TPTP Problem File: SLH0837^1.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_00546_024011__13707294_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1762 ( 741 unt; 480 typ;   0 def)
%            Number of atoms       : 3271 (2137 equ;   0 cnn)
%            Maximal formula atoms :   15 (   2 avg)
%            Number of connectives : 11758 ( 505   ~;  44   |; 453   &;9595   @)
%                                         (   0 <=>;1161  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   6 avg)
%            Number of types       :   88 (  87 usr)
%            Number of type conns  : 1594 (1594   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  396 ( 393 usr;  25 con; 0-7 aty)
%            Number of variables   : 4349 ( 355   ^;3556   !; 438   ?;4349   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:29:40.265
%------------------------------------------------------------------------------
% Could-be-implicit typings (87)
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    map_f_list_nat: ( f > list_nat ) > list_f > list_list_nat ).

thf(sy_c_List_Olist_Omap_001tf__f_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,
    map_f_3293025520963524473ropVar: ( f > list_P8791008543742759928ropVar ) > list_f > list_l3415868495287753214ropVar ).

thf(sy_c_List_Olist_Omap_001tf__f_001t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__List__Olist_Itf__f_J_J_J,type,
    map_f_4304301305749555956list_f: ( f > list_P2466969118621037811list_f ) > list_f > list_l4748583716277862649list_f ).

thf(sy_c_List_Olist_Omap_001tf__f_001t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    map_f_2110070304149036999ropVar: ( f > list_P7636664087304740678ropVar ) > list_f > list_l3293480169019542860ropVar ).

thf(sy_c_List_Olist_Omap_001tf__f_001t__List__Olist_It__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J_J,type,
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thf(sy_c_List_Olist_Omap_001tf__f_001t__List__Olist_It__Product____Type__Oprod_Itf__f_Mtf__f_J_J,type,
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thf(sy_c_List_Olist_Omap_001tf__f_001t__List__Olist_Itf__f_J,type,
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thf(sy_c_List_Olist_Omap_001tf__f_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Olist_Omap_001tf__f_001t__Nat__Onat,type,
    map_f_nat: ( f > nat ) > list_f > list_nat ).

thf(sy_c_List_Olist_Omap_001tf__f_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,
    map_f_2905954635146393971ropVar: ( f > produc2642885279816024434ropVar ) > list_f > list_P8791008543742759928ropVar ).

thf(sy_c_List_Olist_Omap_001tf__f_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mtf__f_J,type,
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thf(sy_c_List_Olist_Omap_001tf__f_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J,type,
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thf(sy_c_List_Olist_Omap_001tf__f_001t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
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thf(sy_c_List_Olist_Omap_001tf__f_001tf__f,type,
    map_f_f: ( f > f ) > list_f > list_f ).

thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__List__Olist_Itf__f_J_J_001tf__f,type,
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thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_001tf__f,type,
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thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Nat__Onat_J_001t__Nat__Onat,type,
    rec_li7516600145284979816at_nat: list_nat > ( nat > list_nat > list_nat > list_nat ) > list_nat > list_nat ).

thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Nat__Onat_J_001tf__f,type,
    rec_list_list_nat_f: list_nat > ( f > list_f > list_nat > list_nat ) > list_f > list_nat ).

thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Orec__list_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_001tf__f,type,
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thf(sy_c_List_Olist_Orec__list_001t__List__Olist_Itf__f_J_001tf__f,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_Itf__f_J,type,
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thf(sy_c_List_Olist_Oset_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Olist_Oset_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__List__Olist_Itf__f_J_Mt__List__Olist_Itf__f_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mtf__f_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Nat__Onat_Mtf__f_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mt__List__Olist_Itf__f_J_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Product____Type__Oprod_Itf__f_Mtf__f_J,type,
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thf(sy_c_List_Olist_Oset_001tf__f,type,
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thf(sy_c_List_Olistrel1_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Olistrel1_001t__Nat__Onat,type,
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thf(sy_c_List_Olistrel1_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
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thf(sy_c_List_Olistrel1_001tf__f,type,
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thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_List_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat,type,
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    product_lists_f: list_list_f > list_list_f ).

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_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_001tf__f,type,
    subseqs_f: list_f > list_list_f ).

thf(sy_c_List_Otranspose_001t__List__Olist_Itf__f_J,type,
    transpose_list_f: list_list_list_f > list_list_list_f ).

thf(sy_c_List_Otranspose_001t__Multiset____Ordering____in____NP__OPropVar,type,
    transp2038539320897558508ropVar: list_l3041241048860003048ropVar > list_l3041241048860003048ropVar ).

thf(sy_c_List_Otranspose_001t__Nat__Onat,type,
    transpose_nat: list_list_nat > list_list_nat ).

thf(sy_c_List_Otranspose_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,
    transp9078802294292144386ropVar: list_l3415868495287753214ropVar > list_l3415868495287753214ropVar ).

thf(sy_c_List_Otranspose_001tf__f,type,
    transpose_f: list_list_f > list_list_f ).

thf(sy_c_List_Otranspose__rel_001t__Nat__Onat,type,
    transpose_rel_nat: list_list_nat > list_list_nat > $o ).

thf(sy_c_List_Otranspose__rel_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,
    transp6848717065427184039ropVar: list_l3415868495287753214ropVar > list_l3415868495287753214ropVar > $o ).

thf(sy_c_List_Otranspose__rel_001tf__f,type,
    transpose_rel_f: list_list_f > list_list_f > $o ).

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

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

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxOneIJ,type,
    multis6646701651571498855xOneIJ: nat > nat > multis3193088007478089820ropVar ).

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_OAuxOneJI,type,
    multis6646701651571564453xOneJI: nat > nat > multis3193088007478089820ropVar ).

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

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

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

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

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_Ocase__PropVar_001_Eo,type,
    multis8641292176345589959pVar_o: ( nat > nat > $o ) > ( nat > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > ( nat > nat > $o ) > multis3193088007478089820ropVar > $o ).

thf(sy_c_Multiset__Ordering__in__NP_OPropVar_Osize__PropVar,type,
    multis2955979900537361535ropVar: multis3193088007478089820ropVar > nat ).

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__Num__Onum,type,
    size_size_num: num > nat ).

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

thf(sy_c_Num_Onum_OBit1,type,
    bit1: num > num ).

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

thf(sy_c_Num_Onum_Osize__num,type,
    size_num: num > 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_001t__Num__Onum,type,
    ord_less_num: num > num > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    ord_le8651388201640276840ropVar: set_Mu5658373669842198716ropVar > set_Mu5658373669842198716ropVar > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_It__Nat__Onat_J,type,
    ord_less_set_nat: set_nat > set_nat > $o ).

thf(sy_c_Orderings_Oord__class_Oless_001t__Set__Oset_Itf__f_J,type,
    ord_less_set_f: set_f > set_f > $o ).

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

thf(sy_c_Product__Type_OPair_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    produc72220940542539688at_nat: ( nat > nat ) > nat > produc8199716216217303280at_nat ).

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

thf(sy_c_Product__Type_OPair_001_062_Itf__f_M_062_Itf__f_M_Eo_J_J_001t__List__Olist_Itf__f_J,type,
    produc8519404107117762418list_f: ( f > f > $o ) > list_f > produc5795555978541011586list_f ).

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

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

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

thf(sy_c_Product__Type_OPair_001t__List__Olist_Itf__f_J_001t__List__Olist_Itf__f_J,type,
    produc8144748701140958423list_f: list_f > list_f > produc1604256039088728679list_f ).

thf(sy_c_Product__Type_OPair_001t__Multiset____Ordering____in____NP__OPropVar_001t__Multiset____Ordering____in____NP__OPropVar,type,
    produc6936306168532358295ropVar: multis3193088007478089820ropVar > multis3193088007478089820ropVar > produc7679832412844898215ropVar ).

thf(sy_c_Product__Type_OPair_001t__Multiset____Ordering____in____NP__OPropVar_001t__Nat__Onat,type,
    produc1224776155581759701ar_nat: multis3193088007478089820ropVar > nat > produc7433502520361543131ar_nat ).

thf(sy_c_Product__Type_OPair_001t__Multiset____Ordering____in____NP__OPropVar_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    produc6194868611276206818ropVar: multis3193088007478089820ropVar > produc7679832412844898215ropVar > produc2642885279816024434ropVar ).

thf(sy_c_Product__Type_OPair_001t__Multiset____Ordering____in____NP__OPropVar_001tf__f,type,
    produc3140662905487486270pVar_f: multis3193088007478089820ropVar > f > produc3819737991106469582pVar_f ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Multiset____Ordering____in____NP__OPropVar,type,
    produc1720817013404193843ropVar: nat > multis3193088007478089820ropVar > produc440817675466425857ropVar ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
    product_Pair_nat_nat: nat > nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001tf__f,type,
    product_Pair_nat_f: nat > f > product_prod_nat_f ).

thf(sy_c_Product__Type_OPair_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    produc6842008660352832535ropVar: produc2642885279816024434ropVar > produc2642885279816024434ropVar > produc7137084944585861159ropVar ).

thf(sy_c_Product__Type_OPair_001tf__f_001t__List__Olist_Itf__f_J,type,
    produc4886901243613182557list_f: f > list_f > produc7257588745650887789list_f ).

thf(sy_c_Product__Type_OPair_001tf__f_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Product__Type_OPair_001tf__f_001t__Nat__Onat,type,
    product_Pair_f_nat: f > nat > product_prod_f_nat ).

thf(sy_c_Product__Type_OPair_001tf__f_001tf__f,type,
    product_Pair_f_f: f > f > product_prod_f_f ).

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    produc2626176000494625587at_nat: ( nat > nat > product_prod_nat_nat ) > product_prod_nat_nat > product_prod_nat_nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
    produc6156676138143019412at_nat: produc8199716216217303280at_nat > nat > nat ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Multiset____Ordering____in____NP__OPropVar_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Product__Type_Oprod_Ofst_001t__Multiset____Ordering____in____NP__OPropVar_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J,type,
    produc6918815732823278798ropVar: produc2642885279816024434ropVar > multis3193088007478089820ropVar ).

thf(sy_c_Product__Type_Oprod_Ofst_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oproduct_001t__Multiset____Ordering____in____NP__OPropVar_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Product__Type_Oproduct_001t__Multiset____Ordering____in____NP__OPropVar_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_Oproduct_001t__Multiset____Ordering____in____NP__OPropVar_001tf__f,type,
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thf(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__Multiset____Ordering____in____NP__OPropVar,type,
    produc7744834158543177912ropVar: set_nat > set_Mu5658373669842198716ropVar > set_Pr8672156015886137399ropVar ).

thf(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001t__Nat__Onat,type,
    produc929334515565554804at_nat: set_nat > set_nat > set_Pr1261947904930325089at_nat ).

thf(sy_c_Product__Type_Oproduct_001t__Nat__Onat_001tf__f,type,
    produc840569836383677279_nat_f: set_nat > set_f > set_Pr4548512049067879902_nat_f ).

thf(sy_c_Product__Type_Oproduct_001tf__f_001t__List__Olist_Itf__f_J,type,
    produc3374443989276135832list_f: set_f > set_list_f > set_Pr9218602257211810509list_f ).

thf(sy_c_Product__Type_Oproduct_001tf__f_001t__Multiset____Ordering____in____NP__OPropVar,type,
    produc8452876027641305707ropVar: set_f > set_Mu5658373669842198716ropVar > set_Pr4690715939279257120ropVar ).

thf(sy_c_Product__Type_Oproduct_001tf__f_001t__Nat__Onat,type,
    produc2985859979987363329_f_nat: set_f > set_nat > set_Pr5245412377734449720_f_nat ).

thf(sy_c_Product__Type_Oproduct_001tf__f_001tf__f,type,
    product_product_f_f: set_f > set_f > set_Product_prod_f_f ).

thf(sy_c_Relation_OId__on_001t__List__Olist_Itf__f_J,type,
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thf(sy_c_Relation_OId__on_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Relation_OId__on_001t__Nat__Onat,type,
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thf(sy_c_Relation_OId__on_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,
    id_on_4441320969702803471ropVar: set_Pr6405597471078531282ropVar > set_Pr3201966908590682119ropVar ).

thf(sy_c_Relation_OId__on_001tf__f,type,
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thf(sy_c_Seq_Oinf__concat__simple,type,
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thf(sy_c_Set_OBall_001t__Multiset____Ordering____in____NP__OPropVar,type,
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thf(sy_c_Set_OBall_001t__Nat__Onat,type,
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thf(sy_c_Set_OBall_001tf__f,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_It__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J_J,type,
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thf(sy_c_Set_OCollect_001t__List__Olist_Itf__f_J,type,
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thf(sy_c_Set_OCollect_001t__Nat__Onat,type,
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thf(sy_c_Set_OCollect_001tf__f,type,
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thf(sy_c_Set_Oimage_001t__List__Olist_Itf__f_J_001tf__f,type,
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thf(sy_c_Set_Oimage_001t__Nat__Onat_001tf__f,type,
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thf(sy_c_Set_Oimage_001tf__f_001t__List__Olist_Itf__f_J,type,
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thf(sy_c_Set_Oimage_001tf__f_001tf__f,type,
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    the_elem_list_f: set_list_f > list_f ).

thf(sy_c_Set_Othe__elem_001t__Multiset____Ordering____in____NP__OPropVar,type,
    the_el4535477323905673297ropVar: set_Mu5658373669842198716ropVar > multis3193088007478089820ropVar ).

thf(sy_c_Set_Othe__elem_001t__Nat__Onat,type,
    the_elem_nat: set_nat > nat ).

thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__Multiset____Ordering____in____NP__OPropVar_J_J,type,
    the_el184889641172005735ropVar: set_Pr6405597471078531282ropVar > produc2642885279816024434ropVar ).

thf(sy_c_Set_Othe__elem_001tf__f,type,
    the_elem_f: set_f > f ).

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

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_It__Nat__Onat_J_J,type,
    accp_list_list_nat: ( list_list_nat > list_list_nat > $o ) > list_list_nat > $o ).

thf(sy_c_Wellfounded_Oaccp_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,
    accp_l8942749945645456647ropVar: ( list_l3415868495287753214ropVar > list_l3415868495287753214ropVar > $o ) > list_l3415868495287753214ropVar > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__List__Olist_Itf__f_J_J,type,
    accp_list_list_f: ( list_list_f > list_list_f > $o ) > list_list_f > $o ).

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

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

thf(sy_c_fChoice_001tf__f,type,
    fChoice_f: ( f > $o ) > f ).

thf(sy_c_member_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J,type,
    member490224382042672267ropVar: list_M6740372356978808802ropVar > set_li3089797549399498434ropVar > $o ).

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

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

thf(sy_c_member_001t__List__Olist_Itf__f_J,type,
    member_list_f: list_f > set_list_f > $o ).

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

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

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Nat__Onat_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member7226740684066999833at_nat: produc8199716216217303280at_nat > set_Pr9093778441882193744at_nat > $o ).

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

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

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

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__f_J_Mt__List__Olist_Itf__f_J_J,type,
    member631280506938425232list_f: produc1604256039088728679list_f > set_Pr3684263487712998727list_f > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset____Ordering____in____NP__OPropVar_Mt__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__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_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_Itf__f_Mtf__f_J,type,
    member5935997662887750672od_f_f: product_prod_f_f > set_Product_prod_f_f > $o ).

thf(sy_c_member_001tf__f,type,
    member_f: f > set_f > $o ).

thf(sy_v_cands____,type,
    cands: list_f ).

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

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

thf(sy_v_enc__conj,type,
    enc_conj: list_f > f ).

thf(sy_v_eval,type,
    eval: ( multis3193088007478089820ropVar > $o ) > f > $o ).

thf(sy_v_f____,type,
    f2: f ).

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

thf(sy_v_j____,type,
    j: nat ).

thf(sy_v_m,type,
    m: nat ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_one____,type,
    one2: f ).

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

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

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

thf(sy_v_xs1____,type,
    xs1: list_nat ).

thf(sy_v_xs2____,type,
    xs2: list_nat ).

thf(sy_v_ys1____,type,
    ys1: list_nat ).

thf(sy_v_ys2____,type,
    ys2: list_nat ).

% Relevant facts (1263)
thf(fact_0_PropVar_Oinject_I1_J,axiom,
    ! [X11: nat,X12: nat,Y11: nat,Y12: nat] :
      ( ( ( multis387687052011358179_Gamma @ X11 @ X12 )
        = ( multis387687052011358179_Gamma @ Y11 @ Y12 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 ) ) ) ).

% PropVar.inject(1)
thf(fact_1_PropVar_Oinject_I4_J,axiom,
    ! [X41: nat,X42: nat,Y41: nat,Y42: nat] :
      ( ( ( multis6646701651571564453xOneJI @ X41 @ X42 )
        = ( multis6646701651571564453xOneJI @ Y41 @ Y42 ) )
      = ( ( X41 = Y41 )
        & ( X42 = Y42 ) ) ) ).

% PropVar.inject(4)
thf(fact_2_PropVar_Oinject_I3_J,axiom,
    ! [X31: nat,X32: nat,Y31: nat,Y32: nat] :
      ( ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
        = ( multis2983220944385521703ZeroJI @ Y31 @ Y32 ) )
      = ( ( X31 = Y31 )
        & ( X32 = Y32 ) ) ) ).

% PropVar.inject(3)
thf(fact_3_PropVar_Odistinct_I19_J,axiom,
    ! [X31: nat,X32: nat,X41: nat,X42: nat] :
      ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
     != ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).

% PropVar.distinct(19)
thf(fact_4_PropVar_Odistinct_I5_J,axiom,
    ! [X11: nat,X12: nat,X41: nat,X42: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).

% PropVar.distinct(5)
thf(fact_5_PropVar_Odistinct_I3_J,axiom,
    ! [X11: nat,X12: nat,X31: nat,X32: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis2983220944385521703ZeroJI @ X31 @ X32 ) ) ).

% PropVar.distinct(3)
thf(fact_6_list_Oset__map,axiom,
    ! [F: f > f,V: list_f] :
      ( ( set_f2 @ ( map_f_f @ F @ V ) )
      = ( image_f_f @ F @ ( set_f2 @ V ) ) ) ).

% list.set_map
thf(fact_7_list_Oset__map,axiom,
    ! [F: nat > f,V: list_nat] :
      ( ( set_f2 @ ( map_nat_f @ F @ V ) )
      = ( image_nat_f @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_8_list_Oset__map,axiom,
    ! [F: multis3193088007478089820ropVar > f,V: list_M6740372356978808802ropVar] :
      ( ( set_f2 @ ( map_Mu6818335725640936427pVar_f @ F @ V ) )
      = ( image_7076402817453355436pVar_f @ F @ ( set_Mu146256361055586289ropVar @ V ) ) ) ).

% list.set_map
thf(fact_9_list_Oset__map,axiom,
    ! [F: f > nat,V: list_f] :
      ( ( set_nat2 @ ( map_f_nat @ F @ V ) )
      = ( image_f_nat @ F @ ( set_f2 @ V ) ) ) ).

% list.set_map
thf(fact_10_list_Oset__map,axiom,
    ! [F: nat > nat,V: list_nat] :
      ( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
      = ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_11_list_Oset__map,axiom,
    ! [F: multis3193088007478089820ropVar > nat,V: list_M6740372356978808802ropVar] :
      ( ( set_nat2 @ ( map_Mu1970517791673442536ar_nat @ F @ V ) )
      = ( image_3439306068482956519ar_nat @ F @ ( set_Mu146256361055586289ropVar @ V ) ) ) ).

% list.set_map
thf(fact_12_list_Oset__map,axiom,
    ! [F: f > multis3193088007478089820ropVar,V: list_f] :
      ( ( set_Mu146256361055586289ropVar @ ( map_f_4134023513460405853ropVar @ F @ V ) )
      = ( image_4392090605272824862ropVar @ F @ ( set_f2 @ V ) ) ) ).

% list.set_map
thf(fact_13_list_Oset__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,V: list_nat] :
      ( ( set_Mu146256361055586289ropVar @ ( map_na2466558649495876678ropVar @ F @ V ) )
      = ( image_3935346926305390661ropVar @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_14_list_Oset__map,axiom,
    ! [F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,V: list_M6740372356978808802ropVar] :
      ( ( set_Mu146256361055586289ropVar @ ( map_Mu2265489581466636612ropVar @ F @ V ) )
      = ( image_7661484744434814213ropVar @ F @ ( set_Mu146256361055586289ropVar @ V ) ) ) ).

% list.set_map
thf(fact_15_list_Oset__map,axiom,
    ! [F: list_f > f,V: list_list_f] :
      ( ( set_f2 @ ( map_list_f_f @ F @ V ) )
      = ( image_list_f_f @ F @ ( set_list_f2 @ V ) ) ) ).

% list.set_map
thf(fact_16_f,axiom,
    member_f @ f2 @ ( set_f2 @ ( cons_f @ one2 @ cands ) ) ).

% f
thf(fact_17_map__eq__conv,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f,G: f > multis3193088007478089820ropVar] :
      ( ( ( map_f_4134023513460405853ropVar @ F @ Xs )
        = ( map_f_4134023513460405853ropVar @ G @ Xs ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_18_map__eq__conv,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f,G: f > produc2642885279816024434ropVar] :
      ( ( ( map_f_2905954635146393971ropVar @ F @ Xs )
        = ( map_f_2905954635146393971ropVar @ G @ Xs ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_19_map__eq__conv,axiom,
    ! [F: f > list_f,Xs: list_f,G: f > list_f] :
      ( ( ( map_f_list_f @ F @ Xs )
        = ( map_f_list_f @ G @ Xs ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_20_map__eq__conv,axiom,
    ! [F: f > nat,Xs: list_f,G: f > nat] :
      ( ( ( map_f_nat @ F @ Xs )
        = ( map_f_nat @ G @ Xs ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_21_map__eq__conv,axiom,
    ! [F: f > f,Xs: list_f,G: f > f] :
      ( ( ( map_f_f @ F @ Xs )
        = ( map_f_f @ G @ Xs ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_22_map__eq__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,G: nat > produc2642885279816024434ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_23_map__eq__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_24_map__eq__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,G: nat > multis3193088007478089820ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Xs ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( ( F @ X )
              = ( G @ X ) ) ) ) ) ).

% map_eq_conv
thf(fact_25_eval__all__maps,axiom,
    ! [F: f > list_f,Fs: list_f,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( maps_f_f @ F @ Fs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Fs ) )
           => ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ ( F @ X ) ) )
               => ( eval @ Alpha @ Y ) ) ) ) ) ).

% eval_all_maps
thf(fact_26_eval__all__maps,axiom,
    ! [F: produc2642885279816024434ropVar > list_f,Fs: list_P8791008543742759928ropVar,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( maps_P2527380396400262994pVar_f @ F @ Fs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ! [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Fs ) )
           => ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ ( F @ X ) ) )
               => ( eval @ Alpha @ Y ) ) ) ) ) ).

% eval_all_maps
thf(fact_27_eval__all__maps,axiom,
    ! [F: nat > list_f,Fs: list_nat,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( maps_nat_f @ F @ Fs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Fs ) )
           => ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ ( F @ X ) ) )
               => ( eval @ Alpha @ Y ) ) ) ) ) ).

% eval_all_maps
thf(fact_28_eval__all__maps,axiom,
    ! [F: list_f > list_f,Fs: list_list_f,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( maps_list_f_f @ F @ Fs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ! [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Fs ) )
           => ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ ( F @ X ) ) )
               => ( eval @ Alpha @ Y ) ) ) ) ) ).

% eval_all_maps
thf(fact_29_eval__all__maps,axiom,
    ! [F: multis3193088007478089820ropVar > list_f,Fs: list_M6740372356978808802ropVar,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( maps_M425433082187236072pVar_f @ F @ Fs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ! [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Fs ) )
           => ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ ( F @ X ) ) )
               => ( eval @ Alpha @ Y ) ) ) ) ) ).

% eval_all_maps
thf(fact_30_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X: nat] : X )
    = ( ^ [Xs2: list_nat] : Xs2 ) ) ).

% map_ident
thf(fact_31_map__ident,axiom,
    ( ( map_f_f
      @ ^ [X: f] : X )
    = ( ^ [Xs2: list_f] : Xs2 ) ) ).

% map_ident
thf(fact_32_image__ident,axiom,
    ! [Y2: set_nat] :
      ( ( image_nat_nat
        @ ^ [X: nat] : X
        @ Y2 )
      = Y2 ) ).

% image_ident
thf(fact_33_image__ident,axiom,
    ! [Y2: set_f] :
      ( ( image_f_f
        @ ^ [X: f] : X
        @ Y2 )
      = Y2 ) ).

% image_ident
thf(fact_34_image__set,axiom,
    ! [F: f > f,Xs: list_f] :
      ( ( image_f_f @ F @ ( set_f2 @ Xs ) )
      = ( set_f2 @ ( map_f_f @ F @ Xs ) ) ) ).

% image_set
thf(fact_35_image__set,axiom,
    ! [F: f > nat,Xs: list_f] :
      ( ( image_f_nat @ F @ ( set_f2 @ Xs ) )
      = ( set_nat2 @ ( map_f_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_36_image__set,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f] :
      ( ( image_4392090605272824862ropVar @ F @ ( set_f2 @ Xs ) )
      = ( set_Mu146256361055586289ropVar @ ( map_f_4134023513460405853ropVar @ F @ Xs ) ) ) ).

% image_set
thf(fact_37_image__set,axiom,
    ! [F: nat > f,Xs: list_nat] :
      ( ( image_nat_f @ F @ ( set_nat2 @ Xs ) )
      = ( set_f2 @ ( map_nat_f @ F @ Xs ) ) ) ).

% image_set
thf(fact_38_image__set,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
      = ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_39_image__set,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( image_3935346926305390661ropVar @ F @ ( set_nat2 @ Xs ) )
      = ( set_Mu146256361055586289ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) ) ) ).

% image_set
thf(fact_40_image__set,axiom,
    ! [F: multis3193088007478089820ropVar > f,Xs: list_M6740372356978808802ropVar] :
      ( ( image_7076402817453355436pVar_f @ F @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( set_f2 @ ( map_Mu6818335725640936427pVar_f @ F @ Xs ) ) ) ).

% image_set
thf(fact_41_image__set,axiom,
    ! [F: multis3193088007478089820ropVar > nat,Xs: list_M6740372356978808802ropVar] :
      ( ( image_3439306068482956519ar_nat @ F @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( set_nat2 @ ( map_Mu1970517791673442536ar_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_42_image__set,axiom,
    ! [F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( image_7661484744434814213ropVar @ F @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( set_Mu146256361055586289ropVar @ ( map_Mu2265489581466636612ropVar @ F @ Xs ) ) ) ).

% image_set
thf(fact_43_image__set,axiom,
    ! [F: f > list_f,Xs: list_f] :
      ( ( image_f_list_f @ F @ ( set_f2 @ Xs ) )
      = ( set_list_f2 @ ( map_f_list_f @ F @ Xs ) ) ) ).

% image_set
thf(fact_44_enc__sound_I13_J,axiom,
    ! [Alpha: multis3193088007478089820ropVar > $o,Fs: list_f] :
      ( ( eval @ Alpha @ ( enc_conj @ Fs ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Fs ) )
           => ( eval @ Alpha @ X ) ) ) ) ).

% enc_sound(13)
thf(fact_45_image__eqI,axiom,
    ! [B: multis3193088007478089820ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member8421201611525043589ropVar @ X2 @ A )
       => ( member8421201611525043589ropVar @ B @ ( image_7661484744434814213ropVar @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_46_image__eqI,axiom,
    ! [B: f,F: multis3193088007478089820ropVar > f,X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member8421201611525043589ropVar @ X2 @ A )
       => ( member_f @ B @ ( image_7076402817453355436pVar_f @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_47_image__eqI,axiom,
    ! [B: nat,F: multis3193088007478089820ropVar > nat,X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member8421201611525043589ropVar @ X2 @ A )
       => ( member_nat @ B @ ( image_3439306068482956519ar_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_48_image__eqI,axiom,
    ! [B: multis3193088007478089820ropVar,F: f > multis3193088007478089820ropVar,X2: f,A: set_f] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_f @ X2 @ A )
       => ( member8421201611525043589ropVar @ B @ ( image_4392090605272824862ropVar @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_49_image__eqI,axiom,
    ! [B: f,F: f > f,X2: f,A: set_f] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_f @ X2 @ A )
       => ( member_f @ B @ ( image_f_f @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_50_image__eqI,axiom,
    ! [B: nat,F: f > nat,X2: f,A: set_f] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_f @ X2 @ A )
       => ( member_nat @ B @ ( image_f_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_51_image__eqI,axiom,
    ! [B: multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar,X2: nat,A: set_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member8421201611525043589ropVar @ B @ ( image_3935346926305390661ropVar @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_52_image__eqI,axiom,
    ! [B: f,F: nat > f,X2: nat,A: set_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member_f @ B @ ( image_nat_f @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_53_image__eqI,axiom,
    ! [B: nat,F: nat > nat,X2: nat,A: set_nat] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_nat @ X2 @ A )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_54_image__eqI,axiom,
    ! [B: list_f,F: f > list_f,X2: f,A: set_f] :
      ( ( B
        = ( F @ X2 ) )
     => ( ( member_f @ X2 @ A )
       => ( member_list_f @ B @ ( image_f_list_f @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_55_prod_Oinject,axiom,
    ! [X1: multis3193088007478089820ropVar,X22: produc7679832412844898215ropVar,Y1: multis3193088007478089820ropVar,Y22: produc7679832412844898215ropVar] :
      ( ( ( produc6194868611276206818ropVar @ X1 @ X22 )
        = ( produc6194868611276206818ropVar @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_56_prod_Oinject,axiom,
    ! [X1: multis3193088007478089820ropVar,X22: multis3193088007478089820ropVar,Y1: multis3193088007478089820ropVar,Y22: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ X1 @ X22 )
        = ( produc6936306168532358295ropVar @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_57_prod_Oinject,axiom,
    ! [X1: nat > nat,X22: nat,Y1: nat > nat,Y22: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X22 )
        = ( produc72220940542539688at_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_58_prod_Oinject,axiom,
    ! [X1: nat,X22: nat,Y1: nat,Y22: nat] :
      ( ( ( product_Pair_nat_nat @ X1 @ X22 )
        = ( product_Pair_nat_nat @ Y1 @ Y22 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y22 ) ) ) ).

% prod.inject
thf(fact_59_old_Oprod_Oinject,axiom,
    ! [A2: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar,A3: multis3193088007478089820ropVar,B2: produc7679832412844898215ropVar] :
      ( ( ( produc6194868611276206818ropVar @ A2 @ B )
        = ( produc6194868611276206818ropVar @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_60_old_Oprod_Oinject,axiom,
    ! [A2: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A3: multis3193088007478089820ropVar,B2: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ A2 @ B )
        = ( produc6936306168532358295ropVar @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_61_old_Oprod_Oinject,axiom,
    ! [A2: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A2 @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_62_old_Oprod_Oinject,axiom,
    ! [A2: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_63_list_Oinject,axiom,
    ! [X21: f,X222: list_f,Y21: f,Y222: list_f] :
      ( ( ( cons_f @ X21 @ X222 )
        = ( cons_f @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_64_list_Oinject,axiom,
    ! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar,Y21: produc2642885279816024434ropVar,Y222: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X21 @ X222 )
        = ( cons_P1504208554336013608ropVar @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_65_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y222: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y222 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y222 ) ) ) ).

% list.inject
thf(fact_66_eval__all__Cons,axiom,
    ! [F: f,Gs: list_f,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( cons_f @ F @ Gs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ( eval @ Alpha @ F )
        & ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Gs ) )
           => ( eval @ Alpha @ X ) ) ) ) ).

% eval_all_Cons
thf(fact_67__C1_C_I1_J,axiom,
    member_nat @ j @ ( set_nat2 @ ( upt @ zero_zero_nat @ m ) ) ).

% "1"(1)
thf(fact_68_not__Cons__self2,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( cons_f @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_69_not__Cons__self2,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_70_not__Cons__self2,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( cons_nat @ X2 @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_71_set__ConsD,axiom,
    ! [Y3: list_f,X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ Y3 @ ( set_list_f2 @ ( cons_list_f @ X2 @ Xs ) ) )
     => ( ( Y3 = X2 )
        | ( member_list_f @ Y3 @ ( set_list_f2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_72_set__ConsD,axiom,
    ! [Y3: multis3193088007478089820ropVar,X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X2 @ Xs ) ) )
     => ( ( Y3 = X2 )
        | ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_73_set__ConsD,axiom,
    ! [Y3: f,X2: f,Xs: list_f] :
      ( ( member_f @ Y3 @ ( set_f2 @ ( cons_f @ X2 @ Xs ) ) )
     => ( ( Y3 = X2 )
        | ( member_f @ Y3 @ ( set_f2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_74_set__ConsD,axiom,
    ! [Y3: produc2642885279816024434ropVar,X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) ) )
     => ( ( Y3 = X2 )
        | ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_75_set__ConsD,axiom,
    ! [Y3: nat,X2: nat,Xs: list_nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X2 @ Xs ) ) )
     => ( ( Y3 = X2 )
        | ( member_nat @ Y3 @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_76_list_Oset__cases,axiom,
    ! [E: list_f,A2: list_list_f] :
      ( ( member_list_f @ E @ ( set_list_f2 @ A2 ) )
     => ( ! [Z2: list_list_f] :
            ( A2
           != ( cons_list_f @ E @ Z2 ) )
       => ~ ! [Z1: list_f,Z2: list_list_f] :
              ( ( A2
                = ( cons_list_f @ Z1 @ Z2 ) )
             => ~ ( member_list_f @ E @ ( set_list_f2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_77_list_Oset__cases,axiom,
    ! [E: multis3193088007478089820ropVar,A2: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ E @ ( set_Mu146256361055586289ropVar @ A2 ) )
     => ( ! [Z2: list_M6740372356978808802ropVar] :
            ( A2
           != ( cons_M4277272395645347858ropVar @ E @ Z2 ) )
       => ~ ! [Z1: multis3193088007478089820ropVar,Z2: list_M6740372356978808802ropVar] :
              ( ( A2
                = ( cons_M4277272395645347858ropVar @ Z1 @ Z2 ) )
             => ~ ( member8421201611525043589ropVar @ E @ ( set_Mu146256361055586289ropVar @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_78_list_Oset__cases,axiom,
    ! [E: f,A2: list_f] :
      ( ( member_f @ E @ ( set_f2 @ A2 ) )
     => ( ! [Z2: list_f] :
            ( A2
           != ( cons_f @ E @ Z2 ) )
       => ~ ! [Z1: f,Z2: list_f] :
              ( ( A2
                = ( cons_f @ Z1 @ Z2 ) )
             => ~ ( member_f @ E @ ( set_f2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_79_list_Oset__cases,axiom,
    ! [E: produc2642885279816024434ropVar,A2: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ E @ ( set_Pr3946994337432900359ropVar @ A2 ) )
     => ( ! [Z2: list_P8791008543742759928ropVar] :
            ( A2
           != ( cons_P1504208554336013608ropVar @ E @ Z2 ) )
       => ~ ! [Z1: produc2642885279816024434ropVar,Z2: list_P8791008543742759928ropVar] :
              ( ( A2
                = ( cons_P1504208554336013608ropVar @ Z1 @ Z2 ) )
             => ~ ( member9210975342120487067ropVar @ E @ ( set_Pr3946994337432900359ropVar @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_80_list_Oset__cases,axiom,
    ! [E: nat,A2: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A2 ) )
     => ( ! [Z2: list_nat] :
            ( A2
           != ( cons_nat @ E @ Z2 ) )
       => ~ ! [Z1: nat,Z2: list_nat] :
              ( ( A2
                = ( cons_nat @ Z1 @ Z2 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_81_list_Oset__intros_I1_J,axiom,
    ! [X21: list_f,X222: list_list_f] : ( member_list_f @ X21 @ ( set_list_f2 @ ( cons_list_f @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_82_list_Oset__intros_I1_J,axiom,
    ! [X21: multis3193088007478089820ropVar,X222: list_M6740372356978808802ropVar] : ( member8421201611525043589ropVar @ X21 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_83_list_Oset__intros_I1_J,axiom,
    ! [X21: f,X222: list_f] : ( member_f @ X21 @ ( set_f2 @ ( cons_f @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_84_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] : ( member9210975342120487067ropVar @ X21 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_85_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_86_list_Oset__intros_I2_J,axiom,
    ! [Y3: list_f,X222: list_list_f,X21: list_f] :
      ( ( member_list_f @ Y3 @ ( set_list_f2 @ X222 ) )
     => ( member_list_f @ Y3 @ ( set_list_f2 @ ( cons_list_f @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_87_list_Oset__intros_I2_J,axiom,
    ! [Y3: multis3193088007478089820ropVar,X222: list_M6740372356978808802ropVar,X21: multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ X222 ) )
     => ( member8421201611525043589ropVar @ Y3 @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_88_list_Oset__intros_I2_J,axiom,
    ! [Y3: f,X222: list_f,X21: f] :
      ( ( member_f @ Y3 @ ( set_f2 @ X222 ) )
     => ( member_f @ Y3 @ ( set_f2 @ ( cons_f @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_89_list_Oset__intros_I2_J,axiom,
    ! [Y3: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar,X21: produc2642885279816024434ropVar] :
      ( ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ X222 ) )
     => ( member9210975342120487067ropVar @ Y3 @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_90_list_Oset__intros_I2_J,axiom,
    ! [Y3: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y3 @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y3 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_91_map__eq__Cons__conv,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( map_f_4134023513460405853ropVar @ F @ Xs )
        = ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Xs
              = ( cons_f @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_f_4134023513460405853ropVar @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_92_map__eq__Cons__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_na2466558649495876678ropVar @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_93_map__eq__Cons__conv,axiom,
    ! [F: f > f,Xs: list_f,Y3: f,Ys: list_f] :
      ( ( ( map_f_f @ F @ Xs )
        = ( cons_f @ Y3 @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Xs
              = ( cons_f @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_f_f @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_94_map__eq__Cons__conv,axiom,
    ! [F: nat > f,Xs: list_nat,Y3: f,Ys: list_f] :
      ( ( ( map_nat_f @ F @ Xs )
        = ( cons_f @ Y3 @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_nat_f @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_95_map__eq__Cons__conv,axiom,
    ! [F: f > nat,Xs: list_f,Y3: nat,Ys: list_nat] :
      ( ( ( map_f_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Xs
              = ( cons_f @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_f_nat @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_96_map__eq__Cons__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_nat_nat @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_97_map__eq__Cons__conv,axiom,
    ! [F: f > list_f,Xs: list_f,Y3: list_f,Ys: list_list_f] :
      ( ( ( map_f_list_f @ F @ Xs )
        = ( cons_list_f @ Y3 @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Xs
              = ( cons_f @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_f_list_f @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_98_map__eq__Cons__conv,axiom,
    ! [F: produc2642885279816024434ropVar > f,Xs: list_P8791008543742759928ropVar,Y3: f,Ys: list_f] :
      ( ( ( map_Pr1064736048566318677pVar_f @ F @ Xs )
        = ( cons_f @ Y3 @ Ys ) )
      = ( ? [Z: produc2642885279816024434ropVar,Zs: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_Pr1064736048566318677pVar_f @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_99_map__eq__Cons__conv,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_f_2905954635146393971ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Xs
              = ( cons_f @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_f_2905954635146393971ropVar @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_100_map__eq__Cons__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Xs
              = ( cons_nat @ Z @ Zs ) )
            & ( ( F @ Z )
              = Y3 )
            & ( ( map_na2145702811007361372ropVar @ F @ Zs )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_101_Cons__eq__map__conv,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,F: f > multis3193088007478089820ropVar,Ys: list_f] :
      ( ( ( cons_M4277272395645347858ropVar @ X2 @ Xs )
        = ( map_f_4134023513460405853ropVar @ F @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Ys
              = ( cons_f @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_f_4134023513460405853ropVar @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_102_Cons__eq__map__conv,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( cons_M4277272395645347858ropVar @ X2 @ Xs )
        = ( map_na2466558649495876678ropVar @ F @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_na2466558649495876678ropVar @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_103_Cons__eq__map__conv,axiom,
    ! [X2: f,Xs: list_f,F: f > f,Ys: list_f] :
      ( ( ( cons_f @ X2 @ Xs )
        = ( map_f_f @ F @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Ys
              = ( cons_f @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_f_f @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_104_Cons__eq__map__conv,axiom,
    ! [X2: f,Xs: list_f,F: nat > f,Ys: list_nat] :
      ( ( ( cons_f @ X2 @ Xs )
        = ( map_nat_f @ F @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_nat_f @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_105_Cons__eq__map__conv,axiom,
    ! [X2: nat,Xs: list_nat,F: f > nat,Ys: list_f] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_f_nat @ F @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Ys
              = ( cons_f @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_f_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_106_Cons__eq__map__conv,axiom,
    ! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_nat_nat @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_107_Cons__eq__map__conv,axiom,
    ! [X2: list_f,Xs: list_list_f,F: f > list_f,Ys: list_f] :
      ( ( ( cons_list_f @ X2 @ Xs )
        = ( map_f_list_f @ F @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Ys
              = ( cons_f @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_f_list_f @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_108_Cons__eq__map__conv,axiom,
    ! [X2: f,Xs: list_f,F: produc2642885279816024434ropVar > f,Ys: list_P8791008543742759928ropVar] :
      ( ( ( cons_f @ X2 @ Xs )
        = ( map_Pr1064736048566318677pVar_f @ F @ Ys ) )
      = ( ? [Z: produc2642885279816024434ropVar,Zs: list_P8791008543742759928ropVar] :
            ( ( Ys
              = ( cons_P1504208554336013608ropVar @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_Pr1064736048566318677pVar_f @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_109_Cons__eq__map__conv,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: f > produc2642885279816024434ropVar,Ys: list_f] :
      ( ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
        = ( map_f_2905954635146393971ropVar @ F @ Ys ) )
      = ( ? [Z: f,Zs: list_f] :
            ( ( Ys
              = ( cons_f @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_f_2905954635146393971ropVar @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_110_Cons__eq__map__conv,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
        = ( map_na2145702811007361372ropVar @ F @ Ys ) )
      = ( ? [Z: nat,Zs: list_nat] :
            ( ( Ys
              = ( cons_nat @ Z @ Zs ) )
            & ( X2
              = ( F @ Z ) )
            & ( Xs
              = ( map_na2145702811007361372ropVar @ F @ Zs ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_111_map__eq__Cons__D,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( map_f_4134023513460405853ropVar @ F @ Xs )
        = ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Xs
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_f_4134023513460405853ropVar @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_112_map__eq__Cons__D,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_na2466558649495876678ropVar @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_113_map__eq__Cons__D,axiom,
    ! [F: f > f,Xs: list_f,Y3: f,Ys: list_f] :
      ( ( ( map_f_f @ F @ Xs )
        = ( cons_f @ Y3 @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Xs
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_f_f @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_114_map__eq__Cons__D,axiom,
    ! [F: nat > f,Xs: list_nat,Y3: f,Ys: list_f] :
      ( ( ( map_nat_f @ F @ Xs )
        = ( cons_f @ Y3 @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_nat_f @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_115_map__eq__Cons__D,axiom,
    ! [F: f > nat,Xs: list_f,Y3: nat,Ys: list_nat] :
      ( ( ( map_f_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Xs
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_f_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_116_map__eq__Cons__D,axiom,
    ! [F: nat > nat,Xs: list_nat,Y3: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y3 @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_nat_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_117_map__eq__Cons__D,axiom,
    ! [F: f > list_f,Xs: list_f,Y3: list_f,Ys: list_list_f] :
      ( ( ( map_f_list_f @ F @ Xs )
        = ( cons_list_f @ Y3 @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Xs
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_f_list_f @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_118_map__eq__Cons__D,axiom,
    ! [F: produc2642885279816024434ropVar > f,Xs: list_P8791008543742759928ropVar,Y3: f,Ys: list_f] :
      ( ( ( map_Pr1064736048566318677pVar_f @ F @ Xs )
        = ( cons_f @ Y3 @ Ys ) )
     => ? [Z3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( cons_P1504208554336013608ropVar @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_Pr1064736048566318677pVar_f @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_119_map__eq__Cons__D,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_f_2905954635146393971ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Xs
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_f_2905954635146393971ropVar @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_120_map__eq__Cons__D,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y3 )
          & ( ( map_na2145702811007361372ropVar @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_121_Cons__eq__map__D,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,F: f > multis3193088007478089820ropVar,Ys: list_f] :
      ( ( ( cons_M4277272395645347858ropVar @ X2 @ Xs )
        = ( map_f_4134023513460405853ropVar @ F @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Ys
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_f_4134023513460405853ropVar @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_122_Cons__eq__map__D,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Ys: list_nat] :
      ( ( ( cons_M4277272395645347858ropVar @ X2 @ Xs )
        = ( map_na2466558649495876678ropVar @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_na2466558649495876678ropVar @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_123_Cons__eq__map__D,axiom,
    ! [X2: f,Xs: list_f,F: f > f,Ys: list_f] :
      ( ( ( cons_f @ X2 @ Xs )
        = ( map_f_f @ F @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Ys
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_f_f @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_124_Cons__eq__map__D,axiom,
    ! [X2: f,Xs: list_f,F: nat > f,Ys: list_nat] :
      ( ( ( cons_f @ X2 @ Xs )
        = ( map_nat_f @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_nat_f @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_125_Cons__eq__map__D,axiom,
    ! [X2: nat,Xs: list_nat,F: f > nat,Ys: list_f] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_f_nat @ F @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Ys
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_f_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_126_Cons__eq__map__D,axiom,
    ! [X2: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_127_Cons__eq__map__D,axiom,
    ! [X2: list_f,Xs: list_list_f,F: f > list_f,Ys: list_f] :
      ( ( ( cons_list_f @ X2 @ Xs )
        = ( map_f_list_f @ F @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Ys
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_f_list_f @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_128_Cons__eq__map__D,axiom,
    ! [X2: f,Xs: list_f,F: produc2642885279816024434ropVar > f,Ys: list_P8791008543742759928ropVar] :
      ( ( ( cons_f @ X2 @ Xs )
        = ( map_Pr1064736048566318677pVar_f @ F @ Ys ) )
     => ? [Z3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Ys
            = ( cons_P1504208554336013608ropVar @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr1064736048566318677pVar_f @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_129_Cons__eq__map__D,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: f > produc2642885279816024434ropVar,Ys: list_f] :
      ( ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
        = ( map_f_2905954635146393971ropVar @ F @ Ys ) )
     => ? [Z3: f,Zs2: list_f] :
          ( ( Ys
            = ( cons_f @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_f_2905954635146393971ropVar @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_130_Cons__eq__map__D,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar,Ys: list_nat] :
      ( ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
        = ( map_na2145702811007361372ropVar @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X2
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_na2145702811007361372ropVar @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_131_list_Osimps_I9_J,axiom,
    ! [F: f > multis3193088007478089820ropVar,X21: f,X222: list_f] :
      ( ( map_f_4134023513460405853ropVar @ F @ ( cons_f @ X21 @ X222 ) )
      = ( cons_M4277272395645347858ropVar @ ( F @ X21 ) @ ( map_f_4134023513460405853ropVar @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_132_list_Osimps_I9_J,axiom,
    ! [F: f > f,X21: f,X222: list_f] :
      ( ( map_f_f @ F @ ( cons_f @ X21 @ X222 ) )
      = ( cons_f @ ( F @ X21 ) @ ( map_f_f @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_133_list_Osimps_I9_J,axiom,
    ! [F: f > nat,X21: f,X222: list_f] :
      ( ( map_f_nat @ F @ ( cons_f @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_f_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_134_list_Osimps_I9_J,axiom,
    ! [F: nat > multis3193088007478089820ropVar,X21: nat,X222: list_nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_M4277272395645347858ropVar @ ( F @ X21 ) @ ( map_na2466558649495876678ropVar @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_135_list_Osimps_I9_J,axiom,
    ! [F: nat > f,X21: nat,X222: list_nat] :
      ( ( map_nat_f @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_f @ ( F @ X21 ) @ ( map_nat_f @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_136_list_Osimps_I9_J,axiom,
    ! [F: nat > nat,X21: nat,X222: list_nat] :
      ( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_137_list_Osimps_I9_J,axiom,
    ! [F: f > list_f,X21: f,X222: list_f] :
      ( ( map_f_list_f @ F @ ( cons_f @ X21 @ X222 ) )
      = ( cons_list_f @ ( F @ X21 ) @ ( map_f_list_f @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_138_list_Osimps_I9_J,axiom,
    ! [F: f > produc2642885279816024434ropVar,X21: f,X222: list_f] :
      ( ( map_f_2905954635146393971ropVar @ F @ ( cons_f @ X21 @ X222 ) )
      = ( cons_P1504208554336013608ropVar @ ( F @ X21 ) @ ( map_f_2905954635146393971ropVar @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_139_list_Osimps_I9_J,axiom,
    ! [F: produc2642885279816024434ropVar > f,X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( ( map_Pr1064736048566318677pVar_f @ F @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) )
      = ( cons_f @ ( F @ X21 ) @ ( map_Pr1064736048566318677pVar_f @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_140_list_Osimps_I9_J,axiom,
    ! [F: produc2642885279816024434ropVar > nat,X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( ( map_Pr7810381211709419518ar_nat @ F @ ( cons_P1504208554336013608ropVar @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_Pr7810381211709419518ar_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_141_zero__reorient,axiom,
    ! [X2: nat] :
      ( ( zero_zero_nat = X2 )
      = ( X2 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_142_Pair__inject,axiom,
    ! [A2: multis3193088007478089820ropVar,B: produc7679832412844898215ropVar,A3: multis3193088007478089820ropVar,B2: produc7679832412844898215ropVar] :
      ( ( ( produc6194868611276206818ropVar @ A2 @ B )
        = ( produc6194868611276206818ropVar @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_143_Pair__inject,axiom,
    ! [A2: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A3: multis3193088007478089820ropVar,B2: multis3193088007478089820ropVar] :
      ( ( ( produc6936306168532358295ropVar @ A2 @ B )
        = ( produc6936306168532358295ropVar @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_144_Pair__inject,axiom,
    ! [A2: nat > nat,B: nat,A3: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A2 @ B )
        = ( produc72220940542539688at_nat @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_145_Pair__inject,axiom,
    ! [A2: nat,B: nat,A3: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A2 @ B )
        = ( product_Pair_nat_nat @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_146_prod__cases,axiom,
    ! [P: produc2642885279816024434ropVar > $o,P2: produc2642885279816024434ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: produc7679832412844898215ropVar] : ( P @ ( produc6194868611276206818ropVar @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_147_prod__cases,axiom,
    ! [P: produc7679832412844898215ropVar > $o,P2: produc7679832412844898215ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] : ( P @ ( produc6936306168532358295ropVar @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_148_prod__cases,axiom,
    ! [P: produc8199716216217303280at_nat > $o,P2: produc8199716216217303280at_nat] :
      ( ! [A4: nat > nat,B3: nat] : ( P @ ( produc72220940542539688at_nat @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_149_prod__cases,axiom,
    ! [P: product_prod_nat_nat > $o,P2: product_prod_nat_nat] :
      ( ! [A4: nat,B3: nat] : ( P @ ( product_Pair_nat_nat @ A4 @ B3 ) )
     => ( P @ P2 ) ) ).

% prod_cases
thf(fact_150_surj__pair,axiom,
    ! [P2: produc2642885279816024434ropVar] :
    ? [X3: multis3193088007478089820ropVar,Y4: produc7679832412844898215ropVar] :
      ( P2
      = ( produc6194868611276206818ropVar @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_151_surj__pair,axiom,
    ! [P2: produc7679832412844898215ropVar] :
    ? [X3: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
      ( P2
      = ( produc6936306168532358295ropVar @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_152_surj__pair,axiom,
    ! [P2: produc8199716216217303280at_nat] :
    ? [X3: nat > nat,Y4: nat] :
      ( P2
      = ( produc72220940542539688at_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_153_surj__pair,axiom,
    ! [P2: product_prod_nat_nat] :
    ? [X3: nat,Y4: nat] :
      ( P2
      = ( product_Pair_nat_nat @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_154_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2642885279816024434ropVar] :
      ~ ! [A4: multis3193088007478089820ropVar,B3: produc7679832412844898215ropVar] :
          ( Y3
         != ( produc6194868611276206818ropVar @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_155_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc7679832412844898215ropVar] :
      ~ ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar] :
          ( Y3
         != ( produc6936306168532358295ropVar @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_156_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8199716216217303280at_nat] :
      ~ ! [A4: nat > nat,B3: nat] :
          ( Y3
         != ( produc72220940542539688at_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_157_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_nat_nat] :
      ~ ! [A4: nat,B3: nat] :
          ( Y3
         != ( product_Pair_nat_nat @ A4 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_158_rev__image__eqI,axiom,
    ! [X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B: multis3193088007478089820ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member8421201611525043589ropVar @ B @ ( image_7661484744434814213ropVar @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_159_rev__image__eqI,axiom,
    ! [X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B: f,F: multis3193088007478089820ropVar > f] :
      ( ( member8421201611525043589ropVar @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_f @ B @ ( image_7076402817453355436pVar_f @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_160_rev__image__eqI,axiom,
    ! [X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B: nat,F: multis3193088007478089820ropVar > nat] :
      ( ( member8421201611525043589ropVar @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_nat @ B @ ( image_3439306068482956519ar_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_161_rev__image__eqI,axiom,
    ! [X2: f,A: set_f,B: multis3193088007478089820ropVar,F: f > multis3193088007478089820ropVar] :
      ( ( member_f @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member8421201611525043589ropVar @ B @ ( image_4392090605272824862ropVar @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_162_rev__image__eqI,axiom,
    ! [X2: f,A: set_f,B: f,F: f > f] :
      ( ( member_f @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_f @ B @ ( image_f_f @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_163_rev__image__eqI,axiom,
    ! [X2: f,A: set_f,B: nat,F: f > nat] :
      ( ( member_f @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_nat @ B @ ( image_f_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_164_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B: multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar] :
      ( ( member_nat @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member8421201611525043589ropVar @ B @ ( image_3935346926305390661ropVar @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_165_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B: f,F: nat > f] :
      ( ( member_nat @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_f @ B @ ( image_nat_f @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_166_rev__image__eqI,axiom,
    ! [X2: nat,A: set_nat,B: nat,F: nat > nat] :
      ( ( member_nat @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_167_rev__image__eqI,axiom,
    ! [X2: f,A: set_f,B: list_f,F: f > list_f] :
      ( ( member_f @ X2 @ A )
     => ( ( B
          = ( F @ X2 ) )
       => ( member_list_f @ B @ ( image_f_list_f @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_168_ball__imageD,axiom,
    ! [F: produc2642885279816024434ropVar > multis3193088007478089820ropVar,A: set_Pr6405597471078531282ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ! [X3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X3 @ ( image_4513341045867859311ropVar @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_169_ball__imageD,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( image_nat_nat @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: nat] :
          ( ( member_nat @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_170_ball__imageD,axiom,
    ! [F: f > multis3193088007478089820ropVar,A: set_f,P: multis3193088007478089820ropVar > $o] :
      ( ! [X3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X3 @ ( image_4392090605272824862ropVar @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: f] :
          ( ( member_f @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_171_ball__imageD,axiom,
    ! [F: f > produc2642885279816024434ropVar,A: set_f,P: produc2642885279816024434ropVar > $o] :
      ( ! [X3: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X3 @ ( image_2710159631831406900ropVar @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: f] :
          ( ( member_f @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_172_ball__imageD,axiom,
    ! [F: f > list_f,A: set_f,P: list_f > $o] :
      ( ! [X3: list_f] :
          ( ( member_list_f @ X3 @ ( image_f_list_f @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: f] :
          ( ( member_f @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_173_ball__imageD,axiom,
    ! [F: f > nat,A: set_f,P: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( image_f_nat @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: f] :
          ( ( member_f @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_174_ball__imageD,axiom,
    ! [F: f > f,A: set_f,P: f > $o] :
      ( ! [X3: f] :
          ( ( member_f @ X3 @ ( image_f_f @ F @ A ) )
         => ( P @ X3 ) )
     => ! [X4: f] :
          ( ( member_f @ X4 @ A )
         => ( P @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_175_image__cong,axiom,
    ! [M: set_Pr6405597471078531282ropVar,N: set_Pr6405597471078531282ropVar,F: produc2642885279816024434ropVar > multis3193088007478089820ropVar,G: produc2642885279816024434ropVar > multis3193088007478089820ropVar] :
      ( ( M = N )
     => ( ! [X3: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_4513341045867859311ropVar @ F @ M )
          = ( image_4513341045867859311ropVar @ G @ N ) ) ) ) ).

% image_cong
thf(fact_176_image__cong,axiom,
    ! [M: set_f,N: set_f,F: f > multis3193088007478089820ropVar,G: f > multis3193088007478089820ropVar] :
      ( ( M = N )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_4392090605272824862ropVar @ F @ M )
          = ( image_4392090605272824862ropVar @ G @ N ) ) ) ) ).

% image_cong
thf(fact_177_image__cong,axiom,
    ! [M: set_f,N: set_f,F: f > produc2642885279816024434ropVar,G: f > produc2642885279816024434ropVar] :
      ( ( M = N )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_2710159631831406900ropVar @ F @ M )
          = ( image_2710159631831406900ropVar @ G @ N ) ) ) ) ).

% image_cong
thf(fact_178_image__cong,axiom,
    ! [M: set_f,N: set_f,F: f > list_f,G: f > list_f] :
      ( ( M = N )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_f_list_f @ F @ M )
          = ( image_f_list_f @ G @ N ) ) ) ) ).

% image_cong
thf(fact_179_image__cong,axiom,
    ! [M: set_f,N: set_f,F: f > nat,G: f > nat] :
      ( ( M = N )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_f_nat @ F @ M )
          = ( image_f_nat @ G @ N ) ) ) ) ).

% image_cong
thf(fact_180_image__cong,axiom,
    ! [M: set_f,N: set_f,F: f > f,G: f > f] :
      ( ( M = N )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_f_f @ F @ M )
          = ( image_f_f @ G @ N ) ) ) ) ).

% image_cong
thf(fact_181_image__cong,axiom,
    ! [M: set_nat,N: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M = N )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_nat @ F @ M )
          = ( image_nat_nat @ G @ N ) ) ) ) ).

% image_cong
thf(fact_182_bex__imageD,axiom,
    ! [F: produc2642885279816024434ropVar > multis3193088007478089820ropVar,A: set_Pr6405597471078531282ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( image_4513341045867859311ropVar @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_183_bex__imageD,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_184_bex__imageD,axiom,
    ! [F: f > multis3193088007478089820ropVar,A: set_f,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( image_4392090605272824862ropVar @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: f] :
          ( ( member_f @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_185_bex__imageD,axiom,
    ! [F: f > produc2642885279816024434ropVar,A: set_f,P: produc2642885279816024434ropVar > $o] :
      ( ? [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( image_2710159631831406900ropVar @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: f] :
          ( ( member_f @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_186_bex__imageD,axiom,
    ! [F: f > list_f,A: set_f,P: list_f > $o] :
      ( ? [X4: list_f] :
          ( ( member_list_f @ X4 @ ( image_f_list_f @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: f] :
          ( ( member_f @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_187_bex__imageD,axiom,
    ! [F: f > nat,A: set_f,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( image_f_nat @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: f] :
          ( ( member_f @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_188_bex__imageD,axiom,
    ! [F: f > f,A: set_f,P: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ ( image_f_f @ F @ A ) )
          & ( P @ X4 ) )
     => ? [X3: f] :
          ( ( member_f @ X3 @ A )
          & ( P @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_189_image__iff,axiom,
    ! [Z4: produc2642885279816024434ropVar,F: f > produc2642885279816024434ropVar,A: set_f] :
      ( ( member9210975342120487067ropVar @ Z4 @ ( image_2710159631831406900ropVar @ F @ A ) )
      = ( ? [X: f] :
            ( ( member_f @ X @ A )
            & ( Z4
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_190_image__iff,axiom,
    ! [Z4: list_f,F: f > list_f,A: set_f] :
      ( ( member_list_f @ Z4 @ ( image_f_list_f @ F @ A ) )
      = ( ? [X: f] :
            ( ( member_f @ X @ A )
            & ( Z4
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_191_image__iff,axiom,
    ! [Z4: multis3193088007478089820ropVar,F: produc2642885279816024434ropVar > multis3193088007478089820ropVar,A: set_Pr6405597471078531282ropVar] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( image_4513341045867859311ropVar @ F @ A ) )
      = ( ? [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ A )
            & ( Z4
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_192_image__iff,axiom,
    ! [Z4: multis3193088007478089820ropVar,F: f > multis3193088007478089820ropVar,A: set_f] :
      ( ( member8421201611525043589ropVar @ Z4 @ ( image_4392090605272824862ropVar @ F @ A ) )
      = ( ? [X: f] :
            ( ( member_f @ X @ A )
            & ( Z4
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_193_image__iff,axiom,
    ! [Z4: f,F: f > f,A: set_f] :
      ( ( member_f @ Z4 @ ( image_f_f @ F @ A ) )
      = ( ? [X: f] :
            ( ( member_f @ X @ A )
            & ( Z4
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_194_image__iff,axiom,
    ! [Z4: nat,F: nat > nat,A: set_nat] :
      ( ( member_nat @ Z4 @ ( image_nat_nat @ F @ A ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ A )
            & ( Z4
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_195_image__iff,axiom,
    ! [Z4: nat,F: f > nat,A: set_f] :
      ( ( member_nat @ Z4 @ ( image_f_nat @ F @ A ) )
      = ( ? [X: f] :
            ( ( member_f @ X @ A )
            & ( Z4
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_196_imageI,axiom,
    ! [X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ A )
     => ( member8421201611525043589ropVar @ ( F @ X2 ) @ ( image_7661484744434814213ropVar @ F @ A ) ) ) ).

% imageI
thf(fact_197_imageI,axiom,
    ! [X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > f] :
      ( ( member8421201611525043589ropVar @ X2 @ A )
     => ( member_f @ ( F @ X2 ) @ ( image_7076402817453355436pVar_f @ F @ A ) ) ) ).

% imageI
thf(fact_198_imageI,axiom,
    ! [X2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,F: multis3193088007478089820ropVar > nat] :
      ( ( member8421201611525043589ropVar @ X2 @ A )
     => ( member_nat @ ( F @ X2 ) @ ( image_3439306068482956519ar_nat @ F @ A ) ) ) ).

% imageI
thf(fact_199_imageI,axiom,
    ! [X2: f,A: set_f,F: f > multis3193088007478089820ropVar] :
      ( ( member_f @ X2 @ A )
     => ( member8421201611525043589ropVar @ ( F @ X2 ) @ ( image_4392090605272824862ropVar @ F @ A ) ) ) ).

% imageI
thf(fact_200_imageI,axiom,
    ! [X2: f,A: set_f,F: f > f] :
      ( ( member_f @ X2 @ A )
     => ( member_f @ ( F @ X2 ) @ ( image_f_f @ F @ A ) ) ) ).

% imageI
thf(fact_201_imageI,axiom,
    ! [X2: f,A: set_f,F: f > nat] :
      ( ( member_f @ X2 @ A )
     => ( member_nat @ ( F @ X2 ) @ ( image_f_nat @ F @ A ) ) ) ).

% imageI
thf(fact_202_imageI,axiom,
    ! [X2: nat,A: set_nat,F: nat > multis3193088007478089820ropVar] :
      ( ( member_nat @ X2 @ A )
     => ( member8421201611525043589ropVar @ ( F @ X2 ) @ ( image_3935346926305390661ropVar @ F @ A ) ) ) ).

% imageI
thf(fact_203_imageI,axiom,
    ! [X2: nat,A: set_nat,F: nat > f] :
      ( ( member_nat @ X2 @ A )
     => ( member_f @ ( F @ X2 ) @ ( image_nat_f @ F @ A ) ) ) ).

% imageI
thf(fact_204_imageI,axiom,
    ! [X2: nat,A: set_nat,F: nat > nat] :
      ( ( member_nat @ X2 @ A )
     => ( member_nat @ ( F @ X2 ) @ ( image_nat_nat @ F @ A ) ) ) ).

% imageI
thf(fact_205_imageI,axiom,
    ! [X2: f,A: set_f,F: f > list_f] :
      ( ( member_f @ X2 @ A )
     => ( member_list_f @ ( F @ X2 ) @ ( image_f_list_f @ F @ A ) ) ) ).

% imageI
thf(fact_206_Ball__def,axiom,
    ( ball_M3160559173508560390ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,P3: multis3193088007478089820ropVar > $o] :
        ! [X: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X @ A5 )
         => ( P3 @ X ) ) ) ) ).

% Ball_def
thf(fact_207_Ball__def,axiom,
    ( ball_f
    = ( ^ [A5: set_f,P3: f > $o] :
        ! [X: f] :
          ( ( member_f @ X @ A5 )
         => ( P3 @ X ) ) ) ) ).

% Ball_def
thf(fact_208_Ball__def,axiom,
    ( ball_nat
    = ( ^ [A5: set_nat,P3: nat > $o] :
        ! [X: nat] :
          ( ( member_nat @ X @ A5 )
         => ( P3 @ X ) ) ) ) ).

% Ball_def
thf(fact_209_Compr__image__eq,axiom,
    ! [F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( collec8658135508005558727ropVar
        @ ^ [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( image_7661484744434814213ropVar @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_7661484744434814213ropVar @ F
        @ ( collec8658135508005558727ropVar
          @ ^ [X: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_210_Compr__image__eq,axiom,
    ! [F: f > multis3193088007478089820ropVar,A: set_f,P: multis3193088007478089820ropVar > $o] :
      ( ( collec8658135508005558727ropVar
        @ ^ [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( image_4392090605272824862ropVar @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_4392090605272824862ropVar @ F
        @ ( collect_f
          @ ^ [X: f] :
              ( ( member_f @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_211_Compr__image__eq,axiom,
    ! [F: nat > multis3193088007478089820ropVar,A: set_nat,P: multis3193088007478089820ropVar > $o] :
      ( ( collec8658135508005558727ropVar
        @ ^ [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( image_3935346926305390661ropVar @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_3935346926305390661ropVar @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_212_Compr__image__eq,axiom,
    ! [F: multis3193088007478089820ropVar > f,A: set_Mu5658373669842198716ropVar,P: f > $o] :
      ( ( collect_f
        @ ^ [X: f] :
            ( ( member_f @ X @ ( image_7076402817453355436pVar_f @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_7076402817453355436pVar_f @ F
        @ ( collec8658135508005558727ropVar
          @ ^ [X: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_213_Compr__image__eq,axiom,
    ! [F: f > f,A: set_f,P: f > $o] :
      ( ( collect_f
        @ ^ [X: f] :
            ( ( member_f @ X @ ( image_f_f @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_f_f @ F
        @ ( collect_f
          @ ^ [X: f] :
              ( ( member_f @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_214_Compr__image__eq,axiom,
    ! [F: nat > f,A: set_nat,P: f > $o] :
      ( ( collect_f
        @ ^ [X: f] :
            ( ( member_f @ X @ ( image_nat_f @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_f @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_215_Compr__image__eq,axiom,
    ! [F: multis3193088007478089820ropVar > nat,A: set_Mu5658373669842198716ropVar,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_3439306068482956519ar_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_3439306068482956519ar_nat @ F
        @ ( collec8658135508005558727ropVar
          @ ^ [X: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_216_Compr__image__eq,axiom,
    ! [F: f > nat,A: set_f,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_f_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_f_nat @ F
        @ ( collect_f
          @ ^ [X: f] :
              ( ( member_f @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_217_Compr__image__eq,axiom,
    ! [F: nat > nat,A: set_nat,P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ ( image_nat_nat @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_218_Compr__image__eq,axiom,
    ! [F: f > list_f,A: set_f,P: list_f > $o] :
      ( ( collect_list_f
        @ ^ [X: list_f] :
            ( ( member_list_f @ X @ ( image_f_list_f @ F @ A ) )
            & ( P @ X ) ) )
      = ( image_f_list_f @ F
        @ ( collect_f
          @ ^ [X: f] :
              ( ( member_f @ X @ A )
              & ( P @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_219_mem__Collect__eq,axiom,
    ! [A2: multis3193088007478089820ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( member8421201611525043589ropVar @ A2 @ ( collec8658135508005558727ropVar @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_220_mem__Collect__eq,axiom,
    ! [A2: f,P: f > $o] :
      ( ( member_f @ A2 @ ( collect_f @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_221_mem__Collect__eq,axiom,
    ! [A2: nat,P: nat > $o] :
      ( ( member_nat @ A2 @ ( collect_nat @ P ) )
      = ( P @ A2 ) ) ).

% mem_Collect_eq
thf(fact_222_Collect__mem__eq,axiom,
    ! [A: set_Mu5658373669842198716ropVar] :
      ( ( collec8658135508005558727ropVar
        @ ^ [X: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_223_Collect__mem__eq,axiom,
    ! [A: set_f] :
      ( ( collect_f
        @ ^ [X: f] : ( member_f @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_224_Collect__mem__eq,axiom,
    ! [A: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_225_image__image,axiom,
    ! [F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,G: f > multis3193088007478089820ropVar,A: set_f] :
      ( ( image_7661484744434814213ropVar @ F @ ( image_4392090605272824862ropVar @ G @ A ) )
      = ( image_4392090605272824862ropVar
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_226_image__image,axiom,
    ! [F: multis3193088007478089820ropVar > nat,G: f > multis3193088007478089820ropVar,A: set_f] :
      ( ( image_3439306068482956519ar_nat @ F @ ( image_4392090605272824862ropVar @ G @ A ) )
      = ( image_f_nat
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_227_image__image,axiom,
    ! [F: multis3193088007478089820ropVar > f,G: f > multis3193088007478089820ropVar,A: set_f] :
      ( ( image_7076402817453355436pVar_f @ F @ ( image_4392090605272824862ropVar @ G @ A ) )
      = ( image_f_f
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_228_image__image,axiom,
    ! [F: nat > multis3193088007478089820ropVar,G: f > nat,A: set_f] :
      ( ( image_3935346926305390661ropVar @ F @ ( image_f_nat @ G @ A ) )
      = ( image_4392090605272824862ropVar
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_229_image__image,axiom,
    ! [F: nat > f,G: f > nat,A: set_f] :
      ( ( image_nat_f @ F @ ( image_f_nat @ G @ A ) )
      = ( image_f_f
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_230_image__image,axiom,
    ! [F: nat > nat,G: nat > nat,A: set_nat] :
      ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A ) )
      = ( image_nat_nat
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_231_image__image,axiom,
    ! [F: nat > nat,G: f > nat,A: set_f] :
      ( ( image_nat_nat @ F @ ( image_f_nat @ G @ A ) )
      = ( image_f_nat
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_232_image__image,axiom,
    ! [F: f > multis3193088007478089820ropVar,G: f > f,A: set_f] :
      ( ( image_4392090605272824862ropVar @ F @ ( image_f_f @ G @ A ) )
      = ( image_4392090605272824862ropVar
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_233_image__image,axiom,
    ! [F: f > nat,G: nat > f,A: set_nat] :
      ( ( image_f_nat @ F @ ( image_nat_f @ G @ A ) )
      = ( image_nat_nat
        @ ^ [X: nat] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_234_image__image,axiom,
    ! [F: f > nat,G: f > f,A: set_f] :
      ( ( image_f_nat @ F @ ( image_f_f @ G @ A ) )
      = ( image_f_nat
        @ ^ [X: f] : ( F @ ( G @ X ) )
        @ A ) ) ).

% image_image
thf(fact_235_imageE,axiom,
    ! [B: multis3193088007478089820ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ B @ ( image_7661484744434814213ropVar @ F @ A ) )
     => ~ ! [X3: multis3193088007478089820ropVar] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member8421201611525043589ropVar @ X3 @ A ) ) ) ).

% imageE
thf(fact_236_imageE,axiom,
    ! [B: multis3193088007478089820ropVar,F: f > multis3193088007478089820ropVar,A: set_f] :
      ( ( member8421201611525043589ropVar @ B @ ( image_4392090605272824862ropVar @ F @ A ) )
     => ~ ! [X3: f] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_f @ X3 @ A ) ) ) ).

% imageE
thf(fact_237_imageE,axiom,
    ! [B: multis3193088007478089820ropVar,F: nat > multis3193088007478089820ropVar,A: set_nat] :
      ( ( member8421201611525043589ropVar @ B @ ( image_3935346926305390661ropVar @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_238_imageE,axiom,
    ! [B: f,F: multis3193088007478089820ropVar > f,A: set_Mu5658373669842198716ropVar] :
      ( ( member_f @ B @ ( image_7076402817453355436pVar_f @ F @ A ) )
     => ~ ! [X3: multis3193088007478089820ropVar] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member8421201611525043589ropVar @ X3 @ A ) ) ) ).

% imageE
thf(fact_239_imageE,axiom,
    ! [B: f,F: f > f,A: set_f] :
      ( ( member_f @ B @ ( image_f_f @ F @ A ) )
     => ~ ! [X3: f] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_f @ X3 @ A ) ) ) ).

% imageE
thf(fact_240_imageE,axiom,
    ! [B: f,F: nat > f,A: set_nat] :
      ( ( member_f @ B @ ( image_nat_f @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_241_imageE,axiom,
    ! [B: nat,F: multis3193088007478089820ropVar > nat,A: set_Mu5658373669842198716ropVar] :
      ( ( member_nat @ B @ ( image_3439306068482956519ar_nat @ F @ A ) )
     => ~ ! [X3: multis3193088007478089820ropVar] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member8421201611525043589ropVar @ X3 @ A ) ) ) ).

% imageE
thf(fact_242_imageE,axiom,
    ! [B: nat,F: f > nat,A: set_f] :
      ( ( member_nat @ B @ ( image_f_nat @ F @ A ) )
     => ~ ! [X3: f] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_f @ X3 @ A ) ) ) ).

% imageE
thf(fact_243_imageE,axiom,
    ! [B: nat,F: nat > nat,A: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ F @ A ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A ) ) ) ).

% imageE
thf(fact_244_imageE,axiom,
    ! [B: list_f,F: f > list_f,A: set_f] :
      ( ( member_list_f @ B @ ( image_f_list_f @ F @ A ) )
     => ~ ! [X3: f] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_f @ X3 @ A ) ) ) ).

% imageE
thf(fact_245_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X: nat] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_246_list_Omap__ident,axiom,
    ! [T: list_f] :
      ( ( map_f_f
        @ ^ [X: f] : X
        @ T )
      = T ) ).

% list.map_ident
thf(fact_247_fst__conv,axiom,
    ! [X1: multis3193088007478089820ropVar,X22: multis3193088007478089820ropVar] :
      ( ( produc3432889363288992387ropVar @ ( produc6936306168532358295ropVar @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_248_fst__conv,axiom,
    ! [X1: nat > nat,X22: nat] :
      ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_249_fst__conv,axiom,
    ! [X1: nat,X22: nat] :
      ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_250_fst__conv,axiom,
    ! [X1: multis3193088007478089820ropVar,X22: produc7679832412844898215ropVar] :
      ( ( produc6918815732823278798ropVar @ ( produc6194868611276206818ropVar @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_251_fst__eqD,axiom,
    ! [X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,A2: multis3193088007478089820ropVar] :
      ( ( ( produc3432889363288992387ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_252_fst__eqD,axiom,
    ! [X2: nat > nat,Y3: nat,A2: nat > nat] :
      ( ( ( produc6156676138143019412at_nat @ ( produc72220940542539688at_nat @ X2 @ Y3 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_253_fst__eqD,axiom,
    ! [X2: nat,Y3: nat,A2: nat] :
      ( ( ( product_fst_nat_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_254_fst__eqD,axiom,
    ! [X2: multis3193088007478089820ropVar,Y3: produc7679832412844898215ropVar,A2: multis3193088007478089820ropVar] :
      ( ( ( produc6918815732823278798ropVar @ ( produc6194868611276206818ropVar @ X2 @ Y3 ) )
        = A2 )
     => ( X2 = A2 ) ) ).

% fst_eqD
thf(fact_255_prod__induct3,axiom,
    ! [P: produc2642885279816024434ropVar > $o,X2: produc2642885279816024434ropVar] :
      ( ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] : ( P @ ( produc6194868611276206818ropVar @ A4 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) )
     => ( P @ X2 ) ) ).

% prod_induct3
thf(fact_256_prod__cases3,axiom,
    ! [Y3: produc2642885279816024434ropVar] :
      ~ ! [A4: multis3193088007478089820ropVar,B3: multis3193088007478089820ropVar,C: multis3193088007478089820ropVar] :
          ( Y3
         != ( produc6194868611276206818ropVar @ A4 @ ( produc6936306168532358295ropVar @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_257_ex__map__conv,axiom,
    ! [Ys: list_f,F: f > f] :
      ( ( ? [Xs2: list_f] :
            ( Ys
            = ( map_f_f @ F @ Xs2 ) ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Ys ) )
           => ? [Y: f] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_258_ex__map__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_na2145702811007361372ropVar @ F @ Xs2 ) ) )
      = ( ! [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Ys ) )
           => ? [Y: nat] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_259_ex__map__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,F: f > produc2642885279816024434ropVar] :
      ( ( ? [Xs2: list_f] :
            ( Ys
            = ( map_f_2905954635146393971ropVar @ F @ Xs2 ) ) )
      = ( ! [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Ys ) )
           => ? [Y: f] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_260_ex__map__conv,axiom,
    ! [Ys: list_nat,F: nat > nat] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_nat_nat @ F @ Xs2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ? [Y: nat] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_261_ex__map__conv,axiom,
    ! [Ys: list_nat,F: f > nat] :
      ( ( ? [Xs2: list_f] :
            ( Ys
            = ( map_f_nat @ F @ Xs2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Ys ) )
           => ? [Y: f] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_262_ex__map__conv,axiom,
    ! [Ys: list_list_f,F: f > list_f] :
      ( ( ? [Xs2: list_f] :
            ( Ys
            = ( map_f_list_f @ F @ Xs2 ) ) )
      = ( ! [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Ys ) )
           => ? [Y: f] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_263_ex__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_na2466558649495876678ropVar @ F @ Xs2 ) ) )
      = ( ! [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Ys ) )
           => ? [Y: nat] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_264_ex__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,F: f > multis3193088007478089820ropVar] :
      ( ( ? [Xs2: list_f] :
            ( Ys
            = ( map_f_4134023513460405853ropVar @ F @ Xs2 ) ) )
      = ( ! [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Ys ) )
           => ? [Y: f] :
                ( X
                = ( F @ Y ) ) ) ) ) ).

% ex_map_conv
thf(fact_265_map__cong,axiom,
    ! [Xs: list_f,Ys: list_f,F: f > multis3193088007478089820ropVar,G: f > multis3193088007478089820ropVar] :
      ( ( Xs = Ys )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ ( set_f2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_f_4134023513460405853ropVar @ F @ Xs )
          = ( map_f_4134023513460405853ropVar @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_266_map__cong,axiom,
    ! [Xs: list_f,Ys: list_f,F: f > produc2642885279816024434ropVar,G: f > produc2642885279816024434ropVar] :
      ( ( Xs = Ys )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ ( set_f2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_f_2905954635146393971ropVar @ F @ Xs )
          = ( map_f_2905954635146393971ropVar @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_267_map__cong,axiom,
    ! [Xs: list_f,Ys: list_f,F: f > list_f,G: f > list_f] :
      ( ( Xs = Ys )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ ( set_f2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_f_list_f @ F @ Xs )
          = ( map_f_list_f @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_268_map__cong,axiom,
    ! [Xs: list_f,Ys: list_f,F: f > nat,G: f > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ ( set_f2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_f_nat @ F @ Xs )
          = ( map_f_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_269_map__cong,axiom,
    ! [Xs: list_f,Ys: list_f,F: f > f,G: f > f] :
      ( ( Xs = Ys )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ ( set_f2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_f_f @ F @ Xs )
          = ( map_f_f @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_270_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_na2145702811007361372ropVar @ F @ Xs )
          = ( map_na2145702811007361372ropVar @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_271_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_nat @ F @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_272_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ Xs )
          = ( map_na2466558649495876678ropVar @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_273_map__idI,axiom,
    ! [Xs: list_f,F: f > f] :
      ( ! [X3: f] :
          ( ( member_f @ X3 @ ( set_f2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_f_f @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_274_map__idI,axiom,
    ! [Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > produc2642885279816024434ropVar] :
      ( ! [X3: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X3 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr3565048146863404228ropVar @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_275_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_276_map__idI,axiom,
    ! [Xs: list_list_f,F: list_f > list_f] :
      ( ! [X3: list_f] :
          ( ( member_list_f @ X3 @ ( set_list_f2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_list_f_list_f @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_277_map__idI,axiom,
    ! [Xs: list_M6740372356978808802ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ! [X3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X3 @ ( set_Mu146256361055586289ropVar @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Mu2265489581466636612ropVar @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_278_map__ext,axiom,
    ! [Xs: list_f,F: f > multis3193088007478089820ropVar,G: f > multis3193088007478089820ropVar] :
      ( ! [X3: f] :
          ( ( member_f @ X3 @ ( set_f2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_f_4134023513460405853ropVar @ F @ Xs )
        = ( map_f_4134023513460405853ropVar @ G @ Xs ) ) ) ).

% map_ext
thf(fact_279_map__ext,axiom,
    ! [Xs: list_f,F: f > produc2642885279816024434ropVar,G: f > produc2642885279816024434ropVar] :
      ( ! [X3: f] :
          ( ( member_f @ X3 @ ( set_f2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_f_2905954635146393971ropVar @ F @ Xs )
        = ( map_f_2905954635146393971ropVar @ G @ Xs ) ) ) ).

% map_ext
thf(fact_280_map__ext,axiom,
    ! [Xs: list_f,F: f > list_f,G: f > list_f] :
      ( ! [X3: f] :
          ( ( member_f @ X3 @ ( set_f2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_f_list_f @ F @ Xs )
        = ( map_f_list_f @ G @ Xs ) ) ) ).

% map_ext
thf(fact_281_map__ext,axiom,
    ! [Xs: list_f,F: f > nat,G: f > nat] :
      ( ! [X3: f] :
          ( ( member_f @ X3 @ ( set_f2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_f_nat @ F @ Xs )
        = ( map_f_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_282_map__ext,axiom,
    ! [Xs: list_f,F: f > f,G: f > f] :
      ( ! [X3: f] :
          ( ( member_f @ X3 @ ( set_f2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_f_f @ F @ Xs )
        = ( map_f_f @ G @ Xs ) ) ) ).

% map_ext
thf(fact_283_map__ext,axiom,
    ! [Xs: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( map_na2145702811007361372ropVar @ G @ Xs ) ) ) ).

% map_ext
thf(fact_284_map__ext,axiom,
    ! [Xs: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_285_map__ext,axiom,
    ! [Xs: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( map_na2466558649495876678ropVar @ G @ Xs ) ) ) ).

% map_ext
thf(fact_286_list_Omap__ident__strong,axiom,
    ! [T: list_f,F: f > f] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_f_f @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_287_list_Omap__ident__strong,axiom,
    ! [T: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > produc2642885279816024434ropVar] :
      ( ! [Z3: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ Z3 @ ( set_Pr3946994337432900359ropVar @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr3565048146863404228ropVar @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_288_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_nat_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_289_list_Omap__ident__strong,axiom,
    ! [T: list_list_f,F: list_f > list_f] :
      ( ! [Z3: list_f] :
          ( ( member_list_f @ Z3 @ ( set_list_f2 @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_list_f_list_f @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_290_list_Omap__ident__strong,axiom,
    ! [T: list_M6740372356978808802ropVar,F: multis3193088007478089820ropVar > multis3193088007478089820ropVar] :
      ( ! [Z3: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ Z3 @ ( set_Mu146256361055586289ropVar @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Mu2265489581466636612ropVar @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_291_list_Oinj__map__strong,axiom,
    ! [X2: list_f,Xa: list_f,F: f > multis3193088007478089820ropVar,Fa: f > multis3193088007478089820ropVar] :
      ( ! [Z3: f,Za: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( member_f @ Za @ ( set_f2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_f_4134023513460405853ropVar @ F @ X2 )
          = ( map_f_4134023513460405853ropVar @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_292_list_Oinj__map__strong,axiom,
    ! [X2: list_f,Xa: list_f,F: f > produc2642885279816024434ropVar,Fa: f > produc2642885279816024434ropVar] :
      ( ! [Z3: f,Za: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( member_f @ Za @ ( set_f2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_f_2905954635146393971ropVar @ F @ X2 )
          = ( map_f_2905954635146393971ropVar @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_293_list_Oinj__map__strong,axiom,
    ! [X2: list_f,Xa: list_f,F: f > list_f,Fa: f > list_f] :
      ( ! [Z3: f,Za: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( member_f @ Za @ ( set_f2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_f_list_f @ F @ X2 )
          = ( map_f_list_f @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_294_list_Oinj__map__strong,axiom,
    ! [X2: list_f,Xa: list_f,F: f > nat,Fa: f > nat] :
      ( ! [Z3: f,Za: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( member_f @ Za @ ( set_f2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_f_nat @ F @ X2 )
          = ( map_f_nat @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_295_list_Oinj__map__strong,axiom,
    ! [X2: list_f,Xa: list_f,F: f > f,Fa: f > f] :
      ( ! [Z3: f,Za: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( member_f @ Za @ ( set_f2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_f_f @ F @ X2 )
          = ( map_f_f @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_296_list_Oinj__map__strong,axiom,
    ! [X2: list_nat,Xa: list_nat,F: nat > produc2642885279816024434ropVar,Fa: nat > produc2642885279816024434ropVar] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_na2145702811007361372ropVar @ F @ X2 )
          = ( map_na2145702811007361372ropVar @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_297_list_Oinj__map__strong,axiom,
    ! [X2: list_nat,Xa: list_nat,F: nat > nat,Fa: nat > nat] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F @ X2 )
          = ( map_nat_nat @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_298_list_Oinj__map__strong,axiom,
    ! [X2: list_nat,Xa: list_nat,F: nat > multis3193088007478089820ropVar,Fa: nat > multis3193088007478089820ropVar] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_na2466558649495876678ropVar @ F @ X2 )
          = ( map_na2466558649495876678ropVar @ Fa @ Xa ) )
       => ( X2 = Xa ) ) ) ).

% list.inj_map_strong
thf(fact_299_list_Omap__cong0,axiom,
    ! [X2: list_f,F: f > multis3193088007478089820ropVar,G: f > multis3193088007478089820ropVar] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_f_4134023513460405853ropVar @ F @ X2 )
        = ( map_f_4134023513460405853ropVar @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_300_list_Omap__cong0,axiom,
    ! [X2: list_f,F: f > produc2642885279816024434ropVar,G: f > produc2642885279816024434ropVar] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_f_2905954635146393971ropVar @ F @ X2 )
        = ( map_f_2905954635146393971ropVar @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_301_list_Omap__cong0,axiom,
    ! [X2: list_f,F: f > list_f,G: f > list_f] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_f_list_f @ F @ X2 )
        = ( map_f_list_f @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_302_list_Omap__cong0,axiom,
    ! [X2: list_f,F: f > nat,G: f > nat] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_f_nat @ F @ X2 )
        = ( map_f_nat @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_303_list_Omap__cong0,axiom,
    ! [X2: list_f,F: f > f,G: f > f] :
      ( ! [Z3: f] :
          ( ( member_f @ Z3 @ ( set_f2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_f_f @ F @ X2 )
        = ( map_f_f @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_304_list_Omap__cong0,axiom,
    ! [X2: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_na2145702811007361372ropVar @ F @ X2 )
        = ( map_na2145702811007361372ropVar @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_305_list_Omap__cong0,axiom,
    ! [X2: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_nat_nat @ F @ X2 )
        = ( map_nat_nat @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_306_list_Omap__cong0,axiom,
    ! [X2: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X2 ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_na2466558649495876678ropVar @ F @ X2 )
        = ( map_na2466558649495876678ropVar @ G @ X2 ) ) ) ).

% list.map_cong0
thf(fact_307_list_Omap__cong,axiom,
    ! [X2: list_f,Ya: list_f,F: f > multis3193088007478089820ropVar,G: f > multis3193088007478089820ropVar] :
      ( ( X2 = Ya )
     => ( ! [Z3: f] :
            ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_f_4134023513460405853ropVar @ F @ X2 )
          = ( map_f_4134023513460405853ropVar @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_308_list_Omap__cong,axiom,
    ! [X2: list_f,Ya: list_f,F: f > produc2642885279816024434ropVar,G: f > produc2642885279816024434ropVar] :
      ( ( X2 = Ya )
     => ( ! [Z3: f] :
            ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_f_2905954635146393971ropVar @ F @ X2 )
          = ( map_f_2905954635146393971ropVar @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_309_list_Omap__cong,axiom,
    ! [X2: list_f,Ya: list_f,F: f > list_f,G: f > list_f] :
      ( ( X2 = Ya )
     => ( ! [Z3: f] :
            ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_f_list_f @ F @ X2 )
          = ( map_f_list_f @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_310_list_Omap__cong,axiom,
    ! [X2: list_f,Ya: list_f,F: f > nat,G: f > nat] :
      ( ( X2 = Ya )
     => ( ! [Z3: f] :
            ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_f_nat @ F @ X2 )
          = ( map_f_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_311_list_Omap__cong,axiom,
    ! [X2: list_f,Ya: list_f,F: f > f,G: f > f] :
      ( ( X2 = Ya )
     => ( ! [Z3: f] :
            ( ( member_f @ Z3 @ ( set_f2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_f_f @ F @ X2 )
          = ( map_f_f @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_312_list_Omap__cong,axiom,
    ! [X2: list_nat,Ya: list_nat,F: nat > produc2642885279816024434ropVar,G: nat > produc2642885279816024434ropVar] :
      ( ( X2 = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_na2145702811007361372ropVar @ F @ X2 )
          = ( map_na2145702811007361372ropVar @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_313_list_Omap__cong,axiom,
    ! [X2: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
      ( ( X2 = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_nat_nat @ F @ X2 )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_314_list_Omap__cong,axiom,
    ! [X2: list_nat,Ya: list_nat,F: nat > multis3193088007478089820ropVar,G: nat > multis3193088007478089820ropVar] :
      ( ( X2 = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_na2466558649495876678ropVar @ F @ X2 )
          = ( map_na2466558649495876678ropVar @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_315_eq__fst__iff,axiom,
    ! [A2: multis3193088007478089820ropVar,P2: produc7679832412844898215ropVar] :
      ( ( A2
        = ( produc3432889363288992387ropVar @ P2 ) )
      = ( ? [B4: multis3193088007478089820ropVar] :
            ( P2
            = ( produc6936306168532358295ropVar @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_316_eq__fst__iff,axiom,
    ! [A2: nat > nat,P2: produc8199716216217303280at_nat] :
      ( ( A2
        = ( produc6156676138143019412at_nat @ P2 ) )
      = ( ? [B4: nat] :
            ( P2
            = ( produc72220940542539688at_nat @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_317_eq__fst__iff,axiom,
    ! [A2: nat,P2: product_prod_nat_nat] :
      ( ( A2
        = ( product_fst_nat_nat @ P2 ) )
      = ( ? [B4: nat] :
            ( P2
            = ( product_Pair_nat_nat @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_318_eq__fst__iff,axiom,
    ! [A2: multis3193088007478089820ropVar,P2: produc2642885279816024434ropVar] :
      ( ( A2
        = ( produc6918815732823278798ropVar @ P2 ) )
      = ( ? [B4: produc7679832412844898215ropVar] :
            ( P2
            = ( produc6194868611276206818ropVar @ A2 @ B4 ) ) ) ) ).

% eq_fst_iff
thf(fact_319_fstI,axiom,
    ! [X2: produc7679832412844898215ropVar,Y3: multis3193088007478089820ropVar,Z4: multis3193088007478089820ropVar] :
      ( ( X2
        = ( produc6936306168532358295ropVar @ Y3 @ Z4 ) )
     => ( ( produc3432889363288992387ropVar @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_320_fstI,axiom,
    ! [X2: produc8199716216217303280at_nat,Y3: nat > nat,Z4: nat] :
      ( ( X2
        = ( produc72220940542539688at_nat @ Y3 @ Z4 ) )
     => ( ( produc6156676138143019412at_nat @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_321_fstI,axiom,
    ! [X2: product_prod_nat_nat,Y3: nat,Z4: nat] :
      ( ( X2
        = ( product_Pair_nat_nat @ Y3 @ Z4 ) )
     => ( ( product_fst_nat_nat @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_322_fstI,axiom,
    ! [X2: produc2642885279816024434ropVar,Y3: multis3193088007478089820ropVar,Z4: produc7679832412844898215ropVar] :
      ( ( X2
        = ( produc6194868611276206818ropVar @ Y3 @ Z4 ) )
     => ( ( produc6918815732823278798ropVar @ X2 )
        = Y3 ) ) ).

% fstI
thf(fact_323_Id__on__set,axiom,
    ! [Xs: list_f] :
      ( ( id_on_f @ ( set_f2 @ Xs ) )
      = ( set_Product_prod_f_f2
        @ ( map_f_9172316846603814120od_f_f
          @ ^ [X: f] : ( product_Pair_f_f @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_324_Id__on__set,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( id_on_4441320969702803471ropVar @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( set_Pr1712790080554429628ropVar
        @ ( map_Pr2157319202780338553ropVar
          @ ^ [X: produc2642885279816024434ropVar] : ( produc6842008660352832535ropVar @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_325_Id__on__set,axiom,
    ! [Xs: list_list_f] :
      ( ( id_on_list_f @ ( set_list_f2 @ Xs ) )
      = ( set_Pr4569077758535692796list_f
        @ ( map_li2361663134577129826list_f
          @ ^ [X: list_f] : ( produc8144748701140958423list_f @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_326_Id__on__set,axiom,
    ! [Xs: list_M6740372356978808802ropVar] :
      ( ( id_on_8850091095513083129ropVar @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( set_Pr4075413451802982460ropVar
        @ ( map_Mu7320851716970821775ropVar
          @ ^ [X: multis3193088007478089820ropVar] : ( produc6936306168532358295ropVar @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_327_Id__on__set,axiom,
    ! [Xs: list_nat] :
      ( ( id_on_nat @ ( set_nat2 @ Xs ) )
      = ( set_Pr5648618587558075414at_nat
        @ ( map_na7298421622053143531at_nat
          @ ^ [X: nat] : ( product_Pair_nat_nat @ X @ X )
          @ Xs ) ) ) ).

% Id_on_set
thf(fact_328_eval__all__concat,axiom,
    ! [Fs: list_list_f,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( concat_f @ Fs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ! [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Fs ) )
           => ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ X ) )
               => ( eval @ Alpha @ Y ) ) ) ) ) ).

% eval_all_concat
thf(fact_329_PropVar_Osize__gen_I3_J,axiom,
    ! [X31: nat,X32: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis2983220944385521703ZeroJI @ X31 @ X32 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(3)
thf(fact_330_PropVar_Osize__gen_I4_J,axiom,
    ! [X41: nat,X42: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis6646701651571564453xOneJI @ X41 @ X42 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(4)
thf(fact_331_PropVar_Osize__gen_I1_J,axiom,
    ! [X11: nat,X12: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(1)
thf(fact_332_eval__all__append,axiom,
    ! [Fs: list_f,Gs: list_f,Alpha: multis3193088007478089820ropVar > $o] :
      ( ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ ( append_f @ Fs @ Gs ) ) )
           => ( eval @ Alpha @ X ) ) )
      = ( ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Fs ) )
           => ( eval @ Alpha @ X ) )
        & ! [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Gs ) )
           => ( eval @ Alpha @ X ) ) ) ) ).

% eval_all_append
thf(fact_333_not__in__set__insert,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ~ ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
     => ( ( insert_list_f @ X2 @ Xs )
        = ( cons_list_f @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_334_not__in__set__insert,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ( ( insert215289443796111736ropVar @ X2 @ Xs )
        = ( cons_M4277272395645347858ropVar @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_335_not__in__set__insert,axiom,
    ! [X2: f,Xs: list_f] :
      ( ~ ( member_f @ X2 @ ( set_f2 @ Xs ) )
     => ( ( insert_f @ X2 @ Xs )
        = ( cons_f @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_336_not__in__set__insert,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ( ( insert214406736284816014ropVar @ X2 @ Xs )
        = ( cons_P1504208554336013608ropVar @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_337_not__in__set__insert,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X2 @ Xs )
        = ( cons_nat @ X2 @ Xs ) ) ) ).

% not_in_set_insert
thf(fact_338_map__eq__map__tailrec,axiom,
    map_na2145702811007361372ropVar = map_ta2661035124953310354ropVar ).

% map_eq_map_tailrec
thf(fact_339_map__eq__map__tailrec,axiom,
    map_nat_nat = map_tailrec_nat_nat ).

% map_eq_map_tailrec
thf(fact_340_map__eq__map__tailrec,axiom,
    map_na2466558649495876678ropVar = map_ta2313973780298087292ropVar ).

% map_eq_map_tailrec
thf(fact_341_map__eq__map__tailrec,axiom,
    map_f_4134023513460405853ropVar = map_ta1837181631030923431ropVar ).

% map_eq_map_tailrec
thf(fact_342_map__eq__map__tailrec,axiom,
    map_f_2905954635146393971ropVar = map_ta4245493162775648189ropVar ).

% map_eq_map_tailrec
thf(fact_343_map__eq__map__tailrec,axiom,
    map_f_list_f = map_tailrec_f_list_f ).

% map_eq_map_tailrec
thf(fact_344_map__eq__map__tailrec,axiom,
    map_f_nat = map_tailrec_f_nat ).

% map_eq_map_tailrec
thf(fact_345_map__eq__map__tailrec,axiom,
    map_f_f = map_tailrec_f_f ).

% map_eq_map_tailrec
thf(fact_346_append_Oassoc,axiom,
    ! [A2: list_f,B: list_f,C2: list_f] :
      ( ( append_f @ ( append_f @ A2 @ B ) @ C2 )
      = ( append_f @ A2 @ ( append_f @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_347_append_Oassoc,axiom,
    ! [A2: list_nat,B: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A2 @ B ) @ C2 )
      = ( append_nat @ A2 @ ( append_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_348_append__assoc,axiom,
    ! [Xs: list_f,Ys: list_f,Zs3: list_f] :
      ( ( append_f @ ( append_f @ Xs @ Ys ) @ Zs3 )
      = ( append_f @ Xs @ ( append_f @ Ys @ Zs3 ) ) ) ).

% append_assoc
thf(fact_349_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs3 )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs3 ) ) ) ).

% append_assoc
thf(fact_350_append__same__eq,axiom,
    ! [Ys: list_f,Xs: list_f,Zs3: list_f] :
      ( ( ( append_f @ Ys @ Xs )
        = ( append_f @ Zs3 @ Xs ) )
      = ( Ys = Zs3 ) ) ).

% append_same_eq
thf(fact_351_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs3 @ Xs ) )
      = ( Ys = Zs3 ) ) ).

% append_same_eq
thf(fact_352_same__append__eq,axiom,
    ! [Xs: list_f,Ys: list_f,Zs3: list_f] :
      ( ( ( append_f @ Xs @ Ys )
        = ( append_f @ Xs @ Zs3 ) )
      = ( Ys = Zs3 ) ) ).

% same_append_eq
thf(fact_353_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs3 ) )
      = ( Ys = Zs3 ) ) ).

% same_append_eq
thf(fact_354_map__append,axiom,
    ! [F: nat > f,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_f @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_f @ ( map_nat_f @ F @ Xs ) @ ( map_nat_f @ F @ Ys ) ) ) ).

% map_append
thf(fact_355_map__append,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Ys: list_nat] :
      ( ( map_na2145702811007361372ropVar @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append4352292717583037453ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ ( map_na2145702811007361372ropVar @ F @ Ys ) ) ) ).

% map_append
thf(fact_356_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_357_map__append,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append2684835082924936439ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ ( map_na2466558649495876678ropVar @ F @ Ys ) ) ) ).

% map_append
thf(fact_358_map__append,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f,Ys: list_f] :
      ( ( map_f_4134023513460405853ropVar @ F @ ( append_f @ Xs @ Ys ) )
      = ( append2684835082924936439ropVar @ ( map_f_4134023513460405853ropVar @ F @ Xs ) @ ( map_f_4134023513460405853ropVar @ F @ Ys ) ) ) ).

% map_append
thf(fact_359_map__append,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f,Ys: list_f] :
      ( ( map_f_2905954635146393971ropVar @ F @ ( append_f @ Xs @ Ys ) )
      = ( append4352292717583037453ropVar @ ( map_f_2905954635146393971ropVar @ F @ Xs ) @ ( map_f_2905954635146393971ropVar @ F @ Ys ) ) ) ).

% map_append
thf(fact_360_map__append,axiom,
    ! [F: f > list_f,Xs: list_f,Ys: list_f] :
      ( ( map_f_list_f @ F @ ( append_f @ Xs @ Ys ) )
      = ( append_list_f @ ( map_f_list_f @ F @ Xs ) @ ( map_f_list_f @ F @ Ys ) ) ) ).

% map_append
thf(fact_361_map__append,axiom,
    ! [F: f > nat,Xs: list_f,Ys: list_f] :
      ( ( map_f_nat @ F @ ( append_f @ Xs @ Ys ) )
      = ( append_nat @ ( map_f_nat @ F @ Xs ) @ ( map_f_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_362_map__append,axiom,
    ! [F: f > f,Xs: list_f,Ys: list_f] :
      ( ( map_f_f @ F @ ( append_f @ Xs @ Ys ) )
      = ( append_f @ ( map_f_f @ F @ Xs ) @ ( map_f_f @ F @ Ys ) ) ) ).

% map_append
thf(fact_363_concat__append,axiom,
    ! [Xs: list_list_nat,Ys: list_list_nat] :
      ( ( concat_nat @ ( append_list_nat @ Xs @ Ys ) )
      = ( append_nat @ ( concat_nat @ Xs ) @ ( concat_nat @ Ys ) ) ) ).

% concat_append
thf(fact_364_concat__append,axiom,
    ! [Xs: list_list_f,Ys: list_list_f] :
      ( ( concat_f @ ( append_list_f @ Xs @ Ys ) )
      = ( append_f @ ( concat_f @ Xs ) @ ( concat_f @ Ys ) ) ) ).

% concat_append
thf(fact_365_in__set__insert,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( member_f @ X2 @ ( set_f2 @ Xs ) )
     => ( ( insert_f @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_366_in__set__insert,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ( ( insert214406736284816014ropVar @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_367_in__set__insert,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ( insert_nat @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_368_in__set__insert,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
     => ( ( insert_list_f @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_369_in__set__insert,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ( ( insert215289443796111736ropVar @ X2 @ Xs )
        = Xs ) ) ).

% in_set_insert
thf(fact_370_j,axiom,
    ord_less_nat @ j @ m ).

% j
thf(fact_371_concat_Osimps_I2_J,axiom,
    ! [X2: list_nat,Xs: list_list_nat] :
      ( ( concat_nat @ ( cons_list_nat @ X2 @ Xs ) )
      = ( append_nat @ X2 @ ( concat_nat @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_372_concat_Osimps_I2_J,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( concat_f @ ( cons_list_f @ X2 @ Xs ) )
      = ( append_f @ X2 @ ( concat_f @ Xs ) ) ) ).

% concat.simps(2)
thf(fact_373_append__eq__appendI,axiom,
    ! [Xs: list_f,Xs1: list_f,Zs3: list_f,Ys: list_f,Us: list_f] :
      ( ( ( append_f @ Xs @ Xs1 )
        = Zs3 )
     => ( ( Ys
          = ( append_f @ Xs1 @ Us ) )
       => ( ( append_f @ Xs @ Ys )
          = ( append_f @ Zs3 @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_374_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs3: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs3 )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs3 @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_375_append__eq__append__conv2,axiom,
    ! [Xs: list_f,Ys: list_f,Zs3: list_f,Ts: list_f] :
      ( ( ( append_f @ Xs @ Ys )
        = ( append_f @ Zs3 @ Ts ) )
      = ( ? [Us2: list_f] :
            ( ( ( Xs
                = ( append_f @ Zs3 @ Us2 ) )
              & ( ( append_f @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_f @ Xs @ Us2 )
                = Zs3 )
              & ( Ys
                = ( append_f @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_376_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs3: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs3 @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs3 @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs3 )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_377_Cons__eq__appendI,axiom,
    ! [X2: f,Xs1: list_f,Ys: list_f,Xs: list_f,Zs3: list_f] :
      ( ( ( cons_f @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_f @ Xs1 @ Zs3 ) )
       => ( ( cons_f @ X2 @ Xs )
          = ( append_f @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_378_Cons__eq__appendI,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs1: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append4352292717583037453ropVar @ Xs1 @ Zs3 ) )
       => ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
          = ( append4352292717583037453ropVar @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_379_Cons__eq__appendI,axiom,
    ! [X2: nat,Xs1: list_nat,Ys: list_nat,Xs: list_nat,Zs3: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append_nat @ Xs1 @ Zs3 ) )
       => ( ( cons_nat @ X2 @ Xs )
          = ( append_nat @ Ys @ Zs3 ) ) ) ) ).

% Cons_eq_appendI
thf(fact_380_append__Cons,axiom,
    ! [X2: f,Xs: list_f,Ys: list_f] :
      ( ( append_f @ ( cons_f @ X2 @ Xs ) @ Ys )
      = ( cons_f @ X2 @ ( append_f @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_381_append__Cons,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ Ys )
      = ( cons_P1504208554336013608ropVar @ X2 @ ( append4352292717583037453ropVar @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_382_append__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( append_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( cons_nat @ X2 @ ( append_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_383_append__eq__map__conv,axiom,
    ! [Ys: list_f,Zs3: list_f,F: nat > f,Xs: list_nat] :
      ( ( ( append_f @ Ys @ Zs3 )
        = ( map_nat_f @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_f @ F @ Us2 ) )
            & ( Zs3
              = ( map_nat_f @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_384_append__eq__map__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar,F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( ( append4352292717583037453ropVar @ Ys @ Zs3 )
        = ( map_na2145702811007361372ropVar @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_na2145702811007361372ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_na2145702811007361372ropVar @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_385_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs3 )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs3
              = ( map_nat_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_386_append__eq__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar,F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( ( append2684835082924936439ropVar @ Ys @ Zs3 )
        = ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_na2466558649495876678ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_na2466558649495876678ropVar @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_387_append__eq__map__conv,axiom,
    ! [Ys: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar,F: f > multis3193088007478089820ropVar,Xs: list_f] :
      ( ( ( append2684835082924936439ropVar @ Ys @ Zs3 )
        = ( map_f_4134023513460405853ropVar @ F @ Xs ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_4134023513460405853ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_4134023513460405853ropVar @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_388_append__eq__map__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar,F: f > produc2642885279816024434ropVar,Xs: list_f] :
      ( ( ( append4352292717583037453ropVar @ Ys @ Zs3 )
        = ( map_f_2905954635146393971ropVar @ F @ Xs ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_2905954635146393971ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_2905954635146393971ropVar @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_389_append__eq__map__conv,axiom,
    ! [Ys: list_list_f,Zs3: list_list_f,F: f > list_f,Xs: list_f] :
      ( ( ( append_list_f @ Ys @ Zs3 )
        = ( map_f_list_f @ F @ Xs ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_list_f @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_list_f @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_390_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,F: f > nat,Xs: list_f] :
      ( ( ( append_nat @ Ys @ Zs3 )
        = ( map_f_nat @ F @ Xs ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_nat @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_nat @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_391_append__eq__map__conv,axiom,
    ! [Ys: list_f,Zs3: list_f,F: f > f,Xs: list_f] :
      ( ( ( append_f @ Ys @ Zs3 )
        = ( map_f_f @ F @ Xs ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_f @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_f @ F @ Vs ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_392_map__eq__append__conv,axiom,
    ! [F: nat > f,Xs: list_nat,Ys: list_f,Zs3: list_f] :
      ( ( ( map_nat_f @ F @ Xs )
        = ( append_f @ Ys @ Zs3 ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_f @ F @ Us2 ) )
            & ( Zs3
              = ( map_nat_f @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_393_map__eq__append__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,Ys: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = ( append4352292717583037453ropVar @ Ys @ Zs3 ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_na2145702811007361372ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_na2145702811007361372ropVar @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_394_map__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs3
              = ( map_nat_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_395_map__eq__append__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,Ys: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = ( append2684835082924936439ropVar @ Ys @ Zs3 ) )
      = ( ? [Us2: list_nat,Vs: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs ) )
            & ( Ys
              = ( map_na2466558649495876678ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_na2466558649495876678ropVar @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_396_map__eq__append__conv,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f,Ys: list_M6740372356978808802ropVar,Zs3: list_M6740372356978808802ropVar] :
      ( ( ( map_f_4134023513460405853ropVar @ F @ Xs )
        = ( append2684835082924936439ropVar @ Ys @ Zs3 ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_4134023513460405853ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_4134023513460405853ropVar @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_397_map__eq__append__conv,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f,Ys: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
      ( ( ( map_f_2905954635146393971ropVar @ F @ Xs )
        = ( append4352292717583037453ropVar @ Ys @ Zs3 ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_2905954635146393971ropVar @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_2905954635146393971ropVar @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_398_map__eq__append__conv,axiom,
    ! [F: f > list_f,Xs: list_f,Ys: list_list_f,Zs3: list_list_f] :
      ( ( ( map_f_list_f @ F @ Xs )
        = ( append_list_f @ Ys @ Zs3 ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_list_f @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_list_f @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_399_map__eq__append__conv,axiom,
    ! [F: f > nat,Xs: list_f,Ys: list_nat,Zs3: list_nat] :
      ( ( ( map_f_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_nat @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_nat @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_400_map__eq__append__conv,axiom,
    ! [F: f > f,Xs: list_f,Ys: list_f,Zs3: list_f] :
      ( ( ( map_f_f @ F @ Xs )
        = ( append_f @ Ys @ Zs3 ) )
      = ( ? [Us2: list_f,Vs: list_f] :
            ( ( Xs
              = ( append_f @ Us2 @ Vs ) )
            & ( Ys
              = ( map_f_f @ F @ Us2 ) )
            & ( Zs3
              = ( map_f_f @ F @ Vs ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_401_map__concat,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_list_nat] :
      ( ( map_na2145702811007361372ropVar @ F @ ( concat_nat @ Xs ) )
      = ( concat1830633707478507187ropVar @ ( map_li4979107708160704082ropVar @ ( map_na2145702811007361372ropVar @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_402_map__concat,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( map_nat_nat @ F @ ( concat_nat @ Xs ) )
      = ( concat_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_403_map__concat,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_list_nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( concat_nat @ Xs ) )
      = ( concat2729506608927398813ropVar @ ( map_li5946103070670478908ropVar @ ( map_na2466558649495876678ropVar @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_404_map__concat,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_list_f] :
      ( ( map_f_4134023513460405853ropVar @ F @ ( concat_f @ Xs ) )
      = ( concat2729506608927398813ropVar @ ( map_li1265598969789384029ropVar @ ( map_f_4134023513460405853ropVar @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_405_map__concat,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_list_f] :
      ( ( map_f_2905954635146393971ropVar @ F @ ( concat_f @ Xs ) )
      = ( concat1830633707478507187ropVar @ ( map_li676030948805672819ropVar @ ( map_f_2905954635146393971ropVar @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_406_map__concat,axiom,
    ! [F: f > list_f,Xs: list_list_f] :
      ( ( map_f_list_f @ F @ ( concat_f @ Xs ) )
      = ( concat_list_f @ ( map_li8627645367811812490list_f @ ( map_f_list_f @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_407_map__concat,axiom,
    ! [F: f > nat,Xs: list_list_f] :
      ( ( map_f_nat @ F @ ( concat_f @ Xs ) )
      = ( concat_nat @ ( map_list_f_list_nat @ ( map_f_nat @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_408_map__concat,axiom,
    ! [F: f > f,Xs: list_list_f] :
      ( ( map_f_f @ F @ ( concat_f @ Xs ) )
      = ( concat_f @ ( map_list_f_list_f @ ( map_f_f @ F ) @ Xs ) ) ) ).

% map_concat
thf(fact_409_concat__map__maps,axiom,
    ! [F: multis3193088007478089820ropVar > list_f,Xs: list_M6740372356978808802ropVar] :
      ( ( concat_f @ ( map_Mu2633154768359244273list_f @ F @ Xs ) )
      = ( maps_M425433082187236072pVar_f @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_410_concat__map__maps,axiom,
    ! [F: list_f > list_f,Xs: list_list_f] :
      ( ( concat_f @ ( map_list_f_list_f @ F @ Xs ) )
      = ( maps_list_f_f @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_411_concat__map__maps,axiom,
    ! [F: nat > list_f,Xs: list_nat] :
      ( ( concat_f @ ( map_nat_list_f @ F @ Xs ) )
      = ( maps_nat_f @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_412_concat__map__maps,axiom,
    ! [F: produc2642885279816024434ropVar > list_f,Xs: list_P8791008543742759928ropVar] :
      ( ( concat_f @ ( map_Pr1732182123645423451list_f @ F @ Xs ) )
      = ( maps_P2527380396400262994pVar_f @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_413_concat__map__maps,axiom,
    ! [F: f > list_f,Xs: list_f] :
      ( ( concat_f @ ( map_f_list_f @ F @ Xs ) )
      = ( maps_f_f @ F @ Xs ) ) ).

% concat_map_maps
thf(fact_414_maps__def,axiom,
    ( maps_M425433082187236072pVar_f
    = ( ^ [F2: multis3193088007478089820ropVar > list_f,Xs2: list_M6740372356978808802ropVar] : ( concat_f @ ( map_Mu2633154768359244273list_f @ F2 @ Xs2 ) ) ) ) ).

% maps_def
thf(fact_415_maps__def,axiom,
    ( maps_list_f_f
    = ( ^ [F2: list_f > list_f,Xs2: list_list_f] : ( concat_f @ ( map_list_f_list_f @ F2 @ Xs2 ) ) ) ) ).

% maps_def
thf(fact_416_maps__def,axiom,
    ( maps_nat_f
    = ( ^ [F2: nat > list_f,Xs2: list_nat] : ( concat_f @ ( map_nat_list_f @ F2 @ Xs2 ) ) ) ) ).

% maps_def
thf(fact_417_maps__def,axiom,
    ( maps_P2527380396400262994pVar_f
    = ( ^ [F2: produc2642885279816024434ropVar > list_f,Xs2: list_P8791008543742759928ropVar] : ( concat_f @ ( map_Pr1732182123645423451list_f @ F2 @ Xs2 ) ) ) ) ).

% maps_def
thf(fact_418_maps__def,axiom,
    ( maps_f_f
    = ( ^ [F2: f > list_f,Xs2: list_f] : ( concat_f @ ( map_f_list_f @ F2 @ Xs2 ) ) ) ) ).

% maps_def
thf(fact_419_split__list__first__prop__iff,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ( ? [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_list_f,X: list_f] :
            ( ? [Zs: list_list_f] :
                ( Xs
                = ( append_list_f @ Ys2 @ ( cons_list_f @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: list_f] :
                ( ( member_list_f @ Y @ ( set_list_f2 @ Ys2 ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_420_split__list__first__prop__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,X: multis3193088007478089820ropVar] :
            ( ? [Zs: list_M6740372356978808802ropVar] :
                ( Xs
                = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ Ys2 ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_421_split__list__first__prop__iff,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ( ? [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_f,X: f] :
            ( ? [Zs: list_f] :
                ( Xs
                = ( append_f @ Ys2 @ ( cons_f @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ Ys2 ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_422_split__list__first__prop__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ? [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,X: produc2642885279816024434ropVar] :
            ( ? [Zs: list_P8791008543742759928ropVar] :
                ( Xs
                = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: produc2642885279816024434ropVar] :
                ( ( member9210975342120487067ropVar @ Y @ ( set_Pr3946994337432900359ropVar @ Ys2 ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_423_split__list__first__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_nat,X: nat] :
            ( ? [Zs: list_nat] :
                ( Xs
                = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: nat] :
                ( ( member_nat @ Y @ ( set_nat2 @ Ys2 ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_424_split__list__last__prop__iff,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ( ? [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_list_f,X: list_f,Zs: list_list_f] :
            ( ( Xs
              = ( append_list_f @ Ys2 @ ( cons_list_f @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: list_f] :
                ( ( member_list_f @ Y @ ( set_list_f2 @ Zs ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_425_split__list__last__prop__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ( ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,X: multis3193088007478089820ropVar,Zs: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ Zs ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_426_split__list__last__prop__iff,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ( ? [X: f] :
            ( ( member_f @ X @ ( set_f2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_f,X: f,Zs: list_f] :
            ( ( Xs
              = ( append_f @ Ys2 @ ( cons_f @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: f] :
                ( ( member_f @ Y @ ( set_f2 @ Zs ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_427_split__list__last__prop__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ( ? [X: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,X: produc2642885279816024434ropVar,Zs: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: produc2642885279816024434ropVar] :
                ( ( member9210975342120487067ropVar @ Y @ ( set_Pr3946994337432900359ropVar @ Zs ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_428_split__list__last__prop__iff,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
            & ( P @ X ) ) )
      = ( ? [Ys2: list_nat,X: nat,Zs: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X @ Zs ) ) )
            & ( P @ X )
            & ! [Y: nat] :
                ( ( member_nat @ Y @ ( set_nat2 @ Zs ) )
               => ~ ( P @ Y ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_429_in__set__conv__decomp__first,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
      = ( ? [Ys2: list_list_f,Zs: list_list_f] :
            ( ( Xs
              = ( append_list_f @ Ys2 @ ( cons_list_f @ X2 @ Zs ) ) )
            & ~ ( member_list_f @ X2 @ ( set_list_f2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_430_in__set__conv__decomp__first,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X2 @ Zs ) ) )
            & ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_431_in__set__conv__decomp__first,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( member_f @ X2 @ ( set_f2 @ Xs ) )
      = ( ? [Ys2: list_f,Zs: list_f] :
            ( ( Xs
              = ( append_f @ Ys2 @ ( cons_f @ X2 @ Zs ) ) )
            & ~ ( member_f @ X2 @ ( set_f2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_432_in__set__conv__decomp__first,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X2 @ Zs ) ) )
            & ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_433_in__set__conv__decomp__first,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
            & ~ ( member_nat @ X2 @ ( set_nat2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_434_in__set__conv__decomp__last,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
      = ( ? [Ys2: list_list_f,Zs: list_list_f] :
            ( ( Xs
              = ( append_list_f @ Ys2 @ ( cons_list_f @ X2 @ Zs ) ) )
            & ~ ( member_list_f @ X2 @ ( set_list_f2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_435_in__set__conv__decomp__last,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X2 @ Zs ) ) )
            & ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_436_in__set__conv__decomp__last,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( member_f @ X2 @ ( set_f2 @ Xs ) )
      = ( ? [Ys2: list_f,Zs: list_f] :
            ( ( Xs
              = ( append_f @ Ys2 @ ( cons_f @ X2 @ Zs ) ) )
            & ~ ( member_f @ X2 @ ( set_f2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_437_in__set__conv__decomp__last,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X2 @ Zs ) ) )
            & ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_438_in__set__conv__decomp__last,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) )
            & ~ ( member_nat @ X2 @ ( set_nat2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_439_split__list__first__propE,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ? [X4: list_f] :
          ( ( member_list_f @ X4 @ ( set_list_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_list_f,X3: list_f] :
            ( ? [Zs2: list_list_f] :
                ( Xs
                = ( append_list_f @ Ys3 @ ( cons_list_f @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_f] :
                    ( ( member_list_f @ Xa2 @ ( set_list_f2 @ Ys3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_440_split__list__first__propE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_M6740372356978808802ropVar,X3: multis3193088007478089820ropVar] :
            ( ? [Zs2: list_M6740372356978808802ropVar] :
                ( Xs
                = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Ys3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_441_split__list__first__propE,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ ( set_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_f,X3: f] :
            ( ? [Zs2: list_f] :
                ( Xs
                = ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: f] :
                    ( ( member_f @ Xa2 @ ( set_f2 @ Ys3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_442_split__list__first__propE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar] :
            ( ? [Zs2: list_P8791008543742759928ropVar] :
                ( Xs
                = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Ys3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_443_split__list__first__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_nat,X3: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs
                = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_first_propE
thf(fact_444_split__list__last__propE,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ? [X4: list_f] :
          ( ( member_list_f @ X4 @ ( set_list_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_list_f,X3: list_f,Zs2: list_list_f] :
            ( ( Xs
              = ( append_list_f @ Ys3 @ ( cons_list_f @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: list_f] :
                    ( ( member_list_f @ Xa2 @ ( set_list_f2 @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_445_split__list__last__propE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_M6740372356978808802ropVar,X3: multis3193088007478089820ropVar,Zs2: list_M6740372356978808802ropVar] :
            ( ( Xs
              = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: multis3193088007478089820ropVar] :
                    ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_446_split__list__last__propE,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ ( set_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_f,X3: f,Zs2: list_f] :
            ( ( Xs
              = ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: f] :
                    ( ( member_f @ Xa2 @ ( set_f2 @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_447_split__list__last__propE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar] :
            ( ( Xs
              = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: produc2642885279816024434ropVar] :
                    ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_448_split__list__last__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_nat,X3: nat,Zs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
           => ( ( P @ X3 )
             => ~ ! [Xa2: nat] :
                    ( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
                   => ~ ( P @ Xa2 ) ) ) ) ) ).

% split_list_last_propE
thf(fact_449_split__list__first__prop,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ? [X4: list_f] :
          ( ( member_list_f @ X4 @ ( set_list_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_list_f,X3: list_f] :
          ( ? [Zs2: list_list_f] :
              ( Xs
              = ( append_list_f @ Ys3 @ ( cons_list_f @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_f] :
              ( ( member_list_f @ Xa2 @ ( set_list_f2 @ Ys3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_450_split__list__first__prop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_M6740372356978808802ropVar,X3: multis3193088007478089820ropVar] :
          ( ? [Zs2: list_M6740372356978808802ropVar] :
              ( Xs
              = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Ys3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_451_split__list__first__prop,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ ( set_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_f,X3: f] :
          ( ? [Zs2: list_f] :
              ( Xs
              = ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: f] :
              ( ( member_f @ Xa2 @ ( set_f2 @ Ys3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_452_split__list__first__prop,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar] :
          ( ? [Zs2: list_P8791008543742759928ropVar] :
              ( Xs
              = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: produc2642885279816024434ropVar] :
              ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Ys3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_453_split__list__first__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_nat,X3: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( set_nat2 @ Ys3 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_first_prop
thf(fact_454_split__list__last__prop,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ? [X4: list_f] :
          ( ( member_list_f @ X4 @ ( set_list_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_list_f,X3: list_f,Zs2: list_list_f] :
          ( ( Xs
            = ( append_list_f @ Ys3 @ ( cons_list_f @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: list_f] :
              ( ( member_list_f @ Xa2 @ ( set_list_f2 @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_455_split__list__last__prop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_M6740372356978808802ropVar,X3: multis3193088007478089820ropVar,Zs2: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Xa2 @ ( set_Mu146256361055586289ropVar @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_456_split__list__last__prop,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ ( set_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_f,X3: f,Zs2: list_f] :
          ( ( Xs
            = ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: f] :
              ( ( member_f @ Xa2 @ ( set_f2 @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_457_split__list__last__prop,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: produc2642885279816024434ropVar] :
              ( ( member9210975342120487067ropVar @ Xa2 @ ( set_Pr3946994337432900359ropVar @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_458_split__list__last__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_nat,X3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ( P @ X3 )
          & ! [Xa2: nat] :
              ( ( member_nat @ Xa2 @ ( set_nat2 @ Zs2 ) )
             => ~ ( P @ Xa2 ) ) ) ) ).

% split_list_last_prop
thf(fact_459_in__set__conv__decomp,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
      = ( ? [Ys2: list_list_f,Zs: list_list_f] :
            ( Xs
            = ( append_list_f @ Ys2 @ ( cons_list_f @ X2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_460_in__set__conv__decomp,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
      = ( ? [Ys2: list_M6740372356978808802ropVar,Zs: list_M6740372356978808802ropVar] :
            ( Xs
            = ( append2684835082924936439ropVar @ Ys2 @ ( cons_M4277272395645347858ropVar @ X2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_461_in__set__conv__decomp,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( member_f @ X2 @ ( set_f2 @ Xs ) )
      = ( ? [Ys2: list_f,Zs: list_f] :
            ( Xs
            = ( append_f @ Ys2 @ ( cons_f @ X2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_462_in__set__conv__decomp,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
      = ( ? [Ys2: list_P8791008543742759928ropVar,Zs: list_P8791008543742759928ropVar] :
            ( Xs
            = ( append4352292717583037453ropVar @ Ys2 @ ( cons_P1504208554336013608ropVar @ X2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_463_in__set__conv__decomp,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
      = ( ? [Ys2: list_nat,Zs: list_nat] :
            ( Xs
            = ( append_nat @ Ys2 @ ( cons_nat @ X2 @ Zs ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_464_append__Cons__eq__iff,axiom,
    ! [X2: list_f,Xs: list_list_f,Ys: list_list_f,Xs3: list_list_f,Ys4: list_list_f] :
      ( ~ ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
     => ( ~ ( member_list_f @ X2 @ ( set_list_f2 @ Ys ) )
       => ( ( ( append_list_f @ Xs @ ( cons_list_f @ X2 @ Ys ) )
            = ( append_list_f @ Xs3 @ ( cons_list_f @ X2 @ Ys4 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_465_append__Cons__eq__iff,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,Xs3: list_M6740372356978808802ropVar,Ys4: list_M6740372356978808802ropVar] :
      ( ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ( ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Ys ) )
       => ( ( ( append2684835082924936439ropVar @ Xs @ ( cons_M4277272395645347858ropVar @ X2 @ Ys ) )
            = ( append2684835082924936439ropVar @ Xs3 @ ( cons_M4277272395645347858ropVar @ X2 @ Ys4 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_466_append__Cons__eq__iff,axiom,
    ! [X2: f,Xs: list_f,Ys: list_f,Xs3: list_f,Ys4: list_f] :
      ( ~ ( member_f @ X2 @ ( set_f2 @ Xs ) )
     => ( ~ ( member_f @ X2 @ ( set_f2 @ Ys ) )
       => ( ( ( append_f @ Xs @ ( cons_f @ X2 @ Ys ) )
            = ( append_f @ Xs3 @ ( cons_f @ X2 @ Ys4 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_467_append__Cons__eq__iff,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Xs3: list_P8791008543742759928ropVar,Ys4: list_P8791008543742759928ropVar] :
      ( ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ( ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Ys ) )
       => ( ( ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X2 @ Ys ) )
            = ( append4352292717583037453ropVar @ Xs3 @ ( cons_P1504208554336013608ropVar @ X2 @ Ys4 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_468_append__Cons__eq__iff,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,Xs3: list_nat,Ys4: list_nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ( ~ ( member_nat @ X2 @ ( set_nat2 @ Ys ) )
       => ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ Ys ) )
            = ( append_nat @ Xs3 @ ( cons_nat @ X2 @ Ys4 ) ) )
          = ( ( Xs = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_469_split__list__propE,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ? [X4: list_f] :
          ( ( member_list_f @ X4 @ ( set_list_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_list_f,X3: list_f] :
            ( ? [Zs2: list_list_f] :
                ( Xs
                = ( append_list_f @ Ys3 @ ( cons_list_f @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_470_split__list__propE,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_M6740372356978808802ropVar,X3: multis3193088007478089820ropVar] :
            ( ? [Zs2: list_M6740372356978808802ropVar] :
                ( Xs
                = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_471_split__list__propE,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ ( set_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_f,X3: f] :
            ( ? [Zs2: list_f] :
                ( Xs
                = ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_472_split__list__propE,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar] :
            ( ? [Zs2: list_P8791008543742759928ropVar] :
                ( Xs
                = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_473_split__list__propE,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
          & ( P @ X4 ) )
     => ~ ! [Ys3: list_nat,X3: nat] :
            ( ? [Zs2: list_nat] :
                ( Xs
                = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
           => ~ ( P @ X3 ) ) ) ).

% split_list_propE
thf(fact_474_split__list__first,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
     => ? [Ys3: list_list_f,Zs2: list_list_f] :
          ( ( Xs
            = ( append_list_f @ Ys3 @ ( cons_list_f @ X2 @ Zs2 ) ) )
          & ~ ( member_list_f @ X2 @ ( set_list_f2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_475_split__list__first,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ? [Ys3: list_M6740372356978808802ropVar,Zs2: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X2 @ Zs2 ) ) )
          & ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_476_split__list__first,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( member_f @ X2 @ ( set_f2 @ Xs ) )
     => ? [Ys3: list_f,Zs2: list_f] :
          ( ( Xs
            = ( append_f @ Ys3 @ ( cons_f @ X2 @ Zs2 ) ) )
          & ~ ( member_f @ X2 @ ( set_f2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_477_split__list__first,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ? [Ys3: list_P8791008543742759928ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
          & ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_478_split__list__first,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
          & ~ ( member_nat @ X2 @ ( set_nat2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_479_split__list__prop,axiom,
    ! [Xs: list_list_f,P: list_f > $o] :
      ( ? [X4: list_f] :
          ( ( member_list_f @ X4 @ ( set_list_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_list_f,X3: list_f] :
          ( ? [Zs2: list_list_f] :
              ( Xs
              = ( append_list_f @ Ys3 @ ( cons_list_f @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_480_split__list__prop,axiom,
    ! [Xs: list_M6740372356978808802ropVar,P: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ ( set_Mu146256361055586289ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_M6740372356978808802ropVar,X3: multis3193088007478089820ropVar] :
          ( ? [Zs2: list_M6740372356978808802ropVar] :
              ( Xs
              = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_481_split__list__prop,axiom,
    ! [Xs: list_f,P: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ ( set_f2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_f,X3: f] :
          ( ? [Zs2: list_f] :
              ( Xs
              = ( append_f @ Ys3 @ ( cons_f @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_482_split__list__prop,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: produc2642885279816024434ropVar > $o] :
      ( ? [X4: produc2642885279816024434ropVar] :
          ( ( member9210975342120487067ropVar @ X4 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_P8791008543742759928ropVar,X3: produc2642885279816024434ropVar] :
          ( ? [Zs2: list_P8791008543742759928ropVar] :
              ( Xs
              = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_483_split__list__prop,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
          & ( P @ X4 ) )
     => ? [Ys3: list_nat,X3: nat] :
          ( ? [Zs2: list_nat] :
              ( Xs
              = ( append_nat @ Ys3 @ ( cons_nat @ X3 @ Zs2 ) ) )
          & ( P @ X3 ) ) ) ).

% split_list_prop
thf(fact_484_split__list__last,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
     => ? [Ys3: list_list_f,Zs2: list_list_f] :
          ( ( Xs
            = ( append_list_f @ Ys3 @ ( cons_list_f @ X2 @ Zs2 ) ) )
          & ~ ( member_list_f @ X2 @ ( set_list_f2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_485_split__list__last,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ? [Ys3: list_M6740372356978808802ropVar,Zs2: list_M6740372356978808802ropVar] :
          ( ( Xs
            = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X2 @ Zs2 ) ) )
          & ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_486_split__list__last,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( member_f @ X2 @ ( set_f2 @ Xs ) )
     => ? [Ys3: list_f,Zs2: list_f] :
          ( ( Xs
            = ( append_f @ Ys3 @ ( cons_f @ X2 @ Zs2 ) ) )
          & ~ ( member_f @ X2 @ ( set_f2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_487_split__list__last,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ? [Ys3: list_P8791008543742759928ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( ( Xs
            = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) )
          & ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_488_split__list__last,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs2: list_nat] :
          ( ( Xs
            = ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) )
          & ~ ( member_nat @ X2 @ ( set_nat2 @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_489_split__list,axiom,
    ! [X2: list_f,Xs: list_list_f] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
     => ? [Ys3: list_list_f,Zs2: list_list_f] :
          ( Xs
          = ( append_list_f @ Ys3 @ ( cons_list_f @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_490_split__list,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ? [Ys3: list_M6740372356978808802ropVar,Zs2: list_M6740372356978808802ropVar] :
          ( Xs
          = ( append2684835082924936439ropVar @ Ys3 @ ( cons_M4277272395645347858ropVar @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_491_split__list,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( member_f @ X2 @ ( set_f2 @ Xs ) )
     => ? [Ys3: list_f,Zs2: list_f] :
          ( Xs
          = ( append_f @ Ys3 @ ( cons_f @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_492_split__list,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ? [Ys3: list_P8791008543742759928ropVar,Zs2: list_P8791008543742759928ropVar] :
          ( Xs
          = ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_493_split__list,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ? [Ys3: list_nat,Zs2: list_nat] :
          ( Xs
          = ( append_nat @ Ys3 @ ( cons_nat @ X2 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_494_maps__simps_I1_J,axiom,
    ! [F: multis3193088007478089820ropVar > list_f,X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( maps_M425433082187236072pVar_f @ F @ ( cons_M4277272395645347858ropVar @ X2 @ Xs ) )
      = ( append_f @ ( F @ X2 ) @ ( maps_M425433082187236072pVar_f @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_495_maps__simps_I1_J,axiom,
    ! [F: list_f > list_f,X2: list_f,Xs: list_list_f] :
      ( ( maps_list_f_f @ F @ ( cons_list_f @ X2 @ Xs ) )
      = ( append_f @ ( F @ X2 ) @ ( maps_list_f_f @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_496_maps__simps_I1_J,axiom,
    ! [F: f > list_f,X2: f,Xs: list_f] :
      ( ( maps_f_f @ F @ ( cons_f @ X2 @ Xs ) )
      = ( append_f @ ( F @ X2 ) @ ( maps_f_f @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_497_maps__simps_I1_J,axiom,
    ! [F: f > list_nat,X2: f,Xs: list_f] :
      ( ( maps_f_nat @ F @ ( cons_f @ X2 @ Xs ) )
      = ( append_nat @ ( F @ X2 ) @ ( maps_f_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_498_maps__simps_I1_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_f,X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( maps_P2527380396400262994pVar_f @ F @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
      = ( append_f @ ( F @ X2 ) @ ( maps_P2527380396400262994pVar_f @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_499_maps__simps_I1_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_nat,X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( maps_P2882123423910056385ar_nat @ F @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
      = ( append_nat @ ( F @ X2 ) @ ( maps_P2882123423910056385ar_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_500_maps__simps_I1_J,axiom,
    ! [F: nat > list_f,X2: nat,Xs: list_nat] :
      ( ( maps_nat_f @ F @ ( cons_nat @ X2 @ Xs ) )
      = ( append_f @ ( F @ X2 ) @ ( maps_nat_f @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_501_maps__simps_I1_J,axiom,
    ! [F: nat > list_nat,X2: nat,Xs: list_nat] :
      ( ( maps_nat_nat @ F @ ( cons_nat @ X2 @ Xs ) )
      = ( append_nat @ ( F @ X2 ) @ ( maps_nat_nat @ F @ Xs ) ) ) ).

% maps_simps(1)
thf(fact_502_Inf_OINF__cong,axiom,
    ! [A: set_Pr6405597471078531282ropVar,B5: set_Pr6405597471078531282ropVar,C3: produc2642885279816024434ropVar > multis3193088007478089820ropVar,D: produc2642885279816024434ropVar > multis3193088007478089820ropVar,Inf: set_Mu5658373669842198716ropVar > multis3193088007478089820ropVar] :
      ( ( A = B5 )
     => ( ! [X3: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_4513341045867859311ropVar @ C3 @ A ) )
          = ( Inf @ ( image_4513341045867859311ropVar @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_503_Inf_OINF__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > multis3193088007478089820ropVar,D: f > multis3193088007478089820ropVar,Inf: set_Mu5658373669842198716ropVar > multis3193088007478089820ropVar] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_4392090605272824862ropVar @ C3 @ A ) )
          = ( Inf @ ( image_4392090605272824862ropVar @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_504_Inf_OINF__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > produc2642885279816024434ropVar,D: f > produc2642885279816024434ropVar,Inf: set_Pr6405597471078531282ropVar > produc2642885279816024434ropVar] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_2710159631831406900ropVar @ C3 @ A ) )
          = ( Inf @ ( image_2710159631831406900ropVar @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_505_Inf_OINF__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > list_f,D: f > list_f,Inf: set_list_f > list_f] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_f_list_f @ C3 @ A ) )
          = ( Inf @ ( image_f_list_f @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_506_Inf_OINF__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > nat,D: f > nat,Inf: set_nat > nat] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_f_nat @ C3 @ A ) )
          = ( Inf @ ( image_f_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_507_Inf_OINF__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > f,D: f > f,Inf: set_f > f] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_f_f @ C3 @ A ) )
          = ( Inf @ ( image_f_f @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_508_Inf_OINF__cong,axiom,
    ! [A: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat,Inf: set_nat > nat] :
      ( ( A = B5 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_nat_nat @ C3 @ A ) )
          = ( Inf @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_509_Sup_OSUP__cong,axiom,
    ! [A: set_Pr6405597471078531282ropVar,B5: set_Pr6405597471078531282ropVar,C3: produc2642885279816024434ropVar > multis3193088007478089820ropVar,D: produc2642885279816024434ropVar > multis3193088007478089820ropVar,Sup: set_Mu5658373669842198716ropVar > multis3193088007478089820ropVar] :
      ( ( A = B5 )
     => ( ! [X3: produc2642885279816024434ropVar] :
            ( ( member9210975342120487067ropVar @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_4513341045867859311ropVar @ C3 @ A ) )
          = ( Sup @ ( image_4513341045867859311ropVar @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_510_Sup_OSUP__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > multis3193088007478089820ropVar,D: f > multis3193088007478089820ropVar,Sup: set_Mu5658373669842198716ropVar > multis3193088007478089820ropVar] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_4392090605272824862ropVar @ C3 @ A ) )
          = ( Sup @ ( image_4392090605272824862ropVar @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_511_Sup_OSUP__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > produc2642885279816024434ropVar,D: f > produc2642885279816024434ropVar,Sup: set_Pr6405597471078531282ropVar > produc2642885279816024434ropVar] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_2710159631831406900ropVar @ C3 @ A ) )
          = ( Sup @ ( image_2710159631831406900ropVar @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_512_Sup_OSUP__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > list_f,D: f > list_f,Sup: set_list_f > list_f] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_f_list_f @ C3 @ A ) )
          = ( Sup @ ( image_f_list_f @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_513_Sup_OSUP__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > nat,D: f > nat,Sup: set_nat > nat] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_f_nat @ C3 @ A ) )
          = ( Sup @ ( image_f_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_514_Sup_OSUP__cong,axiom,
    ! [A: set_f,B5: set_f,C3: f > f,D: f > f,Sup: set_f > f] :
      ( ( A = B5 )
     => ( ! [X3: f] :
            ( ( member_f @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_f_f @ C3 @ A ) )
          = ( Sup @ ( image_f_f @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_515_Sup_OSUP__cong,axiom,
    ! [A: set_nat,B5: set_nat,C3: nat > nat,D: nat > nat,Sup: set_nat > nat] :
      ( ( A = B5 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ B5 )
           => ( ( C3 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_nat_nat @ C3 @ A ) )
          = ( Sup @ ( image_nat_nat @ D @ B5 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_516_List_Oinsert__def,axiom,
    ( insert_list_f
    = ( ^ [X: list_f,Xs2: list_list_f] : ( if_list_list_f @ ( member_list_f @ X @ ( set_list_f2 @ Xs2 ) ) @ Xs2 @ ( cons_list_f @ X @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_517_List_Oinsert__def,axiom,
    ( insert215289443796111736ropVar
    = ( ^ [X: multis3193088007478089820ropVar,Xs2: list_M6740372356978808802ropVar] : ( if_lis9215432529438213544ropVar @ ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ Xs2 ) ) @ Xs2 @ ( cons_M4277272395645347858ropVar @ X @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_518_List_Oinsert__def,axiom,
    ( insert_f
    = ( ^ [X: f,Xs2: list_f] : ( if_list_f @ ( member_f @ X @ ( set_f2 @ Xs2 ) ) @ Xs2 @ ( cons_f @ X @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_519_List_Oinsert__def,axiom,
    ( insert214406736284816014ropVar
    = ( ^ [X: produc2642885279816024434ropVar,Xs2: list_P8791008543742759928ropVar] : ( if_lis1975434623467068862ropVar @ ( member9210975342120487067ropVar @ X @ ( set_Pr3946994337432900359ropVar @ Xs2 ) ) @ Xs2 @ ( cons_P1504208554336013608ropVar @ X @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_520_List_Oinsert__def,axiom,
    ( insert_nat
    = ( ^ [X: nat,Xs2: list_nat] : ( if_list_nat @ ( member_nat @ X @ ( set_nat2 @ Xs2 ) ) @ Xs2 @ ( cons_nat @ X @ Xs2 ) ) ) ) ).

% List.insert_def
thf(fact_521_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_nat > nat,A: set_nat] :
      ( ( Inf
        @ ( image_nat_nat
          @ ^ [X: nat] : X
          @ A ) )
      = ( Inf @ A ) ) ).

% Inf.INF_identity_eq
thf(fact_522_Inf_OINF__identity__eq,axiom,
    ! [Inf: set_f > f,A: set_f] :
      ( ( Inf
        @ ( image_f_f
          @ ^ [X: f] : X
          @ A ) )
      = ( Inf @ A ) ) ).

% Inf.INF_identity_eq
thf(fact_523_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_nat > nat,A: set_nat] :
      ( ( Sup
        @ ( image_nat_nat
          @ ^ [X: nat] : X
          @ A ) )
      = ( Sup @ A ) ) ).

% Sup.SUP_identity_eq
thf(fact_524_Sup_OSUP__identity__eq,axiom,
    ! [Sup: set_f > f,A: set_f] :
      ( ( Sup
        @ ( image_f_f
          @ ^ [X: f] : X
          @ A ) )
      = ( Sup @ A ) ) ).

% Sup.SUP_identity_eq
thf(fact_525_Id__onI,axiom,
    ! [A2: f,A: set_f] :
      ( ( member_f @ A2 @ A )
     => ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A2 @ A2 ) @ ( id_on_f @ A ) ) ) ).

% Id_onI
thf(fact_526_Id__onI,axiom,
    ! [A2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ A2 @ A )
     => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A2 @ A2 ) @ ( id_on_8850091095513083129ropVar @ A ) ) ) ).

% Id_onI
thf(fact_527_Id__onI,axiom,
    ! [A2: nat,A: set_nat] :
      ( ( member_nat @ A2 @ A )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ A2 ) @ ( id_on_nat @ A ) ) ) ).

% Id_onI
thf(fact_528_encode__sum__0__1__main_Ocases,axiom,
    ! [X2: list_P8791008543742759928ropVar] :
      ( ! [X3: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar] :
          ( X2
         != ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X3 @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ nil_Pr6119456312066593144ropVar ) )
     => ( ! [X3: multis3193088007478089820ropVar,Zero: multis3193088007478089820ropVar,One: multis3193088007478089820ropVar,V2: produc2642885279816024434ropVar,Va: list_P8791008543742759928ropVar] :
            ( X2
           != ( cons_P1504208554336013608ropVar @ ( produc6194868611276206818ropVar @ X3 @ ( produc6936306168532358295ropVar @ Zero @ One ) ) @ ( cons_P1504208554336013608ropVar @ V2 @ Va ) ) )
       => ( X2 = nil_Pr6119456312066593144ropVar ) ) ) ).

% encode_sum_0_1_main.cases
thf(fact_529_Id__onE,axiom,
    ! [C2: product_prod_f_f,A: set_f] :
      ( ( member5935997662887750672od_f_f @ C2 @ ( id_on_f @ A ) )
     => ~ ! [X3: f] :
            ( ( member_f @ X3 @ A )
           => ( C2
             != ( product_Pair_f_f @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_530_Id__onE,axiom,
    ! [C2: produc7679832412844898215ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( member2619728555115997648ropVar @ C2 @ ( id_on_8850091095513083129ropVar @ A ) )
     => ~ ! [X3: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X3 @ A )
           => ( C2
             != ( produc6936306168532358295ropVar @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_531_Id__onE,axiom,
    ! [C2: product_prod_nat_nat,A: set_nat] :
      ( ( member8440522571783428010at_nat @ C2 @ ( id_on_nat @ A ) )
     => ~ ! [X3: nat] :
            ( ( member_nat @ X3 @ A )
           => ( C2
             != ( product_Pair_nat_nat @ X3 @ X3 ) ) ) ) ).

% Id_onE
thf(fact_532_Id__on__eqI,axiom,
    ! [A2: f,B: f,A: set_f] :
      ( ( A2 = B )
     => ( ( member_f @ A2 @ A )
       => ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ A2 @ B ) @ ( id_on_f @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_533_Id__on__eqI,axiom,
    ! [A2: multis3193088007478089820ropVar,B: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( A2 = B )
     => ( ( member8421201611525043589ropVar @ A2 @ A )
       => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ A2 @ B ) @ ( id_on_8850091095513083129ropVar @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_534_Id__on__eqI,axiom,
    ! [A2: nat,B: nat,A: set_nat] :
      ( ( A2 = B )
     => ( ( member_nat @ A2 @ A )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A2 @ B ) @ ( id_on_nat @ A ) ) ) ) ).

% Id_on_eqI
thf(fact_535_Id__on__iff,axiom,
    ! [X2: f,Y3: f,A: set_f] :
      ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ ( id_on_f @ A ) )
      = ( ( X2 = Y3 )
        & ( member_f @ X2 @ A ) ) ) ).

% Id_on_iff
thf(fact_536_Id__on__iff,axiom,
    ! [X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ ( id_on_8850091095513083129ropVar @ A ) )
      = ( ( X2 = Y3 )
        & ( member8421201611525043589ropVar @ X2 @ A ) ) ) ).

% Id_on_iff
thf(fact_537_Id__on__iff,axiom,
    ! [X2: nat,Y3: nat,A: set_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( id_on_nat @ A ) )
      = ( ( X2 = Y3 )
        & ( member_nat @ X2 @ A ) ) ) ).

% Id_on_iff
thf(fact_538_bind__simps_I2_J,axiom,
    ! [X2: f,Xs: list_f,F: f > list_f] :
      ( ( bind_f_f @ ( cons_f @ X2 @ Xs ) @ F )
      = ( append_f @ ( F @ X2 ) @ ( bind_f_f @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_539_bind__simps_I2_J,axiom,
    ! [X2: f,Xs: list_f,F: f > list_nat] :
      ( ( bind_f_nat @ ( cons_f @ X2 @ Xs ) @ F )
      = ( append_nat @ ( F @ X2 ) @ ( bind_f_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_540_bind__simps_I2_J,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > list_f] :
      ( ( bind_P8681900402095835032pVar_f @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ F )
      = ( append_f @ ( F @ X2 ) @ ( bind_P8681900402095835032pVar_f @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_541_bind__simps_I2_J,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > list_nat] :
      ( ( bind_P7182508892803130363ar_nat @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ F )
      = ( append_nat @ ( F @ X2 ) @ ( bind_P7182508892803130363ar_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_542_bind__simps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,F: nat > list_f] :
      ( ( bind_nat_f @ ( cons_nat @ X2 @ Xs ) @ F )
      = ( append_f @ ( F @ X2 ) @ ( bind_nat_f @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_543_bind__simps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,F: nat > list_nat] :
      ( ( bind_nat_nat @ ( cons_nat @ X2 @ Xs ) @ F )
      = ( append_nat @ ( F @ X2 ) @ ( bind_nat_nat @ Xs @ F ) ) ) ).

% bind_simps(2)
thf(fact_544_j__ys2_I3_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys2 ) )
     => ( v2 @ ( multis387687052011358179_Gamma @ ( i_of_j2 @ J ) @ J ) ) ) ).

% j_ys2(3)
thf(fact_545_map__upt__Suc,axiom,
    ! [F: nat > multis3193088007478089820ropVar,N2: nat] :
      ( ( map_na2466558649495876678ropVar @ F @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( cons_M4277272395645347858ropVar @ ( F @ zero_zero_nat )
        @ ( map_na2466558649495876678ropVar
          @ ^ [I: nat] : ( F @ ( suc @ I ) )
          @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% map_upt_Suc
thf(fact_546_map__upt__Suc,axiom,
    ! [F: nat > f,N2: nat] :
      ( ( map_nat_f @ F @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( cons_f @ ( F @ zero_zero_nat )
        @ ( map_nat_f
          @ ^ [I: nat] : ( F @ ( suc @ I ) )
          @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% map_upt_Suc
thf(fact_547_map__upt__Suc,axiom,
    ! [F: nat > produc2642885279816024434ropVar,N2: nat] :
      ( ( map_na2145702811007361372ropVar @ F @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( cons_P1504208554336013608ropVar @ ( F @ zero_zero_nat )
        @ ( map_na2145702811007361372ropVar
          @ ^ [I: nat] : ( F @ ( suc @ I ) )
          @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% map_upt_Suc
thf(fact_548_map__upt__Suc,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( map_nat_nat @ F @ ( upt @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( cons_nat @ ( F @ zero_zero_nat )
        @ ( map_nat_nat
          @ ^ [I: nat] : ( F @ ( suc @ I ) )
          @ ( upt @ zero_zero_nat @ N2 ) ) ) ) ).

% map_upt_Suc
thf(fact_549_concat__map__singleton,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( concat2729506608927398813ropVar
        @ ( map_na5381558106008680140ropVar
          @ ^ [X: nat] : ( cons_M4277272395645347858ropVar @ ( F @ X ) @ nil_Mu4132399263956683874ropVar )
          @ Xs ) )
      = ( map_na2466558649495876678ropVar @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_550_concat__map__singleton,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f] :
      ( ( concat2729506608927398813ropVar
        @ ( map_f_7737976587276452195ropVar
          @ ^ [X: f] : ( cons_M4277272395645347858ropVar @ ( F @ X ) @ nil_Mu4132399263956683874ropVar )
          @ Xs ) )
      = ( map_f_4134023513460405853ropVar @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_551_concat__map__singleton,axiom,
    ! [F: f > list_f,Xs: list_f] :
      ( ( concat_list_f
        @ ( map_f_list_list_f
          @ ^ [X: f] : ( cons_list_f @ ( F @ X ) @ nil_list_f )
          @ Xs ) )
      = ( map_f_list_f @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_552_concat__map__singleton,axiom,
    ! [F: f > f,Xs: list_f] :
      ( ( concat_f
        @ ( map_f_list_f
          @ ^ [X: f] : ( cons_f @ ( F @ X ) @ nil_f )
          @ Xs ) )
      = ( map_f_f @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_553_concat__map__singleton,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( concat1830633707478507187ropVar
        @ ( map_na2676018863266313954ropVar
          @ ^ [X: nat] : ( cons_P1504208554336013608ropVar @ ( F @ X ) @ nil_Pr6119456312066593144ropVar )
          @ Xs ) )
      = ( map_na2145702811007361372ropVar @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_554_concat__map__singleton,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f] :
      ( ( concat1830633707478507187ropVar
        @ ( map_f_3293025520963524473ropVar
          @ ^ [X: f] : ( cons_P1504208554336013608ropVar @ ( F @ X ) @ nil_Pr6119456312066593144ropVar )
          @ Xs ) )
      = ( map_f_2905954635146393971ropVar @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_555_concat__map__singleton,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( concat_nat
        @ ( map_nat_list_nat
          @ ^ [X: nat] : ( cons_nat @ ( F @ X ) @ nil_nat )
          @ Xs ) )
      = ( map_nat_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_556_concat__map__singleton,axiom,
    ! [F: f > nat,Xs: list_f] :
      ( ( concat_nat
        @ ( map_f_list_nat
          @ ^ [X: f] : ( cons_nat @ ( F @ X ) @ nil_nat )
          @ Xs ) )
      = ( map_f_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_557_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_558_map__is__Nil__conv,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ Xs )
        = nil_Pr6119456312066593144ropVar )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% map_is_Nil_conv
thf(fact_559_map__is__Nil__conv,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% map_is_Nil_conv
thf(fact_560_map__is__Nil__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( ( map_na2145702811007361372ropVar @ F @ Xs )
        = nil_Pr6119456312066593144ropVar )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_561_map__is__Nil__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_562_map__is__Nil__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ Xs )
        = nil_Mu4132399263956683874ropVar )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_563_map__is__Nil__conv,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f] :
      ( ( ( map_f_4134023513460405853ropVar @ F @ Xs )
        = nil_Mu4132399263956683874ropVar )
      = ( Xs = nil_f ) ) ).

% map_is_Nil_conv
thf(fact_564_map__is__Nil__conv,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f] :
      ( ( ( map_f_2905954635146393971ropVar @ F @ Xs )
        = nil_Pr6119456312066593144ropVar )
      = ( Xs = nil_f ) ) ).

% map_is_Nil_conv
thf(fact_565_map__is__Nil__conv,axiom,
    ! [F: f > list_f,Xs: list_f] :
      ( ( ( map_f_list_f @ F @ Xs )
        = nil_list_f )
      = ( Xs = nil_f ) ) ).

% map_is_Nil_conv
thf(fact_566_map__is__Nil__conv,axiom,
    ! [F: f > nat,Xs: list_f] :
      ( ( ( map_f_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_f ) ) ).

% map_is_Nil_conv
thf(fact_567_map__is__Nil__conv,axiom,
    ! [F: f > f,Xs: list_f] :
      ( ( ( map_f_f @ F @ Xs )
        = nil_f )
      = ( Xs = nil_f ) ) ).

% map_is_Nil_conv
thf(fact_568_Nil__is__map__conv,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( map_Pr3565048146863404228ropVar @ F @ Xs ) )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% Nil_is_map_conv
thf(fact_569_Nil__is__map__conv,axiom,
    ! [F: produc2642885279816024434ropVar > nat,Xs: list_P8791008543742759928ropVar] :
      ( ( nil_nat
        = ( map_Pr7810381211709419518ar_nat @ F @ Xs ) )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% Nil_is_map_conv
thf(fact_570_Nil__is__map__conv,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( map_na2145702811007361372ropVar @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_571_Nil__is__map__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_572_Nil__is__map__conv,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat] :
      ( ( nil_Mu4132399263956683874ropVar
        = ( map_na2466558649495876678ropVar @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_573_Nil__is__map__conv,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f] :
      ( ( nil_Mu4132399263956683874ropVar
        = ( map_f_4134023513460405853ropVar @ F @ Xs ) )
      = ( Xs = nil_f ) ) ).

% Nil_is_map_conv
thf(fact_574_Nil__is__map__conv,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( map_f_2905954635146393971ropVar @ F @ Xs ) )
      = ( Xs = nil_f ) ) ).

% Nil_is_map_conv
thf(fact_575_Nil__is__map__conv,axiom,
    ! [F: f > list_f,Xs: list_f] :
      ( ( nil_list_f
        = ( map_f_list_f @ F @ Xs ) )
      = ( Xs = nil_f ) ) ).

% Nil_is_map_conv
thf(fact_576_Nil__is__map__conv,axiom,
    ! [F: f > nat,Xs: list_f] :
      ( ( nil_nat
        = ( map_f_nat @ F @ Xs ) )
      = ( Xs = nil_f ) ) ).

% Nil_is_map_conv
thf(fact_577_Nil__is__map__conv,axiom,
    ! [F: f > f,Xs: list_f] :
      ( ( nil_f
        = ( map_f_f @ F @ Xs ) )
      = ( Xs = nil_f ) ) ).

% Nil_is_map_conv
thf(fact_578_list_Omap__disc__iff,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar,A2: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr3565048146863404228ropVar @ F @ A2 )
        = nil_Pr6119456312066593144ropVar )
      = ( A2 = nil_Pr6119456312066593144ropVar ) ) ).

% list.map_disc_iff
thf(fact_579_list_Omap__disc__iff,axiom,
    ! [F: produc2642885279816024434ropVar > nat,A2: list_P8791008543742759928ropVar] :
      ( ( ( map_Pr7810381211709419518ar_nat @ F @ A2 )
        = nil_nat )
      = ( A2 = nil_Pr6119456312066593144ropVar ) ) ).

% list.map_disc_iff
thf(fact_580_list_Omap__disc__iff,axiom,
    ! [F: nat > produc2642885279816024434ropVar,A2: list_nat] :
      ( ( ( map_na2145702811007361372ropVar @ F @ A2 )
        = nil_Pr6119456312066593144ropVar )
      = ( A2 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_581_list_Omap__disc__iff,axiom,
    ! [F: nat > nat,A2: list_nat] :
      ( ( ( map_nat_nat @ F @ A2 )
        = nil_nat )
      = ( A2 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_582_list_Omap__disc__iff,axiom,
    ! [F: nat > multis3193088007478089820ropVar,A2: list_nat] :
      ( ( ( map_na2466558649495876678ropVar @ F @ A2 )
        = nil_Mu4132399263956683874ropVar )
      = ( A2 = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_583_list_Omap__disc__iff,axiom,
    ! [F: f > multis3193088007478089820ropVar,A2: list_f] :
      ( ( ( map_f_4134023513460405853ropVar @ F @ A2 )
        = nil_Mu4132399263956683874ropVar )
      = ( A2 = nil_f ) ) ).

% list.map_disc_iff
thf(fact_584_list_Omap__disc__iff,axiom,
    ! [F: f > produc2642885279816024434ropVar,A2: list_f] :
      ( ( ( map_f_2905954635146393971ropVar @ F @ A2 )
        = nil_Pr6119456312066593144ropVar )
      = ( A2 = nil_f ) ) ).

% list.map_disc_iff
thf(fact_585_list_Omap__disc__iff,axiom,
    ! [F: f > list_f,A2: list_f] :
      ( ( ( map_f_list_f @ F @ A2 )
        = nil_list_f )
      = ( A2 = nil_f ) ) ).

% list.map_disc_iff
thf(fact_586_list_Omap__disc__iff,axiom,
    ! [F: f > nat,A2: list_f] :
      ( ( ( map_f_nat @ F @ A2 )
        = nil_nat )
      = ( A2 = nil_f ) ) ).

% list.map_disc_iff
thf(fact_587_list_Omap__disc__iff,axiom,
    ! [F: f > f,A2: list_f] :
      ( ( ( map_f_f @ F @ A2 )
        = nil_f )
      = ( A2 = nil_f ) ) ).

% list.map_disc_iff
thf(fact_588_append__is__Nil__conv,axiom,
    ! [Xs: list_f,Ys: list_f] :
      ( ( ( append_f @ Xs @ Ys )
        = nil_f )
      = ( ( Xs = nil_f )
        & ( Ys = nil_f ) ) ) ).

% append_is_Nil_conv
thf(fact_589_append__is__Nil__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ Ys )
        = nil_Pr6119456312066593144ropVar )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        & ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% append_is_Nil_conv
thf(fact_590_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_591_Nil__is__append__conv,axiom,
    ! [Xs: list_f,Ys: list_f] :
      ( ( nil_f
        = ( append_f @ Xs @ Ys ) )
      = ( ( Xs = nil_f )
        & ( Ys = nil_f ) ) ) ).

% Nil_is_append_conv
thf(fact_592_Nil__is__append__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( append4352292717583037453ropVar @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr6119456312066593144ropVar )
        & ( Ys = nil_Pr6119456312066593144ropVar ) ) ) ).

% Nil_is_append_conv
thf(fact_593_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_594_self__append__conv2,axiom,
    ! [Y3: list_f,Xs: list_f] :
      ( ( Y3
        = ( append_f @ Xs @ Y3 ) )
      = ( Xs = nil_f ) ) ).

% self_append_conv2
thf(fact_595_self__append__conv2,axiom,
    ! [Y3: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( Y3
        = ( append4352292717583037453ropVar @ Xs @ Y3 ) )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% self_append_conv2
thf(fact_596_self__append__conv2,axiom,
    ! [Y3: list_nat,Xs: list_nat] :
      ( ( Y3
        = ( append_nat @ Xs @ Y3 ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_597_append__self__conv2,axiom,
    ! [Xs: list_f,Ys: list_f] :
      ( ( ( append_f @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_f ) ) ).

% append_self_conv2
thf(fact_598_append__self__conv2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr6119456312066593144ropVar ) ) ).

% append_self_conv2
thf(fact_599_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_600_self__append__conv,axiom,
    ! [Y3: list_f,Ys: list_f] :
      ( ( Y3
        = ( append_f @ Y3 @ Ys ) )
      = ( Ys = nil_f ) ) ).

% self_append_conv
thf(fact_601_self__append__conv,axiom,
    ! [Y3: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( Y3
        = ( append4352292717583037453ropVar @ Y3 @ Ys ) )
      = ( Ys = nil_Pr6119456312066593144ropVar ) ) ).

% self_append_conv
thf(fact_602_self__append__conv,axiom,
    ! [Y3: list_nat,Ys: list_nat] :
      ( ( Y3
        = ( append_nat @ Y3 @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_603_append__self__conv,axiom,
    ! [Xs: list_f,Ys: list_f] :
      ( ( ( append_f @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_f ) ) ).

% append_self_conv
thf(fact_604_append__self__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr6119456312066593144ropVar ) ) ).

% append_self_conv
thf(fact_605_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_606_append__Nil2,axiom,
    ! [Xs: list_f] :
      ( ( append_f @ Xs @ nil_f )
      = Xs ) ).

% append_Nil2
thf(fact_607_append__Nil2,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ Xs @ nil_Pr6119456312066593144ropVar )
      = Xs ) ).

% append_Nil2
thf(fact_608_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_609_append_Oright__neutral,axiom,
    ! [A2: list_f] :
      ( ( append_f @ A2 @ nil_f )
      = A2 ) ).

% append.right_neutral
thf(fact_610_append_Oright__neutral,axiom,
    ! [A2: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ A2 @ nil_Pr6119456312066593144ropVar )
      = A2 ) ).

% append.right_neutral
thf(fact_611_append_Oright__neutral,axiom,
    ! [A2: list_nat] :
      ( ( append_nat @ A2 @ nil_nat )
      = A2 ) ).

% append.right_neutral
thf(fact_612_bind__simps_I1_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_P8791008543742759928ropVar] :
      ( ( bind_P5656827158300702215ropVar @ nil_Pr6119456312066593144ropVar @ F )
      = nil_Pr6119456312066593144ropVar ) ).

% bind_simps(1)
thf(fact_613_bind__simps_I1_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_nat] :
      ( ( bind_P7182508892803130363ar_nat @ nil_Pr6119456312066593144ropVar @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_614_bind__simps_I1_J,axiom,
    ! [F: nat > list_P8791008543742759928ropVar] :
      ( ( bind_n1517830492101072217ropVar @ nil_nat @ F )
      = nil_Pr6119456312066593144ropVar ) ).

% bind_simps(1)
thf(fact_615_bind__simps_I1_J,axiom,
    ! [F: nat > list_nat] :
      ( ( bind_nat_nat @ nil_nat @ F )
      = nil_nat ) ).

% bind_simps(1)
thf(fact_616_append1__eq__conv,axiom,
    ! [Xs: list_f,X2: f,Ys: list_f,Y3: f] :
      ( ( ( append_f @ Xs @ ( cons_f @ X2 @ nil_f ) )
        = ( append_f @ Ys @ ( cons_f @ Y3 @ nil_f ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_617_append1__eq__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar] :
      ( ( ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) )
        = ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ Y3 @ nil_Pr6119456312066593144ropVar ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_618_append1__eq__conv,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X2 = Y3 ) ) ) ).

% append1_eq_conv
thf(fact_619_insert__Nil,axiom,
    ! [X2: f] :
      ( ( insert_f @ X2 @ nil_f )
      = ( cons_f @ X2 @ nil_f ) ) ).

% insert_Nil
thf(fact_620_insert__Nil,axiom,
    ! [X2: produc2642885279816024434ropVar] :
      ( ( insert214406736284816014ropVar @ X2 @ nil_Pr6119456312066593144ropVar )
      = ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) ).

% insert_Nil
thf(fact_621_insert__Nil,axiom,
    ! [X2: nat] :
      ( ( insert_nat @ X2 @ nil_nat )
      = ( cons_nat @ X2 @ nil_nat ) ) ).

% insert_Nil
thf(fact_622_concat__eq__Nil__conv,axiom,
    ! [Xss: list_l3415868495287753214ropVar] :
      ( ( ( concat1830633707478507187ropVar @ Xss )
        = nil_Pr6119456312066593144ropVar )
      = ( ! [X: list_P8791008543742759928ropVar] :
            ( ( member6236339133593825953ropVar @ X @ ( set_li6509544621384488717ropVar @ Xss ) )
           => ( X = nil_Pr6119456312066593144ropVar ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_623_concat__eq__Nil__conv,axiom,
    ! [Xss: list_list_nat] :
      ( ( ( concat_nat @ Xss )
        = nil_nat )
      = ( ! [X: list_nat] :
            ( ( member_list_nat @ X @ ( set_list_nat2 @ Xss ) )
           => ( X = nil_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_624_concat__eq__Nil__conv,axiom,
    ! [Xss: list_list_f] :
      ( ( ( concat_f @ Xss )
        = nil_f )
      = ( ! [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Xss ) )
           => ( X = nil_f ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_625_Nil__eq__concat__conv,axiom,
    ! [Xss: list_l3415868495287753214ropVar] :
      ( ( nil_Pr6119456312066593144ropVar
        = ( concat1830633707478507187ropVar @ Xss ) )
      = ( ! [X: list_P8791008543742759928ropVar] :
            ( ( member6236339133593825953ropVar @ X @ ( set_li6509544621384488717ropVar @ Xss ) )
           => ( X = nil_Pr6119456312066593144ropVar ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_626_Nil__eq__concat__conv,axiom,
    ! [Xss: list_list_nat] :
      ( ( nil_nat
        = ( concat_nat @ Xss ) )
      = ( ! [X: list_nat] :
            ( ( member_list_nat @ X @ ( set_list_nat2 @ Xss ) )
           => ( X = nil_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_627_Nil__eq__concat__conv,axiom,
    ! [Xss: list_list_f] :
      ( ( nil_f
        = ( concat_f @ Xss ) )
      = ( ! [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Xss ) )
           => ( X = nil_f ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_628_upt__rec,axiom,
    ( upt
    = ( ^ [I: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I @ J2 ) @ ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_629_transpose_Ocases,axiom,
    ! [X2: list_list_f] :
      ( ( X2 != nil_list_f )
     => ( ! [Xss2: list_list_f] :
            ( X2
           != ( cons_list_f @ nil_f @ Xss2 ) )
       => ~ ! [X3: f,Xs4: list_f,Xss2: list_list_f] :
              ( X2
             != ( cons_list_f @ ( cons_f @ X3 @ Xs4 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_630_transpose_Ocases,axiom,
    ! [X2: list_l3415868495287753214ropVar] :
      ( ( X2 != nil_li2679230897895549182ropVar )
     => ( ! [Xss2: list_l3415868495287753214ropVar] :
            ( X2
           != ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss2 ) )
       => ~ ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Xss2: list_l3415868495287753214ropVar] :
              ( X2
             != ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_631_transpose_Ocases,axiom,
    ! [X2: list_list_nat] :
      ( ( X2 != nil_list_nat )
     => ( ! [Xss2: list_list_nat] :
            ( X2
           != ( cons_list_nat @ nil_nat @ Xss2 ) )
       => ~ ! [X3: nat,Xs4: list_nat,Xss2: list_list_nat] :
              ( X2
             != ( cons_list_nat @ ( cons_nat @ X3 @ Xs4 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_632_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( upt @ I2 @ J )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_633_concat_Osimps_I1_J,axiom,
    ( ( concat_f @ nil_list_f )
    = nil_f ) ).

% concat.simps(1)
thf(fact_634_concat_Osimps_I1_J,axiom,
    ( ( concat1830633707478507187ropVar @ nil_li2679230897895549182ropVar )
    = nil_Pr6119456312066593144ropVar ) ).

% concat.simps(1)
thf(fact_635_concat_Osimps_I1_J,axiom,
    ( ( concat_nat @ nil_list_nat )
    = nil_nat ) ).

% concat.simps(1)
thf(fact_636_concat__eq__append__conv,axiom,
    ! [Xss: list_list_f,Ys: list_f,Zs3: list_f] :
      ( ( ( concat_f @ Xss )
        = ( append_f @ Ys @ Zs3 ) )
      = ( ( ( Xss = nil_list_f )
         => ( ( Ys = nil_f )
            & ( Zs3 = nil_f ) ) )
        & ( ( Xss != nil_list_f )
         => ? [Xss1: list_list_f,Xs2: list_f,Xs5: list_f,Xss22: list_list_f] :
              ( ( Xss
                = ( append_list_f @ Xss1 @ ( cons_list_f @ ( append_f @ Xs2 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_f @ ( concat_f @ Xss1 ) @ Xs2 ) )
              & ( Zs3
                = ( append_f @ Xs5 @ ( concat_f @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_637_concat__eq__append__conv,axiom,
    ! [Xss: list_l3415868495287753214ropVar,Ys: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
      ( ( ( concat1830633707478507187ropVar @ Xss )
        = ( append4352292717583037453ropVar @ Ys @ Zs3 ) )
      = ( ( ( Xss = nil_li2679230897895549182ropVar )
         => ( ( Ys = nil_Pr6119456312066593144ropVar )
            & ( Zs3 = nil_Pr6119456312066593144ropVar ) ) )
        & ( ( Xss != nil_li2679230897895549182ropVar )
         => ? [Xss1: list_l3415868495287753214ropVar,Xs2: list_P8791008543742759928ropVar,Xs5: list_P8791008543742759928ropVar,Xss22: list_l3415868495287753214ropVar] :
              ( ( Xss
                = ( append4316637162825710867ropVar @ Xss1 @ ( cons_l5280711761816980654ropVar @ ( append4352292717583037453ropVar @ Xs2 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append4352292717583037453ropVar @ ( concat1830633707478507187ropVar @ Xss1 ) @ Xs2 ) )
              & ( Zs3
                = ( append4352292717583037453ropVar @ Xs5 @ ( concat1830633707478507187ropVar @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_638_concat__eq__append__conv,axiom,
    ! [Xss: list_list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( concat_nat @ Xss )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ( ( Xss = nil_list_nat )
         => ( ( Ys = nil_nat )
            & ( Zs3 = nil_nat ) ) )
        & ( ( Xss != nil_list_nat )
         => ? [Xss1: list_list_nat,Xs2: list_nat,Xs5: list_nat,Xss22: list_list_nat] :
              ( ( Xss
                = ( append_list_nat @ Xss1 @ ( cons_list_nat @ ( append_nat @ Xs2 @ Xs5 ) @ Xss22 ) ) )
              & ( Ys
                = ( append_nat @ ( concat_nat @ Xss1 ) @ Xs2 ) )
              & ( Zs3
                = ( append_nat @ Xs5 @ ( concat_nat @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_639_zero__less__iff__neq__zero,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( N2 != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_640_gr__implies__not__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_641_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_642_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_643_list_Odistinct_I1_J,axiom,
    ! [X21: f,X222: list_f] :
      ( nil_f
     != ( cons_f @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_644_list_Odistinct_I1_J,axiom,
    ! [X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( nil_Pr6119456312066593144ropVar
     != ( cons_P1504208554336013608ropVar @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_645_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_646_list_OdiscI,axiom,
    ! [List: list_f,X21: f,X222: list_f] :
      ( ( List
        = ( cons_f @ X21 @ X222 ) )
     => ( List != nil_f ) ) ).

% list.discI
thf(fact_647_list_OdiscI,axiom,
    ! [List: list_P8791008543742759928ropVar,X21: produc2642885279816024434ropVar,X222: list_P8791008543742759928ropVar] :
      ( ( List
        = ( cons_P1504208554336013608ropVar @ X21 @ X222 ) )
     => ( List != nil_Pr6119456312066593144ropVar ) ) ).

% list.discI
thf(fact_648_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_649_list_Oexhaust,axiom,
    ! [Y3: list_f] :
      ( ( Y3 != nil_f )
     => ~ ! [X212: f,X223: list_f] :
            ( Y3
           != ( cons_f @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_650_list_Oexhaust,axiom,
    ! [Y3: list_P8791008543742759928ropVar] :
      ( ( Y3 != nil_Pr6119456312066593144ropVar )
     => ~ ! [X212: produc2642885279816024434ropVar,X223: list_P8791008543742759928ropVar] :
            ( Y3
           != ( cons_P1504208554336013608ropVar @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_651_list_Oexhaust,axiom,
    ! [Y3: list_nat] :
      ( ( Y3 != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y3
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_652_min__list_Ocases,axiom,
    ! [X2: list_nat] :
      ( ! [X3: nat,Xs4: list_nat] :
          ( X2
         != ( cons_nat @ X3 @ Xs4 ) )
     => ( X2 = nil_nat ) ) ).

% min_list.cases
thf(fact_653_shuffles_Ocases,axiom,
    ! [X2: produc1604256039088728679list_f] :
      ( ! [Ys3: list_f] :
          ( X2
         != ( produc8144748701140958423list_f @ nil_f @ Ys3 ) )
     => ( ! [Xs4: list_f] :
            ( X2
           != ( produc8144748701140958423list_f @ Xs4 @ nil_f ) )
       => ~ ! [X3: f,Xs4: list_f,Y4: f,Ys3: list_f] :
              ( X2
             != ( produc8144748701140958423list_f @ ( cons_f @ X3 @ Xs4 ) @ ( cons_f @ Y4 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_654_shuffles_Ocases,axiom,
    ! [X2: produc3638977808798937127ropVar] :
      ( ! [Ys3: list_P8791008543742759928ropVar] :
          ( X2
         != ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys3 ) )
     => ( ! [Xs4: list_P8791008543742759928ropVar] :
            ( X2
           != ( produc5333478016634530327ropVar @ Xs4 @ nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] :
              ( X2
             != ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_655_shuffles_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Ys3: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
     => ( ! [Xs4: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ Xs4 @ nil_nat ) )
       => ~ ! [X3: nat,Xs4: list_nat,Y4: nat,Ys3: list_nat] :
              ( X2
             != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs4 ) @ ( cons_nat @ Y4 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_656_sorted__wrt_Ocases,axiom,
    ! [X2: produc5795555978541011586list_f] :
      ( ! [P4: f > f > $o] :
          ( X2
         != ( produc8519404107117762418list_f @ P4 @ nil_f ) )
     => ~ ! [P4: f > f > $o,X3: f,Ys3: list_f] :
            ( X2
           != ( produc8519404107117762418list_f @ P4 @ ( cons_f @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_657_sorted__wrt_Ocases,axiom,
    ! [X2: produc5533280449304468145ropVar] :
      ( ! [P4: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
          ( X2
         != ( produc3887254912111605537ropVar @ P4 @ nil_Pr6119456312066593144ropVar ) )
     => ~ ! [P4: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X3: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] :
            ( X2
           != ( produc3887254912111605537ropVar @ P4 @ ( cons_P1504208554336013608ropVar @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_658_sorted__wrt_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [P4: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
     => ~ ! [P4: nat > nat > $o,X3: nat,Ys3: list_nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_659_remdups__adj_Ocases,axiom,
    ! [X2: list_f] :
      ( ( X2 != nil_f )
     => ( ! [X3: f] :
            ( X2
           != ( cons_f @ X3 @ nil_f ) )
       => ~ ! [X3: f,Y4: f,Xs4: list_f] :
              ( X2
             != ( cons_f @ X3 @ ( cons_f @ Y4 @ Xs4 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_660_remdups__adj_Ocases,axiom,
    ! [X2: list_P8791008543742759928ropVar] :
      ( ( X2 != nil_Pr6119456312066593144ropVar )
     => ( ! [X3: produc2642885279816024434ropVar] :
            ( X2
           != ( cons_P1504208554336013608ropVar @ X3 @ nil_Pr6119456312066593144ropVar ) )
       => ~ ! [X3: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] :
              ( X2
             != ( cons_P1504208554336013608ropVar @ X3 @ ( cons_P1504208554336013608ropVar @ Y4 @ Xs4 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_661_remdups__adj_Ocases,axiom,
    ! [X2: list_nat] :
      ( ( X2 != nil_nat )
     => ( ! [X3: nat] :
            ( X2
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y4: nat,Xs4: list_nat] :
              ( X2
             != ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs4 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_662_successively_Ocases,axiom,
    ! [X2: produc5795555978541011586list_f] :
      ( ! [P4: f > f > $o] :
          ( X2
         != ( produc8519404107117762418list_f @ P4 @ nil_f ) )
     => ( ! [P4: f > f > $o,X3: f] :
            ( X2
           != ( produc8519404107117762418list_f @ P4 @ ( cons_f @ X3 @ nil_f ) ) )
       => ~ ! [P4: f > f > $o,X3: f,Y4: f,Xs4: list_f] :
              ( X2
             != ( produc8519404107117762418list_f @ P4 @ ( cons_f @ X3 @ ( cons_f @ Y4 @ Xs4 ) ) ) ) ) ) ).

% successively.cases
thf(fact_663_successively_Ocases,axiom,
    ! [X2: produc5533280449304468145ropVar] :
      ( ! [P4: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o] :
          ( X2
         != ( produc3887254912111605537ropVar @ P4 @ nil_Pr6119456312066593144ropVar ) )
     => ( ! [P4: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X3: produc2642885279816024434ropVar] :
            ( X2
           != ( produc3887254912111605537ropVar @ P4 @ ( cons_P1504208554336013608ropVar @ X3 @ nil_Pr6119456312066593144ropVar ) ) )
       => ~ ! [P4: produc2642885279816024434ropVar > produc2642885279816024434ropVar > $o,X3: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] :
              ( X2
             != ( produc3887254912111605537ropVar @ P4 @ ( cons_P1504208554336013608ropVar @ X3 @ ( cons_P1504208554336013608ropVar @ Y4 @ Xs4 ) ) ) ) ) ) ).

% successively.cases
thf(fact_664_successively_Ocases,axiom,
    ! [X2: produc254973753779126261st_nat] :
      ( ! [P4: nat > nat > $o] :
          ( X2
         != ( produc4727192421694094319st_nat @ P4 @ nil_nat ) )
     => ( ! [P4: nat > nat > $o,X3: nat] :
            ( X2
           != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [P4: nat > nat > $o,X3: nat,Y4: nat,Xs4: list_nat] :
              ( X2
             != ( produc4727192421694094319st_nat @ P4 @ ( cons_nat @ X3 @ ( cons_nat @ Y4 @ Xs4 ) ) ) ) ) ) ).

% successively.cases
thf(fact_665_neq__Nil__conv,axiom,
    ! [Xs: list_f] :
      ( ( Xs != nil_f )
      = ( ? [Y: f,Ys2: list_f] :
            ( Xs
            = ( cons_f @ Y @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_666_neq__Nil__conv,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
      = ( ? [Y: produc2642885279816024434ropVar,Ys2: list_P8791008543742759928ropVar] :
            ( Xs
            = ( cons_P1504208554336013608ropVar @ Y @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_667_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y: nat,Ys2: list_nat] :
            ( Xs
            = ( cons_nat @ Y @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_668_list__induct2_H,axiom,
    ! [P: list_f > list_f > $o,Xs: list_f,Ys: list_f] :
      ( ( P @ nil_f @ nil_f )
     => ( ! [X3: f,Xs4: list_f] : ( P @ ( cons_f @ X3 @ Xs4 ) @ nil_f )
       => ( ! [Y4: f,Ys3: list_f] : ( P @ nil_f @ ( cons_f @ Y4 @ Ys3 ) )
         => ( ! [X3: f,Xs4: list_f,Y4: f,Ys3: list_f] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_f @ X3 @ Xs4 ) @ ( cons_f @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_669_list__induct2_H,axiom,
    ! [P: list_f > list_P8791008543742759928ropVar > $o,Xs: list_f,Ys: list_P8791008543742759928ropVar] :
      ( ( P @ nil_f @ nil_Pr6119456312066593144ropVar )
     => ( ! [X3: f,Xs4: list_f] : ( P @ ( cons_f @ X3 @ Xs4 ) @ nil_Pr6119456312066593144ropVar )
       => ( ! [Y4: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] : ( P @ nil_f @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys3 ) )
         => ( ! [X3: f,Xs4: list_f,Y4: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_f @ X3 @ Xs4 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_670_list__induct2_H,axiom,
    ! [P: list_f > list_nat > $o,Xs: list_f,Ys: list_nat] :
      ( ( P @ nil_f @ nil_nat )
     => ( ! [X3: f,Xs4: list_f] : ( P @ ( cons_f @ X3 @ Xs4 ) @ nil_nat )
       => ( ! [Y4: nat,Ys3: list_nat] : ( P @ nil_f @ ( cons_nat @ Y4 @ Ys3 ) )
         => ( ! [X3: f,Xs4: list_f,Y4: nat,Ys3: list_nat] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_f @ X3 @ Xs4 ) @ ( cons_nat @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_671_list__induct2_H,axiom,
    ! [P: list_P8791008543742759928ropVar > list_f > $o,Xs: list_P8791008543742759928ropVar,Ys: list_f] :
      ( ( P @ nil_Pr6119456312066593144ropVar @ nil_f )
     => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ nil_f )
       => ( ! [Y4: f,Ys3: list_f] : ( P @ nil_Pr6119456312066593144ropVar @ ( cons_f @ Y4 @ Ys3 ) )
         => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Y4: f,Ys3: list_f] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ ( cons_f @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_672_list__induct2_H,axiom,
    ! [P: list_P8791008543742759928ropVar > list_P8791008543742759928ropVar > $o,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( P @ nil_Pr6119456312066593144ropVar @ nil_Pr6119456312066593144ropVar )
     => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ nil_Pr6119456312066593144ropVar )
       => ( ! [Y4: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] : ( P @ nil_Pr6119456312066593144ropVar @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys3 ) )
         => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_673_list__induct2_H,axiom,
    ! [P: list_P8791008543742759928ropVar > list_nat > $o,Xs: list_P8791008543742759928ropVar,Ys: list_nat] :
      ( ( P @ nil_Pr6119456312066593144ropVar @ nil_nat )
     => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ nil_nat )
       => ( ! [Y4: nat,Ys3: list_nat] : ( P @ nil_Pr6119456312066593144ropVar @ ( cons_nat @ Y4 @ Ys3 ) )
         => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Y4: nat,Ys3: list_nat] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ ( cons_nat @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_674_list__induct2_H,axiom,
    ! [P: list_nat > list_f > $o,Xs: list_nat,Ys: list_f] :
      ( ( P @ nil_nat @ nil_f )
     => ( ! [X3: nat,Xs4: list_nat] : ( P @ ( cons_nat @ X3 @ Xs4 ) @ nil_f )
       => ( ! [Y4: f,Ys3: list_f] : ( P @ nil_nat @ ( cons_f @ Y4 @ Ys3 ) )
         => ( ! [X3: nat,Xs4: list_nat,Y4: f,Ys3: list_f] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_nat @ X3 @ Xs4 ) @ ( cons_f @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_675_list__induct2_H,axiom,
    ! [P: list_nat > list_P8791008543742759928ropVar > $o,Xs: list_nat,Ys: list_P8791008543742759928ropVar] :
      ( ( P @ nil_nat @ nil_Pr6119456312066593144ropVar )
     => ( ! [X3: nat,Xs4: list_nat] : ( P @ ( cons_nat @ X3 @ Xs4 ) @ nil_Pr6119456312066593144ropVar )
       => ( ! [Y4: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] : ( P @ nil_nat @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys3 ) )
         => ( ! [X3: nat,Xs4: list_nat,Y4: produc2642885279816024434ropVar,Ys3: list_P8791008543742759928ropVar] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_nat @ X3 @ Xs4 ) @ ( cons_P1504208554336013608ropVar @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_676_list__induct2_H,axiom,
    ! [P: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P @ nil_nat @ nil_nat )
     => ( ! [X3: nat,Xs4: list_nat] : ( P @ ( cons_nat @ X3 @ Xs4 ) @ nil_nat )
       => ( ! [Y4: nat,Ys3: list_nat] : ( P @ nil_nat @ ( cons_nat @ Y4 @ Ys3 ) )
         => ( ! [X3: nat,Xs4: list_nat,Y4: nat,Ys3: list_nat] :
                ( ( P @ Xs4 @ Ys3 )
               => ( P @ ( cons_nat @ X3 @ Xs4 ) @ ( cons_nat @ Y4 @ Ys3 ) ) )
           => ( P @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_677_list__nonempty__induct,axiom,
    ! [Xs: list_f,P: list_f > $o] :
      ( ( Xs != nil_f )
     => ( ! [X3: f] : ( P @ ( cons_f @ X3 @ nil_f ) )
       => ( ! [X3: f,Xs4: list_f] :
              ( ( Xs4 != nil_f )
             => ( ( P @ Xs4 )
               => ( P @ ( cons_f @ X3 @ Xs4 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_678_list__nonempty__induct,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > $o] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ( ! [X3: produc2642885279816024434ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X3 @ nil_Pr6119456312066593144ropVar ) )
       => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] :
              ( ( Xs4 != nil_Pr6119456312066593144ropVar )
             => ( ( P @ Xs4 )
               => ( P @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_679_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs4: list_nat] :
              ( ( Xs4 != nil_nat )
             => ( ( P @ Xs4 )
               => ( P @ ( cons_nat @ X3 @ Xs4 ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_680_list_Osimps_I8_J,axiom,
    ! [F: produc2642885279816024434ropVar > produc2642885279816024434ropVar] :
      ( ( map_Pr3565048146863404228ropVar @ F @ nil_Pr6119456312066593144ropVar )
      = nil_Pr6119456312066593144ropVar ) ).

% list.simps(8)
thf(fact_681_list_Osimps_I8_J,axiom,
    ! [F: produc2642885279816024434ropVar > nat] :
      ( ( map_Pr7810381211709419518ar_nat @ F @ nil_Pr6119456312066593144ropVar )
      = nil_nat ) ).

% list.simps(8)
thf(fact_682_list_Osimps_I8_J,axiom,
    ! [F: nat > produc2642885279816024434ropVar] :
      ( ( map_na2145702811007361372ropVar @ F @ nil_nat )
      = nil_Pr6119456312066593144ropVar ) ).

% list.simps(8)
thf(fact_683_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_684_list_Osimps_I8_J,axiom,
    ! [F: nat > multis3193088007478089820ropVar] :
      ( ( map_na2466558649495876678ropVar @ F @ nil_nat )
      = nil_Mu4132399263956683874ropVar ) ).

% list.simps(8)
thf(fact_685_list_Osimps_I8_J,axiom,
    ! [F: f > multis3193088007478089820ropVar] :
      ( ( map_f_4134023513460405853ropVar @ F @ nil_f )
      = nil_Mu4132399263956683874ropVar ) ).

% list.simps(8)
thf(fact_686_list_Osimps_I8_J,axiom,
    ! [F: f > produc2642885279816024434ropVar] :
      ( ( map_f_2905954635146393971ropVar @ F @ nil_f )
      = nil_Pr6119456312066593144ropVar ) ).

% list.simps(8)
thf(fact_687_list_Osimps_I8_J,axiom,
    ! [F: f > list_f] :
      ( ( map_f_list_f @ F @ nil_f )
      = nil_list_f ) ).

% list.simps(8)
thf(fact_688_list_Osimps_I8_J,axiom,
    ! [F: f > nat] :
      ( ( map_f_nat @ F @ nil_f )
      = nil_nat ) ).

% list.simps(8)
thf(fact_689_list_Osimps_I8_J,axiom,
    ! [F: f > f] :
      ( ( map_f_f @ F @ nil_f )
      = nil_f ) ).

% list.simps(8)
thf(fact_690_eq__Nil__appendI,axiom,
    ! [Xs: list_f,Ys: list_f] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_f @ nil_f @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_691_eq__Nil__appendI,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_692_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_693_append_Oleft__neutral,axiom,
    ! [A2: list_f] :
      ( ( append_f @ nil_f @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_694_append_Oleft__neutral,axiom,
    ! [A2: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_695_append_Oleft__neutral,axiom,
    ! [A2: list_nat] :
      ( ( append_nat @ nil_nat @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_696_append__Nil,axiom,
    ! [Ys: list_f] :
      ( ( append_f @ nil_f @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_697_append__Nil,axiom,
    ! [Ys: list_P8791008543742759928ropVar] :
      ( ( append4352292717583037453ropVar @ nil_Pr6119456312066593144ropVar @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_698_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_699_map__Suc__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N2 ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_700_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N2: nat,Ns: list_nat,Q: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M2 @ Q ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_701_maps__simps_I2_J,axiom,
    ! [F: multis3193088007478089820ropVar > list_f] :
      ( ( maps_M425433082187236072pVar_f @ F @ nil_Mu4132399263956683874ropVar )
      = nil_f ) ).

% maps_simps(2)
thf(fact_702_maps__simps_I2_J,axiom,
    ! [F: list_f > list_f] :
      ( ( maps_list_f_f @ F @ nil_list_f )
      = nil_f ) ).

% maps_simps(2)
thf(fact_703_maps__simps_I2_J,axiom,
    ! [F: f > list_f] :
      ( ( maps_f_f @ F @ nil_f )
      = nil_f ) ).

% maps_simps(2)
thf(fact_704_maps__simps_I2_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_f] :
      ( ( maps_P2527380396400262994pVar_f @ F @ nil_Pr6119456312066593144ropVar )
      = nil_f ) ).

% maps_simps(2)
thf(fact_705_maps__simps_I2_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_P8791008543742759928ropVar] :
      ( ( maps_P2685787764321406401ropVar @ F @ nil_Pr6119456312066593144ropVar )
      = nil_Pr6119456312066593144ropVar ) ).

% maps_simps(2)
thf(fact_706_maps__simps_I2_J,axiom,
    ! [F: produc2642885279816024434ropVar > list_nat] :
      ( ( maps_P2882123423910056385ar_nat @ F @ nil_Pr6119456312066593144ropVar )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_707_maps__simps_I2_J,axiom,
    ! [F: nat > list_f] :
      ( ( maps_nat_f @ F @ nil_nat )
      = nil_f ) ).

% maps_simps(2)
thf(fact_708_maps__simps_I2_J,axiom,
    ! [F: nat > list_P8791008543742759928ropVar] :
      ( ( maps_n6440817060062774047ropVar @ F @ nil_nat )
      = nil_Pr6119456312066593144ropVar ) ).

% maps_simps(2)
thf(fact_709_maps__simps_I2_J,axiom,
    ! [F: nat > list_nat] :
      ( ( maps_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% maps_simps(2)
thf(fact_710_rev__induct,axiom,
    ! [P: list_f > $o,Xs: list_f] :
      ( ( P @ nil_f )
     => ( ! [X3: f,Xs4: list_f] :
            ( ( P @ Xs4 )
           => ( P @ ( append_f @ Xs4 @ ( cons_f @ X3 @ nil_f ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_711_rev__induct,axiom,
    ! [P: list_P8791008543742759928ropVar > $o,Xs: list_P8791008543742759928ropVar] :
      ( ( P @ nil_Pr6119456312066593144ropVar )
     => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] :
            ( ( P @ Xs4 )
           => ( P @ ( append4352292717583037453ropVar @ Xs4 @ ( cons_P1504208554336013608ropVar @ X3 @ nil_Pr6119456312066593144ropVar ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_712_rev__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ( P @ nil_nat )
     => ( ! [X3: nat,Xs4: list_nat] :
            ( ( P @ Xs4 )
           => ( P @ ( append_nat @ Xs4 @ ( cons_nat @ X3 @ nil_nat ) ) ) )
       => ( P @ Xs ) ) ) ).

% rev_induct
thf(fact_713_rev__exhaust,axiom,
    ! [Xs: list_f] :
      ( ( Xs != nil_f )
     => ~ ! [Ys3: list_f,Y4: f] :
            ( Xs
           != ( append_f @ Ys3 @ ( cons_f @ Y4 @ nil_f ) ) ) ) ).

% rev_exhaust
thf(fact_714_rev__exhaust,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ~ ! [Ys3: list_P8791008543742759928ropVar,Y4: produc2642885279816024434ropVar] :
            ( Xs
           != ( append4352292717583037453ropVar @ Ys3 @ ( cons_P1504208554336013608ropVar @ Y4 @ nil_Pr6119456312066593144ropVar ) ) ) ) ).

% rev_exhaust
thf(fact_715_rev__exhaust,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ~ ! [Ys3: list_nat,Y4: nat] :
            ( Xs
           != ( append_nat @ Ys3 @ ( cons_nat @ Y4 @ nil_nat ) ) ) ) ).

% rev_exhaust
thf(fact_716_Cons__eq__append__conv,axiom,
    ! [X2: f,Xs: list_f,Ys: list_f,Zs3: list_f] :
      ( ( ( cons_f @ X2 @ Xs )
        = ( append_f @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_f )
          & ( ( cons_f @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_f] :
            ( ( ( cons_f @ X2 @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_f @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_717_Cons__eq__append__conv,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar] :
      ( ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
        = ( append4352292717583037453ropVar @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_Pr6119456312066593144ropVar )
          & ( ( cons_P1504208554336013608ropVar @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_P8791008543742759928ropVar] :
            ( ( ( cons_P1504208554336013608ropVar @ X2 @ Ys5 )
              = Ys )
            & ( Xs
              = ( append4352292717583037453ropVar @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_718_Cons__eq__append__conv,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( cons_nat @ X2 @ Xs )
        = ( append_nat @ Ys @ Zs3 ) )
      = ( ( ( Ys = nil_nat )
          & ( ( cons_nat @ X2 @ Xs )
            = Zs3 ) )
        | ? [Ys5: list_nat] :
            ( ( ( cons_nat @ X2 @ Ys5 )
              = Ys )
            & ( Xs
              = ( append_nat @ Ys5 @ Zs3 ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_719_append__eq__Cons__conv,axiom,
    ! [Ys: list_f,Zs3: list_f,X2: f,Xs: list_f] :
      ( ( ( append_f @ Ys @ Zs3 )
        = ( cons_f @ X2 @ Xs ) )
      = ( ( ( Ys = nil_f )
          & ( Zs3
            = ( cons_f @ X2 @ Xs ) ) )
        | ? [Ys5: list_f] :
            ( ( Ys
              = ( cons_f @ X2 @ Ys5 ) )
            & ( ( append_f @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_720_append__eq__Cons__conv,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Zs3: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ( append4352292717583037453ropVar @ Ys @ Zs3 )
        = ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
      = ( ( ( Ys = nil_Pr6119456312066593144ropVar )
          & ( Zs3
            = ( cons_P1504208554336013608ropVar @ X2 @ Xs ) ) )
        | ? [Ys5: list_P8791008543742759928ropVar] :
            ( ( Ys
              = ( cons_P1504208554336013608ropVar @ X2 @ Ys5 ) )
            & ( ( append4352292717583037453ropVar @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_721_append__eq__Cons__conv,axiom,
    ! [Ys: list_nat,Zs3: list_nat,X2: nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs3 )
        = ( cons_nat @ X2 @ Xs ) )
      = ( ( ( Ys = nil_nat )
          & ( Zs3
            = ( cons_nat @ X2 @ Xs ) ) )
        | ? [Ys5: list_nat] :
            ( ( Ys
              = ( cons_nat @ X2 @ Ys5 ) )
            & ( ( append_nat @ Ys5 @ Zs3 )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_722_rev__nonempty__induct,axiom,
    ! [Xs: list_f,P: list_f > $o] :
      ( ( Xs != nil_f )
     => ( ! [X3: f] : ( P @ ( cons_f @ X3 @ nil_f ) )
       => ( ! [X3: f,Xs4: list_f] :
              ( ( Xs4 != nil_f )
             => ( ( P @ Xs4 )
               => ( P @ ( append_f @ Xs4 @ ( cons_f @ X3 @ nil_f ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_723_rev__nonempty__induct,axiom,
    ! [Xs: list_P8791008543742759928ropVar,P: list_P8791008543742759928ropVar > $o] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ( ! [X3: produc2642885279816024434ropVar] : ( P @ ( cons_P1504208554336013608ropVar @ X3 @ nil_Pr6119456312066593144ropVar ) )
       => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar] :
              ( ( Xs4 != nil_Pr6119456312066593144ropVar )
             => ( ( P @ Xs4 )
               => ( P @ ( append4352292717583037453ropVar @ Xs4 @ ( cons_P1504208554336013608ropVar @ X3 @ nil_Pr6119456312066593144ropVar ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_724_rev__nonempty__induct,axiom,
    ! [Xs: list_nat,P: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs4: list_nat] :
              ( ( Xs4 != nil_nat )
             => ( ( P @ Xs4 )
               => ( P @ ( append_nat @ Xs4 @ ( cons_nat @ X3 @ nil_nat ) ) ) ) )
         => ( P @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_725_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_726_concat__eq__appendD,axiom,
    ! [Xss: list_list_nat,Ys: list_nat,Zs3: list_nat] :
      ( ( ( concat_nat @ Xss )
        = ( append_nat @ Ys @ Zs3 ) )
     => ( ( Xss != nil_list_nat )
       => ? [Xss12: list_list_nat,Xs4: list_nat,Xs6: list_nat,Xss23: list_list_nat] :
            ( ( Xss
              = ( append_list_nat @ Xss12 @ ( cons_list_nat @ ( append_nat @ Xs4 @ Xs6 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_nat @ ( concat_nat @ Xss12 ) @ Xs4 ) )
            & ( Zs3
              = ( append_nat @ Xs6 @ ( concat_nat @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_727_concat__eq__appendD,axiom,
    ! [Xss: list_list_f,Ys: list_f,Zs3: list_f] :
      ( ( ( concat_f @ Xss )
        = ( append_f @ Ys @ Zs3 ) )
     => ( ( Xss != nil_list_f )
       => ? [Xss12: list_list_f,Xs4: list_f,Xs6: list_f,Xss23: list_list_f] :
            ( ( Xss
              = ( append_list_f @ Xss12 @ ( cons_list_f @ ( append_f @ Xs4 @ Xs6 ) @ Xss23 ) ) )
            & ( Ys
              = ( append_f @ ( concat_f @ Xss12 ) @ Xs4 ) )
            & ( Zs3
              = ( append_f @ Xs6 @ ( concat_f @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_728_List_Obind__def,axiom,
    ( bind_f_f
    = ( ^ [Xs2: list_f,F2: f > list_f] : ( concat_f @ ( map_f_list_f @ F2 @ Xs2 ) ) ) ) ).

% List.bind_def
thf(fact_729_pred__equals__eq2,axiom,
    ! [R: set_Pr6405597471078531282ropVar,S: set_Pr6405597471078531282ropVar] :
      ( ( ( ^ [X: multis3193088007478089820ropVar,Y: produc7679832412844898215ropVar] : ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X @ Y ) @ R ) )
        = ( ^ [X: multis3193088007478089820ropVar,Y: produc7679832412844898215ropVar] : ( member9210975342120487067ropVar @ ( produc6194868611276206818ropVar @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_730_pred__equals__eq2,axiom,
    ! [R: set_Pr7799659934693882759ropVar,S: set_Pr7799659934693882759ropVar] :
      ( ( ( ^ [X: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar] : ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y ) @ R ) )
        = ( ^ [X: multis3193088007478089820ropVar,Y: multis3193088007478089820ropVar] : ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_731_pred__equals__eq2,axiom,
    ! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ( ( ^ [X: nat > nat,Y: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ R ) )
        = ( ^ [X: nat > nat,Y: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_732_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S ) ) )
      = ( R = S ) ) ).

% pred_equals_eq2
thf(fact_733_j__ys2_I2_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys2 ) )
     => ( cs @ ( i_of_j2 @ J ) @ J ) ) ).

% j_ys2(2)
thf(fact_734_j__ys2_I1_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys2 ) )
     => ( member_nat @ ( i_of_j2 @ J ) @ ( set_nat2 @ xs2 ) ) ) ).

% j_ys2(1)
thf(fact_735_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_736_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_737_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_738_Suc__mono,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_739_Suc__less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_less_eq
thf(fact_740_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_741_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_742_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_743_nat_Oinject,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% nat.inject
thf(fact_744_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A2: nat] :
      ( ( A2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A2 ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_745__092_060open_062i__of__j2_A_092_060equiv_062_A_092_060lambda_062j_O_ASOME_Ai_O_Ai_A_092_060in_062_Aset_Axs2_A_092_060and_062_Acs_Ai_Aj_092_060close_062,axiom,
    ( i_of_j2
    = ( ^ [J2: nat] :
          ( fChoice_nat
          @ ^ [I: nat] :
              ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
              & ( cs @ I @ J2 ) ) ) ) ) ).

% \<open>i_of_j2 \<equiv> \<lambda>j. SOME i. i \<in> set xs2 \<and> cs i j\<close>
thf(fact_746_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_747_Suc__inject,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( ( suc @ X2 )
        = ( suc @ Y3 ) )
     => ( X2 = Y3 ) ) ).

% Suc_inject
thf(fact_748_linorder__neqE__nat,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( X2 != Y3 )
     => ( ~ ( ord_less_nat @ X2 @ Y3 )
       => ( ord_less_nat @ Y3 @ X2 ) ) ) ).

% linorder_neqE_nat
thf(fact_749_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
              & ~ ( P @ M3 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_750_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N3 )
             => ( P @ M3 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_751_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_752_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_753_less__not__refl2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( M2 != N2 ) ) ).

% less_not_refl2
thf(fact_754_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_755_nat__neq__iff,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 != N2 )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( ord_less_nat @ N2 @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_756_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_757_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_758_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_759_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_760_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_761_nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( P @ N3 )
           => ( P @ ( suc @ N3 ) ) )
       => ( P @ N2 ) ) ) ).

% nat_induct
thf(fact_762_diff__induct,axiom,
    ! [P: nat > nat > $o,M2: nat,N2: nat] :
      ( ! [X3: nat] : ( P @ X3 @ zero_zero_nat )
     => ( ! [Y4: nat] : ( P @ zero_zero_nat @ ( suc @ Y4 ) )
       => ( ! [X3: nat,Y4: nat] :
              ( ( P @ X3 @ Y4 )
             => ( P @ ( suc @ X3 ) @ ( suc @ Y4 ) ) )
         => ( P @ M2 @ N2 ) ) ) ) ).

% diff_induct
thf(fact_763_zero__induct,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ zero_zero_nat ) ) ) ).

% zero_induct
thf(fact_764_Suc__neq__Zero,axiom,
    ! [M2: nat] :
      ( ( suc @ M2 )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_765_Zero__neq__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_neq_Suc
thf(fact_766_Zero__not__Suc,axiom,
    ! [M2: nat] :
      ( zero_zero_nat
     != ( suc @ M2 ) ) ).

% Zero_not_Suc
thf(fact_767_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M4: nat] :
          ( N2
          = ( suc @ M4 ) ) ) ).

% not0_implies_Suc
thf(fact_768_bot__nat__0_Oextremum__strict,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_769_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_770_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_771_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_772_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_773_gr__implies__not0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_774_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N3 )
                  & ~ ( P @ M3 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_775_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N2 @ M2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
        = ( N2 = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_776_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_777_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P @ I3 @ J3 )
                 => ( ( P @ J3 @ K2 )
                   => ( P @ I3 @ K2 ) ) ) ) )
         => ( P @ I2 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_778_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_779_Suc__less__SucD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_less_SucD
thf(fact_780_less__antisym,axiom,
    ! [N2: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N2 @ M2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M2 ) )
       => ( M2 = N2 ) ) ) ).

% less_antisym
thf(fact_781_Suc__less__eq2,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M2 )
      = ( ? [M5: nat] :
            ( ( M2
              = ( suc @ M5 ) )
            & ( ord_less_nat @ N2 @ M5 ) ) ) ) ).

% Suc_less_eq2
thf(fact_782_All__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
           => ( P @ I ) ) )
      = ( ( P @ N2 )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
           => ( P @ I ) ) ) ) ).

% All_less_Suc
thf(fact_783_not__less__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_784_less__Suc__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M2 @ N2 )
        | ( M2 = N2 ) ) ) ).

% less_Suc_eq
thf(fact_785_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
            & ( P @ I ) ) )
      = ( ( P @ N2 )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
            & ( P @ I ) ) ) ) ).

% Ex_less_Suc
thf(fact_786_less__SucI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ord_less_nat @ M2 @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_787_less__SucE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M2 @ N2 )
       => ( M2 = N2 ) ) ) ).

% less_SucE
thf(fact_788_Suc__lessI,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ( suc @ M2 )
         != N2 )
       => ( ord_less_nat @ ( suc @ M2 ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_789_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I2 @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_790_Suc__lessD,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N2 )
     => ( ord_less_nat @ M2 @ N2 ) ) ).

% Suc_lessD
thf(fact_791_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ K )
     => ( ( K
         != ( suc @ I2 ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I2 @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_792_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M2: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M2 ) )
        = ( ord_less_nat @ N2 @ M2 ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_793_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_794_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
            & ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
            & ( P @ ( suc @ I ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_795_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M6: nat] :
            ( N2
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_796_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( suc @ N2 ) )
           => ( P @ I ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I: nat] :
            ( ( ord_less_nat @ I @ N2 )
           => ( P @ ( suc @ I ) ) ) ) ) ).

% All_less_Suc2
thf(fact_797_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M4: nat] :
          ( N2
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_798_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N2 ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M2
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_799_i__of__j2__def,axiom,
    ! [J: nat] :
      ( ( i_of_j2 @ J )
      = ( fChoice_nat
        @ ^ [I: nat] :
            ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
            & ( cs @ I @ J ) ) ) ) ).

% i_of_j2_def
thf(fact_800_product__code,axiom,
    ! [Xs: list_f,Ys: list_f] :
      ( ( product_product_f_f @ ( set_f2 @ Xs ) @ ( set_f2 @ Ys ) )
      = ( set_Product_prod_f_f2
        @ ( concat1358092678215723560od_f_f
          @ ( map_f_3881494738060473710od_f_f
            @ ^ [X: f] : ( map_f_9172316846603814120od_f_f @ ( product_Pair_f_f @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_801_product__code,axiom,
    ! [Xs: list_f,Ys: list_nat] :
      ( ( produc2985859979987363329_f_nat @ ( set_f2 @ Xs ) @ ( set_nat2 @ Ys ) )
      = ( set_Pr7102205019285007021_f_nat
        @ ( concat3891831345114404865_f_nat
          @ ( map_f_1703457111857288913_f_nat
            @ ^ [X: f] : ( map_na1633831712172314136_f_nat @ ( product_Pair_f_nat @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_802_product__code,axiom,
    ! [Xs: list_f,Ys: list_M6740372356978808802ropVar] :
      ( ( produc8452876027641305707ropVar @ ( set_f2 @ Xs ) @ ( set_Mu146256361055586289ropVar @ Ys ) )
      = ( set_Pr1673467227197847381ropVar
        @ ( concat435164138041048833ropVar
          @ ( map_f_2110070304149036999ropVar
            @ ^ [X: f] : ( map_Mu6485471659106350504ropVar @ ( produc456350693306955696ropVar @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_803_product__code,axiom,
    ! [Xs: list_nat,Ys: list_f] :
      ( ( produc840569836383677279_nat_f @ ( set_nat2 @ Xs ) @ ( set_f2 @ Ys ) )
      = ( set_Pr4163146859742855507_nat_f
        @ ( concat952773185572253351_nat_f
          @ ( map_na5891172834313282510_nat_f
            @ ^ [X: nat] : ( map_f_6874299659675018215_nat_f @ ( product_Pair_nat_f @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_804_product__code,axiom,
    ! [Xs: list_nat,Ys: list_M6740372356978808802ropVar] :
      ( ( produc7744834158543177912ropVar @ ( set_nat2 @ Xs ) @ ( set_Mu146256361055586289ropVar @ Ys ) )
      = ( set_Pr1649027773232293932ropVar
        @ ( concat7022562035907497344ropVar
          @ ( map_na2537330344972504615ropVar
            @ ^ [X: nat] : ( map_Mu6026073089519504153ropVar @ ( produc1720817013404193843ropVar @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_805_product__code,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_f] :
      ( ( produc1913816202967060473pVar_f @ ( set_Mu146256361055586289ropVar @ Xs ) @ ( set_f2 @ Ys ) )
      = ( set_Pr1829365089302688099pVar_f
        @ ( concat591062000145889551pVar_f
          @ ( map_Mu5645816228276465468pVar_f
            @ ^ [X: multis3193088007478089820ropVar] : ( map_f_5353762226182116303pVar_f @ ( produc3140662905487486270pVar_f @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_806_product__code,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_nat] :
      ( ( produc7248793300720743770ar_nat @ ( set_Mu146256361055586289ropVar @ Xs ) @ ( set_nat2 @ Ys ) )
      = ( set_Pr8641712618127411206ar_nat
        @ ( concat4791874843947838810ar_nat
          @ ( map_Mu4852493151219557763ar_nat
            @ ^ [X: multis3193088007478089820ropVar] : ( map_na6525913470057495921ar_nat @ ( produc1224776155581759701ar_nat @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_807_product__code,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( produc6510128442871834962ropVar @ ( set_Mu146256361055586289ropVar @ Xs ) @ ( set_Mu146256361055586289ropVar @ Ys ) )
      = ( set_Pr4075413451802982460ropVar
        @ ( concat461548690737154024ropVar
          @ ( map_Mu5426375011618868629ropVar
            @ ^ [X: multis3193088007478089820ropVar] : ( map_Mu7320851716970821775ropVar @ ( produc6936306168532358295ropVar @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_808_product__code,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( produc929334515565554804at_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) )
      = ( set_Pr5648618587558075414at_nat
        @ ( concat7691415812945658306at_nat
          @ ( map_na4561905831291441265at_nat
            @ ^ [X: nat] : ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_809_product__code,axiom,
    ! [Xs: list_f,Ys: list_list_f] :
      ( ( produc3374443989276135832list_f @ ( set_f2 @ Xs ) @ ( set_list_f2 @ Ys ) )
      = ( set_Pr1611525768661774978list_f
        @ ( concat5266873820675140654list_f
          @ ( map_f_4304301305749555956list_f
            @ ^ [X: f] : ( map_li6773247824414581992list_f @ ( produc4886901243613182557list_f @ X ) @ Ys )
            @ Xs ) ) ) ) ).

% product_code
thf(fact_810_n__lists__Nil,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( n_list6927042657299049856ropVar @ N2 @ nil_Pr6119456312066593144ropVar )
          = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( n_list6927042657299049856ropVar @ N2 @ nil_Pr6119456312066593144ropVar )
          = nil_li2679230897895549182ropVar ) ) ) ).

% n_lists_Nil
thf(fact_811_n__lists__Nil,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( n_lists_nat @ N2 @ nil_nat )
          = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( n_lists_nat @ N2 @ nil_nat )
          = nil_list_nat ) ) ) ).

% n_lists_Nil
thf(fact_812_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc1604256039088728679list_f] :
      ( ! [Ys3: list_f] :
          ( X2
         != ( produc8144748701140958423list_f @ nil_f @ Ys3 ) )
     => ~ ! [X3: f,Xs4: list_f,Ys3: list_f] :
            ( X2
           != ( produc8144748701140958423list_f @ ( cons_f @ X3 @ Xs4 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_813_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc3638977808798937127ropVar] :
      ( ! [Ys3: list_P8791008543742759928ropVar] :
          ( X2
         != ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Ys3 ) )
     => ~ ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Ys3: list_P8791008543742759928ropVar] :
            ( X2
           != ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_814_subset__eq__mset__impl_Ocases,axiom,
    ! [X2: produc1828647624359046049st_nat] :
      ( ! [Ys3: list_nat] :
          ( X2
         != ( produc2694037385005941721st_nat @ nil_nat @ Ys3 ) )
     => ~ ! [X3: nat,Xs4: list_nat,Ys3: list_nat] :
            ( X2
           != ( produc2694037385005941721st_nat @ ( cons_nat @ X3 @ Xs4 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_815_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( n_list6927042657299049856ropVar @ zero_zero_nat @ Xs )
      = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) ).

% n_lists.simps(1)
thf(fact_816_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_nat] :
      ( ( n_lists_nat @ zero_zero_nat @ Xs )
      = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% n_lists.simps(1)
thf(fact_817_the__elem__set,axiom,
    ! [X2: list_f] :
      ( ( the_elem_list_f @ ( set_list_f2 @ ( cons_list_f @ X2 @ nil_list_f ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_818_the__elem__set,axiom,
    ! [X2: multis3193088007478089820ropVar] :
      ( ( the_el4535477323905673297ropVar @ ( set_Mu146256361055586289ropVar @ ( cons_M4277272395645347858ropVar @ X2 @ nil_Mu4132399263956683874ropVar ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_819_the__elem__set,axiom,
    ! [X2: f] :
      ( ( the_elem_f @ ( set_f2 @ ( cons_f @ X2 @ nil_f ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_820_the__elem__set,axiom,
    ! [X2: produc2642885279816024434ropVar] :
      ( ( the_el184889641172005735ropVar @ ( set_Pr3946994337432900359ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_821_the__elem__set,axiom,
    ! [X2: nat] :
      ( ( the_elem_nat @ ( set_nat2 @ ( cons_nat @ X2 @ nil_nat ) ) )
      = X2 ) ).

% the_elem_set
thf(fact_822_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_f,Xss: list_list_f] :
      ( ( product_lists_f @ ( cons_list_f @ Xs @ Xss ) )
      = ( concat_list_f
        @ ( map_f_list_list_f
          @ ^ [X: f] : ( map_list_f_list_f @ ( cons_f @ X ) @ ( product_lists_f @ Xss ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_823_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Xss: list_l3415868495287753214ropVar] :
      ( ( produc2956654850250905953ropVar @ ( cons_l5280711761816980654ropVar @ Xs @ Xss ) )
      = ( concat4944697247486958777ropVar
        @ ( map_Pr879394845742082896ropVar
          @ ^ [X: produc2642885279816024434ropVar] : ( map_li926741104209434052ropVar @ ( cons_P1504208554336013608ropVar @ X ) @ ( produc2956654850250905953ropVar @ Xss ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_824_product__lists_Osimps_I2_J,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( ( product_lists_nat @ ( cons_list_nat @ Xs @ Xss ) )
      = ( concat_list_nat
        @ ( map_na6205611841492582150st_nat
          @ ^ [X: nat] : ( map_li7225945977422193158st_nat @ ( cons_nat @ X ) @ ( product_lists_nat @ Xss ) )
          @ Xs ) ) ) ).

% product_lists.simps(2)
thf(fact_825_upt__rec__numeral,axiom,
    ! [M2: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_826_product__lists_Osimps_I1_J,axiom,
    ( ( produc2956654850250905953ropVar @ nil_li2679230897895549182ropVar )
    = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) ).

% product_lists.simps(1)
thf(fact_827_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_828_n__lists_Osimps_I2_J,axiom,
    ! [N2: nat,Xs: list_f] :
      ( ( n_lists_f @ ( suc @ N2 ) @ Xs )
      = ( concat_list_f
        @ ( map_li8627645367811812490list_f
          @ ^ [Ys2: list_f] :
              ( map_f_list_f
              @ ^ [Y: f] : ( cons_f @ Y @ Ys2 )
              @ Xs )
          @ ( n_lists_f @ N2 @ Xs ) ) ) ) ).

% n_lists.simps(2)
thf(fact_829_n__lists_Osimps_I2_J,axiom,
    ! [N2: nat,Xs: list_P8791008543742759928ropVar] :
      ( ( n_list6927042657299049856ropVar @ ( suc @ N2 ) @ Xs )
      = ( concat4944697247486958777ropVar
        @ ( map_li2030941865990407626ropVar
          @ ^ [Ys2: list_P8791008543742759928ropVar] :
              ( map_Pr8780118474983044682ropVar
              @ ^ [Y: produc2642885279816024434ropVar] : ( cons_P1504208554336013608ropVar @ Y @ Ys2 )
              @ Xs )
          @ ( n_list6927042657299049856ropVar @ N2 @ Xs ) ) ) ) ).

% n_lists.simps(2)
thf(fact_830_n__lists_Osimps_I2_J,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( n_lists_nat @ ( suc @ N2 ) @ Xs )
      = ( concat_list_nat
        @ ( map_li960784813134754710st_nat
          @ ^ [Ys2: list_nat] :
              ( map_nat_list_nat
              @ ^ [Y: nat] : ( cons_nat @ Y @ Ys2 )
              @ Xs )
          @ ( n_lists_nat @ N2 @ Xs ) ) ) ) ).

% n_lists.simps(2)
thf(fact_831_numeral__less__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M2 @ N2 ) ) ).

% numeral_less_iff
thf(fact_832_some__equality,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ( P @ A2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( X3 = A2 ) )
       => ( ( fChoice_nat @ P )
          = A2 ) ) ) ).

% some_equality
thf(fact_833_some__eq__trivial,axiom,
    ! [X2: nat] :
      ( ( fChoice_nat
        @ ^ [Y: nat] : ( Y = X2 ) )
      = X2 ) ).

% some_eq_trivial
thf(fact_834_some__sym__eq__trivial,axiom,
    ! [X2: nat] :
      ( ( fChoice_nat
        @ ( ^ [Y5: nat,Z5: nat] : ( Y5 = Z5 )
          @ X2 ) )
      = X2 ) ).

% some_sym_eq_trivial
thf(fact_835_product__concat__map,axiom,
    ( produc4762506738712241808ropVar
    = ( ^ [Xs2: list_M6740372356978808802ropVar,Ys2: list_P444016040222843053ropVar] :
          ( concat1830633707478507187ropVar
          @ ( map_Mu7917431645227469024ropVar
            @ ^ [X: multis3193088007478089820ropVar] : ( map_Pr6673589034486973647ropVar @ ( produc6194868611276206818ropVar @ X ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_836_product__concat__map,axiom,
    ( produc3968825954260087365ropVar
    = ( ^ [Xs2: list_M6740372356978808802ropVar,Ys2: list_M6740372356978808802ropVar] :
          ( concat461548690737154024ropVar
          @ ( map_Mu5426375011618868629ropVar
            @ ^ [X: multis3193088007478089820ropVar] : ( map_Mu7320851716970821775ropVar @ ( produc6936306168532358295ropVar @ X ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_837_product__concat__map,axiom,
    ( product_nat_nat_nat
    = ( ^ [Xs2: list_nat_nat,Ys2: list_nat] :
          ( concat821400627202602033at_nat
          @ ( map_na1011612288921435825at_nat
            @ ^ [X: nat > nat] : ( map_na6722963242033808602at_nat @ ( produc72220940542539688at_nat @ X ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_838_product__concat__map,axiom,
    ( product_nat_nat
    = ( ^ [Xs2: list_nat,Ys2: list_nat] :
          ( concat7691415812945658306at_nat
          @ ( map_na4561905831291441265at_nat
            @ ^ [X: nat] : ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X ) @ Ys2 )
            @ Xs2 ) ) ) ) ).

% product_concat_map
thf(fact_839_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_840_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_841_numeral__eq__iff,axiom,
    ! [M2: num,N2: num] :
      ( ( ( numeral_numeral_nat @ M2 )
        = ( numeral_numeral_nat @ N2 ) )
      = ( M2 = N2 ) ) ).

% numeral_eq_iff
thf(fact_842_some__eq__imp,axiom,
    ! [P: nat > $o,A2: nat,B: nat] :
      ( ( ( fChoice_nat @ P )
        = A2 )
     => ( ( P @ B )
       => ( P @ A2 ) ) ) ).

% some_eq_imp
thf(fact_843_tfl__some,axiom,
    ! [P5: nat > $o,X4: nat] :
      ( ( P5 @ X4 )
     => ( P5 @ ( fChoice_nat @ P5 ) ) ) ).

% tfl_some
thf(fact_844_Eps__cong,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ! [X3: nat] :
          ( ( P @ X3 )
          = ( Q2 @ X3 ) )
     => ( ( fChoice_nat @ P )
        = ( fChoice_nat @ Q2 ) ) ) ).

% Eps_cong
thf(fact_845_someI,axiom,
    ! [P: nat > $o,X2: nat] :
      ( ( P @ X2 )
     => ( P @ ( fChoice_nat @ P ) ) ) ).

% someI
thf(fact_846_product_Osimps_I1_J,axiom,
    ! [Uu: list_P444016040222843053ropVar] :
      ( ( produc4762506738712241808ropVar @ nil_Mu4132399263956683874ropVar @ Uu )
      = nil_Pr6119456312066593144ropVar ) ).

% product.simps(1)
thf(fact_847_some1__equality,axiom,
    ! [P: nat > $o,A2: nat] :
      ( ? [X4: nat] :
          ( ( P @ X4 )
          & ! [Y4: nat] :
              ( ( P @ Y4 )
             => ( Y4 = X4 ) ) )
     => ( ( P @ A2 )
       => ( ( fChoice_nat @ P )
          = A2 ) ) ) ).

% some1_equality
thf(fact_848_some__eq__ex,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X5: nat] : ( P @ X5 ) ) ) ).

% some_eq_ex
thf(fact_849_someI2__bex,axiom,
    ! [A: set_Mu5658373669842198716ropVar,P: multis3193088007478089820ropVar > $o,Q2: multis3193088007478089820ropVar > $o] :
      ( ? [X4: multis3193088007478089820ropVar] :
          ( ( member8421201611525043589ropVar @ X4 @ A )
          & ( P @ X4 ) )
     => ( ! [X3: multis3193088007478089820ropVar] :
            ( ( ( member8421201611525043589ropVar @ X3 @ A )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoic539809714188224432ropVar
            @ ^ [X: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_850_someI2__bex,axiom,
    ! [A: set_f,P: f > $o,Q2: f > $o] :
      ( ? [X4: f] :
          ( ( member_f @ X4 @ A )
          & ( P @ X4 ) )
     => ( ! [X3: f] :
            ( ( ( member_f @ X3 @ A )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoice_f
            @ ^ [X: f] :
                ( ( member_f @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_851_someI2__bex,axiom,
    ! [A: set_nat,P: nat > $o,Q2: nat > $o] :
      ( ? [X4: nat] :
          ( ( member_nat @ X4 @ A )
          & ( P @ X4 ) )
     => ( ! [X3: nat] :
            ( ( ( member_nat @ X3 @ A )
              & ( P @ X3 ) )
           => ( Q2 @ X3 ) )
       => ( Q2
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A )
                & ( P @ X ) ) ) ) ) ) ).

% someI2_bex
thf(fact_852_someI2__ex,axiom,
    ! [P: nat > $o,Q2: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( fChoice_nat @ P ) ) ) ) ).

% someI2_ex
thf(fact_853_someI__ex,axiom,
    ! [P: nat > $o] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( P @ ( fChoice_nat @ P ) ) ) ).

% someI_ex
thf(fact_854_someI2,axiom,
    ! [P: nat > $o,A2: nat,Q2: nat > $o] :
      ( ( P @ A2 )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
           => ( Q2 @ X3 ) )
       => ( Q2 @ ( fChoice_nat @ P ) ) ) ) ).

% someI2
thf(fact_855_product_Osimps_I2_J,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_P444016040222843053ropVar] :
      ( ( produc4762506738712241808ropVar @ ( cons_M4277272395645347858ropVar @ X2 @ Xs ) @ Ys )
      = ( append4352292717583037453ropVar @ ( map_Pr6673589034486973647ropVar @ ( produc6194868611276206818ropVar @ X2 ) @ Ys ) @ ( produc4762506738712241808ropVar @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_856_product_Osimps_I2_J,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( produc3968825954260087365ropVar @ ( cons_M4277272395645347858ropVar @ X2 @ Xs ) @ Ys )
      = ( append5590948776005684034ropVar @ ( map_Mu7320851716970821775ropVar @ ( produc6936306168532358295ropVar @ X2 ) @ Ys ) @ ( produc3968825954260087365ropVar @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_857_product_Osimps_I2_J,axiom,
    ! [X2: nat > nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( product_nat_nat_nat @ ( cons_nat_nat @ X2 @ Xs ) @ Ys )
      = ( append8880802010960057739at_nat @ ( map_na6722963242033808602at_nat @ ( produc72220940542539688at_nat @ X2 ) @ Ys ) @ ( product_nat_nat_nat @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_858_product_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat] :
      ( ( product_nat_nat @ ( cons_nat @ X2 @ Xs ) @ Ys )
      = ( append985823374593552924at_nat @ ( map_na7298421622053143531at_nat @ ( product_Pair_nat_nat @ X2 ) @ Ys ) @ ( product_nat_nat @ Xs @ Ys ) ) ) ).

% product.simps(2)
thf(fact_859_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_860_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_861_subseqs_Osimps_I2_J,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( subseqs_f @ ( cons_f @ X2 @ Xs ) )
      = ( append_list_f @ ( map_list_f_list_f @ ( cons_f @ X2 ) @ ( subseqs_f @ Xs ) ) @ ( subseqs_f @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_862_subseqs_Osimps_I2_J,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( subseq3825174504735880391ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
      = ( append4316637162825710867ropVar @ ( map_li926741104209434052ropVar @ ( cons_P1504208554336013608ropVar @ X2 ) @ ( subseq3825174504735880391ropVar @ Xs ) ) @ ( subseq3825174504735880391ropVar @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_863_subseqs_Osimps_I2_J,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( subseqs_nat @ ( cons_nat @ X2 @ Xs ) )
      = ( append_list_nat @ ( map_li7225945977422193158st_nat @ ( cons_nat @ X2 ) @ ( subseqs_nat @ Xs ) ) @ ( subseqs_nat @ Xs ) ) ) ).

% subseqs.simps(2)
thf(fact_864_zero__notin__Suc__image,axiom,
    ! [A: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A ) ) ).

% zero_notin_Suc_image
thf(fact_865_subseqs__refl,axiom,
    ! [Xs: list_f] : ( member_list_f @ Xs @ ( set_list_f2 @ ( subseqs_f @ Xs ) ) ) ).

% subseqs_refl
thf(fact_866_Cons__in__subseqsD,axiom,
    ! [Y3: f,Ys: list_f,Xs: list_f] :
      ( ( member_list_f @ ( cons_f @ Y3 @ Ys ) @ ( set_list_f2 @ ( subseqs_f @ Xs ) ) )
     => ( member_list_f @ Ys @ ( set_list_f2 @ ( subseqs_f @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_867_Cons__in__subseqsD,axiom,
    ! [Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member6236339133593825953ropVar @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) @ ( set_li6509544621384488717ropVar @ ( subseq3825174504735880391ropVar @ Xs ) ) )
     => ( member6236339133593825953ropVar @ Ys @ ( set_li6509544621384488717ropVar @ ( subseq3825174504735880391ropVar @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_868_Cons__in__subseqsD,axiom,
    ! [Y3: nat,Ys: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ ( cons_nat @ Y3 @ Ys ) @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_869_subseqs_Osimps_I1_J,axiom,
    ( ( subseq3825174504735880391ropVar @ nil_Pr6119456312066593144ropVar )
    = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ nil_li2679230897895549182ropVar ) ) ).

% subseqs.simps(1)
thf(fact_870_subseqs_Osimps_I1_J,axiom,
    ( ( subseqs_nat @ nil_nat )
    = ( cons_list_nat @ nil_nat @ nil_list_nat ) ) ).

% subseqs.simps(1)
thf(fact_871_inf__concat__simple_Ocases,axiom,
    ! [X2: produc8199716216217303280at_nat] :
      ( ! [F3: nat > nat] :
          ( X2
         != ( produc72220940542539688at_nat @ F3 @ zero_zero_nat ) )
     => ~ ! [F3: nat > nat,N3: nat] :
            ( X2
           != ( produc72220940542539688at_nat @ F3 @ ( suc @ N3 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_872_transpose_Osimps_I3_J,axiom,
    ! [X2: f,Xs: list_f,Xss: list_list_f] :
      ( ( transpose_f @ ( cons_list_f @ ( cons_f @ X2 @ Xs ) @ Xss ) )
      = ( cons_list_f
        @ ( cons_f @ X2
          @ ( concat_f
            @ ( map_list_f_list_f
              @ ( case_list_list_f_f @ nil_f
                @ ^ [H: f,T2: list_f] : ( cons_f @ H @ nil_f ) )
              @ Xss ) ) )
        @ ( transpose_f
          @ ( cons_list_f @ Xs
            @ ( concat_list_f
              @ ( map_li8627645367811812490list_f
                @ ( case_l3215012710394848538st_f_f @ nil_list_f
                  @ ^ [H: f,T2: list_f] : ( cons_list_f @ T2 @ nil_list_f ) )
                @ Xss ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_873_transpose_Osimps_I3_J,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Xss: list_l3415868495287753214ropVar] :
      ( ( transp9078802294292144386ropVar @ ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ Xss ) )
      = ( cons_l5280711761816980654ropVar
        @ ( cons_P1504208554336013608ropVar @ X2
          @ ( concat1830633707478507187ropVar
            @ ( map_li926741104209434052ropVar
              @ ( case_l7238206891529332192ropVar @ nil_Pr6119456312066593144ropVar
                @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_P1504208554336013608ropVar @ H @ nil_Pr6119456312066593144ropVar ) )
              @ Xss ) ) )
        @ ( transp9078802294292144386ropVar
          @ ( cons_l5280711761816980654ropVar @ Xs
            @ ( concat4944697247486958777ropVar
              @ ( map_li2030941865990407626ropVar
                @ ( case_l7660754118040321498ropVar @ nil_li2679230897895549182ropVar
                  @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_l5280711761816980654ropVar @ T2 @ nil_li2679230897895549182ropVar ) )
                @ Xss ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_874_transpose_Osimps_I3_J,axiom,
    ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
      ( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) )
      = ( cons_list_nat
        @ ( cons_nat @ X2
          @ ( concat_nat
            @ ( map_li7225945977422193158st_nat
              @ ( case_l2340614614379431832at_nat @ nil_nat
                @ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
              @ Xss ) ) )
        @ ( transpose_nat
          @ ( cons_list_nat @ Xs
            @ ( concat_list_nat
              @ ( map_li960784813134754710st_nat
                @ ( case_l3331202209248957608at_nat @ nil_list_nat
                  @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                @ Xss ) ) ) ) ) ) ).

% transpose.simps(3)
thf(fact_875_transpose_Oelims,axiom,
    ! [X2: list_list_f,Y3: list_list_f] :
      ( ( ( transpose_f @ X2 )
        = Y3 )
     => ( ( ( X2 = nil_list_f )
         => ( Y3 != nil_list_f ) )
       => ( ! [Xss2: list_list_f] :
              ( ( X2
                = ( cons_list_f @ nil_f @ Xss2 ) )
             => ( Y3
               != ( transpose_f @ Xss2 ) ) )
         => ~ ! [X3: f,Xs4: list_f,Xss2: list_list_f] :
                ( ( X2
                  = ( cons_list_f @ ( cons_f @ X3 @ Xs4 ) @ Xss2 ) )
               => ( Y3
                 != ( cons_list_f
                    @ ( cons_f @ X3
                      @ ( concat_f
                        @ ( map_list_f_list_f
                          @ ( case_list_list_f_f @ nil_f
                            @ ^ [H: f,T2: list_f] : ( cons_f @ H @ nil_f ) )
                          @ Xss2 ) ) )
                    @ ( transpose_f
                      @ ( cons_list_f @ Xs4
                        @ ( concat_list_f
                          @ ( map_li8627645367811812490list_f
                            @ ( case_l3215012710394848538st_f_f @ nil_list_f
                              @ ^ [H: f,T2: list_f] : ( cons_list_f @ T2 @ nil_list_f ) )
                            @ Xss2 ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_876_transpose_Oelims,axiom,
    ! [X2: list_l3415868495287753214ropVar,Y3: list_l3415868495287753214ropVar] :
      ( ( ( transp9078802294292144386ropVar @ X2 )
        = Y3 )
     => ( ( ( X2 = nil_li2679230897895549182ropVar )
         => ( Y3 != nil_li2679230897895549182ropVar ) )
       => ( ! [Xss2: list_l3415868495287753214ropVar] :
              ( ( X2
                = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss2 ) )
             => ( Y3
               != ( transp9078802294292144386ropVar @ Xss2 ) ) )
         => ~ ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Xss2: list_l3415868495287753214ropVar] :
                ( ( X2
                  = ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ Xss2 ) )
               => ( Y3
                 != ( cons_l5280711761816980654ropVar
                    @ ( cons_P1504208554336013608ropVar @ X3
                      @ ( concat1830633707478507187ropVar
                        @ ( map_li926741104209434052ropVar
                          @ ( case_l7238206891529332192ropVar @ nil_Pr6119456312066593144ropVar
                            @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_P1504208554336013608ropVar @ H @ nil_Pr6119456312066593144ropVar ) )
                          @ Xss2 ) ) )
                    @ ( transp9078802294292144386ropVar
                      @ ( cons_l5280711761816980654ropVar @ Xs4
                        @ ( concat4944697247486958777ropVar
                          @ ( map_li2030941865990407626ropVar
                            @ ( case_l7660754118040321498ropVar @ nil_li2679230897895549182ropVar
                              @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_l5280711761816980654ropVar @ T2 @ nil_li2679230897895549182ropVar ) )
                            @ Xss2 ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_877_transpose_Oelims,axiom,
    ! [X2: list_list_nat,Y3: list_list_nat] :
      ( ( ( transpose_nat @ X2 )
        = Y3 )
     => ( ( ( X2 = nil_list_nat )
         => ( Y3 != nil_list_nat ) )
       => ( ! [Xss2: list_list_nat] :
              ( ( X2
                = ( cons_list_nat @ nil_nat @ Xss2 ) )
             => ( Y3
               != ( transpose_nat @ Xss2 ) ) )
         => ~ ! [X3: nat,Xs4: list_nat,Xss2: list_list_nat] :
                ( ( X2
                  = ( cons_list_nat @ ( cons_nat @ X3 @ Xs4 ) @ Xss2 ) )
               => ( Y3
                 != ( cons_list_nat
                    @ ( cons_nat @ X3
                      @ ( concat_nat
                        @ ( map_li7225945977422193158st_nat
                          @ ( case_l2340614614379431832at_nat @ nil_nat
                            @ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
                          @ Xss2 ) ) )
                    @ ( transpose_nat
                      @ ( cons_list_nat @ Xs4
                        @ ( concat_list_nat
                          @ ( map_li960784813134754710st_nat
                            @ ( case_l3331202209248957608at_nat @ nil_list_nat
                              @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                            @ Xss2 ) ) ) ) ) ) ) ) ) ) ).

% transpose.elims
thf(fact_878_nths__Cons,axiom,
    ! [X2: f,L: list_f,A: set_nat] :
      ( ( nths_f @ ( cons_f @ X2 @ L ) @ A )
      = ( append_f @ ( if_list_f @ ( member_nat @ zero_zero_nat @ A ) @ ( cons_f @ X2 @ nil_f ) @ nil_f )
        @ ( nths_f @ L
          @ ( collect_nat
            @ ^ [J2: nat] : ( member_nat @ ( suc @ J2 ) @ A ) ) ) ) ) ).

% nths_Cons
thf(fact_879_nths__Cons,axiom,
    ! [X2: produc2642885279816024434ropVar,L: list_P8791008543742759928ropVar,A: set_nat] :
      ( ( nths_P6330350825811437174ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ L ) @ A )
      = ( append4352292717583037453ropVar @ ( if_lis1975434623467068862ropVar @ ( member_nat @ zero_zero_nat @ A ) @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) @ nil_Pr6119456312066593144ropVar )
        @ ( nths_P6330350825811437174ropVar @ L
          @ ( collect_nat
            @ ^ [J2: nat] : ( member_nat @ ( suc @ J2 ) @ A ) ) ) ) ) ).

% nths_Cons
thf(fact_880_nths__Cons,axiom,
    ! [X2: nat,L: list_nat,A: set_nat] :
      ( ( nths_nat @ ( cons_nat @ X2 @ L ) @ A )
      = ( append_nat @ ( if_list_nat @ ( member_nat @ zero_zero_nat @ A ) @ ( cons_nat @ X2 @ nil_nat ) @ nil_nat )
        @ ( nths_nat @ L
          @ ( collect_nat
            @ ^ [J2: nat] : ( member_nat @ ( suc @ J2 ) @ A ) ) ) ) ) ).

% nths_Cons
thf(fact_881_map__rec,axiom,
    ( map_na2466558649495876678ropVar
    = ( ^ [F2: nat > multis3193088007478089820ropVar] :
          ( rec_li3635176538394059004ar_nat @ nil_Mu4132399263956683874ropVar
          @ ^ [X: nat,Uu2: list_nat] : ( cons_M4277272395645347858ropVar @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_882_map__rec,axiom,
    ( map_f_4134023513460405853ropVar
    = ( ^ [F2: f > multis3193088007478089820ropVar] :
          ( rec_li7324671433992849623pVar_f @ nil_Mu4132399263956683874ropVar
          @ ^ [X: f,Uu2: list_f] : ( cons_M4277272395645347858ropVar @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_883_map__rec,axiom,
    ( map_f_list_f
    = ( ^ [F2: f > list_f] :
          ( rec_li320335195676633578st_f_f @ nil_list_f
          @ ^ [X: f,Uu2: list_f] : ( cons_list_f @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_884_map__rec,axiom,
    ( map_f_f
    = ( ^ [F2: f > f] :
          ( rec_list_list_f_f @ nil_f
          @ ^ [X: f,Uu2: list_f] : ( cons_f @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_885_map__rec,axiom,
    ( map_na2145702811007361372ropVar
    = ( ^ [F2: nat > produc2642885279816024434ropVar] :
          ( rec_li2530905255354511122ar_nat @ nil_Pr6119456312066593144ropVar
          @ ^ [X: nat,Uu2: list_nat] : ( cons_P1504208554336013608ropVar @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_886_map__rec,axiom,
    ( map_f_2905954635146393971ropVar
    = ( ^ [F2: f > produc2642885279816024434ropVar] :
          ( rec_li5971325897304429633pVar_f @ nil_Pr6119456312066593144ropVar
          @ ^ [X: f,Uu2: list_f] : ( cons_P1504208554336013608ropVar @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_887_map__rec,axiom,
    ( map_nat_nat
    = ( ^ [F2: nat > nat] :
          ( rec_li7516600145284979816at_nat @ nil_nat
          @ ^ [X: nat,Uu2: list_nat] : ( cons_nat @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_888_map__rec,axiom,
    ( map_f_nat
    = ( ^ [F2: f > nat] :
          ( rec_list_list_nat_f @ nil_nat
          @ ^ [X: f,Uu2: list_f] : ( cons_nat @ ( F2 @ X ) ) ) ) ) ).

% map_rec
thf(fact_889_in__measures_I2_J,axiom,
    ! [X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,F: multis3193088007478089820ropVar > nat,Fs: list_M2713250183505059747ar_nat] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ ( measur2342621812062629212ropVar @ ( cons_M4455541548838542291ar_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
        | ( ( ( F @ X2 )
            = ( F @ Y3 ) )
          & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ ( measur2342621812062629212ropVar @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_890_in__measures_I2_J,axiom,
    ! [X2: nat,Y3: nat,F: nat > nat,Fs: list_nat_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
        | ( ( ( F @ X2 )
            = ( F @ Y3 ) )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measures_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_891_transpose_Opinduct,axiom,
    ! [A0: list_list_f,P: list_list_f > $o] :
      ( ( accp_list_list_f @ transpose_rel_f @ A0 )
     => ( ( ( accp_list_list_f @ transpose_rel_f @ nil_list_f )
         => ( P @ nil_list_f ) )
       => ( ! [Xss2: list_list_f] :
              ( ( accp_list_list_f @ transpose_rel_f @ ( cons_list_f @ nil_f @ Xss2 ) )
             => ( ( P @ Xss2 )
               => ( P @ ( cons_list_f @ nil_f @ Xss2 ) ) ) )
         => ( ! [X3: f,Xs4: list_f,Xss2: list_list_f] :
                ( ( accp_list_list_f @ transpose_rel_f @ ( cons_list_f @ ( cons_f @ X3 @ Xs4 ) @ Xss2 ) )
               => ( ( P
                    @ ( cons_list_f @ Xs4
                      @ ( concat_list_f
                        @ ( map_li8627645367811812490list_f
                          @ ( case_l3215012710394848538st_f_f @ nil_list_f
                            @ ^ [H: f,T2: list_f] : ( cons_list_f @ T2 @ nil_list_f ) )
                          @ Xss2 ) ) ) )
                 => ( P @ ( cons_list_f @ ( cons_f @ X3 @ Xs4 ) @ Xss2 ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_892_transpose_Opinduct,axiom,
    ! [A0: list_l3415868495287753214ropVar,P: list_l3415868495287753214ropVar > $o] :
      ( ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ A0 )
     => ( ( ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ nil_li2679230897895549182ropVar )
         => ( P @ nil_li2679230897895549182ropVar ) )
       => ( ! [Xss2: list_l3415868495287753214ropVar] :
              ( ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss2 ) )
             => ( ( P @ Xss2 )
               => ( P @ ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss2 ) ) ) )
         => ( ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Xss2: list_l3415868495287753214ropVar] :
                ( ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ Xss2 ) )
               => ( ( P
                    @ ( cons_l5280711761816980654ropVar @ Xs4
                      @ ( concat4944697247486958777ropVar
                        @ ( map_li2030941865990407626ropVar
                          @ ( case_l7660754118040321498ropVar @ nil_li2679230897895549182ropVar
                            @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_l5280711761816980654ropVar @ T2 @ nil_li2679230897895549182ropVar ) )
                          @ Xss2 ) ) ) )
                 => ( P @ ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ Xss2 ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_893_transpose_Opinduct,axiom,
    ! [A0: list_list_nat,P: list_list_nat > $o] :
      ( ( accp_list_list_nat @ transpose_rel_nat @ A0 )
     => ( ( ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat )
         => ( P @ nil_list_nat ) )
       => ( ! [Xss2: list_list_nat] :
              ( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) )
             => ( ( P @ Xss2 )
               => ( P @ ( cons_list_nat @ nil_nat @ Xss2 ) ) ) )
         => ( ! [X3: nat,Xs4: list_nat,Xss2: list_list_nat] :
                ( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs4 ) @ Xss2 ) )
               => ( ( P
                    @ ( cons_list_nat @ Xs4
                      @ ( concat_list_nat
                        @ ( map_li960784813134754710st_nat
                          @ ( case_l3331202209248957608at_nat @ nil_list_nat
                            @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                          @ Xss2 ) ) ) )
                 => ( P @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs4 ) @ Xss2 ) ) ) )
           => ( P @ A0 ) ) ) ) ) ).

% transpose.pinduct
thf(fact_894_nths__nil,axiom,
    ! [A: set_nat] :
      ( ( nths_P6330350825811437174ropVar @ nil_Pr6119456312066593144ropVar @ A )
      = nil_Pr6119456312066593144ropVar ) ).

% nths_nil
thf(fact_895_nths__nil,axiom,
    ! [A: set_nat] :
      ( ( nths_nat @ nil_nat @ A )
      = nil_nat ) ).

% nths_nil
thf(fact_896_in__measures_I1_J,axiom,
    ! [X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar] :
      ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ ( measur2342621812062629212ropVar @ nil_Mu330590394950748195ar_nat ) ) ).

% in_measures(1)
thf(fact_897_in__measures_I1_J,axiom,
    ! [X2: nat,Y3: nat] :
      ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measures_nat @ nil_nat_nat ) ) ).

% in_measures(1)
thf(fact_898_nths__singleton,axiom,
    ! [A: set_nat,X2: f] :
      ( ( ( member_nat @ zero_zero_nat @ A )
       => ( ( nths_f @ ( cons_f @ X2 @ nil_f ) @ A )
          = ( cons_f @ X2 @ nil_f ) ) )
      & ( ~ ( member_nat @ zero_zero_nat @ A )
       => ( ( nths_f @ ( cons_f @ X2 @ nil_f ) @ A )
          = nil_f ) ) ) ).

% nths_singleton
thf(fact_899_nths__singleton,axiom,
    ! [A: set_nat,X2: produc2642885279816024434ropVar] :
      ( ( ( member_nat @ zero_zero_nat @ A )
       => ( ( nths_P6330350825811437174ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) @ A )
          = ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) )
      & ( ~ ( member_nat @ zero_zero_nat @ A )
       => ( ( nths_P6330350825811437174ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) @ A )
          = nil_Pr6119456312066593144ropVar ) ) ) ).

% nths_singleton
thf(fact_900_nths__singleton,axiom,
    ! [A: set_nat,X2: nat] :
      ( ( ( member_nat @ zero_zero_nat @ A )
       => ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A )
          = ( cons_nat @ X2 @ nil_nat ) ) )
      & ( ~ ( member_nat @ zero_zero_nat @ A )
       => ( ( nths_nat @ ( cons_nat @ X2 @ nil_nat ) @ A )
          = nil_nat ) ) ) ).

% nths_singleton
thf(fact_901_transpose_Opsimps_I2_J,axiom,
    ! [Xss: list_l3415868495287753214ropVar] :
      ( ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss ) )
     => ( ( transp9078802294292144386ropVar @ ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss ) )
        = ( transp9078802294292144386ropVar @ Xss ) ) ) ).

% transpose.psimps(2)
thf(fact_902_transpose_Opsimps_I2_J,axiom,
    ! [Xss: list_list_nat] :
      ( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
     => ( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
        = ( transpose_nat @ Xss ) ) ) ).

% transpose.psimps(2)
thf(fact_903_in__set__nthsD,axiom,
    ! [X2: f,Xs: list_f,I4: set_nat] :
      ( ( member_f @ X2 @ ( set_f2 @ ( nths_f @ Xs @ I4 ) ) )
     => ( member_f @ X2 @ ( set_f2 @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_904_in__set__nthsD,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,I4: set_nat] :
      ( ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ ( nths_P6330350825811437174ropVar @ Xs @ I4 ) ) )
     => ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_905_in__set__nthsD,axiom,
    ! [X2: nat,Xs: list_nat,I4: set_nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) )
     => ( member_nat @ X2 @ ( set_nat2 @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_906_in__set__nthsD,axiom,
    ! [X2: list_f,Xs: list_list_f,I4: set_nat] :
      ( ( member_list_f @ X2 @ ( set_list_f2 @ ( nths_list_f @ Xs @ I4 ) ) )
     => ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_907_in__set__nthsD,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,I4: set_nat] :
      ( ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ ( nths_M3372179620484912480ropVar @ Xs @ I4 ) ) )
     => ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) ) ) ).

% in_set_nthsD
thf(fact_908_notin__set__nthsI,axiom,
    ! [X2: f,Xs: list_f,I4: set_nat] :
      ( ~ ( member_f @ X2 @ ( set_f2 @ Xs ) )
     => ~ ( member_f @ X2 @ ( set_f2 @ ( nths_f @ Xs @ I4 ) ) ) ) ).

% notin_set_nthsI
thf(fact_909_notin__set__nthsI,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,I4: set_nat] :
      ( ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ Xs ) )
     => ~ ( member9210975342120487067ropVar @ X2 @ ( set_Pr3946994337432900359ropVar @ ( nths_P6330350825811437174ropVar @ Xs @ I4 ) ) ) ) ).

% notin_set_nthsI
thf(fact_910_notin__set__nthsI,axiom,
    ! [X2: nat,Xs: list_nat,I4: set_nat] :
      ( ~ ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
     => ~ ( member_nat @ X2 @ ( set_nat2 @ ( nths_nat @ Xs @ I4 ) ) ) ) ).

% notin_set_nthsI
thf(fact_911_notin__set__nthsI,axiom,
    ! [X2: list_f,Xs: list_list_f,I4: set_nat] :
      ( ~ ( member_list_f @ X2 @ ( set_list_f2 @ Xs ) )
     => ~ ( member_list_f @ X2 @ ( set_list_f2 @ ( nths_list_f @ Xs @ I4 ) ) ) ) ).

% notin_set_nthsI
thf(fact_912_notin__set__nthsI,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,I4: set_nat] :
      ( ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ Xs ) )
     => ~ ( member8421201611525043589ropVar @ X2 @ ( set_Mu146256361055586289ropVar @ ( nths_M3372179620484912480ropVar @ Xs @ I4 ) ) ) ) ).

% notin_set_nthsI
thf(fact_913_nths__map,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_nat,I4: set_nat] :
      ( ( nths_P6330350825811437174ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) @ I4 )
      = ( map_na2145702811007361372ropVar @ F @ ( nths_nat @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_914_nths__map,axiom,
    ! [F: nat > nat,Xs: list_nat,I4: set_nat] :
      ( ( nths_nat @ ( map_nat_nat @ F @ Xs ) @ I4 )
      = ( map_nat_nat @ F @ ( nths_nat @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_915_nths__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_nat,I4: set_nat] :
      ( ( nths_M3372179620484912480ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) @ I4 )
      = ( map_na2466558649495876678ropVar @ F @ ( nths_nat @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_916_nths__map,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_f,I4: set_nat] :
      ( ( nths_M3372179620484912480ropVar @ ( map_f_4134023513460405853ropVar @ F @ Xs ) @ I4 )
      = ( map_f_4134023513460405853ropVar @ F @ ( nths_f @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_917_nths__map,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_f,I4: set_nat] :
      ( ( nths_P6330350825811437174ropVar @ ( map_f_2905954635146393971ropVar @ F @ Xs ) @ I4 )
      = ( map_f_2905954635146393971ropVar @ F @ ( nths_f @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_918_nths__map,axiom,
    ! [F: f > list_f,Xs: list_f,I4: set_nat] :
      ( ( nths_list_f @ ( map_f_list_f @ F @ Xs ) @ I4 )
      = ( map_f_list_f @ F @ ( nths_f @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_919_nths__map,axiom,
    ! [F: f > nat,Xs: list_f,I4: set_nat] :
      ( ( nths_nat @ ( map_f_nat @ F @ Xs ) @ I4 )
      = ( map_f_nat @ F @ ( nths_f @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_920_nths__map,axiom,
    ! [F: f > f,Xs: list_f,I4: set_nat] :
      ( ( nths_f @ ( map_f_f @ F @ Xs ) @ I4 )
      = ( map_f_f @ F @ ( nths_f @ Xs @ I4 ) ) ) ).

% nths_map
thf(fact_921_list_Odisc__eq__case_I2_J,axiom,
    ! [List: list_P8791008543742759928ropVar] :
      ( ( List != nil_Pr6119456312066593144ropVar )
      = ( case_l5945071740722813248ropVar @ $false
        @ ^ [Uu2: produc2642885279816024434ropVar,Uv: list_P8791008543742759928ropVar] : $true
        @ List ) ) ).

% list.disc_eq_case(2)
thf(fact_922_list_Odisc__eq__case_I2_J,axiom,
    ! [List: list_nat] :
      ( ( List != nil_nat )
      = ( case_list_o_nat @ $false
        @ ^ [Uu2: nat,Uv: list_nat] : $true
        @ List ) ) ).

% list.disc_eq_case(2)
thf(fact_923_list_Odisc__eq__case_I1_J,axiom,
    ! [List: list_P8791008543742759928ropVar] :
      ( ( List = nil_Pr6119456312066593144ropVar )
      = ( case_l5945071740722813248ropVar @ $true
        @ ^ [Uu2: produc2642885279816024434ropVar,Uv: list_P8791008543742759928ropVar] : $false
        @ List ) ) ).

% list.disc_eq_case(1)
thf(fact_924_list_Odisc__eq__case_I1_J,axiom,
    ! [List: list_nat] :
      ( ( List = nil_nat )
      = ( case_list_o_nat @ $true
        @ ^ [Uu2: nat,Uv: list_nat] : $false
        @ List ) ) ).

% list.disc_eq_case(1)
thf(fact_925_transpose_Osimps_I2_J,axiom,
    ! [Xss: list_l3415868495287753214ropVar] :
      ( ( transp9078802294292144386ropVar @ ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss ) )
      = ( transp9078802294292144386ropVar @ Xss ) ) ).

% transpose.simps(2)
thf(fact_926_transpose_Osimps_I2_J,axiom,
    ! [Xss: list_list_nat] :
      ( ( transpose_nat @ ( cons_list_nat @ nil_nat @ Xss ) )
      = ( transpose_nat @ Xss ) ) ).

% transpose.simps(2)
thf(fact_927_transpose__map__map,axiom,
    ! [F: nat > produc2642885279816024434ropVar,Xs: list_list_nat] :
      ( ( transp9078802294292144386ropVar @ ( map_li4979107708160704082ropVar @ ( map_na2145702811007361372ropVar @ F ) @ Xs ) )
      = ( map_li4979107708160704082ropVar @ ( map_na2145702811007361372ropVar @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_928_transpose__map__map,axiom,
    ! [F: nat > nat,Xs: list_list_nat] :
      ( ( transpose_nat @ ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ Xs ) )
      = ( map_li7225945977422193158st_nat @ ( map_nat_nat @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_929_transpose__map__map,axiom,
    ! [F: nat > multis3193088007478089820ropVar,Xs: list_list_nat] :
      ( ( transp2038539320897558508ropVar @ ( map_li5946103070670478908ropVar @ ( map_na2466558649495876678ropVar @ F ) @ Xs ) )
      = ( map_li5946103070670478908ropVar @ ( map_na2466558649495876678ropVar @ F ) @ ( transpose_nat @ Xs ) ) ) ).

% transpose_map_map
thf(fact_930_transpose__map__map,axiom,
    ! [F: f > multis3193088007478089820ropVar,Xs: list_list_f] :
      ( ( transp2038539320897558508ropVar @ ( map_li1265598969789384029ropVar @ ( map_f_4134023513460405853ropVar @ F ) @ Xs ) )
      = ( map_li1265598969789384029ropVar @ ( map_f_4134023513460405853ropVar @ F ) @ ( transpose_f @ Xs ) ) ) ).

% transpose_map_map
thf(fact_931_transpose__map__map,axiom,
    ! [F: f > produc2642885279816024434ropVar,Xs: list_list_f] :
      ( ( transp9078802294292144386ropVar @ ( map_li676030948805672819ropVar @ ( map_f_2905954635146393971ropVar @ F ) @ Xs ) )
      = ( map_li676030948805672819ropVar @ ( map_f_2905954635146393971ropVar @ F ) @ ( transpose_f @ Xs ) ) ) ).

% transpose_map_map
thf(fact_932_transpose__map__map,axiom,
    ! [F: f > list_f,Xs: list_list_f] :
      ( ( transpose_list_f @ ( map_li8627645367811812490list_f @ ( map_f_list_f @ F ) @ Xs ) )
      = ( map_li8627645367811812490list_f @ ( map_f_list_f @ F ) @ ( transpose_f @ Xs ) ) ) ).

% transpose_map_map
thf(fact_933_transpose__map__map,axiom,
    ! [F: f > nat,Xs: list_list_f] :
      ( ( transpose_nat @ ( map_list_f_list_nat @ ( map_f_nat @ F ) @ Xs ) )
      = ( map_list_f_list_nat @ ( map_f_nat @ F ) @ ( transpose_f @ Xs ) ) ) ).

% transpose_map_map
thf(fact_934_transpose__map__map,axiom,
    ! [F: f > f,Xs: list_list_f] :
      ( ( transpose_f @ ( map_list_f_list_f @ ( map_f_f @ F ) @ Xs ) )
      = ( map_list_f_list_f @ ( map_f_f @ F ) @ ( transpose_f @ Xs ) ) ) ).

% transpose_map_map
thf(fact_935_measures__less,axiom,
    ! [F: multis3193088007478089820ropVar > nat,X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,Fs: list_M2713250183505059747ar_nat] :
      ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
     => ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ ( measur2342621812062629212ropVar @ ( cons_M4455541548838542291ar_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_936_measures__less,axiom,
    ! [F: nat > nat,X2: nat,Y3: nat,Fs: list_nat_nat] :
      ( ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) )
     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ ( measures_nat @ ( cons_nat_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_937_transpose__empty,axiom,
    ! [Xs: list_l3415868495287753214ropVar] :
      ( ( ( transp9078802294292144386ropVar @ Xs )
        = nil_li2679230897895549182ropVar )
      = ( ! [X: list_P8791008543742759928ropVar] :
            ( ( member6236339133593825953ropVar @ X @ ( set_li6509544621384488717ropVar @ Xs ) )
           => ( X = nil_Pr6119456312066593144ropVar ) ) ) ) ).

% transpose_empty
thf(fact_938_transpose__empty,axiom,
    ! [Xs: list_list_nat] :
      ( ( ( transpose_nat @ Xs )
        = nil_list_nat )
      = ( ! [X: list_nat] :
            ( ( member_list_nat @ X @ ( set_list_nat2 @ Xs ) )
           => ( X = nil_nat ) ) ) ) ).

% transpose_empty
thf(fact_939_transpose__empty,axiom,
    ! [Xs: list_list_f] :
      ( ( ( transpose_f @ Xs )
        = nil_list_f )
      = ( ! [X: list_f] :
            ( ( member_list_f @ X @ ( set_list_f2 @ Xs ) )
           => ( X = nil_f ) ) ) ) ).

% transpose_empty
thf(fact_940_transpose_Opelims,axiom,
    ! [X2: list_list_f,Y3: list_list_f] :
      ( ( ( transpose_f @ X2 )
        = Y3 )
     => ( ( accp_list_list_f @ transpose_rel_f @ X2 )
       => ( ( ( X2 = nil_list_f )
           => ( ( Y3 = nil_list_f )
             => ~ ( accp_list_list_f @ transpose_rel_f @ nil_list_f ) ) )
         => ( ! [Xss2: list_list_f] :
                ( ( X2
                  = ( cons_list_f @ nil_f @ Xss2 ) )
               => ( ( Y3
                    = ( transpose_f @ Xss2 ) )
                 => ~ ( accp_list_list_f @ transpose_rel_f @ ( cons_list_f @ nil_f @ Xss2 ) ) ) )
           => ~ ! [X3: f,Xs4: list_f,Xss2: list_list_f] :
                  ( ( X2
                    = ( cons_list_f @ ( cons_f @ X3 @ Xs4 ) @ Xss2 ) )
                 => ( ( Y3
                      = ( cons_list_f
                        @ ( cons_f @ X3
                          @ ( concat_f
                            @ ( map_list_f_list_f
                              @ ( case_list_list_f_f @ nil_f
                                @ ^ [H: f,T2: list_f] : ( cons_f @ H @ nil_f ) )
                              @ Xss2 ) ) )
                        @ ( transpose_f
                          @ ( cons_list_f @ Xs4
                            @ ( concat_list_f
                              @ ( map_li8627645367811812490list_f
                                @ ( case_l3215012710394848538st_f_f @ nil_list_f
                                  @ ^ [H: f,T2: list_f] : ( cons_list_f @ T2 @ nil_list_f ) )
                                @ Xss2 ) ) ) ) ) )
                   => ~ ( accp_list_list_f @ transpose_rel_f @ ( cons_list_f @ ( cons_f @ X3 @ Xs4 ) @ Xss2 ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_941_transpose_Opelims,axiom,
    ! [X2: list_l3415868495287753214ropVar,Y3: list_l3415868495287753214ropVar] :
      ( ( ( transp9078802294292144386ropVar @ X2 )
        = Y3 )
     => ( ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ X2 )
       => ( ( ( X2 = nil_li2679230897895549182ropVar )
           => ( ( Y3 = nil_li2679230897895549182ropVar )
             => ~ ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ nil_li2679230897895549182ropVar ) ) )
         => ( ! [Xss2: list_l3415868495287753214ropVar] :
                ( ( X2
                  = ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss2 ) )
               => ( ( Y3
                    = ( transp9078802294292144386ropVar @ Xss2 ) )
                 => ~ ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ ( cons_l5280711761816980654ropVar @ nil_Pr6119456312066593144ropVar @ Xss2 ) ) ) )
           => ~ ! [X3: produc2642885279816024434ropVar,Xs4: list_P8791008543742759928ropVar,Xss2: list_l3415868495287753214ropVar] :
                  ( ( X2
                    = ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ Xss2 ) )
                 => ( ( Y3
                      = ( cons_l5280711761816980654ropVar
                        @ ( cons_P1504208554336013608ropVar @ X3
                          @ ( concat1830633707478507187ropVar
                            @ ( map_li926741104209434052ropVar
                              @ ( case_l7238206891529332192ropVar @ nil_Pr6119456312066593144ropVar
                                @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_P1504208554336013608ropVar @ H @ nil_Pr6119456312066593144ropVar ) )
                              @ Xss2 ) ) )
                        @ ( transp9078802294292144386ropVar
                          @ ( cons_l5280711761816980654ropVar @ Xs4
                            @ ( concat4944697247486958777ropVar
                              @ ( map_li2030941865990407626ropVar
                                @ ( case_l7660754118040321498ropVar @ nil_li2679230897895549182ropVar
                                  @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_l5280711761816980654ropVar @ T2 @ nil_li2679230897895549182ropVar ) )
                                @ Xss2 ) ) ) ) ) )
                   => ~ ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X3 @ Xs4 ) @ Xss2 ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_942_transpose_Opelims,axiom,
    ! [X2: list_list_nat,Y3: list_list_nat] :
      ( ( ( transpose_nat @ X2 )
        = Y3 )
     => ( ( accp_list_list_nat @ transpose_rel_nat @ X2 )
       => ( ( ( X2 = nil_list_nat )
           => ( ( Y3 = nil_list_nat )
             => ~ ( accp_list_list_nat @ transpose_rel_nat @ nil_list_nat ) ) )
         => ( ! [Xss2: list_list_nat] :
                ( ( X2
                  = ( cons_list_nat @ nil_nat @ Xss2 ) )
               => ( ( Y3
                    = ( transpose_nat @ Xss2 ) )
                 => ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ nil_nat @ Xss2 ) ) ) )
           => ~ ! [X3: nat,Xs4: list_nat,Xss2: list_list_nat] :
                  ( ( X2
                    = ( cons_list_nat @ ( cons_nat @ X3 @ Xs4 ) @ Xss2 ) )
                 => ( ( Y3
                      = ( cons_list_nat
                        @ ( cons_nat @ X3
                          @ ( concat_nat
                            @ ( map_li7225945977422193158st_nat
                              @ ( case_l2340614614379431832at_nat @ nil_nat
                                @ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
                              @ Xss2 ) ) )
                        @ ( transpose_nat
                          @ ( cons_list_nat @ Xs4
                            @ ( concat_list_nat
                              @ ( map_li960784813134754710st_nat
                                @ ( case_l3331202209248957608at_nat @ nil_list_nat
                                  @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                                @ Xss2 ) ) ) ) ) )
                   => ~ ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X3 @ Xs4 ) @ Xss2 ) ) ) ) ) ) ) ) ).

% transpose.pelims
thf(fact_943_transpose_Opsimps_I3_J,axiom,
    ! [X2: f,Xs: list_f,Xss: list_list_f] :
      ( ( accp_list_list_f @ transpose_rel_f @ ( cons_list_f @ ( cons_f @ X2 @ Xs ) @ Xss ) )
     => ( ( transpose_f @ ( cons_list_f @ ( cons_f @ X2 @ Xs ) @ Xss ) )
        = ( cons_list_f
          @ ( cons_f @ X2
            @ ( concat_f
              @ ( map_list_f_list_f
                @ ( case_list_list_f_f @ nil_f
                  @ ^ [H: f,T2: list_f] : ( cons_f @ H @ nil_f ) )
                @ Xss ) ) )
          @ ( transpose_f
            @ ( cons_list_f @ Xs
              @ ( concat_list_f
                @ ( map_li8627645367811812490list_f
                  @ ( case_l3215012710394848538st_f_f @ nil_list_f
                    @ ^ [H: f,T2: list_f] : ( cons_list_f @ T2 @ nil_list_f ) )
                  @ Xss ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_944_transpose_Opsimps_I3_J,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Xss: list_l3415868495287753214ropVar] :
      ( ( accp_l8942749945645456647ropVar @ transp6848717065427184039ropVar @ ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ Xss ) )
     => ( ( transp9078802294292144386ropVar @ ( cons_l5280711761816980654ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ Xss ) )
        = ( cons_l5280711761816980654ropVar
          @ ( cons_P1504208554336013608ropVar @ X2
            @ ( concat1830633707478507187ropVar
              @ ( map_li926741104209434052ropVar
                @ ( case_l7238206891529332192ropVar @ nil_Pr6119456312066593144ropVar
                  @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_P1504208554336013608ropVar @ H @ nil_Pr6119456312066593144ropVar ) )
                @ Xss ) ) )
          @ ( transp9078802294292144386ropVar
            @ ( cons_l5280711761816980654ropVar @ Xs
              @ ( concat4944697247486958777ropVar
                @ ( map_li2030941865990407626ropVar
                  @ ( case_l7660754118040321498ropVar @ nil_li2679230897895549182ropVar
                    @ ^ [H: produc2642885279816024434ropVar,T2: list_P8791008543742759928ropVar] : ( cons_l5280711761816980654ropVar @ T2 @ nil_li2679230897895549182ropVar ) )
                  @ Xss ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_945_transpose_Opsimps_I3_J,axiom,
    ! [X2: nat,Xs: list_nat,Xss: list_list_nat] :
      ( ( accp_list_list_nat @ transpose_rel_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) )
     => ( ( transpose_nat @ ( cons_list_nat @ ( cons_nat @ X2 @ Xs ) @ Xss ) )
        = ( cons_list_nat
          @ ( cons_nat @ X2
            @ ( concat_nat
              @ ( map_li7225945977422193158st_nat
                @ ( case_l2340614614379431832at_nat @ nil_nat
                  @ ^ [H: nat,T2: list_nat] : ( cons_nat @ H @ nil_nat ) )
                @ Xss ) ) )
          @ ( transpose_nat
            @ ( cons_list_nat @ Xs
              @ ( concat_list_nat
                @ ( map_li960784813134754710st_nat
                  @ ( case_l3331202209248957608at_nat @ nil_list_nat
                    @ ^ [H: nat,T2: list_nat] : ( cons_list_nat @ T2 @ nil_list_nat ) )
                  @ Xss ) ) ) ) ) ) ) ).

% transpose.psimps(3)
thf(fact_946_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_f,X2: f,Ys: list_f,Y3: f,R2: set_Product_prod_f_f] :
      ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( append_f @ Xs @ ( cons_f @ X2 @ nil_f ) ) @ ( append_f @ Ys @ ( cons_f @ Y3 @ nil_f ) ) ) @ ( listrel1_f @ R2 ) )
      = ( ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys ) @ ( listrel1_f @ R2 ) )
          & ( X2 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ R2 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_947_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,R2: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) @ ( append4352292717583037453ropVar @ Ys @ ( cons_P1504208554336013608ropVar @ Y3 @ nil_Pr6119456312066593144ropVar ) ) ) @ ( listre6655839188755619153ropVar @ R2 ) )
      = ( ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R2 ) )
          & ( X2 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y3 ) @ R2 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_948_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_M6740372356978808802ropVar,X2: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,R2: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( append2684835082924936439ropVar @ Xs @ ( cons_M4277272395645347858ropVar @ X2 @ nil_Mu4132399263956683874ropVar ) ) @ ( append2684835082924936439ropVar @ Ys @ ( cons_M4277272395645347858ropVar @ Y3 @ nil_Mu4132399263956683874ropVar ) ) ) @ ( listre3736066117146447419ropVar @ R2 ) )
      = ( ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R2 ) )
          & ( X2 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ R2 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_949_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_nat,X2: nat,Ys: list_nat,Y3: nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) @ ( append_nat @ Ys @ ( cons_nat @ Y3 @ nil_nat ) ) ) @ ( listrel1_nat @ R2 ) )
      = ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
          & ( X2 = Y3 ) )
        | ( ( Xs = Ys )
          & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_950_map__decr__upt,axiom,
    ! [M2: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N5: nat] : ( minus_minus_nat @ N5 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N2 ) ) )
      = ( upt @ M2 @ N2 ) ) ).

% map_decr_upt
thf(fact_951_verit__sko__forall__indirect2,axiom,
    ! [X2: nat,P: nat > $o,P6: nat > $o] :
      ( ( X2
        = ( fChoice_nat
          @ ^ [X: nat] :
              ~ ( P @ X ) ) )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
            = ( P6 @ X3 ) )
       => ( ( ! [X5: nat] : ( P6 @ X5 ) )
          = ( P @ X2 ) ) ) ) ).

% verit_sko_forall_indirect2
thf(fact_952_verit__sko__forall__indirect,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( X2
        = ( fChoice_nat
          @ ^ [X: nat] :
              ~ ( P @ X ) ) )
     => ( ( ! [X5: nat] : ( P @ X5 ) )
        = ( P @ X2 ) ) ) ).

% verit_sko_forall_indirect
thf(fact_953_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ A2 )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_954_diff__zero,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ A2 @ zero_zero_nat )
      = A2 ) ).

% diff_zero
thf(fact_955_zero__diff,axiom,
    ! [A2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A2 )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_956_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_957_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_958_diff__Suc__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M2 ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_959_Suc__diff__diff,axiom,
    ! [M2: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M2 ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_960_zero__less__diff,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% zero_less_diff
thf(fact_961_Suc__pred,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) )
        = N2 ) ) ).

% Suc_pred
thf(fact_962_Cons__listrel1__Cons,axiom,
    ! [X2: f,Xs: list_f,Y3: f,Ys: list_f,R2: set_Product_prod_f_f] :
      ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Ys ) ) @ ( listrel1_f @ R2 ) )
      = ( ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X2 = Y3 )
          & ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys ) @ ( listrel1_f @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_963_Cons__listrel1__Cons,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,R2: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre6655839188755619153ropVar @ R2 ) )
      = ( ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y3 ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X2 = Y3 )
          & ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_964_Cons__listrel1__Cons,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,R2: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X2 @ Xs ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) ) @ ( listre3736066117146447419ropVar @ R2 ) )
      = ( ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X2 = Y3 )
          & ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_965_Cons__listrel1__Cons,axiom,
    ! [X2: nat,Xs: list_nat,Y3: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R2 ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
          & ( Xs = Ys ) )
        | ( ( X2 = Y3 )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_966_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).

% diff_commute
thf(fact_967_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A2: nat,C2: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A2 @ C2 ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A2 @ B ) @ C2 ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_968_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_969_diffs0__imp__equal,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M2 )
          = zero_zero_nat )
       => ( M2 = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_970_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I2: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).

% zero_induct_lemma
thf(fact_971_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N2 ) @ K ) ) ).

% less_imp_diff_less
thf(fact_972_diff__less__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ( ( ord_less_nat @ M2 @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_973_diff__less,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ) ) ).

% diff_less
thf(fact_974_Suc__diff__Suc,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ N2 @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M2 @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_975_diff__less__Suc,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N2 ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_976_listrel1I2,axiom,
    ! [Xs: list_f,Ys: list_f,R2: set_Product_prod_f_f,X2: f] :
      ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys ) @ ( listrel1_f @ R2 ) )
     => ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ X2 @ Ys ) ) @ ( listrel1_f @ R2 ) ) ) ).

% listrel1I2
thf(fact_977_listrel1I2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R2: set_Pr3201966908590682119ropVar,X2: produc2642885279816024434ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R2 ) )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ X2 @ Ys ) ) @ ( listre6655839188755619153ropVar @ R2 ) ) ) ).

% listrel1I2
thf(fact_978_listrel1I2,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,X2: nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ X2 @ Ys ) ) @ ( listrel1_nat @ R2 ) ) ) ).

% listrel1I2
thf(fact_979_not__Nil__listrel1,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R2: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ nil_Pr6119456312066593144ropVar @ Xs ) @ ( listre6655839188755619153ropVar @ R2 ) ) ).

% not_Nil_listrel1
thf(fact_980_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R2 ) ) ).

% not_Nil_listrel1
thf(fact_981_not__listrel1__Nil,axiom,
    ! [Xs: list_P8791008543742759928ropVar,R2: set_Pr3201966908590682119ropVar] :
      ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ nil_Pr6119456312066593144ropVar ) @ ( listre6655839188755619153ropVar @ R2 ) ) ).

% not_listrel1_Nil
thf(fact_982_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R2 ) ) ).

% not_listrel1_Nil
thf(fact_983_append__listrel1I,axiom,
    ! [Xs: list_f,Ys: list_f,R2: set_Product_prod_f_f,Us: list_f,Vs2: list_f] :
      ( ( ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys ) @ ( listrel1_f @ R2 ) )
          & ( Us = Vs2 ) )
        | ( ( Xs = Ys )
          & ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Us @ Vs2 ) @ ( listrel1_f @ R2 ) ) ) )
     => ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( append_f @ Xs @ Us ) @ ( append_f @ Ys @ Vs2 ) ) @ ( listrel1_f @ R2 ) ) ) ).

% append_listrel1I
thf(fact_984_append__listrel1I,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat,Us: list_nat,Vs2: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
          & ( Us = Vs2 ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs2 ) @ ( listrel1_nat @ R2 ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs2 ) ) @ ( listrel1_nat @ R2 ) ) ) ).

% append_listrel1I
thf(fact_985_diff__Suc__less,axiom,
    ! [N2: nat,I2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_986_verit__comp__simplify1_I1_J,axiom,
    ! [A2: nat] :
      ~ ( ord_less_nat @ A2 @ A2 ) ).

% verit_comp_simplify1(1)
thf(fact_987_listrel1I1,axiom,
    ! [X2: f,Y3: f,R2: set_Product_prod_f_f,Xs: list_f] :
      ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ R2 )
     => ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X2 @ Xs ) @ ( cons_f @ Y3 @ Xs ) ) @ ( listrel1_f @ R2 ) ) ) ).

% listrel1I1
thf(fact_988_listrel1I1,axiom,
    ! [X2: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,R2: set_Pr3201966908590682119ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y3 ) @ R2 )
     => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( cons_P1504208554336013608ropVar @ Y3 @ Xs ) ) @ ( listre6655839188755619153ropVar @ R2 ) ) ) ).

% listrel1I1
thf(fact_989_listrel1I1,axiom,
    ! [X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R2: set_Pr7799659934693882759ropVar,Xs: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ R2 )
     => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X2 @ Xs ) @ ( cons_M4277272395645347858ropVar @ Y3 @ Xs ) ) @ ( listre3736066117146447419ropVar @ R2 ) ) ) ).

% listrel1I1
thf(fact_990_listrel1I1,axiom,
    ! [X2: nat,Y3: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ ( cons_nat @ Y3 @ Xs ) ) @ ( listrel1_nat @ R2 ) ) ) ).

% listrel1I1
thf(fact_991_Cons__listrel1E1,axiom,
    ! [X2: f,Xs: list_f,Ys: list_f,R2: set_Product_prod_f_f] :
      ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ ( cons_f @ X2 @ Xs ) @ Ys ) @ ( listrel1_f @ R2 ) )
     => ( ! [Y4: f] :
            ( ( Ys
              = ( cons_f @ Y4 @ Xs ) )
           => ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y4 ) @ R2 ) )
       => ~ ! [Zs2: list_f] :
              ( ( Ys
                = ( cons_f @ X2 @ Zs2 ) )
             => ~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Zs2 ) @ ( listrel1_f @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_992_Cons__listrel1E1,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R2: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ Ys ) @ ( listre6655839188755619153ropVar @ R2 ) )
     => ( ! [Y4: produc2642885279816024434ropVar] :
            ( ( Ys
              = ( cons_P1504208554336013608ropVar @ Y4 @ Xs ) )
           => ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y4 ) @ R2 ) )
       => ~ ! [Zs2: list_P8791008543742759928ropVar] :
              ( ( Ys
                = ( cons_P1504208554336013608ropVar @ X2 @ Zs2 ) )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Zs2 ) @ ( listre6655839188755619153ropVar @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_993_Cons__listrel1E1,axiom,
    ! [X2: multis3193088007478089820ropVar,Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,R2: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ ( cons_M4277272395645347858ropVar @ X2 @ Xs ) @ Ys ) @ ( listre3736066117146447419ropVar @ R2 ) )
     => ( ! [Y4: multis3193088007478089820ropVar] :
            ( ( Ys
              = ( cons_M4277272395645347858ropVar @ Y4 @ Xs ) )
           => ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y4 ) @ R2 ) )
       => ~ ! [Zs2: list_M6740372356978808802ropVar] :
              ( ( Ys
                = ( cons_M4277272395645347858ropVar @ X2 @ Zs2 ) )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Zs2 ) @ ( listre3736066117146447419ropVar @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_994_Cons__listrel1E1,axiom,
    ! [X2: nat,Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X2 @ Xs ) @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ( ! [Y4: nat] :
            ( ( Ys
              = ( cons_nat @ Y4 @ Xs ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y4 ) @ R2 ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Ys
                = ( cons_nat @ X2 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Zs2 ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_995_Cons__listrel1E2,axiom,
    ! [Xs: list_f,Y3: f,Ys: list_f,R2: set_Product_prod_f_f] :
      ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ ( cons_f @ Y3 @ Ys ) ) @ ( listrel1_f @ R2 ) )
     => ( ! [X3: f] :
            ( ( Xs
              = ( cons_f @ X3 @ Ys ) )
           => ~ ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y3 ) @ R2 ) )
       => ~ ! [Zs2: list_f] :
              ( ( Xs
                = ( cons_f @ Y3 @ Zs2 ) )
             => ~ ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Zs2 @ Ys ) @ ( listrel1_f @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_996_Cons__listrel1E2,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Y3: produc2642885279816024434ropVar,Ys: list_P8791008543742759928ropVar,R2: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ Y3 @ Ys ) ) @ ( listre6655839188755619153ropVar @ R2 ) )
     => ( ! [X3: produc2642885279816024434ropVar] :
            ( ( Xs
              = ( cons_P1504208554336013608ropVar @ X3 @ Ys ) )
           => ~ ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X3 @ Y3 ) @ R2 ) )
       => ~ ! [Zs2: list_P8791008543742759928ropVar] :
              ( ( Xs
                = ( cons_P1504208554336013608ropVar @ Y3 @ Zs2 ) )
             => ~ ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Zs2 @ Ys ) @ ( listre6655839188755619153ropVar @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_997_Cons__listrel1E2,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Y3: multis3193088007478089820ropVar,Ys: list_M6740372356978808802ropVar,R2: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ ( cons_M4277272395645347858ropVar @ Y3 @ Ys ) ) @ ( listre3736066117146447419ropVar @ R2 ) )
     => ( ! [X3: multis3193088007478089820ropVar] :
            ( ( Xs
              = ( cons_M4277272395645347858ropVar @ X3 @ Ys ) )
           => ~ ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X3 @ Y3 ) @ R2 ) )
       => ~ ! [Zs2: list_M6740372356978808802ropVar] :
              ( ( Xs
                = ( cons_M4277272395645347858ropVar @ Y3 @ Zs2 ) )
             => ~ ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Zs2 @ Ys ) @ ( listre3736066117146447419ropVar @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_998_Cons__listrel1E2,axiom,
    ! [Xs: list_nat,Y3: nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ ( cons_nat @ Y3 @ Ys ) ) @ ( listrel1_nat @ R2 ) )
     => ( ! [X3: nat] :
            ( ( Xs
              = ( cons_nat @ X3 @ Ys ) )
           => ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ R2 ) )
       => ~ ! [Zs2: list_nat] :
              ( ( Xs
                = ( cons_nat @ Y3 @ Zs2 ) )
             => ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Zs2 @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_999_listrel1E,axiom,
    ! [Xs: list_f,Ys: list_f,R2: set_Product_prod_f_f] :
      ( ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys ) @ ( listrel1_f @ R2 ) )
     => ~ ! [X3: f,Y4: f] :
            ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X3 @ Y4 ) @ R2 )
           => ! [Us3: list_f,Vs3: list_f] :
                ( ( Xs
                  = ( append_f @ Us3 @ ( cons_f @ X3 @ Vs3 ) ) )
               => ( Ys
                 != ( append_f @ Us3 @ ( cons_f @ Y4 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1000_listrel1E,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar,R2: set_Pr3201966908590682119ropVar] :
      ( ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R2 ) )
     => ~ ! [X3: produc2642885279816024434ropVar,Y4: produc2642885279816024434ropVar] :
            ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X3 @ Y4 ) @ R2 )
           => ! [Us3: list_P8791008543742759928ropVar,Vs3: list_P8791008543742759928ropVar] :
                ( ( Xs
                  = ( append4352292717583037453ropVar @ Us3 @ ( cons_P1504208554336013608ropVar @ X3 @ Vs3 ) ) )
               => ( Ys
                 != ( append4352292717583037453ropVar @ Us3 @ ( cons_P1504208554336013608ropVar @ Y4 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1001_listrel1E,axiom,
    ! [Xs: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar,R2: set_Pr7799659934693882759ropVar] :
      ( ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R2 ) )
     => ~ ! [X3: multis3193088007478089820ropVar,Y4: multis3193088007478089820ropVar] :
            ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X3 @ Y4 ) @ R2 )
           => ! [Us3: list_M6740372356978808802ropVar,Vs3: list_M6740372356978808802ropVar] :
                ( ( Xs
                  = ( append2684835082924936439ropVar @ Us3 @ ( cons_M4277272395645347858ropVar @ X3 @ Vs3 ) ) )
               => ( Ys
                 != ( append2684835082924936439ropVar @ Us3 @ ( cons_M4277272395645347858ropVar @ Y4 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1002_listrel1E,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
     => ~ ! [X3: nat,Y4: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y4 ) @ R2 )
           => ! [Us3: list_nat,Vs3: list_nat] :
                ( ( Xs
                  = ( append_nat @ Us3 @ ( cons_nat @ X3 @ Vs3 ) ) )
               => ( Ys
                 != ( append_nat @ Us3 @ ( cons_nat @ Y4 @ Vs3 ) ) ) ) ) ) ).

% listrel1E
thf(fact_1003_listrel1I,axiom,
    ! [X2: f,Y3: f,R2: set_Product_prod_f_f,Xs: list_f,Us: list_f,Vs2: list_f,Ys: list_f] :
      ( ( member5935997662887750672od_f_f @ ( product_Pair_f_f @ X2 @ Y3 ) @ R2 )
     => ( ( Xs
          = ( append_f @ Us @ ( cons_f @ X2 @ Vs2 ) ) )
       => ( ( Ys
            = ( append_f @ Us @ ( cons_f @ Y3 @ Vs2 ) ) )
         => ( member631280506938425232list_f @ ( produc8144748701140958423list_f @ Xs @ Ys ) @ ( listrel1_f @ R2 ) ) ) ) ) ).

% listrel1I
thf(fact_1004_listrel1I,axiom,
    ! [X2: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,R2: set_Pr3201966908590682119ropVar,Xs: list_P8791008543742759928ropVar,Us: list_P8791008543742759928ropVar,Vs2: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
      ( ( member4092153873655836240ropVar @ ( produc6842008660352832535ropVar @ X2 @ Y3 ) @ R2 )
     => ( ( Xs
          = ( append4352292717583037453ropVar @ Us @ ( cons_P1504208554336013608ropVar @ X2 @ Vs2 ) ) )
       => ( ( Ys
            = ( append4352292717583037453ropVar @ Us @ ( cons_P1504208554336013608ropVar @ Y3 @ Vs2 ) ) )
         => ( member6468787482982540624ropVar @ ( produc5333478016634530327ropVar @ Xs @ Ys ) @ ( listre6655839188755619153ropVar @ R2 ) ) ) ) ) ).

% listrel1I
thf(fact_1005_listrel1I,axiom,
    ! [X2: multis3193088007478089820ropVar,Y3: multis3193088007478089820ropVar,R2: set_Pr7799659934693882759ropVar,Xs: list_M6740372356978808802ropVar,Us: list_M6740372356978808802ropVar,Vs2: list_M6740372356978808802ropVar,Ys: list_M6740372356978808802ropVar] :
      ( ( member2619728555115997648ropVar @ ( produc6936306168532358295ropVar @ X2 @ Y3 ) @ R2 )
     => ( ( Xs
          = ( append2684835082924936439ropVar @ Us @ ( cons_M4277272395645347858ropVar @ X2 @ Vs2 ) ) )
       => ( ( Ys
            = ( append2684835082924936439ropVar @ Us @ ( cons_M4277272395645347858ropVar @ Y3 @ Vs2 ) ) )
         => ( member7927977717557115344ropVar @ ( produc2076565587003802519ropVar @ Xs @ Ys ) @ ( listre3736066117146447419ropVar @ R2 ) ) ) ) ) ).

% listrel1I
thf(fact_1006_listrel1I,axiom,
    ! [X2: nat,Y3: nat,R2: set_Pr1261947904930325089at_nat,Xs: list_nat,Us: list_nat,Vs2: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y3 ) @ R2 )
     => ( ( Xs
          = ( append_nat @ Us @ ( cons_nat @ X2 @ Vs2 ) ) )
       => ( ( Ys
            = ( append_nat @ Us @ ( cons_nat @ Y3 @ Vs2 ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) ) ) ) ) ).

% listrel1I
thf(fact_1007_verit__sko__ex_H,axiom,
    ! [P: nat > $o,A: $o] :
      ( ( ( P @ ( fChoice_nat @ P ) )
        = A )
     => ( ( ? [X5: nat] : ( P @ X5 ) )
        = A ) ) ).

% verit_sko_ex'
thf(fact_1008_verit__sko__forall,axiom,
    ( ( ^ [P7: nat > $o] :
        ! [X6: nat] : ( P7 @ X6 ) )
    = ( ^ [P3: nat > $o] :
          ( P3
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ~ ( P3 @ X ) ) ) ) ) ).

% verit_sko_forall
thf(fact_1009_verit__sko__forall_H,axiom,
    ! [P: nat > $o,A: $o] :
      ( ( ( P
          @ ( fChoice_nat
            @ ^ [X: nat] :
                ~ ( P @ X ) ) )
        = A )
     => ( ( ! [X5: nat] : ( P @ X5 ) )
        = A ) ) ).

% verit_sko_forall'
thf(fact_1010_verit__sko__forall_H_H,axiom,
    ! [B5: nat,A: nat,P: nat > $o] :
      ( ( B5 = A )
     => ( ( ( fChoice_nat @ P )
          = A )
        = ( ( fChoice_nat @ P )
          = B5 ) ) ) ).

% verit_sko_forall''
thf(fact_1011_verit__sko__ex__indirect,axiom,
    ! [X2: nat,P: nat > $o] :
      ( ( X2
        = ( fChoice_nat @ P ) )
     => ( ( ? [X5: nat] : ( P @ X5 ) )
        = ( P @ X2 ) ) ) ).

% verit_sko_ex_indirect
thf(fact_1012_verit__sko__ex__indirect2,axiom,
    ! [X2: nat,P: nat > $o,P6: nat > $o] :
      ( ( X2
        = ( fChoice_nat @ P ) )
     => ( ! [X3: nat] :
            ( ( P @ X3 )
            = ( P6 @ X3 ) )
       => ( ( ? [X5: nat] : ( P6 @ X5 ) )
          = ( P @ X2 ) ) ) ) ).

% verit_sko_ex_indirect2
thf(fact_1013_Succ__def,axiom,
    ( bNF_Greatest_Succ_f
    = ( ^ [Kl: set_list_f,Kl2: list_f] :
          ( collect_f
          @ ^ [K3: f] : ( member_list_f @ ( append_f @ Kl2 @ ( cons_f @ K3 @ nil_f ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_1014_Succ__def,axiom,
    ( bNF_Gr2655190671417346957ropVar
    = ( ^ [Kl: set_li8998331099879477720ropVar,Kl2: list_P8791008543742759928ropVar] :
          ( collec3724222472359928029ropVar
          @ ^ [K3: produc2642885279816024434ropVar] : ( member6236339133593825953ropVar @ ( append4352292717583037453ropVar @ Kl2 @ ( cons_P1504208554336013608ropVar @ K3 @ nil_Pr6119456312066593144ropVar ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_1015_Succ__def,axiom,
    ( bNF_Gr6352880689984616693cc_nat
    = ( ^ [Kl: set_list_nat,Kl2: list_nat] :
          ( collect_nat
          @ ^ [K3: nat] : ( member_list_nat @ ( append_nat @ Kl2 @ ( cons_nat @ K3 @ nil_nat ) ) @ Kl ) ) ) ) ).

% Succ_def
thf(fact_1016_SuccI,axiom,
    ! [Kl3: list_M6740372356978808802ropVar,K: multis3193088007478089820ropVar,Kl4: set_li3089797549399498434ropVar] :
      ( ( member490224382042672267ropVar @ ( append2684835082924936439ropVar @ Kl3 @ ( cons_M4277272395645347858ropVar @ K @ nil_Mu4132399263956683874ropVar ) ) @ Kl4 )
     => ( member8421201611525043589ropVar @ K @ ( bNF_Gr6307721265529545847ropVar @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1017_SuccI,axiom,
    ! [Kl3: list_f,K: f,Kl4: set_list_f] :
      ( ( member_list_f @ ( append_f @ Kl3 @ ( cons_f @ K @ nil_f ) ) @ Kl4 )
     => ( member_f @ K @ ( bNF_Greatest_Succ_f @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1018_SuccI,axiom,
    ! [Kl3: list_P8791008543742759928ropVar,K: produc2642885279816024434ropVar,Kl4: set_li8998331099879477720ropVar] :
      ( ( member6236339133593825953ropVar @ ( append4352292717583037453ropVar @ Kl3 @ ( cons_P1504208554336013608ropVar @ K @ nil_Pr6119456312066593144ropVar ) ) @ Kl4 )
     => ( member9210975342120487067ropVar @ K @ ( bNF_Gr2655190671417346957ropVar @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1019_SuccI,axiom,
    ! [Kl3: list_nat,K: nat,Kl4: set_list_nat] :
      ( ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K @ nil_nat ) ) @ Kl4 )
     => ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) ) ) ).

% SuccI
thf(fact_1020_SuccD,axiom,
    ! [K: multis3193088007478089820ropVar,Kl4: set_li3089797549399498434ropVar,Kl3: list_M6740372356978808802ropVar] :
      ( ( member8421201611525043589ropVar @ K @ ( bNF_Gr6307721265529545847ropVar @ Kl4 @ Kl3 ) )
     => ( member490224382042672267ropVar @ ( append2684835082924936439ropVar @ Kl3 @ ( cons_M4277272395645347858ropVar @ K @ nil_Mu4132399263956683874ropVar ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1021_SuccD,axiom,
    ! [K: f,Kl4: set_list_f,Kl3: list_f] :
      ( ( member_f @ K @ ( bNF_Greatest_Succ_f @ Kl4 @ Kl3 ) )
     => ( member_list_f @ ( append_f @ Kl3 @ ( cons_f @ K @ nil_f ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1022_SuccD,axiom,
    ! [K: produc2642885279816024434ropVar,Kl4: set_li8998331099879477720ropVar,Kl3: list_P8791008543742759928ropVar] :
      ( ( member9210975342120487067ropVar @ K @ ( bNF_Gr2655190671417346957ropVar @ Kl4 @ Kl3 ) )
     => ( member6236339133593825953ropVar @ ( append4352292717583037453ropVar @ Kl3 @ ( cons_P1504208554336013608ropVar @ K @ nil_Pr6119456312066593144ropVar ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1023_SuccD,axiom,
    ! [K: nat,Kl4: set_list_nat,Kl3: list_nat] :
      ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ Kl3 ) )
     => ( member_list_nat @ ( append_nat @ Kl3 @ ( cons_nat @ K @ nil_nat ) ) @ Kl4 ) ) ).

% SuccD
thf(fact_1024_Suc__diff__1,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) )
        = N2 ) ) ).

% Suc_diff_1
thf(fact_1025_Diff__iff,axiom,
    ! [C2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C2 @ ( minus_2621998875110806563ropVar @ A @ B5 ) )
      = ( ( member8421201611525043589ropVar @ C2 @ A )
        & ~ ( member8421201611525043589ropVar @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_1026_Diff__iff,axiom,
    ! [C2: f,A: set_f,B5: set_f] :
      ( ( member_f @ C2 @ ( minus_minus_set_f @ A @ B5 ) )
      = ( ( member_f @ C2 @ A )
        & ~ ( member_f @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_1027_Diff__iff,axiom,
    ! [C2: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B5 ) )
      = ( ( member_nat @ C2 @ A )
        & ~ ( member_nat @ C2 @ B5 ) ) ) ).

% Diff_iff
thf(fact_1028_DiffI,axiom,
    ! [C2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C2 @ A )
     => ( ~ ( member8421201611525043589ropVar @ C2 @ B5 )
       => ( member8421201611525043589ropVar @ C2 @ ( minus_2621998875110806563ropVar @ A @ B5 ) ) ) ) ).

% DiffI
thf(fact_1029_DiffI,axiom,
    ! [C2: f,A: set_f,B5: set_f] :
      ( ( member_f @ C2 @ A )
     => ( ~ ( member_f @ C2 @ B5 )
       => ( member_f @ C2 @ ( minus_minus_set_f @ A @ B5 ) ) ) ) ).

% DiffI
thf(fact_1030_DiffI,axiom,
    ! [C2: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ A )
     => ( ~ ( member_nat @ C2 @ B5 )
       => ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B5 ) ) ) ) ).

% DiffI
thf(fact_1031_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_1032_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_1033_psubset__imp__ex__mem,axiom,
    ! [A: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
      ( ( ord_le8651388201640276840ropVar @ A @ B5 )
     => ? [B3: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ B3 @ ( minus_2621998875110806563ropVar @ B5 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1034_psubset__imp__ex__mem,axiom,
    ! [A: set_f,B5: set_f] :
      ( ( ord_less_set_f @ A @ B5 )
     => ? [B3: f] : ( member_f @ B3 @ ( minus_minus_set_f @ B5 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1035_psubset__imp__ex__mem,axiom,
    ! [A: set_nat,B5: set_nat] :
      ( ( ord_less_set_nat @ A @ B5 )
     => ? [B3: nat] : ( member_nat @ B3 @ ( minus_minus_set_nat @ B5 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1036_minus__set__def,axiom,
    ( minus_2621998875110806563ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,B6: set_Mu5658373669842198716ropVar] :
          ( collec8658135508005558727ropVar
          @ ( minus_1245740275806398586pVar_o
            @ ^ [X: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X @ A5 )
            @ ^ [X: multis3193088007478089820ropVar] : ( member8421201611525043589ropVar @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1037_minus__set__def,axiom,
    ( minus_minus_set_f
    = ( ^ [A5: set_f,B6: set_f] :
          ( collect_f
          @ ( minus_minus_f_o
            @ ^ [X: f] : ( member_f @ X @ A5 )
            @ ^ [X: f] : ( member_f @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1038_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B6: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A5 )
            @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1039_DiffD2,axiom,
    ! [C2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C2 @ ( minus_2621998875110806563ropVar @ A @ B5 ) )
     => ~ ( member8421201611525043589ropVar @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_1040_DiffD2,axiom,
    ! [C2: f,A: set_f,B5: set_f] :
      ( ( member_f @ C2 @ ( minus_minus_set_f @ A @ B5 ) )
     => ~ ( member_f @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_1041_DiffD2,axiom,
    ! [C2: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B5 ) )
     => ~ ( member_nat @ C2 @ B5 ) ) ).

% DiffD2
thf(fact_1042_DiffD1,axiom,
    ! [C2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C2 @ ( minus_2621998875110806563ropVar @ A @ B5 ) )
     => ( member8421201611525043589ropVar @ C2 @ A ) ) ).

% DiffD1
thf(fact_1043_DiffD1,axiom,
    ! [C2: f,A: set_f,B5: set_f] :
      ( ( member_f @ C2 @ ( minus_minus_set_f @ A @ B5 ) )
     => ( member_f @ C2 @ A ) ) ).

% DiffD1
thf(fact_1044_DiffD1,axiom,
    ! [C2: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B5 ) )
     => ( member_nat @ C2 @ A ) ) ).

% DiffD1
thf(fact_1045_DiffE,axiom,
    ! [C2: multis3193088007478089820ropVar,A: set_Mu5658373669842198716ropVar,B5: set_Mu5658373669842198716ropVar] :
      ( ( member8421201611525043589ropVar @ C2 @ ( minus_2621998875110806563ropVar @ A @ B5 ) )
     => ~ ( ( member8421201611525043589ropVar @ C2 @ A )
         => ( member8421201611525043589ropVar @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_1046_DiffE,axiom,
    ! [C2: f,A: set_f,B5: set_f] :
      ( ( member_f @ C2 @ ( minus_minus_set_f @ A @ B5 ) )
     => ~ ( ( member_f @ C2 @ A )
         => ( member_f @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_1047_DiffE,axiom,
    ! [C2: nat,A: set_nat,B5: set_nat] :
      ( ( member_nat @ C2 @ ( minus_minus_set_nat @ A @ B5 ) )
     => ~ ( ( member_nat @ C2 @ A )
         => ( member_nat @ C2 @ B5 ) ) ) ).

% DiffE
thf(fact_1048_set__diff__eq,axiom,
    ( minus_2621998875110806563ropVar
    = ( ^ [A5: set_Mu5658373669842198716ropVar,B6: set_Mu5658373669842198716ropVar] :
          ( collec8658135508005558727ropVar
          @ ^ [X: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X @ A5 )
              & ~ ( member8421201611525043589ropVar @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1049_set__diff__eq,axiom,
    ( minus_minus_set_f
    = ( ^ [A5: set_f,B6: set_f] :
          ( collect_f
          @ ^ [X: f] :
              ( ( member_f @ X @ A5 )
              & ~ ( member_f @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1050_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B6: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              & ~ ( member_nat @ X @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1051_one__reorient,axiom,
    ! [X2: nat] :
      ( ( one_one_nat = X2 )
      = ( X2 = one_one_nat ) ) ).

% one_reorient
thf(fact_1052_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_1053_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1054_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_1055_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1056_diff__Suc__eq__diff__pred,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ M2 @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_1057_nat__induct__non__zero,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( P @ one_one_nat )
       => ( ! [N3: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1058_Suc__pred_H,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( N2
        = ( suc @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1059_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N2 )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1060_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1061_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1062_empty__Shift,axiom,
    ! [Kl4: set_li3089797549399498434ropVar,K: multis3193088007478089820ropVar] :
      ( ( member490224382042672267ropVar @ nil_Mu4132399263956683874ropVar @ Kl4 )
     => ( ( member8421201611525043589ropVar @ K @ ( bNF_Gr6307721265529545847ropVar @ Kl4 @ nil_Mu4132399263956683874ropVar ) )
       => ( member490224382042672267ropVar @ nil_Mu4132399263956683874ropVar @ ( bNF_Gr2525381062296485627ropVar @ Kl4 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1063_empty__Shift,axiom,
    ! [Kl4: set_list_f,K: f] :
      ( ( member_list_f @ nil_f @ Kl4 )
     => ( ( member_f @ K @ ( bNF_Greatest_Succ_f @ Kl4 @ nil_f ) )
       => ( member_list_f @ nil_f @ ( bNF_Greatest_Shift_f @ Kl4 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1064_empty__Shift,axiom,
    ! [Kl4: set_li8998331099879477720ropVar,K: produc2642885279816024434ropVar] :
      ( ( member6236339133593825953ropVar @ nil_Pr6119456312066593144ropVar @ Kl4 )
     => ( ( member9210975342120487067ropVar @ K @ ( bNF_Gr2655190671417346957ropVar @ Kl4 @ nil_Pr6119456312066593144ropVar ) )
       => ( member6236339133593825953ropVar @ nil_Pr6119456312066593144ropVar @ ( bNF_Gr249481558315355665ropVar @ Kl4 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1065_empty__Shift,axiom,
    ! [Kl4: set_list_nat,K: nat] :
      ( ( member_list_nat @ nil_nat @ Kl4 )
     => ( ( member_nat @ K @ ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ nil_nat ) )
       => ( member_list_nat @ nil_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K ) ) ) ) ).

% empty_Shift
thf(fact_1066_Succ__Shift,axiom,
    ! [Kl4: set_list_f,K: f,Kl3: list_f] :
      ( ( bNF_Greatest_Succ_f @ ( bNF_Greatest_Shift_f @ Kl4 @ K ) @ Kl3 )
      = ( bNF_Greatest_Succ_f @ Kl4 @ ( cons_f @ K @ Kl3 ) ) ) ).

% Succ_Shift
thf(fact_1067_Succ__Shift,axiom,
    ! [Kl4: set_li8998331099879477720ropVar,K: produc2642885279816024434ropVar,Kl3: list_P8791008543742759928ropVar] :
      ( ( bNF_Gr2655190671417346957ropVar @ ( bNF_Gr249481558315355665ropVar @ Kl4 @ K ) @ Kl3 )
      = ( bNF_Gr2655190671417346957ropVar @ Kl4 @ ( cons_P1504208554336013608ropVar @ K @ Kl3 ) ) ) ).

% Succ_Shift
thf(fact_1068_Succ__Shift,axiom,
    ! [Kl4: set_list_nat,K: nat,Kl3: list_nat] :
      ( ( bNF_Gr6352880689984616693cc_nat @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K ) @ Kl3 )
      = ( bNF_Gr6352880689984616693cc_nat @ Kl4 @ ( cons_nat @ K @ Kl3 ) ) ) ).

% Succ_Shift
thf(fact_1069_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1070_ShiftD,axiom,
    ! [Kl3: list_f,Kl4: set_list_f,K: f] :
      ( ( member_list_f @ Kl3 @ ( bNF_Greatest_Shift_f @ Kl4 @ K ) )
     => ( member_list_f @ ( cons_f @ K @ Kl3 ) @ Kl4 ) ) ).

% ShiftD
thf(fact_1071_ShiftD,axiom,
    ! [Kl3: list_P8791008543742759928ropVar,Kl4: set_li8998331099879477720ropVar,K: produc2642885279816024434ropVar] :
      ( ( member6236339133593825953ropVar @ Kl3 @ ( bNF_Gr249481558315355665ropVar @ Kl4 @ K ) )
     => ( member6236339133593825953ropVar @ ( cons_P1504208554336013608ropVar @ K @ Kl3 ) @ Kl4 ) ) ).

% ShiftD
thf(fact_1072_ShiftD,axiom,
    ! [Kl3: list_nat,Kl4: set_list_nat,K: nat] :
      ( ( member_list_nat @ Kl3 @ ( bNF_Gr1872714664788909425ft_nat @ Kl4 @ K ) )
     => ( member_list_nat @ ( cons_nat @ K @ Kl3 ) @ Kl4 ) ) ).

% ShiftD
thf(fact_1073_Shift__def,axiom,
    ( bNF_Greatest_Shift_f
    = ( ^ [Kl: set_list_f,K3: f] :
          ( collect_list_f
          @ ^ [Kl2: list_f] : ( member_list_f @ ( cons_f @ K3 @ Kl2 ) @ Kl ) ) ) ) ).

% Shift_def
thf(fact_1074_Shift__def,axiom,
    ( bNF_Gr249481558315355665ropVar
    = ( ^ [Kl: set_li8998331099879477720ropVar,K3: produc2642885279816024434ropVar] :
          ( collec6020084784975397347ropVar
          @ ^ [Kl2: list_P8791008543742759928ropVar] : ( member6236339133593825953ropVar @ ( cons_P1504208554336013608ropVar @ K3 @ Kl2 ) @ Kl ) ) ) ) ).

% Shift_def
thf(fact_1075_Shift__def,axiom,
    ( bNF_Gr1872714664788909425ft_nat
    = ( ^ [Kl: set_list_nat,K3: nat] :
          ( collect_list_nat
          @ ^ [Kl2: list_nat] : ( member_list_nat @ ( cons_nat @ K3 @ Kl2 ) @ Kl ) ) ) ) ).

% Shift_def
thf(fact_1076_nth__Cons__pos,axiom,
    ! [N2: nat,X2: f,Xs: list_f] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( nth_f @ ( cons_f @ X2 @ Xs ) @ N2 )
        = ( nth_f @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_1077_nth__Cons__pos,axiom,
    ! [N2: nat,X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ N2 )
        = ( nth_Pr227365692153740281ropVar @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_1078_nth__Cons__pos,axiom,
    ! [N2: nat,X2: nat,Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
        = ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% nth_Cons_pos
thf(fact_1079_last__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( last_nat @ ( upt @ I2 @ J ) )
        = ( minus_minus_nat @ J @ one_one_nat ) ) ) ).

% last_upt
thf(fact_1080_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_1081_gen__length__code_I2_J,axiom,
    ! [N2: nat,X2: f,Xs: list_f] :
      ( ( gen_length_f @ N2 @ ( cons_f @ X2 @ Xs ) )
      = ( gen_length_f @ ( suc @ N2 ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_1082_gen__length__code_I2_J,axiom,
    ! [N2: nat,X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( gen_le3870443462522697605ropVar @ N2 @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
      = ( gen_le3870443462522697605ropVar @ ( suc @ N2 ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_1083_gen__length__code_I2_J,axiom,
    ! [N2: nat,X2: nat,Xs: list_nat] :
      ( ( gen_length_nat @ N2 @ ( cons_nat @ X2 @ Xs ) )
      = ( gen_length_nat @ ( suc @ N2 ) @ Xs ) ) ).

% gen_length_code(2)
thf(fact_1084_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_1085_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_1086_nth__Cons__0,axiom,
    ! [X2: f,Xs: list_f] :
      ( ( nth_f @ ( cons_f @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1087_nth__Cons__0,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1088_nth__Cons__0,axiom,
    ! [X2: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ zero_zero_nat )
      = X2 ) ).

% nth_Cons_0
thf(fact_1089_nth__Cons__Suc,axiom,
    ! [X2: f,Xs: list_f,N2: nat] :
      ( ( nth_f @ ( cons_f @ X2 @ Xs ) @ ( suc @ N2 ) )
      = ( nth_f @ Xs @ N2 ) ) ).

% nth_Cons_Suc
thf(fact_1090_nth__Cons__Suc,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,N2: nat] :
      ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( suc @ N2 ) )
      = ( nth_Pr227365692153740281ropVar @ Xs @ N2 ) ) ).

% nth_Cons_Suc
thf(fact_1091_nth__Cons__Suc,axiom,
    ! [X2: nat,Xs: list_nat,N2: nat] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( suc @ N2 ) )
      = ( nth_nat @ Xs @ N2 ) ) ).

% nth_Cons_Suc
thf(fact_1092_last__appendL,axiom,
    ! [Ys: list_f,Xs: list_f] :
      ( ( Ys = nil_f )
     => ( ( last_f @ ( append_f @ Xs @ Ys ) )
        = ( last_f @ Xs ) ) ) ).

% last_appendL
thf(fact_1093_last__appendL,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( Ys = nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) )
        = ( last_P5278392773003955985ropVar @ Xs ) ) ) ).

% last_appendL
thf(fact_1094_last__appendL,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys = nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Xs ) ) ) ).

% last_appendL
thf(fact_1095_last__appendR,axiom,
    ! [Ys: list_f,Xs: list_f] :
      ( ( Ys != nil_f )
     => ( ( last_f @ ( append_f @ Xs @ Ys ) )
        = ( last_f @ Ys ) ) ) ).

% last_appendR
thf(fact_1096_last__appendR,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( Ys != nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) )
        = ( last_P5278392773003955985ropVar @ Ys ) ) ) ).

% last_appendR
thf(fact_1097_last__appendR,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( Ys != nil_nat )
     => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
        = ( last_nat @ Ys ) ) ) ).

% last_appendR
thf(fact_1098_last__snoc,axiom,
    ! [Xs: list_f,X2: f] :
      ( ( last_f @ ( append_f @ Xs @ ( cons_f @ X2 @ nil_f ) ) )
      = X2 ) ).

% last_snoc
thf(fact_1099_last__snoc,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
      ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs @ ( cons_P1504208554336013608ropVar @ X2 @ nil_Pr6119456312066593144ropVar ) ) )
      = X2 ) ).

% last_snoc
thf(fact_1100_last__snoc,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( last_nat @ ( append_nat @ Xs @ ( cons_nat @ X2 @ nil_nat ) ) )
      = X2 ) ).

% last_snoc
thf(fact_1101_nth__Cons__numeral,axiom,
    ! [X2: f,Xs: list_f,V: num] :
      ( ( nth_f @ ( cons_f @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_f @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_1102_nth__Cons__numeral,axiom,
    ! [X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,V: num] :
      ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_Pr227365692153740281ropVar @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_1103_nth__Cons__numeral,axiom,
    ! [X2: nat,Xs: list_nat,V: num] :
      ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ ( numeral_numeral_nat @ V ) )
      = ( nth_nat @ Xs @ ( minus_minus_nat @ ( numeral_numeral_nat @ V ) @ one_one_nat ) ) ) ).

% nth_Cons_numeral
thf(fact_1104_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_1105_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_1106_last__ConsR,axiom,
    ! [Xs: list_f,X2: f] :
      ( ( Xs != nil_f )
     => ( ( last_f @ ( cons_f @ X2 @ Xs ) )
        = ( last_f @ Xs ) ) ) ).

% last_ConsR
thf(fact_1107_last__ConsR,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
        = ( last_P5278392773003955985ropVar @ Xs ) ) ) ).

% last_ConsR
thf(fact_1108_last__ConsR,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
        = ( last_nat @ Xs ) ) ) ).

% last_ConsR
thf(fact_1109_last__ConsL,axiom,
    ! [Xs: list_f,X2: f] :
      ( ( Xs = nil_f )
     => ( ( last_f @ ( cons_f @ X2 @ Xs ) )
        = X2 ) ) ).

% last_ConsL
thf(fact_1110_last__ConsL,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
      ( ( Xs = nil_Pr6119456312066593144ropVar )
     => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
        = X2 ) ) ).

% last_ConsL
thf(fact_1111_last__ConsL,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( Xs = nil_nat )
     => ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
        = X2 ) ) ).

% last_ConsL
thf(fact_1112_last_Osimps,axiom,
    ! [Xs: list_f,X2: f] :
      ( ( ( Xs = nil_f )
       => ( ( last_f @ ( cons_f @ X2 @ Xs ) )
          = X2 ) )
      & ( ( Xs != nil_f )
       => ( ( last_f @ ( cons_f @ X2 @ Xs ) )
          = ( last_f @ Xs ) ) ) ) ).

% last.simps
thf(fact_1113_last_Osimps,axiom,
    ! [Xs: list_P8791008543742759928ropVar,X2: produc2642885279816024434ropVar] :
      ( ( ( Xs = nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
          = X2 ) )
      & ( ( Xs != nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) )
          = ( last_P5278392773003955985ropVar @ Xs ) ) ) ) ).

% last.simps
thf(fact_1114_last_Osimps,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( ( Xs = nil_nat )
       => ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
          = X2 ) )
      & ( ( Xs != nil_nat )
       => ( ( last_nat @ ( cons_nat @ X2 @ Xs ) )
          = ( last_nat @ Xs ) ) ) ) ).

% last.simps
thf(fact_1115_last__in__set,axiom,
    ! [As: list_f] :
      ( ( As != nil_f )
     => ( member_f @ ( last_f @ As ) @ ( set_f2 @ As ) ) ) ).

% last_in_set
thf(fact_1116_last__in__set,axiom,
    ! [As: list_P8791008543742759928ropVar] :
      ( ( As != nil_Pr6119456312066593144ropVar )
     => ( member9210975342120487067ropVar @ ( last_P5278392773003955985ropVar @ As ) @ ( set_Pr3946994337432900359ropVar @ As ) ) ) ).

% last_in_set
thf(fact_1117_last__in__set,axiom,
    ! [As: list_nat] :
      ( ( As != nil_nat )
     => ( member_nat @ ( last_nat @ As ) @ ( set_nat2 @ As ) ) ) ).

% last_in_set
thf(fact_1118_last__in__set,axiom,
    ! [As: list_list_f] :
      ( ( As != nil_list_f )
     => ( member_list_f @ ( last_list_f @ As ) @ ( set_list_f2 @ As ) ) ) ).

% last_in_set
thf(fact_1119_last__in__set,axiom,
    ! [As: list_M6740372356978808802ropVar] :
      ( ( As != nil_Mu4132399263956683874ropVar )
     => ( member8421201611525043589ropVar @ ( last_M5158658609589141499ropVar @ As ) @ ( set_Mu146256361055586289ropVar @ As ) ) ) ).

% last_in_set
thf(fact_1120_last__map,axiom,
    ! [Xs: list_P8791008543742759928ropVar,F: produc2642885279816024434ropVar > nat] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ( ( last_nat @ ( map_Pr7810381211709419518ar_nat @ F @ Xs ) )
        = ( F @ ( last_P5278392773003955985ropVar @ Xs ) ) ) ) ).

% last_map
thf(fact_1121_last__map,axiom,
    ! [Xs: list_nat,F: nat > produc2642885279816024434ropVar] :
      ( ( Xs != nil_nat )
     => ( ( last_P5278392773003955985ropVar @ ( map_na2145702811007361372ropVar @ F @ Xs ) )
        = ( F @ ( last_nat @ Xs ) ) ) ) ).

% last_map
thf(fact_1122_last__map,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ( Xs != nil_nat )
     => ( ( last_nat @ ( map_nat_nat @ F @ Xs ) )
        = ( F @ ( last_nat @ Xs ) ) ) ) ).

% last_map
thf(fact_1123_last__map,axiom,
    ! [Xs: list_nat,F: nat > multis3193088007478089820ropVar] :
      ( ( Xs != nil_nat )
     => ( ( last_M5158658609589141499ropVar @ ( map_na2466558649495876678ropVar @ F @ Xs ) )
        = ( F @ ( last_nat @ Xs ) ) ) ) ).

% last_map
thf(fact_1124_last__map,axiom,
    ! [Xs: list_f,F: f > multis3193088007478089820ropVar] :
      ( ( Xs != nil_f )
     => ( ( last_M5158658609589141499ropVar @ ( map_f_4134023513460405853ropVar @ F @ Xs ) )
        = ( F @ ( last_f @ Xs ) ) ) ) ).

% last_map
thf(fact_1125_last__map,axiom,
    ! [Xs: list_f,F: f > produc2642885279816024434ropVar] :
      ( ( Xs != nil_f )
     => ( ( last_P5278392773003955985ropVar @ ( map_f_2905954635146393971ropVar @ F @ Xs ) )
        = ( F @ ( last_f @ Xs ) ) ) ) ).

% last_map
thf(fact_1126_last__map,axiom,
    ! [Xs: list_f,F: f > list_f] :
      ( ( Xs != nil_f )
     => ( ( last_list_f @ ( map_f_list_f @ F @ Xs ) )
        = ( F @ ( last_f @ Xs ) ) ) ) ).

% last_map
thf(fact_1127_last__map,axiom,
    ! [Xs: list_f,F: f > nat] :
      ( ( Xs != nil_f )
     => ( ( last_nat @ ( map_f_nat @ F @ Xs ) )
        = ( F @ ( last_f @ Xs ) ) ) ) ).

% last_map
thf(fact_1128_last__map,axiom,
    ! [Xs: list_f,F: f > f] :
      ( ( Xs != nil_f )
     => ( ( last_f @ ( map_f_f @ F @ Xs ) )
        = ( F @ ( last_f @ Xs ) ) ) ) ).

% last_map
thf(fact_1129_last__append,axiom,
    ! [Ys: list_f,Xs: list_f] :
      ( ( ( Ys = nil_f )
       => ( ( last_f @ ( append_f @ Xs @ Ys ) )
          = ( last_f @ Xs ) ) )
      & ( ( Ys != nil_f )
       => ( ( last_f @ ( append_f @ Xs @ Ys ) )
          = ( last_f @ Ys ) ) ) ) ).

% last_append
thf(fact_1130_last__append,axiom,
    ! [Ys: list_P8791008543742759928ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ( Ys = nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) )
          = ( last_P5278392773003955985ropVar @ Xs ) ) )
      & ( ( Ys != nil_Pr6119456312066593144ropVar )
       => ( ( last_P5278392773003955985ropVar @ ( append4352292717583037453ropVar @ Xs @ Ys ) )
          = ( last_P5278392773003955985ropVar @ Ys ) ) ) ) ).

% last_append
thf(fact_1131_last__append,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( ( Ys = nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Xs ) ) )
      & ( ( Ys != nil_nat )
       => ( ( last_nat @ ( append_nat @ Xs @ Ys ) )
          = ( last_nat @ Ys ) ) ) ) ).

% last_append
thf(fact_1132_longest__common__suffix,axiom,
    ! [Xs: list_f,Ys: list_f] :
    ? [Ss: list_f,Xs6: list_f,Ys6: list_f] :
      ( ( Xs
        = ( append_f @ Xs6 @ Ss ) )
      & ( Ys
        = ( append_f @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_f )
        | ( Ys6 = nil_f )
        | ( ( last_f @ Xs6 )
         != ( last_f @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1133_longest__common__suffix,axiom,
    ! [Xs: list_P8791008543742759928ropVar,Ys: list_P8791008543742759928ropVar] :
    ? [Ss: list_P8791008543742759928ropVar,Xs6: list_P8791008543742759928ropVar,Ys6: list_P8791008543742759928ropVar] :
      ( ( Xs
        = ( append4352292717583037453ropVar @ Xs6 @ Ss ) )
      & ( Ys
        = ( append4352292717583037453ropVar @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_Pr6119456312066593144ropVar )
        | ( Ys6 = nil_Pr6119456312066593144ropVar )
        | ( ( last_P5278392773003955985ropVar @ Xs6 )
         != ( last_P5278392773003955985ropVar @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1134_longest__common__suffix,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
    ? [Ss: list_nat,Xs6: list_nat,Ys6: list_nat] :
      ( ( Xs
        = ( append_nat @ Xs6 @ Ss ) )
      & ( Ys
        = ( append_nat @ Ys6 @ Ss ) )
      & ( ( Xs6 = nil_nat )
        | ( Ys6 = nil_nat )
        | ( ( last_nat @ Xs6 )
         != ( last_nat @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_1135_gen__length__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( gen_le3870443462522697605ropVar @ N2 @ nil_Pr6119456312066593144ropVar )
      = N2 ) ).

% gen_length_code(1)
thf(fact_1136_gen__length__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( gen_length_nat @ N2 @ nil_nat )
      = N2 ) ).

% gen_length_code(1)
thf(fact_1137_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_1138_nth__Cons_H,axiom,
    ! [N2: nat,X2: f,Xs: list_f] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( nth_f @ ( cons_f @ X2 @ Xs ) @ N2 )
          = X2 ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( nth_f @ ( cons_f @ X2 @ Xs ) @ N2 )
          = ( nth_f @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_1139_nth__Cons_H,axiom,
    ! [N2: nat,X2: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ N2 )
          = X2 ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ N2 )
          = ( nth_Pr227365692153740281ropVar @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_1140_nth__Cons_H,axiom,
    ! [N2: nat,X2: nat,Xs: list_nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
          = X2 ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
          = ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% nth_Cons'
thf(fact_1141_nth__non__equal__first__eq,axiom,
    ! [X2: f,Y3: f,Xs: list_f,N2: nat] :
      ( ( X2 != Y3 )
     => ( ( ( nth_f @ ( cons_f @ X2 @ Xs ) @ N2 )
          = Y3 )
        = ( ( ( nth_f @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_1142_nth__non__equal__first__eq,axiom,
    ! [X2: produc2642885279816024434ropVar,Y3: produc2642885279816024434ropVar,Xs: list_P8791008543742759928ropVar,N2: nat] :
      ( ( X2 != Y3 )
     => ( ( ( nth_Pr227365692153740281ropVar @ ( cons_P1504208554336013608ropVar @ X2 @ Xs ) @ N2 )
          = Y3 )
        = ( ( ( nth_Pr227365692153740281ropVar @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_1143_nth__non__equal__first__eq,axiom,
    ! [X2: nat,Y3: nat,Xs: list_nat,N2: nat] :
      ( ( X2 != Y3 )
     => ( ( ( nth_nat @ ( cons_nat @ X2 @ Xs ) @ N2 )
          = Y3 )
        = ( ( ( nth_nat @ Xs @ ( minus_minus_nat @ N2 @ one_one_nat ) )
            = Y3 )
          & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ) ).

% nth_non_equal_first_eq
thf(fact_1144_append__butlast__last__id,axiom,
    ! [Xs: list_f] :
      ( ( Xs != nil_f )
     => ( ( append_f @ ( butlast_f @ Xs ) @ ( cons_f @ ( last_f @ Xs ) @ nil_f ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1145_append__butlast__last__id,axiom,
    ! [Xs: list_P8791008543742759928ropVar] :
      ( ( Xs != nil_Pr6119456312066593144ropVar )
     => ( ( append4352292717583037453ropVar @ ( butlas5098402688846288580ropVar @ Xs ) @ ( cons_P1504208554336013608ropVar @ ( last_P5278392773003955985ropVar @ Xs ) @ nil_Pr6119456312066593144ropVar ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1146_append__butlast__last__id,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
     => ( ( append_nat @ ( butlast_nat @ Xs ) @ ( cons_nat @ ( last_nat @ Xs ) @ nil_nat ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1147_nth__map__upt,axiom,
    ! [I2: nat,N2: nat,M2: nat,F: nat > produc2642885279816024434ropVar] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N2 @ M2 ) )
     => ( ( nth_Pr227365692153740281ropVar @ ( map_na2145702811007361372ropVar @ F @ ( upt @ M2 @ N2 ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_1148_nth__map__upt,axiom,
    ! [I2: nat,N2: nat,M2: nat,F: nat > nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N2 @ M2 ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ ( upt @ M2 @ N2 ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_1149_nth__map__upt,axiom,
    ! [I2: nat,N2: nat,M2: nat,F: nat > multis3193088007478089820ropVar] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ N2 @ M2 ) )
     => ( ( nth_Mu4537670848183934179ropVar @ ( map_na2466558649495876678ropVar @ F @ ( upt @ M2 @ N2 ) ) @ I2 )
        = ( F @ ( plus_plus_nat @ M2 @ I2 ) ) ) ) ).

% nth_map_upt
thf(fact_1150_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_1151_add__is__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ( M2 = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_1152_add__Suc__right,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ M2 @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc_right
thf(fact_1153_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_1154_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_1155_add__gr__0,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_1156_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_1157_Nat_Odiff__cancel,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_1158_diff__cancel2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M2 @ N2 ) ) ).

% diff_cancel2
thf(fact_1159_diff__add__inverse,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse
thf(fact_1160_diff__add__inverse2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse2
thf(fact_1161_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
     => ( ord_less_nat @ I2 @ K ) ) ).

% add_lessD1
thf(fact_1162_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_1163_not__add__less1,axiom,
    ! [I2: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).

% not_add_less1
thf(fact_1164_not__add__less2,axiom,
    ! [J: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_1165_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_1166_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_1167_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_1168_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M2: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M2 @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M2 @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_1169_nat__arith_Osuc1,axiom,
    ! [A: nat,K: nat,A2: nat] :
      ( ( A
        = ( plus_plus_nat @ K @ A2 ) )
     => ( ( suc @ A )
        = ( plus_plus_nat @ K @ ( suc @ A2 ) ) ) ) ).

% nat_arith.suc1
thf(fact_1170_add__Suc,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% add_Suc
thf(fact_1171_add__Suc__shift,axiom,
    ! [M2: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M2 ) @ N2 )
      = ( plus_plus_nat @ M2 @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_1172_add__eq__self__zero,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = M2 )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_1173_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_1174_Suc3__eq__add__3,axiom,
    ! [N2: nat] :
      ( ( suc @ ( suc @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ).

% Suc3_eq_add_3
thf(fact_1175_one__is__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M2 @ N2 ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_1176_add__is__1,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M2 @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M2
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M2 = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_1177_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I2 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1178_less__natE,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ~ ! [Q3: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1179_less__add__Suc1,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_1180_less__add__Suc2,axiom,
    ! [I2: nat,M2: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M2 @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_1181_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N5: nat] :
        ? [K3: nat] :
          ( N5
          = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1182_less__imp__Suc__add,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_nat @ M2 @ N2 )
     => ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1183_diff__add__0,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_1184_add__diff__inverse__nat,axiom,
    ! [M2: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M2 @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M2 @ N2 ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_1185_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1186_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N5: nat] : ( plus_plus_nat @ N5 @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_1187_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_1188_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_1189_nat__diff__split,axiom,
    ! [P: nat > $o,A2: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B ) )
      = ( ( ( ord_less_nat @ A2 @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D2: nat] :
            ( ( A2
              = ( plus_plus_nat @ B @ D2 ) )
           => ( P @ D2 ) ) ) ) ).

% nat_diff_split
thf(fact_1190_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A2: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A2 @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A2 @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D2: nat] :
                ( ( A2
                  = ( plus_plus_nat @ B @ D2 ) )
                & ~ ( P @ D2 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1191_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_1192_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M6: nat,N5: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N5 @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N5 ) ) ) ) ) ).

% add_eq_if
thf(fact_1193_map__add__upt,axiom,
    ! [N2: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I: nat] : ( plus_plus_nat @ I @ N2 )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ) ).

% map_add_upt
thf(fact_1194_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X2: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J )
        = ( cons_nat @ X2 @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J )
        & ( I2 = X2 )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_1195_num_Osize__gen_I3_J,axiom,
    ! [X33: num] :
      ( ( size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size_gen(3)
thf(fact_1196_num_Osize_I6_J,axiom,
    ! [X33: num] :
      ( ( size_size_num @ ( bit1 @ X33 ) )
      = ( plus_plus_nat @ ( size_size_num @ X33 ) @ ( suc @ zero_zero_nat ) ) ) ).

% num.size(6)
thf(fact_1197_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A2: nat,B: nat] :
      ( ! [A4: nat,B3: nat] :
          ( ( P @ A4 @ B3 )
          = ( P @ B3 @ A4 ) )
     => ( ! [A4: nat] : ( P @ A4 @ zero_zero_nat )
       => ( ! [A4: nat,B3: nat] :
              ( ( P @ A4 @ B3 )
             => ( P @ A4 @ ( plus_plus_nat @ A4 @ B3 ) ) )
         => ( P @ A2 @ B ) ) ) ) ).

% Euclid_induct
thf(fact_1198_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_1199_Suc__nat__number__of__add,axiom,
    ! [V: num,N2: nat] :
      ( ( suc @ ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ N2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ one ) ) @ N2 ) ) ).

% Suc_nat_number_of_add
thf(fact_1200_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1201_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_1202_length__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J ) )
      = ( minus_minus_nat @ J @ I2 ) ) ).

% length_upt
thf(fact_1203_PropVar_Osize_I7_J,axiom,
    ! [X11: nat,X12: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis387687052011358179_Gamma @ X11 @ X12 ) )
      = zero_zero_nat ) ).

% PropVar.size(7)
thf(fact_1204_PropVar_Osize_I9_J,axiom,
    ! [X31: nat,X32: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis2983220944385521703ZeroJI @ X31 @ X32 ) )
      = zero_zero_nat ) ).

% PropVar.size(9)
thf(fact_1205_PropVar_Osize_I10_J,axiom,
    ! [X41: nat,X42: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis6646701651571564453xOneJI @ X41 @ X42 ) )
      = zero_zero_nat ) ).

% PropVar.size(10)
thf(fact_1206_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C4: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_1207_pos_I3_J,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X3 )
              = X2 )
            & ! [Y6: nat] :
                ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y6 )
                    = X2 ) )
               => ( Y6 = X3 ) ) ) ) ) ).

% pos(3)
thf(fact_1208_p__ys_I3_J,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ys1 ) )
     => ? [X3: nat] :
          ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ ys1 ) )
          & ( ( nth_nat @ ys1 @ X3 )
            = X2 )
          & ! [Y6: nat] :
              ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ ys1 ) )
                & ( ( nth_nat @ ys1 @ Y6 )
                  = X2 ) )
             => ( Y6 = X3 ) ) ) ) ).

% p_ys(3)
thf(fact_1209__092_060open_062distinct_Ays1_092_060close_062,axiom,
    distinct_nat @ ys1 ).

% \<open>distinct ys1\<close>
thf(fact_1210_dist__ys,axiom,
    distinct_nat @ ( append_nat @ ys1 @ ys2 ) ).

% dist_ys
thf(fact_1211_ys,axiom,
    ( ( mset_nat @ ( upt @ zero_zero_nat @ m ) )
    = ( plus_p6334493942879108393et_nat @ ( mset_nat @ ys1 ) @ ( mset_nat @ ys2 ) ) ) ).

% ys
thf(fact_1212_distinct__upt,axiom,
    ! [I2: nat,J: nat] : ( distinct_nat @ ( upt @ I2 @ J ) ) ).

% distinct_upt
thf(fact_1213_p__ys_I1_J,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ys1 ) )
     => ( ord_less_nat @ ( pos_of @ ys1 @ X2 ) @ ( size_size_list_nat @ ys1 ) ) ) ).

% p_ys(1)
thf(fact_1214_pos_I1_J,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( ord_less_nat @ ( pos_of @ Xs @ X2 ) @ ( size_size_list_nat @ Xs ) ) ) ) ).

% pos(1)
thf(fact_1215_pos_I2_J,axiom,
    ! [Xs: list_nat,X2: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X2 @ ( set_nat2 @ Xs ) )
       => ( ( nth_nat @ Xs @ ( pos_of @ Xs @ X2 ) )
          = X2 ) ) ) ).

% pos(2)
thf(fact_1216_p__ys_I2_J,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ ys1 ) )
     => ( ( nth_nat @ ys1 @ ( pos_of @ ys1 @ X2 ) )
        = X2 ) ) ).

% p_ys(2)
thf(fact_1217__092_060open_062pos__of_A_092_060equiv_062_A_092_060lambda_062xs_Ai_O_ATHE_Ap_O_Ap_A_060_Alength_Axs_A_092_060and_062_Axs_A_B_Ap_A_061_Ai_092_060close_062,axiom,
    ( pos_of
    = ( ^ [Xs2: list_nat,I: nat] :
          ( the_nat
          @ ^ [P8: nat] :
              ( ( ord_less_nat @ P8 @ ( size_size_list_nat @ Xs2 ) )
              & ( ( nth_nat @ Xs2 @ P8 )
                = I ) ) ) ) ) ).

% \<open>pos_of \<equiv> \<lambda>xs i. THE p. p < length xs \<and> xs ! p = i\<close>
thf(fact_1218_xs,axiom,
    ( ( mset_nat @ ( upt @ zero_zero_nat @ n ) )
    = ( plus_p6334493942879108393et_nat @ ( mset_nat @ xs1 ) @ ( mset_nat @ xs2 ) ) ) ).

% xs
thf(fact_1219_j__ys1_I3_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys1 ) )
     => ( v2 @ ( multis387687052011358179_Gamma @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) @ J ) ) ) ).

% j_ys1(3)
thf(fact_1220_pos__of__def,axiom,
    ! [Xs: list_nat,I2: nat] :
      ( ( pos_of @ Xs @ I2 )
      = ( the_nat
        @ ^ [P8: nat] :
            ( ( ord_less_nat @ P8 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ P8 )
              = I2 ) ) ) ) ).

% pos_of_def
thf(fact_1221__092_060open_062distinct_Axs1_092_060close_062,axiom,
    distinct_nat @ xs1 ).

% \<open>distinct xs1\<close>
thf(fact_1222_p__xs_I2_J,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ xs1 ) )
     => ( ( nth_nat @ xs1 @ ( pos_of @ xs1 @ X2 ) )
        = X2 ) ) ).

% p_xs(2)
thf(fact_1223_p__xs_I3_J,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ xs1 ) )
     => ? [X3: nat] :
          ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ xs1 ) )
          & ( ( nth_nat @ xs1 @ X3 )
            = X2 )
          & ! [Y6: nat] :
              ( ( ( ord_less_nat @ Y6 @ ( size_size_list_nat @ xs1 ) )
                & ( ( nth_nat @ xs1 @ Y6 )
                  = X2 ) )
             => ( Y6 = X3 ) ) ) ) ).

% p_xs(3)
thf(fact_1224_dist__xs,axiom,
    distinct_nat @ ( append_nat @ xs1 @ xs2 ) ).

% dist_xs
thf(fact_1225_p__xs_I1_J,axiom,
    ! [X2: nat] :
      ( ( member_nat @ X2 @ ( set_nat2 @ xs1 ) )
     => ( ord_less_nat @ ( pos_of @ xs1 @ X2 ) @ ( size_size_list_nat @ xs1 ) ) ) ).

% p_xs(1)
thf(fact_1226_j__ys1_I1_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys1 ) )
     => ( member_nat @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) @ ( set_nat2 @ xs1 ) ) ) ).

% j_ys1(1)
thf(fact_1227_j__ys1_I2_J,axiom,
    ! [J: nat] :
      ( ( member_nat @ J @ ( set_nat2 @ ys1 ) )
     => ( cns @ ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J ) ) @ J ) ) ).

% j_ys1(2)
thf(fact_1228_size_H__char__eq__0,axiom,
    ( size_char
    = ( ^ [C4: char] : zero_zero_nat ) ) ).

% size'_char_eq_0
thf(fact_1229__092_060open_062v_H_A_092_060equiv_062_A_092_060lambda_062x_O_Acase_Ax_Aof_AGamma_Ai_Aj_A_092_060Rightarrow_062_Ai_A_092_060in_062_Aset_Axs1_A_092_060and_062_Aj_A_092_060in_062_Aset_Ays1_A_092_060and_062_Ai_A_061_Axs1_A_B_Apos__of_Ays1_Aj_A_092_060or_062_Ai_A_092_060in_062_Aset_Axs2_A_092_060and_062_Aj_A_092_060in_062_Aset_Ays2_A_092_060and_062_Ai_A_061_Ai__of__j2_Aj_A_124_AEpsilon_Ai_A_092_060Rightarrow_062_Ai_A_092_060in_062_Aset_Axs1_092_060close_062,axiom,
    ( v
    = ( multis8641292176345589959pVar_o
      @ ^ [I: nat,J2: nat] :
          ( ( ( member_nat @ I @ ( set_nat2 @ xs1 ) )
            & ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
            & ( I
              = ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) ) )
          | ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
            & ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
            & ( I
              = ( i_of_j2 @ J2 ) ) ) )
      @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs1 ) )
      @ ^ [A6: nat,B4: nat] : undefined_o
      @ ^ [A6: nat,B4: nat] : undefined_o
      @ ^ [A6: nat,B4: nat] : undefined_o
      @ ^ [A6: nat,B4: nat] : undefined_o ) ) ).

% \<open>v' \<equiv> \<lambda>x. case x of Gamma i j \<Rightarrow> i \<in> set xs1 \<and> j \<in> set ys1 \<and> i = xs1 ! pos_of ys1 j \<or> i \<in> set xs2 \<and> j \<in> set ys2 \<and> i = i_of_j2 j | Epsilon i \<Rightarrow> i \<in> set xs1\<close>
thf(fact_1230_v_H__def,axiom,
    ! [X2: multis3193088007478089820ropVar] :
      ( ( v @ X2 )
      = ( multis8641292176345589959pVar_o
        @ ^ [I: nat,J2: nat] :
            ( ( ( member_nat @ I @ ( set_nat2 @ xs1 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys1 ) )
              & ( I
                = ( nth_nat @ xs1 @ ( pos_of @ ys1 @ J2 ) ) ) )
            | ( ( member_nat @ I @ ( set_nat2 @ xs2 ) )
              & ( member_nat @ J2 @ ( set_nat2 @ ys2 ) )
              & ( I
                = ( i_of_j2 @ J2 ) ) ) )
        @ ^ [I: nat] : ( member_nat @ I @ ( set_nat2 @ xs1 ) )
        @ ^ [A6: nat,B4: nat] : undefined_o
        @ ^ [A6: nat,B4: nat] : undefined_o
        @ ^ [A6: nat,B4: nat] : undefined_o
        @ ^ [A6: nat,B4: nat] : undefined_o
        @ X2 ) ) ).

% v'_def
thf(fact_1231_PropVar_Oinject_I2_J,axiom,
    ! [X22: nat,Y22: nat] :
      ( ( ( multis2544335231667181926psilon @ X22 )
        = ( multis2544335231667181926psilon @ Y22 ) )
      = ( X22 = Y22 ) ) ).

% PropVar.inject(2)
thf(fact_1232_PropVar_Odistinct_I11_J,axiom,
    ! [X22: nat,X31: nat,X32: nat] :
      ( ( multis2544335231667181926psilon @ X22 )
     != ( multis2983220944385521703ZeroJI @ X31 @ X32 ) ) ).

% PropVar.distinct(11)
thf(fact_1233_PropVar_Odistinct_I13_J,axiom,
    ! [X22: nat,X41: nat,X42: nat] :
      ( ( multis2544335231667181926psilon @ X22 )
     != ( multis6646701651571564453xOneJI @ X41 @ X42 ) ) ).

% PropVar.distinct(13)
thf(fact_1234_PropVar_Odistinct_I1_J,axiom,
    ! [X11: nat,X12: nat,X22: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis2544335231667181926psilon @ X22 ) ) ).

% PropVar.distinct(1)
thf(fact_1235_PropVar_Osize_I8_J,axiom,
    ! [X22: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis2544335231667181926psilon @ X22 ) )
      = zero_zero_nat ) ).

% PropVar.size(8)
thf(fact_1236_PropVar_Osize__gen_I2_J,axiom,
    ! [X22: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis2544335231667181926psilon @ X22 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(2)
thf(fact_1237_PropVar_Oinject_I5_J,axiom,
    ! [X51: nat,X52: nat,Y51: nat,Y52: nat] :
      ( ( ( multis2983220944385456105ZeroIJ @ X51 @ X52 )
        = ( multis2983220944385456105ZeroIJ @ Y51 @ Y52 ) )
      = ( ( X51 = Y51 )
        & ( X52 = Y52 ) ) ) ).

% PropVar.inject(5)
thf(fact_1238_PropVar_Oinject_I6_J,axiom,
    ! [X61: nat,X62: nat,Y61: nat,Y62: nat] :
      ( ( ( multis6646701651571498855xOneIJ @ X61 @ X62 )
        = ( multis6646701651571498855xOneIJ @ Y61 @ Y62 ) )
      = ( ( X61 = Y61 )
        & ( X62 = Y62 ) ) ) ).

% PropVar.inject(6)
thf(fact_1239_PropVar_Odistinct_I21_J,axiom,
    ! [X31: nat,X32: nat,X51: nat,X52: nat] :
      ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(21)
thf(fact_1240_PropVar_Odistinct_I23_J,axiom,
    ! [X31: nat,X32: nat,X61: nat,X62: nat] :
      ( ( multis2983220944385521703ZeroJI @ X31 @ X32 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(23)
thf(fact_1241_PropVar_Odistinct_I25_J,axiom,
    ! [X41: nat,X42: nat,X51: nat,X52: nat] :
      ( ( multis6646701651571564453xOneJI @ X41 @ X42 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(25)
thf(fact_1242_PropVar_Odistinct_I27_J,axiom,
    ! [X41: nat,X42: nat,X61: nat,X62: nat] :
      ( ( multis6646701651571564453xOneJI @ X41 @ X42 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(27)
thf(fact_1243_PropVar_Odistinct_I7_J,axiom,
    ! [X11: nat,X12: nat,X51: nat,X52: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(7)
thf(fact_1244_PropVar_Odistinct_I9_J,axiom,
    ! [X11: nat,X12: nat,X61: nat,X62: nat] :
      ( ( multis387687052011358179_Gamma @ X11 @ X12 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(9)
thf(fact_1245_PropVar_Odistinct_I29_J,axiom,
    ! [X51: nat,X52: nat,X61: nat,X62: nat] :
      ( ( multis2983220944385456105ZeroIJ @ X51 @ X52 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(29)
thf(fact_1246_PropVar_Odistinct_I17_J,axiom,
    ! [X22: nat,X61: nat,X62: nat] :
      ( ( multis2544335231667181926psilon @ X22 )
     != ( multis6646701651571498855xOneIJ @ X61 @ X62 ) ) ).

% PropVar.distinct(17)
thf(fact_1247_PropVar_Odistinct_I15_J,axiom,
    ! [X22: nat,X51: nat,X52: nat] :
      ( ( multis2544335231667181926psilon @ X22 )
     != ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) ) ).

% PropVar.distinct(15)
thf(fact_1248_PropVar_Oexhaust,axiom,
    ! [Y3: multis3193088007478089820ropVar] :
      ( ! [X112: nat,X122: nat] :
          ( Y3
         != ( multis387687052011358179_Gamma @ X112 @ X122 ) )
     => ( ! [X23: nat] :
            ( Y3
           != ( multis2544335231667181926psilon @ X23 ) )
       => ( ! [X312: nat,X322: nat] :
              ( Y3
             != ( multis2983220944385521703ZeroJI @ X312 @ X322 ) )
         => ( ! [X412: nat,X422: nat] :
                ( Y3
               != ( multis6646701651571564453xOneJI @ X412 @ X422 ) )
           => ( ! [X512: nat,X522: nat] :
                  ( Y3
                 != ( multis2983220944385456105ZeroIJ @ X512 @ X522 ) )
             => ~ ! [X612: nat,X622: nat] :
                    ( Y3
                   != ( multis6646701651571498855xOneIJ @ X612 @ X622 ) ) ) ) ) ) ) ).

% PropVar.exhaust
thf(fact_1249_PropVar_Osize_I12_J,axiom,
    ! [X61: nat,X62: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis6646701651571498855xOneIJ @ X61 @ X62 ) )
      = zero_zero_nat ) ).

% PropVar.size(12)
thf(fact_1250_PropVar_Osize_I11_J,axiom,
    ! [X51: nat,X52: nat] :
      ( ( size_s6253272723116879048ropVar @ ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) )
      = zero_zero_nat ) ).

% PropVar.size(11)
thf(fact_1251_PropVar_Osize__gen_I6_J,axiom,
    ! [X61: nat,X62: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis6646701651571498855xOneIJ @ X61 @ X62 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(6)
thf(fact_1252_PropVar_Osize__gen_I5_J,axiom,
    ! [X51: nat,X52: nat] :
      ( ( multis2955979900537361535ropVar @ ( multis2983220944385456105ZeroIJ @ X51 @ X52 ) )
      = zero_zero_nat ) ).

% PropVar.size_gen(5)
thf(fact_1253_drop__upt,axiom,
    ! [M2: nat,I2: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I2 @ J ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_1254__092_060open_062v_A_092_060equiv_062_A_092_060lambda_062x_O_Acase_Ax_Aof_AAuxZeroJI_Ai_Aj_A_092_060Rightarrow_062_A_092_060not_062_ABex_A_Iset_A_Idrop_Ai_A_Imap_A_I_092_060lambda_062i_O_AGamma_Ai_Aj_J_A_0910_O_O_060n_093_J_J_J_Av_H_A_124_AAuxOneJI_Ai_Aj_A_092_060Rightarrow_062_A_092_060exists_062_By_O_Ay_A_092_060in_062_Aset_A_Idrop_Ai_A_Imap_A_I_092_060lambda_062i_O_AGamma_Ai_Aj_J_A_0910_O_O_060n_093_J_J_A_092_060and_062_Av_H_Ay_A_124_AAuxZeroIJ_Ai_Aj_A_092_060Rightarrow_062_A_092_060not_062_ABex_A_Iset_A_Idrop_Aj_A_Imap_A_IGamma_Ai_J_A_0910_O_O_060m_093_J_J_J_Av_H_A_124_AAuxOneIJ_Ai_Aj_A_092_060Rightarrow_062_A_092_060exists_062_By_O_Ay_A_092_060in_062_Aset_A_Idrop_Aj_A_Imap_A_IGamma_Ai_J_A_0910_O_O_060m_093_J_J_A_092_060and_062_Av_H_Ay_A_124_A___A_092_060Rightarrow_062_Av_H_Ax_092_060close_062,axiom,
    ( v2
    = ( ^ [X: multis3193088007478089820ropVar] :
          ( multis8641292176345589959pVar_o
          @ ^ [Nat1: nat,Nat22: nat] : ( v @ X )
          @ ^ [Nat4: nat] : ( v @ X )
          @ ^ [I: nat,J2: nat] :
              ~ ? [Y: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y
                    @ ( set_Mu146256361055586289ropVar
                      @ ( drop_M2293212096379261346ropVar @ I
                        @ ( map_na2466558649495876678ropVar
                          @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J2 )
                          @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                  & ( v @ Y ) )
          @ ^ [I: nat,J2: nat] :
            ? [Y: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Y
                @ ( set_Mu146256361055586289ropVar
                  @ ( drop_M2293212096379261346ropVar @ I
                    @ ( map_na2466558649495876678ropVar
                      @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J2 )
                      @ ( upt @ zero_zero_nat @ n ) ) ) ) )
              & ( v @ Y )
              & ! [Z: multis3193088007478089820ropVar] :
                  ( ( ( member8421201611525043589ropVar @ Z
                      @ ( set_Mu146256361055586289ropVar
                        @ ( drop_M2293212096379261346ropVar @ I
                          @ ( map_na2466558649495876678ropVar
                            @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J2 )
                            @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                    & ( v @ Z ) )
                 => ( Z = Y ) ) )
          @ ^ [I: nat,J2: nat] :
              ~ ? [Y: multis3193088007478089820ropVar] :
                  ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                  & ( v @ Y ) )
          @ ^ [I: nat,J2: nat] :
            ? [Y: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
              & ( v @ Y )
              & ! [Z: multis3193088007478089820ropVar] :
                  ( ( ( member8421201611525043589ropVar @ Z @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                    & ( v @ Z ) )
                 => ( Z = Y ) ) )
          @ X ) ) ) ).

% \<open>v \<equiv> \<lambda>x. case x of AuxZeroJI i j \<Rightarrow> \<not> Bex (set (drop i (map (\<lambda>i. Gamma i j) [0..<n]))) v' | AuxOneJI i j \<Rightarrow> \<exists>!y. y \<in> set (drop i (map (\<lambda>i. Gamma i j) [0..<n])) \<and> v' y | AuxZeroIJ i j \<Rightarrow> \<not> Bex (set (drop j (map (Gamma i) [0..<m]))) v' | AuxOneIJ i j \<Rightarrow> \<exists>!y. y \<in> set (drop j (map (Gamma i) [0..<m])) \<and> v' y | _ \<Rightarrow> v' x\<close>
thf(fact_1255_v__def,axiom,
    ! [X2: multis3193088007478089820ropVar] :
      ( ( v2 @ X2 )
      = ( multis8641292176345589959pVar_o
        @ ^ [Nat1: nat,Nat22: nat] : ( v @ X2 )
        @ ^ [Nat4: nat] : ( v @ X2 )
        @ ^ [I: nat,J2: nat] :
            ~ ? [X: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X
                  @ ( set_Mu146256361055586289ropVar
                    @ ( drop_M2293212096379261346ropVar @ I
                      @ ( map_na2466558649495876678ropVar
                        @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J2 )
                        @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                & ( v @ X ) )
        @ ^ [I: nat,J2: nat] :
          ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X
              @ ( set_Mu146256361055586289ropVar
                @ ( drop_M2293212096379261346ropVar @ I
                  @ ( map_na2466558649495876678ropVar
                    @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J2 )
                    @ ( upt @ zero_zero_nat @ n ) ) ) ) )
            & ( v @ X )
            & ! [Y: multis3193088007478089820ropVar] :
                ( ( ( member8421201611525043589ropVar @ Y
                    @ ( set_Mu146256361055586289ropVar
                      @ ( drop_M2293212096379261346ropVar @ I
                        @ ( map_na2466558649495876678ropVar
                          @ ^ [K3: nat] : ( multis387687052011358179_Gamma @ K3 @ J2 )
                          @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                  & ( v @ Y ) )
               => ( Y = X ) ) )
        @ ^ [I: nat,J2: nat] :
            ~ ? [X: multis3193088007478089820ropVar] :
                ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                & ( v @ X ) )
        @ ^ [I: nat,J2: nat] :
          ? [X: multis3193088007478089820ropVar] :
            ( ( member8421201611525043589ropVar @ X @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
            & ( v @ X )
            & ! [Y: multis3193088007478089820ropVar] :
                ( ( ( member8421201611525043589ropVar @ Y @ ( set_Mu146256361055586289ropVar @ ( drop_M2293212096379261346ropVar @ J2 @ ( map_na2466558649495876678ropVar @ ( multis387687052011358179_Gamma @ I ) @ ( upt @ zero_zero_nat @ m ) ) ) ) )
                  & ( v @ Y ) )
               => ( Y = X ) ) )
        @ X2 ) ) ).

% v_def
thf(fact_1256_Euclidean__Division_Odivmod__nat__if,axiom,
    ( euclidean_divmod_nat
    = ( ^ [M6: nat,N5: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N5 = zero_zero_nat )
            | ( ord_less_nat @ M6 @ N5 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( euclidean_divmod_nat @ ( minus_minus_nat @ M6 @ N5 ) @ N5 ) ) ) ) ) ).

% Euclidean_Division.divmod_nat_if
thf(fact_1257_inf__concat__simple_Osimps_I2_J,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( inf_concat_simple @ F @ ( suc @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [I: nat,J2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J2 ) @ ( F @ I ) ) @ ( product_Pair_nat_nat @ I @ ( suc @ J2 ) ) @ ( product_Pair_nat_nat @ ( suc @ I ) @ zero_zero_nat ) )
        @ ( inf_concat_simple @ F @ N2 ) ) ) ).

% inf_concat_simple.simps(2)
thf(fact_1258_inf__concat__simple_Osimps_I1_J,axiom,
    ! [F: nat > nat] :
      ( ( inf_concat_simple @ F @ zero_zero_nat )
      = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).

% inf_concat_simple.simps(1)
thf(fact_1259_inf__concat__simple__surj__zero,axiom,
    ! [F: nat > nat,I2: nat] :
    ? [K2: nat] :
      ( ( inf_concat_simple @ F @ K2 )
      = ( product_Pair_nat_nat @ I2 @ zero_zero_nat ) ) ).

% inf_concat_simple_surj_zero
thf(fact_1260_inf__concat__simple__surj,axiom,
    ! [J: nat,F: nat > nat,I2: nat] :
      ( ( ord_less_nat @ J @ ( F @ I2 ) )
     => ? [K2: nat] :
          ( ( inf_concat_simple @ F @ K2 )
          = ( product_Pair_nat_nat @ I2 @ J ) ) ) ).

% inf_concat_simple_surj
thf(fact_1261_inf__concat__simple__add,axiom,
    ! [F: nat > nat,K: nat,I2: nat,J: nat,L: nat] :
      ( ( ( inf_concat_simple @ F @ K )
        = ( product_Pair_nat_nat @ I2 @ J ) )
     => ( ( ord_less_nat @ ( plus_plus_nat @ J @ L ) @ ( F @ I2 ) )
       => ( ( inf_concat_simple @ F @ ( plus_plus_nat @ K @ L ) )
          = ( product_Pair_nat_nat @ I2 @ ( plus_plus_nat @ J @ L ) ) ) ) ) ).

% inf_concat_simple_add
thf(fact_1262_inf__concat__simple_Oelims,axiom,
    ! [X2: nat > nat,Xa: nat,Y3: product_prod_nat_nat] :
      ( ( ( inf_concat_simple @ X2 @ Xa )
        = Y3 )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y3
           != ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) )
       => ~ ! [N3: nat] :
              ( ( Xa
                = ( suc @ N3 ) )
             => ( Y3
               != ( produc2626176000494625587at_nat
                  @ ^ [I: nat,J2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J2 ) @ ( X2 @ I ) ) @ ( product_Pair_nat_nat @ I @ ( suc @ J2 ) ) @ ( product_Pair_nat_nat @ ( suc @ I ) @ zero_zero_nat ) )
                  @ ( inf_concat_simple @ X2 @ N3 ) ) ) ) ) ) ).

% inf_concat_simple.elims

% Helper facts (18)
thf(help_fChoice_1_1_fChoice_001tf__f_T,axiom,
    ! [P: f > $o] :
      ( ( P @ ( fChoice_f @ P ) )
      = ( ? [X5: f] : ( P @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( if_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X2: nat,Y3: nat] :
      ( ( if_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_fChoice_1_1_fChoice_001t__Nat__Onat_T,axiom,
    ! [P: nat > $o] :
      ( ( P @ ( fChoice_nat @ P ) )
      = ( ? [X5: nat] : ( P @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__List__Olist_Itf__f_J_T,axiom,
    ! [X2: list_f,Y3: list_f] :
      ( ( if_list_f @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__f_J_T,axiom,
    ! [X2: list_f,Y3: list_f] :
      ( ( if_list_f @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X2: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__List__Olist_Itf__f_J_J_T,axiom,
    ! [X2: list_list_f,Y3: list_list_f] :
      ( ( if_list_list_f @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__List__Olist_Itf__f_J_J_T,axiom,
    ! [X2: list_list_f,Y3: list_list_f] :
      ( ( if_list_list_f @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_fChoice_1_1_fChoice_001t__Multiset____Ordering____in____NP__OPropVar_T,axiom,
    ! [P: multis3193088007478089820ropVar > $o] :
      ( ( P @ ( fChoic539809714188224432ropVar @ P ) )
      = ( ? [X5: multis3193088007478089820ropVar] : ( P @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X2: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X2 @ Y3 )
      = X2 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_T,axiom,
    ! [X2: list_M6740372356978808802ropVar,Y3: list_M6740372356978808802ropVar] :
      ( ( if_lis9215432529438213544ropVar @ $false @ X2 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Multiset____Ordering____in____NP__OPropVar_J_T,axiom,
    ! [X2: list_M6740372356978808802ropVar,Y3: list_M6740372356978808802ropVar] :
      ( ( if_lis9215432529438213544ropVar @ $true @ X2 @ Y3 )
      = X2 ) ).

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,
    ! [X2: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( if_lis1975434623467068862ropVar @ $false @ X2 @ 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,
    ! [X2: list_P8791008543742759928ropVar,Y3: list_P8791008543742759928ropVar] :
      ( ( if_lis1975434623467068862ropVar @ $true @ X2 @ Y3 )
      = X2 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ( ! [X3: f] :
        ( ( member_f @ X3 @ ( set_f2 @ cands ) )
       => ( eval @ v2 @ X3 ) )
    & ( ( eval @ v2 @ one2 )
     != ( ~ ? [X: multis3193088007478089820ropVar] :
              ( ( member8421201611525043589ropVar @ X
                @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar
                  @ ( set_Pr3946994337432900359ropVar
                    @ ( map_na2145702811007361372ropVar
                      @ ^ [I: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ I @ j ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385521703ZeroJI @ I @ j ) @ ( multis6646701651571564453xOneJI @ I @ j ) ) )
                      @ ( upt @ zero_zero_nat @ n ) ) ) ) )
              & ( v2 @ X )
              & ! [Y: multis3193088007478089820ropVar] :
                  ( ( ( member8421201611525043589ropVar @ Y
                      @ ( image_4513341045867859311ropVar @ produc6918815732823278798ropVar
                        @ ( set_Pr3946994337432900359ropVar
                          @ ( map_na2145702811007361372ropVar
                            @ ^ [I: nat] : ( produc6194868611276206818ropVar @ ( multis387687052011358179_Gamma @ I @ j ) @ ( produc6936306168532358295ropVar @ ( multis2983220944385521703ZeroJI @ I @ j ) @ ( multis6646701651571564453xOneJI @ I @ j ) ) )
                            @ ( upt @ zero_zero_nat @ n ) ) ) ) )
                    & ( v2 @ Y ) )
                 => ( Y = X ) ) ) ) ) ) ).

%------------------------------------------------------------------------------