(fact_598_rev__induct,axiom,
    ! [P: list_P8791008543742759928ropVar > $o,Xs2: list_P8791008543742759928ropVar] :
      ( ( P @ nil_Pr6119456312066593144ropVar )
     => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
            ( ( P @ Xs )
           => ( P @ ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) ) )
       => ( P @ Xs2 ) ) ) ).

% rev_induct
thf(fact_599_rev__induct,axiom,
    ! [P: list_P1665994083459302249pVar_o > $o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( P @ nil_Pr6413368191065476585pVar_o )
     => ( ! [X2: produc8999643685545340131pVar_o,Xs: list_P1665994083459302249pVar_o] :
            ( ( P @ Xs )
           => ( P @ ( append1087384436348988414pVar_o @ Xs @ ( cons_P6860203055867761049pVar_o @ X2 @ nil_Pr6413368191065476585pVar_o ) ) ) )
       => ( P @ Xs2 ) ) ) ).

% rev_induct
thf(fact_600_rev__induct,axiom,
    ! [P: list_l1255154591254203759pVar_o > $o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( P @ nil_li1310946081133177199pVar_o )
     => ( ! [X2: list_P1665994083459302249pVar_o,Xs: list_l1255154591254203759pVar_o] :
            ( ( P @ Xs )
           => ( P @ ( append8069948826146371844pVar_o @ Xs @ ( cons_l3452452106213086495pVar_o @ X2 @ nil_li1310946081133177199pVar_o ) ) ) )
       => ( P @ Xs2 ) ) ) ).

% rev_induct
thf(fact_601_rev__induct,axiom,
    ! [P: list_l985434930288970485pVar_o > $o,Xs2: list_l985434930288970485pVar_o] :
      ( ( P @ nil_li590035233778055285pVar_o )
     => ( ! [X2: list_l1255154591254203759pVar_o,Xs: list_l985434930288970485pVar_o] :
            ( ( P @ Xs )
           => ( P @ ( append8959015459476584842pVar_o @ Xs @ ( cons_l7316790070876032037pVar_o @ X2 @ nil_li590035233778055285pVar_o ) ) ) )
       => ( P @ Xs2 ) ) ) ).

% rev_induct
thf(fact_602_rev__induct,axiom,
    ! [P: list_nat > $o,Xs2: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X2: nat,Xs: list_nat] :
            ( ( P @ Xs )
           => ( P @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) ) )
       => ( P @ Xs2 ) ) ) ).

% rev_induct
thf(fact_603_lexord__append__leftI,axiom,
    ! [U: list_l985434930288970485pVar_o,V2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o,X4: list_l985434930288970485pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ U @ V2 ) @ ( lexord4213358897963926754pVar_o @ R ) )
     => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ X4 @ U ) @ ( append8959015459476584842pVar_o @ X4 @ V2 ) ) @ ( lexord4213358897963926754pVar_o @ R ) ) ) ).

% lexord_append_leftI
thf(fact_604_lexord__append__leftI,axiom,
    ! [U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat,X4: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X4 @ U ) @ ( append_nat @ X4 @ V2 ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_leftI
thf(fact_605_lexord__append__leftI,axiom,
    ! [U: list_P723798329145367999ropVar,V2: list_P723798329145367999ropVar,R: set_Pr572148484142614943ropVar,X4: list_P723798329145367999ropVar] :
      ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ U @ V2 ) @ ( lexord5395303177630463356ropVar @ R ) )
     => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ X4 @ U ) @ ( append7708039934470209236ropVar @ X4 @ V2 ) ) @ ( lexord5395303177630463356ropVar @ R ) ) ) ).

% lexord_append_leftI
thf(fact_606_listrel1I2,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar,X4: produc2642885279816024434ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( listre6655839188755619153ropVar @ R ) )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ X4 @ Ys2 ) ) @ ( listre6655839188755619153ropVar @ R ) ) ) ).

% listrel1I2
thf(fact_607_listrel1I2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o,X4: produc8999643685545340131pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Ys2 ) @ ( listre2444918065581779778pVar_o @ R ) )
     => ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ X4 @ Ys2 ) ) @ ( listre2444918065581779778pVar_o @ R ) ) ) ).

% listrel1I2
thf(fact_608_listrel1I2,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o,X4: list_P1665994083459302249pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ Ys2 ) @ ( listre4950778487325574728pVar_o @ R ) )
     => ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ X4 @ Ys2 ) ) @ ( listre4950778487325574728pVar_o @ R ) ) ) ).

% listrel1I2
thf(fact_609_listrel1I2,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o,X4: list_l1255154591254203759pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) )
     => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ X4 @ Ys2 ) ) @ ( listre8274924349387117774pVar_o @ R ) ) ) ).

% listrel1I2
thf(fact_610_listrel1I2,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat,X4: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( listrel1_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ X4 @ Ys2 ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I2
thf(fact_611_not__Nil__listrel1,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Xs2 ) @ ( listre6655839188755619153ropVar @ R ) ) ).

% not_Nil_listrel1
thf(fact_612_not__Nil__listrel1,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Xs2 ) @ ( listre2444918065581779778pVar_o @ R ) ) ).

% not_Nil_listrel1
thf(fact_613_not__Nil__listrel1,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Xs2 ) @ ( listre4950778487325574728pVar_o @ R ) ) ).

% not_Nil_listrel1
thf(fact_614_not__Nil__listrel1,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Xs2 ) @ ( listre8274924349387117774pVar_o @ R ) ) ).

% not_Nil_listrel1
thf(fact_615_not__Nil__listrel1,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs2 ) @ ( listrel1_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_616_not__listrel1__Nil,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ nil_Pr6119456312066593144ropVar ) @ ( listre6655839188755619153ropVar @ R ) ) ).

% not_listrel1_Nil
thf(fact_617_not__listrel1__Nil,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ nil_Pr6413368191065476585pVar_o ) @ ( listre2444918065581779778pVar_o @ R ) ) ).

% not_listrel1_Nil
thf(fact_618_not__listrel1__Nil,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ nil_li1310946081133177199pVar_o ) @ ( listre4950778487325574728pVar_o @ R ) ) ).

% not_listrel1_Nil
thf(fact_619_not__listrel1__Nil,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ nil_li590035233778055285pVar_o ) @ ( listre8274924349387117774pVar_o @ R ) ) ).

% not_listrel1_Nil
thf(fact_620_not__listrel1__Nil,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( listrel1_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_621_lenlex__irreflexive,axiom,
    ! [R: set_Product_prod_o_o,Xs2: list_o] :
      ( ! [X2: $o] :
          ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
     => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Xs2 ) @ ( lenlex_o @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_622_lenlex__irreflexive,axiom,
    ! [R: set_Pr7799659934693882759ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ! [X2: multis3193088007478089820ropVar] :
          ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ X2 ) @ R )
     => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Xs2 ) @ ( lenlex5770254724618424167ropVar @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_623_Nil__lenlex__iff2,axiom,
    ! [Ns: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Ns @ nil_Pr6119456312066593144ropVar ) @ ( lenlex46774552128460413ropVar @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_624_Nil__lenlex__iff2,axiom,
    ! [Ns: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Ns @ nil_Pr6413368191065476585pVar_o ) @ ( lenlex2260575179428607598pVar_o @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_625_Nil__lenlex__iff2,axiom,
    ! [Ns: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Ns @ nil_li1310946081133177199pVar_o ) @ ( lenlex4014889434604538740pVar_o @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_626_Nil__lenlex__iff2,axiom,
    ! [Ns: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Ns @ nil_li590035233778055285pVar_o ) @ ( lenlex3313702719058564090pVar_o @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_627_Nil__lenlex__iff2,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ns @ nil_nat ) @ ( lenlex_nat @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_628_lexord__append__leftD,axiom,
    ! [X4: list_l985434930288970485pVar_o,U: list_l985434930288970485pVar_o,V2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ X4 @ U ) @ ( append8959015459476584842pVar_o @ X4 @ V2 ) ) @ ( lexord4213358897963926754pVar_o @ R ) )
     => ( ! [A3: list_l1255154591254203759pVar_o] :
            ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ A3 @ A3 ) @ R )
       => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ U @ V2 ) @ ( lexord4213358897963926754pVar_o @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_629_lexord__append__leftD,axiom,
    ! [X4: list_nat,U: list_nat,V2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X4 @ U ) @ ( append_nat @ X4 @ V2 ) ) @ ( lexord_nat @ R ) )
     => ( ! [A3: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ A3 ) @ R )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V2 ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_630_lexord__append__leftD,axiom,
    ! [X4: list_P723798329145367999ropVar,U: list_P723798329145367999ropVar,V2: list_P723798329145367999ropVar,R: set_Pr572148484142614943ropVar] :
      ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ X4 @ U ) @ ( append7708039934470209236ropVar @ X4 @ V2 ) ) @ ( lexord5395303177630463356ropVar @ R ) )
     => ( ! [A3: propos3600597766819905199ropVar] :
            ~ ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ A3 @ A3 ) @ R )
       => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ U @ V2 ) @ ( lexord5395303177630463356ropVar @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_631_lexord__append__leftD,axiom,
    ! [X4: list_o,U: list_o,V2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ X4 @ U ) @ ( append_o @ X4 @ V2 ) ) @ ( lexord_o @ R ) )
     => ( ! [A3: $o] :
            ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A3 @ A3 ) @ R )
       => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ U @ V2 ) @ ( lexord_o @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_632_lexord__append__leftD,axiom,
    ! [X4: list_M6740372356978808802ropVar,U: list_M6740372356978808802ropVar,V2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ X4 @ U ) @ ( append2684835082924936439ropVar @ X4 @ V2 ) ) @ ( lexord1374882220397993039ropVar @ R ) )
     => ( ! [A3: multis3193088007478089820ropVar] :
            ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A3 @ A3 ) @ R )
       => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ U @ V2 ) @ ( lexord1374882220397993039ropVar @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_633_lexord__append__rightI,axiom,
    ! [Y3: list_P723798329145367999ropVar,X4: list_P723798329145367999ropVar,R: set_Pr572148484142614943ropVar] :
      ( ? [B4: propos3600597766819905199ropVar,Z2: list_P723798329145367999ropVar] :
          ( Y3
          = ( cons_P6757824865275295865ropVar @ B4 @ Z2 ) )
     => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ X4 @ ( append7708039934470209236ropVar @ X4 @ Y3 ) ) @ ( lexord5395303177630463356ropVar @ R ) ) ) ).

% lexord_append_rightI
thf(fact_634_lexord__append__rightI,axiom,
    ! [Y3: list_P8791008543742759928ropVar,X4: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ? [B4: produc2642885279816024434ropVar,Z2: list_P8791008543742759928ropVar] :
          ( Y3
          = ( cons_P1504208554336013608ropVar @ B4 @ Z2 ) )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ X4 @ ( append4352292717583037453ropVar @ X4 @ Y3 ) ) @ ( lexord832287969131410277ropVar @ R ) ) ) ).

% lexord_append_rightI
thf(fact_635_lexord__append__rightI,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,X4: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ? [B4: produc8999643685545340131pVar_o,Z2: list_P1665994083459302249pVar_o] :
          ( Y3
          = ( cons_P6860203055867761049pVar_o @ B4 @ Z2 ) )
     => ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ ( append1087384436348988414pVar_o @ X4 @ Y3 ) ) @ ( lexord9212727487545282390pVar_o @ R ) ) ) ).

% lexord_append_rightI
thf(fact_636_lexord__append__rightI,axiom,
    ! [Y3: list_l1255154591254203759pVar_o,X4: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ? [B4: list_P1665994083459302249pVar_o,Z2: list_l1255154591254203759pVar_o] :
          ( Y3
          = ( cons_l3452452106213086495pVar_o @ B4 @ Z2 ) )
     => ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ ( append8069948826146371844pVar_o @ X4 @ Y3 ) ) @ ( lexord8675461415035850844pVar_o @ R ) ) ) ).

% lexord_append_rightI
thf(fact_637_lexord__append__rightI,axiom,
    ! [Y3: list_l985434930288970485pVar_o,X4: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ? [B4: list_l1255154591254203759pVar_o,Z2: list_l985434930288970485pVar_o] :
          ( Y3
          = ( cons_l7316790070876032037pVar_o @ B4 @ Z2 ) )
     => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ X4 @ ( append8959015459476584842pVar_o @ X4 @ Y3 ) ) @ ( lexord4213358897963926754pVar_o @ R ) ) ) ).

% lexord_append_rightI
thf(fact_638_lexord__append__rightI,axiom,
    ! [Y3: list_nat,X4: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ? [B4: nat,Z2: list_nat] :
          ( Y3
          = ( cons_nat @ B4 @ Z2 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ ( append_nat @ X4 @ Y3 ) ) @ ( lexord_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_639_Cons__listrel1E2,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) ) @ ( listre6655839188755619153ropVar @ R ) )
     => ( ! [X2: produc2642885279816024434ropVar] :
            ( ( Xs2
              = ( cons_P1504208554336013608ropVar @ X2 @ Ys2 ) )
           => ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_P8791008543742759928ropVar] :
              ( ( Xs2
                = ( cons_P1504208554336013608ropVar @ Y3 @ Zs2 ) )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Zs2 @ Ys2 ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_640_Cons__listrel1E2,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) @ ( listre2444918065581779778pVar_o @ R ) )
     => ( ! [X2: produc8999643685545340131pVar_o] :
            ( ( Xs2
              = ( cons_P6860203055867761049pVar_o @ X2 @ Ys2 ) )
           => ~ ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_P1665994083459302249pVar_o] :
              ( ( Xs2
                = ( cons_P6860203055867761049pVar_o @ Y3 @ Zs2 ) )
             => ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Zs2 @ Ys2 ) @ ( listre2444918065581779778pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_641_Cons__listrel1E2,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) @ ( listre4950778487325574728pVar_o @ R ) )
     => ( ! [X2: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( cons_l3452452106213086495pVar_o @ X2 @ Ys2 ) )
           => ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_l1255154591254203759pVar_o] :
              ( ( Xs2
                = ( cons_l3452452106213086495pVar_o @ Y3 @ Zs2 ) )
             => ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Zs2 @ Ys2 ) @ ( listre4950778487325574728pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_642_Cons__listrel1E2,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) ) @ ( listre8274924349387117774pVar_o @ R ) )
     => ( ! [X2: list_l1255154591254203759pVar_o] :
            ( ( Xs2
              = ( cons_l7316790070876032037pVar_o @ X2 @ Ys2 ) )
           => ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_l985434930288970485pVar_o] :
              ( ( Xs2
                = ( cons_l7316790070876032037pVar_o @ Y3 @ Zs2 ) )
             => ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Zs2 @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_643_Cons__listrel1E2,axiom,
    ! [Xs2: list_nat,Y3: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( listrel1_nat @ R ) )
     => ( ! [X2: nat] :
            ( ( Xs2
              = ( cons_nat @ X2 @ Ys2 ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs2
                = ( cons_nat @ Y3 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_644_Cons__listrel1E2,axiom,
    ! [Xs2: list_o,Y3: $o,Ys2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ ( cons_o @ Y3 @ Ys2 ) ) @ ( listrel1_o @ R ) )
     => ( ! [X2: $o] :
            ( ( Xs2
              = ( cons_o @ X2 @ Ys2 ) )
           => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_o] :
              ( ( Xs2
                = ( cons_o @ Y3 @ Zs2 ) )
             => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Zs2 @ Ys2 ) @ ( listrel1_o @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_645_Cons__listrel1E2,axiom,
    ! [Xs2: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys2 ) ) @ ( listre3736066117146447419ropVar @ R ) )
     => ( ! [X2: multis3193088007478089820ropVar] :
            ( ( Xs2
              = ( cons_M4277272395645347858ropVar @ X2 @ Ys2 ) )
           => ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_M6740372356978808802ropVar] :
              ( ( Xs2
                = ( cons_M4277272395645347858ropVar @ Y3 @ Zs2 ) )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Zs2 @ Ys2 ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_646_Cons__listrel1E1,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ Ys2 ) @ ( listre6655839188755619153ropVar @ R ) )
     => ( ! [Y4: produc2642885279816024434ropVar] :
            ( ( Ys2
              = ( cons_P1504208554336013608ropVar @ Y4 @ Xs2 ) )
           => ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_P8791008543742759928ropVar] :
              ( ( Ys2
                = ( cons_P1504208554336013608ropVar @ X4 @ Zs2 ) )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Zs2 ) @ ( listre6655839188755619153ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_647_Cons__listrel1E1,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ Ys2 ) @ ( listre2444918065581779778pVar_o @ R ) )
     => ( ! [Y4: produc8999643685545340131pVar_o] :
            ( ( Ys2
              = ( cons_P6860203055867761049pVar_o @ Y4 @ Xs2 ) )
           => ~ ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X4 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_P1665994083459302249pVar_o] :
              ( ( Ys2
                = ( cons_P6860203055867761049pVar_o @ X4 @ Zs2 ) )
             => ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Zs2 ) @ ( listre2444918065581779778pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_648_Cons__listrel1E1,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ Ys2 ) @ ( listre4950778487325574728pVar_o @ R ) )
     => ( ! [Y4: list_P1665994083459302249pVar_o] :
            ( ( Ys2
              = ( cons_l3452452106213086495pVar_o @ Y4 @ Xs2 ) )
           => ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_l1255154591254203759pVar_o] :
              ( ( Ys2
                = ( cons_l3452452106213086495pVar_o @ X4 @ Zs2 ) )
             => ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ Zs2 ) @ ( listre4950778487325574728pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_649_Cons__listrel1E1,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ Ys2 ) @ ( listre8274924349387117774pVar_o @ R ) )
     => ( ! [Y4: list_l1255154591254203759pVar_o] :
            ( ( Ys2
              = ( cons_l7316790070876032037pVar_o @ Y4 @ Xs2 ) )
           => ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_l985434930288970485pVar_o] :
              ( ( Ys2
                = ( cons_l7316790070876032037pVar_o @ X4 @ Zs2 ) )
             => ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Zs2 ) @ ( listre8274924349387117774pVar_o @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_650_Cons__listrel1E1,axiom,
    ! [X4: nat,Xs2: list_nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ Ys2 ) @ ( listrel1_nat @ R ) )
     => ( ! [Y4: nat] :
            ( ( Ys2
              = ( cons_nat @ Y4 @ Xs2 ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys2
                = ( cons_nat @ X4 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs2 ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_651_Cons__listrel1E1,axiom,
    ! [X4: $o,Xs2: list_o,Ys2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X4 @ Xs2 ) @ Ys2 ) @ ( listrel1_o @ R ) )
     => ( ! [Y4: $o] :
            ( ( Ys2
              = ( cons_o @ Y4 @ Xs2 ) )
           => ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_o] :
              ( ( Ys2
                = ( cons_o @ X4 @ Zs2 ) )
             => ~ ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Zs2 ) @ ( listrel1_o @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_652_Cons__listrel1E1,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X4 @ Xs2 ) @ Ys2 ) @ ( listre3736066117146447419ropVar @ R ) )
     => ( ! [Y4: multis3193088007478089820ropVar] :
            ( ( Ys2
              = ( cons_M4277272395645347858ropVar @ Y4 @ Xs2 ) )
           => ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y4 ) @ R ) )
       => ~ ! [Zs2: list_M6740372356978808802ropVar] :
              ( ( Ys2
                = ( cons_M4277272395645347858ropVar @ X4 @ Zs2 ) )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Zs2 ) @ ( listre3736066117146447419ropVar @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_653_listrel1I1,axiom,
    ! [X4: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,R: set_Pr3201966908590682119ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ R )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Xs2 ) ) @ ( listre6655839188755619153ropVar @ R ) ) ) ).

% listrel1I1
thf(fact_654_listrel1I1,axiom,
    ! [X4: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,R: set_Pr4574656050966626247pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X4 @ Y3 ) @ R )
     => ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Xs2 ) ) @ ( listre2444918065581779778pVar_o @ R ) ) ) ).

% listrel1I1
thf(fact_655_listrel1I1,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,R: set_Pr8855275280618702407pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ Y3 ) @ R )
     => ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Xs2 ) ) @ ( listre4950778487325574728pVar_o @ R ) ) ) ).

% listrel1I1
thf(fact_656_listrel1I1,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o,R: set_Pr3139130378708280007pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ Y3 ) @ R )
     => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Xs2 ) ) @ ( listre8274924349387117774pVar_o @ R ) ) ) ).

% listrel1I1
thf(fact_657_listrel1I1,axiom,
    ! [X4: nat,Y3: nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Xs2 ) ) @ ( listrel1_nat @ R ) ) ) ).

% listrel1I1
thf(fact_658_listrel1I1,axiom,
    ! [X4: $o,Y3: $o,R: set_Product_prod_o_o,Xs2: list_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R )
     => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X4 @ Xs2 ) @ ( cons_o @ Y3 @ Xs2 ) ) @ ( listrel1_o @ R ) ) ) ).

% listrel1I1
thf(fact_659_listrel1I1,axiom,
    ! [X4: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R: set_Pr7799659934693882759ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R )
     => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X4 @ Xs2 ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Xs2 ) ) @ ( listre3736066117146447419ropVar @ R ) ) ) ).

% listrel1I1
thf(fact_660_encoder_OencodeGammaCond_Opelims,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_neg: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_impl: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,X4: multis3193088007478089820ropVar,Xa: multis3193088007478089820ropVar,Xb: $o,Xc: $o,Y3: list_l1255154591254203759pVar_o] :
      ( ( multis1610358401135282399pVar_o @ Eval @ 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 )
     => ( ( ( multis6017319109170748100ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ X4 @ Xa @ Xb @ Xc )
          = Y3 )
       => ( ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ Xb @ Xc ) ) ) )
         => ( ( Xb
             => ( Xc
               => ( ( Y3 = Enc_True )
                 => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $true @ $true ) ) ) ) ) ) )
           => ( ( ~ Xb
               => ( ~ Xc
                 => ( ( Y3
                      = ( Enc_neg @ X4 ) )
                   => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $false @ $false ) ) ) ) ) ) )
             => ( ( ~ Xb
                 => ( Xc
                   => ( ( Y3
                        = ( Enc_var_impl @ X4 @ Xa ) )
                     => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $false @ $true ) ) ) ) ) ) )
               => ~ ( Xb
                   => ( ~ Xc
                     => ( ( Y3
                          = ( Enc_not_and @ X4 @ Xa ) )
                       => ~ ( accp_P8208393750683106630od_o_o @ multis3344368520953930393ropVar @ ( produc4931073445691921453od_o_o @ X4 @ ( produc9033520932313852770od_o_o @ Xa @ ( product_Pair_o_o @ $true @ $false ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% encoder.encodeGammaCond.pelims
thf(fact_661_maps__simps_I1_J,axiom,
    ! [F2: nat > list_nat,X4: nat,Xs2: list_nat] :
      ( ( maps_nat_nat @ F2 @ ( cons_nat @ X4 @ Xs2 ) )
      = ( append_nat @ ( F2 @ X4 ) @ ( maps_nat_nat @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_662_maps__simps_I1_J,axiom,
    ! [F2: nat > list_P723798329145367999ropVar,X4: nat,Xs2: list_nat] :
      ( ( maps_n2360302221952498754ropVar @ F2 @ ( cons_nat @ X4 @ Xs2 ) )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( maps_n2360302221952498754ropVar @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_663_maps__simps_I1_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > list_nat,X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( maps_P3684901262545219122_o_nat @ F2 @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
      = ( append_nat @ ( F2 @ X4 ) @ ( maps_P3684901262545219122_o_nat @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_664_maps__simps_I1_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > list_P723798329145367999ropVar,X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( maps_P2265790774297355281ropVar @ F2 @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( maps_P2265790774297355281ropVar @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_665_maps__simps_I1_J,axiom,
    ! [F2: list_P1665994083459302249pVar_o > list_nat,X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( maps_l127622714395944376_o_nat @ F2 @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
      = ( append_nat @ ( F2 @ X4 ) @ ( maps_l127622714395944376_o_nat @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_666_maps__simps_I1_J,axiom,
    ! [F2: produc2642885279816024434ropVar > list_nat,X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( maps_P2882123423910056385ar_nat @ F2 @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
      = ( append_nat @ ( F2 @ X4 ) @ ( maps_P2882123423910056385ar_nat @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_667_maps__simps_I1_J,axiom,
    ! [F2: list_P1665994083459302249pVar_o > list_P723798329145367999ropVar,X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( maps_l3134634397932293271ropVar @ F2 @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( maps_l3134634397932293271ropVar @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_668_maps__simps_I1_J,axiom,
    ! [F2: list_l1255154591254203759pVar_o > list_nat,X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( maps_l865014085272113342_o_nat @ F2 @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
      = ( append_nat @ ( F2 @ X4 ) @ ( maps_l865014085272113342_o_nat @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_669_maps__simps_I1_J,axiom,
    ! [F2: nat > list_l985434930288970485pVar_o,X4: nat,Xs2: list_nat] :
      ( ( maps_n2783302837639436700pVar_o @ F2 @ ( cons_nat @ X4 @ Xs2 ) )
      = ( append8959015459476584842pVar_o @ ( F2 @ X4 ) @ ( maps_n2783302837639436700pVar_o @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_670_maps__simps_I1_J,axiom,
    ! [F2: produc2642885279816024434ropVar > list_P723798329145367999ropVar,X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( maps_P4109729386520289824ropVar @ F2 @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
      = ( append7708039934470209236ropVar @ ( F2 @ X4 ) @ ( maps_P4109729386520289824ropVar @ F2 @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_671_butlast__snoc,axiom,
    ! [Xs2: list_P723798329145367999ropVar,X4: propos3600597766819905199ropVar] :
      ( ( butlas7628565150644405725ropVar @ ( append7708039934470209236ropVar @ Xs2 @ ( cons_P6757824865275295865ropVar @ X4 @ nil_Pr7606595143512524841ropVar ) ) )
      = Xs2 ) ).

% butlast_snoc
thf(fact_672_butlast__snoc,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
      ( ( butlas5098402688846288580ropVar @ ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) )
      = Xs2 ) ).

% butlast_snoc
thf(fact_673_butlast__snoc,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o] :
      ( ( butlas7924156930066306357pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) ) )
      = Xs2 ) ).

% butlast_snoc
thf(fact_674_butlast__snoc,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o] :
      ( ( butlas54256187331189435pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) ) )
      = Xs2 ) ).

% butlast_snoc
thf(fact_675_butlast__snoc,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o] :
      ( ( butlas1161285084838040513pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) ) )
      = Xs2 ) ).

% butlast_snoc
thf(fact_676_butlast__snoc,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( butlast_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) )
      = Xs2 ) ).

% butlast_snoc
thf(fact_677_concat__eq__append__conv,axiom,
    ! [Xss2: list_l1644708917474426447ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ( ( concat4611077035641296302ropVar @ Xss2 )
        = ( append7708039934470209236ropVar @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_li6797326493501628985ropVar )
         => ( ( Ys2 = nil_Pr7606595143512524841ropVar )
            & ( Zs = nil_Pr7606595143512524841ropVar ) ) )
        & ( ( Xss2 != nil_li6797326493501628985ropVar )
         => ? [Xss1: list_l1644708917474426447ropVar,Xs3: list_P723798329145367999ropVar,Xs4: list_P723798329145367999ropVar,Xss22: list_l1644708917474426447ropVar] :
              ( ( Xss2
                = ( append2757213247338973028ropVar @ Xss1 @ ( cons_l2454487807130495625ropVar @ ( append7708039934470209236ropVar @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append7708039934470209236ropVar @ ( concat4611077035641296302ropVar @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append7708039934470209236ropVar @ Xs4 @ ( concat4611077035641296302ropVar @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_678_concat__eq__append__conv,axiom,
    ! [Xss2: list_l3415868495287753214ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
      ( ( ( concat1830633707478507187ropVar @ Xss2 )
        = ( append4352292717583037453ropVar @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_li2679230897895549182ropVar )
         => ( ( Ys2 = nil_Pr6119456312066593144ropVar )
            & ( Zs = nil_Pr6119456312066593144ropVar ) ) )
        & ( ( Xss2 != nil_li2679230897895549182ropVar )
         => ? [Xss1: list_l3415868495287753214ropVar,Xs3: list_P8791008543742759928ropVar,Xs4: list_P8791008543742759928ropVar,Xss22: list_l3415868495287753214ropVar] :
              ( ( Xss2
                = ( append4316637162825710867ropVar @ Xss1 @ ( cons_l5280711761816980654ropVar @ ( append4352292717583037453ropVar @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append4352292717583037453ropVar @ ( concat1830633707478507187ropVar @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append4352292717583037453ropVar @ Xs4 @ ( concat1830633707478507187ropVar @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_679_concat__eq__append__conv,axiom,
    ! [Xss2: list_l6341520962544925691pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ( ( concat8126406963686051632pVar_o @ Xss2 )
        = ( append8959015459476584842pVar_o @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_li5579170522155219707pVar_o )
         => ( ( Ys2 = nil_li590035233778055285pVar_o )
            & ( Zs = nil_li590035233778055285pVar_o ) ) )
        & ( ( Xss2 != nil_li5579170522155219707pVar_o )
         => ? [Xss1: list_l6341520962544925691pVar_o,Xs3: list_l985434930288970485pVar_o,Xs4: list_l985434930288970485pVar_o,Xss22: list_l6341520962544925691pVar_o] :
              ( ( Xss2
                = ( append9215492216909589904pVar_o @ Xss1 @ ( cons_l8096070148660426411pVar_o @ ( append8959015459476584842pVar_o @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append8959015459476584842pVar_o @ ( concat8126406963686051632pVar_o @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append8959015459476584842pVar_o @ Xs4 @ ( concat8126406963686051632pVar_o @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_680_concat__eq__append__conv,axiom,
    ! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_list_nat )
         => ( ( Ys2 = nil_nat )
            & ( Zs = nil_nat ) ) )
        & ( ( Xss2 != nil_list_nat )
         => ? [Xss1: list_list_nat,Xs3: list_nat,Xs4: list_nat,Xss22: list_list_nat] :
              ( ( Xss2
                = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append_nat @ Xs4 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_681_concat__eq__append__conv,axiom,
    ! [Xss2: list_l1255154591254203759pVar_o,Ys2: list_P1665994083459302249pVar_o,Zs: list_P1665994083459302249pVar_o] :
      ( ( ( concat7668843100863076260pVar_o @ Xss2 )
        = ( append1087384436348988414pVar_o @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_li1310946081133177199pVar_o )
         => ( ( Ys2 = nil_Pr6413368191065476585pVar_o )
            & ( Zs = nil_Pr6413368191065476585pVar_o ) ) )
        & ( ( Xss2 != nil_li1310946081133177199pVar_o )
         => ? [Xss1: list_l1255154591254203759pVar_o,Xs3: list_P1665994083459302249pVar_o,Xs4: list_P1665994083459302249pVar_o,Xss22: list_l1255154591254203759pVar_o] :
              ( ( Xss2
                = ( append8069948826146371844pVar_o @ Xss1 @ ( cons_l3452452106213086495pVar_o @ ( append1087384436348988414pVar_o @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append1087384436348988414pVar_o @ ( concat7668843100863076260pVar_o @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append1087384436348988414pVar_o @ Xs4 @ ( concat7668843100863076260pVar_o @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_682_concat__eq__append__conv,axiom,
    ! [Xss2: list_l985434930288970485pVar_o,Ys2: list_l1255154591254203759pVar_o,Zs: list_l1255154591254203759pVar_o] :
      ( ( ( concat8635278358871234986pVar_o @ Xss2 )
        = ( append8069948826146371844pVar_o @ Ys2 @ Zs ) )
      = ( ( ( Xss2 = nil_li590035233778055285pVar_o )
         => ( ( Ys2 = nil_li1310946081133177199pVar_o )
            & ( Zs = nil_li1310946081133177199pVar_o ) ) )
        & ( ( Xss2 != nil_li590035233778055285pVar_o )
         => ? [Xss1: list_l985434930288970485pVar_o,Xs3: list_l1255154591254203759pVar_o,Xs4: list_l1255154591254203759pVar_o,Xss22: list_l985434930288970485pVar_o] :
              ( ( Xss2
                = ( append8959015459476584842pVar_o @ Xss1 @ ( cons_l7316790070876032037pVar_o @ ( append8069948826146371844pVar_o @ Xs3 @ Xs4 ) @ Xss22 ) ) )
              & ( Ys2
                = ( append8069948826146371844pVar_o @ ( concat8635278358871234986pVar_o @ Xss1 ) @ Xs3 ) )
              & ( Zs
                = ( append8069948826146371844pVar_o @ Xs4 @ ( concat8635278358871234986pVar_o @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_683_lexord__same__pref__iff,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar,R: set_Pr572148484142614943ropVar] :
      ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) @ ( append7708039934470209236ropVar @ Xs2 @ Zs ) ) @ ( lexord5395303177630463356ropVar @ R ) )
      = ( ? [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
            & ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ X3 @ X3 ) @ R ) )
        | ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Ys2 @ Zs ) @ ( lexord5395303177630463356ropVar @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_684_lexord__same__pref__iff,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) @ ( append4352292717583037453ropVar @ Xs2 @ Zs ) ) @ ( lexord832287969131410277ropVar @ R ) )
      = ( ? [X3: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X3 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
            & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X3 @ X3 ) @ R ) )
        | ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Ys2 @ Zs ) @ ( lexord832287969131410277ropVar @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_685_lexord__same__pref__iff,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,Ys2: list_l1644708917474426447ropVar,Zs: list_l1644708917474426447ropVar,R: set_Pr3948510271449578431ropVar] :
      ( ( member8130401662325377320ropVar @ ( produc7125445329903452983ropVar @ ( append2757213247338973028ropVar @ Xs2 @ Ys2 ) @ ( append2757213247338973028ropVar @ Xs2 @ Zs ) ) @ ( lexord1232427612563343884ropVar @ R ) )
      = ( ? [X3: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
            & ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ X3 @ X3 ) @ R ) )
        | ( member8130401662325377320ropVar @ ( produc7125445329903452983ropVar @ Ys2 @ Zs ) @ ( lexord1232427612563343884ropVar @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_686_lexord__same__pref__iff,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) @ ( append8959015459476584842pVar_o @ Xs2 @ Zs ) ) @ ( lexord4213358897963926754pVar_o @ R ) )
      = ( ? [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
            & ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X3 @ X3 ) @ R ) )
        | ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Ys2 @ Zs ) @ ( lexord4213358897963926754pVar_o @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_687_lexord__same__pref__iff,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,Ys2: list_l6341520962544925691pVar_o,Zs: list_l6341520962544925691pVar_o,R: set_Pr7475081812231853383pVar_o] :
      ( ( member5210069903039637520pVar_o @ ( produc504187628453117527pVar_o @ ( append9215492216909589904pVar_o @ Xs2 @ Ys2 ) @ ( append9215492216909589904pVar_o @ Xs2 @ Zs ) ) @ ( lexord8641875953490177256pVar_o @ R ) )
      = ( ? [X3: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
            & ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ X3 @ X3 ) @ R ) )
        | ( member5210069903039637520pVar_o @ ( produc504187628453117527pVar_o @ Ys2 @ Zs ) @ ( lexord8641875953490177256pVar_o @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_688_lexord__same__pref__iff,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lexord_nat @ R ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R ) )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_689_lexord__same__pref__iff,axiom,
    ! [Xs2: list_o,Ys2: list_o,Zs: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ Ys2 ) @ ( append_o @ Xs2 @ Zs ) ) @ ( lexord_o @ R ) )
      = ( ? [X3: $o] :
            ( ( member_o @ X3 @ ( set_o2 @ Xs2 ) )
            & ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ X3 ) @ R ) )
        | ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs ) @ ( lexord_o @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_690_lexord__same__pref__iff,axiom,
    ! [Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ Xs2 @ Ys2 ) @ ( append2684835082924936439ropVar @ Xs2 @ Zs ) ) @ ( lexord1374882220397993039ropVar @ R ) )
      = ( ? [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
            & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X3 @ X3 ) @ R ) )
        | ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Ys2 @ Zs ) @ ( lexord1374882220397993039ropVar @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_691_last__snoc,axiom,
    ! [Xs2: list_P723798329145367999ropVar,X4: propos3600597766819905199ropVar] :
      ( ( last_P3456477928032711888ropVar @ ( append7708039934470209236ropVar @ Xs2 @ ( cons_P6757824865275295865ropVar @ X4 @ nil_Pr7606595143512524841ropVar ) ) )
      = X4 ) ).

% last_snoc
thf(fact_692_last__snoc,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
      ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) )
      = X4 ) ).

% last_snoc
thf(fact_693_last__snoc,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o] :
      ( ( last_P703351216231693570pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) ) )
      = X4 ) ).

% last_snoc
thf(fact_694_last__snoc,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o] :
      ( ( last_l9069138231800379400pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) ) )
      = X4 ) ).

% last_snoc
thf(fact_695_last__snoc,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o] :
      ( ( last_l30396060243306126pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) ) )
      = X4 ) ).

% last_snoc
thf(fact_696_last__snoc,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( last_nat @ ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) )
      = X4 ) ).

% last_snoc
thf(fact_697_lex__append__leftD,axiom,
    ! [R: set_Pr3139130378708280007pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ! [X2: list_l1255154591254203759pVar_o] :
          ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X2 @ X2 ) @ R )
     => ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) @ ( append8959015459476584842pVar_o @ Xs2 @ Zs ) ) @ ( lex_li5300212533296449753pVar_o @ R ) )
       => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Ys2 @ Zs ) @ ( lex_li5300212533296449753pVar_o @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_698_lex__append__leftD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ! [X2: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_699_lex__append__leftD,axiom,
    ! [R: set_Pr572148484142614943ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ! [X2: propos3600597766819905199ropVar] :
          ~ ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ X2 @ X2 ) @ R )
     => ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) @ ( append7708039934470209236ropVar @ Xs2 @ Zs ) ) @ ( lex_Pr8947373434141532869ropVar @ R ) )
       => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Ys2 @ Zs ) @ ( lex_Pr8947373434141532869ropVar @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_700_lex__append__leftD,axiom,
    ! [R: set_Product_prod_o_o,Xs2: list_o,Ys2: list_o,Zs: list_o] :
      ( ! [X2: $o] :
          ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
     => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ Ys2 ) @ ( append_o @ Xs2 @ Zs ) ) @ ( lex_o @ R ) )
       => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs ) @ ( lex_o @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_701_lex__append__leftD,axiom,
    ! [R: set_Pr7799659934693882759ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
      ( ! [X2: multis3193088007478089820ropVar] :
          ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ X2 ) @ R )
     => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ Xs2 @ Ys2 ) @ ( append2684835082924936439ropVar @ Xs2 @ Zs ) ) @ ( lex_Mu6218539406383542982ropVar @ R ) )
       => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Ys2 @ Zs ) @ ( lex_Mu6218539406383542982ropVar @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_702_lex__append__left__iff,axiom,
    ! [R: set_Pr3139130378708280007pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ! [X2: list_l1255154591254203759pVar_o] :
          ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X2 @ X2 ) @ R )
     => ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) @ ( append8959015459476584842pVar_o @ Xs2 @ Zs ) ) @ ( lex_li5300212533296449753pVar_o @ R ) )
        = ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Ys2 @ Zs ) @ ( lex_li5300212533296449753pVar_o @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_703_lex__append__left__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs2: list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ! [X2: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ X2 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_704_lex__append__left__iff,axiom,
    ! [R: set_Pr572148484142614943ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ! [X2: propos3600597766819905199ropVar] :
          ~ ( member8566377481635068392ropVar @ ( produc2531927925724607479ropVar @ X2 @ X2 ) @ R )
     => ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) @ ( append7708039934470209236ropVar @ Xs2 @ Zs ) ) @ ( lex_Pr8947373434141532869ropVar @ R ) )
        = ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Ys2 @ Zs ) @ ( lex_Pr8947373434141532869ropVar @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_705_lex__append__left__iff,axiom,
    ! [R: set_Product_prod_o_o,Xs2: list_o,Ys2: list_o,Zs: list_o] :
      ( ! [X2: $o] :
          ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ X2 ) @ R )
     => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( append_o @ Xs2 @ Ys2 ) @ ( append_o @ Xs2 @ Zs ) ) @ ( lex_o @ R ) )
        = ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs ) @ ( lex_o @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_706_lex__append__left__iff,axiom,
    ! [R: set_Pr7799659934693882759ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
      ( ! [X2: multis3193088007478089820ropVar] :
          ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ X2 ) @ R )
     => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ Xs2 @ Ys2 ) @ ( append2684835082924936439ropVar @ Xs2 @ Zs ) ) @ ( lex_Mu6218539406383542982ropVar @ R ) )
        = ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Ys2 @ Zs ) @ ( lex_Mu6218539406383542982ropVar @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_707_rotate1_Osimps_I2_J,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( rotate7455714379189909404ropVar @ ( cons_P6757824865275295865ropVar @ X4 @ Xs2 ) )
      = ( append7708039934470209236ropVar @ Xs2 @ ( cons_P6757824865275295865ropVar @ X4 @ nil_Pr7606595143512524841ropVar ) ) ) ).

% rotate1.simps(2)
thf(fact_708_rotate1_Osimps_I2_J,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( rotate7816721141519739205ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
      = ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) ) ).

% rotate1.simps(2)
thf(fact_709_rotate1_Osimps_I2_J,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( rotate3041276922869945654pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
      = ( append1087384436348988414pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) ) ) ).

% rotate1.simps(2)
thf(fact_710_rotate1_Osimps_I2_J,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( rotate4853913444325910076pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
      = ( append8069948826146371844pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) ) ) ).

% rotate1.simps(2)
thf(fact_711_rotate1_Osimps_I2_J,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( rotate8080253354684623554pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
      = ( append8959015459476584842pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) ) ) ).

% rotate1.simps(2)
thf(fact_712_rotate1_Osimps_I2_J,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( rotate1_nat @ ( cons_nat @ X4 @ Xs2 ) )
      = ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) ) ) ).

% rotate1.simps(2)
thf(fact_713_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l3415868495287753214ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( concat1830633707478507187ropVar @ Xss2 ) )
      = ( ! [X3: list_P8791008543742759928ropVar] :
            ( ( member6236339133593825953ropVar @ X3 @ ( set_li6509544621384488717ropVar @ Xss2 ) )
           => ( X3 = nil_Pr6119456312066593144ropVar ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_714_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l1255154591254203759pVar_o] :
      ( ( nil_Pr6413368191065476585pVar_o
        = ( concat7668843100863076260pVar_o @ Xss2 ) )
      = ( ! [X3: list_P1665994083459302249pVar_o] :
            ( ( member7042831661676915602pVar_o @ X3 @ ( set_li5575011780610867710pVar_o @ Xss2 ) )
           => ( X3 = nil_Pr6413368191065476585pVar_o ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_715_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_list_nat] :
      ( ( nil_nat
        = ( concat_nat @ Xss2 ) )
      = ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
           => ( X3 = nil_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_716_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l1644708917474426447ropVar] :
      ( ( nil_Pr7606595143512524841ropVar
        = ( concat4611077035641296302ropVar @ Xss2 ) )
      = ( ! [X3: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Xss2 ) )
           => ( X3 = nil_Pr7606595143512524841ropVar ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_717_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l985434930288970485pVar_o] :
      ( ( nil_li1310946081133177199pVar_o
        = ( concat8635278358871234986pVar_o @ Xss2 ) )
      = ( ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Xss2 ) )
           => ( X3 = nil_li1310946081133177199pVar_o ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_718_Nil__eq__concat__conv,axiom,
    ! [Xss2: list_l6341520962544925691pVar_o] :
      ( ( nil_li590035233778055285pVar_o
        = ( concat8126406963686051632pVar_o @ Xss2 ) )
      = ( ! [X3: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Xss2 ) )
           => ( X3 = nil_li590035233778055285pVar_o ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_719_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l3415868495287753214ropVar] :
      ( ( ( concat1830633707478507187ropVar @ Xss2 )
        = nil_Pr6119456312066593144ropVar )
      = ( ! [X3: list_P8791008543742759928ropVar] :
            ( ( member6236339133593825953ropVar @ X3 @ ( set_li6509544621384488717ropVar @ Xss2 ) )
           => ( X3 = nil_Pr6119456312066593144ropVar ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_720_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l1255154591254203759pVar_o] :
      ( ( ( concat7668843100863076260pVar_o @ Xss2 )
        = nil_Pr6413368191065476585pVar_o )
      = ( ! [X3: list_P1665994083459302249pVar_o] :
            ( ( member7042831661676915602pVar_o @ X3 @ ( set_li5575011780610867710pVar_o @ Xss2 ) )
           => ( X3 = nil_Pr6413368191065476585pVar_o ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_721_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = nil_nat )
      = ( ! [X3: list_nat] :
            ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss2 ) )
           => ( X3 = nil_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_722_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l1644708917474426447ropVar] :
      ( ( ( concat4611077035641296302ropVar @ Xss2 )
        = nil_Pr7606595143512524841ropVar )
      = ( ! [X3: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Xss2 ) )
           => ( X3 = nil_Pr7606595143512524841ropVar ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_723_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l985434930288970485pVar_o] :
      ( ( ( concat8635278358871234986pVar_o @ Xss2 )
        = nil_li1310946081133177199pVar_o )
      = ( ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Xss2 ) )
           => ( X3 = nil_li1310946081133177199pVar_o ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_724_concat__eq__Nil__conv,axiom,
    ! [Xss2: list_l6341520962544925691pVar_o] :
      ( ( ( concat8126406963686051632pVar_o @ Xss2 )
        = nil_li590035233778055285pVar_o )
      = ( ! [X3: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Xss2 ) )
           => ( X3 = nil_li590035233778055285pVar_o ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_725_rotate1__is__Nil__conv,axiom,
    ! [Xs2: list_P8791008543742759928ropVar] :
      ( ( ( rotate7816721141519739205ropVar @ Xs2 )
        = nil_Pr6119456312066593144ropVar )
      = ( Xs2 = nil_Pr6119456312066593144ropVar ) ) ).

% rotate1_is_Nil_conv
thf(fact_726_rotate1__is__Nil__conv,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o] :
      ( ( ( rotate3041276922869945654pVar_o @ Xs2 )
        = nil_Pr6413368191065476585pVar_o )
      = ( Xs2 = nil_Pr6413368191065476585pVar_o ) ) ).

% rotate1_is_Nil_conv
thf(fact_727_rotate1__is__Nil__conv,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o] :
      ( ( ( rotate4853913444325910076pVar_o @ Xs2 )
        = nil_li1310946081133177199pVar_o )
      = ( Xs2 = nil_li1310946081133177199pVar_o ) ) ).

% rotate1_is_Nil_conv
thf(fact_728_rotate1__is__Nil__conv,axiom,
    ! [Xs2: list_l985434930288970485pVar_o] :
      ( ( ( rotate8080253354684623554pVar_o @ Xs2 )
        = nil_li590035233778055285pVar_o )
      = ( Xs2 = nil_li590035233778055285pVar_o ) ) ).

% rotate1_is_Nil_conv
thf(fact_729_rotate1__is__Nil__conv,axiom,
    ! [Xs2: list_nat] :
      ( ( ( rotate1_nat @ Xs2 )
        = nil_nat )
      = ( Xs2 = nil_nat ) ) ).

% rotate1_is_Nil_conv
thf(fact_730_set__rotate1,axiom,
    ! [Xs2: list_P8791008543742759928ropVar] :
      ( ( set_Pr3946994337432900359ropVar @ ( rotate7816721141519739205ropVar @ Xs2 ) )
      = ( set_Pr3946994337432900359ropVar @ Xs2 ) ) ).

% set_rotate1
thf(fact_731_set__rotate1,axiom,
    ! [Xs2: list_l1644708917474426447ropVar] :
      ( ( set_li8604685205718225898ropVar @ ( rotate3763493056403258412ropVar @ Xs2 ) )
      = ( set_li8604685205718225898ropVar @ Xs2 ) ) ).

% set_rotate1
thf(fact_732_set__rotate1,axiom,
    ! [Xs2: list_l985434930288970485pVar_o] :
      ( ( set_li3549506505754852740pVar_o @ ( rotate8080253354684623554pVar_o @ Xs2 ) )
      = ( set_li3549506505754852740pVar_o @ Xs2 ) ) ).

% set_rotate1
thf(fact_733_set__rotate1,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o] :
      ( ( set_li4369674465895099530pVar_o @ ( rotate6464143096120402632pVar_o @ Xs2 ) )
      = ( set_li4369674465895099530pVar_o @ Xs2 ) ) ).

% set_rotate1
thf(fact_734_set__rotate1,axiom,
    ! [Xs2: list_nat] :
      ( ( set_nat2 @ ( rotate1_nat @ Xs2 ) )
      = ( set_nat2 @ Xs2 ) ) ).

% set_rotate1
thf(fact_735_in__set__insert,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ( ( insert215289443796111736ropVar @ X4 @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_736_in__set__insert,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ( ( insert8896757123549915715od_o_o @ X4 @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_737_in__set__insert,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ( ( insert214406736284816014ropVar @ X4 @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_738_in__set__insert,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ( ( insert6886557058002715043ropVar @ X4 @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_739_in__set__insert,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ( ( insert7135957767216192651pVar_o @ X4 @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_740_in__set__insert,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ( ( insert6473862492288621585pVar_o @ X4 @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_741_in__set__insert,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ( insert_nat @ X4 @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_742_last__appendR,axiom,
    ! [Ys2: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( Ys2 != nil_Pr7606595143512524841ropVar )
     => ( ( last_P3456477928032711888ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) )
        = ( last_P3456477928032711888ropVar @ Ys2 ) ) ) ).

% last_appendR
thf(fact_743_last__appendR,axiom,
    ! [Ys2: list_P8791008543742759928ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( Ys2 != nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) )
        = ( last_P5278392773003955985ropVar @ Ys2 ) ) ) ).

% last_appendR
thf(fact_744_last__appendR,axiom,
    ! [Ys2: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( Ys2 != nil_Pr6413368191065476585pVar_o )
     => ( ( last_P703351216231693570pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) )
        = ( last_P703351216231693570pVar_o @ Ys2 ) ) ) ).

% last_appendR
thf(fact_745_last__appendR,axiom,
    ! [Ys2: list_l1255154591254203759pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( Ys2 != nil_li1310946081133177199pVar_o )
     => ( ( last_l9069138231800379400pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) )
        = ( last_l9069138231800379400pVar_o @ Ys2 ) ) ) ).

% last_appendR
thf(fact_746_last__appendR,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( Ys2 != nil_li590035233778055285pVar_o )
     => ( ( last_l30396060243306126pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
        = ( last_l30396060243306126pVar_o @ Ys2 ) ) ) ).

% last_appendR
thf(fact_747_last__appendR,axiom,
    ! [Ys2: list_nat,Xs2: list_nat] :
      ( ( Ys2 != nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
        = ( last_nat @ Ys2 ) ) ) ).

% last_appendR
thf(fact_748_last__appendL,axiom,
    ! [Ys2: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( Ys2 = nil_Pr7606595143512524841ropVar )
     => ( ( last_P3456477928032711888ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) )
        = ( last_P3456477928032711888ropVar @ Xs2 ) ) ) ).

% last_appendL
thf(fact_749_last__appendL,axiom,
    ! [Ys2: list_P8791008543742759928ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( Ys2 = nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) )
        = ( last_P5278392773003955985ropVar @ Xs2 ) ) ) ).

% last_appendL
thf(fact_750_last__appendL,axiom,
    ! [Ys2: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( Ys2 = nil_Pr6413368191065476585pVar_o )
     => ( ( last_P703351216231693570pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) )
        = ( last_P703351216231693570pVar_o @ Xs2 ) ) ) ).

% last_appendL
thf(fact_751_last__appendL,axiom,
    ! [Ys2: list_l1255154591254203759pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( Ys2 = nil_li1310946081133177199pVar_o )
     => ( ( last_l9069138231800379400pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) )
        = ( last_l9069138231800379400pVar_o @ Xs2 ) ) ) ).

% last_appendL
thf(fact_752_last__appendL,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( Ys2 = nil_li590035233778055285pVar_o )
     => ( ( last_l30396060243306126pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
        = ( last_l30396060243306126pVar_o @ Xs2 ) ) ) ).

% last_appendL
thf(fact_753_last__appendL,axiom,
    ! [Ys2: list_nat,Xs2: list_nat] :
      ( ( Ys2 = nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
        = ( last_nat @ Xs2 ) ) ) ).

% last_appendL
thf(fact_754_not__in__set__insert,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ( ( insert215289443796111736ropVar @ X4 @ Xs2 )
        = ( cons_M4277272395645347858ropVar @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_755_not__in__set__insert,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ( ( insert8896757123549915715od_o_o @ X4 @ Xs2 )
        = ( cons_P8766293264717362397od_o_o @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_756_not__in__set__insert,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ( ( insert6886557058002715043ropVar @ X4 @ Xs2 )
        = ( cons_l2454487807130495625ropVar @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_757_not__in__set__insert,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ( ( insert6473862492288621585pVar_o @ X4 @ Xs2 )
        = ( cons_l8096070148660426411pVar_o @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_758_not__in__set__insert,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ( ( insert214406736284816014ropVar @ X4 @ Xs2 )
        = ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_759_not__in__set__insert,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ~ ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
     => ( ( insert5031086936754616319pVar_o @ X4 @ Xs2 )
        = ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_760_not__in__set__insert,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ~ ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
     => ( ( insert8144517603577680005pVar_o @ X4 @ Xs2 )
        = ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_761_not__in__set__insert,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ( ( insert7135957767216192651pVar_o @ X4 @ Xs2 )
        = ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_762_not__in__set__insert,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ~ ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ( insert_nat @ X4 @ Xs2 )
        = ( cons_nat @ X4 @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_763_concat__append,axiom,
    ! [Xs2: list_list_nat,Ys2: list_list_nat] :
      ( ( concat_nat @ ( append_list_nat @ Xs2 @ Ys2 ) )
      = ( append_nat @ ( concat_nat @ Xs2 ) @ ( concat_nat @ Ys2 ) ) ) ).

% concat_append
thf(fact_764_concat__append,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,Ys2: list_l6341520962544925691pVar_o] :
      ( ( concat8126406963686051632pVar_o @ ( append9215492216909589904pVar_o @ Xs2 @ Ys2 ) )
      = ( append8959015459476584842pVar_o @ ( concat8126406963686051632pVar_o @ Xs2 ) @ ( concat8126406963686051632pVar_o @ Ys2 ) ) ) ).

% concat_append
thf(fact_765_concat__append,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( concat8635278358871234986pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
      = ( append8069948826146371844pVar_o @ ( concat8635278358871234986pVar_o @ Xs2 ) @ ( concat8635278358871234986pVar_o @ Ys2 ) ) ) ).

% concat_append
thf(fact_766_concat__append,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,Ys2: list_l1644708917474426447ropVar] :
      ( ( concat4611077035641296302ropVar @ ( append2757213247338973028ropVar @ Xs2 @ Ys2 ) )
      = ( append7708039934470209236ropVar @ ( concat4611077035641296302ropVar @ Xs2 ) @ ( concat4611077035641296302ropVar @ Ys2 ) ) ) ).

% concat_append
thf(fact_767_append__butlast__last__id,axiom,
    ! [Xs2: list_P723798329145367999ropVar] :
      ( ( Xs2 != nil_Pr7606595143512524841ropVar )
     => ( ( append7708039934470209236ropVar @ ( butlas7628565150644405725ropVar @ Xs2 ) @ ( cons_P6757824865275295865ropVar @ ( last_P3456477928032711888ropVar @ Xs2 ) @ nil_Pr7606595143512524841ropVar ) )
        = Xs2 ) ) ).

% append_butlast_last_id
thf(fact_768_append__butlast__last__id,axiom,
    ! [Xs2: list_P8791008543742759928ropVar] :
      ( ( Xs2 != nil_Pr6119456312066593144ropVar )
     => ( ( append4352292717583037453ropVar @ ( butlas5098402688846288580ropVar @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ ( last_P5278392773003955985ropVar @ Xs2 ) @ nil_Pr6119456312066593144ropVar ) )
        = Xs2 ) ) ).

% append_butlast_last_id
thf(fact_769_append__butlast__last__id,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o] :
      ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
     => ( ( append1087384436348988414pVar_o @ ( butlas7924156930066306357pVar_o @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ ( last_P703351216231693570pVar_o @ Xs2 ) @ nil_Pr6413368191065476585pVar_o ) )
        = Xs2 ) ) ).

% append_butlast_last_id
thf(fact_770_append__butlast__last__id,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o] :
      ( ( Xs2 != nil_li1310946081133177199pVar_o )
     => ( ( append8069948826146371844pVar_o @ ( butlas54256187331189435pVar_o @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ ( last_l9069138231800379400pVar_o @ Xs2 ) @ nil_li1310946081133177199pVar_o ) )
        = Xs2 ) ) ).

% append_butlast_last_id
thf(fact_771_append__butlast__last__id,axiom,
    ! [Xs2: list_l985434930288970485pVar_o] :
      ( ( Xs2 != nil_li590035233778055285pVar_o )
     => ( ( append8959015459476584842pVar_o @ ( butlas1161285084838040513pVar_o @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ ( last_l30396060243306126pVar_o @ Xs2 ) @ nil_li590035233778055285pVar_o ) )
        = Xs2 ) ) ).

% append_butlast_last_id
thf(fact_772_append__butlast__last__id,axiom,
    ! [Xs2: list_nat] :
      ( ( Xs2 != nil_nat )
     => ( ( append_nat @ ( butlast_nat @ Xs2 ) @ ( cons_nat @ ( last_nat @ Xs2 ) @ nil_nat ) )
        = Xs2 ) ) ).

% append_butlast_last_id
thf(fact_773_in__set__butlast__appendI,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
      ( ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( butlas2616725375618095534ropVar @ Xs2 ) ) )
        | ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( butlas2616725375618095534ropVar @ Ys2 ) ) ) )
     => ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( butlas2616725375618095534ropVar @ ( append2684835082924936439ropVar @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_774_in__set__butlast__appendI,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o,Ys2: list_P4002435161011370285od_o_o] :
      ( ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ ( butlas6593100082972991097od_o_o @ Xs2 ) ) )
        | ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ ( butlas6593100082972991097od_o_o @ Ys2 ) ) ) )
     => ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ ( butlas6593100082972991097od_o_o @ ( append2614242729457001410od_o_o @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_775_in__set__butlast__appendI,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ ( butlas7628565150644405725ropVar @ Xs2 ) ) )
        | ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ ( butlas7628565150644405725ropVar @ Ys2 ) ) ) )
     => ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ ( butlas7628565150644405725ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_776_in__set__butlast__appendI,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ ( butlas5098402688846288580ropVar @ Xs2 ) ) )
        | ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ ( butlas5098402688846288580ropVar @ Ys2 ) ) ) )
     => ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ ( butlas5098402688846288580ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_777_in__set__butlast__appendI,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar,Ys2: list_l1644708917474426447ropVar] :
      ( ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ ( butlas4686232535363925485ropVar @ Xs2 ) ) )
        | ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ ( butlas4686232535363925485ropVar @ Ys2 ) ) ) )
     => ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ ( butlas4686232535363925485ropVar @ ( append2757213247338973028ropVar @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_778_in__set__butlast__appendI,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ ( butlas1161285084838040513pVar_o @ Xs2 ) ) )
        | ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ ( butlas1161285084838040513pVar_o @ Ys2 ) ) ) )
     => ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ ( butlas1161285084838040513pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_779_in__set__butlast__appendI,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o,Ys2: list_l6341520962544925691pVar_o] :
      ( ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ ( butlas4398145439059492935pVar_o @ Xs2 ) ) )
        | ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ ( butlas4398145439059492935pVar_o @ Ys2 ) ) ) )
     => ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ ( butlas4398145439059492935pVar_o @ ( append9215492216909589904pVar_o @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_780_in__set__butlast__appendI,axiom,
    ! [X4: nat,Xs2: list_nat,Ys2: list_nat] :
      ( ( ( member_nat @ X4 @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
        | ( member_nat @ X4 @ ( set_nat2 @ ( butlast_nat @ Ys2 ) ) ) )
     => ( member_nat @ X4 @ ( set_nat2 @ ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_781_last__in__set,axiom,
    ! [As2: list_M6740372356978808802ropVar] :
      ( ( As2 != nil_Mu4132399263956683874ropVar )
     => ( member8421201611525043589ropVar @ ( last_M5158658609589141499ropVar @ As2 ) @ ( set_Mu146256361055586289ropVar @ As2 ) ) ) ).

% last_in_set
thf(fact_782_last__in__set,axiom,
    ! [As2: list_P4002435161011370285od_o_o] :
      ( ( As2 != nil_Product_prod_o_o )
     => ( member7466972457876170832od_o_o @ ( last_P7913258175222030022od_o_o @ As2 ) @ ( set_Product_prod_o_o2 @ As2 ) ) ) ).

% last_in_set
thf(fact_783_last__in__set,axiom,
    ! [As2: list_P1665994083459302249pVar_o] :
      ( ( As2 != nil_Pr6413368191065476585pVar_o )
     => ( member1729440322722938252pVar_o @ ( last_P703351216231693570pVar_o @ As2 ) @ ( set_Pr2838981755425241592pVar_o @ As2 ) ) ) ).

% last_in_set
thf(fact_784_last__in__set,axiom,
    ! [As2: list_l1255154591254203759pVar_o] :
      ( ( As2 != nil_li1310946081133177199pVar_o )
     => ( member7042831661676915602pVar_o @ ( last_l9069138231800379400pVar_o @ As2 ) @ ( set_li5575011780610867710pVar_o @ As2 ) ) ) ).

% last_in_set
thf(fact_785_last__in__set,axiom,
    ! [As2: list_P8791008543742759928ropVar] :
      ( ( As2 != nil_Pr6119456312066593144ropVar )
     => ( member9210975342120487067ropVar @ ( last_P5278392773003955985ropVar @ As2 ) @ ( set_Pr3946994337432900359ropVar @ As2 ) ) ) ).

% last_in_set
thf(fact_786_last__in__set,axiom,
    ! [As2: list_l1644708917474426447ropVar] :
      ( ( As2 != nil_li6797326493501628985ropVar )
     => ( member7375955847912140118ropVar @ ( last_l7140692191916183392ropVar @ As2 ) @ ( set_li8604685205718225898ropVar @ As2 ) ) ) ).

% last_in_set
thf(fact_787_last__in__set,axiom,
    ! [As2: list_l985434930288970485pVar_o] :
      ( ( As2 != nil_li590035233778055285pVar_o )
     => ( member6267526602072755480pVar_o @ ( last_l30396060243306126pVar_o @ As2 ) @ ( set_li3549506505754852740pVar_o @ As2 ) ) ) ).

% last_in_set
thf(fact_788_last__in__set,axiom,
    ! [As2: list_l6341520962544925691pVar_o] :
      ( ( As2 != nil_li5579170522155219707pVar_o )
     => ( member9046334641704116766pVar_o @ ( last_l6170502382654556308pVar_o @ As2 ) @ ( set_li4369674465895099530pVar_o @ As2 ) ) ) ).

% last_in_set
thf(fact_789_last__in__set,axiom,
    ! [As2: list_nat] :
      ( ( As2 != nil_nat )
     => ( member_nat @ ( last_nat @ As2 ) @ ( set_nat2 @ As2 ) ) ) ).

% last_in_set
thf(fact_790_in__set__butlastD,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( butlas2616725375618095534ropVar @ Xs2 ) ) )
     => ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_791_in__set__butlastD,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ ( butlas6593100082972991097od_o_o @ Xs2 ) ) )
     => ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_792_in__set__butlastD,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ ( butlas5098402688846288580ropVar @ Xs2 ) ) )
     => ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_793_in__set__butlastD,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ ( butlas4686232535363925485ropVar @ Xs2 ) ) )
     => ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_794_in__set__butlastD,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ ( butlas1161285084838040513pVar_o @ Xs2 ) ) )
     => ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_795_in__set__butlastD,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ ( butlas4398145439059492935pVar_o @ Xs2 ) ) )
     => ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_796_in__set__butlastD,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ ( butlast_nat @ Xs2 ) ) )
     => ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_797_encoder_OencodeGammaCond_Ocong,axiom,
    multis6017319109170748100ropVar = multis6017319109170748100ropVar ).

% encoder.encodeGammaCond.cong
thf(fact_798_list_Oset__intros_I2_J,axiom,
    ! [Y3: multis3193088007478089820ropVar,X22: list_M6740372356978808802ropVar,X21: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ X22 ) )
     => ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_799_list_Oset__intros_I2_J,axiom,
    ! [Y3: product_prod_o_o,X22: list_P4002435161011370285od_o_o,X21: product_prod_o_o] :
      ( ( member7466972457876170832od_o_o @ Y3 @ ( set_Product_prod_o_o2 @ X22 ) )
     => ( member7466972457876170832od_o_o @ Y3 @ ( set_Product_prod_o_o2 @ ( cons_P8766293264717362397od_o_o @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_800_list_Oset__intros_I2_J,axiom,
    ! [Y3: list_P723798329145367999ropVar,X22: list_l1644708917474426447ropVar,X21: list_P723798329145367999ropVar] :
      ( ( member7375955847912140118ropVar @ Y3 @ ( set_li8604685205718225898ropVar @ X22 ) )
     => ( member7375955847912140118ropVar @ Y3 @ ( set_li8604685205718225898ropVar @ ( cons_l2454487807130495625ropVar @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_801_list_Oset__intros_I2_J,axiom,
    ! [Y3: list_l985434930288970485pVar_o,X22: list_l6341520962544925691pVar_o,X21: list_l985434930288970485pVar_o] :
      ( ( member9046334641704116766pVar_o @ Y3 @ ( set_li4369674465895099530pVar_o @ X22 ) )
     => ( member9046334641704116766pVar_o @ Y3 @ ( set_li4369674465895099530pVar_o @ ( cons_l8096070148660426411pVar_o @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_802_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc2642885279816024434ropVar,X22: list_P8791008543742759928ropVar,X21: produc2642885279816024434ropVar] :
      ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ X22 ) )
     => ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_803_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc8999643685545340131pVar_o,X22: list_P1665994083459302249pVar_o,X21: produc8999643685545340131pVar_o] :
      ( ( member1729440322722938252pVar_o @ Y3 @ ( set_Pr2838981755425241592pVar_o @ X22 ) )
     => ( member1729440322722938252pVar_o @ Y3 @ ( set_Pr2838981755425241592pVar_o @ ( cons_P6860203055867761049pVar_o @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_804_list_Oset__intros_I2_J,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,X22: list_l1255154591254203759pVar_o,X21: list_P1665994083459302249pVar_o] :
      ( ( member7042831661676915602pVar_o @ Y3 @ ( set_li5575011780610867710pVar_o @ X22 ) )
     => ( member7042831661676915602pVar_o @ Y3 @ ( set_li5575011780610867710pVar_o @ ( cons_l3452452106213086495pVar_o @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_805_list_Oset__intros_I2_J,axiom,
    ! [Y3: list_l1255154591254203759pVar_o,X22: list_l985434930288970485pVar_o,X21: list_l1255154591254203759pVar_o] :
      ( ( member6267526602072755480pVar_o @ Y3 @ ( set_li3549506505754852740pVar_o @ X22 ) )
     => ( member6267526602072755480pVar_o @ Y3 @ ( set_li3549506505754852740pVar_o @ ( cons_l7316790070876032037pVar_o @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_806_list_Oset__intros_I2_J,axiom,
    ! [Y3: nat,X22: list_nat,X21: nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ X22 ) )
     => ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ) ).

% list.set_intros(2)
thf(fact_807_list_Oset__intros_I1_J,axiom,
    ! [X21: multis3193088007478089820ropVar,X22: list_M6740372356978808802ropVar] : ( member8421201611525043589ropVar @ X21 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_808_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_o_o,X22: list_P4002435161011370285od_o_o] : ( member7466972457876170832od_o_o @ X21 @ ( set_Product_prod_o_o2 @ ( cons_P8766293264717362397od_o_o @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_809_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P723798329145367999ropVar,X22: list_l1644708917474426447ropVar] : ( member7375955847912140118ropVar @ X21 @ ( set_li8604685205718225898ropVar @ ( cons_l2454487807130495625ropVar @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_810_list_Oset__intros_I1_J,axiom,
    ! [X21: list_l985434930288970485pVar_o,X22: list_l6341520962544925691pVar_o] : ( member9046334641704116766pVar_o @ X21 @ ( set_li4369674465895099530pVar_o @ ( cons_l8096070148660426411pVar_o @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_811_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2642885279816024434ropVar,X22: list_P8791008543742759928ropVar] : ( member9210975342120487067ropVar @ X21 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_812_list_Oset__intros_I1_J,axiom,
    ! [X21: produc8999643685545340131pVar_o,X22: list_P1665994083459302249pVar_o] : ( member1729440322722938252pVar_o @ X21 @ ( set_Pr2838981755425241592pVar_o @ ( cons_P6860203055867761049pVar_o @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_813_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P1665994083459302249pVar_o,X22: list_l1255154591254203759pVar_o] : ( member7042831661676915602pVar_o @ X21 @ ( set_li5575011780610867710pVar_o @ ( cons_l3452452106213086495pVar_o @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_814_list_Oset__intros_I1_J,axiom,
    ! [X21: list_l1255154591254203759pVar_o,X22: list_l985434930288970485pVar_o] : ( member6267526602072755480pVar_o @ X21 @ ( set_li3549506505754852740pVar_o @ ( cons_l7316790070876032037pVar_o @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_815_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X22: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X22 ) ) ) ).

% list.set_intros(1)
thf(fact_816_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_817_list_Oset__cases,axiom,
    ! [E: product_prod_o_o,A: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ E @ ( set_Product_prod_o_o2 @ A ) )
     => ( ! [Z22: list_P4002435161011370285od_o_o] :
            ( A
           != ( cons_P8766293264717362397od_o_o @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_o_o,Z22: list_P4002435161011370285od_o_o] :
              ( ( A
                = ( cons_P8766293264717362397od_o_o @ Z1 @ Z22 ) )
             => ~ ( member7466972457876170832od_o_o @ E @ ( set_Product_prod_o_o2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_818_list_Oset__cases,axiom,
    ! [E: list_P723798329145367999ropVar,A: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ E @ ( set_li8604685205718225898ropVar @ A ) )
     => ( ! [Z22: list_l1644708917474426447ropVar] :
            ( A
           != ( cons_l2454487807130495625ropVar @ E @ Z22 ) )
       => ~ ! [Z1: list_P723798329145367999ropVar,Z22: list_l1644708917474426447ropVar] :
              ( ( A
                = ( cons_l2454487807130495625ropVar @ Z1 @ Z22 ) )
             => ~ ( member7375955847912140118ropVar @ E @ ( set_li8604685205718225898ropVar @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_819_list_Oset__cases,axiom,
    ! [E: list_l985434930288970485pVar_o,A: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ E @ ( set_li4369674465895099530pVar_o @ A ) )
     => ( ! [Z22: list_l6341520962544925691pVar_o] :
            ( A
           != ( cons_l8096070148660426411pVar_o @ E @ Z22 ) )
       => ~ ! [Z1: list_l985434930288970485pVar_o,Z22: list_l6341520962544925691pVar_o] :
              ( ( A
                = ( cons_l8096070148660426411pVar_o @ Z1 @ Z22 ) )
             => ~ ( member9046334641704116766pVar_o @ E @ ( set_li4369674465895099530pVar_o @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_820_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_821_list_Oset__cases,axiom,
    ! [E: produc8999643685545340131pVar_o,A: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ E @ ( set_Pr2838981755425241592pVar_o @ A ) )
     => ( ! [Z22: list_P1665994083459302249pVar_o] :
            ( A
           != ( cons_P6860203055867761049pVar_o @ E @ Z22 ) )
       => ~ ! [Z1: produc8999643685545340131pVar_o,Z22: list_P1665994083459302249pVar_o] :
              ( ( A
                = ( cons_P6860203055867761049pVar_o @ Z1 @ Z22 ) )
             => ~ ( member1729440322722938252pVar_o @ E @ ( set_Pr2838981755425241592pVar_o @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_822_list_Oset__cases,axiom,
    ! [E: list_P1665994083459302249pVar_o,A: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ E @ ( set_li5575011780610867710pVar_o @ A ) )
     => ( ! [Z22: list_l1255154591254203759pVar_o] :
            ( A
           != ( cons_l3452452106213086495pVar_o @ E @ Z22 ) )
       => ~ ! [Z1: list_P1665994083459302249pVar_o,Z22: list_l1255154591254203759pVar_o] :
              ( ( A
                = ( cons_l3452452106213086495pVar_o @ Z1 @ Z22 ) )
             => ~ ( member7042831661676915602pVar_o @ E @ ( set_li5575011780610867710pVar_o @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_823_list_Oset__cases,axiom,
    ! [E: list_l1255154591254203759pVar_o,A: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ E @ ( set_li3549506505754852740pVar_o @ A ) )
     => ( ! [Z22: list_l985434930288970485pVar_o] :
            ( A
           != ( cons_l7316790070876032037pVar_o @ E @ Z22 ) )
       => ~ ! [Z1: list_l1255154591254203759pVar_o,Z22: list_l985434930288970485pVar_o] :
              ( ( A
                = ( cons_l7316790070876032037pVar_o @ Z1 @ Z22 ) )
             => ~ ( member6267526602072755480pVar_o @ E @ ( set_li3549506505754852740pVar_o @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_824_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_825_set__ConsD,axiom,
    ! [Y3: multis3193088007478089820ropVar,X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_826_set__ConsD,axiom,
    ! [Y3: product_prod_o_o,X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ Y3 @ ( set_Product_prod_o_o2 @ ( cons_P8766293264717362397od_o_o @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member7466972457876170832od_o_o @ Y3 @ ( set_Product_prod_o_o2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_827_set__ConsD,axiom,
    ! [Y3: list_P723798329145367999ropVar,X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ Y3 @ ( set_li8604685205718225898ropVar @ ( cons_l2454487807130495625ropVar @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member7375955847912140118ropVar @ Y3 @ ( set_li8604685205718225898ropVar @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_828_set__ConsD,axiom,
    ! [Y3: list_l985434930288970485pVar_o,X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ Y3 @ ( set_li4369674465895099530pVar_o @ ( cons_l8096070148660426411pVar_o @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member9046334641704116766pVar_o @ Y3 @ ( set_li4369674465895099530pVar_o @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_829_set__ConsD,axiom,
    ! [Y3: produc2642885279816024434ropVar,X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_830_set__ConsD,axiom,
    ! [Y3: produc8999643685545340131pVar_o,X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ Y3 @ ( set_Pr2838981755425241592pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member1729440322722938252pVar_o @ Y3 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_831_set__ConsD,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ Y3 @ ( set_li5575011780610867710pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member7042831661676915602pVar_o @ Y3 @ ( set_li5575011780610867710pVar_o @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_832_set__ConsD,axiom,
    ! [Y3: list_l1255154591254203759pVar_o,X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ Y3 @ ( set_li3549506505754852740pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member6267526602072755480pVar_o @ Y3 @ ( set_li3549506505754852740pVar_o @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_833_set__ConsD,axiom,
    ! [Y3: nat,X4: nat,Xs2: list_nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X4 @ Xs2 ) ) )
     => ( ( Y3 = X4 )
        | ( member_nat @ Y3 @ ( set_nat2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_834_butlast_Osimps_I1_J,axiom,
    ( ( butlas5098402688846288580ropVar @ nil_Pr6119456312066593144ropVar )
    = nil_Pr6119456312066593144ropVar ) ).

% butlast.simps(1)
thf(fact_835_butlast_Osimps_I1_J,axiom,
    ( ( butlas7924156930066306357pVar_o @ nil_Pr6413368191065476585pVar_o )
    = nil_Pr6413368191065476585pVar_o ) ).

% butlast.simps(1)
thf(fact_836_butlast_Osimps_I1_J,axiom,
    ( ( butlas54256187331189435pVar_o @ nil_li1310946081133177199pVar_o )
    = nil_li1310946081133177199pVar_o ) ).

% butlast.simps(1)
thf(fact_837_butlast_Osimps_I1_J,axiom,
    ( ( butlas1161285084838040513pVar_o @ nil_li590035233778055285pVar_o )
    = nil_li590035233778055285pVar_o ) ).

% butlast.simps(1)
thf(fact_838_butlast_Osimps_I1_J,axiom,
    ( ( butlast_nat @ nil_nat )
    = nil_nat ) ).

% butlast.simps(1)
thf(fact_839_subseqs__refl,axiom,
    ! [Xs2: list_P723798329145367999ropVar] : ( member7375955847912140118ropVar @ Xs2 @ ( set_li8604685205718225898ropVar @ ( subseq8000409699144225946ropVar @ Xs2 ) ) ) ).

% subseqs_refl
thf(fact_840_subseqs__refl,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o] : ( member6267526602072755480pVar_o @ Xs2 @ ( set_li3549506505754852740pVar_o @ ( subseq2663478470768632766pVar_o @ Xs2 ) ) ) ).

% subseqs_refl
thf(fact_841_subseqs__refl,axiom,
    ! [Xs2: list_l985434930288970485pVar_o] : ( member9046334641704116766pVar_o @ Xs2 @ ( set_li4369674465895099530pVar_o @ ( subseq7541546836001800004pVar_o @ Xs2 ) ) ) ).

% subseqs_refl
thf(fact_842_rotate1_Osimps_I1_J,axiom,
    ( ( rotate7816721141519739205ropVar @ nil_Pr6119456312066593144ropVar )
    = nil_Pr6119456312066593144ropVar ) ).

% rotate1.simps(1)
thf(fact_843_rotate1_Osimps_I1_J,axiom,
    ( ( rotate3041276922869945654pVar_o @ nil_Pr6413368191065476585pVar_o )
    = nil_Pr6413368191065476585pVar_o ) ).

% rotate1.simps(1)
thf(fact_844_rotate1_Osimps_I1_J,axiom,
    ( ( rotate4853913444325910076pVar_o @ nil_li1310946081133177199pVar_o )
    = nil_li1310946081133177199pVar_o ) ).

% rotate1.simps(1)
thf(fact_845_rotate1_Osimps_I1_J,axiom,
    ( ( rotate8080253354684623554pVar_o @ nil_li590035233778055285pVar_o )
    = nil_li590035233778055285pVar_o ) ).

% rotate1.simps(1)
thf(fact_846_rotate1_Osimps_I1_J,axiom,
    ( ( rotate1_nat @ nil_nat )
    = nil_nat ) ).

% rotate1.simps(1)
thf(fact_847_snoc__eq__iff__butlast,axiom,
    ! [Xs2: list_P723798329145367999ropVar,X4: propos3600597766819905199ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( ( append7708039934470209236ropVar @ Xs2 @ ( cons_P6757824865275295865ropVar @ X4 @ nil_Pr7606595143512524841ropVar ) )
        = Ys2 )
      = ( ( Ys2 != nil_Pr7606595143512524841ropVar )
        & ( ( butlas7628565150644405725ropVar @ Ys2 )
          = Xs2 )
        & ( ( last_P3456477928032711888ropVar @ Ys2 )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_848_snoc__eq__iff__butlast,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) )
        = Ys2 )
      = ( ( Ys2 != nil_Pr6119456312066593144ropVar )
        & ( ( butlas5098402688846288580ropVar @ Ys2 )
          = Xs2 )
        & ( ( last_P5278392773003955985ropVar @ Ys2 )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_849_snoc__eq__iff__butlast,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( ( append1087384436348988414pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) )
        = Ys2 )
      = ( ( Ys2 != nil_Pr6413368191065476585pVar_o )
        & ( ( butlas7924156930066306357pVar_o @ Ys2 )
          = Xs2 )
        & ( ( last_P703351216231693570pVar_o @ Ys2 )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_850_snoc__eq__iff__butlast,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( ( append8069948826146371844pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) )
        = Ys2 )
      = ( ( Ys2 != nil_li1310946081133177199pVar_o )
        & ( ( butlas54256187331189435pVar_o @ Ys2 )
          = Xs2 )
        & ( ( last_l9069138231800379400pVar_o @ Ys2 )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_851_snoc__eq__iff__butlast,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( ( append8959015459476584842pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) )
        = Ys2 )
      = ( ( Ys2 != nil_li590035233778055285pVar_o )
        & ( ( butlas1161285084838040513pVar_o @ Ys2 )
          = Xs2 )
        & ( ( last_l30396060243306126pVar_o @ Ys2 )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_852_snoc__eq__iff__butlast,axiom,
    ! [Xs2: list_nat,X4: nat,Ys2: list_nat] :
      ( ( ( append_nat @ Xs2 @ ( cons_nat @ X4 @ nil_nat ) )
        = Ys2 )
      = ( ( Ys2 != nil_nat )
        & ( ( butlast_nat @ Ys2 )
          = Xs2 )
        & ( ( last_nat @ Ys2 )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_853_list__ex1__iff,axiom,
    ( list_e4360549196154024869ropVar
    = ( ^ [P4: multis3193088007478089820ropVar > $o,Xs3: list_M6740372356978808802ropVar] :
        ? [X3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs3 ) )
          & ( P4 @ X3 )
          & ! [Y2: multis3193088007478089820ropVar] :
              ( ( ( member8421201611525043589ropVar @ Y2 @ ( set_Mu146256361055586289ropVar @ Xs3 ) )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_854_list__ex1__iff,axiom,
    ( list_e8715582737019594352od_o_o
    = ( ^ [P4: product_prod_o_o > $o,Xs3: list_P4002435161011370285od_o_o] :
        ? [X3: product_prod_o_o] :
          ( ( member7466972457876170832od_o_o @ X3 @ ( set_Product_prod_o_o2 @ Xs3 ) )
          & ( P4 @ X3 )
          & ! [Y2: product_prod_o_o] :
              ( ( ( member7466972457876170832od_o_o @ Y2 @ ( set_Product_prod_o_o2 @ Xs3 ) )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_855_list__ex1__iff,axiom,
    ( list_e1423168564539623099ropVar
    = ( ^ [P4: produc2642885279816024434ropVar > $o,Xs3: list_P8791008543742759928ropVar] :
        ? [X3: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X3 @ ( set_Pr3946994337432900359ropVar @ Xs3 ) )
          & ( P4 @ X3 )
          & ! [Y2: produc2642885279816024434ropVar] :
              ( ( ( member9210975342120487067ropVar @ Y2 @ ( set_Pr3946994337432900359ropVar @ Xs3 ) )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_856_list__ex1__iff,axiom,
    ( list_e8541863376151676726ropVar
    = ( ^ [P4: list_P723798329145367999ropVar > $o,Xs3: list_l1644708917474426447ropVar] :
        ? [X3: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Xs3 ) )
          & ( P4 @ X3 )
          & ! [Y2: list_P723798329145367999ropVar] :
              ( ( ( member7375955847912140118ropVar @ Y2 @ ( set_li8604685205718225898ropVar @ Xs3 ) )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_857_list__ex1__iff,axiom,
    ( list_e5196591910180150072pVar_o
    = ( ^ [P4: list_l1255154591254203759pVar_o > $o,Xs3: list_l985434930288970485pVar_o] :
        ? [X3: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Xs3 ) )
          & ( P4 @ X3 )
          & ! [Y2: list_l1255154591254203759pVar_o] :
              ( ( ( member6267526602072755480pVar_o @ Y2 @ ( set_li3549506505754852740pVar_o @ Xs3 ) )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_858_list__ex1__iff,axiom,
    ( list_e115186782557003838pVar_o
    = ( ^ [P4: list_l985434930288970485pVar_o > $o,Xs3: list_l6341520962544925691pVar_o] :
        ? [X3: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Xs3 ) )
          & ( P4 @ X3 )
          & ! [Y2: list_l985434930288970485pVar_o] :
              ( ( ( member9046334641704116766pVar_o @ Y2 @ ( set_li4369674465895099530pVar_o @ Xs3 ) )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_859_list__ex1__iff,axiom,
    ( list_ex1_nat
    = ( ^ [P4: nat > $o,Xs3: list_nat] :
        ? [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs3 ) )
          & ( P4 @ X3 )
          & ! [Y2: nat] :
              ( ( ( member_nat @ Y2 @ ( set_nat2 @ Xs3 ) )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_860_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ( ? [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_P723798329145367999ropVar,X3: propos3600597766819905199ropVar] :
            ( ? [Zs3: list_P723798329145367999ropVar] :
                ( Xs2
                = ( append7708039934470209236ropVar @ Ys3 @ ( cons_P6757824865275295865ropVar @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: propos3600597766819905199ropVar] :
                ( ( member1796676888578033094ropVar @ Y2 @ ( set_Pr342157451780195930ropVar @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_861_split__list__first__prop__iff,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ( ? [X3: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l1644708917474426447ropVar,X3: list_P723798329145367999ropVar] :
            ( ? [Zs3: list_l1644708917474426447ropVar] :
                ( Xs2
                = ( append2757213247338973028ropVar @ Ys3 @ ( cons_l2454487807130495625ropVar @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_P723798329145367999ropVar] :
                ( ( member7375955847912140118ropVar @ Y2 @ ( set_li8604685205718225898ropVar @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_862_split__list__first__prop__iff,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ( ? [X3: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l6341520962544925691pVar_o,X3: list_l985434930288970485pVar_o] :
            ( ? [Zs3: list_l6341520962544925691pVar_o] :
                ( Xs2
                = ( append9215492216909589904pVar_o @ Ys3 @ ( cons_l8096070148660426411pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_l985434930288970485pVar_o] :
                ( ( member9046334641704116766pVar_o @ Y2 @ ( set_li4369674465895099530pVar_o @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_863_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ? [X3: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X3 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar] :
            ( ? [Zs3: list_P8791008543742759928ropVar] :
                ( Xs2
                = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: produc2642885279816024434ropVar] :
                ( ( member9210975342120487067ropVar @ Y2 @ ( set_Pr3946994337432900359ropVar @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_864_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ( ? [X3: produc8999643685545340131pVar_o] :
            ( ( member1729440322722938252pVar_o @ X3 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_P1665994083459302249pVar_o,X3: produc8999643685545340131pVar_o] :
            ( ? [Zs3: list_P1665994083459302249pVar_o] :
                ( Xs2
                = ( append1087384436348988414pVar_o @ Ys3 @ ( cons_P6860203055867761049pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: produc8999643685545340131pVar_o] :
                ( ( member1729440322722938252pVar_o @ Y2 @ ( set_Pr2838981755425241592pVar_o @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_865_split__list__first__prop__iff,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ( ? [X3: list_P1665994083459302249pVar_o] :
            ( ( member7042831661676915602pVar_o @ X3 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l1255154591254203759pVar_o,X3: list_P1665994083459302249pVar_o] :
            ( ? [Zs3: list_l1255154591254203759pVar_o] :
                ( Xs2
                = ( append8069948826146371844pVar_o @ Ys3 @ ( cons_l3452452106213086495pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_P1665994083459302249pVar_o] :
                ( ( member7042831661676915602pVar_o @ Y2 @ ( set_li5575011780610867710pVar_o @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_866_split__list__first__prop__iff,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ( ? [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l985434930288970485pVar_o,X3: list_l1255154591254203759pVar_o] :
            ( ? [Zs3: list_l985434930288970485pVar_o] :
                ( Xs2
                = ( append8959015459476584842pVar_o @ Ys3 @ ( cons_l7316790070876032037pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_l1255154591254203759pVar_o] :
                ( ( member6267526602072755480pVar_o @ Y2 @ ( set_li3549506505754852740pVar_o @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_867_split__list__first__prop__iff,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_nat,X3: nat] :
            ( ? [Zs3: list_nat] :
                ( Xs2
                = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: nat] :
                ( ( member_nat @ Y2 @ ( set_nat2 @ Ys3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_868_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ( ? [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_P723798329145367999ropVar,X3: propos3600597766819905199ropVar,Zs3: list_P723798329145367999ropVar] :
            ( ( Xs2
              = ( append7708039934470209236ropVar @ Ys3 @ ( cons_P6757824865275295865ropVar @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: propos3600597766819905199ropVar] :
                ( ( member1796676888578033094ropVar @ Y2 @ ( set_Pr342157451780195930ropVar @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_869_split__list__last__prop__iff,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ( ? [X3: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l1644708917474426447ropVar,X3: list_P723798329145367999ropVar,Zs3: list_l1644708917474426447ropVar] :
            ( ( Xs2
              = ( append2757213247338973028ropVar @ Ys3 @ ( cons_l2454487807130495625ropVar @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_P723798329145367999ropVar] :
                ( ( member7375955847912140118ropVar @ Y2 @ ( set_li8604685205718225898ropVar @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_870_split__list__last__prop__iff,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ( ? [X3: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l6341520962544925691pVar_o,X3: list_l985434930288970485pVar_o,Zs3: list_l6341520962544925691pVar_o] :
            ( ( Xs2
              = ( append9215492216909589904pVar_o @ Ys3 @ ( cons_l8096070148660426411pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_l985434930288970485pVar_o] :
                ( ( member9046334641704116766pVar_o @ Y2 @ ( set_li4369674465895099530pVar_o @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_871_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ? [X3: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X3 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs2
              = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: produc2642885279816024434ropVar] :
                ( ( member9210975342120487067ropVar @ Y2 @ ( set_Pr3946994337432900359ropVar @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_872_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ( ? [X3: produc8999643685545340131pVar_o] :
            ( ( member1729440322722938252pVar_o @ X3 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_P1665994083459302249pVar_o,X3: produc8999643685545340131pVar_o,Zs3: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( append1087384436348988414pVar_o @ Ys3 @ ( cons_P6860203055867761049pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: produc8999643685545340131pVar_o] :
                ( ( member1729440322722938252pVar_o @ Y2 @ ( set_Pr2838981755425241592pVar_o @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_873_split__list__last__prop__iff,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ( ? [X3: list_P1665994083459302249pVar_o] :
            ( ( member7042831661676915602pVar_o @ X3 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l1255154591254203759pVar_o,X3: list_P1665994083459302249pVar_o,Zs3: list_l1255154591254203759pVar_o] :
            ( ( Xs2
              = ( append8069948826146371844pVar_o @ Ys3 @ ( cons_l3452452106213086495pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_P1665994083459302249pVar_o] :
                ( ( member7042831661676915602pVar_o @ Y2 @ ( set_li5575011780610867710pVar_o @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_874_split__list__last__prop__iff,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ( ? [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_l985434930288970485pVar_o,X3: list_l1255154591254203759pVar_o,Zs3: list_l985434930288970485pVar_o] :
            ( ( Xs2
              = ( append8959015459476584842pVar_o @ Ys3 @ ( cons_l7316790070876032037pVar_o @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: list_l1255154591254203759pVar_o] :
                ( ( member6267526602072755480pVar_o @ Y2 @ ( set_li3549506505754852740pVar_o @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_875_split__list__last__prop__iff,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs2 ) )
            & ( P @ X3 ) ) )
      = ( ? [Ys3: list_nat,X3: nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs3 ) ) )
            & ( P @ X3 )
            & ! [Y2: nat] :
                ( ( member_nat @ Y2 @ ( set_nat2 @ Zs3 ) )
               => ~ ( P @ Y2 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_876_in__set__conv__decomp__first,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
      = ( ? [Ys3: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
            ( ( Xs2
              = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_877_in__set__conv__decomp__first,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
      = ( ? [Ys3: list_P4002435161011370285od_o_o,Zs3: list_P4002435161011370285od_o_o] :
            ( ( Xs2
              = ( append2614242729457001410od_o_o @ Ys3 @ ( cons_P8766293264717362397od_o_o @ X4 @ Zs3 ) ) )
            & ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_878_in__set__conv__decomp__first,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
      = ( ? [Ys3: list_P723798329145367999ropVar,Zs3: list_P723798329145367999ropVar] :
            ( ( Xs2
              = ( append7708039934470209236ropVar @ Ys3 @ ( cons_P6757824865275295865ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_879_in__set__conv__decomp__first,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
      = ( ? [Ys3: list_l1644708917474426447ropVar,Zs3: list_l1644708917474426447ropVar] :
            ( ( Xs2
              = ( append2757213247338973028ropVar @ Ys3 @ ( cons_l2454487807130495625ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_880_in__set__conv__decomp__first,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l6341520962544925691pVar_o,Zs3: list_l6341520962544925691pVar_o] :
            ( ( Xs2
              = ( append9215492216909589904pVar_o @ Ys3 @ ( cons_l8096070148660426411pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_881_in__set__conv__decomp__first,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
      = ( ? [Ys3: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs2
              = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_882_in__set__conv__decomp__first,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_P1665994083459302249pVar_o,Zs3: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( append1087384436348988414pVar_o @ Ys3 @ ( cons_P6860203055867761049pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_883_in__set__conv__decomp__first,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l1255154591254203759pVar_o,Zs3: list_l1255154591254203759pVar_o] :
            ( ( Xs2
              = ( append8069948826146371844pVar_o @ Ys3 @ ( cons_l3452452106213086495pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_884_in__set__conv__decomp__first,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l985434930288970485pVar_o,Zs3: list_l985434930288970485pVar_o] :
            ( ( Xs2
              = ( append8959015459476584842pVar_o @ Ys3 @ ( cons_l7316790070876032037pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_885_in__set__conv__decomp__first,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ~ ( member_nat @ X4 @ ( set_nat2 @ Ys3 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_886_in__set__conv__decomp__last,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
      = ( ? [Ys3: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
            ( ( Xs2
              = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_887_in__set__conv__decomp__last,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
      = ( ? [Ys3: list_P4002435161011370285od_o_o,Zs3: list_P4002435161011370285od_o_o] :
            ( ( Xs2
              = ( append2614242729457001410od_o_o @ Ys3 @ ( cons_P8766293264717362397od_o_o @ X4 @ Zs3 ) ) )
            & ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_888_in__set__conv__decomp__last,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
      = ( ? [Ys3: list_P723798329145367999ropVar,Zs3: list_P723798329145367999ropVar] :
            ( ( Xs2
              = ( append7708039934470209236ropVar @ Ys3 @ ( cons_P6757824865275295865ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_889_in__set__conv__decomp__last,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
      = ( ? [Ys3: list_l1644708917474426447ropVar,Zs3: list_l1644708917474426447ropVar] :
            ( ( Xs2
              = ( append2757213247338973028ropVar @ Ys3 @ ( cons_l2454487807130495625ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_890_in__set__conv__decomp__last,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l6341520962544925691pVar_o,Zs3: list_l6341520962544925691pVar_o] :
            ( ( Xs2
              = ( append9215492216909589904pVar_o @ Ys3 @ ( cons_l8096070148660426411pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_891_in__set__conv__decomp__last,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
      = ( ? [Ys3: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( ( Xs2
              = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs3 ) ) )
            & ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_892_in__set__conv__decomp__last,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_P1665994083459302249pVar_o,Zs3: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( append1087384436348988414pVar_o @ Ys3 @ ( cons_P6860203055867761049pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_893_in__set__conv__decomp__last,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l1255154591254203759pVar_o,Zs3: list_l1255154591254203759pVar_o] :
            ( ( Xs2
              = ( append8069948826146371844pVar_o @ Ys3 @ ( cons_l3452452106213086495pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_894_in__set__conv__decomp__last,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l985434930288970485pVar_o,Zs3: list_l985434930288970485pVar_o] :
            ( ( Xs2
              = ( append8959015459476584842pVar_o @ Ys3 @ ( cons_l7316790070876032037pVar_o @ X4 @ Zs3 ) ) )
            & ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_895_in__set__conv__decomp__last,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_nat,Zs3: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) )
            & ~ ( member_nat @ X4 @ ( set_nat2 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_896_split__list__first__propE,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ? [X5: propos3600597766819905199ropVar] :
          ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P723798329145367999ropVar,X2: propos3600597766819905199ropVar] :
            ( ? [Zs2: list_P723798329145367999ropVar] :
                ( Xs2
                = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: propos3600597766819905199ropVar] :
                    ( ( member1796676888578033094ropVar @ Xa2 @ ( set_Pr342157451780195930ropVar @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_897_split__list__first__propE,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ? [X5: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l1644708917474426447ropVar,X2: list_P723798329145367999ropVar] :
            ( ? [Zs2: list_l1644708917474426447ropVar] :
                ( Xs2
                = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_P723798329145367999ropVar] :
                    ( ( member7375955847912140118ropVar @ Xa2 @ ( set_li8604685205718225898ropVar @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_898_split__list__first__propE,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ? [X5: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l6341520962544925691pVar_o,X2: list_l985434930288970485pVar_o] :
            ( ? [Zs2: list_l6341520962544925691pVar_o] :
                ( Xs2
                = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_l985434930288970485pVar_o] :
                    ( ( member9046334641704116766pVar_o @ Xa2 @ ( set_li4369674465895099530pVar_o @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_899_split__list__first__propE,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X5: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
            ( ? [Zs2: list_P8791008543742759928ropVar] :
                ( Xs2
                = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_900_split__list__first__propE,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ? [X5: produc8999643685545340131pVar_o] :
          ( ( member1729440322722938252pVar_o @ X5 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P1665994083459302249pVar_o,X2: produc8999643685545340131pVar_o] :
            ( ? [Zs2: list_P1665994083459302249pVar_o] :
                ( Xs2
                = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: produc8999643685545340131pVar_o] :
                    ( ( member1729440322722938252pVar_o @ Xa2 @ ( set_Pr2838981755425241592pVar_o @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_901_split__list__first__propE,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ? [X5: list_P1665994083459302249pVar_o] :
          ( ( member7042831661676915602pVar_o @ X5 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l1255154591254203759pVar_o,X2: list_P1665994083459302249pVar_o] :
            ( ? [Zs2: list_l1255154591254203759pVar_o] :
                ( Xs2
                = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_P1665994083459302249pVar_o] :
                    ( ( member7042831661676915602pVar_o @ Xa2 @ ( set_li5575011780610867710pVar_o @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_902_split__list__first__propE,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ? [X5: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l985434930288970485pVar_o,X2: list_l1255154591254203759pVar_o] :
            ( ? [Zs2: list_l985434930288970485pVar_o] :
                ( Xs2
                = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_l1255154591254203759pVar_o] :
                    ( ( member6267526602072755480pVar_o @ Xa2 @ ( set_li3549506505754852740pVar_o @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_903_split__list__first__propE,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_nat,X2: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs2
                = ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_904_split__list__last__propE,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ? [X5: propos3600597766819905199ropVar] :
          ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P723798329145367999ropVar,X2: propos3600597766819905199ropVar,Zs2: list_P723798329145367999ropVar] :
            ( ( Xs2
              = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: propos3600597766819905199ropVar] :
                    ( ( member1796676888578033094ropVar @ Xa2 @ ( set_Pr342157451780195930ropVar @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_905_split__list__last__propE,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ? [X5: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l1644708917474426447ropVar,X2: list_P723798329145367999ropVar,Zs2: list_l1644708917474426447ropVar] :
            ( ( Xs2
              = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_P723798329145367999ropVar] :
                    ( ( member7375955847912140118ropVar @ Xa2 @ ( set_li8604685205718225898ropVar @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_906_split__list__last__propE,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ? [X5: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l6341520962544925691pVar_o,X2: list_l985434930288970485pVar_o,Zs2: list_l6341520962544925691pVar_o] :
            ( ( Xs2
              = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_l985434930288970485pVar_o] :
                    ( ( member9046334641704116766pVar_o @ Xa2 @ ( set_li4369674465895099530pVar_o @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_907_split__list__last__propE,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X5: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar] :
            ( ( Xs2
              = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_908_split__list__last__propE,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ? [X5: produc8999643685545340131pVar_o] :
          ( ( member1729440322722938252pVar_o @ X5 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P1665994083459302249pVar_o,X2: produc8999643685545340131pVar_o,Zs2: list_P1665994083459302249pVar_o] :
            ( ( Xs2
              = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: produc8999643685545340131pVar_o] :
                    ( ( member1729440322722938252pVar_o @ Xa2 @ ( set_Pr2838981755425241592pVar_o @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_909_split__list__last__propE,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ? [X5: list_P1665994083459302249pVar_o] :
          ( ( member7042831661676915602pVar_o @ X5 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l1255154591254203759pVar_o,X2: list_P1665994083459302249pVar_o,Zs2: list_l1255154591254203759pVar_o] :
            ( ( Xs2
              = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_P1665994083459302249pVar_o] :
                    ( ( member7042831661676915602pVar_o @ Xa2 @ ( set_li5575011780610867710pVar_o @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_910_split__list__last__propE,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ? [X5: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l985434930288970485pVar_o,X2: list_l1255154591254203759pVar_o,Zs2: list_l985434930288970485pVar_o] :
            ( ( Xs2
              = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: list_l1255154591254203759pVar_o] :
                    ( ( member6267526602072755480pVar_o @ Xa2 @ ( set_li3549506505754852740pVar_o @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_911_split__list__last__propE,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_nat,X2: nat,Zs2: list_nat] :
            ( ( Xs2
              = ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
           => ( ( P @ X2 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_912_split__list__first__prop,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ? [X5: propos3600597766819905199ropVar] :
          ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P723798329145367999ropVar,X2: propos3600597766819905199ropVar] :
          ( ? [Zs2: list_P723798329145367999ropVar] :
              ( Xs2
              = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: propos3600597766819905199ropVar] :
              ( ( member1796676888578033094ropVar @ Xa2 @ ( set_Pr342157451780195930ropVar @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_913_split__list__first__prop,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ? [X5: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l1644708917474426447ropVar,X2: list_P723798329145367999ropVar] :
          ( ? [Zs2: list_l1644708917474426447ropVar] :
              ( Xs2
              = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_P723798329145367999ropVar] :
              ( ( member7375955847912140118ropVar @ Xa2 @ ( set_li8604685205718225898ropVar @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_914_split__list__first__prop,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ? [X5: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l6341520962544925691pVar_o,X2: list_l985434930288970485pVar_o] :
          ( ? [Zs2: list_l6341520962544925691pVar_o] :
              ( Xs2
              = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_l985434930288970485pVar_o] :
              ( ( member9046334641704116766pVar_o @ Xa2 @ ( set_li4369674465895099530pVar_o @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_915_split__list__first__prop,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X5: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
          ( ? [Zs2: list_P8791008543742759928ropVar] :
              ( Xs2
              = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: produc2642885279816024434ropVar] :
              ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_916_split__list__first__prop,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ? [X5: produc8999643685545340131pVar_o] :
          ( ( member1729440322722938252pVar_o @ X5 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P1665994083459302249pVar_o,X2: produc8999643685545340131pVar_o] :
          ( ? [Zs2: list_P1665994083459302249pVar_o] :
              ( Xs2
              = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: produc8999643685545340131pVar_o] :
              ( ( member1729440322722938252pVar_o @ Xa2 @ ( set_Pr2838981755425241592pVar_o @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_917_split__list__first__prop,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ? [X5: list_P1665994083459302249pVar_o] :
          ( ( member7042831661676915602pVar_o @ X5 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l1255154591254203759pVar_o,X2: list_P1665994083459302249pVar_o] :
          ( ? [Zs2: list_l1255154591254203759pVar_o] :
              ( Xs2
              = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_P1665994083459302249pVar_o] :
              ( ( member7042831661676915602pVar_o @ Xa2 @ ( set_li5575011780610867710pVar_o @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_918_split__list__first__prop,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ? [X5: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l985434930288970485pVar_o,X2: list_l1255154591254203759pVar_o] :
          ( ? [Zs2: list_l985434930288970485pVar_o] :
              ( Xs2
              = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_l1255154591254203759pVar_o] :
              ( ( member6267526602072755480pVar_o @ Xa2 @ ( set_li3549506505754852740pVar_o @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_919_split__list__first__prop,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_nat,X2: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs2
              = ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_920_split__list__last__prop,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ? [X5: propos3600597766819905199ropVar] :
          ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P723798329145367999ropVar,X2: propos3600597766819905199ropVar,Zs2: list_P723798329145367999ropVar] :
          ( ( Xs2
            = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: propos3600597766819905199ropVar] :
              ( ( member1796676888578033094ropVar @ Xa2 @ ( set_Pr342157451780195930ropVar @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_921_split__list__last__prop,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ? [X5: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l1644708917474426447ropVar,X2: list_P723798329145367999ropVar,Zs2: list_l1644708917474426447ropVar] :
          ( ( Xs2
            = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_P723798329145367999ropVar] :
              ( ( member7375955847912140118ropVar @ Xa2 @ ( set_li8604685205718225898ropVar @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_922_split__list__last__prop,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ? [X5: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l6341520962544925691pVar_o,X2: list_l985434930288970485pVar_o,Zs2: list_l6341520962544925691pVar_o] :
          ( ( Xs2
            = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_l985434930288970485pVar_o] :
              ( ( member9046334641704116766pVar_o @ Xa2 @ ( set_li4369674465895099530pVar_o @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_923_split__list__last__prop,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X5: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Xs2
            = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: produc2642885279816024434ropVar] :
              ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_924_split__list__last__prop,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ? [X5: produc8999643685545340131pVar_o] :
          ( ( member1729440322722938252pVar_o @ X5 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P1665994083459302249pVar_o,X2: produc8999643685545340131pVar_o,Zs2: list_P1665994083459302249pVar_o] :
          ( ( Xs2
            = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: produc8999643685545340131pVar_o] :
              ( ( member1729440322722938252pVar_o @ Xa2 @ ( set_Pr2838981755425241592pVar_o @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_925_split__list__last__prop,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ? [X5: list_P1665994083459302249pVar_o] :
          ( ( member7042831661676915602pVar_o @ X5 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l1255154591254203759pVar_o,X2: list_P1665994083459302249pVar_o,Zs2: list_l1255154591254203759pVar_o] :
          ( ( Xs2
            = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_P1665994083459302249pVar_o] :
              ( ( member7042831661676915602pVar_o @ Xa2 @ ( set_li5575011780610867710pVar_o @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_926_split__list__last__prop,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ? [X5: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l985434930288970485pVar_o,X2: list_l1255154591254203759pVar_o,Zs2: list_l985434930288970485pVar_o] :
          ( ( Xs2
            = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: list_l1255154591254203759pVar_o] :
              ( ( member6267526602072755480pVar_o @ Xa2 @ ( set_li3549506505754852740pVar_o @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_927_split__list__last__prop,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_nat,X2: nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
          & ( P @ X2 )
          & ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_928_in__set__conv__decomp,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
      = ( ? [Ys3: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
            ( Xs2
            = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_929_in__set__conv__decomp,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
      = ( ? [Ys3: list_P4002435161011370285od_o_o,Zs3: list_P4002435161011370285od_o_o] :
            ( Xs2
            = ( append2614242729457001410od_o_o @ Ys3 @ ( cons_P8766293264717362397od_o_o @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_930_in__set__conv__decomp,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
      = ( ? [Ys3: list_P723798329145367999ropVar,Zs3: list_P723798329145367999ropVar] :
            ( Xs2
            = ( append7708039934470209236ropVar @ Ys3 @ ( cons_P6757824865275295865ropVar @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_931_in__set__conv__decomp,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
      = ( ? [Ys3: list_l1644708917474426447ropVar,Zs3: list_l1644708917474426447ropVar] :
            ( Xs2
            = ( append2757213247338973028ropVar @ Ys3 @ ( cons_l2454487807130495625ropVar @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_932_in__set__conv__decomp,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l6341520962544925691pVar_o,Zs3: list_l6341520962544925691pVar_o] :
            ( Xs2
            = ( append9215492216909589904pVar_o @ Ys3 @ ( cons_l8096070148660426411pVar_o @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_933_in__set__conv__decomp,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
      = ( ? [Ys3: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
            ( Xs2
            = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_934_in__set__conv__decomp,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_P1665994083459302249pVar_o,Zs3: list_P1665994083459302249pVar_o] :
            ( Xs2
            = ( append1087384436348988414pVar_o @ Ys3 @ ( cons_P6860203055867761049pVar_o @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_935_in__set__conv__decomp,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l1255154591254203759pVar_o,Zs3: list_l1255154591254203759pVar_o] :
            ( Xs2
            = ( append8069948826146371844pVar_o @ Ys3 @ ( cons_l3452452106213086495pVar_o @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_936_in__set__conv__decomp,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
      = ( ? [Ys3: list_l985434930288970485pVar_o,Zs3: list_l985434930288970485pVar_o] :
            ( Xs2
            = ( append8959015459476584842pVar_o @ Ys3 @ ( cons_l7316790070876032037pVar_o @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_937_in__set__conv__decomp,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
      = ( ? [Ys3: list_nat,Zs3: list_nat] :
            ( Xs2
            = ( append_nat @ Ys3 @ ( cons_nat @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_938_append__Cons__eq__iff,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar,Xs5: list_M6740372356978808802ropVar,Ys5: list_M6740372356978808802ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ( ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Ys2 ) )
       => ( ( ( append2684835082924936439ropVar @ Xs2 @ ( cons_M4277272395645347858ropVar @ X4 @ Ys2 ) )
            = ( append2684835082924936439ropVar @ Xs5 @ ( cons_M4277272395645347858ropVar @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_939_append__Cons__eq__iff,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o,Ys2: list_P4002435161011370285od_o_o,Xs5: list_P4002435161011370285od_o_o,Ys5: list_P4002435161011370285od_o_o] :
      ( ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ( ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Ys2 ) )
       => ( ( ( append2614242729457001410od_o_o @ Xs2 @ ( cons_P8766293264717362397od_o_o @ X4 @ Ys2 ) )
            = ( append2614242729457001410od_o_o @ Xs5 @ ( cons_P8766293264717362397od_o_o @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_940_append__Cons__eq__iff,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Xs5: list_P723798329145367999ropVar,Ys5: list_P723798329145367999ropVar] :
      ( ~ ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
     => ( ~ ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Ys2 ) )
       => ( ( ( append7708039934470209236ropVar @ Xs2 @ ( cons_P6757824865275295865ropVar @ X4 @ Ys2 ) )
            = ( append7708039934470209236ropVar @ Xs5 @ ( cons_P6757824865275295865ropVar @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_941_append__Cons__eq__iff,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar,Ys2: list_l1644708917474426447ropVar,Xs5: list_l1644708917474426447ropVar,Ys5: list_l1644708917474426447ropVar] :
      ( ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ( ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Ys2 ) )
       => ( ( ( append2757213247338973028ropVar @ Xs2 @ ( cons_l2454487807130495625ropVar @ X4 @ Ys2 ) )
            = ( append2757213247338973028ropVar @ Xs5 @ ( cons_l2454487807130495625ropVar @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_942_append__Cons__eq__iff,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o,Ys2: list_l6341520962544925691pVar_o,Xs5: list_l6341520962544925691pVar_o,Ys5: list_l6341520962544925691pVar_o] :
      ( ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ( ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Ys2 ) )
       => ( ( ( append9215492216909589904pVar_o @ Xs2 @ ( cons_l8096070148660426411pVar_o @ X4 @ Ys2 ) )
            = ( append9215492216909589904pVar_o @ Xs5 @ ( cons_l8096070148660426411pVar_o @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_943_append__Cons__eq__iff,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Xs5: list_P8791008543742759928ropVar,Ys5: list_P8791008543742759928ropVar] :
      ( ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ( ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Ys2 ) )
       => ( ( ( append4352292717583037453ropVar @ Xs2 @ ( cons_P1504208554336013608ropVar @ X4 @ Ys2 ) )
            = ( append4352292717583037453ropVar @ Xs5 @ ( cons_P1504208554336013608ropVar @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_944_append__Cons__eq__iff,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o,Xs5: list_P1665994083459302249pVar_o,Ys5: list_P1665994083459302249pVar_o] :
      ( ~ ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
     => ( ~ ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Ys2 ) )
       => ( ( ( append1087384436348988414pVar_o @ Xs2 @ ( cons_P6860203055867761049pVar_o @ X4 @ Ys2 ) )
            = ( append1087384436348988414pVar_o @ Xs5 @ ( cons_P6860203055867761049pVar_o @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_945_append__Cons__eq__iff,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o,Xs5: list_l1255154591254203759pVar_o,Ys5: list_l1255154591254203759pVar_o] :
      ( ~ ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
     => ( ~ ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Ys2 ) )
       => ( ( ( append8069948826146371844pVar_o @ Xs2 @ ( cons_l3452452106213086495pVar_o @ X4 @ Ys2 ) )
            = ( append8069948826146371844pVar_o @ Xs5 @ ( cons_l3452452106213086495pVar_o @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_946_append__Cons__eq__iff,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Xs5: list_l985434930288970485pVar_o,Ys5: list_l985434930288970485pVar_o] :
      ( ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ( ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Ys2 ) )
       => ( ( ( append8959015459476584842pVar_o @ Xs2 @ ( cons_l7316790070876032037pVar_o @ X4 @ Ys2 ) )
            = ( append8959015459476584842pVar_o @ Xs5 @ ( cons_l7316790070876032037pVar_o @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_947_append__Cons__eq__iff,axiom,
    ! [X4: nat,Xs2: list_nat,Ys2: list_nat,Xs5: list_nat,Ys5: list_nat] :
      ( ~ ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ~ ( member_nat @ X4 @ ( set_nat2 @ Ys2 ) )
       => ( ( ( append_nat @ Xs2 @ ( cons_nat @ X4 @ Ys2 ) )
            = ( append_nat @ Xs5 @ ( cons_nat @ X4 @ Ys5 ) ) )
          = ( ( Xs2 = Xs5 )
            & ( Ys2 = Ys5 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_948_split__list__propE,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ? [X5: propos3600597766819905199ropVar] :
          ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P723798329145367999ropVar,X2: propos3600597766819905199ropVar] :
            ( ? [Zs2: list_P723798329145367999ropVar] :
                ( Xs2
                = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_949_split__list__propE,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ? [X5: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l1644708917474426447ropVar,X2: list_P723798329145367999ropVar] :
            ( ? [Zs2: list_l1644708917474426447ropVar] :
                ( Xs2
                = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_950_split__list__propE,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ? [X5: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l6341520962544925691pVar_o,X2: list_l985434930288970485pVar_o] :
            ( ? [Zs2: list_l6341520962544925691pVar_o] :
                ( Xs2
                = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_951_split__list__propE,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X5: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
            ( ? [Zs2: list_P8791008543742759928ropVar] :
                ( Xs2
                = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_952_split__list__propE,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ? [X5: produc8999643685545340131pVar_o] :
          ( ( member1729440322722938252pVar_o @ X5 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_P1665994083459302249pVar_o,X2: produc8999643685545340131pVar_o] :
            ( ? [Zs2: list_P1665994083459302249pVar_o] :
                ( Xs2
                = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_953_split__list__propE,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ? [X5: list_P1665994083459302249pVar_o] :
          ( ( member7042831661676915602pVar_o @ X5 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l1255154591254203759pVar_o,X2: list_P1665994083459302249pVar_o] :
            ( ? [Zs2: list_l1255154591254203759pVar_o] :
                ( Xs2
                = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_954_split__list__propE,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ? [X5: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_l985434930288970485pVar_o,X2: list_l1255154591254203759pVar_o] :
            ( ? [Zs2: list_l985434930288970485pVar_o] :
                ( Xs2
                = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_955_split__list__propE,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ~ ! [Ys: list_nat,X2: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs2
                = ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
           => ~ ( P @ X2 ) ) ) ).

% split_list_propE
thf(fact_956_split__list__first,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ? [Ys: list_M6740372356978808802ropVar,Zs2: list_M6740372356978808802ropVar] :
          ( ( Xs2
            = ( append2684835082924936439ropVar @ Ys @ ( cons_M4277272395645347858ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Ys ) ) ) ) ).

% split_list_first
thf(fact_957_split__list__first,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ? [Ys: list_P4002435161011370285od_o_o,Zs2: list_P4002435161011370285od_o_o] :
          ( ( Xs2
            = ( append2614242729457001410od_o_o @ Ys @ ( cons_P8766293264717362397od_o_o @ X4 @ Zs2 ) ) )
          & ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Ys ) ) ) ) ).

% split_list_first
thf(fact_958_split__list__first,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
     => ? [Ys: list_P723798329145367999ropVar,Zs2: list_P723798329145367999ropVar] :
          ( ( Xs2
            = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Ys ) ) ) ) ).

% split_list_first
thf(fact_959_split__list__first,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ? [Ys: list_l1644708917474426447ropVar,Zs2: list_l1644708917474426447ropVar] :
          ( ( Xs2
            = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Ys ) ) ) ) ).

% split_list_first
thf(fact_960_split__list__first,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ? [Ys: list_l6341520962544925691pVar_o,Zs2: list_l6341520962544925691pVar_o] :
          ( ( Xs2
            = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Ys ) ) ) ) ).

% split_list_first
thf(fact_961_split__list__first,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ? [Ys: list_P8791008543742759928ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Xs2
            = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Ys ) ) ) ) ).

% split_list_first
thf(fact_962_split__list__first,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
     => ? [Ys: list_P1665994083459302249pVar_o,Zs2: list_P1665994083459302249pVar_o] :
          ( ( Xs2
            = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Ys ) ) ) ) ).

% split_list_first
thf(fact_963_split__list__first,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
     => ? [Ys: list_l1255154591254203759pVar_o,Zs2: list_l1255154591254203759pVar_o] :
          ( ( Xs2
            = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Ys ) ) ) ) ).

% split_list_first
thf(fact_964_split__list__first,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ? [Ys: list_l985434930288970485pVar_o,Zs2: list_l985434930288970485pVar_o] :
          ( ( Xs2
            = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Ys ) ) ) ) ).

% split_list_first
thf(fact_965_split__list__first,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ? [Ys: list_nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys @ ( cons_nat @ X4 @ Zs2 ) ) )
          & ~ ( member_nat @ X4 @ ( set_nat2 @ Ys ) ) ) ) ).

% split_list_first
thf(fact_966_split__list__prop,axiom,
    ! [Xs2: list_P723798329145367999ropVar,P: propos3600597766819905199ropVar > $o] :
      ( ? [X5: propos3600597766819905199ropVar] :
          ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P723798329145367999ropVar,X2: propos3600597766819905199ropVar] :
          ( ? [Zs2: list_P723798329145367999ropVar] :
              ( Xs2
              = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_967_split__list__prop,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,P: list_P723798329145367999ropVar > $o] :
      ( ? [X5: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l1644708917474426447ropVar,X2: list_P723798329145367999ropVar] :
          ( ? [Zs2: list_l1644708917474426447ropVar] :
              ( Xs2
              = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_968_split__list__prop,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,P: list_l985434930288970485pVar_o > $o] :
      ( ? [X5: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l6341520962544925691pVar_o,X2: list_l985434930288970485pVar_o] :
          ( ? [Zs2: list_l6341520962544925691pVar_o] :
              ( Xs2
              = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_969_split__list__prop,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X5: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
          ( ? [Zs2: list_P8791008543742759928ropVar] :
              ( Xs2
              = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_970_split__list__prop,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,P: produc8999643685545340131pVar_o > $o] :
      ( ? [X5: produc8999643685545340131pVar_o] :
          ( ( member1729440322722938252pVar_o @ X5 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_P1665994083459302249pVar_o,X2: produc8999643685545340131pVar_o] :
          ( ? [Zs2: list_P1665994083459302249pVar_o] :
              ( Xs2
              = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_971_split__list__prop,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,P: list_P1665994083459302249pVar_o > $o] :
      ( ? [X5: list_P1665994083459302249pVar_o] :
          ( ( member7042831661676915602pVar_o @ X5 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l1255154591254203759pVar_o,X2: list_P1665994083459302249pVar_o] :
          ( ? [Zs2: list_l1255154591254203759pVar_o] :
              ( Xs2
              = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_972_split__list__prop,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,P: list_l1255154591254203759pVar_o > $o] :
      ( ? [X5: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_l985434930288970485pVar_o,X2: list_l1255154591254203759pVar_o] :
          ( ? [Zs2: list_l985434930288970485pVar_o] :
              ( Xs2
              = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_973_split__list__prop,axiom,
    ! [Xs2: list_nat,P: nat > $o] :
      ( ? [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs2 ) )
          & ( P @ X5 ) )
     => ? [Ys: list_nat,X2: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs2
              = ( append_nat @ Ys @ ( cons_nat @ X2 @ Zs2 ) ) )
          & ( P @ X2 ) ) ) ).

% split_list_prop
thf(fact_974_split__list__last,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ? [Ys: list_M6740372356978808802ropVar,Zs2: list_M6740372356978808802ropVar] :
          ( ( Xs2
            = ( append2684835082924936439ropVar @ Ys @ ( cons_M4277272395645347858ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_975_split__list__last,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ? [Ys: list_P4002435161011370285od_o_o,Zs2: list_P4002435161011370285od_o_o] :
          ( ( Xs2
            = ( append2614242729457001410od_o_o @ Ys @ ( cons_P8766293264717362397od_o_o @ X4 @ Zs2 ) ) )
          & ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_976_split__list__last,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
     => ? [Ys: list_P723798329145367999ropVar,Zs2: list_P723798329145367999ropVar] :
          ( ( Xs2
            = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_977_split__list__last,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ? [Ys: list_l1644708917474426447ropVar,Zs2: list_l1644708917474426447ropVar] :
          ( ( Xs2
            = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_978_split__list__last,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ? [Ys: list_l6341520962544925691pVar_o,Zs2: list_l6341520962544925691pVar_o] :
          ( ( Xs2
            = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_979_split__list__last,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ? [Ys: list_P8791008543742759928ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Xs2
            = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X4 @ Zs2 ) ) )
          & ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_980_split__list__last,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
     => ? [Ys: list_P1665994083459302249pVar_o,Zs2: list_P1665994083459302249pVar_o] :
          ( ( Xs2
            = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_981_split__list__last,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
     => ? [Ys: list_l1255154591254203759pVar_o,Zs2: list_l1255154591254203759pVar_o] :
          ( ( Xs2
            = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_982_split__list__last,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ? [Ys: list_l985434930288970485pVar_o,Zs2: list_l985434930288970485pVar_o] :
          ( ( Xs2
            = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X4 @ Zs2 ) ) )
          & ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_983_split__list__last,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ? [Ys: list_nat,Zs2: list_nat] :
          ( ( Xs2
            = ( append_nat @ Ys @ ( cons_nat @ X4 @ Zs2 ) ) )
          & ~ ( member_nat @ X4 @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_984_split__list,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ? [Ys: list_M6740372356978808802ropVar,Zs2: list_M6740372356978808802ropVar] :
          ( Xs2
          = ( append2684835082924936439ropVar @ Ys @ ( cons_M4277272395645347858ropVar @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_985_split__list,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ? [Ys: list_P4002435161011370285od_o_o,Zs2: list_P4002435161011370285od_o_o] :
          ( Xs2
          = ( append2614242729457001410od_o_o @ Ys @ ( cons_P8766293264717362397od_o_o @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_986_split__list,axiom,
    ! [X4: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ X4 @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
     => ? [Ys: list_P723798329145367999ropVar,Zs2: list_P723798329145367999ropVar] :
          ( Xs2
          = ( append7708039934470209236ropVar @ Ys @ ( cons_P6757824865275295865ropVar @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_987_split__list,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ? [Ys: list_l1644708917474426447ropVar,Zs2: list_l1644708917474426447ropVar] :
          ( Xs2
          = ( append2757213247338973028ropVar @ Ys @ ( cons_l2454487807130495625ropVar @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_988_split__list,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ? [Ys: list_l6341520962544925691pVar_o,Zs2: list_l6341520962544925691pVar_o] :
          ( Xs2
          = ( append9215492216909589904pVar_o @ Ys @ ( cons_l8096070148660426411pVar_o @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_989_split__list,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ? [Ys: list_P8791008543742759928ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( Xs2
          = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_990_split__list,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ X4 @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
     => ? [Ys: list_P1665994083459302249pVar_o,Zs2: list_P1665994083459302249pVar_o] :
          ( Xs2
          = ( append1087384436348988414pVar_o @ Ys @ ( cons_P6860203055867761049pVar_o @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_991_split__list,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ X4 @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
     => ? [Ys: list_l1255154591254203759pVar_o,Zs2: list_l1255154591254203759pVar_o] :
          ( Xs2
          = ( append8069948826146371844pVar_o @ Ys @ ( cons_l3452452106213086495pVar_o @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_992_split__list,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ? [Ys: list_l985434930288970485pVar_o,Zs2: list_l985434930288970485pVar_o] :
          ( Xs2
          = ( append8959015459476584842pVar_o @ Ys @ ( cons_l7316790070876032037pVar_o @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_993_split__list,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ? [Ys: list_nat,Zs2: list_nat] :
          ( Xs2
          = ( append_nat @ Ys @ ( cons_nat @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_994_concat_Osimps_I1_J,axiom,
    ( ( concat4611077035641296302ropVar @ nil_li6797326493501628985ropVar )
    = nil_Pr7606595143512524841ropVar ) ).

% concat.simps(1)
thf(fact_995_concat_Osimps_I1_J,axiom,
    ( ( concat1830633707478507187ropVar @ nil_li2679230897895549182ropVar )
    = nil_Pr6119456312066593144ropVar ) ).

% concat.simps(1)
thf(fact_996_concat_Osimps_I1_J,axiom,
    ( ( concat8126406963686051632pVar_o @ nil_li5579170522155219707pVar_o )
    = nil_li590035233778055285pVar_o ) ).

% concat.simps(1)
thf(fact_997_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_998_concat_Osimps_I1_J,axiom,
    ( ( concat7668843100863076260pVar_o @ nil_li1310946081133177199pVar_o )
    = nil_Pr6413368191065476585pVar_o ) ).

% concat.simps(1)
thf(fact_999_concat_Osimps_I1_J,axiom,
    ( ( concat8635278358871234986pVar_o @ nil_li590035233778055285pVar_o )
    = nil_li1310946081133177199pVar_o ) ).

% concat.simps(1)
thf(fact_1000_concat_Osimps_I2_J,axiom,
    ! [X4: list_nat,Xs2: list_list_nat] :
      ( ( concat_nat @ ( cons_list_nat @ X4 @ Xs2 ) )
      = ( append_nat @ X4 @ ( concat_nat @ Xs2 ) ) ) ).

% concat.simps(2)
thf(fact_1001_concat_Osimps_I2_J,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( concat8126406963686051632pVar_o @ ( cons_l8096070148660426411pVar_o @ X4 @ Xs2 ) )
      = ( append8959015459476584842pVar_o @ X4 @ ( concat8126406963686051632pVar_o @ Xs2 ) ) ) ).

% concat.simps(2)
thf(fact_1002_concat_Osimps_I2_J,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( concat4611077035641296302ropVar @ ( cons_l2454487807130495625ropVar @ X4 @ Xs2 ) )
      = ( append7708039934470209236ropVar @ X4 @ ( concat4611077035641296302ropVar @ Xs2 ) ) ) ).

% concat.simps(2)
thf(fact_1003_concat_Osimps_I2_J,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( concat7668843100863076260pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
      = ( append1087384436348988414pVar_o @ X4 @ ( concat7668843100863076260pVar_o @ Xs2 ) ) ) ).

% concat.simps(2)
thf(fact_1004_concat_Osimps_I2_J,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( concat8635278358871234986pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
      = ( append8069948826146371844pVar_o @ X4 @ ( concat8635278358871234986pVar_o @ Xs2 ) ) ) ).

% concat.simps(2)
thf(fact_1005_last_Osimps,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
      ( ( ( Xs2 = nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
          = X4 ) )
      & ( ( Xs2 != nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
          = ( last_P5278392773003955985ropVar @ Xs2 ) ) ) ) ).

% last.simps
thf(fact_1006_last_Osimps,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o] :
      ( ( ( Xs2 = nil_Pr6413368191065476585pVar_o )
       => ( ( last_P703351216231693570pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
          = X4 ) )
      & ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
       => ( ( last_P703351216231693570pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
          = ( last_P703351216231693570pVar_o @ Xs2 ) ) ) ) ).

% last.simps
thf(fact_1007_last_Osimps,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o] :
      ( ( ( Xs2 = nil_li1310946081133177199pVar_o )
       => ( ( last_l9069138231800379400pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
          = X4 ) )
      & ( ( Xs2 != nil_li1310946081133177199pVar_o )
       => ( ( last_l9069138231800379400pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
          = ( last_l9069138231800379400pVar_o @ Xs2 ) ) ) ) ).

% last.simps
thf(fact_1008_last_Osimps,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o] :
      ( ( ( Xs2 = nil_li590035233778055285pVar_o )
       => ( ( last_l30396060243306126pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
          = X4 ) )
      & ( ( Xs2 != nil_li590035233778055285pVar_o )
       => ( ( last_l30396060243306126pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
          = ( last_l30396060243306126pVar_o @ Xs2 ) ) ) ) ).

% last.simps
thf(fact_1009_last_Osimps,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( ( Xs2 = nil_nat )
       => ( ( last_nat @ ( cons_nat @ X4 @ Xs2 ) )
          = X4 ) )
      & ( ( Xs2 != nil_nat )
       => ( ( last_nat @ ( cons_nat @ X4 @ Xs2 ) )
          = ( last_nat @ Xs2 ) ) ) ) ).

% last.simps
thf(fact_1010_last__ConsL,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
      ( ( Xs2 = nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
        = X4 ) ) ).

% last_ConsL
thf(fact_1011_last__ConsL,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o] :
      ( ( Xs2 = nil_Pr6413368191065476585pVar_o )
     => ( ( last_P703351216231693570pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
        = X4 ) ) ).

% last_ConsL
thf(fact_1012_last__ConsL,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o] :
      ( ( Xs2 = nil_li1310946081133177199pVar_o )
     => ( ( last_l9069138231800379400pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
        = X4 ) ) ).

% last_ConsL
thf(fact_1013_last__ConsL,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o] :
      ( ( Xs2 = nil_li590035233778055285pVar_o )
     => ( ( last_l30396060243306126pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
        = X4 ) ) ).

% last_ConsL
thf(fact_1014_last__ConsL,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( Xs2 = nil_nat )
     => ( ( last_nat @ ( cons_nat @ X4 @ Xs2 ) )
        = X4 ) ) ).

% last_ConsL
thf(fact_1015_last__ConsR,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
      ( ( Xs2 != nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
        = ( last_P5278392773003955985ropVar @ Xs2 ) ) ) ).

% last_ConsR
thf(fact_1016_last__ConsR,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o] :
      ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
     => ( ( last_P703351216231693570pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
        = ( last_P703351216231693570pVar_o @ Xs2 ) ) ) ).

% last_ConsR
thf(fact_1017_last__ConsR,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o] :
      ( ( Xs2 != nil_li1310946081133177199pVar_o )
     => ( ( last_l9069138231800379400pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
        = ( last_l9069138231800379400pVar_o @ Xs2 ) ) ) ).

% last_ConsR
thf(fact_1018_last__ConsR,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o] :
      ( ( Xs2 != nil_li590035233778055285pVar_o )
     => ( ( last_l30396060243306126pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
        = ( last_l30396060243306126pVar_o @ Xs2 ) ) ) ).

% last_ConsR
thf(fact_1019_last__ConsR,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( Xs2 != nil_nat )
     => ( ( last_nat @ ( cons_nat @ X4 @ Xs2 ) )
        = ( last_nat @ Xs2 ) ) ) ).

% last_ConsR
thf(fact_1020_last__append,axiom,
    ! [Ys2: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( ( Ys2 = nil_Pr7606595143512524841ropVar )
       => ( ( last_P3456477928032711888ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) )
          = ( last_P3456477928032711888ropVar @ Xs2 ) ) )
      & ( ( Ys2 != nil_Pr7606595143512524841ropVar )
       => ( ( last_P3456477928032711888ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) )
          = ( last_P3456477928032711888ropVar @ Ys2 ) ) ) ) ).

% last_append
thf(fact_1021_last__append,axiom,
    ! [Ys2: list_P8791008543742759928ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( ( Ys2 = nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) )
          = ( last_P5278392773003955985ropVar @ Xs2 ) ) )
      & ( ( Ys2 != nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) )
          = ( last_P5278392773003955985ropVar @ Ys2 ) ) ) ) ).

% last_append
thf(fact_1022_last__append,axiom,
    ! [Ys2: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( ( Ys2 = nil_Pr6413368191065476585pVar_o )
       => ( ( last_P703351216231693570pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) )
          = ( last_P703351216231693570pVar_o @ Xs2 ) ) )
      & ( ( Ys2 != nil_Pr6413368191065476585pVar_o )
       => ( ( last_P703351216231693570pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) )
          = ( last_P703351216231693570pVar_o @ Ys2 ) ) ) ) ).

% last_append
thf(fact_1023_last__append,axiom,
    ! [Ys2: list_l1255154591254203759pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( ( Ys2 = nil_li1310946081133177199pVar_o )
       => ( ( last_l9069138231800379400pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) )
          = ( last_l9069138231800379400pVar_o @ Xs2 ) ) )
      & ( ( Ys2 != nil_li1310946081133177199pVar_o )
       => ( ( last_l9069138231800379400pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) )
          = ( last_l9069138231800379400pVar_o @ Ys2 ) ) ) ) ).

% last_append
thf(fact_1024_last__append,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( ( Ys2 = nil_li590035233778055285pVar_o )
       => ( ( last_l30396060243306126pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
          = ( last_l30396060243306126pVar_o @ Xs2 ) ) )
      & ( ( Ys2 != nil_li590035233778055285pVar_o )
       => ( ( last_l30396060243306126pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
          = ( last_l30396060243306126pVar_o @ Ys2 ) ) ) ) ).

% last_append
thf(fact_1025_last__append,axiom,
    ! [Ys2: list_nat,Xs2: list_nat] :
      ( ( ( Ys2 = nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
          = ( last_nat @ Xs2 ) ) )
      & ( ( Ys2 != nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs2 @ Ys2 ) )
          = ( last_nat @ Ys2 ) ) ) ) ).

% last_append
thf(fact_1026_longest__common__suffix,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
    ? [Ss: list_P723798329145367999ropVar,Xs6: list_P723798329145367999ropVar,Ys6: list_P723798329145367999ropVar] :
      ( ( Xs2
        = ( append7708039934470209236ropVar @ Xs6 @ Ss ) )
      & ( Ys2
        = ( append7708039934470209236ropVar @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_Pr7606595143512524841ropVar )
        | ( Ys6 = nil_Pr7606595143512524841ropVar )
        | ( ( last_P3456477928032711888ropVar @ Xs6 )
         != ( last_P3456477928032711888ropVar @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1027_longest__common__suffix,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
    ? [Ss: list_P8791008543742759928ropVar,Xs6: list_P8791008543742759928ropVar,Ys6: list_P8791008543742759928ropVar] :
      ( ( Xs2
        = ( append4352292717583037453ropVar @ Xs6 @ Ss ) )
      & ( Ys2
        = ( append4352292717583037453ropVar @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_Pr6119456312066593144ropVar )
        | ( Ys6 = nil_Pr6119456312066593144ropVar )
        | ( ( last_P5278392773003955985ropVar @ Xs6 )
         != ( last_P5278392773003955985ropVar @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1028_longest__common__suffix,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
    ? [Ss: list_P1665994083459302249pVar_o,Xs6: list_P1665994083459302249pVar_o,Ys6: list_P1665994083459302249pVar_o] :
      ( ( Xs2
        = ( append1087384436348988414pVar_o @ Xs6 @ Ss ) )
      & ( Ys2
        = ( append1087384436348988414pVar_o @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_Pr6413368191065476585pVar_o )
        | ( Ys6 = nil_Pr6413368191065476585pVar_o )
        | ( ( last_P703351216231693570pVar_o @ Xs6 )
         != ( last_P703351216231693570pVar_o @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1029_longest__common__suffix,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
    ? [Ss: list_l1255154591254203759pVar_o,Xs6: list_l1255154591254203759pVar_o,Ys6: list_l1255154591254203759pVar_o] :
      ( ( Xs2
        = ( append8069948826146371844pVar_o @ Xs6 @ Ss ) )
      & ( Ys2
        = ( append8069948826146371844pVar_o @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_li1310946081133177199pVar_o )
        | ( Ys6 = nil_li1310946081133177199pVar_o )
        | ( ( last_l9069138231800379400pVar_o @ Xs6 )
         != ( last_l9069138231800379400pVar_o @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1030_longest__common__suffix,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
    ? [Ss: list_l985434930288970485pVar_o,Xs6: list_l985434930288970485pVar_o,Ys6: list_l985434930288970485pVar_o] :
      ( ( Xs2
        = ( append8959015459476584842pVar_o @ Xs6 @ Ss ) )
      & ( Ys2
        = ( append8959015459476584842pVar_o @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_li590035233778055285pVar_o )
        | ( Ys6 = nil_li590035233778055285pVar_o )
        | ( ( last_l30396060243306126pVar_o @ Xs6 )
         != ( last_l30396060243306126pVar_o @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1031_longest__common__suffix,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
    ? [Ss: list_nat,Xs6: list_nat,Ys6: list_nat] :
      ( ( Xs2
        = ( append_nat @ Xs6 @ Ss ) )
      & ( Ys2
        = ( append_nat @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_nat )
        | ( Ys6 = nil_nat )
        | ( ( last_nat @ Xs6 )
         != ( last_nat @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1032_List_Oinsert__def,axiom,
    ( insert215289443796111736ropVar
    = ( ^ [X3: multis3193088007478089820ropVar,Xs3: list_M6740372356978808802ropVar] : ( if_lis9215432529438213544ropVar @ ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs3 ) ) @ Xs3 @ ( cons_M4277272395645347858ropVar @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1033_List_Oinsert__def,axiom,
    ( insert8896757123549915715od_o_o
    = ( ^ [X3: product_prod_o_o,Xs3: list_P4002435161011370285od_o_o] : ( if_lis8143501808108548851od_o_o @ ( member7466972457876170832od_o_o @ X3 @ ( set_Product_prod_o_o2 @ Xs3 ) ) @ Xs3 @ ( cons_P8766293264717362397od_o_o @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1034_List_Oinsert__def,axiom,
    ( insert6886557058002715043ropVar
    = ( ^ [X3: list_P723798329145367999ropVar,Xs3: list_l1644708917474426447ropVar] : ( if_lis5532466218389838089ropVar @ ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Xs3 ) ) @ Xs3 @ ( cons_l2454487807130495625ropVar @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1035_List_Oinsert__def,axiom,
    ( insert6473862492288621585pVar_o
    = ( ^ [X3: list_l985434930288970485pVar_o,Xs3: list_l6341520962544925691pVar_o] : ( if_lis1723474115809273281pVar_o @ ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Xs3 ) ) @ Xs3 @ ( cons_l8096070148660426411pVar_o @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1036_List_Oinsert__def,axiom,
    ( insert214406736284816014ropVar
    = ( ^ [X3: produc2642885279816024434ropVar,Xs3: list_P8791008543742759928ropVar] : ( if_lis1975434623467068862ropVar @ ( member9210975342120487067ropVar @ X3 @ ( set_Pr3946994337432900359ropVar @ Xs3 ) ) @ Xs3 @ ( cons_P1504208554336013608ropVar @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1037_List_Oinsert__def,axiom,
    ( insert5031086936754616319pVar_o
    = ( ^ [X3: produc8999643685545340131pVar_o,Xs3: list_P1665994083459302249pVar_o] : ( if_lis4159902583309516335pVar_o @ ( member1729440322722938252pVar_o @ X3 @ ( set_Pr2838981755425241592pVar_o @ Xs3 ) ) @ Xs3 @ ( cons_P6860203055867761049pVar_o @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1038_List_Oinsert__def,axiom,
    ( insert8144517603577680005pVar_o
    = ( ^ [X3: list_P1665994083459302249pVar_o,Xs3: list_l1255154591254203759pVar_o] : ( if_lis6453251472562810165pVar_o @ ( member7042831661676915602pVar_o @ X3 @ ( set_li5575011780610867710pVar_o @ Xs3 ) ) @ Xs3 @ ( cons_l3452452106213086495pVar_o @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1039_List_Oinsert__def,axiom,
    ( insert7135957767216192651pVar_o
    = ( ^ [X3: list_l1255154591254203759pVar_o,Xs3: list_l985434930288970485pVar_o] : ( if_lis463867677060124603pVar_o @ ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Xs3 ) ) @ Xs3 @ ( cons_l7316790070876032037pVar_o @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1040_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X3: nat,Xs3: list_nat] : ( if_list_nat @ ( member_nat @ X3 @ ( set_nat2 @ Xs3 ) ) @ Xs3 @ ( cons_nat @ X3 @ Xs3 ) ) ) ) ).

% List.insert_def
thf(fact_1041_encoder_OencodeGammaCond_Oelims,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_neg: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_impl: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,X4: multis3193088007478089820ropVar,Xa: multis3193088007478089820ropVar,Xb: $o,Xc: $o,Y3: list_l1255154591254203759pVar_o] :
      ( ( multis1610358401135282399pVar_o @ Eval @ 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 )
     => ( ( ( multis6017319109170748100ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ X4 @ Xa @ Xb @ Xc )
          = Y3 )
       => ( ( Xb
           => ( Xc
             => ( Y3 != Enc_True ) ) )
         => ( ( ~ Xb
             => ( ~ Xc
               => ( Y3
                 != ( Enc_neg @ X4 ) ) ) )
           => ( ( ~ Xb
               => ( Xc
                 => ( Y3
                   != ( Enc_var_impl @ X4 @ Xa ) ) ) )
             => ~ ( Xb
                 => ( ~ Xc
                   => ( Y3
                     != ( Enc_not_and @ X4 @ Xa ) ) ) ) ) ) ) ) ) ).

% encoder.encodeGammaCond.elims
thf(fact_1042_encoder_OencodeGammaCond,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_neg: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_impl: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,Alpha: multis3193088007478089820ropVar > $o,Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar,S: $o,Ns: $o] :
      ( ( multis1610358401135282399pVar_o @ Eval @ 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 @ Alpha @ ( multis6017319109170748100ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ Gam2 @ Eps2 @ S @ Ns ) )
        = ( ( Alpha @ Gam2 )
         => ( ( ( Alpha @ Eps2 )
             => Ns )
            & ( ~ ( Alpha @ Eps2 )
             => S ) ) ) ) ) ).

% encoder.encodeGammaCond
thf(fact_1043_encoder_OencodeGammaCond_Osimps_I1_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_neg: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_impl: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis1610358401135282399pVar_o @ Eval @ 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 )
     => ( ( multis6017319109170748100ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ Gam2 @ Eps2 @ $true @ $true )
        = Enc_True ) ) ).

% encoder.encodeGammaCond.simps(1)
thf(fact_1044_encoder_OencodeGammaCond_Osimps_I2_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_neg: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_impl: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis1610358401135282399pVar_o @ Eval @ 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 )
     => ( ( multis6017319109170748100ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ Gam2 @ Eps2 @ $false @ $false )
        = ( Enc_neg @ Gam2 ) ) ) ).

% encoder.encodeGammaCond.simps(2)
thf(fact_1045_encoder_OencodeGammaCond_Osimps_I3_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_neg: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_impl: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis1610358401135282399pVar_o @ Eval @ 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 )
     => ( ( multis6017319109170748100ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ Gam2 @ Eps2 @ $false @ $true )
        = ( Enc_var_impl @ Gam2 @ Eps2 ) ) ) ).

% encoder.encodeGammaCond.simps(3)
thf(fact_1046_encoder_OencodeGammaCond_Osimps_I4_J,axiom,
    ! [Eval: ( multis3193088007478089820ropVar > $o ) > list_l1255154591254203759pVar_o > $o,Enc_False: list_l1255154591254203759pVar_o,Enc_True: list_l1255154591254203759pVar_o,Enc_pos: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_neg: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_different: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_and_not: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_equiv_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_ite: multis3193088007478089820ropVar > multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_impl: multis3193088007478089820ropVar > list_l1255154591254203759pVar_o > list_l1255154591254203759pVar_o,Enc_var_impl: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_and: multis3193088007478089820ropVar > multis3193088007478089820ropVar > list_l1255154591254203759pVar_o,Enc_not_all: list_M6740372356978808802ropVar > list_l1255154591254203759pVar_o,Enc_conj: list_l985434930288970485pVar_o > list_l1255154591254203759pVar_o,Gam2: multis3193088007478089820ropVar,Eps2: multis3193088007478089820ropVar] :
      ( ( multis1610358401135282399pVar_o @ Eval @ 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 )
     => ( ( multis6017319109170748100ropVar @ Enc_True @ Enc_neg @ Enc_var_impl @ Enc_not_and @ Gam2 @ Eps2 @ $true @ $false )
        = ( Enc_not_and @ Gam2 @ Eps2 ) ) ) ).

% encoder.encodeGammaCond.simps(4)
thf(fact_1047_Cons__in__subseqsD,axiom,
    ! [Y3: propos3600597766819905199ropVar,Ys2: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member7375955847912140118ropVar @ ( cons_P6757824865275295865ropVar @ Y3 @ Ys2 ) @ ( set_li8604685205718225898ropVar @ ( subseq8000409699144225946ropVar @ Xs2 ) ) )
     => ( member7375955847912140118ropVar @ Ys2 @ ( set_li8604685205718225898ropVar @ ( subseq8000409699144225946ropVar @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_1048_Cons__in__subseqsD,axiom,
    ! [Y3: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( member6236339133593825953ropVar @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) @ ( set_li6509544621384488717ropVar @ ( subseq3825174504735880391ropVar @ Xs2 ) ) )
     => ( member6236339133593825953ropVar @ Ys2 @ ( set_li6509544621384488717ropVar @ ( subseq3825174504735880391ropVar @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_1049_Cons__in__subseqsD,axiom,
    ! [Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( member7042831661676915602pVar_o @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) @ ( set_li5575011780610867710pVar_o @ ( subseq8245697508244618168pVar_o @ Xs2 ) ) )
     => ( member7042831661676915602pVar_o @ Ys2 @ ( set_li5575011780610867710pVar_o @ ( subseq8245697508244618168pVar_o @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_1050_Cons__in__subseqsD,axiom,
    ! [Y3: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( member6267526602072755480pVar_o @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) @ ( set_li3549506505754852740pVar_o @ ( subseq2663478470768632766pVar_o @ Xs2 ) ) )
     => ( member6267526602072755480pVar_o @ Ys2 @ ( set_li3549506505754852740pVar_o @ ( subseq2663478470768632766pVar_o @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_1051_Cons__in__subseqsD,axiom,
    ! [Y3: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member9046334641704116766pVar_o @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) @ ( set_li4369674465895099530pVar_o @ ( subseq7541546836001800004pVar_o @ Xs2 ) ) )
     => ( member9046334641704116766pVar_o @ Ys2 @ ( set_li4369674465895099530pVar_o @ ( subseq7541546836001800004pVar_o @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_1052_Cons__in__subseqsD,axiom,
    ! [Y3: nat,Ys2: list_nat,Xs2: list_nat] :
      ( ( member_list_nat @ ( cons_nat @ Y3 @ Ys2 ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) )
     => ( member_list_nat @ Ys2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_1053_butlast_Osimps_I2_J,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,X4: produc2642885279816024434ropVar] :
      ( ( ( Xs2 = nil_Pr6119456312066593144ropVar )
       => ( ( butlas5098402688846288580ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
          = nil_Pr6119456312066593144ropVar ) )
      & ( ( Xs2 != nil_Pr6119456312066593144ropVar )
       => ( ( butlas5098402688846288580ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) )
          = ( cons_P1504208554336013608ropVar @ X4 @ ( butlas5098402688846288580ropVar @ Xs2 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1054_butlast_Osimps_I2_J,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,X4: produc8999643685545340131pVar_o] :
      ( ( ( Xs2 = nil_Pr6413368191065476585pVar_o )
       => ( ( butlas7924156930066306357pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
          = nil_Pr6413368191065476585pVar_o ) )
      & ( ( Xs2 != nil_Pr6413368191065476585pVar_o )
       => ( ( butlas7924156930066306357pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) )
          = ( cons_P6860203055867761049pVar_o @ X4 @ ( butlas7924156930066306357pVar_o @ Xs2 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1055_butlast_Osimps_I2_J,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,X4: list_P1665994083459302249pVar_o] :
      ( ( ( Xs2 = nil_li1310946081133177199pVar_o )
       => ( ( butlas54256187331189435pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
          = nil_li1310946081133177199pVar_o ) )
      & ( ( Xs2 != nil_li1310946081133177199pVar_o )
       => ( ( butlas54256187331189435pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) )
          = ( cons_l3452452106213086495pVar_o @ X4 @ ( butlas54256187331189435pVar_o @ Xs2 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1056_butlast_Osimps_I2_J,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,X4: list_l1255154591254203759pVar_o] :
      ( ( ( Xs2 = nil_li590035233778055285pVar_o )
       => ( ( butlas1161285084838040513pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
          = nil_li590035233778055285pVar_o ) )
      & ( ( Xs2 != nil_li590035233778055285pVar_o )
       => ( ( butlas1161285084838040513pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) )
          = ( cons_l7316790070876032037pVar_o @ X4 @ ( butlas1161285084838040513pVar_o @ Xs2 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1057_butlast_Osimps_I2_J,axiom,
    ! [Xs2: list_nat,X4: nat] :
      ( ( ( Xs2 = nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X4 @ Xs2 ) )
          = nil_nat ) )
      & ( ( Xs2 != nil_nat )
       => ( ( butlast_nat @ ( cons_nat @ X4 @ Xs2 ) )
          = ( cons_nat @ X4 @ ( butlast_nat @ Xs2 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1058_butlast__append,axiom,
    ! [Ys2: list_P723798329145367999ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( ( Ys2 = nil_Pr7606595143512524841ropVar )
       => ( ( butlas7628565150644405725ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) )
          = ( butlas7628565150644405725ropVar @ Xs2 ) ) )
      & ( ( Ys2 != nil_Pr7606595143512524841ropVar )
       => ( ( butlas7628565150644405725ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) )
          = ( append7708039934470209236ropVar @ Xs2 @ ( butlas7628565150644405725ropVar @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_1059_butlast__append,axiom,
    ! [Ys2: list_P8791008543742759928ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( ( Ys2 = nil_Pr6119456312066593144ropVar )
       => ( ( butlas5098402688846288580ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) )
          = ( butlas5098402688846288580ropVar @ Xs2 ) ) )
      & ( ( Ys2 != nil_Pr6119456312066593144ropVar )
       => ( ( butlas5098402688846288580ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Ys2 ) )
          = ( append4352292717583037453ropVar @ Xs2 @ ( butlas5098402688846288580ropVar @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_1060_butlast__append,axiom,
    ! [Ys2: list_P1665994083459302249pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( ( Ys2 = nil_Pr6413368191065476585pVar_o )
       => ( ( butlas7924156930066306357pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) )
          = ( butlas7924156930066306357pVar_o @ Xs2 ) ) )
      & ( ( Ys2 != nil_Pr6413368191065476585pVar_o )
       => ( ( butlas7924156930066306357pVar_o @ ( append1087384436348988414pVar_o @ Xs2 @ Ys2 ) )
          = ( append1087384436348988414pVar_o @ Xs2 @ ( butlas7924156930066306357pVar_o @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_1061_butlast__append,axiom,
    ! [Ys2: list_l1255154591254203759pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( ( Ys2 = nil_li1310946081133177199pVar_o )
       => ( ( butlas54256187331189435pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) )
          = ( butlas54256187331189435pVar_o @ Xs2 ) ) )
      & ( ( Ys2 != nil_li1310946081133177199pVar_o )
       => ( ( butlas54256187331189435pVar_o @ ( append8069948826146371844pVar_o @ Xs2 @ Ys2 ) )
          = ( append8069948826146371844pVar_o @ Xs2 @ ( butlas54256187331189435pVar_o @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_1062_butlast__append,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( ( Ys2 = nil_li590035233778055285pVar_o )
       => ( ( butlas1161285084838040513pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
          = ( butlas1161285084838040513pVar_o @ Xs2 ) ) )
      & ( ( Ys2 != nil_li590035233778055285pVar_o )
       => ( ( butlas1161285084838040513pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) )
          = ( append8959015459476584842pVar_o @ Xs2 @ ( butlas1161285084838040513pVar_o @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_1063_butlast__append,axiom,
    ! [Ys2: list_nat,Xs2: list_nat] :
      ( ( ( Ys2 = nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
          = ( butlast_nat @ Xs2 ) ) )
      & ( ( Ys2 != nil_nat )
       => ( ( butlast_nat @ ( append_nat @ Xs2 @ Ys2 ) )
          = ( append_nat @ Xs2 @ ( butlast_nat @ Ys2 ) ) ) ) ) ).

% butlast_append
thf(fact_1064_maps__simps_I2_J,axiom,
    ! [F2: nat > list_nat] :
      ( ( maps_nat_nat @ F2 @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1065_maps__simps_I2_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > list_nat] :
      ( ( maps_P3684901262545219122_o_nat @ F2 @ nil_Pr6413368191065476585pVar_o )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1066_maps__simps_I2_J,axiom,
    ! [F2: nat > list_P1665994083459302249pVar_o] :
      ( ( maps_n168770212195511824pVar_o @ F2 @ nil_nat )
      = nil_Pr6413368191065476585pVar_o ) ).

% maps_simps(2)
thf(fact_1067_maps__simps_I2_J,axiom,
    ! [F2: list_P1665994083459302249pVar_o > list_nat] :
      ( ( maps_l127622714395944376_o_nat @ F2 @ nil_li1310946081133177199pVar_o )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1068_maps__simps_I2_J,axiom,
    ! [F2: nat > list_l1255154591254203759pVar_o] :
      ( ( maps_n1657913043809323542pVar_o @ F2 @ nil_nat )
      = nil_li1310946081133177199pVar_o ) ).

% maps_simps(2)
thf(fact_1069_maps__simps_I2_J,axiom,
    ! [F2: produc2642885279816024434ropVar > list_nat] :
      ( ( maps_P2882123423910056385ar_nat @ F2 @ nil_Pr6119456312066593144ropVar )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1070_maps__simps_I2_J,axiom,
    ! [F2: produc8999643685545340131pVar_o > list_P1665994083459302249pVar_o] :
      ( ( maps_P7682357790657813505pVar_o @ F2 @ nil_Pr6413368191065476585pVar_o )
      = nil_Pr6413368191065476585pVar_o ) ).

% maps_simps(2)
thf(fact_1071_maps__simps_I2_J,axiom,
    ! [F2: list_l1255154591254203759pVar_o > list_nat] :
      ( ( maps_l865014085272113342_o_nat @ F2 @ nil_li590035233778055285pVar_o )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_1072_maps__simps_I2_J,axiom,
    ! [F2: nat > list_P8791008543742759928ropVar] :
      ( ( maps_n6440817060062774047ropVar @ F2 @ nil_nat )
      = nil_Pr6119456312066593144ropVar ) ).

% maps_simps(2)
thf(fact_1073_maps__simps_I2_J,axiom,
    ! [F2: nat > list_l985434930288970485pVar_o] :
      ( ( maps_n2783302837639436700pVar_o @ F2 @ nil_nat )
      = nil_li590035233778055285pVar_o ) ).

% maps_simps(2)
thf(fact_1074_partition__P,axiom,
    ! [P: produc2642885279816024434ropVar > $o,Xs2: list_P8791008543742759928ropVar,Yes: list_P8791008543742759928ropVar,No: list_P8791008543742759928ropVar] :
      ( ( ( partit1257260948940242929ropVar @ P @ Xs2 )
        = ( produc5333478016634530327ropVar @ Yes @ No ) )
     => ( ! [X5: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ Yes ) )
           => ( P @ X5 ) )
        & ! [X5: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X5 @ ( set_Pr3946994337432900359ropVar @ No ) )
           => ~ ( P @ X5 ) ) ) ) ).

% partition_P
thf(fact_1075_partition__P,axiom,
    ! [P: list_P723798329145367999ropVar > $o,Xs2: list_l1644708917474426447ropVar,Yes: list_l1644708917474426447ropVar,No: list_l1644708917474426447ropVar] :
      ( ( ( partit3168477439376328320ropVar @ P @ Xs2 )
        = ( produc7125445329903452983ropVar @ Yes @ No ) )
     => ( ! [X5: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ Yes ) )
           => ( P @ X5 ) )
        & ! [X5: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X5 @ ( set_li8604685205718225898ropVar @ No ) )
           => ~ ( P @ X5 ) ) ) ) ).

% partition_P
thf(fact_1076_partition__P,axiom,
    ! [P: list_l1255154591254203759pVar_o > $o,Xs2: list_l985434930288970485pVar_o,Yes: list_l985434930288970485pVar_o,No: list_l985434930288970485pVar_o] :
      ( ( ( partit2223932869413013358pVar_o @ P @ Xs2 )
        = ( produc2080502776689584855pVar_o @ Yes @ No ) )
     => ( ! [X5: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Yes ) )
           => ( P @ X5 ) )
        & ! [X5: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ No ) )
           => ~ ( P @ X5 ) ) ) ) ).

% partition_P
thf(fact_1077_partition__P,axiom,
    ! [P: list_l985434930288970485pVar_o > $o,Xs2: list_l6341520962544925691pVar_o,Yes: list_l6341520962544925691pVar_o,No: list_l6341520962544925691pVar_o] :
      ( ( ( partit6292427544759859572pVar_o @ P @ Xs2 )
        = ( produc504187628453117527pVar_o @ Yes @ No ) )
     => ( ! [X5: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ Yes ) )
           => ( P @ X5 ) )
        & ! [X5: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X5 @ ( set_li4369674465895099530pVar_o @ No ) )
           => ~ ( P @ X5 ) ) ) ) ).

% partition_P
thf(fact_1078_partition__P,axiom,
    ! [P: nat > $o,Xs2: list_nat,Yes: list_nat,No: list_nat] :
      ( ( ( partition_nat @ P @ Xs2 )
        = ( produc2694037385005941721st_nat @ Yes @ No ) )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Yes ) )
           => ( P @ X5 ) )
        & ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ No ) )
           => ~ ( P @ X5 ) ) ) ) ).

% partition_P
thf(fact_1079_Nil2__notin__lex,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ nil_Pr6119456312066593144ropVar ) @ ( lex_Pr821843617573478364ropVar @ R ) ) ).

% Nil2_notin_lex
thf(fact_1080_Nil2__notin__lex,axiom,
    ! [Xs2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ nil_Pr6413368191065476585pVar_o ) @ ( lex_Pr4269406784341172813pVar_o @ R ) ) ).

% Nil2_notin_lex
thf(fact_1081_Nil2__notin__lex,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ nil_li1310946081133177199pVar_o ) @ ( lex_li7886316385931344851pVar_o @ R ) ) ).

% Nil2_notin_lex
thf(fact_1082_Nil2__notin__lex,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ nil_li590035233778055285pVar_o ) @ ( lex_li5300212533296449753pVar_o @ R ) ) ).

% Nil2_notin_lex
thf(fact_1083_Nil2__notin__lex,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ nil_nat ) @ ( lex_nat @ R ) ) ).

% Nil2_notin_lex
thf(fact_1084_Nil__notin__lex,axiom,
    ! [Ys2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys2 ) @ ( lex_Pr821843617573478364ropVar @ R ) ) ).

% Nil_notin_lex
thf(fact_1085_Nil__notin__lex,axiom,
    ! [Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ~ ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ nil_Pr6413368191065476585pVar_o @ Ys2 ) @ ( lex_Pr4269406784341172813pVar_o @ R ) ) ).

% Nil_notin_lex
thf(fact_1086_Nil__notin__lex,axiom,
    ! [Ys2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ~ ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ nil_li1310946081133177199pVar_o @ Ys2 ) @ ( lex_li7886316385931344851pVar_o @ R ) ) ).

% Nil_notin_lex
thf(fact_1087_Nil__notin__lex,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ~ ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ nil_li590035233778055285pVar_o @ Ys2 ) @ ( lex_li5300212533296449753pVar_o @ R ) ) ).

% Nil_notin_lex
thf(fact_1088_Nil__notin__lex,axiom,
    ! [Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ys2 ) @ ( lex_nat @ R ) ) ).

% Nil_notin_lex
thf(fact_1089_lex__append__leftI,axiom,
    ! [Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Ys2 @ Zs ) @ ( lex_li5300212533296449753pVar_o @ R ) )
     => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( append8959015459476584842pVar_o @ Xs2 @ Ys2 ) @ ( append8959015459476584842pVar_o @ Xs2 @ Zs ) ) @ ( lex_li5300212533296449753pVar_o @ R ) ) ) ).

% lex_append_leftI
thf(fact_1090_lex__append__leftI,axiom,
    ! [Ys2: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat,Xs2: list_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lex_nat @ R ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs2 @ Ys2 ) @ ( append_nat @ Xs2 @ Zs ) ) @ ( lex_nat @ R ) ) ) ).

% lex_append_leftI
thf(fact_1091_lex__append__leftI,axiom,
    ! [Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar,R: set_Pr572148484142614943ropVar,Xs2: list_P723798329145367999ropVar] :
      ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Ys2 @ Zs ) @ ( lex_Pr8947373434141532869ropVar @ R ) )
     => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ ( append7708039934470209236ropVar @ Xs2 @ Ys2 ) @ ( append7708039934470209236ropVar @ Xs2 @ Zs ) ) @ ( lex_Pr8947373434141532869ropVar @ R ) ) ) ).

% lex_append_leftI
thf(fact_1092_lexord__partial__trans,axiom,
    ! [Xs2: list_P4002435161011370285od_o_o,R: set_Pr7011910401827753159od_o_o,Ys2: list_P4002435161011370285od_o_o,Zs: list_P4002435161011370285od_o_o] :
      ( ! [X2: product_prod_o_o,Y4: product_prod_o_o,Z3: product_prod_o_o] :
          ( ( member7466972457876170832od_o_o @ X2 @ ( set_Product_prod_o_o2 @ Xs2 ) )
         => ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ Y4 ) @ R )
           => ( ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ Y4 @ Z3 ) @ R )
             => ( member3169104723136097040od_o_o @ ( produc3331439552374987607od_o_o @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member8820170678258176144od_o_o @ ( produc3440030253097803735od_o_o @ Xs2 @ Ys2 ) @ ( lexord2884012491495875354od_o_o @ R ) )
       => ( ( member8820170678258176144od_o_o @ ( produc3440030253097803735od_o_o @ Ys2 @ Zs ) @ ( lexord2884012491495875354od_o_o @ R ) )
         => ( member8820170678258176144od_o_o @ ( produc3440030253097803735od_o_o @ Xs2 @ Zs ) @ ( lexord2884012491495875354od_o_o @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1093_lexord__partial__trans,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
      ( ! [X2: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Z3: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
         => ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y4 ) @ R )
           => ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ Y4 @ Z3 ) @ R )
             => ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( lexord832287969131410277ropVar @ R ) )
       => ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Ys2 @ Zs ) @ ( lexord832287969131410277ropVar @ R ) )
         => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Zs ) @ ( lexord832287969131410277ropVar @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1094_lexord__partial__trans,axiom,
    ! [Xs2: list_l1644708917474426447ropVar,R: set_Pr3948510271449578431ropVar,Ys2: list_l1644708917474426447ropVar,Zs: list_l1644708917474426447ropVar] :
      ( ! [X2: list_P723798329145367999ropVar,Y4: list_P723798329145367999ropVar,Z3: list_P723798329145367999ropVar] :
          ( ( member7375955847912140118ropVar @ X2 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
         => ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ X2 @ Y4 ) @ R )
           => ( ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ Y4 @ Z3 ) @ R )
             => ( member4835615740757207048ropVar @ ( produc4840566532856537623ropVar @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member8130401662325377320ropVar @ ( produc7125445329903452983ropVar @ Xs2 @ Ys2 ) @ ( lexord1232427612563343884ropVar @ R ) )
       => ( ( member8130401662325377320ropVar @ ( produc7125445329903452983ropVar @ Ys2 @ Zs ) @ ( lexord1232427612563343884ropVar @ R ) )
         => ( member8130401662325377320ropVar @ ( produc7125445329903452983ropVar @ Xs2 @ Zs ) @ ( lexord1232427612563343884ropVar @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1095_lexord__partial__trans,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ! [X2: list_l1255154591254203759pVar_o,Y4: list_l1255154591254203759pVar_o,Z3: list_l1255154591254203759pVar_o] :
          ( ( member6267526602072755480pVar_o @ X2 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
         => ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X2 @ Y4 ) @ R )
           => ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Y4 @ Z3 ) @ R )
             => ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( lexord4213358897963926754pVar_o @ R ) )
       => ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Ys2 @ Zs ) @ ( lexord4213358897963926754pVar_o @ R ) )
         => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Zs ) @ ( lexord4213358897963926754pVar_o @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1096_lexord__partial__trans,axiom,
    ! [Xs2: list_l6341520962544925691pVar_o,R: set_Pr7475081812231853383pVar_o,Ys2: list_l6341520962544925691pVar_o,Zs: list_l6341520962544925691pVar_o] :
      ( ! [X2: list_l985434930288970485pVar_o,Y4: list_l985434930288970485pVar_o,Z3: list_l985434930288970485pVar_o] :
          ( ( member9046334641704116766pVar_o @ X2 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
         => ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ X2 @ Y4 ) @ R )
           => ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Y4 @ Z3 ) @ R )
             => ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member5210069903039637520pVar_o @ ( produc504187628453117527pVar_o @ Xs2 @ Ys2 ) @ ( lexord8641875953490177256pVar_o @ R ) )
       => ( ( member5210069903039637520pVar_o @ ( produc504187628453117527pVar_o @ Ys2 @ Zs ) @ ( lexord8641875953490177256pVar_o @ R ) )
         => ( member5210069903039637520pVar_o @ ( produc504187628453117527pVar_o @ Xs2 @ Zs ) @ ( lexord8641875953490177256pVar_o @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1097_lexord__partial__trans,axiom,
    ! [Xs2: list_nat,R: set_Pr1261947904930325089at_nat,Ys2: list_nat,Zs: list_nat] :
      ( ! [X2: nat,Y4: nat,Z3: nat] :
          ( ( member_nat @ X2 @ ( set_nat2 @ Xs2 ) )
         => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R )
           => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ R )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lexord_nat @ R ) )
       => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys2 @ Zs ) @ ( lexord_nat @ R ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Zs ) @ ( lexord_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1098_lexord__partial__trans,axiom,
    ! [Xs2: list_o,R: set_Product_prod_o_o,Ys2: list_o,Zs: list_o] :
      ( ! [X2: $o,Y4: $o,Z3: $o] :
          ( ( member_o @ X2 @ ( set_o2 @ Xs2 ) )
         => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y4 ) @ R )
           => ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ Y4 @ Z3 ) @ R )
             => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( lexord_o @ R ) )
       => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Ys2 @ Zs ) @ ( lexord_o @ R ) )
         => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Zs ) @ ( lexord_o @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1099_lexord__partial__trans,axiom,
    ! [Xs2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar,Ys2: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
      ( ! [X2: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar,Z3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
         => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y4 ) @ R )
           => ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ Y4 @ Z3 ) @ R )
             => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Z3 ) @ R ) ) ) )
     => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( lexord1374882220397993039ropVar @ R ) )
       => ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Ys2 @ Zs ) @ ( lexord1374882220397993039ropVar @ R ) )
         => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Zs ) @ ( lexord1374882220397993039ropVar @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1100_concat__eq__appendD,axiom,
    ! [Xss2: list_list_nat,Ys2: list_nat,Zs: list_nat] :
      ( ( ( concat_nat @ Xss2 )
        = ( append_nat @ Ys2 @ Zs ) )
     => ( ( Xss2 != nil_list_nat )
       => ? [Xss12: list_list_nat,Xs: list_nat,Xs6: list_nat,Xss23: list_list_nat] :
            ( ( Xss2
              = ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs @ Xs6 ) @ Xss23 ) ) )
            & ( Ys2
              = ( append_nat @ ( concat_nat @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append_nat @ Xs6 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1101_concat__eq__appendD,axiom,
    ! [Xss2: list_l6341520962544925691pVar_o,Ys2: list_l985434930288970485pVar_o,Zs: list_l985434930288970485pVar_o] :
      ( ( ( concat8126406963686051632pVar_o @ Xss2 )
        = ( append8959015459476584842pVar_o @ Ys2 @ Zs ) )
     => ( ( Xss2 != nil_li5579170522155219707pVar_o )
       => ? [Xss12: list_l6341520962544925691pVar_o,Xs: list_l985434930288970485pVar_o,Xs6: list_l985434930288970485pVar_o,Xss23: list_l6341520962544925691pVar_o] :
            ( ( Xss2
              = ( append9215492216909589904pVar_o @ Xss12 @ ( cons_l8096070148660426411pVar_o @ ( append8959015459476584842pVar_o @ Xs @ Xs6 ) @ Xss23 ) ) )
            & ( Ys2
              = ( append8959015459476584842pVar_o @ ( concat8126406963686051632pVar_o @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append8959015459476584842pVar_o @ Xs6 @ ( concat8126406963686051632pVar_o @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1102_concat__eq__appendD,axiom,
    ! [Xss2: list_l1644708917474426447ropVar,Ys2: list_P723798329145367999ropVar,Zs: list_P723798329145367999ropVar] :
      ( ( ( concat4611077035641296302ropVar @ Xss2 )
        = ( append7708039934470209236ropVar @ Ys2 @ Zs ) )
     => ( ( Xss2 != nil_li6797326493501628985ropVar )
       => ? [Xss12: list_l1644708917474426447ropVar,Xs: list_P723798329145367999ropVar,Xs6: list_P723798329145367999ropVar,Xss23: list_l1644708917474426447ropVar] :
            ( ( Xss2
              = ( append2757213247338973028ropVar @ Xss12 @ ( cons_l2454487807130495625ropVar @ ( append7708039934470209236ropVar @ Xs @ Xs6 ) @ Xss23 ) ) )
            & ( Ys2
              = ( append7708039934470209236ropVar @ ( concat4611077035641296302ropVar @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append7708039934470209236ropVar @ Xs6 @ ( concat4611077035641296302ropVar @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1103_concat__eq__appendD,axiom,
    ! [Xss2: list_l1255154591254203759pVar_o,Ys2: list_P1665994083459302249pVar_o,Zs: list_P1665994083459302249pVar_o] :
      ( ( ( concat7668843100863076260pVar_o @ Xss2 )
        = ( append1087384436348988414pVar_o @ Ys2 @ Zs ) )
     => ( ( Xss2 != nil_li1310946081133177199pVar_o )
       => ? [Xss12: list_l1255154591254203759pVar_o,Xs: list_P1665994083459302249pVar_o,Xs6: list_P1665994083459302249pVar_o,Xss23: list_l1255154591254203759pVar_o] :
            ( ( Xss2
              = ( append8069948826146371844pVar_o @ Xss12 @ ( cons_l3452452106213086495pVar_o @ ( append1087384436348988414pVar_o @ Xs @ Xs6 ) @ Xss23 ) ) )
            & ( Ys2
              = ( append1087384436348988414pVar_o @ ( concat7668843100863076260pVar_o @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append1087384436348988414pVar_o @ Xs6 @ ( concat7668843100863076260pVar_o @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1104_concat__eq__appendD,axiom,
    ! [Xss2: list_l985434930288970485pVar_o,Ys2: list_l1255154591254203759pVar_o,Zs: list_l1255154591254203759pVar_o] :
      ( ( ( concat8635278358871234986pVar_o @ Xss2 )
        = ( append8069948826146371844pVar_o @ Ys2 @ Zs ) )
     => ( ( Xss2 != nil_li590035233778055285pVar_o )
       => ? [Xss12: list_l985434930288970485pVar_o,Xs: list_l1255154591254203759pVar_o,Xs6: list_l1255154591254203759pVar_o,Xss23: list_l985434930288970485pVar_o] :
            ( ( Xss2
              = ( append8959015459476584842pVar_o @ Xss12 @ ( cons_l7316790070876032037pVar_o @ ( append8069948826146371844pVar_o @ Xs @ Xs6 ) @ Xss23 ) ) )
            & ( Ys2
              = ( append8069948826146371844pVar_o @ ( concat8635278358871234986pVar_o @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append8069948826146371844pVar_o @ Xs6 @ ( concat8635278358871234986pVar_o @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1105_can__select__set__list__ex1,axiom,
    ! [P: produc2642885279816024434ropVar > $o,A4: list_P8791008543742759928ropVar] :
      ( ( can_se3660272072692622319ropVar @ P @ ( set_Pr3946994337432900359ropVar @ A4 ) )
      = ( list_e1423168564539623099ropVar @ P @ A4 ) ) ).

% can_select_set_list_ex1
thf(fact_1106_can__select__set__list__ex1,axiom,
    ! [P: list_P723798329145367999ropVar > $o,A4: list_l1644708917474426447ropVar] :
      ( ( can_se8035874654641615106ropVar @ P @ ( set_li8604685205718225898ropVar @ A4 ) )
      = ( list_e8541863376151676726ropVar @ P @ A4 ) ) ).

% can_select_set_list_ex1
thf(fact_1107_can__select__set__list__ex1,axiom,
    ! [P: list_l1255154591254203759pVar_o > $o,A4: list_l985434930288970485pVar_o] :
      ( ( can_se537254311553704556pVar_o @ P @ ( set_li3549506505754852740pVar_o @ A4 ) )
      = ( list_e5196591910180150072pVar_o @ P @ A4 ) ) ).

% can_select_set_list_ex1
thf(fact_1108_can__select__set__list__ex1,axiom,
    ! [P: list_l985434930288970485pVar_o > $o,A4: list_l6341520962544925691pVar_o] :
      ( ( can_se5210340557449723250pVar_o @ P @ ( set_li4369674465895099530pVar_o @ A4 ) )
      = ( list_e115186782557003838pVar_o @ P @ A4 ) ) ).

% can_select_set_list_ex1
thf(fact_1109_can__select__set__list__ex1,axiom,
    ! [P: nat > $o,A4: list_nat] :
      ( ( can_select_nat @ P @ ( set_nat2 @ A4 ) )
      = ( list_ex1_nat @ P @ A4 ) ) ).

% can_select_set_list_ex1
thf(fact_1110_the__elem__set,axiom,
    ! [X4: list_P723798329145367999ropVar] :
      ( ( the_el6047765288001715082ropVar @ ( set_li8604685205718225898ropVar @ ( cons_l2454487807130495625ropVar @ X4 @ nil_li6797326493501628985ropVar ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_1111_the__elem__set,axiom,
    ! [X4: list_l985434930288970485pVar_o] :
      ( ( the_el8232945750843460842pVar_o @ ( set_li4369674465895099530pVar_o @ ( cons_l8096070148660426411pVar_o @ X4 @ nil_li5579170522155219707pVar_o ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_1112_the__elem__set,axiom,
    ! [X4: produc2642885279816024434ropVar] :
      ( ( the_el184889641172005735ropVar @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ nil_Pr6119456312066593144ropVar ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_1113_the__elem__set,axiom,
    ! [X4: produc8999643685545340131pVar_o] :
      ( ( the_el2703590303065123416pVar_o @ ( set_Pr2838981755425241592pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_1114_the__elem__set,axiom,
    ! [X4: list_P1665994083459302249pVar_o] :
      ( ( the_el4167730460182282846pVar_o @ ( set_li5575011780610867710pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ nil_li1310946081133177199pVar_o ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_1115_the__elem__set,axiom,
    ! [X4: list_l1255154591254203759pVar_o] :
      ( ( the_el3116185405049327076pVar_o @ ( set_li3549506505754852740pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ nil_li590035233778055285pVar_o ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_1116_the__elem__set,axiom,
    ! [X4: nat] :
      ( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X4 @ nil_nat ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_1117_Cons__in__lex,axiom,
    ! [X4: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Y3: produc8999643685545340131pVar_o,Ys2: list_P1665994083459302249pVar_o,R: set_Pr4574656050966626247pVar_o] :
      ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ ( cons_P6860203055867761049pVar_o @ X4 @ Xs2 ) @ ( cons_P6860203055867761049pVar_o @ Y3 @ Ys2 ) ) @ ( lex_Pr4269406784341172813pVar_o @ R ) )
      = ( ( ( member693507358219557392pVar_o @ ( produc2166305868615414871pVar_o @ X4 @ Y3 ) @ R )
          & ( ( size_s2015474115514096597pVar_o @ Xs2 )
            = ( size_s2015474115514096597pVar_o @ Ys2 ) ) )
        | ( ( X4 = Y3 )
          & ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ Xs2 @ Ys2 ) @ ( lex_Pr4269406784341172813pVar_o @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1118_Cons__in__lex,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Y3: list_P1665994083459302249pVar_o,Ys2: list_l1255154591254203759pVar_o,R: set_Pr8855275280618702407pVar_o] :
      ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ ( cons_l3452452106213086495pVar_o @ X4 @ Xs2 ) @ ( cons_l3452452106213086495pVar_o @ Y3 @ Ys2 ) ) @ ( lex_li7886316385931344851pVar_o @ R ) )
      = ( ( ( member9103552303253245072pVar_o @ ( produc9064796567173055959pVar_o @ X4 @ Y3 ) @ R )
          & ( ( size_s1343900380416901083pVar_o @ Xs2 )
            = ( size_s1343900380416901083pVar_o @ Ys2 ) ) )
        | ( ( X4 = Y3 )
          & ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ Xs2 @ Ys2 ) @ ( lex_li7886316385931344851pVar_o @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1119_Cons__in__lex,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o,Ys2: list_l985434930288970485pVar_o,R: set_Pr3139130378708280007pVar_o] :
      ( ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ ( cons_l7316790070876032037pVar_o @ X4 @ Xs2 ) @ ( cons_l7316790070876032037pVar_o @ Y3 @ Ys2 ) ) @ ( lex_li5300212533296449753pVar_o @ R ) )
      = ( ( ( member8387026561508807952pVar_o @ ( produc5036572322992794967pVar_o @ X4 @ Y3 ) @ R )
          & ( ( size_s6265379629157799777pVar_o @ Xs2 )
            = ( size_s6265379629157799777pVar_o @ Ys2 ) ) )
        | ( ( X4 = Y3 )
          & ( member2119952159244120976pVar_o @ ( produc2080502776689584855pVar_o @ Xs2 @ Ys2 ) @ ( lex_li5300212533296449753pVar_o @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1120_Cons__in__lex,axiom,
    ! [X4: $o,Xs2: list_o,Y3: $o,Ys2: list_o,R: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( cons_o @ X4 @ Xs2 ) @ ( cons_o @ Y3 @ Ys2 ) ) @ ( lex_o @ R ) )
      = ( ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ R )
          & ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Ys2 ) ) )
        | ( ( X4 = Y3 )
          & ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs2 @ Ys2 ) @ ( lex_o @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1121_Cons__in__lex,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys2: list_M6740372356978808802ropVar,R: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X4 @ Xs2 ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys2 ) ) @ ( lex_Mu6218539406383542982ropVar @ R ) )
      = ( ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X4 @ Y3 ) @ R )
          & ( ( size_s1943289338788136526ropVar @ Xs2 )
            = ( size_s1943289338788136526ropVar @ Ys2 ) ) )
        | ( ( X4 = Y3 )
          & ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs2 @ Ys2 ) @ ( lex_Mu6218539406383542982ropVar @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1122_Cons__in__lex,axiom,
    ! [X4: nat,Xs2: list_nat,Y3: nat,Ys2: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X4 @ Xs2 ) @ ( cons_nat @ Y3 @ Ys2 ) ) @ ( lex_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ R )
          & ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys2 ) ) )
        | ( ( X4 = Y3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs2 @ Ys2 ) @ ( lex_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1123_Cons__in__lex,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X4 @ Xs2 ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys2 ) ) @ ( lex_Pr821843617573478364ropVar @ R ) )
      = ( ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X4 @ Y3 ) @ R )
          & ( ( size_s6752116297430514276ropVar @ Xs2 )
            = ( size_s6752116297430514276ropVar @ Ys2 ) ) )
        | ( ( X4 = Y3 )
          & ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs2 @ Ys2 ) @ ( lex_Pr821843617573478364ropVar @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1124_concat__conv__foldr,axiom,
    ( concat4611077035641296302ropVar
    = ( ^ [Xss3: list_l1644708917474426447ropVar] : ( foldr_3739391264446828839ropVar @ append7708039934470209236ropVar @ Xss3 @ nil_Pr7606595143512524841ropVar ) ) ) ).

% concat_conv_foldr
thf(fact_1125_concat__conv__foldr,axiom,
    ( concat1830633707478507187ropVar
    = ( ^ [Xss3: list_l3415868495287753214ropVar] : ( foldr_8624739855404409127ropVar @ append4352292717583037453ropVar @ Xss3 @ nil_Pr6119456312066593144ropVar ) ) ) ).

% concat_conv_foldr
thf(fact_1126_concat__conv__foldr,axiom,
    ( concat7668843100863076260pVar_o
    = ( ^ [Xss3: list_l1255154591254203759pVar_o] : ( foldr_5453194195076639975pVar_o @ append1087384436348988414pVar_o @ Xss3 @ nil_Pr6413368191065476585pVar_o ) ) ) ).

% concat_conv_foldr
thf(fact_1127_concat__conv__foldr,axiom,
    ( concat8635278358871234986pVar_o
    = ( ^ [Xss3: list_l985434930288970485pVar_o] : ( foldr_5719581397002175591pVar_o @ append8069948826146371844pVar_o @ Xss3 @ nil_li1310946081133177199pVar_o ) ) ) ).

% concat_conv_foldr
thf(fact_1128_concat__conv__foldr,axiom,
    ( concat8126406963686051632pVar_o
    = ( ^ [Xss3: list_l6341520962544925691pVar_o] : ( foldr_8054164109660582375pVar_o @ append8959015459476584842pVar_o @ Xss3 @ nil_li590035233778055285pVar_o ) ) ) ).

% concat_conv_foldr
thf(fact_1129_concat__conv__foldr,axiom,
    ( concat_nat
    = ( ^ [Xss3: list_list_nat] : ( foldr_6871341030409798377st_nat @ append_nat @ Xss3 @ nil_nat ) ) ) ).

% concat_conv_foldr
thf(fact_1130_SuccI,axiom,
    ! [Kl: list_M6740372356978808802ropVar,K: multis3193088007478089820ropVar,Kl2: set_li3089797549399498434ropVar] :
      ( ( member490224382042672267ropVar @ ( append2684835082924936439ropVar @ Kl @ ( cons_M4277272395645347858ropVar @ K @ nil_Mu4132399263956683874ropVar ) ) @ Kl2 )
     => ( member8421201611525043589ropVar @ K @ ( bNF_Gr6307721265529545847ropVar @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1131_SuccI,axiom,
    ! [Kl: list_P4002435161011370285od_o_o,K: product_prod_o_o,Kl2: set_li6534384796922688653od_o_o] :
      ( ( member1154424878073987030od_o_o @ ( append2614242729457001410od_o_o @ Kl @ ( cons_P8766293264717362397od_o_o @ K @ nil_Product_prod_o_o ) ) @ Kl2 )
     => ( member7466972457876170832od_o_o @ K @ ( bNF_Gr7432766864382442306od_o_o @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1132_SuccI,axiom,
    ! [Kl: list_P723798329145367999ropVar,K: propos3600597766819905199ropVar,Kl2: set_li9140175473581139189ropVar] :
      ( ( member7375955847912140118ropVar @ ( append7708039934470209236ropVar @ Kl @ ( cons_P6757824865275295865ropVar @ K @ nil_Pr7606595143512524841ropVar ) ) @ Kl2 )
     => ( member1796676888578033094ropVar @ K @ ( bNF_Gr6265355799493510996ropVar @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1133_SuccI,axiom,
    ! [Kl: list_P8791008543742759928ropVar,K: produc2642885279816024434ropVar,Kl2: set_li8998331099879477720ropVar] :
      ( ( member6236339133593825953ropVar @ ( append4352292717583037453ropVar @ Kl @ ( cons_P1504208554336013608ropVar @ K @ nil_Pr6119456312066593144ropVar ) ) @ Kl2 )
     => ( member9210975342120487067ropVar @ K @ ( bNF_Gr2655190671417346957ropVar @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1134_SuccI,axiom,
    ! [Kl: list_P1665994083459302249pVar_o,K: produc8999643685545340131pVar_o,Kl2: set_li254716981354726473pVar_o] :
      ( ( member7042831661676915602pVar_o @ ( append1087384436348988414pVar_o @ Kl @ ( cons_P6860203055867761049pVar_o @ K @ nil_Pr6413368191065476585pVar_o ) ) @ Kl2 )
     => ( member1729440322722938252pVar_o @ K @ ( bNF_Gr3337489131723850622pVar_o @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1135_SuccI,axiom,
    ! [Kl: list_l1255154591254203759pVar_o,K: list_P1665994083459302249pVar_o,Kl2: set_li3954596911114931407pVar_o] :
      ( ( member6267526602072755480pVar_o @ ( append8069948826146371844pVar_o @ Kl @ ( cons_l3452452106213086495pVar_o @ K @ nil_li1310946081133177199pVar_o ) ) @ Kl2 )
     => ( member7042831661676915602pVar_o @ K @ ( bNF_Gr5940263876698902660pVar_o @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1136_SuccI,axiom,
    ! [Kl: list_l985434930288970485pVar_o,K: list_l1255154591254203759pVar_o,Kl2: set_li5493982441917991125pVar_o] :
      ( ( member9046334641704116766pVar_o @ ( append8959015459476584842pVar_o @ Kl @ ( cons_l7316790070876032037pVar_o @ K @ nil_li590035233778055285pVar_o ) ) @ Kl2 )
     => ( member6267526602072755480pVar_o @ K @ ( bNF_Gr5278975096840786186pVar_o @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1137_SuccI,axiom,
    ! [Kl: list_nat,K: nat,Kl2: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 )
     => ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_1138_SuccD,axiom,
    ! [K: multis3193088007478089820ropVar,Kl2: set_li3089797549399498434ropVar,Kl: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ K @ ( bNF_Gr6307721265529545847ropVar @ Kl2 @ Kl ) )
     => ( member490224382042672267ropVar @ ( append2684835082924936439ropVar @ Kl @ ( cons_M4277272395645347858ropVar @ K @ nil_Mu4132399263956683874ropVar ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1139_SuccD,axiom,
    ! [K: product_prod_o_o,Kl2: set_li6534384796922688653od_o_o,Kl: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ K @ ( bNF_Gr7432766864382442306od_o_o @ Kl2 @ Kl ) )
     => ( member1154424878073987030od_o_o @ ( append2614242729457001410od_o_o @ Kl @ ( cons_P8766293264717362397od_o_o @ K @ nil_Product_prod_o_o ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1140_SuccD,axiom,
    ! [K: propos3600597766819905199ropVar,Kl2: set_li9140175473581139189ropVar,Kl: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ K @ ( bNF_Gr6265355799493510996ropVar @ Kl2 @ Kl ) )
     => ( member7375955847912140118ropVar @ ( append7708039934470209236ropVar @ Kl @ ( cons_P6757824865275295865ropVar @ K @ nil_Pr7606595143512524841ropVar ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1141_SuccD,axiom,
    ! [K: produc2642885279816024434ropVar,Kl2: set_li8998331099879477720ropVar,Kl: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ K @ ( bNF_Gr2655190671417346957ropVar @ Kl2 @ Kl ) )
     => ( member6236339133593825953ropVar @ ( append4352292717583037453ropVar @ Kl @ ( cons_P1504208554336013608ropVar @ K @ nil_Pr6119456312066593144ropVar ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1142_SuccD,axiom,
    ! [K: produc8999643685545340131pVar_o,Kl2: set_li254716981354726473pVar_o,Kl: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ K @ ( bNF_Gr3337489131723850622pVar_o @ Kl2 @ Kl ) )
     => ( member7042831661676915602pVar_o @ ( append1087384436348988414pVar_o @ Kl @ ( cons_P6860203055867761049pVar_o @ K @ nil_Pr6413368191065476585pVar_o ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1143_SuccD,axiom,
    ! [K: list_P1665994083459302249pVar_o,Kl2: set_li3954596911114931407pVar_o,Kl: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ K @ ( bNF_Gr5940263876698902660pVar_o @ Kl2 @ Kl ) )
     => ( member6267526602072755480pVar_o @ ( append8069948826146371844pVar_o @ Kl @ ( cons_l3452452106213086495pVar_o @ K @ nil_li1310946081133177199pVar_o ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1144_SuccD,axiom,
    ! [K: list_l1255154591254203759pVar_o,Kl2: set_li5493982441917991125pVar_o,Kl: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ K @ ( bNF_Gr5278975096840786186pVar_o @ Kl2 @ Kl ) )
     => ( member9046334641704116766pVar_o @ ( append8959015459476584842pVar_o @ Kl @ ( cons_l7316790070876032037pVar_o @ K @ nil_li590035233778055285pVar_o ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1145_SuccD,axiom,
    ! [K: nat,Kl2: set_list_nat,Kl: list_nat] :
      ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl2 @ Kl ) )
     => ( member_list_nat @ ( append_nat @ Kl @ ( cons_nat @ K @ nil_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_1146_remove1__split,axiom,
    ! [A: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ A @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ( ( ( remove5793944696144199672ropVar @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_M6740372356978808802ropVar,Rs: list_M6740372356978808802ropVar] :
              ( ( Xs2
                = ( append2684835082924936439ropVar @ Ls @ ( cons_M4277272395645347858ropVar @ A @ Rs ) ) )
              & ~ ( member8421201611525043589ropVar @ A @ ( set_Mu146256361055586289ropVar @ Ls ) )
              & ( Ys2
                = ( append2684835082924936439ropVar @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1147_remove1__split,axiom,
    ! [A: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o,Ys2: list_P4002435161011370285od_o_o] :
      ( ( member7466972457876170832od_o_o @ A @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ( ( ( remove8878500798450800835od_o_o @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_P4002435161011370285od_o_o,Rs: list_P4002435161011370285od_o_o] :
              ( ( Xs2
                = ( append2614242729457001410od_o_o @ Ls @ ( cons_P8766293264717362397od_o_o @ A @ Rs ) ) )
              & ~ ( member7466972457876170832od_o_o @ A @ ( set_Product_prod_o_o2 @ Ls ) )
              & ( Ys2
                = ( append2614242729457001410od_o_o @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1148_remove1__split,axiom,
    ! [A: propos3600597766819905199ropVar,Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar] :
      ( ( member1796676888578033094ropVar @ A @ ( set_Pr342157451780195930ropVar @ Xs2 ) )
     => ( ( ( remove251651604638340627ropVar @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_P723798329145367999ropVar,Rs: list_P723798329145367999ropVar] :
              ( ( Xs2
                = ( append7708039934470209236ropVar @ Ls @ ( cons_P6757824865275295865ropVar @ A @ Rs ) ) )
              & ~ ( member1796676888578033094ropVar @ A @ ( set_Pr342157451780195930ropVar @ Ls ) )
              & ( Ys2
                = ( append7708039934470209236ropVar @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1149_remove1__split,axiom,
    ! [A: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar,Ys2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ A @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ( ( ( remove4743378409927258915ropVar @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_l1644708917474426447ropVar,Rs: list_l1644708917474426447ropVar] :
              ( ( Xs2
                = ( append2757213247338973028ropVar @ Ls @ ( cons_l2454487807130495625ropVar @ A @ Rs ) ) )
              & ~ ( member7375955847912140118ropVar @ A @ ( set_li8604685205718225898ropVar @ Ls ) )
              & ( Ys2
                = ( append2757213247338973028ropVar @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1150_remove1__split,axiom,
    ! [A: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o,Ys2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ A @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ( ( ( remove8904596938398812817pVar_o @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_l6341520962544925691pVar_o,Rs: list_l6341520962544925691pVar_o] :
              ( ( Xs2
                = ( append9215492216909589904pVar_o @ Ls @ ( cons_l8096070148660426411pVar_o @ A @ Rs ) ) )
              & ~ ( member9046334641704116766pVar_o @ A @ ( set_li4369674465895099530pVar_o @ Ls ) )
              & ( Ys2
                = ( append9215492216909589904pVar_o @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1151_remove1__split,axiom,
    ! [A: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ A @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ( ( ( remove1876521362034606350ropVar @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_P8791008543742759928ropVar,Rs: list_P8791008543742759928ropVar] :
              ( ( Xs2
                = ( append4352292717583037453ropVar @ Ls @ ( cons_P1504208554336013608ropVar @ A @ Rs ) ) )
              & ~ ( member9210975342120487067ropVar @ A @ ( set_Pr3946994337432900359ropVar @ Ls ) )
              & ( Ys2
                = ( append4352292717583037453ropVar @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1152_remove1__split,axiom,
    ! [A: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o,Ys2: list_P1665994083459302249pVar_o] :
      ( ( member1729440322722938252pVar_o @ A @ ( set_Pr2838981755425241592pVar_o @ Xs2 ) )
     => ( ( ( remove4799796591832458879pVar_o @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_P1665994083459302249pVar_o,Rs: list_P1665994083459302249pVar_o] :
              ( ( Xs2
                = ( append1087384436348988414pVar_o @ Ls @ ( cons_P6860203055867761049pVar_o @ A @ Rs ) ) )
              & ~ ( member1729440322722938252pVar_o @ A @ ( set_Pr2838981755425241592pVar_o @ Ls ) )
              & ( Ys2
                = ( append1087384436348988414pVar_o @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1153_remove1__split,axiom,
    ! [A: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o,Ys2: list_l1255154591254203759pVar_o] :
      ( ( member7042831661676915602pVar_o @ A @ ( set_li5575011780610867710pVar_o @ Xs2 ) )
     => ( ( ( remove5341357285405916933pVar_o @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_l1255154591254203759pVar_o,Rs: list_l1255154591254203759pVar_o] :
              ( ( Xs2
                = ( append8069948826146371844pVar_o @ Ls @ ( cons_l3452452106213086495pVar_o @ A @ Rs ) ) )
              & ~ ( member7042831661676915602pVar_o @ A @ ( set_li5575011780610867710pVar_o @ Ls ) )
              & ( Ys2
                = ( append8069948826146371844pVar_o @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1154_remove1__split,axiom,
    ! [A: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ A @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ( ( ( remove3721140173947915019pVar_o @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_l985434930288970485pVar_o,Rs: list_l985434930288970485pVar_o] :
              ( ( Xs2
                = ( append8959015459476584842pVar_o @ Ls @ ( cons_l7316790070876032037pVar_o @ A @ Rs ) ) )
              & ~ ( member6267526602072755480pVar_o @ A @ ( set_li3549506505754852740pVar_o @ Ls ) )
              & ( Ys2
                = ( append8959015459476584842pVar_o @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1155_remove1__split,axiom,
    ! [A: nat,Xs2: list_nat,Ys2: list_nat] :
      ( ( member_nat @ A @ ( set_nat2 @ Xs2 ) )
     => ( ( ( remove1_nat @ A @ Xs2 )
          = Ys2 )
        = ( ? [Ls: list_nat,Rs: list_nat] :
              ( ( Xs2
                = ( append_nat @ Ls @ ( cons_nat @ A @ Rs ) ) )
              & ~ ( member_nat @ A @ ( set_nat2 @ Ls ) )
              & ( Ys2
                = ( append_nat @ Ls @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1156_lexord__lex,axiom,
    ! [X4: list_nat,Y3: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ Y3 ) @ ( lex_nat @ R ) )
      = ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ X4 @ Y3 ) @ ( lexord_nat @ R ) )
        & ( ( size_size_list_nat @ X4 )
          = ( size_size_list_nat @ Y3 ) ) ) ) ).

% lexord_lex
thf(fact_1157_lexord__lex,axiom,
    ! [X4: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar,R: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ X4 @ Y3 ) @ ( lex_Pr821843617573478364ropVar @ R ) )
      = ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ X4 @ Y3 ) @ ( lexord832287969131410277ropVar @ R ) )
        & ( ( size_s6752116297430514276ropVar @ X4 )
          = ( size_s6752116297430514276ropVar @ Y3 ) ) ) ) ).

% lexord_lex
thf(fact_1158_append__eq__append__conv,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Ys2: list_l985434930288970485pVar_o,Us: list_l985434930288970485pVar_o,Vs: list_l985434930288970485pVar_o] :
      ( ( ( ( size_s6265379629157799777pVar_o @ Xs2 )
          = ( size_s6265379629157799777pVar_o @ Ys2 ) )
        | ( ( size_s6265379629157799777pVar_o @ Us )
          = ( size_s6265379629157799777pVar_o @ Vs ) ) )
     => ( ( ( append8959015459476584842pVar_o @ Xs2 @ Us )
          = ( append8959015459476584842pVar_o @ Ys2 @ Vs ) )
        = ( ( Xs2 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1159_append__eq__append__conv,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Ys2: list_P723798329145367999ropVar,Us: list_P723798329145367999ropVar,Vs: list_P723798329145367999ropVar] :
      ( ( ( ( size_s8966483601851171411ropVar @ Xs2 )
          = ( size_s8966483601851171411ropVar @ Ys2 ) )
        | ( ( size_s8966483601851171411ropVar @ Us )
          = ( size_s8966483601851171411ropVar @ Vs ) ) )
     => ( ( ( append7708039934470209236ropVar @ Xs2 @ Us )
          = ( append7708039934470209236ropVar @ Ys2 @ Vs ) )
        = ( ( Xs2 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1160_append__eq__append__conv,axiom,
    ! [Xs2: list_nat,Ys2: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs2 )
          = ( size_size_list_nat @ Ys2 ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs2 @ Us )
          = ( append_nat @ Ys2 @ Vs ) )
        = ( ( Xs2 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1161_append__eq__append__conv,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Us: list_P8791008543742759928ropVar,Vs: list_P8791008543742759928ropVar] :
      ( ( ( ( size_s6752116297430514276ropVar @ Xs2 )
          = ( size_s6752116297430514276ropVar @ Ys2 ) )
        | ( ( size_s6752116297430514276ropVar @ Us )
          = ( size_s6752116297430514276ropVar @ Vs ) ) )
     => ( ( ( append4352292717583037453ropVar @ Xs2 @ Us )
          = ( append4352292717583037453ropVar @ Ys2 @ Vs ) )
        = ( ( Xs2 = Ys2 )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1162_in__set__remove1,axiom,
    ! [A: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ( A != B )
     => ( ( member8421201611525043589ropVar @ A @ ( set_Mu146256361055586289ropVar @ ( remove5793944696144199672ropVar @ B @ Xs2 ) ) )
        = ( member8421201611525043589ropVar @ A @ ( set_Mu146256361055586289ropVar @ Xs2 ) ) ) ) ).

% in_set_remove1
thf(fact_1163_in__set__remove1,axiom,
    ! [A: product_prod_o_o,B: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ( A != B )
     => ( ( member7466972457876170832od_o_o @ A @ ( set_Product_prod_o_o2 @ ( remove8878500798450800835od_o_o @ B @ Xs2 ) ) )
        = ( member7466972457876170832od_o_o @ A @ ( set_Product_prod_o_o2 @ Xs2 ) ) ) ) ).

% in_set_remove1
thf(fact_1164_in__set__remove1,axiom,
    ! [A: produc2642885279816024434ropVar,B: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( A != B )
     => ( ( member9210975342120487067ropVar @ A @ ( set_Pr3946994337432900359ropVar @ ( remove1876521362034606350ropVar @ B @ Xs2 ) ) )
        = ( member9210975342120487067ropVar @ A @ ( set_Pr3946994337432900359ropVar @ Xs2 ) ) ) ) ).

% in_set_remove1
thf(fact_1165_in__set__remove1,axiom,
    ! [A: list_P723798329145367999ropVar,B: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ( A != B )
     => ( ( member7375955847912140118ropVar @ A @ ( set_li8604685205718225898ropVar @ ( remove4743378409927258915ropVar @ B @ Xs2 ) ) )
        = ( member7375955847912140118ropVar @ A @ ( set_li8604685205718225898ropVar @ Xs2 ) ) ) ) ).

% in_set_remove1
thf(fact_1166_in__set__remove1,axiom,
    ! [A: list_l1255154591254203759pVar_o,B: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( A != B )
     => ( ( member6267526602072755480pVar_o @ A @ ( set_li3549506505754852740pVar_o @ ( remove3721140173947915019pVar_o @ B @ Xs2 ) ) )
        = ( member6267526602072755480pVar_o @ A @ ( set_li3549506505754852740pVar_o @ Xs2 ) ) ) ) ).

% in_set_remove1
thf(fact_1167_in__set__remove1,axiom,
    ! [A: list_l985434930288970485pVar_o,B: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ( A != B )
     => ( ( member9046334641704116766pVar_o @ A @ ( set_li4369674465895099530pVar_o @ ( remove8904596938398812817pVar_o @ B @ Xs2 ) ) )
        = ( member9046334641704116766pVar_o @ A @ ( set_li4369674465895099530pVar_o @ Xs2 ) ) ) ) ).

% in_set_remove1
thf(fact_1168_in__set__remove1,axiom,
    ! [A: nat,B: nat,Xs2: list_nat] :
      ( ( A != B )
     => ( ( member_nat @ A @ ( set_nat2 @ ( remove1_nat @ B @ Xs2 ) ) )
        = ( member_nat @ A @ ( set_nat2 @ Xs2 ) ) ) ) ).

% in_set_remove1
thf(fact_1169_length__rotate1,axiom,
    ! [Xs2: list_nat] :
      ( ( size_size_list_nat @ ( rotate1_nat @ Xs2 ) )
      = ( size_size_list_nat @ Xs2 ) ) ).

% length_rotate1
thf(fact_1170_length__rotate1,axiom,
    ! [Xs2: list_P8791008543742759928ropVar] :
      ( ( size_s6752116297430514276ropVar @ ( rotate7816721141519739205ropVar @ Xs2 ) )
      = ( size_s6752116297430514276ropVar @ Xs2 ) ) ).

% length_rotate1
thf(fact_1171_neq__if__length__neq,axiom,
    ! [Xs2: list_nat,Ys2: list_nat] :
      ( ( ( size_size_list_nat @ Xs2 )
       != ( size_size_list_nat @ Ys2 ) )
     => ( Xs2 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_1172_neq__if__length__neq,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
       != ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( Xs2 != Ys2 ) ) ).

% neq_if_length_neq
thf(fact_1173_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_nat] :
      ( ( size_size_list_nat @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_1174_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs: list_P8791008543742759928ropVar] :
      ( ( size_s6752116297430514276ropVar @ Xs )
      = N ) ).

% Ex_list_of_length
thf(fact_1175_in__set__product__lists__length,axiom,
    ! [Xs2: list_P723798329145367999ropVar,Xss2: list_l1644708917474426447ropVar] :
      ( ( member7375955847912140118ropVar @ Xs2 @ ( set_li8604685205718225898ropVar @ ( produc6575831517920476800ropVar @ Xss2 ) ) )
     => ( ( size_s8966483601851171411ropVar @ Xs2 )
        = ( size_s2237552474102614243ropVar @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_1176_in__set__product__lists__length,axiom,
    ! [Xs2: list_l1255154591254203759pVar_o,Xss2: list_l985434930288970485pVar_o] :
      ( ( member6267526602072755480pVar_o @ Xs2 @ ( set_li3549506505754852740pVar_o @ ( produc8865489164070834776pVar_o @ Xss2 ) ) )
     => ( ( size_s1343900380416901083pVar_o @ Xs2 )
        = ( size_s6265379629157799777pVar_o @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_1177_in__set__product__lists__length,axiom,
    ! [Xs2: list_l985434930288970485pVar_o,Xss2: list_l6341520962544925691pVar_o] :
      ( ( member9046334641704116766pVar_o @ Xs2 @ ( set_li4369674465895099530pVar_o @ ( produc7231796756852922590pVar_o @ Xss2 ) ) )
     => ( ( size_s6265379629157799777pVar_o @ Xs2 )
        = ( size_s780392743736205927pVar_o @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_1178_in__set__product__lists__length,axiom,
    ! [Xs2: list_nat,Xss2: list_list_nat] :
      ( ( member_list_nat @ Xs2 @ ( set_list_nat2 @ ( product_lists_nat @ Xss2 ) ) )
     => ( ( size_size_list_nat @ Xs2 )
        = ( size_s3023201423986296836st_nat @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_1179_in__set__product__lists__length,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Xss2: list_l3415868495287753214ropVar] :
      ( ( member6236339133593825953ropVar @ Xs2 @ ( set_li6509544621384488717ropVar @ ( produc2956654850250905953ropVar @ Xss2 ) ) )
     => ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6779274088989432426ropVar @ Xss2 ) ) ) ).

% in_set_product_lists_length
thf(fact_1180_can__select__def,axiom,
    ( can_se4489006107803666649ropVar
    = ( ^ [P4: multis3193088007478089820ropVar > $o,A6: set_Mu5658373669842198716ropVar] :
        ? [X3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X3 @ A6 )
          & ( P4 @ X3 )
          & ! [Y2: multis3193088007478089820ropVar] :
              ( ( ( member8421201611525043589ropVar @ Y2 @ A6 )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_1181_can__select__def,axiom,
    ( can_se3332146993876558756od_o_o
    = ( ^ [P4: product_prod_o_o > $o,A6: set_Product_prod_o_o] :
        ? [X3: product_prod_o_o] :
          ( ( member7466972457876170832od_o_o @ X3 @ A6 )
          & ( P4 @ X3 )
          & ! [Y2: product_prod_o_o] :
              ( ( ( member7466972457876170832od_o_o @ Y2 @ A6 )
                & ( P4 @ Y2 ) )
             => ( Y2 = X3 ) ) ) ) ) ).

% can_select_def
thf(fact_1182_remove1_Osimps_I2_J,axiom,
    ! [X4: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ( ( X4 = Y3 )
       => ( ( remove1876521362034606350ropVar @ X4 @ ( cons_P1504208554336013608ropVar @ Y3 @ Xs2 ) )
          = Xs2 ) )
      & ( ( X4 != Y3 )
       => ( ( remove1876521362034606350ropVar @ X4 @ ( cons_P1504208554336013608ropVar @ Y3 @ Xs2 ) )
          = ( cons_P1504208554336013608ropVar @ Y3 @ ( remove1876521362034606350ropVar @ X4 @ Xs2 ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_1183_remove1_Osimps_I2_J,axiom,
    ! [X4: produc8999643685545340131pVar_o,Y3: produc8999643685545340131pVar_o,Xs2: list_P1665994083459302249pVar_o] :
      ( ( ( X4 = Y3 )
       => ( ( remove4799796591832458879pVar_o @ X4 @ ( cons_P6860203055867761049pVar_o @ Y3 @ Xs2 ) )
          = Xs2 ) )
      & ( ( X4 != Y3 )
       => ( ( remove4799796591832458879pVar_o @ X4 @ ( cons_P6860203055867761049pVar_o @ Y3 @ Xs2 ) )
          = ( cons_P6860203055867761049pVar_o @ Y3 @ ( remove4799796591832458879pVar_o @ X4 @ Xs2 ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_1184_remove1_Osimps_I2_J,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o,Xs2: list_l1255154591254203759pVar_o] :
      ( ( ( X4 = Y3 )
       => ( ( remove5341357285405916933pVar_o @ X4 @ ( cons_l3452452106213086495pVar_o @ Y3 @ Xs2 ) )
          = Xs2 ) )
      & ( ( X4 != Y3 )
       => ( ( remove5341357285405916933pVar_o @ X4 @ ( cons_l3452452106213086495pVar_o @ Y3 @ Xs2 ) )
          = ( cons_l3452452106213086495pVar_o @ Y3 @ ( remove5341357285405916933pVar_o @ X4 @ Xs2 ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_1185_remove1_Osimps_I2_J,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ( ( X4 = Y3 )
       => ( ( remove3721140173947915019pVar_o @ X4 @ ( cons_l7316790070876032037pVar_o @ Y3 @ Xs2 ) )
          = Xs2 ) )
      & ( ( X4 != Y3 )
       => ( ( remove3721140173947915019pVar_o @ X4 @ ( cons_l7316790070876032037pVar_o @ Y3 @ Xs2 ) )
          = ( cons_l7316790070876032037pVar_o @ Y3 @ ( remove3721140173947915019pVar_o @ X4 @ Xs2 ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_1186_remove1_Osimps_I2_J,axiom,
    ! [X4: nat,Y3: nat,Xs2: list_nat] :
      ( ( ( X4 = Y3 )
       => ( ( remove1_nat @ X4 @ ( cons_nat @ Y3 @ Xs2 ) )
          = Xs2 ) )
      & ( ( X4 != Y3 )
       => ( ( remove1_nat @ X4 @ ( cons_nat @ Y3 @ Xs2 ) )
          = ( cons_nat @ Y3 @ ( remove1_nat @ X4 @ Xs2 ) ) ) ) ) ).

% remove1.simps(2)
thf(fact_1187_remove1_Osimps_I1_J,axiom,
    ! [X4: produc2642885279816024434ropVar] :
      ( ( remove1876521362034606350ropVar @ X4 @ nil_Pr6119456312066593144ropVar )
      = nil_Pr6119456312066593144ropVar ) ).

% remove1.simps(1)
thf(fact_1188_remove1_Osimps_I1_J,axiom,
    ! [X4: produc8999643685545340131pVar_o] :
      ( ( remove4799796591832458879pVar_o @ X4 @ nil_Pr6413368191065476585pVar_o )
      = nil_Pr6413368191065476585pVar_o ) ).

% remove1.simps(1)
thf(fact_1189_remove1_Osimps_I1_J,axiom,
    ! [X4: list_P1665994083459302249pVar_o] :
      ( ( remove5341357285405916933pVar_o @ X4 @ nil_li1310946081133177199pVar_o )
      = nil_li1310946081133177199pVar_o ) ).

% remove1.simps(1)
thf(fact_1190_remove1_Osimps_I1_J,axiom,
    ! [X4: list_l1255154591254203759pVar_o] :
      ( ( remove3721140173947915019pVar_o @ X4 @ nil_li590035233778055285pVar_o )
      = nil_li590035233778055285pVar_o ) ).

% remove1.simps(1)
thf(fact_1191_remove1_Osimps_I1_J,axiom,
    ! [X4: nat] :
      ( ( remove1_nat @ X4 @ nil_nat )
      = nil_nat ) ).

% remove1.simps(1)
thf(fact_1192_notin__set__remove1,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ ( remove5793944696144199672ropVar @ Y3 @ Xs2 ) ) ) ) ).

% notin_set_remove1
thf(fact_1193_notin__set__remove1,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o,Y3: product_prod_o_o] :
      ( ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ ( remove8878500798450800835od_o_o @ Y3 @ Xs2 ) ) ) ) ).

% notin_set_remove1
thf(fact_1194_notin__set__remove1,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ ( remove1876521362034606350ropVar @ Y3 @ Xs2 ) ) ) ) ).

% notin_set_remove1
thf(fact_1195_notin__set__remove1,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar,Y3: list_P723798329145367999ropVar] :
      ( ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ ( remove4743378409927258915ropVar @ Y3 @ Xs2 ) ) ) ) ).

% notin_set_remove1
thf(fact_1196_notin__set__remove1,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o,Y3: list_l1255154591254203759pVar_o] :
      ( ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ ( remove3721140173947915019pVar_o @ Y3 @ Xs2 ) ) ) ) ).

% notin_set_remove1
thf(fact_1197_notin__set__remove1,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o,Y3: list_l985434930288970485pVar_o] :
      ( ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ ( remove8904596938398812817pVar_o @ Y3 @ Xs2 ) ) ) ) ).

% notin_set_remove1
thf(fact_1198_notin__set__remove1,axiom,
    ! [X4: nat,Xs2: list_nat,Y3: nat] :
      ( ~ ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ~ ( member_nat @ X4 @ ( set_nat2 @ ( remove1_nat @ Y3 @ Xs2 ) ) ) ) ).

% notin_set_remove1
thf(fact_1199_remove1__idem,axiom,
    ! [X4: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
     => ( ( remove5793944696144199672ropVar @ X4 @ Xs2 )
        = Xs2 ) ) ).

% remove1_idem
thf(fact_1200_remove1__idem,axiom,
    ! [X4: product_prod_o_o,Xs2: list_P4002435161011370285od_o_o] :
      ( ~ ( member7466972457876170832od_o_o @ X4 @ ( set_Product_prod_o_o2 @ Xs2 ) )
     => ( ( remove8878500798450800835od_o_o @ X4 @ Xs2 )
        = Xs2 ) ) ).

% remove1_idem
thf(fact_1201_remove1__idem,axiom,
    ! [X4: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] :
      ( ~ ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs2 ) )
     => ( ( remove1876521362034606350ropVar @ X4 @ Xs2 )
        = Xs2 ) ) ).

% remove1_idem
thf(fact_1202_remove1__idem,axiom,
    ! [X4: list_P723798329145367999ropVar,Xs2: list_l1644708917474426447ropVar] :
      ( ~ ( member7375955847912140118ropVar @ X4 @ ( set_li8604685205718225898ropVar @ Xs2 ) )
     => ( ( remove4743378409927258915ropVar @ X4 @ Xs2 )
        = Xs2 ) ) ).

% remove1_idem
thf(fact_1203_remove1__idem,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Xs2: list_l985434930288970485pVar_o] :
      ( ~ ( member6267526602072755480pVar_o @ X4 @ ( set_li3549506505754852740pVar_o @ Xs2 ) )
     => ( ( remove3721140173947915019pVar_o @ X4 @ Xs2 )
        = Xs2 ) ) ).

% remove1_idem
thf(fact_1204_remove1__idem,axiom,
    ! [X4: list_l985434930288970485pVar_o,Xs2: list_l6341520962544925691pVar_o] :
      ( ~ ( member9046334641704116766pVar_o @ X4 @ ( set_li4369674465895099530pVar_o @ Xs2 ) )
     => ( ( remove8904596938398812817pVar_o @ X4 @ Xs2 )
        = Xs2 ) ) ).

% remove1_idem
thf(fact_1205_remove1__idem,axiom,
    ! [X4: nat,Xs2: list_nat] :
      ( ~ ( member_nat @ X4 @ ( set_nat2 @ Xs2 ) )
     => ( ( remove1_nat @ X4 @ Xs2 )
        = Xs2 ) ) ).

% remove1_idem
thf(fact_1206_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_P1665994083459302249pVar_o,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > list_P1665994083459302249pVar_o > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s2015474115514096597pVar_o @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_Pr6413368191065476585pVar_o )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: nat,Zs2: list_nat,W: produc8999643685545340131pVar_o,Ws2: list_P1665994083459302249pVar_o] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s2015474115514096597pVar_o @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_P6860203055867761049pVar_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1207_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_l1255154591254203759pVar_o,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > list_l1255154591254203759pVar_o > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s1343900380416901083pVar_o @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_li1310946081133177199pVar_o )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: nat,Zs2: list_nat,W: list_P1665994083459302249pVar_o,Ws2: list_l1255154591254203759pVar_o] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s1343900380416901083pVar_o @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_l3452452106213086495pVar_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1208_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_l985434930288970485pVar_o,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > list_l985434930288970485pVar_o > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s6265379629157799777pVar_o @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_li590035233778055285pVar_o )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: nat,Zs2: list_nat,W: list_l1255154591254203759pVar_o,Ws2: list_l985434930288970485pVar_o] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s6265379629157799777pVar_o @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_l7316790070876032037pVar_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1209_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_nat,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_nat )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: nat,Zs2: list_nat,W: nat,Ws2: list_nat] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1210_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_nat,Ws: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_size_list_nat @ Zs ) )
       => ( ( ( size_size_list_nat @ Zs )
            = ( size_s6752116297430514276ropVar @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat @ nil_Pr6119456312066593144ropVar )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: nat,Zs2: list_nat,W: produc2642885279816024434ropVar,Ws2: list_P8791008543742759928ropVar] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_size_list_nat @ Zs2 ) )
                   => ( ( ( size_size_list_nat @ Zs2 )
                        = ( size_s6752116297430514276ropVar @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_nat @ Z3 @ Zs2 ) @ ( cons_P1504208554336013608ropVar @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1211_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,Ws: list_P1665994083459302249pVar_o,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P1665994083459302249pVar_o > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_s2015474115514096597pVar_o @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6413368191065476585pVar_o )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar,W: produc8999643685545340131pVar_o,Ws2: list_P1665994083459302249pVar_o] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_s6752116297430514276ropVar @ Zs2 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs2 )
                        = ( size_s2015474115514096597pVar_o @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_P1504208554336013608ropVar @ Z3 @ Zs2 ) @ ( cons_P6860203055867761049pVar_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1212_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,Ws: list_l1255154591254203759pVar_o,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_l1255154591254203759pVar_o > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_s1343900380416901083pVar_o @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_li1310946081133177199pVar_o )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar,W: list_P1665994083459302249pVar_o,Ws2: list_l1255154591254203759pVar_o] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_s6752116297430514276ropVar @ Zs2 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs2 )
                        = ( size_s1343900380416901083pVar_o @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_P1504208554336013608ropVar @ Z3 @ Zs2 ) @ ( cons_l3452452106213086495pVar_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1213_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,Ws: list_l985434930288970485pVar_o,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_l985434930288970485pVar_o > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_s6265379629157799777pVar_o @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_li590035233778055285pVar_o )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar,W: list_l1255154591254203759pVar_o,Ws2: list_l985434930288970485pVar_o] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_s6752116297430514276ropVar @ Zs2 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs2 )
                        = ( size_s6265379629157799777pVar_o @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_P1504208554336013608ropVar @ Z3 @ Zs2 ) @ ( cons_l7316790070876032037pVar_o @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1214_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,Ws: list_nat,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_nat > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_size_list_nat @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_nat )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar,W: nat,Ws2: list_nat] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_s6752116297430514276ropVar @ Zs2 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs2 )
                        = ( size_size_list_nat @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_P1504208554336013608ropVar @ Z3 @ Zs2 ) @ ( cons_nat @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1215_list__induct4,axiom,
    ! [Xs2: list_P8791008543742759928ropVar,Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar,Ws: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > $o] :
      ( ( ( size_s6752116297430514276ropVar @ Xs2 )
        = ( size_s6752116297430514276ropVar @ Ys2 ) )
     => ( ( ( size_s6752116297430514276ropVar @ Ys2 )
          = ( size_s6752116297430514276ropVar @ Zs ) )
       => ( ( ( size_s6752116297430514276ropVar @ Zs )
            = ( size_s6752116297430514276ropVar @ Ws ) )
         => ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar )
           => ( ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Z3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar,W: produc2642885279816024434ropVar,Ws2: list_P8791008543742759928ropVar] :
                  ( ( ( size_s6752116297430514276ropVar @ Xs )
                    = ( size_s6752116297430514276ropVar @ Ys ) )
                 => ( ( ( size_s6752116297430514276ropVar @ Ys )
                      = ( size_s6752116297430514276ropVar @ Zs2 ) )
                   => ( ( ( size_s6752116297430514276ropVar @ Zs2 )
                        = ( size_s6752116297430514276ropVar @ Ws2 ) )
                     => ( ( P @ Xs @ Ys @ Zs2 @ Ws2 )
                       => ( P @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys ) @ ( cons_P1504208554336013608ropVar @ Z3 @ Zs2 ) @ ( cons_P1504208554336013608ropVar @ W @ Ws2 ) ) ) ) ) )
             => ( P @ Xs2 @ Ys2 @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1216_cnf__encoder_Oeval__all__append,axiom,
    ! [Fs: list_l985434930288970485pVar_o,Gs: list_l985434930288970485pVar_o,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ ( append8959015459476584842pVar_o @ Fs @ Gs ) ) )
           => ( propos2885041177628046573ropVar @ Alpha @ X3 ) ) )
      = ( ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Fs ) )
           => ( propos2885041177628046573ropVar @ Alpha @ X3 ) )
        & ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Gs ) )
           => ( propos2885041177628046573ropVar @ Alpha @ X3 ) ) ) ) ).

% cnf_encoder.eval_all_append
thf(fact_1217_cnf__encoder_Oeval__all__concat,axiom,
    ! [Fs: list_l6341520962544925691pVar_o,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ ( concat8126406963686051632pVar_o @ Fs ) ) )
           => ( propos2885041177628046573ropVar @ Alpha @ X3 ) ) )
      = ( ! [X3: list_l985434930288970485pVar_o] :
            ( ( member9046334641704116766pVar_o @ X3 @ ( set_li4369674465895099530pVar_o @ Fs ) )
           => ! [Y2: list_l1255154591254203759pVar_o] :
                ( ( member6267526602072755480pVar_o @ Y2 @ ( set_li3549506505754852740pVar_o @ X3 ) )
               => ( propos2885041177628046573ropVar @ Alpha @ Y2 ) ) ) ) ) ).

% cnf_encoder.eval_all_concat
thf(fact_1218_cnf__encoder_Oenc__sound_I13_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,Fs: list_l985434930288970485pVar_o] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( concat8635278358871234986pVar_o @ Fs ) )
      = ( ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Fs ) )
           => ( propos2885041177628046573ropVar @ Alpha @ X3 ) ) ) ) ).

% cnf_encoder.enc_sound(13)
thf(fact_1219_cnf__encoder_Oeval__all__Cons,axiom,
    ! [F2: list_l1255154591254203759pVar_o,Gs: list_l985434930288970485pVar_o,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ ( cons_l7316790070876032037pVar_o @ F2 @ Gs ) ) )
           => ( propos2885041177628046573ropVar @ Alpha @ X3 ) ) )
      = ( ( propos2885041177628046573ropVar @ Alpha @ F2 )
        & ! [X3: list_l1255154591254203759pVar_o] :
            ( ( member6267526602072755480pVar_o @ X3 @ ( set_li3549506505754852740pVar_o @ Gs ) )
           => ( propos2885041177628046573ropVar @ Alpha @ X3 ) ) ) ) ).

% cnf_encoder.eval_all_Cons
thf(fact_1220_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_1221_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ ( suc @ I2 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_1222_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N: nat,J2: nat,I2: nat] :
      ( ( ord_less_nat @ N @ ( minus_minus_nat @ J2 @ I2 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) ) @ N )
        = ( suc @ ( plus_plus_nat @ I2 @ N ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_1223_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J2 ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_1224_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ J2 )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J2 ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J2 ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_1225_upt__rec__numeral,axiom,
    ! [M: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_1226_tl__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ N ) ) ).

% tl_upt
thf(fact_1227_hd__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( hd_nat @ ( upt @ I2 @ J2 ) )
        = I2 ) ) ).

% hd_upt
thf(fact_1228_drop__upt,axiom,
    ! [M: nat,I2: nat,J2: nat] :
      ( ( drop_nat @ M @ ( upt @ I2 @ J2 ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M ) @ J2 ) ) ).

% drop_upt
thf(fact_1229_take__upt,axiom,
    ! [I2: nat,M: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N )
     => ( ( take_nat @ M @ ( upt @ I2 @ N ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).

% take_upt
thf(fact_1230_upt__conv__Nil,axiom,
    ! [J2: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J2 @ I2 )
     => ( ( upt @ I2 @ J2 )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1231_length__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J2 ) )
      = ( minus_minus_nat @ J2 @ I2 ) ) ).

% length_upt
thf(fact_1232_last__upt,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( last_nat @ ( upt @ I2 @ J2 ) )
        = ( minus_minus_nat @ J2 @ one_one_nat ) ) ) ).

% last_upt
thf(fact_1233_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( upt @ I2 @ J2 )
        = nil_nat )
      = ( ( J2 = zero_zero_nat )
        | ( ord_less_eq_nat @ J2 @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_1234_nth__upt,axiom,
    ! [I2: nat,K: nat,J2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J2 )
     => ( ( nth_nat @ ( upt @ I2 @ J2 ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_1235_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_1236_upt__add__eq__append,axiom,
    ! [I2: nat,J2: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J2 @ K ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( upt @ J2 @ ( plus_plus_nat @ J2 @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_1237_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M @ Q2 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1238_sorted__wrt__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N ) ) ).

% sorted_wrt_upt
thf(fact_1239_sorted__upt,axiom,
    ! [M: nat,N: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N ) ) ).

% sorted_upt
thf(fact_1240_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_1241_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N2: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N2 ) @ M2 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_1242_upt__conv__Cons,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ J2 )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) ) ) ).

% upt_conv_Cons
thf(fact_1243_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1244_upt__Suc__append,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J2 )
     => ( ( upt @ I2 @ ( suc @ J2 ) )
        = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1245_upt__Suc,axiom,
    ! [I2: nat,J2: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = ( append_nat @ ( upt @ I2 @ J2 ) @ ( cons_nat @ J2 @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J2 )
       => ( ( upt @ I2 @ ( suc @ J2 ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1246_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J2: nat,X4: nat,Xs2: list_nat] :
      ( ( ( upt @ I2 @ J2 )
        = ( cons_nat @ X4 @ Xs2 ) )
      = ( ( ord_less_nat @ I2 @ J2 )
        & ( I2 = X4 )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J2 )
          = Xs2 ) ) ) ).

% upt_eq_Cons_conv
thf(fact_1247_sorted__list__of__set__atMost__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_atMost_nat @ K ) ) @ ( cons_nat @ ( suc @ K ) @ nil_nat ) ) ) ).

% sorted_list_of_set_atMost_Suc
thf(fact_1248_sorted__list__of__set__lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ ( suc @ K ) ) )
      = ( append_nat @ ( linord2614967742042102400et_nat @ ( set_ord_lessThan_nat @ K ) ) @ ( cons_nat @ K @ nil_nat ) ) ) ).

% sorted_list_of_set_lessThan_Suc
thf(fact_1249_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% atLeast_upt
thf(fact_1250_map__Suc__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_1251_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N2: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) ) ) ) ).

% atMost_upto
thf(fact_1252_distinct__upt,axiom,
    ! [I2: nat,J2: nat] : ( distinct_nat @ ( upt @ I2 @ J2 ) ) ).

% distinct_upt
thf(fact_1253_cnf__encoder_Oencode__exactly__one__sound,axiom,
    ! [Vars: list_P8791008543742759928ropVar,One2: list_l1255154591254203759pVar_o,Conds: list_l985434930288970485pVar_o,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ( multis8929244183426903067ly_one @ Vars )
        = ( produc4064290870397114077pVar_o @ One2 @ Conds ) )
     => ( ( distin2675581014522817237ropVar @ ( map_Pr3335389669906741166ropVar @ produc6918815732823278798ropVar @ Vars ) )
       => ( ( propos2885041177628046573ropVar @ Alpha @ One2 )
         => ( ! [X2: list_l1255154591254203759pVar_o] :
                ( ( member6267526602072755480pVar_o @ X2 @ ( set_li3549506505754852740pVar_o @ Conds ) )
               => ( propos2885041177628046573ropVar @ Alpha @ X2 ) )
           => ? [X2: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                & ( Alpha @ X2 )
                & ! [Y5: multis3193088007478089820ropVar] :
                    ( ( ( member8421201611525043589ropVar @ Y5 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                      & ( Alpha @ Y5 ) )
                   => ( Y5 = X2 ) ) ) ) ) ) ) ).

% cnf_encoder.encode_exactly_one_sound
thf(fact_1254_cnf__encoder_Oencode__exactly__one__complete,axiom,
    ! [Vars: list_P8791008543742759928ropVar,One2: list_l1255154591254203759pVar_o,Conds: list_l985434930288970485pVar_o,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ( multis8929244183426903067ly_one @ Vars )
        = ( produc4064290870397114077pVar_o @ One2 @ Conds ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Vars ) )
           => ! [X2: multis3193088007478089820ropVar,Ze: multis3193088007478089820ropVar,On: multis3193088007478089820ropVar] :
                ( ( ( nth_Pr227365692153740281ropVar @ Vars @ I )
                  = ( produc6194868611276206818ropVar @ X2 @ ( produc6936306168532358295ropVar @ Ze @ On ) ) )
               => ( ( ( Alpha @ Ze )
                    = ( ~ ? [Y2: multis3193088007478089820ropVar] :
                            ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ ( drop_P4829783966340880056ropVar @ I @ Vars ) ) ) )
                            & ( Alpha @ Y2 ) ) ) )
                  & ( ( Alpha @ On )
                    = ( ? [Y2: multis3193088007478089820ropVar] :
                          ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ ( drop_P4829783966340880056ropVar @ I @ Vars ) ) ) )
                          & ( Alpha @ Y2 )
                          & ! [Z4: multis3193088007478089820ropVar] :
                              ( ( ( member8421201611525043589ropVar @ Z4 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ ( drop_P4829783966340880056ropVar @ I @ Vars ) ) ) )
                                & ( Alpha @ Z4 ) )
                             => ( Z4 = Y2 ) ) ) ) ) ) ) )
       => ( ( distin2675581014522817237ropVar @ ( map_Pr3335389669906741166ropVar @ produc6918815732823278798ropVar @ Vars ) )
         => ( ! [X5: list_l1255154591254203759pVar_o] :
                ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Conds ) )
               => ( propos2885041177628046573ropVar @ Alpha @ X5 ) )
            & ( ( propos2885041177628046573ropVar @ Alpha @ One2 )
              = ( ? [X3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ X3 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                    & ( Alpha @ X3 )
                    & ! [Y2: multis3193088007478089820ropVar] :
                        ( ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                          & ( Alpha @ Y2 ) )
                       => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ).

% cnf_encoder.encode_exactly_one_complete
thf(fact_1255_cnf__encoder_Oencode__sum__0__1__main,axiom,
    ! [Vars: list_P8791008543742759928ropVar,Conds: list_l985434930288970485pVar_o,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar,Prop: $o,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ( multis70479551085940736041_main @ Vars )
        = ( produc2520692699247754633ropVar @ Conds @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) )
     => ( ( Prop
         => ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Vars ) )
             => ! [X2: multis3193088007478089820ropVar,Ze: multis3193088007478089820ropVar,On: multis3193088007478089820ropVar,Re: list_P8791008543742759928ropVar] :
                  ( ( ( drop_P4829783966340880056ropVar @ I @ Vars )
                    = ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X2 @ ( produc6936306168532358295ropVar @ Ze @ On ) ) @ Re ) )
                 => ( ( ( Alpha @ Ze )
                      = ( ~ ? [Y2: multis3193088007478089820ropVar] :
                              ( ( member8421201611525043589ropVar @ Y2 @ ( insert1868789817947358764ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Re ) ) ) )
                              & ( Alpha @ Y2 ) ) ) )
                    & ( ( Alpha @ On )
                      = ( ? [Y2: multis3193088007478089820ropVar] :
                            ( ( member8421201611525043589ropVar @ Y2 @ ( insert1868789817947358764ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Re ) ) ) )
                            & ( Alpha @ Y2 )
                            & ! [Z4: multis3193088007478089820ropVar] :
                                ( ( ( member8421201611525043589ropVar @ Z4 @ ( insert1868789817947358764ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Re ) ) ) )
                                  & ( Alpha @ Z4 ) )
                               => ( Z4 = Y2 ) ) ) ) ) ) ) ) )
       => ( ( ~ Prop
           => ! [X2: list_l1255154591254203759pVar_o] :
                ( ( member6267526602072755480pVar_o @ X2 @ ( set_li3549506505754852740pVar_o @ Conds ) )
               => ( propos2885041177628046573ropVar @ Alpha @ X2 ) ) )
         => ( ( distin2675581014522817237ropVar @ ( map_Pr3335389669906741166ropVar @ produc6918815732823278798ropVar @ Vars ) )
           => ( ( Vars != nil_Pr6119456312066593144ropVar )
             => ( ! [X5: list_l1255154591254203759pVar_o] :
                    ( ( member6267526602072755480pVar_o @ X5 @ ( set_li3549506505754852740pVar_o @ Conds ) )
                   => ( propos2885041177628046573ropVar @ Alpha @ X5 ) )
                & ( ( Alpha @ Zero2 )
                  = ( ~ ? [X3: multis3193088007478089820ropVar] :
                          ( ( member8421201611525043589ropVar @ X3 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                          & ( Alpha @ X3 ) ) ) )
                & ( ( Alpha @ One2 )
                  = ( ? [X3: multis3193088007478089820ropVar] :
                        ( ( member8421201611525043589ropVar @ X3 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                        & ( Alpha @ X3 )
                        & ! [Y2: multis3193088007478089820ropVar] :
                            ( ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                              & ( Alpha @ Y2 ) )
                           => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ) ) ).

% cnf_encoder.encode_sum_0_1_main
thf(fact_1256_pf__encoder_Oencode__sum__0__1__main,axiom,
    ! [Vars: list_P8791008543742759928ropVar,Conds: list_P723798329145367999ropVar,Zero2: multis3193088007478089820ropVar,One2: multis3193088007478089820ropVar,Prop: $o,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ( multis55434767431285815131_main @ Vars )
        = ( produc2463153034239240527ropVar @ Conds @ ( produc6936306168532358295ropVar @ Zero2 @ One2 ) ) )
     => ( ( Prop
         => ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Vars ) )
             => ! [X2: multis3193088007478089820ropVar,Ze: multis3193088007478089820ropVar,On: multis3193088007478089820ropVar,Re: list_P8791008543742759928ropVar] :
                  ( ( ( drop_P4829783966340880056ropVar @ I @ Vars )
                    = ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X2 @ ( produc6936306168532358295ropVar @ Ze @ On ) ) @ Re ) )
                 => ( ( ( Alpha @ Ze )
                      = ( ~ ? [Y2: multis3193088007478089820ropVar] :
                              ( ( member8421201611525043589ropVar @ Y2 @ ( insert1868789817947358764ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Re ) ) ) )
                              & ( Alpha @ Y2 ) ) ) )
                    & ( ( Alpha @ On )
                      = ( ? [Y2: multis3193088007478089820ropVar] :
                            ( ( member8421201611525043589ropVar @ Y2 @ ( insert1868789817947358764ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Re ) ) ) )
                            & ( Alpha @ Y2 )
                            & ! [Z4: multis3193088007478089820ropVar] :
                                ( ( ( member8421201611525043589ropVar @ Z4 @ ( insert1868789817947358764ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Re ) ) ) )
                                  & ( Alpha @ Z4 ) )
                               => ( Z4 = Y2 ) ) ) ) ) ) ) ) )
       => ( ( ~ Prop
           => ! [X2: propos3600597766819905199ropVar] :
                ( ( member1796676888578033094ropVar @ X2 @ ( set_Pr342157451780195930ropVar @ Conds ) )
               => ( propos8617422795459586798ropVar @ Alpha @ X2 ) ) )
         => ( ( distin2675581014522817237ropVar @ ( map_Pr3335389669906741166ropVar @ produc6918815732823278798ropVar @ Vars ) )
           => ( ( Vars != nil_Pr6119456312066593144ropVar )
             => ( ! [X5: propos3600597766819905199ropVar] :
                    ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Conds ) )
                   => ( propos8617422795459586798ropVar @ Alpha @ X5 ) )
                & ( ( Alpha @ Zero2 )
                  = ( ~ ? [X3: multis3193088007478089820ropVar] :
                          ( ( member8421201611525043589ropVar @ X3 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                          & ( Alpha @ X3 ) ) ) )
                & ( ( Alpha @ One2 )
                  = ( ? [X3: multis3193088007478089820ropVar] :
                        ( ( member8421201611525043589ropVar @ X3 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                        & ( Alpha @ X3 )
                        & ! [Y2: multis3193088007478089820ropVar] :
                            ( ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                              & ( Alpha @ Y2 ) )
                           => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ) ) ).

% pf_encoder.encode_sum_0_1_main
thf(fact_1257_pf__encoder_Oeval__all__concat,axiom,
    ! [Fs: list_l1644708917474426447ropVar,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ ( concat4611077035641296302ropVar @ Fs ) ) )
           => ( propos8617422795459586798ropVar @ Alpha @ X3 ) ) )
      = ( ! [X3: list_P723798329145367999ropVar] :
            ( ( member7375955847912140118ropVar @ X3 @ ( set_li8604685205718225898ropVar @ Fs ) )
           => ! [Y2: propos3600597766819905199ropVar] :
                ( ( member1796676888578033094ropVar @ Y2 @ ( set_Pr342157451780195930ropVar @ X3 ) )
               => ( propos8617422795459586798ropVar @ Alpha @ Y2 ) ) ) ) ) ).

% pf_encoder.eval_all_concat
thf(fact_1258_pf__encoder_Oeval__all__append,axiom,
    ! [Fs: list_P723798329145367999ropVar,Gs: list_P723798329145367999ropVar,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ ( append7708039934470209236ropVar @ Fs @ Gs ) ) )
           => ( propos8617422795459586798ropVar @ Alpha @ X3 ) ) )
      = ( ! [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ Fs ) )
           => ( propos8617422795459586798ropVar @ Alpha @ X3 ) )
        & ! [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ Gs ) )
           => ( propos8617422795459586798ropVar @ Alpha @ X3 ) ) ) ) ).

% pf_encoder.eval_all_append
thf(fact_1259_pf__encoder_Oeval__all__Cons,axiom,
    ! [F2: propos3600597766819905199ropVar,Gs: list_P723798329145367999ropVar,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ ( cons_P6757824865275295865ropVar @ F2 @ Gs ) ) )
           => ( propos8617422795459586798ropVar @ Alpha @ X3 ) ) )
      = ( ( propos8617422795459586798ropVar @ Alpha @ F2 )
        & ! [X3: propos3600597766819905199ropVar] :
            ( ( member1796676888578033094ropVar @ X3 @ ( set_Pr342157451780195930ropVar @ Gs ) )
           => ( propos8617422795459586798ropVar @ Alpha @ X3 ) ) ) ) ).

% pf_encoder.eval_all_Cons
thf(fact_1260_pf__encoder_Oencode__exactly__one__complete,axiom,
    ! [Vars: list_P8791008543742759928ropVar,One2: propos3600597766819905199ropVar,Conds: list_P723798329145367999ropVar,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ( multis7914470704280101590ly_one @ Vars )
        = ( produc5136867404843011847ropVar @ One2 @ Conds ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6752116297430514276ropVar @ Vars ) )
           => ! [X2: multis3193088007478089820ropVar,Ze: multis3193088007478089820ropVar,On: multis3193088007478089820ropVar] :
                ( ( ( nth_Pr227365692153740281ropVar @ Vars @ I )
                  = ( produc6194868611276206818ropVar @ X2 @ ( produc6936306168532358295ropVar @ Ze @ On ) ) )
               => ( ( ( Alpha @ Ze )
                    = ( ~ ? [Y2: multis3193088007478089820ropVar] :
                            ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ ( drop_P4829783966340880056ropVar @ I @ Vars ) ) ) )
                            & ( Alpha @ Y2 ) ) ) )
                  & ( ( Alpha @ On )
                    = ( ? [Y2: multis3193088007478089820ropVar] :
                          ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ ( drop_P4829783966340880056ropVar @ I @ Vars ) ) ) )
                          & ( Alpha @ Y2 )
                          & ! [Z4: multis3193088007478089820ropVar] :
                              ( ( ( member8421201611525043589ropVar @ Z4 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ ( drop_P4829783966340880056ropVar @ I @ Vars ) ) ) )
                                & ( Alpha @ Z4 ) )
                             => ( Z4 = Y2 ) ) ) ) ) ) ) )
       => ( ( distin2675581014522817237ropVar @ ( map_Pr3335389669906741166ropVar @ produc6918815732823278798ropVar @ Vars ) )
         => ( ! [X5: propos3600597766819905199ropVar] :
                ( ( member1796676888578033094ropVar @ X5 @ ( set_Pr342157451780195930ropVar @ Conds ) )
               => ( propos8617422795459586798ropVar @ Alpha @ X5 ) )
            & ( ( propos8617422795459586798ropVar @ Alpha @ One2 )
              = ( ? [X3: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ X3 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                    & ( Alpha @ X3 )
                    & ! [Y2: multis3193088007478089820ropVar] :
                        ( ( ( member8421201611525043589ropVar @ Y2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                          & ( Alpha @ Y2 ) )
                       => ( Y2 = X3 ) ) ) ) ) ) ) ) ) ).

% pf_encoder.encode_exactly_one_complete
thf(fact_1261_pf__encoder_Oencode__exactly__one__sound,axiom,
    ! [Vars: list_P8791008543742759928ropVar,One2: propos3600597766819905199ropVar,Conds: list_P723798329145367999ropVar,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ( multis7914470704280101590ly_one @ Vars )
        = ( produc5136867404843011847ropVar @ One2 @ Conds ) )
     => ( ( distin2675581014522817237ropVar @ ( map_Pr3335389669906741166ropVar @ produc6918815732823278798ropVar @ Vars ) )
       => ( ( propos8617422795459586798ropVar @ Alpha @ One2 )
         => ( ! [X2: propos3600597766819905199ropVar] :
                ( ( member1796676888578033094ropVar @ X2 @ ( set_Pr342157451780195930ropVar @ Conds ) )
               => ( propos8617422795459586798ropVar @ Alpha @ X2 ) )
           => ? [X2: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X2 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                & ( Alpha @ X2 )
                & ! [Y5: multis3193088007478089820ropVar] :
                    ( ( ( member8421201611525043589ropVar @ Y5 @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar @ ( set_Pr3946994337432900359ropVar @ Vars ) ) )
                      & ( Alpha @ Y5 ) )
                   => ( Y5 = X2 ) ) ) ) ) ) ) ).

% pf_encoder.encode_exactly_one_sound
thf(fact_1262_sorted__list__of__set__range,axiom,
    ! [M: nat,N: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N ) )
      = ( upt @ M @ N ) ) ).

% sorted_list_of_set_range
thf(fact_1263_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_1264_all__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
           => ( P @ M2 ) ) )
      = ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
           => ( P @ X3 ) ) ) ) ).

% all_nat_less_eq
thf(fact_1265_ex__nat__less__eq,axiom,
    ! [N: nat,P: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_nat @ M2 @ N )
            & ( P @ M2 ) ) )
      = ( ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N ) )
            & ( P @ X3 ) ) ) ) ).

% ex_nat_less_eq
thf(fact_1266_cnf__encoder_Oenc__sound_I9_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,X4: multis3193088007478089820ropVar,F2: list_l1255154591254203759pVar_o] :
      ( ( propos2885041177628046573ropVar @ Alpha @ ( map_li2782605871620812932pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X4 @ $false ) ) @ F2 ) )
      = ( ( Alpha @ X4 )
       => ( propos2885041177628046573ropVar @ Alpha @ F2 ) ) ) ).

% cnf_encoder.enc_sound(9)
thf(fact_1267_cnf__encoder_Oenc__sound_I12_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,Xs2: list_M6740372356978808802ropVar] :
      ( ( propos2885041177628046573ropVar @ Alpha
        @ ( cons_l3452452106213086495pVar_o
          @ ( map_Mu8395235412166537675pVar_o
            @ ^ [X3: multis3193088007478089820ropVar] : ( produc2127521181850112851pVar_o @ X3 @ $false )
            @ Xs2 )
          @ nil_li1310946081133177199pVar_o ) )
      = ( ~ ! [X3: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs2 ) )
             => ( Alpha @ X3 ) ) ) ) ).

% cnf_encoder.enc_sound(12)
thf(fact_1268_map__decr__upt,axiom,
    ! [M: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
      = ( upt @ M @ N ) ) ).

% map_decr_upt
thf(fact_1269_map__add__upt,axiom,
    ! [N: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N @ ( plus_plus_nat @ M @ N ) ) ) ).

% map_add_upt
thf(fact_1270_cnf__encode__sum__0__1__main__def,axiom,
    ( multis70479551085940736041_main
    = ( multis6898674672137113254pVar_o
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Z4: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Z4: multis3193088007478089820ropVar,U2: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) ) ) ) ) ).

% cnf_encode_sum_0_1_main_def
thf(fact_1271_cnf__encodeGammaCond__def,axiom,
    ( multis334970729563149064maCond
    = ( multis6017319109170748100ropVar @ nil_li1310946081133177199pVar_o
      @ ^ [X3: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ) ).

% cnf_encodeGammaCond_def
thf(fact_1272_cnf__encode__exactly__one__def,axiom,
    ( multis8929244183426903067ly_one
    = ( multis1909274777570284281ropVar @ ( cons_l3452452106213086495pVar_o @ nil_Pr6413368191065476585pVar_o @ nil_li1310946081133177199pVar_o )
      @ ^ [X3: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ nil_Pr6413368191065476585pVar_o ) @ nil_li1310946081133177199pVar_o )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Z4: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Z4: multis3193088007478089820ropVar,U2: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U2 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $false ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ U2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) ) @ nil_li1310946081133177199pVar_o ) ) ) )
      @ ^ [X3: multis3193088007478089820ropVar,Y2: multis3193088007478089820ropVar,Z4: multis3193088007478089820ropVar] : ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $true ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Z4 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ ( cons_l3452452106213086495pVar_o @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ X3 @ $false ) @ ( cons_P6860203055867761049pVar_o @ ( produc2127521181850112851pVar_o @ Y2 @ $true ) @ nil_Pr6413368191065476585pVar_o ) ) @ nil_li1310946081133177199pVar_o ) ) ) ) ).

% cnf_encode_exactly_one_def

% Helper facts (19)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_T,axiom,
    ! [X4: list_P4002435161011370285od_o_o,Y3: list_P4002435161011370285od_o_o] :
      ( ( if_lis8143501808108548851od_o_o @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J_T,axiom,
    ! [X4: list_P4002435161011370285od_o_o,Y3: list_P4002435161011370285od_o_o] :
      ( ( if_lis8143501808108548851od_o_o @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_T,axiom,
    ! [X4: list_M6740372356978808802ropVar,Y3: list_M6740372356978808802ropVar] :
      ( ( if_lis9215432529438213544ropVar @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_T,axiom,
    ! [X4: list_M6740372356978808802ropVar,Y3: list_M6740372356978808802ropVar] :
      ( ( if_lis9215432529438213544ropVar @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_T,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( if_lis4159902583309516335pVar_o @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_T,axiom,
    ! [X4: list_P1665994083459302249pVar_o,Y3: list_P1665994083459302249pVar_o] :
      ( ( if_lis4159902583309516335pVar_o @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_T,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o] :
      ( ( if_lis6453251472562810165pVar_o @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_T,axiom,
    ! [X4: list_l1255154591254203759pVar_o,Y3: list_l1255154591254203759pVar_o] :
      ( ( if_lis6453251472562810165pVar_o @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J_J_T,axiom,
    ! [X4: list_l1644708917474426447ropVar,Y3: list_l1644708917474426447ropVar] :
      ( ( if_lis5532466218389838089ropVar @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__Propositional____Formula__Oformula_It__Multiset____Ordering____in____NP__OPropVar_J_J_J_T,axiom,
    ! [X4: list_l1644708917474426447ropVar,Y3: list_l1644708917474426447ropVar] :
      ( ( if_lis5532466218389838089ropVar @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_T,axiom,
    ! [X4: list_l985434930288970485pVar_o,Y3: list_l985434930288970485pVar_o] :
      ( ( if_lis463867677060124603pVar_o @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_T,axiom,
    ! [X4: list_l985434930288970485pVar_o,Y3: list_l985434930288970485pVar_o] :
      ( ( if_lis463867677060124603pVar_o @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_J_T,axiom,
    ! [X4: list_l6341520962544925691pVar_o,Y3: list_l6341520962544925691pVar_o] :
      ( ( if_lis1723474115809273281pVar_o @ $false @ X4 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_It__List__Olist_It__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_M_Eo_J_J_J_J_J_T,axiom,
    ! [X4: list_l6341520962544925691pVar_o,Y3: list_l6341520962544925691pVar_o] :
      ( ( if_lis1723474115809273281pVar_o @ $true @ X4 @ Y3 )
      = X4 ) ).

thf(help_If_3_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,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

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,
    ! [X4: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( if_lis1975434623467068862ropVar @ $false @ X4 @ 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,
    ! [X4: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( if_lis1975434623467068862ropVar @ $true @ X4 @ Y3 )
      = X4 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [X5: multis3193088007478089820ropVar,Ze2: multis3193088007478089820ropVar,On2: multis3193088007478089820ropVar,Vs3: list_P8791008543742759928ropVar] :
      ( ( vars
        = ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X5 @ ( produc6936306168532358295ropVar @ Ze2 @ On2 ) ) @ Vs3 ) )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------