%------------------------------------------------------------------------------
% 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/0004_RPO_NP_Hard/prob_00107_005039__14250790_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1814 ( 627 unt; 542 typ;   0 def)
%            Number of atoms       : 4308 (1349 equ;   0 cnn)
%            Maximal formula atoms :   27 (   3 avg)
%            Number of connectives : 14631 ( 382   ~;  25   |; 325   &;12486   @)
%                                         (   0 <=>;1413  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   24 (   7 avg)
%            Number of types       :  112 ( 111 usr)
%            Number of type conns  : 2496 (2496   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  434 ( 431 usr;  32 con; 0-6 aty)
%            Number of variables   : 4911 (1342   ^;3520   !;  49   ?;4911   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:51.762
%------------------------------------------------------------------------------
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thf(ty_n_t__Multiset__Omultiset_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Set__Oset_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__List__Olist_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
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thf(ty_n_t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__RPO____NP____Hard__OFSyms_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
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thf(ty_n_t__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_Itf__a_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(ty_n_t__Multiset__Omultiset_I_Eo_J,type,
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thf(ty_n_t__WPO__Oorder____tag,type,
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thf(ty_n_t__List__Olist_I_Eo_J,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_tf__a,type,
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% Explicit typings (431)
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thf(sy_c_Groups_Ominus__class_Ominus_001_062_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_M_Eo_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001_062_I_Eo_M_062_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost_001t__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
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thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
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thf(sy_c_Set__Interval_Oord_OgreaterThanAtMost_001t__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Set__Interval_Oord_OgreaterThanLessThan_001t__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
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    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

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    member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $o ).

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    member4585374397974051670on_nat: produc118098540238352959on_nat > set_Pr737724919871395061on_nat > $o ).

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    member2831970296344990928_a_nat: produc9222932313792378791_a_nat > set_Pr6005870677871970951_a_nat > $o ).

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    member460288039655959140on_nat: produc4501289166466631099on_nat > set_Pr4220151837912703003on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Multiset__Omultiset_It__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    member348394700049801936_a_nat: produc3567963958052727207_a_nat > set_Pr2856029803362396295_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    member4278285197320822468on_nat: produc9176241080893687067on_nat > set_Pr3573155178515677307on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J,type,
    member6928026473249828560_a_nat: produc5238312538367656103_a_nat > set_Pr8529752790658994311_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__List__Olist_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    member3390692376801263108on_nat: produc4744472034076538459on_nat > set_Pr8800748304379094971on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    member110630379338885904_a_nat: produc1372005301246316263_a_nat > set_Pr1670232986683020487_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Multiset__Omultiset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    member2295143444729160836on_nat: produc1578829960824226011on_nat > set_Pr3515720519750671931on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__List__Olist_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_Mt__Product____Type__Oprod_It__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_Mt__Multiset__Omultiset_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J_J_J,type,
    member4535363888883972368_a_nat: produc6700810439531315431_a_nat > set_Pr5342889764443927239_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    member5906418179908956708on_nat: produc1313295348606329723on_nat > set_Pr2535564044409765851on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_Mt__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J_J,type,
    member6937586146271652048_a_nat: produc3043957382837978535_a_nat > set_Pr630484882382513799_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_Eo_J,type,
    member2997933407038931562od_a_o: product_prod_a_o > set_Product_prod_a_o > $o ).

thf(sy_c_member_001t__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J,type,
    member8476201367316484836on_nat: sum_su4907927656612468301on_nat > set_Su266273576555847555on_nat > $o ).

thf(sy_c_member_001t__Term__Oterm_It__RPO____NP____Hard__OFSyms_Mt__Sum____Type__Osum_Itf__a_Mt__Nat__Onat_J_J,type,
    member7260319590701762419_a_nat: term_R1747997765655105994_a_nat > set_te1179867837402157482_a_nat > $o ).

thf(sy_v_NS____,type,
    ns: list_P9127870862658512897on_nat ).

thf(sy_v_S____,type,
    s: list_P9127870862658512897on_nat ).

thf(sy_v_as,type,
    as: list_t5668669291736003536_a_nat ).

thf(sy_v_bs,type,
    bs: list_t5668669291736003536_a_nat ).

thf(sy_v_cnf,type,
    cnf: list_l8336264072933094989od_a_o ).

thf(sy_v_xs____,type,
    xs: list_S5253404628753938141on_nat ).

thf(sy_v_ys____,type,
    ys: list_S5253404628753938141on_nat ).

% Relevant facts (1271)
thf(fact_0_trivial__rpo_Oprl,axiom,
    product_snd_o_o @ ( product_Pair_o_o @ $false @ $true ) ).

% trivial_rpo.prl
thf(fact_1_trivial__rpo_Oprl2,axiom,
    ~ ( product_fst_o_o @ ( product_Pair_o_o @ $false @ $true ) ) ).

% trivial_rpo.prl2
thf(fact_2_trivial__rpo_Oprc__compat,axiom,
    ! [S1: $o,Ns1: $o,S2: $o,Ns2: $o,S: $o,Ns: $o] :
      ( ( ( product_Pair_o_o @ $false @ $true )
        = ( product_Pair_o_o @ S1 @ Ns1 ) )
     => ( ( ( product_Pair_o_o @ $false @ $true )
          = ( product_Pair_o_o @ S2 @ Ns2 ) )
       => ( ( ( product_Pair_o_o @ $false @ $true )
            = ( product_Pair_o_o @ S @ Ns ) )
         => ( ( ( Ns1
                & Ns2 )
             => Ns )
            & ( ( Ns1
                & S2 )
             => S )
            & ( ( S1
                & Ns2 )
             => S ) ) ) ) ) ).

% trivial_rpo.prc_compat
thf(fact_3_trivial__rpo_Oprc__stri__imp__nstri,axiom,
    ( ( product_fst_o_o @ ( product_Pair_o_o @ $false @ $true ) )
   => ( product_snd_o_o @ ( product_Pair_o_o @ $false @ $true ) ) ) ).

% trivial_rpo.prc_stri_imp_nstri
thf(fact_4_assms_I1_J,axiom,
    ( ( rPO_NP4415194297383320304_cnf_a @ cnf )
    = ( produc5685097156399411607_a_nat @ as @ bs ) ) ).

% assms(1)
thf(fact_5_prod_Ocollapse,axiom,
    ! [Prod: produc3043957382837978535_a_nat] :
      ( ( produc4880773030396931735_a_nat @ ( produc2126783593110491267_a_nat @ Prod ) @ ( produc2284112530981304773_a_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_6_prod_Ocollapse,axiom,
    ! [Prod: produc118098540238352959on_nat] :
      ( ( produc4176415733022414961on_nat @ ( produc6765150182969198981on_nat @ Prod ) @ ( produc5312734412660135107on_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_7_prod_Ocollapse,axiom,
    ! [Prod: produc8325617224762090749on_nat] :
      ( ( produc1545487600272464053on_nat @ ( produc311592285738829985on_nat @ Prod ) @ ( produc3934153714877691619on_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_8_prod_Ocollapse,axiom,
    ! [Prod: produc5580075136493545115on_nat] :
      ( ( produc2912653624306349395on_nat @ ( produc1822290298201838911on_nat @ Prod ) @ ( produc2374478376588790657on_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_9_prod_Ocollapse,axiom,
    ! [Prod: produc394670695091157671_a_nat] :
      ( ( produc5685097156399411607_a_nat @ ( produc3471587812718774147_a_nat @ Prod ) @ ( produc4727165237793840837_a_nat @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_10_prod_Ocollapse,axiom,
    ! [Prod: product_prod_o_o] :
      ( ( product_Pair_o_o @ ( product_fst_o_o @ Prod ) @ ( product_snd_o_o @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_11_case__prodI,axiom,
    ! [F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat] :
      ( ( F @ A @ B )
     => ( produc1360909077854731642_nat_o @ F @ ( produc4176415733022414961on_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_12_case__prodI,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] :
      ( ( F @ A @ B )
     => ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_13_case__prodI,axiom,
    ! [F: $o > $o > $o,A: $o,B: $o] :
      ( ( F @ A @ B )
     => ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ A @ B ) ) ) ).

% case_prodI
thf(fact_14_case__prodI,axiom,
    ! [F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,A: list_t5668669291736003536_a_nat,B: list_t5668669291736003536_a_nat] :
      ( ( F @ A @ B )
     => ( produc4945827410475597548_nat_o @ F @ ( produc5685097156399411607_a_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_15_case__prodI,axiom,
    ! [F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( F @ A @ B )
     => ( produc4098677624135445996_nat_o @ F @ ( produc4880773030396931735_a_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_16_case__prodI,axiom,
    ! [F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat] :
      ( ( F @ A @ B )
     => ( produc7163990014243126832_nat_o @ F @ ( produc2912653624306349395on_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_17_case__prodI2,axiom,
    ! [P: produc394670695091157671_a_nat,C: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
          ( ( P
            = ( produc5685097156399411607_a_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc4945827410475597548_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_18_case__prodI2,axiom,
    ! [P: produc3043957382837978535_a_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( ( P
            = ( produc4880773030396931735_a_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc4098677624135445996_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_19_case__prodI2,axiom,
    ! [P: produc5580075136493545115on_nat,C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
          ( ( P
            = ( produc2912653624306349395on_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc7163990014243126832_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_20_case__prodI2,axiom,
    ! [P: product_prod_o_o,C: $o > $o > $o] :
      ( ! [A2: $o,B2: $o] :
          ( ( P
            = ( product_Pair_o_o @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc6197397395684419436_o_o_o @ C @ P ) ) ).

% case_prodI2
thf(fact_21_case__prodI2,axiom,
    ! [P: produc8325617224762090749on_nat,C: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
          ( ( P
            = ( produc1545487600272464053on_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc7728289706344822350_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_22_case__prodI2,axiom,
    ! [P: produc118098540238352959on_nat,C: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
          ( ( P
            = ( produc4176415733022414961on_nat @ A2 @ B2 ) )
         => ( C @ A2 @ B2 ) )
     => ( produc1360909077854731642_nat_o @ C @ P ) ) ).

% case_prodI2
thf(fact_23_case__prod__conv,axiom,
    ! [F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,A: list_t5668669291736003536_a_nat,B: list_t5668669291736003536_a_nat] :
      ( ( produc4945827410475597548_nat_o @ F @ ( produc5685097156399411607_a_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_24_case__prod__conv,axiom,
    ! [F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( produc4098677624135445996_nat_o @ F @ ( produc4880773030396931735_a_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_25_case__prod__conv,axiom,
    ! [F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat] :
      ( ( produc7163990014243126832_nat_o @ F @ ( produc2912653624306349395on_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_26_case__prod__conv,axiom,
    ! [F: $o > $o > $o,A: $o,B: $o] :
      ( ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_27_case__prod__conv,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] :
      ( ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_28_case__prod__conv,axiom,
    ! [F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat] :
      ( ( produc1360909077854731642_nat_o @ F @ ( produc4176415733022414961on_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_29_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Prod: produc394670695091157671_a_nat] :
      ( ( P2 @ ( produc4945827410475597548_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( produc5685097156399411607_a_nat @ ( produc3471587812718774147_a_nat @ Prod ) @ ( produc4727165237793840837_a_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( produc3471587812718774147_a_nat @ Prod ) @ ( produc4727165237793840837_a_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_30_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Prod: produc3043957382837978535_a_nat] :
      ( ( P2 @ ( produc4098677624135445996_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( produc4880773030396931735_a_nat @ ( produc2126783593110491267_a_nat @ Prod ) @ ( produc2284112530981304773_a_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( produc2126783593110491267_a_nat @ Prod ) @ ( produc2284112530981304773_a_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_31_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Prod: produc5580075136493545115on_nat] :
      ( ( P2 @ ( produc7163990014243126832_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( produc2912653624306349395on_nat @ ( produc1822290298201838911on_nat @ Prod ) @ ( produc2374478376588790657on_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( produc1822290298201838911on_nat @ Prod ) @ ( produc2374478376588790657on_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_32_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: $o > $o > $o,Prod: product_prod_o_o] :
      ( ( P2 @ ( produc6197397395684419436_o_o_o @ F @ Prod ) )
      = ( ( Prod
          = ( product_Pair_o_o @ ( product_fst_o_o @ Prod ) @ ( product_snd_o_o @ Prod ) ) )
       => ( P2 @ ( F @ ( product_fst_o_o @ Prod ) @ ( product_snd_o_o @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_33_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,Prod: produc8325617224762090749on_nat] :
      ( ( P2 @ ( produc7728289706344822350_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( produc1545487600272464053on_nat @ ( produc311592285738829985on_nat @ Prod ) @ ( produc3934153714877691619on_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( produc311592285738829985on_nat @ Prod ) @ ( produc3934153714877691619on_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_34_prod_Osplit__sel,axiom,
    ! [P2: $o > $o,F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,Prod: produc118098540238352959on_nat] :
      ( ( P2 @ ( produc1360909077854731642_nat_o @ F @ Prod ) )
      = ( ( Prod
          = ( produc4176415733022414961on_nat @ ( produc6765150182969198981on_nat @ Prod ) @ ( produc5312734412660135107on_nat @ Prod ) ) )
       => ( P2 @ ( F @ ( produc6765150182969198981on_nat @ Prod ) @ ( produc5312734412660135107on_nat @ Prod ) ) ) ) ) ).

% prod.split_sel
thf(fact_35_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Prod: produc394670695091157671_a_nat] :
      ( ( P2 @ ( produc4945827410475597548_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc5685097156399411607_a_nat @ ( produc3471587812718774147_a_nat @ Prod ) @ ( produc4727165237793840837_a_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( produc3471587812718774147_a_nat @ Prod ) @ ( produc4727165237793840837_a_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_36_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Prod: produc3043957382837978535_a_nat] :
      ( ( P2 @ ( produc4098677624135445996_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc4880773030396931735_a_nat @ ( produc2126783593110491267_a_nat @ Prod ) @ ( produc2284112530981304773_a_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( produc2126783593110491267_a_nat @ Prod ) @ ( produc2284112530981304773_a_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_37_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Prod: produc5580075136493545115on_nat] :
      ( ( P2 @ ( produc7163990014243126832_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc2912653624306349395on_nat @ ( produc1822290298201838911on_nat @ Prod ) @ ( produc2374478376588790657on_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( produc1822290298201838911on_nat @ Prod ) @ ( produc2374478376588790657on_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_38_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: $o > $o > $o,Prod: product_prod_o_o] :
      ( ( P2 @ ( produc6197397395684419436_o_o_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( product_Pair_o_o @ ( product_fst_o_o @ Prod ) @ ( product_snd_o_o @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( product_fst_o_o @ Prod ) @ ( product_snd_o_o @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_39_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,Prod: produc8325617224762090749on_nat] :
      ( ( P2 @ ( produc7728289706344822350_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc1545487600272464053on_nat @ ( produc311592285738829985on_nat @ Prod ) @ ( produc3934153714877691619on_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( produc311592285738829985on_nat @ Prod ) @ ( produc3934153714877691619on_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_40_prod_Osplit__sel__asm,axiom,
    ! [P2: $o > $o,F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,Prod: produc118098540238352959on_nat] :
      ( ( P2 @ ( produc1360909077854731642_nat_o @ F @ Prod ) )
      = ( ~ ( ( Prod
              = ( produc4176415733022414961on_nat @ ( produc6765150182969198981on_nat @ Prod ) @ ( produc5312734412660135107on_nat @ Prod ) ) )
            & ~ ( P2 @ ( F @ ( produc6765150182969198981on_nat @ Prod ) @ ( produc5312734412660135107on_nat @ Prod ) ) ) ) ) ) ).

% prod.split_sel_asm
thf(fact_41_split__part,axiom,
    ! [P2: $o,Q: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ( produc4945827410475597548_nat_o
        @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] :
            ( P2
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: produc394670695091157671_a_nat] :
            ( P2
            & ( produc4945827410475597548_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_42_split__part,axiom,
    ! [P2: $o,Q: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ( produc4098677624135445996_nat_o
        @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] :
            ( P2
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: produc3043957382837978535_a_nat] :
            ( P2
            & ( produc4098677624135445996_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_43_split__part,axiom,
    ! [P2: $o,Q: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ( produc7163990014243126832_nat_o
        @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] :
            ( P2
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: produc5580075136493545115on_nat] :
            ( P2
            & ( produc7163990014243126832_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_44_split__part,axiom,
    ! [P2: $o,Q: $o > $o > $o] :
      ( ( produc6197397395684419436_o_o_o
        @ ^ [A3: $o,B3: $o] :
            ( P2
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: product_prod_o_o] :
            ( P2
            & ( produc6197397395684419436_o_o_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_45_split__part,axiom,
    ! [P2: $o,Q: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ( produc7728289706344822350_nat_o
        @ ^ [A3: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] :
            ( P2
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: produc8325617224762090749on_nat] :
            ( P2
            & ( produc7728289706344822350_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_46_split__part,axiom,
    ! [P2: $o,Q: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ( produc1360909077854731642_nat_o
        @ ^ [A3: list_S5253404628753938141on_nat,B3: produc1714117091004681831on_nat] :
            ( P2
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: produc118098540238352959on_nat] :
            ( P2
            & ( produc1360909077854731642_nat_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_47_exE__realizer,axiom,
    ! [P2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,P: produc394670695091157671_a_nat,Q: $o > $o,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ( P2 @ ( produc4727165237793840837_a_nat @ P ) @ ( produc3471587812718774147_a_nat @ P ) )
     => ( ! [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
            ( ( P2 @ Y @ X )
           => ( Q @ ( F @ X @ Y ) ) )
       => ( Q @ ( produc4945827410475597548_nat_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_48_exE__realizer,axiom,
    ! [P2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,P: produc3043957382837978535_a_nat,Q: $o > $o,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ( P2 @ ( produc2284112530981304773_a_nat @ P ) @ ( produc2126783593110491267_a_nat @ P ) )
     => ( ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
            ( ( P2 @ Y @ X )
           => ( Q @ ( F @ X @ Y ) ) )
       => ( Q @ ( produc4098677624135445996_nat_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_49_exE__realizer,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,P: produc5580075136493545115on_nat,Q: $o > $o,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ( P2 @ ( produc2374478376588790657on_nat @ P ) @ ( produc1822290298201838911on_nat @ P ) )
     => ( ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
            ( ( P2 @ Y @ X )
           => ( Q @ ( F @ X @ Y ) ) )
       => ( Q @ ( produc7163990014243126832_nat_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_50_exE__realizer,axiom,
    ! [P2: $o > $o > $o,P: product_prod_o_o,Q: $o > $o,F: $o > $o > $o] :
      ( ( P2 @ ( product_snd_o_o @ P ) @ ( product_fst_o_o @ P ) )
     => ( ! [X: $o,Y: $o] :
            ( ( P2 @ Y @ X )
           => ( Q @ ( F @ X @ Y ) ) )
       => ( Q @ ( produc6197397395684419436_o_o_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_51_exE__realizer,axiom,
    ! [P2: produc118098540238352959on_nat > list_S5253404628753938141on_nat > $o,P: produc8325617224762090749on_nat,Q: $o > $o,F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ( P2 @ ( produc3934153714877691619on_nat @ P ) @ ( produc311592285738829985on_nat @ P ) )
     => ( ! [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
            ( ( P2 @ Y @ X )
           => ( Q @ ( F @ X @ Y ) ) )
       => ( Q @ ( produc7728289706344822350_nat_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_52_exE__realizer,axiom,
    ! [P2: produc1714117091004681831on_nat > list_S5253404628753938141on_nat > $o,P: produc118098540238352959on_nat,Q: $o > $o,F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ( P2 @ ( produc5312734412660135107on_nat @ P ) @ ( produc6765150182969198981on_nat @ P ) )
     => ( ! [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
            ( ( P2 @ Y @ X )
           => ( Q @ ( F @ X @ Y ) ) )
       => ( Q @ ( produc1360909077854731642_nat_o @ F @ P ) ) ) ) ).

% exE_realizer
thf(fact_53_case__prod__beta_H,axiom,
    ( produc4945827410475597548_nat_o
    = ( ^ [F2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,X2: produc394670695091157671_a_nat] : ( F2 @ ( produc3471587812718774147_a_nat @ X2 ) @ ( produc4727165237793840837_a_nat @ X2 ) ) ) ) ).

% case_prod_beta'
thf(fact_54_case__prod__beta_H,axiom,
    ( produc4098677624135445996_nat_o
    = ( ^ [F2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,X2: produc3043957382837978535_a_nat] : ( F2 @ ( produc2126783593110491267_a_nat @ X2 ) @ ( produc2284112530981304773_a_nat @ X2 ) ) ) ) ).

% case_prod_beta'
thf(fact_55_case__prod__beta_H,axiom,
    ( produc7163990014243126832_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,X2: produc5580075136493545115on_nat] : ( F2 @ ( produc1822290298201838911on_nat @ X2 ) @ ( produc2374478376588790657on_nat @ X2 ) ) ) ) ).

% case_prod_beta'
thf(fact_56_case__prod__beta_H,axiom,
    ( produc6197397395684419436_o_o_o
    = ( ^ [F2: $o > $o > $o,X2: product_prod_o_o] : ( F2 @ ( product_fst_o_o @ X2 ) @ ( product_snd_o_o @ X2 ) ) ) ) ).

% case_prod_beta'
thf(fact_57_case__prod__beta_H,axiom,
    ( produc7728289706344822350_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,X2: produc8325617224762090749on_nat] : ( F2 @ ( produc311592285738829985on_nat @ X2 ) @ ( produc3934153714877691619on_nat @ X2 ) ) ) ) ).

% case_prod_beta'
thf(fact_58_case__prod__beta_H,axiom,
    ( produc1360909077854731642_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,X2: produc118098540238352959on_nat] : ( F2 @ ( produc6765150182969198981on_nat @ X2 ) @ ( produc5312734412660135107on_nat @ X2 ) ) ) ) ).

% case_prod_beta'
thf(fact_59_case__prod__unfold,axiom,
    ( produc4945827410475597548_nat_o
    = ( ^ [C2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,P3: produc394670695091157671_a_nat] : ( C2 @ ( produc3471587812718774147_a_nat @ P3 ) @ ( produc4727165237793840837_a_nat @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_60_case__prod__unfold,axiom,
    ( produc4098677624135445996_nat_o
    = ( ^ [C2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,P3: produc3043957382837978535_a_nat] : ( C2 @ ( produc2126783593110491267_a_nat @ P3 ) @ ( produc2284112530981304773_a_nat @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_61_case__prod__unfold,axiom,
    ( produc7163990014243126832_nat_o
    = ( ^ [C2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,P3: produc5580075136493545115on_nat] : ( C2 @ ( produc1822290298201838911on_nat @ P3 ) @ ( produc2374478376588790657on_nat @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_62_case__prod__unfold,axiom,
    ( produc6197397395684419436_o_o_o
    = ( ^ [C2: $o > $o > $o,P3: product_prod_o_o] : ( C2 @ ( product_fst_o_o @ P3 ) @ ( product_snd_o_o @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_63_case__prod__unfold,axiom,
    ( produc7728289706344822350_nat_o
    = ( ^ [C2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,P3: produc8325617224762090749on_nat] : ( C2 @ ( produc311592285738829985on_nat @ P3 ) @ ( produc3934153714877691619on_nat @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_64_case__prod__unfold,axiom,
    ( produc1360909077854731642_nat_o
    = ( ^ [C2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,P3: produc118098540238352959on_nat] : ( C2 @ ( produc6765150182969198981on_nat @ P3 ) @ ( produc5312734412660135107on_nat @ P3 ) ) ) ) ).

% case_prod_unfold
thf(fact_65_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat,A4: list_S5253404628753938141on_nat,B4: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B )
        = ( produc4176415733022414961on_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_66_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A4: list_S5253404628753938141on_nat,B4: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_67_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,A4: list_S5253404628753938141on_nat,B4: list_S5253404628753938141on_nat] :
      ( ( ( produc2912653624306349395on_nat @ A @ B )
        = ( produc2912653624306349395on_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_68_old_Oprod_Oinject,axiom,
    ! [A: list_t5668669291736003536_a_nat,B: list_t5668669291736003536_a_nat,A4: list_t5668669291736003536_a_nat,B4: list_t5668669291736003536_a_nat] :
      ( ( ( produc5685097156399411607_a_nat @ A @ B )
        = ( produc5685097156399411607_a_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_69_old_Oprod_Oinject,axiom,
    ! [A: $o,B: $o,A4: $o,B4: $o] :
      ( ( ( product_Pair_o_o @ A @ B )
        = ( product_Pair_o_o @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_70_old_Oprod_Oinject,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,A4: term_R1747997765655105994_a_nat,B4: term_R1747997765655105994_a_nat] :
      ( ( ( produc4880773030396931735_a_nat @ A @ B )
        = ( produc4880773030396931735_a_nat @ A4 @ B4 ) )
      = ( ( A = A4 )
        & ( B = B4 ) ) ) ).

% old.prod.inject
thf(fact_71_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc1714117091004681831on_nat,Y1: list_S5253404628753938141on_nat,Y2: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ X1 @ X22 )
        = ( produc4176415733022414961on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_72_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc118098540238352959on_nat,Y1: list_S5253404628753938141on_nat,Y2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ X1 @ X22 )
        = ( produc1545487600272464053on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_73_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: list_S5253404628753938141on_nat,Y1: list_S5253404628753938141on_nat,Y2: list_S5253404628753938141on_nat] :
      ( ( ( produc2912653624306349395on_nat @ X1 @ X22 )
        = ( produc2912653624306349395on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_74_prod_Oinject,axiom,
    ! [X1: list_t5668669291736003536_a_nat,X22: list_t5668669291736003536_a_nat,Y1: list_t5668669291736003536_a_nat,Y2: list_t5668669291736003536_a_nat] :
      ( ( ( produc5685097156399411607_a_nat @ X1 @ X22 )
        = ( produc5685097156399411607_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_75_prod_Oinject,axiom,
    ! [X1: $o,X22: $o,Y1: $o,Y2: $o] :
      ( ( ( product_Pair_o_o @ X1 @ X22 )
        = ( product_Pair_o_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_76_prod_Oinject,axiom,
    ! [X1: term_R1747997765655105994_a_nat,X22: term_R1747997765655105994_a_nat,Y1: term_R1747997765655105994_a_nat,Y2: term_R1747997765655105994_a_nat] :
      ( ( ( produc4880773030396931735_a_nat @ X1 @ X22 )
        = ( produc4880773030396931735_a_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_77_mem__case__prodI2,axiom,
    ! [P: product_prod_o_o,Z: produc394670695091157671_a_nat,C: $o > $o > set_Pr4991467605864603271_a_nat] :
      ( ! [A2: $o,B2: $o] :
          ( ( P
            = ( product_Pair_o_o @ A2 @ B2 ) )
         => ( member4231271695026834640_a_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member4231271695026834640_a_nat @ Z @ ( produc562102371737676891_a_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_78_mem__case__prodI2,axiom,
    ! [P: product_prod_o_o,Z: produc5238312538367656103_a_nat,C: $o > $o > set_Pr8529752790658994311_a_nat] :
      ( ! [A2: $o,B2: $o] :
          ( ( P
            = ( product_Pair_o_o @ A2 @ B2 ) )
         => ( member6928026473249828560_a_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member6928026473249828560_a_nat @ Z @ ( produc5637684430685687387_a_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_79_mem__case__prodI2,axiom,
    ! [P: product_prod_o_o,Z: produc9176241080893687067on_nat,C: $o > $o > set_Pr3573155178515677307on_nat] :
      ( ! [A2: $o,B2: $o] :
          ( ( P
            = ( product_Pair_o_o @ A2 @ B2 ) )
         => ( member4278285197320822468on_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member4278285197320822468on_nat @ Z @ ( produc7610502994550608975on_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_80_mem__case__prodI2,axiom,
    ! [P: product_prod_o_o,Z: produc5580075136493545115on_nat,C: $o > $o > set_Pr7828957961435280891on_nat] :
      ( ! [A2: $o,B2: $o] :
          ( ( P
            = ( product_Pair_o_o @ A2 @ B2 ) )
         => ( member4221009242412147780on_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member4221009242412147780on_nat @ Z @ ( produc101180125040630223on_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_81_mem__case__prodI2,axiom,
    ! [P: produc3043957382837978535_a_nat,Z: produc394670695091157671_a_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr4991467605864603271_a_nat] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( ( P
            = ( produc4880773030396931735_a_nat @ A2 @ B2 ) )
         => ( member4231271695026834640_a_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member4231271695026834640_a_nat @ Z @ ( produc1335742200159302363_a_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_82_mem__case__prodI2,axiom,
    ! [P: produc3043957382837978535_a_nat,Z: produc5238312538367656103_a_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr8529752790658994311_a_nat] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( ( P
            = ( produc4880773030396931735_a_nat @ A2 @ B2 ) )
         => ( member6928026473249828560_a_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member6928026473249828560_a_nat @ Z @ ( produc142518736041383131_a_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_83_mem__case__prodI2,axiom,
    ! [P: produc3043957382837978535_a_nat,Z: produc9176241080893687067on_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr3573155178515677307on_nat] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( ( P
            = ( produc4880773030396931735_a_nat @ A2 @ B2 ) )
         => ( member4278285197320822468on_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member4278285197320822468on_nat @ Z @ ( produc8016192610482346703on_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_84_mem__case__prodI2,axiom,
    ! [P: produc3043957382837978535_a_nat,Z: produc5580075136493545115on_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr7828957961435280891on_nat] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( ( P
            = ( produc4880773030396931735_a_nat @ A2 @ B2 ) )
         => ( member4221009242412147780on_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member4221009242412147780on_nat @ Z @ ( produc924006725934604367on_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_85_mem__case__prodI2,axiom,
    ! [P: produc5580075136493545115on_nat,Z: produc394670695091157671_a_nat,C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > set_Pr4991467605864603271_a_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
          ( ( P
            = ( produc2912653624306349395on_nat @ A2 @ B2 ) )
         => ( member4231271695026834640_a_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member4231271695026834640_a_nat @ Z @ ( produc5862169952008903967_a_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_86_mem__case__prodI2,axiom,
    ! [P: produc5580075136493545115on_nat,Z: produc5238312538367656103_a_nat,C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > set_Pr8529752790658994311_a_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
          ( ( P
            = ( produc2912653624306349395on_nat @ A2 @ B2 ) )
         => ( member6928026473249828560_a_nat @ Z @ ( C @ A2 @ B2 ) ) )
     => ( member6928026473249828560_a_nat @ Z @ ( produc4811812625541345055_a_nat @ C @ P ) ) ) ).

% mem_case_prodI2
thf(fact_87_mem__case__prodI,axiom,
    ! [Z: produc394670695091157671_a_nat,C: $o > $o > set_Pr4991467605864603271_a_nat,A: $o,B: $o] :
      ( ( member4231271695026834640_a_nat @ Z @ ( C @ A @ B ) )
     => ( member4231271695026834640_a_nat @ Z @ ( produc562102371737676891_a_nat @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_88_mem__case__prodI,axiom,
    ! [Z: produc5238312538367656103_a_nat,C: $o > $o > set_Pr8529752790658994311_a_nat,A: $o,B: $o] :
      ( ( member6928026473249828560_a_nat @ Z @ ( C @ A @ B ) )
     => ( member6928026473249828560_a_nat @ Z @ ( produc5637684430685687387_a_nat @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_89_mem__case__prodI,axiom,
    ! [Z: produc9176241080893687067on_nat,C: $o > $o > set_Pr3573155178515677307on_nat,A: $o,B: $o] :
      ( ( member4278285197320822468on_nat @ Z @ ( C @ A @ B ) )
     => ( member4278285197320822468on_nat @ Z @ ( produc7610502994550608975on_nat @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_90_mem__case__prodI,axiom,
    ! [Z: produc5580075136493545115on_nat,C: $o > $o > set_Pr7828957961435280891on_nat,A: $o,B: $o] :
      ( ( member4221009242412147780on_nat @ Z @ ( C @ A @ B ) )
     => ( member4221009242412147780on_nat @ Z @ ( produc101180125040630223on_nat @ C @ ( product_Pair_o_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_91_mem__case__prodI,axiom,
    ! [Z: produc394670695091157671_a_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr4991467605864603271_a_nat,A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member4231271695026834640_a_nat @ Z @ ( C @ A @ B ) )
     => ( member4231271695026834640_a_nat @ Z @ ( produc1335742200159302363_a_nat @ C @ ( produc4880773030396931735_a_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_92_mem__case__prodI,axiom,
    ! [Z: produc5238312538367656103_a_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr8529752790658994311_a_nat,A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6928026473249828560_a_nat @ Z @ ( C @ A @ B ) )
     => ( member6928026473249828560_a_nat @ Z @ ( produc142518736041383131_a_nat @ C @ ( produc4880773030396931735_a_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_93_mem__case__prodI,axiom,
    ! [Z: produc9176241080893687067on_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr3573155178515677307on_nat,A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member4278285197320822468on_nat @ Z @ ( C @ A @ B ) )
     => ( member4278285197320822468on_nat @ Z @ ( produc8016192610482346703on_nat @ C @ ( produc4880773030396931735_a_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_94_mem__case__prodI,axiom,
    ! [Z: produc5580075136493545115on_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr7828957961435280891on_nat,A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member4221009242412147780on_nat @ Z @ ( C @ A @ B ) )
     => ( member4221009242412147780on_nat @ Z @ ( produc924006725934604367on_nat @ C @ ( produc4880773030396931735_a_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_95_mem__case__prodI,axiom,
    ! [Z: produc394670695091157671_a_nat,C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > set_Pr4991467605864603271_a_nat,A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat] :
      ( ( member4231271695026834640_a_nat @ Z @ ( C @ A @ B ) )
     => ( member4231271695026834640_a_nat @ Z @ ( produc5862169952008903967_a_nat @ C @ ( produc2912653624306349395on_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_96_mem__case__prodI,axiom,
    ! [Z: produc5238312538367656103_a_nat,C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > set_Pr8529752790658994311_a_nat,A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat] :
      ( ( member6928026473249828560_a_nat @ Z @ ( C @ A @ B ) )
     => ( member6928026473249828560_a_nat @ Z @ ( produc4811812625541345055_a_nat @ C @ ( produc2912653624306349395on_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_97_non__triv,axiom,
    cnf != nil_li1372968520647675725od_a_o ).

% non_triv
thf(fact_98__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062xs_Ays_AS_ANS_O_Amultiset__problem__of__cnf_Acnf_A_061_A_Ixs_M_Ays_M_AS_M_ANS_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,S3: list_P9127870862658512897on_nat,NS: list_P9127870862658512897on_nat] :
        ( ( multis4221440410396274297_cnf_a @ cnf )
       != ( produc1545487600272464053on_nat @ Xs @ ( produc4176415733022414961on_nat @ Ys @ ( produc7152764266822961111on_nat @ S3 @ NS ) ) ) ) ).

% \<open>\<And>thesis. (\<And>xs ys S NS. multiset_problem_of_cnf cnf = (xs, ys, S, NS) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_99_mem__case__prodE,axiom,
    ! [Z: produc394670695091157671_a_nat,C: $o > $o > set_Pr4991467605864603271_a_nat,P: product_prod_o_o] :
      ( ( member4231271695026834640_a_nat @ Z @ ( produc562102371737676891_a_nat @ C @ P ) )
     => ~ ! [X: $o,Y: $o] :
            ( ( P
              = ( product_Pair_o_o @ X @ Y ) )
           => ~ ( member4231271695026834640_a_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_100_mem__case__prodE,axiom,
    ! [Z: produc5238312538367656103_a_nat,C: $o > $o > set_Pr8529752790658994311_a_nat,P: product_prod_o_o] :
      ( ( member6928026473249828560_a_nat @ Z @ ( produc5637684430685687387_a_nat @ C @ P ) )
     => ~ ! [X: $o,Y: $o] :
            ( ( P
              = ( product_Pair_o_o @ X @ Y ) )
           => ~ ( member6928026473249828560_a_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_101_mem__case__prodE,axiom,
    ! [Z: produc9176241080893687067on_nat,C: $o > $o > set_Pr3573155178515677307on_nat,P: product_prod_o_o] :
      ( ( member4278285197320822468on_nat @ Z @ ( produc7610502994550608975on_nat @ C @ P ) )
     => ~ ! [X: $o,Y: $o] :
            ( ( P
              = ( product_Pair_o_o @ X @ Y ) )
           => ~ ( member4278285197320822468on_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_102_mem__case__prodE,axiom,
    ! [Z: produc5580075136493545115on_nat,C: $o > $o > set_Pr7828957961435280891on_nat,P: product_prod_o_o] :
      ( ( member4221009242412147780on_nat @ Z @ ( produc101180125040630223on_nat @ C @ P ) )
     => ~ ! [X: $o,Y: $o] :
            ( ( P
              = ( product_Pair_o_o @ X @ Y ) )
           => ~ ( member4221009242412147780on_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_103_mem__case__prodE,axiom,
    ! [Z: produc394670695091157671_a_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr4991467605864603271_a_nat,P: produc3043957382837978535_a_nat] :
      ( ( member4231271695026834640_a_nat @ Z @ ( produc1335742200159302363_a_nat @ C @ P ) )
     => ~ ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
            ( ( P
              = ( produc4880773030396931735_a_nat @ X @ Y ) )
           => ~ ( member4231271695026834640_a_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_104_mem__case__prodE,axiom,
    ! [Z: produc5238312538367656103_a_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr8529752790658994311_a_nat,P: produc3043957382837978535_a_nat] :
      ( ( member6928026473249828560_a_nat @ Z @ ( produc142518736041383131_a_nat @ C @ P ) )
     => ~ ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
            ( ( P
              = ( produc4880773030396931735_a_nat @ X @ Y ) )
           => ~ ( member6928026473249828560_a_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_105_mem__case__prodE,axiom,
    ! [Z: produc9176241080893687067on_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr3573155178515677307on_nat,P: produc3043957382837978535_a_nat] :
      ( ( member4278285197320822468on_nat @ Z @ ( produc8016192610482346703on_nat @ C @ P ) )
     => ~ ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
            ( ( P
              = ( produc4880773030396931735_a_nat @ X @ Y ) )
           => ~ ( member4278285197320822468on_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_106_mem__case__prodE,axiom,
    ! [Z: produc5580075136493545115on_nat,C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > set_Pr7828957961435280891on_nat,P: produc3043957382837978535_a_nat] :
      ( ( member4221009242412147780on_nat @ Z @ ( produc924006725934604367on_nat @ C @ P ) )
     => ~ ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
            ( ( P
              = ( produc4880773030396931735_a_nat @ X @ Y ) )
           => ~ ( member4221009242412147780on_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_107_mem__case__prodE,axiom,
    ! [Z: produc394670695091157671_a_nat,C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > set_Pr4991467605864603271_a_nat,P: produc5580075136493545115on_nat] :
      ( ( member4231271695026834640_a_nat @ Z @ ( produc5862169952008903967_a_nat @ C @ P ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
            ( ( P
              = ( produc2912653624306349395on_nat @ X @ Y ) )
           => ~ ( member4231271695026834640_a_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_108_mem__case__prodE,axiom,
    ! [Z: produc5238312538367656103_a_nat,C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > set_Pr8529752790658994311_a_nat,P: produc5580075136493545115on_nat] :
      ( ( member6928026473249828560_a_nat @ Z @ ( produc4811812625541345055_a_nat @ C @ P ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
            ( ( P
              = ( produc2912653624306349395on_nat @ X @ Y ) )
           => ~ ( member6928026473249828560_a_nat @ Z @ ( C @ X @ Y ) ) ) ) ).

% mem_case_prodE
thf(fact_109_prod__induct4,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X3: produc8325617224762090749on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,C3: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A2 @ ( produc4176415733022414961on_nat @ B2 @ ( produc7152764266822961111on_nat @ C3 @ D ) ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct4
thf(fact_110_prod__induct3,axiom,
    ! [P2: produc118098540238352959on_nat > $o,X3: produc118098540238352959on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_P9127870862658512897on_nat,C3: list_P9127870862658512897on_nat] : ( P2 @ ( produc4176415733022414961on_nat @ A2 @ ( produc7152764266822961111on_nat @ B2 @ C3 ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_111_prod__induct3,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X3: produc8325617224762090749on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,C3: produc1714117091004681831on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A2 @ ( produc4176415733022414961on_nat @ B2 @ C3 ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_112_prod__cases4,axiom,
    ! [Y3: produc8325617224762090749on_nat] :
      ~ ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,C3: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] :
          ( Y3
         != ( produc1545487600272464053on_nat @ A2 @ ( produc4176415733022414961on_nat @ B2 @ ( produc7152764266822961111on_nat @ C3 @ D ) ) ) ) ).

% prod_cases4
thf(fact_113_prod__cases3,axiom,
    ! [Y3: produc118098540238352959on_nat] :
      ~ ! [A2: list_S5253404628753938141on_nat,B2: list_P9127870862658512897on_nat,C3: list_P9127870862658512897on_nat] :
          ( Y3
         != ( produc4176415733022414961on_nat @ A2 @ ( produc7152764266822961111on_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_114_prod__cases3,axiom,
    ! [Y3: produc8325617224762090749on_nat] :
      ~ ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,C3: produc1714117091004681831on_nat] :
          ( Y3
         != ( produc1545487600272464053on_nat @ A2 @ ( produc4176415733022414961on_nat @ B2 @ C3 ) ) ) ).

% prod_cases3
thf(fact_115_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat,A4: list_S5253404628753938141on_nat,B4: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B )
        = ( produc4176415733022414961on_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_116_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A4: list_S5253404628753938141on_nat,B4: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_117_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,A4: list_S5253404628753938141on_nat,B4: list_S5253404628753938141on_nat] :
      ( ( ( produc2912653624306349395on_nat @ A @ B )
        = ( produc2912653624306349395on_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_118_Pair__inject,axiom,
    ! [A: list_t5668669291736003536_a_nat,B: list_t5668669291736003536_a_nat,A4: list_t5668669291736003536_a_nat,B4: list_t5668669291736003536_a_nat] :
      ( ( ( produc5685097156399411607_a_nat @ A @ B )
        = ( produc5685097156399411607_a_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_119_Pair__inject,axiom,
    ! [A: $o,B: $o,A4: $o,B4: $o] :
      ( ( ( product_Pair_o_o @ A @ B )
        = ( product_Pair_o_o @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B = ~ B4 ) ) ) ).

% Pair_inject
thf(fact_120_Pair__inject,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,A4: term_R1747997765655105994_a_nat,B4: term_R1747997765655105994_a_nat] :
      ( ( ( produc4880773030396931735_a_nat @ A @ B )
        = ( produc4880773030396931735_a_nat @ A4 @ B4 ) )
     => ~ ( ( A = A4 )
         => ( B != B4 ) ) ) ).

% Pair_inject
thf(fact_121_prod__cases,axiom,
    ! [P2: produc118098540238352959on_nat > $o,P: produc118098540238352959on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] : ( P2 @ ( produc4176415733022414961on_nat @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_122_prod__cases,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_123_prod__cases,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,P: produc5580075136493545115on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] : ( P2 @ ( produc2912653624306349395on_nat @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_124_prod__cases,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,P: produc394670695091157671_a_nat] :
      ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] : ( P2 @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_125_prod__cases,axiom,
    ! [P2: product_prod_o_o > $o,P: product_prod_o_o] :
      ( ! [A2: $o,B2: $o] : ( P2 @ ( product_Pair_o_o @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_126_prod__cases,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,P: produc3043957382837978535_a_nat] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] : ( P2 @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_127_surj__pair,axiom,
    ! [P: produc118098540238352959on_nat] :
    ? [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
      ( P
      = ( produc4176415733022414961on_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_128_surj__pair,axiom,
    ! [P: produc8325617224762090749on_nat] :
    ? [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
      ( P
      = ( produc1545487600272464053on_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_129_surj__pair,axiom,
    ! [P: produc5580075136493545115on_nat] :
    ? [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
      ( P
      = ( produc2912653624306349395on_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_130_surj__pair,axiom,
    ! [P: produc394670695091157671_a_nat] :
    ? [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
      ( P
      = ( produc5685097156399411607_a_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_131_surj__pair,axiom,
    ! [P: product_prod_o_o] :
    ? [X: $o,Y: $o] :
      ( P
      = ( product_Pair_o_o @ X @ Y ) ) ).

% surj_pair
thf(fact_132_surj__pair,axiom,
    ! [P: produc3043957382837978535_a_nat] :
    ? [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
      ( P
      = ( produc4880773030396931735_a_nat @ X @ Y ) ) ).

% surj_pair
thf(fact_133_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc118098540238352959on_nat] :
      ~ ! [A2: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
          ( Y3
         != ( produc4176415733022414961on_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_134_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8325617224762090749on_nat] :
      ~ ! [A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
          ( Y3
         != ( produc1545487600272464053on_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_135_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc5580075136493545115on_nat] :
      ~ ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
          ( Y3
         != ( produc2912653624306349395on_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_136_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc394670695091157671_a_nat] :
      ~ ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
          ( Y3
         != ( produc5685097156399411607_a_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_137_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_o_o] :
      ~ ! [A2: $o,B2: $o] :
          ( Y3
         != ( product_Pair_o_o @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_138_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc3043957382837978535_a_nat] :
      ~ ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( Y3
         != ( produc4880773030396931735_a_nat @ A2 @ B2 ) ) ).

% old.prod.exhaust
thf(fact_139_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Prod: produc394670695091157671_a_nat] :
      ( ( H @ ( produc4945827410475597548_nat_o @ F @ Prod ) )
      = ( produc4945827410475597548_nat_o
        @ ^ [X12: list_t5668669291736003536_a_nat,X23: list_t5668669291736003536_a_nat] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_140_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Prod: produc3043957382837978535_a_nat] :
      ( ( H @ ( produc4098677624135445996_nat_o @ F @ Prod ) )
      = ( produc4098677624135445996_nat_o
        @ ^ [X12: term_R1747997765655105994_a_nat,X23: term_R1747997765655105994_a_nat] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_141_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Prod: produc5580075136493545115on_nat] :
      ( ( H @ ( produc7163990014243126832_nat_o @ F @ Prod ) )
      = ( produc7163990014243126832_nat_o
        @ ^ [X12: list_S5253404628753938141on_nat,X23: list_S5253404628753938141on_nat] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_142_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: $o > $o > $o,Prod: product_prod_o_o] :
      ( ( H @ ( produc6197397395684419436_o_o_o @ F @ Prod ) )
      = ( produc6197397395684419436_o_o_o
        @ ^ [X12: $o,X23: $o] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_143_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,Prod: produc8325617224762090749on_nat] :
      ( ( H @ ( produc7728289706344822350_nat_o @ F @ Prod ) )
      = ( produc7728289706344822350_nat_o
        @ ^ [X12: list_S5253404628753938141on_nat,X23: produc118098540238352959on_nat] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_144_prod_Ocase__distrib,axiom,
    ! [H: $o > $o,F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,Prod: produc118098540238352959on_nat] :
      ( ( H @ ( produc1360909077854731642_nat_o @ F @ Prod ) )
      = ( produc1360909077854731642_nat_o
        @ ^ [X12: list_S5253404628753938141on_nat,X23: produc1714117091004681831on_nat] : ( H @ ( F @ X12 @ X23 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_145_mem__Collect__eq,axiom,
    ! [A: produc5238312538367656103_a_nat,P2: produc5238312538367656103_a_nat > $o] :
      ( ( member6928026473249828560_a_nat @ A @ ( collec6671004892620940562_a_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_146_mem__Collect__eq,axiom,
    ! [A: produc9176241080893687067on_nat,P2: produc9176241080893687067on_nat > $o] :
      ( ( member4278285197320822468on_nat @ A @ ( collec1886838643678311430on_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_147_mem__Collect__eq,axiom,
    ! [A: produc394670695091157671_a_nat,P2: produc394670695091157671_a_nat > $o] :
      ( ( member4231271695026834640_a_nat @ A @ ( collec8858057866656402194_a_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_148_mem__Collect__eq,axiom,
    ! [A: produc3043957382837978535_a_nat,P2: produc3043957382837978535_a_nat > $o] :
      ( ( member6937586146271652048_a_nat @ A @ ( collec9088346736735320338_a_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_149_mem__Collect__eq,axiom,
    ! [A: produc5580075136493545115on_nat,P2: produc5580075136493545115on_nat > $o] :
      ( ( member4221009242412147780on_nat @ A @ ( collec1293568243156533638on_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_150_Collect__mem__eq,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat] :
      ( ( collec6671004892620940562_a_nat
        @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_151_Collect__mem__eq,axiom,
    ! [A5: set_Pr3573155178515677307on_nat] :
      ( ( collec1886838643678311430on_nat
        @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_152_Collect__mem__eq,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat] :
      ( ( collec8858057866656402194_a_nat
        @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_153_Collect__mem__eq,axiom,
    ! [A5: set_Pr630484882382513799_a_nat] :
      ( ( collec9088346736735320338_a_nat
        @ ^ [X2: produc3043957382837978535_a_nat] : ( member6937586146271652048_a_nat @ X2 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_154_Collect__mem__eq,axiom,
    ! [A5: set_Pr7828957961435280891on_nat] :
      ( ( collec1293568243156533638on_nat
        @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ A5 ) )
      = A5 ) ).

% Collect_mem_eq
thf(fact_155_Collect__cong,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,Q: produc394670695091157671_a_nat > $o] :
      ( ! [X: produc394670695091157671_a_nat] :
          ( ( P2 @ X )
          = ( Q @ X ) )
     => ( ( collec8858057866656402194_a_nat @ P2 )
        = ( collec8858057866656402194_a_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_156_Collect__cong,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ! [X: produc3043957382837978535_a_nat] :
          ( ( P2 @ X )
          = ( Q @ X ) )
     => ( ( collec9088346736735320338_a_nat @ P2 )
        = ( collec9088346736735320338_a_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_157_Collect__cong,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,Q: produc5580075136493545115on_nat > $o] :
      ( ! [X: produc5580075136493545115on_nat] :
          ( ( P2 @ X )
          = ( Q @ X ) )
     => ( ( collec1293568243156533638on_nat @ P2 )
        = ( collec1293568243156533638on_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_158_prod_Odisc__eq__case,axiom,
    ! [Prod: produc394670695091157671_a_nat] :
      ( produc4945827410475597548_nat_o
      @ ^ [Uu: list_t5668669291736003536_a_nat,Uv: list_t5668669291736003536_a_nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_159_prod_Odisc__eq__case,axiom,
    ! [Prod: produc3043957382837978535_a_nat] :
      ( produc4098677624135445996_nat_o
      @ ^ [Uu: term_R1747997765655105994_a_nat,Uv: term_R1747997765655105994_a_nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_160_prod_Odisc__eq__case,axiom,
    ! [Prod: produc5580075136493545115on_nat] :
      ( produc7163990014243126832_nat_o
      @ ^ [Uu: list_S5253404628753938141on_nat,Uv: list_S5253404628753938141on_nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_161_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_o_o] :
      ( produc6197397395684419436_o_o_o
      @ ^ [Uu: $o,Uv: $o] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_162_prod_Odisc__eq__case,axiom,
    ! [Prod: produc8325617224762090749on_nat] :
      ( produc7728289706344822350_nat_o
      @ ^ [Uu: list_S5253404628753938141on_nat,Uv: produc118098540238352959on_nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_163_prod_Odisc__eq__case,axiom,
    ! [Prod: produc118098540238352959on_nat] :
      ( produc1360909077854731642_nat_o
      @ ^ [Uu: list_S5253404628753938141on_nat,Uv: produc1714117091004681831on_nat] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_164_old_Oprod_Ocase,axiom,
    ! [F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,X1: list_t5668669291736003536_a_nat,X22: list_t5668669291736003536_a_nat] :
      ( ( produc4945827410475597548_nat_o @ F @ ( produc5685097156399411607_a_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_165_old_Oprod_Ocase,axiom,
    ! [F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,X1: term_R1747997765655105994_a_nat,X22: term_R1747997765655105994_a_nat] :
      ( ( produc4098677624135445996_nat_o @ F @ ( produc4880773030396931735_a_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_166_old_Oprod_Ocase,axiom,
    ! [F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,X1: list_S5253404628753938141on_nat,X22: list_S5253404628753938141on_nat] :
      ( ( produc7163990014243126832_nat_o @ F @ ( produc2912653624306349395on_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_167_old_Oprod_Ocase,axiom,
    ! [F: $o > $o > $o,X1: $o,X22: $o] :
      ( ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_168_old_Oprod_Ocase,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,X1: list_S5253404628753938141on_nat,X22: produc118098540238352959on_nat] :
      ( ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_169_old_Oprod_Ocase,axiom,
    ! [F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,X1: list_S5253404628753938141on_nat,X22: produc1714117091004681831on_nat] :
      ( ( produc1360909077854731642_nat_o @ F @ ( produc4176415733022414961on_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_170_fst__conv,axiom,
    ! [X1: term_R1747997765655105994_a_nat,X22: term_R1747997765655105994_a_nat] :
      ( ( produc2126783593110491267_a_nat @ ( produc4880773030396931735_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_171_fst__conv,axiom,
    ! [X1: $o,X22: $o] :
      ( ( product_fst_o_o @ ( product_Pair_o_o @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_172_fst__conv,axiom,
    ! [X1: list_t5668669291736003536_a_nat,X22: list_t5668669291736003536_a_nat] :
      ( ( produc3471587812718774147_a_nat @ ( produc5685097156399411607_a_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_173_fst__conv,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: list_S5253404628753938141on_nat] :
      ( ( produc1822290298201838911on_nat @ ( produc2912653624306349395on_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_174_fst__conv,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc118098540238352959on_nat] :
      ( ( produc311592285738829985on_nat @ ( produc1545487600272464053on_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_175_fst__conv,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc1714117091004681831on_nat] :
      ( ( produc6765150182969198981on_nat @ ( produc4176415733022414961on_nat @ X1 @ X22 ) )
      = X1 ) ).

% fst_conv
thf(fact_176_fst__eqD,axiom,
    ! [X3: term_R1747997765655105994_a_nat,Y3: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( ( produc2126783593110491267_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_177_fst__eqD,axiom,
    ! [X3: $o,Y3: $o,A: $o] :
      ( ( ( product_fst_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_178_fst__eqD,axiom,
    ! [X3: list_t5668669291736003536_a_nat,Y3: list_t5668669291736003536_a_nat,A: list_t5668669291736003536_a_nat] :
      ( ( ( produc3471587812718774147_a_nat @ ( produc5685097156399411607_a_nat @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_179_fst__eqD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y3: list_S5253404628753938141on_nat,A: list_S5253404628753938141on_nat] :
      ( ( ( produc1822290298201838911on_nat @ ( produc2912653624306349395on_nat @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_180_fst__eqD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat,A: list_S5253404628753938141on_nat] :
      ( ( ( produc311592285738829985on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_181_fst__eqD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat,A: list_S5253404628753938141on_nat] :
      ( ( ( produc6765150182969198981on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y3 ) )
        = A )
     => ( X3 = A ) ) ).

% fst_eqD
thf(fact_182_snd__conv,axiom,
    ! [X1: term_R1747997765655105994_a_nat,X22: term_R1747997765655105994_a_nat] :
      ( ( produc2284112530981304773_a_nat @ ( produc4880773030396931735_a_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_183_snd__conv,axiom,
    ! [X1: $o,X22: $o] :
      ( ( product_snd_o_o @ ( product_Pair_o_o @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_184_snd__conv,axiom,
    ! [X1: list_t5668669291736003536_a_nat,X22: list_t5668669291736003536_a_nat] :
      ( ( produc4727165237793840837_a_nat @ ( produc5685097156399411607_a_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_185_snd__conv,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: list_S5253404628753938141on_nat] :
      ( ( produc2374478376588790657on_nat @ ( produc2912653624306349395on_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_186_snd__conv,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc118098540238352959on_nat] :
      ( ( produc3934153714877691619on_nat @ ( produc1545487600272464053on_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_187_snd__conv,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc1714117091004681831on_nat] :
      ( ( produc5312734412660135107on_nat @ ( produc4176415733022414961on_nat @ X1 @ X22 ) )
      = X22 ) ).

% snd_conv
thf(fact_188_snd__eqD,axiom,
    ! [X3: term_R1747997765655105994_a_nat,Y3: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( ( produc2284112530981304773_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_189_snd__eqD,axiom,
    ! [X3: $o,Y3: $o,A: $o] :
      ( ( ( product_snd_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_190_snd__eqD,axiom,
    ! [X3: list_t5668669291736003536_a_nat,Y3: list_t5668669291736003536_a_nat,A: list_t5668669291736003536_a_nat] :
      ( ( ( produc4727165237793840837_a_nat @ ( produc5685097156399411607_a_nat @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_191_snd__eqD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y3: list_S5253404628753938141on_nat,A: list_S5253404628753938141on_nat] :
      ( ( ( produc2374478376588790657on_nat @ ( produc2912653624306349395on_nat @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_192_snd__eqD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat,A: produc118098540238352959on_nat] :
      ( ( ( produc3934153714877691619on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_193_snd__eqD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat,A: produc1714117091004681831on_nat] :
      ( ( ( produc5312734412660135107on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y3 ) )
        = A )
     => ( Y3 = A ) ) ).

% snd_eqD
thf(fact_194_prod__eq__iff,axiom,
    ( ( ^ [Y4: product_prod_o_o,Z2: product_prod_o_o] : ( Y4 = Z2 ) )
    = ( ^ [S4: product_prod_o_o,T: product_prod_o_o] :
          ( ( ( product_fst_o_o @ S4 )
            = ( product_fst_o_o @ T ) )
          & ( ( product_snd_o_o @ S4 )
            = ( product_snd_o_o @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_195_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc394670695091157671_a_nat,Z2: produc394670695091157671_a_nat] : ( Y4 = Z2 ) )
    = ( ^ [S4: produc394670695091157671_a_nat,T: produc394670695091157671_a_nat] :
          ( ( ( produc3471587812718774147_a_nat @ S4 )
            = ( produc3471587812718774147_a_nat @ T ) )
          & ( ( produc4727165237793840837_a_nat @ S4 )
            = ( produc4727165237793840837_a_nat @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_196_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc5580075136493545115on_nat,Z2: produc5580075136493545115on_nat] : ( Y4 = Z2 ) )
    = ( ^ [S4: produc5580075136493545115on_nat,T: produc5580075136493545115on_nat] :
          ( ( ( produc1822290298201838911on_nat @ S4 )
            = ( produc1822290298201838911on_nat @ T ) )
          & ( ( produc2374478376588790657on_nat @ S4 )
            = ( produc2374478376588790657on_nat @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_197_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc8325617224762090749on_nat,Z2: produc8325617224762090749on_nat] : ( Y4 = Z2 ) )
    = ( ^ [S4: produc8325617224762090749on_nat,T: produc8325617224762090749on_nat] :
          ( ( ( produc311592285738829985on_nat @ S4 )
            = ( produc311592285738829985on_nat @ T ) )
          & ( ( produc3934153714877691619on_nat @ S4 )
            = ( produc3934153714877691619on_nat @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_198_prod__eq__iff,axiom,
    ( ( ^ [Y4: produc118098540238352959on_nat,Z2: produc118098540238352959on_nat] : ( Y4 = Z2 ) )
    = ( ^ [S4: produc118098540238352959on_nat,T: produc118098540238352959on_nat] :
          ( ( ( produc6765150182969198981on_nat @ S4 )
            = ( produc6765150182969198981on_nat @ T ) )
          & ( ( produc5312734412660135107on_nat @ S4 )
            = ( produc5312734412660135107on_nat @ T ) ) ) ) ) ).

% prod_eq_iff
thf(fact_199_exE__realizer_H,axiom,
    ! [P2: $o > $o > $o,P: product_prod_o_o] :
      ( ( P2 @ ( product_snd_o_o @ P ) @ ( product_fst_o_o @ P ) )
     => ~ ! [X: $o,Y: $o] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_200_exE__realizer_H,axiom,
    ! [P2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,P: produc394670695091157671_a_nat] :
      ( ( P2 @ ( produc4727165237793840837_a_nat @ P ) @ ( produc3471587812718774147_a_nat @ P ) )
     => ~ ! [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_201_exE__realizer_H,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,P: produc5580075136493545115on_nat] :
      ( ( P2 @ ( produc2374478376588790657on_nat @ P ) @ ( produc1822290298201838911on_nat @ P ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_202_exE__realizer_H,axiom,
    ! [P2: produc118098540238352959on_nat > list_S5253404628753938141on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ( P2 @ ( produc3934153714877691619on_nat @ P ) @ ( produc311592285738829985on_nat @ P ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_203_exE__realizer_H,axiom,
    ! [P2: produc1714117091004681831on_nat > list_S5253404628753938141on_nat > $o,P: produc118098540238352959on_nat] :
      ( ( P2 @ ( produc5312734412660135107on_nat @ P ) @ ( produc6765150182969198981on_nat @ P ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
            ~ ( P2 @ Y @ X ) ) ).

% exE_realizer'
thf(fact_204_prod__eqI,axiom,
    ! [P: product_prod_o_o,Q2: product_prod_o_o] :
      ( ( ( product_fst_o_o @ P )
        = ( product_fst_o_o @ Q2 ) )
     => ( ( ( product_snd_o_o @ P )
          = ( product_snd_o_o @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_205_prod__eqI,axiom,
    ! [P: produc394670695091157671_a_nat,Q2: produc394670695091157671_a_nat] :
      ( ( ( produc3471587812718774147_a_nat @ P )
        = ( produc3471587812718774147_a_nat @ Q2 ) )
     => ( ( ( produc4727165237793840837_a_nat @ P )
          = ( produc4727165237793840837_a_nat @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_206_prod__eqI,axiom,
    ! [P: produc5580075136493545115on_nat,Q2: produc5580075136493545115on_nat] :
      ( ( ( produc1822290298201838911on_nat @ P )
        = ( produc1822290298201838911on_nat @ Q2 ) )
     => ( ( ( produc2374478376588790657on_nat @ P )
          = ( produc2374478376588790657on_nat @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_207_prod__eqI,axiom,
    ! [P: produc8325617224762090749on_nat,Q2: produc8325617224762090749on_nat] :
      ( ( ( produc311592285738829985on_nat @ P )
        = ( produc311592285738829985on_nat @ Q2 ) )
     => ( ( ( produc3934153714877691619on_nat @ P )
          = ( produc3934153714877691619on_nat @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_208_prod__eqI,axiom,
    ! [P: produc118098540238352959on_nat,Q2: produc118098540238352959on_nat] :
      ( ( ( produc6765150182969198981on_nat @ P )
        = ( produc6765150182969198981on_nat @ Q2 ) )
     => ( ( ( produc5312734412660135107on_nat @ P )
          = ( produc5312734412660135107on_nat @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_209_prod_Oexpand,axiom,
    ! [Prod: product_prod_o_o,Prod2: product_prod_o_o] :
      ( ( ( ( product_fst_o_o @ Prod )
          = ( product_fst_o_o @ Prod2 ) )
        & ( ( product_snd_o_o @ Prod )
          = ( product_snd_o_o @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_210_prod_Oexpand,axiom,
    ! [Prod: produc394670695091157671_a_nat,Prod2: produc394670695091157671_a_nat] :
      ( ( ( ( produc3471587812718774147_a_nat @ Prod )
          = ( produc3471587812718774147_a_nat @ Prod2 ) )
        & ( ( produc4727165237793840837_a_nat @ Prod )
          = ( produc4727165237793840837_a_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_211_prod_Oexpand,axiom,
    ! [Prod: produc5580075136493545115on_nat,Prod2: produc5580075136493545115on_nat] :
      ( ( ( ( produc1822290298201838911on_nat @ Prod )
          = ( produc1822290298201838911on_nat @ Prod2 ) )
        & ( ( produc2374478376588790657on_nat @ Prod )
          = ( produc2374478376588790657on_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_212_prod_Oexpand,axiom,
    ! [Prod: produc8325617224762090749on_nat,Prod2: produc8325617224762090749on_nat] :
      ( ( ( ( produc311592285738829985on_nat @ Prod )
          = ( produc311592285738829985on_nat @ Prod2 ) )
        & ( ( produc3934153714877691619on_nat @ Prod )
          = ( produc3934153714877691619on_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_213_prod_Oexpand,axiom,
    ! [Prod: produc118098540238352959on_nat,Prod2: produc118098540238352959on_nat] :
      ( ( ( ( produc6765150182969198981on_nat @ Prod )
          = ( produc6765150182969198981on_nat @ Prod2 ) )
        & ( ( produc5312734412660135107on_nat @ Prod )
          = ( produc5312734412660135107on_nat @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_214_cond__case__prod__eta,axiom,
    ! [F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,G: produc394670695091157671_a_nat > $o] :
      ( ! [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
          ( ( F @ X @ Y )
          = ( G @ ( produc5685097156399411607_a_nat @ X @ Y ) ) )
     => ( ( produc4945827410475597548_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_215_cond__case__prod__eta,axiom,
    ! [F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,G: produc3043957382837978535_a_nat > $o] :
      ( ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
          ( ( F @ X @ Y )
          = ( G @ ( produc4880773030396931735_a_nat @ X @ Y ) ) )
     => ( ( produc4098677624135445996_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_216_cond__case__prod__eta,axiom,
    ! [F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,G: produc5580075136493545115on_nat > $o] :
      ( ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
          ( ( F @ X @ Y )
          = ( G @ ( produc2912653624306349395on_nat @ X @ Y ) ) )
     => ( ( produc7163990014243126832_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_217_cond__case__prod__eta,axiom,
    ! [F: $o > $o > $o,G: product_prod_o_o > $o] :
      ( ! [X: $o,Y: $o] :
          ( ( F @ X @ Y )
          = ( G @ ( product_Pair_o_o @ X @ Y ) ) )
     => ( ( produc6197397395684419436_o_o_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_218_cond__case__prod__eta,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,G: produc8325617224762090749on_nat > $o] :
      ( ! [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
          ( ( F @ X @ Y )
          = ( G @ ( produc1545487600272464053on_nat @ X @ Y ) ) )
     => ( ( produc7728289706344822350_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_219_cond__case__prod__eta,axiom,
    ! [F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,G: produc118098540238352959on_nat > $o] :
      ( ! [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
          ( ( F @ X @ Y )
          = ( G @ ( produc4176415733022414961on_nat @ X @ Y ) ) )
     => ( ( produc1360909077854731642_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_220_case__prod__eta,axiom,
    ! [F: produc394670695091157671_a_nat > $o] :
      ( ( produc4945827410475597548_nat_o
        @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( F @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_221_case__prod__eta,axiom,
    ! [F: produc3043957382837978535_a_nat > $o] :
      ( ( produc4098677624135445996_nat_o
        @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( F @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_222_case__prod__eta,axiom,
    ! [F: produc5580075136493545115on_nat > $o] :
      ( ( produc7163990014243126832_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( F @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_223_case__prod__eta,axiom,
    ! [F: product_prod_o_o > $o] :
      ( ( produc6197397395684419436_o_o_o
        @ ^ [X2: $o,Y5: $o] : ( F @ ( product_Pair_o_o @ X2 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_224_case__prod__eta,axiom,
    ! [F: produc8325617224762090749on_nat > $o] :
      ( ( produc7728289706344822350_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( F @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_225_case__prod__eta,axiom,
    ! [F: produc118098540238352959on_nat > $o] :
      ( ( produc1360909077854731642_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( F @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_226_case__prodE2,axiom,
    ! [Q: $o > $o,P2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Z: produc394670695091157671_a_nat] :
      ( ( Q @ ( produc4945827410475597548_nat_o @ P2 @ Z ) )
     => ~ ! [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
            ( ( Z
              = ( produc5685097156399411607_a_nat @ X @ Y ) )
           => ~ ( Q @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_227_case__prodE2,axiom,
    ! [Q: $o > $o,P2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Z: produc3043957382837978535_a_nat] :
      ( ( Q @ ( produc4098677624135445996_nat_o @ P2 @ Z ) )
     => ~ ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
            ( ( Z
              = ( produc4880773030396931735_a_nat @ X @ Y ) )
           => ~ ( Q @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_228_case__prodE2,axiom,
    ! [Q: $o > $o,P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Z: produc5580075136493545115on_nat] :
      ( ( Q @ ( produc7163990014243126832_nat_o @ P2 @ Z ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
            ( ( Z
              = ( produc2912653624306349395on_nat @ X @ Y ) )
           => ~ ( Q @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_229_case__prodE2,axiom,
    ! [Q: $o > $o,P2: $o > $o > $o,Z: product_prod_o_o] :
      ( ( Q @ ( produc6197397395684419436_o_o_o @ P2 @ Z ) )
     => ~ ! [X: $o,Y: $o] :
            ( ( Z
              = ( product_Pair_o_o @ X @ Y ) )
           => ~ ( Q @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_230_case__prodE2,axiom,
    ! [Q: $o > $o,P2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,Z: produc8325617224762090749on_nat] :
      ( ( Q @ ( produc7728289706344822350_nat_o @ P2 @ Z ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
            ( ( Z
              = ( produc1545487600272464053on_nat @ X @ Y ) )
           => ~ ( Q @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_231_case__prodE2,axiom,
    ! [Q: $o > $o,P2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,Z: produc118098540238352959on_nat] :
      ( ( Q @ ( produc1360909077854731642_nat_o @ P2 @ Z ) )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
            ( ( Z
              = ( produc4176415733022414961on_nat @ X @ Y ) )
           => ~ ( Q @ ( P2 @ X @ Y ) ) ) ) ).

% case_prodE2
thf(fact_232_case__prodE,axiom,
    ! [C: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,P: produc394670695091157671_a_nat] :
      ( ( produc4945827410475597548_nat_o @ C @ P )
     => ~ ! [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
            ( ( P
              = ( produc5685097156399411607_a_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_233_case__prodE,axiom,
    ! [C: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,P: produc3043957382837978535_a_nat] :
      ( ( produc4098677624135445996_nat_o @ C @ P )
     => ~ ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
            ( ( P
              = ( produc4880773030396931735_a_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_234_case__prodE,axiom,
    ! [C: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,P: produc5580075136493545115on_nat] :
      ( ( produc7163990014243126832_nat_o @ C @ P )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
            ( ( P
              = ( produc2912653624306349395on_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_235_case__prodE,axiom,
    ! [C: $o > $o > $o,P: product_prod_o_o] :
      ( ( produc6197397395684419436_o_o_o @ C @ P )
     => ~ ! [X: $o,Y: $o] :
            ( ( P
              = ( product_Pair_o_o @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_236_case__prodE,axiom,
    ! [C: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ( produc7728289706344822350_nat_o @ C @ P )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
            ( ( P
              = ( produc1545487600272464053on_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_237_case__prodE,axiom,
    ! [C: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,P: produc118098540238352959on_nat] :
      ( ( produc1360909077854731642_nat_o @ C @ P )
     => ~ ! [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
            ( ( P
              = ( produc4176415733022414961on_nat @ X @ Y ) )
           => ~ ( C @ X @ Y ) ) ) ).

% case_prodE
thf(fact_238_case__prodD,axiom,
    ! [F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,A: list_t5668669291736003536_a_nat,B: list_t5668669291736003536_a_nat] :
      ( ( produc4945827410475597548_nat_o @ F @ ( produc5685097156399411607_a_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_239_case__prodD,axiom,
    ! [F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( produc4098677624135445996_nat_o @ F @ ( produc4880773030396931735_a_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_240_case__prodD,axiom,
    ! [F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat] :
      ( ( produc7163990014243126832_nat_o @ F @ ( produc2912653624306349395on_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_241_case__prodD,axiom,
    ! [F: $o > $o > $o,A: $o,B: $o] :
      ( ( produc6197397395684419436_o_o_o @ F @ ( product_Pair_o_o @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_242_case__prodD,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] :
      ( ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_243_case__prodD,axiom,
    ! [F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat] :
      ( ( produc1360909077854731642_nat_o @ F @ ( produc4176415733022414961on_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_244_fst__def,axiom,
    ( produc3471587812718774147_a_nat
    = ( produc6071536960187114788_a_nat
      @ ^ [X12: list_t5668669291736003536_a_nat,X23: list_t5668669291736003536_a_nat] : X12 ) ) ).

% fst_def
thf(fact_245_fst__def,axiom,
    ( produc1822290298201838911on_nat
    = ( produc6984445701720766021on_nat
      @ ^ [X12: list_S5253404628753938141on_nat,X23: list_S5253404628753938141on_nat] : X12 ) ) ).

% fst_def
thf(fact_246_fst__def,axiom,
    ( produc311592285738829985on_nat
    = ( produc5836180625088671911on_nat
      @ ^ [X12: list_S5253404628753938141on_nat,X23: produc118098540238352959on_nat] : X12 ) ) ).

% fst_def
thf(fact_247_fst__def,axiom,
    ( produc6765150182969198981on_nat
    = ( produc3445967709534052987on_nat
      @ ^ [X12: list_S5253404628753938141on_nat,X23: produc1714117091004681831on_nat] : X12 ) ) ).

% fst_def
thf(fact_248_fst__def,axiom,
    ( product_fst_o_o
    = ( produc6197397395684419436_o_o_o
      @ ^ [X12: $o,X23: $o] : X12 ) ) ).

% fst_def
thf(fact_249_snd__def,axiom,
    ( produc4727165237793840837_a_nat
    = ( produc6071536960187114788_a_nat
      @ ^ [X12: list_t5668669291736003536_a_nat,X23: list_t5668669291736003536_a_nat] : X23 ) ) ).

% snd_def
thf(fact_250_snd__def,axiom,
    ( produc2374478376588790657on_nat
    = ( produc6984445701720766021on_nat
      @ ^ [X12: list_S5253404628753938141on_nat,X23: list_S5253404628753938141on_nat] : X23 ) ) ).

% snd_def
thf(fact_251_snd__def,axiom,
    ( produc3934153714877691619on_nat
    = ( produc1948010502932907017on_nat
      @ ^ [X12: list_S5253404628753938141on_nat,X23: produc118098540238352959on_nat] : X23 ) ) ).

% snd_def
thf(fact_252_snd__def,axiom,
    ( produc5312734412660135107on_nat
    = ( produc7209264684439357769on_nat
      @ ^ [X12: list_S5253404628753938141on_nat,X23: produc1714117091004681831on_nat] : X23 ) ) ).

% snd_def
thf(fact_253_snd__def,axiom,
    ( product_snd_o_o
    = ( produc6197397395684419436_o_o_o
      @ ^ [X12: $o,X23: $o] : X23 ) ) ).

% snd_def
thf(fact_254_surjective__pairing,axiom,
    ! [T2: produc3043957382837978535_a_nat] :
      ( T2
      = ( produc4880773030396931735_a_nat @ ( produc2126783593110491267_a_nat @ T2 ) @ ( produc2284112530981304773_a_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_255_surjective__pairing,axiom,
    ! [T2: product_prod_o_o] :
      ( T2
      = ( product_Pair_o_o @ ( product_fst_o_o @ T2 ) @ ( product_snd_o_o @ T2 ) ) ) ).

% surjective_pairing
thf(fact_256_surjective__pairing,axiom,
    ! [T2: produc394670695091157671_a_nat] :
      ( T2
      = ( produc5685097156399411607_a_nat @ ( produc3471587812718774147_a_nat @ T2 ) @ ( produc4727165237793840837_a_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_257_surjective__pairing,axiom,
    ! [T2: produc5580075136493545115on_nat] :
      ( T2
      = ( produc2912653624306349395on_nat @ ( produc1822290298201838911on_nat @ T2 ) @ ( produc2374478376588790657on_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_258_surjective__pairing,axiom,
    ! [T2: produc8325617224762090749on_nat] :
      ( T2
      = ( produc1545487600272464053on_nat @ ( produc311592285738829985on_nat @ T2 ) @ ( produc3934153714877691619on_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_259_surjective__pairing,axiom,
    ! [T2: produc118098540238352959on_nat] :
      ( T2
      = ( produc4176415733022414961on_nat @ ( produc6765150182969198981on_nat @ T2 ) @ ( produc5312734412660135107on_nat @ T2 ) ) ) ).

% surjective_pairing
thf(fact_260_conjI__realizer,axiom,
    ! [P2: term_R1747997765655105994_a_nat > $o,P: term_R1747997765655105994_a_nat,Q: term_R1747997765655105994_a_nat > $o,Q2: term_R1747997765655105994_a_nat] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc2126783593110491267_a_nat @ ( produc4880773030396931735_a_nat @ P @ Q2 ) ) )
          & ( Q @ ( produc2284112530981304773_a_nat @ ( produc4880773030396931735_a_nat @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_261_conjI__realizer,axiom,
    ! [P2: $o > $o,P: $o,Q: $o > $o,Q2: $o] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( product_fst_o_o @ ( product_Pair_o_o @ P @ Q2 ) ) )
          & ( Q @ ( product_snd_o_o @ ( product_Pair_o_o @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_262_conjI__realizer,axiom,
    ! [P2: list_t5668669291736003536_a_nat > $o,P: list_t5668669291736003536_a_nat,Q: list_t5668669291736003536_a_nat > $o,Q2: list_t5668669291736003536_a_nat] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc3471587812718774147_a_nat @ ( produc5685097156399411607_a_nat @ P @ Q2 ) ) )
          & ( Q @ ( produc4727165237793840837_a_nat @ ( produc5685097156399411607_a_nat @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_263_conjI__realizer,axiom,
    ! [P2: list_S5253404628753938141on_nat > $o,P: list_S5253404628753938141on_nat,Q: list_S5253404628753938141on_nat > $o,Q2: list_S5253404628753938141on_nat] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc1822290298201838911on_nat @ ( produc2912653624306349395on_nat @ P @ Q2 ) ) )
          & ( Q @ ( produc2374478376588790657on_nat @ ( produc2912653624306349395on_nat @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_264_conjI__realizer,axiom,
    ! [P2: list_S5253404628753938141on_nat > $o,P: list_S5253404628753938141on_nat,Q: produc118098540238352959on_nat > $o,Q2: produc118098540238352959on_nat] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc311592285738829985on_nat @ ( produc1545487600272464053on_nat @ P @ Q2 ) ) )
          & ( Q @ ( produc3934153714877691619on_nat @ ( produc1545487600272464053on_nat @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_265_conjI__realizer,axiom,
    ! [P2: list_S5253404628753938141on_nat > $o,P: list_S5253404628753938141on_nat,Q: produc1714117091004681831on_nat > $o,Q2: produc1714117091004681831on_nat] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( produc6765150182969198981on_nat @ ( produc4176415733022414961on_nat @ P @ Q2 ) ) )
          & ( Q @ ( produc5312734412660135107on_nat @ ( produc4176415733022414961on_nat @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_266_exI__realizer,axiom,
    ! [P2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc2284112530981304773_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) ) @ ( produc2126783593110491267_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_267_exI__realizer,axiom,
    ! [P2: $o > $o > $o,Y3: $o,X3: $o] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( product_snd_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) ) @ ( product_fst_o_o @ ( product_Pair_o_o @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_268_exI__realizer,axiom,
    ! [P2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Y3: list_t5668669291736003536_a_nat,X3: list_t5668669291736003536_a_nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc4727165237793840837_a_nat @ ( produc5685097156399411607_a_nat @ X3 @ Y3 ) ) @ ( produc3471587812718774147_a_nat @ ( produc5685097156399411607_a_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_269_exI__realizer,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Y3: list_S5253404628753938141on_nat,X3: list_S5253404628753938141on_nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc2374478376588790657on_nat @ ( produc2912653624306349395on_nat @ X3 @ Y3 ) ) @ ( produc1822290298201838911on_nat @ ( produc2912653624306349395on_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_270_exI__realizer,axiom,
    ! [P2: produc118098540238352959on_nat > list_S5253404628753938141on_nat > $o,Y3: produc118098540238352959on_nat,X3: list_S5253404628753938141on_nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc3934153714877691619on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y3 ) ) @ ( produc311592285738829985on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_271_exI__realizer,axiom,
    ! [P2: produc1714117091004681831on_nat > list_S5253404628753938141on_nat > $o,Y3: produc1714117091004681831on_nat,X3: list_S5253404628753938141on_nat] :
      ( ( P2 @ Y3 @ X3 )
     => ( P2 @ ( produc5312734412660135107on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y3 ) ) @ ( produc6765150182969198981on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y3 ) ) ) ) ).

% exI_realizer
thf(fact_272_prod_Oexhaust__sel,axiom,
    ! [Prod: produc3043957382837978535_a_nat] :
      ( Prod
      = ( produc4880773030396931735_a_nat @ ( produc2126783593110491267_a_nat @ Prod ) @ ( produc2284112530981304773_a_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_273_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_o_o] :
      ( Prod
      = ( product_Pair_o_o @ ( product_fst_o_o @ Prod ) @ ( product_snd_o_o @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_274_prod_Oexhaust__sel,axiom,
    ! [Prod: produc394670695091157671_a_nat] :
      ( Prod
      = ( produc5685097156399411607_a_nat @ ( produc3471587812718774147_a_nat @ Prod ) @ ( produc4727165237793840837_a_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_275_prod_Oexhaust__sel,axiom,
    ! [Prod: produc5580075136493545115on_nat] :
      ( Prod
      = ( produc2912653624306349395on_nat @ ( produc1822290298201838911on_nat @ Prod ) @ ( produc2374478376588790657on_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_276_prod_Oexhaust__sel,axiom,
    ! [Prod: produc8325617224762090749on_nat] :
      ( Prod
      = ( produc1545487600272464053on_nat @ ( produc311592285738829985on_nat @ Prod ) @ ( produc3934153714877691619on_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_277_prod_Oexhaust__sel,axiom,
    ! [Prod: produc118098540238352959on_nat] :
      ( Prod
      = ( produc4176415733022414961on_nat @ ( produc6765150182969198981on_nat @ Prod ) @ ( produc5312734412660135107on_nat @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_278_case__prod__beta,axiom,
    ( produc4945827410475597548_nat_o
    = ( ^ [F2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,P3: produc394670695091157671_a_nat] : ( F2 @ ( produc3471587812718774147_a_nat @ P3 ) @ ( produc4727165237793840837_a_nat @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_279_case__prod__beta,axiom,
    ( produc4098677624135445996_nat_o
    = ( ^ [F2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,P3: produc3043957382837978535_a_nat] : ( F2 @ ( produc2126783593110491267_a_nat @ P3 ) @ ( produc2284112530981304773_a_nat @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_280_case__prod__beta,axiom,
    ( produc7163990014243126832_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,P3: produc5580075136493545115on_nat] : ( F2 @ ( produc1822290298201838911on_nat @ P3 ) @ ( produc2374478376588790657on_nat @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_281_case__prod__beta,axiom,
    ( produc6197397395684419436_o_o_o
    = ( ^ [F2: $o > $o > $o,P3: product_prod_o_o] : ( F2 @ ( product_fst_o_o @ P3 ) @ ( product_snd_o_o @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_282_case__prod__beta,axiom,
    ( produc7728289706344822350_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,P3: produc8325617224762090749on_nat] : ( F2 @ ( produc311592285738829985on_nat @ P3 ) @ ( produc3934153714877691619on_nat @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_283_case__prod__beta,axiom,
    ( produc1360909077854731642_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,P3: produc118098540238352959on_nat] : ( F2 @ ( produc6765150182969198981on_nat @ P3 ) @ ( produc5312734412660135107on_nat @ P3 ) ) ) ) ).

% case_prod_beta
thf(fact_284_split__beta,axiom,
    ( produc4945827410475597548_nat_o
    = ( ^ [F2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Prod3: produc394670695091157671_a_nat] : ( F2 @ ( produc3471587812718774147_a_nat @ Prod3 ) @ ( produc4727165237793840837_a_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_285_split__beta,axiom,
    ( produc4098677624135445996_nat_o
    = ( ^ [F2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Prod3: produc3043957382837978535_a_nat] : ( F2 @ ( produc2126783593110491267_a_nat @ Prod3 ) @ ( produc2284112530981304773_a_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_286_split__beta,axiom,
    ( produc7163990014243126832_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Prod3: produc5580075136493545115on_nat] : ( F2 @ ( produc1822290298201838911on_nat @ Prod3 ) @ ( produc2374478376588790657on_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_287_split__beta,axiom,
    ( produc6197397395684419436_o_o_o
    = ( ^ [F2: $o > $o > $o,Prod3: product_prod_o_o] : ( F2 @ ( product_fst_o_o @ Prod3 ) @ ( product_snd_o_o @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_288_split__beta,axiom,
    ( produc7728289706344822350_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,Prod3: produc8325617224762090749on_nat] : ( F2 @ ( produc311592285738829985on_nat @ Prod3 ) @ ( produc3934153714877691619on_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_289_split__beta,axiom,
    ( produc1360909077854731642_nat_o
    = ( ^ [F2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,Prod3: produc118098540238352959on_nat] : ( F2 @ ( produc6765150182969198981on_nat @ Prod3 ) @ ( produc5312734412660135107on_nat @ Prod3 ) ) ) ) ).

% split_beta
thf(fact_290_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: multis6046679608167549072_a_nat > multis6046679608167549072_a_nat > $o] :
      ( ( member6928026473249828560_a_nat @ X3 @ ( collec6671004892620940562_a_nat @ ( produc5157571406952832236_nat_o @ A5 ) ) )
     => ( A5 @ ( produc5510942010692487555_a_nat @ X3 ) @ ( produc7019377601104156869_a_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_291_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: multis1856558993800614429on_nat > multis1856558993800614429on_nat > $o] :
      ( ( member4278285197320822468on_nat @ X3 @ ( collec1886838643678311430on_nat @ ( produc2086596812675560368_nat_o @ A5 ) ) )
     => ( A5 @ ( produc60446082731974591on_nat @ X3 ) @ ( produc8092638228495552001on_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_292_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ( member4231271695026834640_a_nat @ X3 @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ A5 ) ) )
     => ( A5 @ ( produc3471587812718774147_a_nat @ X3 ) @ ( produc4727165237793840837_a_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_293_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc3043957382837978535_a_nat,A5: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ( member6937586146271652048_a_nat @ X3 @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ A5 ) ) )
     => ( A5 @ ( produc2126783593110491267_a_nat @ X3 ) @ ( produc2284112530981304773_a_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_294_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ( member4221009242412147780on_nat @ X3 @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ A5 ) ) )
     => ( A5 @ ( produc1822290298201838911on_nat @ X3 ) @ ( produc2374478376588790657on_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_295_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: product_prod_o_o,A5: $o > $o > $o] :
      ( ( member7466972457876170832od_o_o @ X3 @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ A5 ) ) )
     => ( A5 @ ( product_fst_o_o @ X3 ) @ ( product_snd_o_o @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_296_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc8325617224762090749on_nat,A5: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ( member3464510148199751334on_nat @ X3 @ ( collec624303909867598824on_nat @ ( produc7728289706344822350_nat_o @ A5 ) ) )
     => ( A5 @ ( produc311592285738829985on_nat @ X3 ) @ ( produc3934153714877691619on_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_297_Product__Type_OCollect__case__prodD,axiom,
    ! [X3: produc118098540238352959on_nat,A5: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ( member4585374397974051670on_nat @ X3 @ ( collec8524684687135612820on_nat @ ( produc1360909077854731642_nat_o @ A5 ) ) )
     => ( A5 @ ( produc6765150182969198981on_nat @ X3 ) @ ( produc5312734412660135107on_nat @ X3 ) ) ) ).

% Product_Type.Collect_case_prodD
thf(fact_298_trivial__rpo_ORPO__SN__order__pair,axiom,
    ( order_1584829357401681496_a_nat
    @ ( collec9088346736735320338_a_nat
      @ ( produc4098677624135445996_nat_o
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
            ( product_fst_o_o
            @ ( rpo_RP6885983473509681171_a_nat
              @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
              @ ^ [F2: produc2248741730625028719ms_nat] : $true
              @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
              @ zero_zero_nat
              @ S4
              @ T ) ) ) )
    @ ( collec9088346736735320338_a_nat
      @ ( produc4098677624135445996_nat_o
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
            ( product_snd_o_o
            @ ( rpo_RP6885983473509681171_a_nat
              @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
              @ ^ [F2: produc2248741730625028719ms_nat] : $true
              @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
              @ zero_zero_nat
              @ S4
              @ T ) ) ) ) ) ).

% trivial_rpo.RPO_SN_order_pair
thf(fact_299_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,X3: term_R1747997765655105994_a_nat,Y3: term_R1747997765655105994_a_nat,A: produc3043957382837978535_a_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc4880773030396931735_a_nat @ X3 @ Y3 ) )
       => ( P2 @ ( produc2126783593110491267_a_nat @ A ) @ ( produc2284112530981304773_a_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_300_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: $o > $o > $o,X3: $o,Y3: $o,A: product_prod_o_o] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( product_Pair_o_o @ X3 @ Y3 ) )
       => ( P2 @ ( product_fst_o_o @ A ) @ ( product_snd_o_o @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_301_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,X3: list_t5668669291736003536_a_nat,Y3: list_t5668669291736003536_a_nat,A: produc394670695091157671_a_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc5685097156399411607_a_nat @ X3 @ Y3 ) )
       => ( P2 @ ( produc3471587812718774147_a_nat @ A ) @ ( produc4727165237793840837_a_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_302_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,X3: list_S5253404628753938141on_nat,Y3: list_S5253404628753938141on_nat,A: produc5580075136493545115on_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc2912653624306349395on_nat @ X3 @ Y3 ) )
       => ( P2 @ ( produc1822290298201838911on_nat @ A ) @ ( produc2374478376588790657on_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_303_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat,A: produc8325617224762090749on_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc1545487600272464053on_nat @ X3 @ Y3 ) )
       => ( P2 @ ( produc311592285738829985on_nat @ A ) @ ( produc3934153714877691619on_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_304_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,X3: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat,A: produc118098540238352959on_nat] :
      ( ( P2 @ X3 @ Y3 )
     => ( ( A
          = ( produc4176415733022414961on_nat @ X3 @ Y3 ) )
       => ( P2 @ ( produc6765150182969198981on_nat @ A ) @ ( produc5312734412660135107on_nat @ A ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_305_trivial__rpo_ORPO__S__subset__RPO__NS,axiom,
    ( ord_le251666825002368551_a_nat
    @ ( collec9088346736735320338_a_nat
      @ ( produc4098677624135445996_nat_o
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
            ( product_fst_o_o
            @ ( rpo_RP6885983473509681171_a_nat
              @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
              @ ^ [F2: produc2248741730625028719ms_nat] : $true
              @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
              @ zero_zero_nat
              @ S4
              @ T ) ) ) )
    @ ( collec9088346736735320338_a_nat
      @ ( produc4098677624135445996_nat_o
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
            ( product_snd_o_o
            @ ( rpo_RP6885983473509681171_a_nat
              @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
              @ ^ [F2: produc2248741730625028719ms_nat] : $true
              @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
              @ zero_zero_nat
              @ S4
              @ T ) ) ) ) ) ).

% trivial_rpo.RPO_S_subset_RPO_NS
thf(fact_306_smulextp__def,axiom,
    ( multis1321808316593418887_a_nat
    = ( ^ [F2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o,M: multis2209918487243275798_a_nat,N: multis2209918487243275798_a_nat] :
          ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ M @ N )
          @ ( multis3722385149336313575_a_nat
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% smulextp_def
thf(fact_307_smulextp__def,axiom,
    ( multis1105153566473756134on_nat
    = ( ^ [F2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o,M: multis3959152272169849901on_nat,N: multis3959152272169849901on_nat] :
          ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ M @ N )
          @ ( multis2514206365584361862on_nat
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% smulextp_def
thf(fact_308_smulextp__def,axiom,
    ( multiset_smulextp_o
    = ( ^ [F2: $o > $o > product_prod_o_o,M: multiset_o,N: multiset_o] :
          ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ M @ N )
          @ ( multiset_s_mul_ext_o
            @ ( collec3167064739498627218od_o_o
              @ ( produc6197397395684419436_o_o_o
                @ ^ [X2: $o,Y5: $o] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec3167064739498627218od_o_o
              @ ( produc6197397395684419436_o_o_o
                @ ^ [X2: $o,Y5: $o] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% smulextp_def
thf(fact_309_smulextp__def,axiom,
    ( multis6604845555486480385_a_nat
    = ( ^ [F2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,M: multis6046679608167549072_a_nat,N: multis6046679608167549072_a_nat] :
          ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ M @ N )
          @ ( multis3666123592011920481_a_nat
            @ ( collec9088346736735320338_a_nat
              @ ( produc4098677624135445996_nat_o
                @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec9088346736735320338_a_nat
              @ ( produc4098677624135445996_nat_o
                @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% smulextp_def
thf(fact_310_smulextp__def,axiom,
    ( multis8482119632275936726on_nat
    = ( ^ [F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,M: multis1856558993800614429on_nat,N: multis1856558993800614429on_nat] :
          ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M @ N )
          @ ( multis1620213760974180726on_nat
            @ ( collec5585564048161825126on_nat
              @ ( produc208325271403635792_nat_o
                @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec5585564048161825126on_nat
              @ ( produc208325271403635792_nat_o
                @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% smulextp_def
thf(fact_311_mul__ext__unfold,axiom,
    ! [X3: list_l3889969465389992278_a_nat,Y3: list_l3889969465389992278_a_nat,G: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o] :
      ( ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ X3 @ Y3 )
        @ ( collec4515946210884203282_a_nat
          @ ( produc2556087688056060396_nat_o
            @ ^ [A3: list_l3889969465389992278_a_nat,B3: list_l3889969465389992278_a_nat] : ( product_fst_o_o @ ( multis2433842936921369738_a_nat @ G @ A3 @ B3 ) ) ) ) )
      = ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ X3 ) @ ( mset_l7063064786757117541_a_nat @ Y3 ) )
        @ ( multis3722385149336313575_a_nat
          @ ( collec8858057866656402194_a_nat
            @ ( produc4945827410475597548_nat_o
              @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] : ( product_snd_o_o @ ( G @ A3 @ B3 ) ) ) )
          @ ( collec8858057866656402194_a_nat
            @ ( produc4945827410475597548_nat_o
              @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( G @ A3 @ B3 ) ) ) ) ) ) ) ).

% mul_ext_unfold
thf(fact_312_mul__ext__unfold,axiom,
    ! [X3: list_l8999537460041914605on_nat,Y3: list_l8999537460041914605on_nat,G: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o] :
      ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ X3 @ Y3 )
        @ ( collec1004640060711064742on_nat
          @ ( produc9222835799945651472_nat_o
            @ ^ [A3: list_l8999537460041914605on_nat,B3: list_l8999537460041914605on_nat] : ( product_fst_o_o @ ( multis5717424201997335715on_nat @ G @ A3 @ B3 ) ) ) ) )
      = ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ X3 ) @ ( mset_l1352240747471131016on_nat @ Y3 ) )
        @ ( multis2514206365584361862on_nat
          @ ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] : ( product_snd_o_o @ ( G @ A3 @ B3 ) ) ) )
          @ ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( G @ A3 @ B3 ) ) ) ) ) ) ) ).

% mul_ext_unfold
thf(fact_313_mul__ext__unfold,axiom,
    ! [X3: list_o,Y3: list_o,G: $o > $o > product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ X3 @ Y3 )
        @ ( collec4435127810633735570list_o
          @ ( produc1052442579958709612st_o_o
            @ ^ [A3: list_o,B3: list_o] : ( product_fst_o_o @ ( multiset_mul_ext_o @ G @ A3 @ B3 ) ) ) ) )
      = ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ X3 ) @ ( mset_o @ Y3 ) )
        @ ( multiset_s_mul_ext_o
          @ ( collec3167064739498627218od_o_o
            @ ( produc6197397395684419436_o_o_o
              @ ^ [A3: $o,B3: $o] : ( product_snd_o_o @ ( G @ A3 @ B3 ) ) ) )
          @ ( collec3167064739498627218od_o_o
            @ ( produc6197397395684419436_o_o_o
              @ ^ [A3: $o,B3: $o] : ( product_fst_o_o @ ( G @ A3 @ B3 ) ) ) ) ) ) ) ).

% mul_ext_unfold
thf(fact_314_mul__ext__unfold,axiom,
    ! [X3: list_t5668669291736003536_a_nat,Y3: list_t5668669291736003536_a_nat,G: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o] :
      ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X3 @ Y3 )
        @ ( collec8858057866656402194_a_nat
          @ ( produc4945827410475597548_nat_o
            @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( multis2940008709989298052_a_nat @ G @ A3 @ B3 ) ) ) ) )
      = ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ X3 ) @ ( mset_t7141044528732312287_a_nat @ Y3 ) )
        @ ( multis3666123592011920481_a_nat
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( G @ A3 @ B3 ) ) ) )
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( G @ A3 @ B3 ) ) ) ) ) ) ) ).

% mul_ext_unfold
thf(fact_315_mul__ext__unfold,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y3: list_S5253404628753938141on_nat,G: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X3 @ Y3 )
        @ ( collec1293568243156533638on_nat
          @ ( produc7163990014243126832_nat_o
            @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( multis6586140943272769299on_nat @ G @ A3 @ B3 ) ) ) ) )
      = ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ X3 ) @ ( mset_S7907735292211298936on_nat @ Y3 ) )
        @ ( multis1620213760974180726on_nat
          @ ( collec5585564048161825126on_nat
            @ ( produc208325271403635792_nat_o
              @ ^ [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( product_snd_o_o @ ( G @ A3 @ B3 ) ) ) )
          @ ( collec5585564048161825126on_nat
            @ ( produc208325271403635792_nat_o
              @ ^ [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( product_fst_o_o @ ( G @ A3 @ B3 ) ) ) ) ) ) ) ).

% mul_ext_unfold
thf(fact_316_eq__snd__iff,axiom,
    ! [B: term_R1747997765655105994_a_nat,P: produc3043957382837978535_a_nat] :
      ( ( B
        = ( produc2284112530981304773_a_nat @ P ) )
      = ( ? [A3: term_R1747997765655105994_a_nat] :
            ( P
            = ( produc4880773030396931735_a_nat @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_317_eq__snd__iff,axiom,
    ! [B: $o,P: product_prod_o_o] :
      ( ( B
        = ( product_snd_o_o @ P ) )
      = ( ? [A3: $o] :
            ( P
            = ( product_Pair_o_o @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_318_eq__snd__iff,axiom,
    ! [B: list_t5668669291736003536_a_nat,P: produc394670695091157671_a_nat] :
      ( ( B
        = ( produc4727165237793840837_a_nat @ P ) )
      = ( ? [A3: list_t5668669291736003536_a_nat] :
            ( P
            = ( produc5685097156399411607_a_nat @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_319_eq__snd__iff,axiom,
    ! [B: list_S5253404628753938141on_nat,P: produc5580075136493545115on_nat] :
      ( ( B
        = ( produc2374478376588790657on_nat @ P ) )
      = ( ? [A3: list_S5253404628753938141on_nat] :
            ( P
            = ( produc2912653624306349395on_nat @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_320_eq__snd__iff,axiom,
    ! [B: produc118098540238352959on_nat,P: produc8325617224762090749on_nat] :
      ( ( B
        = ( produc3934153714877691619on_nat @ P ) )
      = ( ? [A3: list_S5253404628753938141on_nat] :
            ( P
            = ( produc1545487600272464053on_nat @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_321_eq__snd__iff,axiom,
    ! [B: produc1714117091004681831on_nat,P: produc118098540238352959on_nat] :
      ( ( B
        = ( produc5312734412660135107on_nat @ P ) )
      = ( ? [A3: list_S5253404628753938141on_nat] :
            ( P
            = ( produc4176415733022414961on_nat @ A3 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_322_sndI,axiom,
    ! [X3: produc3043957382837978535_a_nat,Y3: term_R1747997765655105994_a_nat,Z: term_R1747997765655105994_a_nat] :
      ( ( X3
        = ( produc4880773030396931735_a_nat @ Y3 @ Z ) )
     => ( ( produc2284112530981304773_a_nat @ X3 )
        = Z ) ) ).

% sndI
thf(fact_323_sndI,axiom,
    ! [X3: product_prod_o_o,Y3: $o,Z: $o] :
      ( ( X3
        = ( product_Pair_o_o @ Y3 @ Z ) )
     => ( ( product_snd_o_o @ X3 )
        = Z ) ) ).

% sndI
thf(fact_324_sndI,axiom,
    ! [X3: produc394670695091157671_a_nat,Y3: list_t5668669291736003536_a_nat,Z: list_t5668669291736003536_a_nat] :
      ( ( X3
        = ( produc5685097156399411607_a_nat @ Y3 @ Z ) )
     => ( ( produc4727165237793840837_a_nat @ X3 )
        = Z ) ) ).

% sndI
thf(fact_325_sndI,axiom,
    ! [X3: produc5580075136493545115on_nat,Y3: list_S5253404628753938141on_nat,Z: list_S5253404628753938141on_nat] :
      ( ( X3
        = ( produc2912653624306349395on_nat @ Y3 @ Z ) )
     => ( ( produc2374478376588790657on_nat @ X3 )
        = Z ) ) ).

% sndI
thf(fact_326_sndI,axiom,
    ! [X3: produc8325617224762090749on_nat,Y3: list_S5253404628753938141on_nat,Z: produc118098540238352959on_nat] :
      ( ( X3
        = ( produc1545487600272464053on_nat @ Y3 @ Z ) )
     => ( ( produc3934153714877691619on_nat @ X3 )
        = Z ) ) ).

% sndI
thf(fact_327_sndI,axiom,
    ! [X3: produc118098540238352959on_nat,Y3: list_S5253404628753938141on_nat,Z: produc1714117091004681831on_nat] :
      ( ( X3
        = ( produc4176415733022414961on_nat @ Y3 @ Z ) )
     => ( ( produc5312734412660135107on_nat @ X3 )
        = Z ) ) ).

% sndI
thf(fact_328_eq__fst__iff,axiom,
    ! [A: term_R1747997765655105994_a_nat,P: produc3043957382837978535_a_nat] :
      ( ( A
        = ( produc2126783593110491267_a_nat @ P ) )
      = ( ? [B3: term_R1747997765655105994_a_nat] :
            ( P
            = ( produc4880773030396931735_a_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_329_eq__fst__iff,axiom,
    ! [A: $o,P: product_prod_o_o] :
      ( ( A
        = ( product_fst_o_o @ P ) )
      = ( ? [B3: $o] :
            ( P
            = ( product_Pair_o_o @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_330_eq__fst__iff,axiom,
    ! [A: list_t5668669291736003536_a_nat,P: produc394670695091157671_a_nat] :
      ( ( A
        = ( produc3471587812718774147_a_nat @ P ) )
      = ( ? [B3: list_t5668669291736003536_a_nat] :
            ( P
            = ( produc5685097156399411607_a_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_331_eq__fst__iff,axiom,
    ! [A: list_S5253404628753938141on_nat,P: produc5580075136493545115on_nat] :
      ( ( A
        = ( produc1822290298201838911on_nat @ P ) )
      = ( ? [B3: list_S5253404628753938141on_nat] :
            ( P
            = ( produc2912653624306349395on_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_332_eq__fst__iff,axiom,
    ! [A: list_S5253404628753938141on_nat,P: produc8325617224762090749on_nat] :
      ( ( A
        = ( produc311592285738829985on_nat @ P ) )
      = ( ? [B3: produc118098540238352959on_nat] :
            ( P
            = ( produc1545487600272464053on_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_333_eq__fst__iff,axiom,
    ! [A: list_S5253404628753938141on_nat,P: produc118098540238352959on_nat] :
      ( ( A
        = ( produc6765150182969198981on_nat @ P ) )
      = ( ? [B3: produc1714117091004681831on_nat] :
            ( P
            = ( produc4176415733022414961on_nat @ A @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_334_fstI,axiom,
    ! [X3: produc3043957382837978535_a_nat,Y3: term_R1747997765655105994_a_nat,Z: term_R1747997765655105994_a_nat] :
      ( ( X3
        = ( produc4880773030396931735_a_nat @ Y3 @ Z ) )
     => ( ( produc2126783593110491267_a_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_335_fstI,axiom,
    ! [X3: product_prod_o_o,Y3: $o,Z: $o] :
      ( ( X3
        = ( product_Pair_o_o @ Y3 @ Z ) )
     => ( ( product_fst_o_o @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_336_fstI,axiom,
    ! [X3: produc394670695091157671_a_nat,Y3: list_t5668669291736003536_a_nat,Z: list_t5668669291736003536_a_nat] :
      ( ( X3
        = ( produc5685097156399411607_a_nat @ Y3 @ Z ) )
     => ( ( produc3471587812718774147_a_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_337_fstI,axiom,
    ! [X3: produc5580075136493545115on_nat,Y3: list_S5253404628753938141on_nat,Z: list_S5253404628753938141on_nat] :
      ( ( X3
        = ( produc2912653624306349395on_nat @ Y3 @ Z ) )
     => ( ( produc1822290298201838911on_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_338_fstI,axiom,
    ! [X3: produc8325617224762090749on_nat,Y3: list_S5253404628753938141on_nat,Z: produc118098540238352959on_nat] :
      ( ( X3
        = ( produc1545487600272464053on_nat @ Y3 @ Z ) )
     => ( ( produc311592285738829985on_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_339_fstI,axiom,
    ! [X3: produc118098540238352959on_nat,Y3: list_S5253404628753938141on_nat,Z: produc1714117091004681831on_nat] :
      ( ( X3
        = ( produc4176415733022414961on_nat @ Y3 @ Z ) )
     => ( ( produc6765150182969198981on_nat @ X3 )
        = Y3 ) ) ).

% fstI
thf(fact_340_split__cong,axiom,
    ! [Q2: produc394670695091157671_a_nat,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,G: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,P: produc394670695091157671_a_nat] :
      ( ! [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
          ( ( ( produc5685097156399411607_a_nat @ X @ Y )
            = Q2 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q2 )
       => ( ( produc4945827410475597548_nat_o @ F @ P )
          = ( produc4945827410475597548_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_341_split__cong,axiom,
    ! [Q2: produc3043957382837978535_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,G: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,P: produc3043957382837978535_a_nat] :
      ( ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
          ( ( ( produc4880773030396931735_a_nat @ X @ Y )
            = Q2 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q2 )
       => ( ( produc4098677624135445996_nat_o @ F @ P )
          = ( produc4098677624135445996_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_342_split__cong,axiom,
    ! [Q2: produc5580075136493545115on_nat,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,G: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,P: produc5580075136493545115on_nat] :
      ( ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
          ( ( ( produc2912653624306349395on_nat @ X @ Y )
            = Q2 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q2 )
       => ( ( produc7163990014243126832_nat_o @ F @ P )
          = ( produc7163990014243126832_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_343_split__cong,axiom,
    ! [Q2: product_prod_o_o,F: $o > $o > $o,G: $o > $o > $o,P: product_prod_o_o] :
      ( ! [X: $o,Y: $o] :
          ( ( ( product_Pair_o_o @ X @ Y )
            = Q2 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q2 )
       => ( ( produc6197397395684419436_o_o_o @ F @ P )
          = ( produc6197397395684419436_o_o_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_344_split__cong,axiom,
    ! [Q2: produc8325617224762090749on_nat,F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,G: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ! [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
          ( ( ( produc1545487600272464053on_nat @ X @ Y )
            = Q2 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q2 )
       => ( ( produc7728289706344822350_nat_o @ F @ P )
          = ( produc7728289706344822350_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_345_split__cong,axiom,
    ! [Q2: produc118098540238352959on_nat,F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,G: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,P: produc118098540238352959on_nat] :
      ( ! [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
          ( ( ( produc4176415733022414961on_nat @ X @ Y )
            = Q2 )
         => ( ( F @ X @ Y )
            = ( G @ X @ Y ) ) )
     => ( ( P = Q2 )
       => ( ( produc1360909077854731642_nat_o @ F @ P )
          = ( produc1360909077854731642_nat_o @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_346_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_347_subset__Collect__iff,axiom,
    ! [B5: set_Pr8529752790658994311_a_nat,A5: set_Pr8529752790658994311_a_nat,P2: produc5238312538367656103_a_nat > $o] :
      ( ( ord_le7248185767425706023_a_nat @ B5 @ A5 )
     => ( ( ord_le7248185767425706023_a_nat @ B5
          @ ( collec6671004892620940562_a_nat
            @ ^ [X2: produc5238312538367656103_a_nat] :
                ( ( member6928026473249828560_a_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: produc5238312538367656103_a_nat] :
              ( ( member6928026473249828560_a_nat @ X2 @ B5 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_348_subset__Collect__iff,axiom,
    ! [B5: set_Pr3573155178515677307on_nat,A5: set_Pr3573155178515677307on_nat,P2: produc9176241080893687067on_nat > $o] :
      ( ( ord_le927452247459997723on_nat @ B5 @ A5 )
     => ( ( ord_le927452247459997723on_nat @ B5
          @ ( collec1886838643678311430on_nat
            @ ^ [X2: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X2 @ B5 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_349_subset__Collect__iff,axiom,
    ! [B5: set_Pr4991467605864603271_a_nat,A5: set_Pr4991467605864603271_a_nat,P2: produc394670695091157671_a_nat > $o] :
      ( ( ord_le998509814859404839_a_nat @ B5 @ A5 )
     => ( ( ord_le998509814859404839_a_nat @ B5
          @ ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( member4231271695026834640_a_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: produc394670695091157671_a_nat] :
              ( ( member4231271695026834640_a_nat @ X2 @ B5 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_350_subset__Collect__iff,axiom,
    ! [B5: set_Pr630484882382513799_a_nat,A5: set_Pr630484882382513799_a_nat,P2: produc3043957382837978535_a_nat > $o] :
      ( ( ord_le251666825002368551_a_nat @ B5 @ A5 )
     => ( ( ord_le251666825002368551_a_nat @ B5
          @ ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( member6937586146271652048_a_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: produc3043957382837978535_a_nat] :
              ( ( member6937586146271652048_a_nat @ X2 @ B5 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_351_subset__Collect__iff,axiom,
    ! [B5: set_Pr7828957961435280891on_nat,A5: set_Pr7828957961435280891on_nat,P2: produc5580075136493545115on_nat > $o] :
      ( ( ord_le6423710186836778395on_nat @ B5 @ A5 )
     => ( ( ord_le6423710186836778395on_nat @ B5
          @ ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( member4221009242412147780on_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) )
        = ( ! [X2: produc5580075136493545115on_nat] :
              ( ( member4221009242412147780on_nat @ X2 @ B5 )
             => ( P2 @ X2 ) ) ) ) ) ).

% subset_Collect_iff
thf(fact_352_subset__CollectI,axiom,
    ! [B5: set_Pr8529752790658994311_a_nat,A5: set_Pr8529752790658994311_a_nat,Q: produc5238312538367656103_a_nat > $o,P2: produc5238312538367656103_a_nat > $o] :
      ( ( ord_le7248185767425706023_a_nat @ B5 @ A5 )
     => ( ! [X: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P2 @ X ) ) )
       => ( ord_le7248185767425706023_a_nat
          @ ( collec6671004892620940562_a_nat
            @ ^ [X2: produc5238312538367656103_a_nat] :
                ( ( member6928026473249828560_a_nat @ X2 @ B5 )
                & ( Q @ X2 ) ) )
          @ ( collec6671004892620940562_a_nat
            @ ^ [X2: produc5238312538367656103_a_nat] :
                ( ( member6928026473249828560_a_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_353_subset__CollectI,axiom,
    ! [B5: set_Pr3573155178515677307on_nat,A5: set_Pr3573155178515677307on_nat,Q: produc9176241080893687067on_nat > $o,P2: produc9176241080893687067on_nat > $o] :
      ( ( ord_le927452247459997723on_nat @ B5 @ A5 )
     => ( ! [X: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P2 @ X ) ) )
       => ( ord_le927452247459997723on_nat
          @ ( collec1886838643678311430on_nat
            @ ^ [X2: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X2 @ B5 )
                & ( Q @ X2 ) ) )
          @ ( collec1886838643678311430on_nat
            @ ^ [X2: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_354_subset__CollectI,axiom,
    ! [B5: set_Pr4991467605864603271_a_nat,A5: set_Pr4991467605864603271_a_nat,Q: produc394670695091157671_a_nat > $o,P2: produc394670695091157671_a_nat > $o] :
      ( ( ord_le998509814859404839_a_nat @ B5 @ A5 )
     => ( ! [X: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P2 @ X ) ) )
       => ( ord_le998509814859404839_a_nat
          @ ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( member4231271695026834640_a_nat @ X2 @ B5 )
                & ( Q @ X2 ) ) )
          @ ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( member4231271695026834640_a_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_355_subset__CollectI,axiom,
    ! [B5: set_Pr630484882382513799_a_nat,A5: set_Pr630484882382513799_a_nat,Q: produc3043957382837978535_a_nat > $o,P2: produc3043957382837978535_a_nat > $o] :
      ( ( ord_le251666825002368551_a_nat @ B5 @ A5 )
     => ( ! [X: produc3043957382837978535_a_nat] :
            ( ( member6937586146271652048_a_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P2 @ X ) ) )
       => ( ord_le251666825002368551_a_nat
          @ ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( member6937586146271652048_a_nat @ X2 @ B5 )
                & ( Q @ X2 ) ) )
          @ ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( member6937586146271652048_a_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_356_subset__CollectI,axiom,
    ! [B5: set_Pr7828957961435280891on_nat,A5: set_Pr7828957961435280891on_nat,Q: produc5580075136493545115on_nat > $o,P2: produc5580075136493545115on_nat > $o] :
      ( ( ord_le6423710186836778395on_nat @ B5 @ A5 )
     => ( ! [X: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X @ B5 )
           => ( ( Q @ X )
             => ( P2 @ X ) ) )
       => ( ord_le6423710186836778395on_nat
          @ ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( member4221009242412147780on_nat @ X2 @ B5 )
                & ( Q @ X2 ) ) )
          @ ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( member4221009242412147780on_nat @ X2 @ A5 )
                & ( P2 @ X2 ) ) ) ) ) ) ).

% subset_CollectI
thf(fact_357_s__mul__ext__mono,axiom,
    ! [Ns: set_Pr630484882382513799_a_nat,Ns3: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat,S5: set_Pr630484882382513799_a_nat] :
      ( ( ord_le251666825002368551_a_nat @ Ns @ Ns3 )
     => ( ( ord_le251666825002368551_a_nat @ S @ S5 )
       => ( ord_le7248185767425706023_a_nat @ ( multis3666123592011920481_a_nat @ Ns @ S ) @ ( multis3666123592011920481_a_nat @ Ns3 @ S5 ) ) ) ) ).

% s_mul_ext_mono
thf(fact_358_s__mul__ext__mono,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat,Ns3: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,S5: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ Ns @ Ns3 )
     => ( ( ord_le6961793561435735931on_nat @ S @ S5 )
       => ( ord_le927452247459997723on_nat @ ( multis1620213760974180726on_nat @ Ns @ S ) @ ( multis1620213760974180726on_nat @ Ns3 @ S5 ) ) ) ) ).

% s_mul_ext_mono
thf(fact_359_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_360_smulextp__mset__code,axiom,
    ! [F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,Xs2: list_t5668669291736003536_a_nat,Ys2: list_t5668669291736003536_a_nat] :
      ( ( multis6604845555486480385_a_nat @ F @ ( mset_t7141044528732312287_a_nat @ Xs2 ) @ ( mset_t7141044528732312287_a_nat @ Ys2 ) )
      = ( product_fst_o_o @ ( multis2940008709989298052_a_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% smulextp_mset_code
thf(fact_361_smulextp__mset__code,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( multis8482119632275936726on_nat @ F @ ( mset_S7907735292211298936on_nat @ Xs2 ) @ ( mset_S7907735292211298936on_nat @ Ys2 ) )
      = ( product_fst_o_o @ ( multis6586140943272769299on_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% smulextp_mset_code
thf(fact_362_s__mul__ext__bottom,axiom,
    ! [A5: multis6046679608167549072_a_nat,Ns: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat] :
      ( ( A5 != zero_z3261727345284792775_a_nat )
     => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A5 @ zero_z3261727345284792775_a_nat ) @ ( multis3666123592011920481_a_nat @ Ns @ S ) ) ) ).

% s_mul_ext_bottom
thf(fact_363_s__mul__ext__bottom,axiom,
    ! [A5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( A5 != zero_z8702567803361615526on_nat )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A5 @ zero_z8702567803361615526on_nat ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ).

% s_mul_ext_bottom
thf(fact_364_s__mul__ext__bottom__strict,axiom,
    ! [A5: multis6046679608167549072_a_nat,Ns: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat] :
      ~ ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ zero_z3261727345284792775_a_nat @ A5 ) @ ( multis3666123592011920481_a_nat @ Ns @ S ) ) ).

% s_mul_ext_bottom_strict
thf(fact_365_s__mul__ext__bottom__strict,axiom,
    ! [A5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ A5 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ).

% s_mul_ext_bottom_strict
thf(fact_366_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
      = ( X3 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_367_mset__s__mul__ext,axiom,
    ! [Xs2: list_l3889969465389992278_a_nat,Ys2: list_l3889969465389992278_a_nat,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o] :
      ( ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ Xs2 ) @ ( mset_l7063064786757117541_a_nat @ Ys2 ) )
        @ ( multis3722385149336313575_a_nat
          @ ( collec8858057866656402194_a_nat
            @ ( produc4945827410475597548_nat_o
              @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec8858057866656402194_a_nat
            @ ( produc4945827410475597548_nat_o
              @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_fst_o_o @ ( multis2433842936921369738_a_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_s_mul_ext
thf(fact_368_mset__s__mul__ext,axiom,
    ! [Xs2: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o] :
      ( ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ Xs2 ) @ ( mset_l1352240747471131016on_nat @ Ys2 ) )
        @ ( multis2514206365584361862on_nat
          @ ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_fst_o_o @ ( multis5717424201997335715on_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_s_mul_ext
thf(fact_369_mset__s__mul__ext,axiom,
    ! [Xs2: list_o,Ys2: list_o,F: $o > $o > product_prod_o_o] :
      ( ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ Xs2 ) @ ( mset_o @ Ys2 ) )
        @ ( multiset_s_mul_ext_o
          @ ( collec3167064739498627218od_o_o
            @ ( produc6197397395684419436_o_o_o
              @ ^ [X2: $o,Y5: $o] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec3167064739498627218od_o_o
            @ ( produc6197397395684419436_o_o_o
              @ ^ [X2: $o,Y5: $o] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_fst_o_o @ ( multiset_mul_ext_o @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_s_mul_ext
thf(fact_370_mset__s__mul__ext,axiom,
    ! [Xs2: list_t5668669291736003536_a_nat,Ys2: list_t5668669291736003536_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o] :
      ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ Xs2 ) @ ( mset_t7141044528732312287_a_nat @ Ys2 ) )
        @ ( multis3666123592011920481_a_nat
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_fst_o_o @ ( multis2940008709989298052_a_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_s_mul_ext
thf(fact_371_mset__s__mul__ext,axiom,
    ! [Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ Xs2 ) @ ( mset_S7907735292211298936on_nat @ Ys2 ) )
        @ ( multis1620213760974180726on_nat
          @ ( collec5585564048161825126on_nat
            @ ( produc208325271403635792_nat_o
              @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec5585564048161825126on_nat
            @ ( produc208325271403635792_nat_o
              @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_fst_o_o @ ( multis6586140943272769299on_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_s_mul_ext
thf(fact_372_mset__zero__iff__right,axiom,
    ! [X3: list_l8336264072933094989od_a_o] :
      ( ( zero_z8905637151584487364od_a_o
        = ( mset_l8139550355095933148od_a_o @ X3 ) )
      = ( X3 = nil_li1372968520647675725od_a_o ) ) ).

% mset_zero_iff_right
thf(fact_373_mset__zero__iff__right,axiom,
    ! [X3: list_t5668669291736003536_a_nat] :
      ( ( zero_z3261727345284792775_a_nat
        = ( mset_t7141044528732312287_a_nat @ X3 ) )
      = ( X3 = nil_te8332778836088660176_a_nat ) ) ).

% mset_zero_iff_right
thf(fact_374_mset__zero__iff__right,axiom,
    ! [X3: list_S5253404628753938141on_nat] :
      ( ( zero_z8702567803361615526on_nat
        = ( mset_S7907735292211298936on_nat @ X3 ) )
      = ( X3 = nil_Su3078906344068544711on_nat ) ) ).

% mset_zero_iff_right
thf(fact_375_mset__zero__iff,axiom,
    ! [X3: list_l8336264072933094989od_a_o] :
      ( ( ( mset_l8139550355095933148od_a_o @ X3 )
        = zero_z8905637151584487364od_a_o )
      = ( X3 = nil_li1372968520647675725od_a_o ) ) ).

% mset_zero_iff
thf(fact_376_mset__zero__iff,axiom,
    ! [X3: list_t5668669291736003536_a_nat] :
      ( ( ( mset_t7141044528732312287_a_nat @ X3 )
        = zero_z3261727345284792775_a_nat )
      = ( X3 = nil_te8332778836088660176_a_nat ) ) ).

% mset_zero_iff
thf(fact_377_mset__zero__iff,axiom,
    ! [X3: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ X3 )
        = zero_z8702567803361615526on_nat )
      = ( X3 = nil_Su3078906344068544711on_nat ) ) ).

% mset_zero_iff
thf(fact_378_mul__ext__def,axiom,
    ( multis2433842936921369738_a_nat
    = ( ^ [F2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o,Xs3: list_l3889969465389992278_a_nat,Ys3: list_l3889969465389992278_a_nat] :
          ( product_Pair_o_o
          @ ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ Xs3 ) @ ( mset_l7063064786757117541_a_nat @ Ys3 ) )
            @ ( multis3722385149336313575_a_nat
              @ ( collec8858057866656402194_a_nat
                @ ( produc4945827410475597548_nat_o
                  @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec8858057866656402194_a_nat
                @ ( produc4945827410475597548_nat_o
                  @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) )
          @ ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ Xs3 ) @ ( mset_l7063064786757117541_a_nat @ Ys3 ) )
            @ ( multis6753421577935756037_a_nat
              @ ( collec8858057866656402194_a_nat
                @ ( produc4945827410475597548_nat_o
                  @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec8858057866656402194_a_nat
                @ ( produc4945827410475597548_nat_o
                  @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ) ).

% mul_ext_def
thf(fact_379_mul__ext__def,axiom,
    ( multis5717424201997335715on_nat
    = ( ^ [F2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o,Xs3: list_l8999537460041914605on_nat,Ys3: list_l8999537460041914605on_nat] :
          ( product_Pair_o_o
          @ ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ Xs3 ) @ ( mset_l1352240747471131016on_nat @ Ys3 ) )
            @ ( multis2514206365584361862on_nat
              @ ( collec1293568243156533638on_nat
                @ ( produc7163990014243126832_nat_o
                  @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec1293568243156533638on_nat
                @ ( produc7163990014243126832_nat_o
                  @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) )
          @ ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ Xs3 ) @ ( mset_l1352240747471131016on_nat @ Ys3 ) )
            @ ( multis1816195516039316712on_nat
              @ ( collec1293568243156533638on_nat
                @ ( produc7163990014243126832_nat_o
                  @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec1293568243156533638on_nat
                @ ( produc7163990014243126832_nat_o
                  @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ) ).

% mul_ext_def
thf(fact_380_mul__ext__def,axiom,
    ( multiset_mul_ext_o
    = ( ^ [F2: $o > $o > product_prod_o_o,Xs3: list_o,Ys3: list_o] :
          ( product_Pair_o_o
          @ ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ Xs3 ) @ ( mset_o @ Ys3 ) )
            @ ( multiset_s_mul_ext_o
              @ ( collec3167064739498627218od_o_o
                @ ( produc6197397395684419436_o_o_o
                  @ ^ [X2: $o,Y5: $o] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec3167064739498627218od_o_o
                @ ( produc6197397395684419436_o_o_o
                  @ ^ [X2: $o,Y5: $o] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) )
          @ ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ Xs3 ) @ ( mset_o @ Ys3 ) )
            @ ( multis1436808229551206221_ext_o
              @ ( collec3167064739498627218od_o_o
                @ ( produc6197397395684419436_o_o_o
                  @ ^ [X2: $o,Y5: $o] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec3167064739498627218od_o_o
                @ ( produc6197397395684419436_o_o_o
                  @ ^ [X2: $o,Y5: $o] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ) ).

% mul_ext_def
thf(fact_381_mul__ext__def,axiom,
    ( multis2940008709989298052_a_nat
    = ( ^ [F2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,Xs3: list_t5668669291736003536_a_nat,Ys3: list_t5668669291736003536_a_nat] :
          ( product_Pair_o_o
          @ ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ Xs3 ) @ ( mset_t7141044528732312287_a_nat @ Ys3 ) )
            @ ( multis3666123592011920481_a_nat
              @ ( collec9088346736735320338_a_nat
                @ ( produc4098677624135445996_nat_o
                  @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec9088346736735320338_a_nat
                @ ( produc4098677624135445996_nat_o
                  @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) )
          @ ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ Xs3 ) @ ( mset_t7141044528732312287_a_nat @ Ys3 ) )
            @ ( multis6945725571414890367_a_nat
              @ ( collec9088346736735320338_a_nat
                @ ( produc4098677624135445996_nat_o
                  @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec9088346736735320338_a_nat
                @ ( produc4098677624135445996_nat_o
                  @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ) ).

% mul_ext_def
thf(fact_382_mul__ext__def,axiom,
    ( multis6586140943272769299on_nat
    = ( ^ [F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,Xs3: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
          ( product_Pair_o_o
          @ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ Xs3 ) @ ( mset_S7907735292211298936on_nat @ Ys3 ) )
            @ ( multis1620213760974180726on_nat
              @ ( collec5585564048161825126on_nat
                @ ( produc208325271403635792_nat_o
                  @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec5585564048161825126on_nat
                @ ( produc208325271403635792_nat_o
                  @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) )
          @ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ Xs3 ) @ ( mset_S7907735292211298936on_nat @ Ys3 ) )
            @ ( multis4183089776494585304on_nat
              @ ( collec5585564048161825126on_nat
                @ ( produc208325271403635792_nat_o
                  @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
              @ ( collec5585564048161825126on_nat
                @ ( produc208325271403635792_nat_o
                  @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ) ).

% mul_ext_def
thf(fact_383_subsetI,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ! [X: produc394670695091157671_a_nat] :
          ( ( member4231271695026834640_a_nat @ X @ A5 )
         => ( member4231271695026834640_a_nat @ X @ B5 ) )
     => ( ord_le998509814859404839_a_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_384_subsetI,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ! [X: produc5238312538367656103_a_nat] :
          ( ( member6928026473249828560_a_nat @ X @ A5 )
         => ( member6928026473249828560_a_nat @ X @ B5 ) )
     => ( ord_le7248185767425706023_a_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_385_subsetI,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ! [X: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X @ A5 )
         => ( member4278285197320822468on_nat @ X @ B5 ) )
     => ( ord_le927452247459997723on_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_386_subsetI,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ! [X: produc5580075136493545115on_nat] :
          ( ( member4221009242412147780on_nat @ X @ A5 )
         => ( member4221009242412147780on_nat @ X @ B5 ) )
     => ( ord_le6423710186836778395on_nat @ A5 @ B5 ) ) ).

% subsetI
thf(fact_387_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_388_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_389_trivial__rpo_Osupteq__subset__RPO__NS,axiom,
    ( ord_le251666825002368551_a_nat @ subter3376206727240021173_a_nat
    @ ( collec9088346736735320338_a_nat
      @ ( produc4098677624135445996_nat_o
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
            ( product_snd_o_o
            @ ( rpo_RP6885983473509681171_a_nat
              @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
              @ ^ [F2: produc2248741730625028719ms_nat] : $true
              @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
              @ zero_zero_nat
              @ S4
              @ T ) ) ) ) ) ).

% trivial_rpo.supteq_subset_RPO_NS
thf(fact_390_trivial__rpo_Osupt__subset__RPO__S,axiom,
    ( ord_le251666825002368551_a_nat @ subter8519896202621964801_a_nat
    @ ( collec9088346736735320338_a_nat
      @ ( produc4098677624135445996_nat_o
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
            ( product_fst_o_o
            @ ( rpo_RP6885983473509681171_a_nat
              @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
              @ ^ [F2: produc2248741730625028719ms_nat] : $true
              @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
              @ zero_zero_nat
              @ S4
              @ T ) ) ) ) ) ).

% trivial_rpo.supt_subset_RPO_S
thf(fact_391_less__eq__set__def,axiom,
    ( ord_le998509814859404839_a_nat
    = ( ^ [A6: set_Pr4991467605864603271_a_nat,B6: set_Pr4991467605864603271_a_nat] :
          ( ord_le2908930255090152822_nat_o
          @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ A6 )
          @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_392_less__eq__set__def,axiom,
    ( ord_le7248185767425706023_a_nat
    = ( ^ [A6: set_Pr8529752790658994311_a_nat,B6: set_Pr8529752790658994311_a_nat] :
          ( ord_le6847517786381931382_nat_o
          @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ A6 )
          @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_393_less__eq__set__def,axiom,
    ( ord_le927452247459997723on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B6: set_Pr3573155178515677307on_nat] :
          ( ord_le3750545489853369986_nat_o
          @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ A6 )
          @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_394_less__eq__set__def,axiom,
    ( ord_le6423710186836778395on_nat
    = ( ^ [A6: set_Pr7828957961435280891on_nat,B6: set_Pr7828957961435280891on_nat] :
          ( ord_le356980763962747138_nat_o
          @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ A6 )
          @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_395_ns__mul__ext__bottom__uniqueness,axiom,
    ! [A5: multis6046679608167549072_a_nat,Ns: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat] :
      ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ zero_z3261727345284792775_a_nat @ A5 ) @ ( multis6945725571414890367_a_nat @ Ns @ S ) )
     => ( A5 = zero_z3261727345284792775_a_nat ) ) ).

% ns_mul_ext_bottom_uniqueness
thf(fact_396_ns__mul__ext__bottom__uniqueness,axiom,
    ! [A5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ A5 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ( A5 = zero_z8702567803361615526on_nat ) ) ).

% ns_mul_ext_bottom_uniqueness
thf(fact_397_ns__mul__ext__bottom2,axiom,
    ! [A5: multis6046679608167549072_a_nat,B5: multis6046679608167549072_a_nat,Ns: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat] :
      ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A5 @ B5 ) @ ( multis6945725571414890367_a_nat @ Ns @ S ) )
     => ( ( B5 != zero_z3261727345284792775_a_nat )
       => ( A5 != zero_z3261727345284792775_a_nat ) ) ) ).

% ns_mul_ext_bottom2
thf(fact_398_ns__mul__ext__bottom2,axiom,
    ! [A5: multis1856558993800614429on_nat,B5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A5 @ B5 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ( ( B5 != zero_z8702567803361615526on_nat )
       => ( A5 != zero_z8702567803361615526on_nat ) ) ) ).

% ns_mul_ext_bottom2
thf(fact_399_ns__mul__ext__bottom,axiom,
    ! [A5: multis6046679608167549072_a_nat,Ns: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A5 @ zero_z3261727345284792775_a_nat ) @ ( multis6945725571414890367_a_nat @ Ns @ S ) ) ).

% ns_mul_ext_bottom
thf(fact_400_ns__mul__ext__bottom,axiom,
    ! [A5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A5 @ zero_z8702567803361615526on_nat ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ).

% ns_mul_ext_bottom
thf(fact_401_Collect__case__prod__mono,axiom,
    ! [A5: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,B5: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ( ord_le5401177226363330958_nat_o @ A5 @ B5 )
     => ( ord_le998509814859404839_a_nat @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ A5 ) ) @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_402_Collect__case__prod__mono,axiom,
    ! [A5: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,B5: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ( ord_le9078739809567865102_nat_o @ A5 @ B5 )
     => ( ord_le251666825002368551_a_nat @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ A5 ) ) @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_403_Collect__case__prod__mono,axiom,
    ! [A5: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,B5: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ( ord_le36159016115119212_nat_o @ A5 @ B5 )
     => ( ord_le6423710186836778395on_nat @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ A5 ) ) @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_404_Collect__case__prod__mono,axiom,
    ! [A5: $o > $o > $o,B5: $o > $o > $o] :
      ( ( ord_less_eq_o_o_o @ A5 @ B5 )
     => ( ord_le192195778054999463od_o_o @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ A5 ) ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_405_Collect__case__prod__mono,axiom,
    ! [A5: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,B5: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ( ord_le291120284566496650_nat_o @ A5 @ B5 )
     => ( ord_le3399424890108663549on_nat @ ( collec624303909867598824on_nat @ ( produc7728289706344822350_nat_o @ A5 ) ) @ ( collec624303909867598824on_nat @ ( produc7728289706344822350_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_406_Collect__case__prod__mono,axiom,
    ! [A5: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,B5: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ( ord_le4954966455738030386_nat_o @ A5 @ B5 )
     => ( ord_le248320493815342421on_nat @ ( collec8524684687135612820on_nat @ ( produc1360909077854731642_nat_o @ A5 ) ) @ ( collec8524684687135612820on_nat @ ( produc1360909077854731642_nat_o @ B5 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_407_s__ns__mul__ext,axiom,
    ! [A5: multis6046679608167549072_a_nat,B5: multis6046679608167549072_a_nat,Ns: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat] :
      ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A5 @ B5 ) @ ( multis3666123592011920481_a_nat @ Ns @ S ) )
     => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A5 @ B5 ) @ ( multis6945725571414890367_a_nat @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_408_s__ns__mul__ext,axiom,
    ! [A5: multis1856558993800614429on_nat,B5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A5 @ B5 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A5 @ B5 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_409_SN__order__pair_Omul__ext__SN__order__pair,axiom,
    ! [S6: set_Pr630484882382513799_a_nat,NS2: set_Pr630484882382513799_a_nat] :
      ( ( order_1584829357401681496_a_nat @ S6 @ NS2 )
     => ( order_3741756632120994334_a_nat @ ( multis3666123592011920481_a_nat @ NS2 @ S6 ) @ ( multis6945725571414890367_a_nat @ NS2 @ S6 ) ) ) ).

% SN_order_pair.mul_ext_SN_order_pair
thf(fact_410_SN__order__pair_Omul__ext__SN__order__pair,axiom,
    ! [S6: set_Pr2535564044409765851on_nat,NS2: set_Pr2535564044409765851on_nat] :
      ( ( order_760306688390729023on_nat @ S6 @ NS2 )
     => ( order_8476787420702163983on_nat @ ( multis1620213760974180726on_nat @ NS2 @ S6 ) @ ( multis4183089776494585304on_nat @ NS2 @ S6 ) ) ) ).

% SN_order_pair.mul_ext_SN_order_pair
thf(fact_411_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_412_le__cases3,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_413_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [X2: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X2 @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X2 ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_414_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_415_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_416_order__antisym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_417_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_418_order__trans,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z )
       => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_419_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A2 @ B2 )
         => ( P2 @ A2 @ B2 ) )
     => ( ! [A2: nat,B2: nat] :
            ( ( P2 @ B2 @ A2 )
           => ( P2 @ A2 @ B2 ) )
       => ( P2 @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_420_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_421_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_422_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_423_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_424_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z2: nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_425_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_426_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_427_order__eq__refl,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_428_linorder__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_429_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_430_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X: nat,Y: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ord_less_eq_nat @ ( F @ X ) @ ( F @ Y ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_431_linorder__le__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_432_order__antisym__conv,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_433_in__mono,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat,X3: produc394670695091157671_a_nat] :
      ( ( ord_le998509814859404839_a_nat @ A5 @ B5 )
     => ( ( member4231271695026834640_a_nat @ X3 @ A5 )
       => ( member4231271695026834640_a_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_434_in__mono,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat,X3: produc5238312538367656103_a_nat] :
      ( ( ord_le7248185767425706023_a_nat @ A5 @ B5 )
     => ( ( member6928026473249828560_a_nat @ X3 @ A5 )
       => ( member6928026473249828560_a_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_435_in__mono,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat,X3: produc9176241080893687067on_nat] :
      ( ( ord_le927452247459997723on_nat @ A5 @ B5 )
     => ( ( member4278285197320822468on_nat @ X3 @ A5 )
       => ( member4278285197320822468on_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_436_in__mono,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat,X3: produc5580075136493545115on_nat] :
      ( ( ord_le6423710186836778395on_nat @ A5 @ B5 )
     => ( ( member4221009242412147780on_nat @ X3 @ A5 )
       => ( member4221009242412147780on_nat @ X3 @ B5 ) ) ) ).

% in_mono
thf(fact_437_subsetD,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat,C: produc394670695091157671_a_nat] :
      ( ( ord_le998509814859404839_a_nat @ A5 @ B5 )
     => ( ( member4231271695026834640_a_nat @ C @ A5 )
       => ( member4231271695026834640_a_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_438_subsetD,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat,C: produc5238312538367656103_a_nat] :
      ( ( ord_le7248185767425706023_a_nat @ A5 @ B5 )
     => ( ( member6928026473249828560_a_nat @ C @ A5 )
       => ( member6928026473249828560_a_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_439_subsetD,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat,C: produc9176241080893687067on_nat] :
      ( ( ord_le927452247459997723on_nat @ A5 @ B5 )
     => ( ( member4278285197320822468on_nat @ C @ A5 )
       => ( member4278285197320822468on_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_440_subsetD,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat,C: produc5580075136493545115on_nat] :
      ( ( ord_le6423710186836778395on_nat @ A5 @ B5 )
     => ( ( member4221009242412147780on_nat @ C @ A5 )
       => ( member4221009242412147780on_nat @ C @ B5 ) ) ) ).

% subsetD
thf(fact_441_subset__eq,axiom,
    ( ord_le998509814859404839_a_nat
    = ( ^ [A6: set_Pr4991467605864603271_a_nat,B6: set_Pr4991467605864603271_a_nat] :
        ! [X2: produc394670695091157671_a_nat] :
          ( ( member4231271695026834640_a_nat @ X2 @ A6 )
         => ( member4231271695026834640_a_nat @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_442_subset__eq,axiom,
    ( ord_le7248185767425706023_a_nat
    = ( ^ [A6: set_Pr8529752790658994311_a_nat,B6: set_Pr8529752790658994311_a_nat] :
        ! [X2: produc5238312538367656103_a_nat] :
          ( ( member6928026473249828560_a_nat @ X2 @ A6 )
         => ( member6928026473249828560_a_nat @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_443_subset__eq,axiom,
    ( ord_le927452247459997723on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B6: set_Pr3573155178515677307on_nat] :
        ! [X2: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X2 @ A6 )
         => ( member4278285197320822468on_nat @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_444_subset__eq,axiom,
    ( ord_le6423710186836778395on_nat
    = ( ^ [A6: set_Pr7828957961435280891on_nat,B6: set_Pr7828957961435280891on_nat] :
        ! [X2: produc5580075136493545115on_nat] :
          ( ( member4221009242412147780on_nat @ X2 @ A6 )
         => ( member4221009242412147780on_nat @ X2 @ B6 ) ) ) ) ).

% subset_eq
thf(fact_445_subset__iff,axiom,
    ( ord_le998509814859404839_a_nat
    = ( ^ [A6: set_Pr4991467605864603271_a_nat,B6: set_Pr4991467605864603271_a_nat] :
        ! [T: produc394670695091157671_a_nat] :
          ( ( member4231271695026834640_a_nat @ T @ A6 )
         => ( member4231271695026834640_a_nat @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_446_subset__iff,axiom,
    ( ord_le7248185767425706023_a_nat
    = ( ^ [A6: set_Pr8529752790658994311_a_nat,B6: set_Pr8529752790658994311_a_nat] :
        ! [T: produc5238312538367656103_a_nat] :
          ( ( member6928026473249828560_a_nat @ T @ A6 )
         => ( member6928026473249828560_a_nat @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_447_subset__iff,axiom,
    ( ord_le927452247459997723on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B6: set_Pr3573155178515677307on_nat] :
        ! [T: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ T @ A6 )
         => ( member4278285197320822468on_nat @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_448_subset__iff,axiom,
    ( ord_le6423710186836778395on_nat
    = ( ^ [A6: set_Pr7828957961435280891on_nat,B6: set_Pr7828957961435280891on_nat] :
        ! [T: produc5580075136493545115on_nat] :
          ( ( member4221009242412147780on_nat @ T @ A6 )
         => ( member4221009242412147780on_nat @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_449_Collect__mono,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,Q: produc394670695091157671_a_nat > $o] :
      ( ! [X: produc394670695091157671_a_nat] :
          ( ( P2 @ X )
         => ( Q @ X ) )
     => ( ord_le998509814859404839_a_nat @ ( collec8858057866656402194_a_nat @ P2 ) @ ( collec8858057866656402194_a_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_450_Collect__mono,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ! [X: produc3043957382837978535_a_nat] :
          ( ( P2 @ X )
         => ( Q @ X ) )
     => ( ord_le251666825002368551_a_nat @ ( collec9088346736735320338_a_nat @ P2 ) @ ( collec9088346736735320338_a_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_451_Collect__mono,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,Q: produc5580075136493545115on_nat > $o] :
      ( ! [X: produc5580075136493545115on_nat] :
          ( ( P2 @ X )
         => ( Q @ X ) )
     => ( ord_le6423710186836778395on_nat @ ( collec1293568243156533638on_nat @ P2 ) @ ( collec1293568243156533638on_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_452_Collect__mono__iff,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,Q: produc394670695091157671_a_nat > $o] :
      ( ( ord_le998509814859404839_a_nat @ ( collec8858057866656402194_a_nat @ P2 ) @ ( collec8858057866656402194_a_nat @ Q ) )
      = ( ! [X2: produc394670695091157671_a_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_453_Collect__mono__iff,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ( ord_le251666825002368551_a_nat @ ( collec9088346736735320338_a_nat @ P2 ) @ ( collec9088346736735320338_a_nat @ Q ) )
      = ( ! [X2: produc3043957382837978535_a_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_454_Collect__mono__iff,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,Q: produc5580075136493545115on_nat > $o] :
      ( ( ord_le6423710186836778395on_nat @ ( collec1293568243156533638on_nat @ P2 ) @ ( collec1293568243156533638on_nat @ Q ) )
      = ( ! [X2: produc5580075136493545115on_nat] :
            ( ( P2 @ X2 )
           => ( Q @ X2 ) ) ) ) ).

% Collect_mono_iff
thf(fact_455_ex__mset,axiom,
    ! [X4: multis6046679608167549072_a_nat] :
    ? [Xs: list_t5668669291736003536_a_nat] :
      ( ( mset_t7141044528732312287_a_nat @ Xs )
      = X4 ) ).

% ex_mset
thf(fact_456_ex__mset,axiom,
    ! [X4: multis1856558993800614429on_nat] :
    ? [Xs: list_S5253404628753938141on_nat] :
      ( ( mset_S7907735292211298936on_nat @ Xs )
      = X4 ) ).

% ex_mset
thf(fact_457_case__prod__Pair__iden,axiom,
    ! [P: produc118098540238352959on_nat] :
      ( ( produc7619365177700836829on_nat @ produc4176415733022414961on_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_458_case__prod__Pair__iden,axiom,
    ! [P: produc8325617224762090749on_nat] :
      ( ( produc4035448539481142195on_nat @ produc1545487600272464053on_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_459_case__prod__Pair__iden,axiom,
    ! [P: produc5580075136493545115on_nat] :
      ( ( produc6113360178064570675on_nat @ produc2912653624306349395on_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_460_case__prod__Pair__iden,axiom,
    ! [P: produc394670695091157671_a_nat] :
      ( ( produc8464362608410685435_a_nat @ produc5685097156399411607_a_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_461_case__prod__Pair__iden,axiom,
    ! [P: product_prod_o_o] :
      ( ( produc7436348682273225467od_o_o @ product_Pair_o_o @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_462_case__prod__Pair__iden,axiom,
    ! [P: produc3043957382837978535_a_nat] :
      ( ( produc6569487705768959483_a_nat @ produc4880773030396931735_a_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_463_Collect__subset,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,P2: produc5238312538367656103_a_nat > $o] :
      ( ord_le7248185767425706023_a_nat
      @ ( collec6671004892620940562_a_nat
        @ ^ [X2: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X2 @ A5 )
            & ( P2 @ X2 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_464_Collect__subset,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,P2: produc9176241080893687067on_nat > $o] :
      ( ord_le927452247459997723on_nat
      @ ( collec1886838643678311430on_nat
        @ ^ [X2: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X2 @ A5 )
            & ( P2 @ X2 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_465_Collect__subset,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,P2: produc394670695091157671_a_nat > $o] :
      ( ord_le998509814859404839_a_nat
      @ ( collec8858057866656402194_a_nat
        @ ^ [X2: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X2 @ A5 )
            & ( P2 @ X2 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_466_Collect__subset,axiom,
    ! [A5: set_Pr630484882382513799_a_nat,P2: produc3043957382837978535_a_nat > $o] :
      ( ord_le251666825002368551_a_nat
      @ ( collec9088346736735320338_a_nat
        @ ^ [X2: produc3043957382837978535_a_nat] :
            ( ( member6937586146271652048_a_nat @ X2 @ A5 )
            & ( P2 @ X2 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_467_Collect__subset,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,P2: produc5580075136493545115on_nat > $o] :
      ( ord_le6423710186836778395on_nat
      @ ( collec1293568243156533638on_nat
        @ ^ [X2: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X2 @ A5 )
            & ( P2 @ X2 ) ) )
      @ A5 ) ).

% Collect_subset
thf(fact_468_mul__ext__list_Oextension,axiom,
    ! [S6: set_Pr630484882382513799_a_nat,NS2: set_Pr630484882382513799_a_nat] :
      ( ( order_1584829357401681496_a_nat @ S6 @ NS2 )
     => ( order_2465994371294916958_a_nat
        @ ( collec8858057866656402194_a_nat
          @ ( produc4945827410475597548_nat_o
            @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis3666123592011920481_a_nat @ NS2 @ S6 ) ) ) )
        @ ( collec8858057866656402194_a_nat
          @ ( produc4945827410475597548_nat_o
            @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis6945725571414890367_a_nat @ NS2 @ S6 ) ) ) ) ) ) ).

% mul_ext_list.extension
thf(fact_469_mul__ext__list_Oextension,axiom,
    ! [S6: set_Pr2535564044409765851on_nat,NS2: set_Pr2535564044409765851on_nat] :
      ( ( order_760306688390729023on_nat @ S6 @ NS2 )
     => ( order_7380660629604060367on_nat
        @ ( collec1293568243156533638on_nat
          @ ( produc7163990014243126832_nat_o
            @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis1620213760974180726on_nat @ NS2 @ S6 ) ) ) )
        @ ( collec1293568243156533638on_nat
          @ ( produc7163990014243126832_nat_o
            @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis4183089776494585304on_nat @ NS2 @ S6 ) ) ) ) ) ) ).

% mul_ext_list.extension
thf(fact_470_mset_Osimps_I1_J,axiom,
    ( ( mset_l8139550355095933148od_a_o @ nil_li1372968520647675725od_a_o )
    = zero_z8905637151584487364od_a_o ) ).

% mset.simps(1)
thf(fact_471_mset_Osimps_I1_J,axiom,
    ( ( mset_t7141044528732312287_a_nat @ nil_te8332778836088660176_a_nat )
    = zero_z3261727345284792775_a_nat ) ).

% mset.simps(1)
thf(fact_472_mset_Osimps_I1_J,axiom,
    ( ( mset_S7907735292211298936on_nat @ nil_Su3078906344068544711on_nat )
    = zero_z8702567803361615526on_nat ) ).

% mset.simps(1)
thf(fact_473_mset__ns__mul__ext,axiom,
    ! [Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ Xs2 ) @ ( mset_S7907735292211298936on_nat @ Ys2 ) )
        @ ( multis4183089776494585304on_nat
          @ ( collec5585564048161825126on_nat
            @ ( produc208325271403635792_nat_o
              @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec5585564048161825126on_nat
            @ ( produc208325271403635792_nat_o
              @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_snd_o_o @ ( multis6586140943272769299on_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_ns_mul_ext
thf(fact_474_mset__ns__mul__ext,axiom,
    ! [Xs2: list_l3889969465389992278_a_nat,Ys2: list_l3889969465389992278_a_nat,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o] :
      ( ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ Xs2 ) @ ( mset_l7063064786757117541_a_nat @ Ys2 ) )
        @ ( multis6753421577935756037_a_nat
          @ ( collec8858057866656402194_a_nat
            @ ( produc4945827410475597548_nat_o
              @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec8858057866656402194_a_nat
            @ ( produc4945827410475597548_nat_o
              @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_snd_o_o @ ( multis2433842936921369738_a_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_ns_mul_ext
thf(fact_475_mset__ns__mul__ext,axiom,
    ! [Xs2: list_t5668669291736003536_a_nat,Ys2: list_t5668669291736003536_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o] :
      ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ Xs2 ) @ ( mset_t7141044528732312287_a_nat @ Ys2 ) )
        @ ( multis6945725571414890367_a_nat
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_snd_o_o @ ( multis2940008709989298052_a_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_ns_mul_ext
thf(fact_476_mset__ns__mul__ext,axiom,
    ! [Xs2: list_l8999537460041914605on_nat,Ys2: list_l8999537460041914605on_nat,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o] :
      ( ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ Xs2 ) @ ( mset_l1352240747471131016on_nat @ Ys2 ) )
        @ ( multis1816195516039316712on_nat
          @ ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_snd_o_o @ ( multis5717424201997335715on_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_ns_mul_ext
thf(fact_477_mset__ns__mul__ext,axiom,
    ! [Xs2: list_o,Ys2: list_o,F: $o > $o > product_prod_o_o] :
      ( ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ Xs2 ) @ ( mset_o @ Ys2 ) )
        @ ( multis1436808229551206221_ext_o
          @ ( collec3167064739498627218od_o_o
            @ ( produc6197397395684419436_o_o_o
              @ ^ [X2: $o,Y5: $o] : ( product_snd_o_o @ ( F @ X2 @ Y5 ) ) ) )
          @ ( collec3167064739498627218od_o_o
            @ ( produc6197397395684419436_o_o_o
              @ ^ [X2: $o,Y5: $o] : ( product_fst_o_o @ ( F @ X2 @ Y5 ) ) ) ) ) )
      = ( product_snd_o_o @ ( multiset_mul_ext_o @ F @ Xs2 @ Ys2 ) ) ) ).

% mset_ns_mul_ext
thf(fact_478_suptI,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( A != B )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat ) ) ) ).

% suptI
thf(fact_479_nsmulextp__def,axiom,
    ( multis5465171036894672884on_nat
    = ( ^ [F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,M: multis1856558993800614429on_nat,N: multis1856558993800614429on_nat] :
          ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M @ N )
          @ ( multis4183089776494585304on_nat
            @ ( collec5585564048161825126on_nat
              @ ( produc208325271403635792_nat_o
                @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec5585564048161825126on_nat
              @ ( produc208325271403635792_nat_o
                @ ^ [X2: sum_su4907927656612468301on_nat,Y5: sum_su4907927656612468301on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% nsmulextp_def
thf(fact_480_nsmulextp__def,axiom,
    ( multis7856080302945509353_a_nat
    = ( ^ [F2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o,M: multis2209918487243275798_a_nat,N: multis2209918487243275798_a_nat] :
          ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ M @ N )
          @ ( multis6753421577935756037_a_nat
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% nsmulextp_def
thf(fact_481_nsmulextp__def,axiom,
    ( multis6902705103113543267_a_nat
    = ( ^ [F2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,M: multis6046679608167549072_a_nat,N: multis6046679608167549072_a_nat] :
          ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ M @ N )
          @ ( multis6945725571414890367_a_nat
            @ ( collec9088346736735320338_a_nat
              @ ( produc4098677624135445996_nat_o
                @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec9088346736735320338_a_nat
              @ ( produc4098677624135445996_nat_o
                @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% nsmulextp_def
thf(fact_482_nsmulextp__def,axiom,
    ( multis8015136079900678404on_nat
    = ( ^ [F2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o,M: multis3959152272169849901on_nat,N: multis3959152272169849901on_nat] :
          ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ M @ N )
          @ ( multis1816195516039316712on_nat
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% nsmulextp_def
thf(fact_483_nsmulextp__def,axiom,
    ( multiset_nsmulextp_o
    = ( ^ [F2: $o > $o > product_prod_o_o,M: multiset_o,N: multiset_o] :
          ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ M @ N )
          @ ( multis1436808229551206221_ext_o
            @ ( collec3167064739498627218od_o_o
              @ ( produc6197397395684419436_o_o_o
                @ ^ [X2: $o,Y5: $o] : ( product_snd_o_o @ ( F2 @ X2 @ Y5 ) ) ) )
            @ ( collec3167064739498627218od_o_o
              @ ( produc6197397395684419436_o_o_o
                @ ^ [X2: $o,Y5: $o] : ( product_fst_o_o @ ( F2 @ X2 @ Y5 ) ) ) ) ) ) ) ) ).

% nsmulextp_def
thf(fact_484_mul__ext__list_Olist__order__extension__axioms,axiom,
    ( list_l9000741177501154326_a_nat
    @ ^ [S4: set_Pr630484882382513799_a_nat,Ns4: set_Pr630484882382513799_a_nat] :
        ( collec8858057866656402194_a_nat
        @ ( produc4945827410475597548_nat_o
          @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis3666123592011920481_a_nat @ Ns4 @ S4 ) ) ) )
    @ ^ [S4: set_Pr630484882382513799_a_nat,Ns4: set_Pr630484882382513799_a_nat] :
        ( collec8858057866656402194_a_nat
        @ ( produc4945827410475597548_nat_o
          @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis6945725571414890367_a_nat @ Ns4 @ S4 ) ) ) ) ) ).

% mul_ext_list.list_order_extension_axioms
thf(fact_485_mul__ext__list_Olist__order__extension__axioms,axiom,
    ( list_l4998409290742832897on_nat
    @ ^ [S4: set_Pr2535564044409765851on_nat,Ns4: set_Pr2535564044409765851on_nat] :
        ( collec1293568243156533638on_nat
        @ ( produc7163990014243126832_nat_o
          @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis1620213760974180726on_nat @ Ns4 @ S4 ) ) ) )
    @ ^ [S4: set_Pr2535564044409765851on_nat,Ns4: set_Pr2535564044409765851on_nat] :
        ( collec1293568243156533638on_nat
        @ ( produc7163990014243126832_nat_o
          @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis4183089776494585304on_nat @ Ns4 @ S4 ) ) ) ) ) ).

% mul_ext_list.list_order_extension_axioms
thf(fact_486_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_487_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_488_pred__subset__eq2,axiom,
    ! [R: set_Pr8529752790658994311_a_nat,S6: set_Pr8529752790658994311_a_nat] :
      ( ( ord_le6897027387178272142_nat_o
        @ ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le7248185767425706023_a_nat @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_489_pred__subset__eq2,axiom,
    ! [R: set_Pr3573155178515677307on_nat,S6: set_Pr3573155178515677307on_nat] :
      ( ( ord_le2626687706590981100_nat_o
        @ ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le927452247459997723on_nat @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_490_pred__subset__eq2,axiom,
    ! [R: set_Pr737724919871395061on_nat,S6: set_Pr737724919871395061on_nat] :
      ( ( ord_le4954966455738030386_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le248320493815342421on_nat @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_491_pred__subset__eq2,axiom,
    ! [R: set_Pr6244131468446592861on_nat,S6: set_Pr6244131468446592861on_nat] :
      ( ( ord_le291120284566496650_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le3399424890108663549on_nat @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_492_pred__subset__eq2,axiom,
    ! [R: set_Pr7828957961435280891on_nat,S6: set_Pr7828957961435280891on_nat] :
      ( ( ord_le36159016115119212_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le6423710186836778395on_nat @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_493_pred__subset__eq2,axiom,
    ! [R: set_Pr4991467605864603271_a_nat,S6: set_Pr4991467605864603271_a_nat] :
      ( ( ord_le5401177226363330958_nat_o
        @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le998509814859404839_a_nat @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_494_pred__subset__eq2,axiom,
    ! [R: set_Product_prod_o_o,S6: set_Product_prod_o_o] :
      ( ( ord_less_eq_o_o_o
        @ ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ R )
        @ ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le192195778054999463od_o_o @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_495_pred__subset__eq2,axiom,
    ! [R: set_Pr630484882382513799_a_nat,S6: set_Pr630484882382513799_a_nat] :
      ( ( ord_le9078739809567865102_nat_o
        @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ord_le251666825002368551_a_nat @ R @ S6 ) ) ).

% pred_subset_eq2
thf(fact_496_trivial__rpo_Orpo__vars__term,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( ( product_fst_o_o
          @ ( rpo_RP6885983473509681171_a_nat
            @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
            @ ^ [F2: produc2248741730625028719ms_nat] : $true
            @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
            @ zero_zero_nat
            @ S
            @ T2 ) )
        | ( product_snd_o_o
          @ ( rpo_RP6885983473509681171_a_nat
            @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
            @ ^ [F2: produc2248741730625028719ms_nat] : $true
            @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
            @ zero_zero_nat
            @ S
            @ T2 ) ) )
     => ( ord_le1325389633284124927_a_nat @ ( vars_t2456970921184301395_a_nat @ T2 ) @ ( vars_t2456970921184301395_a_nat @ S ) ) ) ).

% trivial_rpo.rpo_vars_term
thf(fact_497_subterm_OleD,axiom,
    ! [X3: term_R1747997765655105994_a_nat,Y3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter3376206727240021173_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.leD
thf(fact_498_subterm_Ole__less,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
        | ( X3 = Y3 ) ) ) ).

% subterm.le_less
thf(fact_499_Nat_Oex__has__greatest__nat,axiom,
    ! [P2: nat > $o,K: nat,B: nat] :
      ( ( P2 @ K )
     => ( ! [Y: nat] :
            ( ( P2 @ Y )
           => ( ord_less_eq_nat @ Y @ B ) )
       => ? [X: nat] :
            ( ( P2 @ X )
            & ! [Y6: nat] :
                ( ( P2 @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_500_nat__le__linear,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
      | ( ord_less_eq_nat @ N2 @ M2 ) ) ).

% nat_le_linear
thf(fact_501_le__antisym,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M2 )
       => ( M2 = N2 ) ) ) ).

% le_antisym
thf(fact_502_eq__imp__le,axiom,
    ! [M2: nat,N2: nat] :
      ( ( M2 = N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% eq_imp_le
thf(fact_503_le__trans,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I @ K ) ) ) ).

% le_trans
thf(fact_504_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_505_trivial__rpo_Owpo_Ocases,axiom,
    ! [X3: produc3043957382837978535_a_nat] :
      ~ ! [S7: term_R1747997765655105994_a_nat,T3: term_R1747997765655105994_a_nat] :
          ( X3
         != ( produc4880773030396931735_a_nat @ S7 @ T3 ) ) ).

% trivial_rpo.wpo.cases
thf(fact_506_supteq__imp__vars__term__subset,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
     => ( ord_le1325389633284124927_a_nat @ ( vars_t2456970921184301395_a_nat @ T2 ) @ ( vars_t2456970921184301395_a_nat @ S ) ) ) ).

% supteq_imp_vars_term_subset
thf(fact_507_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_508_bot__nat__0_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_unique
thf(fact_509_bot__nat__0_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( A = zero_zero_nat ) ) ).

% bot_nat_0.extremum_uniqueI
thf(fact_510_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_511_supteq__antisym,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ S ) @ subter3376206727240021173_a_nat )
       => ( S = T2 ) ) ) ).

% supteq_antisym
thf(fact_512_supteq__trans,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ U ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ U ) @ subter3376206727240021173_a_nat ) ) ) ).

% supteq_trans
thf(fact_513_eq__supteq,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( S = T2 )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat ) ) ).

% eq_supteq
thf(fact_514_subterm_Odual__order_Oantisym,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
       => ( A = B ) ) ) ).

% subterm.dual_order.antisym
thf(fact_515_subterm_Odual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: term_R1747997765655105994_a_nat,Z2: term_R1747997765655105994_a_nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] :
          ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A3 @ B3 ) @ subter3376206727240021173_a_nat )
          & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B3 @ A3 ) @ subter3376206727240021173_a_nat ) ) ) ) ).

% subterm.dual_order.eq_iff
thf(fact_516_subterm_Odual__order_Otrans,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ C ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ C ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.dual_order.trans
thf(fact_517_subterm_Oord__le__eq__trans,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
     => ( ( B = C )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.ord_le_eq_trans
thf(fact_518_subterm_Oord__eq__le__trans,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( A = B )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ B ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.ord_eq_le_trans
thf(fact_519_subterm_Odual__order_Orefl,axiom,
    ! [A: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ A ) @ subter3376206727240021173_a_nat ) ).

% subterm.dual_order.refl
thf(fact_520_subterm_Oorder__antisym,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter3376206727240021173_a_nat )
       => ( X3 = Y3 ) ) ) ).

% subterm.order_antisym
thf(fact_521_subterm_Oorder_Oantisym,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
       => ( A = B ) ) ) ).

% subterm.order.antisym
thf(fact_522_subterm_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: term_R1747997765655105994_a_nat,Z2: term_R1747997765655105994_a_nat] : ( Y4 = Z2 ) )
    = ( ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] :
          ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y5 @ X2 ) @ subter3376206727240021173_a_nat )
          & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ subter3376206727240021173_a_nat ) ) ) ) ).

% subterm.order_eq_iff
thf(fact_523_subterm_Oorder_Oeq__iff,axiom,
    ( ( ^ [Y4: term_R1747997765655105994_a_nat,Z2: term_R1747997765655105994_a_nat] : ( Y4 = Z2 ) )
    = ( ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] :
          ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B3 @ A3 ) @ subter3376206727240021173_a_nat )
          & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A3 @ B3 ) @ subter3376206727240021173_a_nat ) ) ) ) ).

% subterm.order.eq_iff
thf(fact_524_subterm_Oantisym__conv,axiom,
    ! [X3: term_R1747997765655105994_a_nat,Y3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
        = ( X3 = Y3 ) ) ) ).

% subterm.antisym_conv
thf(fact_525_subterm_Oorder__trans,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat,Z: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ Y3 ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ X3 ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.order_trans
thf(fact_526_subterm_Oorder_Otrans,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ B ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.order.trans
thf(fact_527_subterm_Oorder__refl,axiom,
    ! [X3: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ X3 ) @ subter3376206727240021173_a_nat ) ).

% subterm.order_refl
thf(fact_528_subterm_Oeq__refl,axiom,
    ! [X3: term_R1747997765655105994_a_nat,Y3: term_R1747997765655105994_a_nat] :
      ( ( X3 = Y3 )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat ) ) ).

% subterm.eq_refl
thf(fact_529_supteq_Orefl,axiom,
    ! [T2: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ T2 ) @ subter3376206727240021173_a_nat ) ).

% supteq.refl
thf(fact_530_subterm__induct,axiom,
    ! [P2: term_R1747997765655105994_a_nat > $o,T2: term_R1747997765655105994_a_nat] :
      ( ! [T3: term_R1747997765655105994_a_nat] :
          ( ! [S8: term_R1747997765655105994_a_nat] :
              ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T3 @ S8 ) @ subter8519896202621964801_a_nat )
             => ( P2 @ S8 ) )
         => ( P2 @ T3 ) )
     => ( P2 @ T2 ) ) ).

% subterm_induct
thf(fact_531_supt__not__refl,axiom,
    ! [T2: term_R1747997765655105994_a_nat] :
      ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ T2 ) @ subter8519896202621964801_a_nat ) ).

% supt_not_refl
thf(fact_532_supt__not__sym,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ S ) @ subter8519896202621964801_a_nat ) ) ).

% supt_not_sym
thf(fact_533_supt__irrefl,axiom,
    ! [T2: term_R1747997765655105994_a_nat] :
      ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ T2 ) @ subter8519896202621964801_a_nat ) ).

% supt_irrefl
thf(fact_534_supt__trans,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ U ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ U ) @ subter8519896202621964801_a_nat ) ) ) ).

% supt_trans
thf(fact_535_supt__neqD,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
     => ( S != T2 ) ) ).

% supt_neqD
thf(fact_536_subterm_Odual__order_Ostrict__implies__not__eq,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
     => ( A != B ) ) ).

% subterm.dual_order.strict_implies_not_eq
thf(fact_537_subterm_Oorder_Ostrict__implies__not__eq,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
     => ( A != B ) ) ).

% subterm.order.strict_implies_not_eq
thf(fact_538_subterm_Odual__order_Ostrict__trans,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ C ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ C ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.dual_order.strict_trans
thf(fact_539_subterm_Oorder_Ostrict__trans,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ B ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.order.strict_trans
thf(fact_540_subterm_Oord__less__eq__trans,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
     => ( ( B = C )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.ord_less_eq_trans
thf(fact_541_subterm_Oord__eq__less__trans,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( A = B )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ B ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.ord_eq_less_trans
thf(fact_542_subterm_Oless__imp__not__less,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.less_imp_not_less
thf(fact_543_subterm_Odual__order_Oirrefl,axiom,
    ! [A: term_R1747997765655105994_a_nat] :
      ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ A ) @ subter8519896202621964801_a_nat ) ).

% subterm.dual_order.irrefl
thf(fact_544_subterm_Oless__imp__not__eq2,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( Y3 != X3 ) ) ).

% subterm.less_imp_not_eq2
thf(fact_545_subterm_Oless__imp__not__eq,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( X3 != Y3 ) ) ).

% subterm.less_imp_not_eq
thf(fact_546_subterm_Odual__order_Oasym,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.dual_order.asym
thf(fact_547_subterm_Oless__imp__triv,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat,P2: $o] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter8519896202621964801_a_nat )
       => P2 ) ) ).

% subterm.less_imp_triv
thf(fact_548_subterm_Oless__not__sym,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.less_not_sym
thf(fact_549_subterm_Oless__imp__neq,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( X3 != Y3 ) ) ).

% subterm.less_imp_neq
thf(fact_550_subterm_Oless__irrefl,axiom,
    ! [X3: term_R1747997765655105994_a_nat] :
      ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ X3 ) @ subter8519896202621964801_a_nat ) ).

% subterm.less_irrefl
thf(fact_551_subterm_Oorder_Oasym,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.order.asym
thf(fact_552_subterm_Oless__trans,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat,Z: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ Y3 ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ X3 ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.less_trans
thf(fact_553_subterm_Oless__asym_H,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.less_asym'
thf(fact_554_subterm_Oless__asym,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.less_asym
thf(fact_555_pred__equals__eq2,axiom,
    ! [R: set_Pr8529752790658994311_a_nat,S6: set_Pr8529752790658994311_a_nat] :
      ( ( ( ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_556_pred__equals__eq2,axiom,
    ! [R: set_Pr3573155178515677307on_nat,S6: set_Pr3573155178515677307on_nat] :
      ( ( ( ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_557_pred__equals__eq2,axiom,
    ! [R: set_Pr737724919871395061on_nat,S6: set_Pr737724919871395061on_nat] :
      ( ( ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_558_pred__equals__eq2,axiom,
    ! [R: set_Pr6244131468446592861on_nat,S6: set_Pr6244131468446592861on_nat] :
      ( ( ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_559_pred__equals__eq2,axiom,
    ! [R: set_Pr7828957961435280891on_nat,S6: set_Pr7828957961435280891on_nat] :
      ( ( ( ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_560_pred__equals__eq2,axiom,
    ! [R: set_Pr4991467605864603271_a_nat,S6: set_Pr4991467605864603271_a_nat] :
      ( ( ( ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_561_pred__equals__eq2,axiom,
    ! [R: set_Product_prod_o_o,S6: set_Product_prod_o_o] :
      ( ( ( ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_562_pred__equals__eq2,axiom,
    ! [R: set_Pr630484882382513799_a_nat,S6: set_Pr630484882382513799_a_nat] :
      ( ( ( ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ R ) )
        = ( ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ S6 ) ) )
      = ( R = S6 ) ) ).

% pred_equals_eq2
thf(fact_563_pred__subset__eq,axiom,
    ! [R: set_Pr4991467605864603271_a_nat,S6: set_Pr4991467605864603271_a_nat] :
      ( ( ord_le2908930255090152822_nat_o
        @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ R )
        @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ S6 ) )
      = ( ord_le998509814859404839_a_nat @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_564_pred__subset__eq,axiom,
    ! [R: set_Pr8529752790658994311_a_nat,S6: set_Pr8529752790658994311_a_nat] :
      ( ( ord_le6847517786381931382_nat_o
        @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ R )
        @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ S6 ) )
      = ( ord_le7248185767425706023_a_nat @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_565_pred__subset__eq,axiom,
    ! [R: set_Pr3573155178515677307on_nat,S6: set_Pr3573155178515677307on_nat] :
      ( ( ord_le3750545489853369986_nat_o
        @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ R )
        @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ S6 ) )
      = ( ord_le927452247459997723on_nat @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_566_pred__subset__eq,axiom,
    ! [R: set_Pr7828957961435280891on_nat,S6: set_Pr7828957961435280891on_nat] :
      ( ( ord_le356980763962747138_nat_o
        @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ R )
        @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ S6 ) )
      = ( ord_le6423710186836778395on_nat @ R @ S6 ) ) ).

% pred_subset_eq
thf(fact_567_subrelI,axiom,
    ! [R2: set_Pr8529752790658994311_a_nat,S: set_Pr8529752790658994311_a_nat] :
      ( ! [X: multis6046679608167549072_a_nat,Y: multis6046679608167549072_a_nat] :
          ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X @ Y ) @ R2 )
         => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X @ Y ) @ S ) )
     => ( ord_le7248185767425706023_a_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_568_subrelI,axiom,
    ! [R2: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ! [X: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat] :
          ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y ) @ R2 )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y ) @ S ) )
     => ( ord_le927452247459997723on_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_569_subrelI,axiom,
    ! [R2: set_Pr737724919871395061on_nat,S: set_Pr737724919871395061on_nat] :
      ( ! [X: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
          ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y ) @ R2 )
         => ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y ) @ S ) )
     => ( ord_le248320493815342421on_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_570_subrelI,axiom,
    ! [R2: set_Pr6244131468446592861on_nat,S: set_Pr6244131468446592861on_nat] :
      ( ! [X: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
          ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y ) @ R2 )
         => ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y ) @ S ) )
     => ( ord_le3399424890108663549on_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_571_subrelI,axiom,
    ! [R2: set_Pr7828957961435280891on_nat,S: set_Pr7828957961435280891on_nat] :
      ( ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
          ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X @ Y ) @ R2 )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X @ Y ) @ S ) )
     => ( ord_le6423710186836778395on_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_572_subrelI,axiom,
    ! [R2: set_Pr4991467605864603271_a_nat,S: set_Pr4991467605864603271_a_nat] :
      ( ! [X: list_t5668669291736003536_a_nat,Y: list_t5668669291736003536_a_nat] :
          ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X @ Y ) @ R2 )
         => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X @ Y ) @ S ) )
     => ( ord_le998509814859404839_a_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_573_subrelI,axiom,
    ! [R2: set_Product_prod_o_o,S: set_Product_prod_o_o] :
      ( ! [X: $o,Y: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ R2 )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X @ Y ) @ S ) )
     => ( ord_le192195778054999463od_o_o @ R2 @ S ) ) ).

% subrelI
thf(fact_574_subrelI,axiom,
    ! [R2: set_Pr630484882382513799_a_nat,S: set_Pr630484882382513799_a_nat] :
      ( ! [X: term_R1747997765655105994_a_nat,Y: term_R1747997765655105994_a_nat] :
          ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X @ Y ) @ R2 )
         => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X @ Y ) @ S ) )
     => ( ord_le251666825002368551_a_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_575_nsmulextp__mset__code,axiom,
    ! [F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,Xs2: list_t5668669291736003536_a_nat,Ys2: list_t5668669291736003536_a_nat] :
      ( ( multis6902705103113543267_a_nat @ F @ ( mset_t7141044528732312287_a_nat @ Xs2 ) @ ( mset_t7141044528732312287_a_nat @ Ys2 ) )
      = ( product_snd_o_o @ ( multis2940008709989298052_a_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% nsmulextp_mset_code
thf(fact_576_nsmulextp__mset__code,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( multis5465171036894672884on_nat @ F @ ( mset_S7907735292211298936on_nat @ Xs2 ) @ ( mset_S7907735292211298936on_nat @ Ys2 ) )
      = ( product_snd_o_o @ ( multis6586140943272769299on_nat @ F @ Xs2 @ Ys2 ) ) ) ).

% nsmulextp_mset_code
thf(fact_577_supt__supteq__not__supteq,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
        & ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ S ) @ subter3376206727240021173_a_nat ) ) ) ).

% supt_supteq_not_supteq
thf(fact_578_supteq__supt__trans,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ U ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ U ) @ subter8519896202621964801_a_nat ) ) ) ).

% supteq_supt_trans
thf(fact_579_supt__supteq__trans,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T2 @ U ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ U ) @ subter8519896202621964801_a_nat ) ) ) ).

% supt_supteq_trans
thf(fact_580_supteq__supt__conv,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
        | ( S = T2 ) ) ) ).

% supteq_supt_conv
thf(fact_581_supt__supteq__conv,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
        & ( S != T2 ) ) ) ).

% supt_supteq_conv
thf(fact_582_supteq__not__supt,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
     => ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
       => ( S = T2 ) ) ) ).

% supteq_not_supt
thf(fact_583_supt__imp__supteq,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat ) ) ).

% supt_imp_supteq
thf(fact_584_suptE,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
     => ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
       => ( S != T2 ) ) ) ).

% suptE
thf(fact_585_subterm_Oorder_Onot__eq__order__implies__strict,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( A != B )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.order.not_eq_order_implies_strict
thf(fact_586_subterm_Odual__order_Ostrict__implies__order,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat ) ) ).

% subterm.dual_order.strict_implies_order
thf(fact_587_subterm_Odual__order_Ostrict__iff__order,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
        & ( A != B ) ) ) ).

% subterm.dual_order.strict_iff_order
thf(fact_588_subterm_Odual__order_Oorder__iff__strict,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
        | ( A = B ) ) ) ).

% subterm.dual_order.order_iff_strict
thf(fact_589_subterm_Oorder_Ostrict__implies__order,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat ) ) ).

% subterm.order.strict_implies_order
thf(fact_590_subterm_Odual__order_Ostrict__iff__not,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
        & ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.dual_order.strict_iff_not
thf(fact_591_subterm_Odual__order_Ostrict__trans2,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ C ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ C ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.dual_order.strict_trans2
thf(fact_592_subterm_Odual__order_Ostrict__trans1,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ C ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ C ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.dual_order.strict_trans1
thf(fact_593_subterm_Oorder_Ostrict__iff__order,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
        & ( A != B ) ) ) ).

% subterm.order.strict_iff_order
thf(fact_594_subterm_Oorder_Oorder__iff__strict,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
        | ( A = B ) ) ) ).

% subterm.order.order_iff_strict
thf(fact_595_subterm_Oorder_Ostrict__iff__not,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
        & ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.order.strict_iff_not
thf(fact_596_subterm_Oorder_Ostrict__trans2,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ B ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.order.strict_trans2
thf(fact_597_subterm_Oorder_Ostrict__trans1,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ B ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ C @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.order.strict_trans1
thf(fact_598_subterm_Ole__imp__less__or__eq,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
        | ( X3 = Y3 ) ) ) ).

% subterm.le_imp_less_or_eq
thf(fact_599_subterm_Oless__le__not__le,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
        & ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.less_le_not_le
thf(fact_600_subterm_Oless__le__trans,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat,Z: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ Y3 ) @ subter3376206727240021173_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ X3 ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.less_le_trans
thf(fact_601_subterm_Ole__less__trans,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat,Z: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ Y3 ) @ subter8519896202621964801_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Z @ X3 ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.le_less_trans
thf(fact_602_subterm_Oantisym__conv2,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
     => ( ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat ) )
        = ( X3 = Y3 ) ) ) ).

% subterm.antisym_conv2
thf(fact_603_subterm_Oantisym__conv1,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
        = ( X3 = Y3 ) ) ) ).

% subterm.antisym_conv1
thf(fact_604_subterm_Ole__neq__trans,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
     => ( ( A != B )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.le_neq_trans
thf(fact_605_subterm_Oless__imp__le,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat ) ) ).

% subterm.less_imp_le
thf(fact_606_subterm_Onless__le,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat ) )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
        | ( A = B ) ) ) ).

% subterm.nless_le
thf(fact_607_subterm_Oless__le,axiom,
    ! [Y3: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter8519896202621964801_a_nat )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ X3 ) @ subter3376206727240021173_a_nat )
        & ( X3 != Y3 ) ) ) ).

% subterm.less_le
thf(fact_608_rpo__with__assms_Orpo__vars__term,axiom,
    ! [Prc: produc2248741730625028719ms_nat > produc2248741730625028719ms_nat > product_prod_o_o,Prl: produc2248741730625028719ms_nat > $o,C: produc2248741730625028719ms_nat > order_tag,N2: nat,S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat] :
      ( ( rpo_wi5375545839774465844_FSyms @ Prc @ Prl )
     => ( ( ( product_fst_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S @ T2 ) )
          | ( product_snd_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S @ T2 ) ) )
       => ( ord_le1325389633284124927_a_nat @ ( vars_t2456970921184301395_a_nat @ T2 ) @ ( vars_t2456970921184301395_a_nat @ S ) ) ) ) ).

% rpo_with_assms.rpo_vars_term
thf(fact_609_subterm_OgreaterThanAtMost__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_gr6446970317771492622_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ L
          @ U ) )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ I @ L ) @ subter8519896202621964801_a_nat )
        & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ U @ I ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.greaterThanAtMost_iff
thf(fact_610_subterm_OatLeastLessThan__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_at5605147574288828367_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ L
          @ U ) )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ I @ L ) @ subter3376206727240021173_a_nat )
        & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ U @ I ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.atLeastLessThan_iff
thf(fact_611_trivial__rpo_ORPO__NS__subst,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,Sigma: sum_sum_a_nat > term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 )
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
                ( product_snd_o_o
                @ ( rpo_RP6885983473509681171_a_nat
                  @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
                  @ ^ [F2: produc2248741730625028719ms_nat] : $true
                  @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
                  @ zero_zero_nat
                  @ S4
                  @ T ) ) ) ) )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( subst_4241131627535844496_a_nat @ S @ Sigma ) @ ( subst_4241131627535844496_a_nat @ T2 @ Sigma ) )
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
                ( product_snd_o_o
                @ ( rpo_RP6885983473509681171_a_nat
                  @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
                  @ ^ [F2: produc2248741730625028719ms_nat] : $true
                  @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
                  @ zero_zero_nat
                  @ S4
                  @ T ) ) ) ) ) ) ).

% trivial_rpo.RPO_NS_subst
thf(fact_612_trivial__rpo_ORPO__S__subst,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,Sigma: sum_sum_a_nat > term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 )
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
                ( product_fst_o_o
                @ ( rpo_RP6885983473509681171_a_nat
                  @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
                  @ ^ [F2: produc2248741730625028719ms_nat] : $true
                  @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
                  @ zero_zero_nat
                  @ S4
                  @ T ) ) ) ) )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( subst_4241131627535844496_a_nat @ S @ Sigma ) @ ( subst_4241131627535844496_a_nat @ T2 @ Sigma ) )
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
                ( product_fst_o_o
                @ ( rpo_RP6885983473509681171_a_nat
                  @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
                  @ ^ [F2: produc2248741730625028719ms_nat] : $true
                  @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
                  @ zero_zero_nat
                  @ S4
                  @ T ) ) ) ) ) ) ).

% trivial_rpo.RPO_S_subst
thf(fact_613_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_614_supteq__subst,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,Sigma: sum_sum_a_nat > term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter3376206727240021173_a_nat )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( subst_4241131627535844496_a_nat @ S @ Sigma ) @ ( subst_4241131627535844496_a_nat @ T2 @ Sigma ) ) @ subter3376206727240021173_a_nat ) ) ).

% supteq_subst
thf(fact_615_supt__subst,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,Sigma: sum_sum_a_nat > term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ subter8519896202621964801_a_nat )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( subst_4241131627535844496_a_nat @ S @ Sigma ) @ ( subst_4241131627535844496_a_nat @ T2 @ Sigma ) ) @ subter8519896202621964801_a_nat ) ) ).

% supt_subst
thf(fact_616_rpo__with__assms_ORPO__SN__order__pair,axiom,
    ! [Prc: produc2248741730625028719ms_nat > produc2248741730625028719ms_nat > product_prod_o_o,Prl: produc2248741730625028719ms_nat > $o,C: produc2248741730625028719ms_nat > order_tag,N2: nat] :
      ( ( rpo_wi5375545839774465844_FSyms @ Prc @ Prl )
     => ( order_1584829357401681496_a_nat
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) )
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) ) ) ).

% rpo_with_assms.RPO_SN_order_pair
thf(fact_617_rpo__with__assms_ORPO__S__subset__RPO__NS,axiom,
    ! [Prc: produc2248741730625028719ms_nat > produc2248741730625028719ms_nat > product_prod_o_o,Prl: produc2248741730625028719ms_nat > $o,C: produc2248741730625028719ms_nat > order_tag,N2: nat] :
      ( ( rpo_wi5375545839774465844_FSyms @ Prc @ Prl )
     => ( ord_le251666825002368551_a_nat
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) )
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) ) ) ).

% rpo_with_assms.RPO_S_subset_RPO_NS
thf(fact_618_rpo__with__assms_Osupt__subset__RPO__S,axiom,
    ! [Prc: produc2248741730625028719ms_nat > produc2248741730625028719ms_nat > product_prod_o_o,Prl: produc2248741730625028719ms_nat > $o,C: produc2248741730625028719ms_nat > order_tag,N2: nat] :
      ( ( rpo_wi5375545839774465844_FSyms @ Prc @ Prl )
     => ( ord_le251666825002368551_a_nat @ subter8519896202621964801_a_nat
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) ) ) ).

% rpo_with_assms.supt_subset_RPO_S
thf(fact_619_rpo__with__assms_Osupteq__subset__RPO__NS,axiom,
    ! [Prc: produc2248741730625028719ms_nat > produc2248741730625028719ms_nat > product_prod_o_o,Prl: produc2248741730625028719ms_nat > $o,C: produc2248741730625028719ms_nat > order_tag,N2: nat] :
      ( ( rpo_wi5375545839774465844_FSyms @ Prc @ Prl )
     => ( ord_le251666825002368551_a_nat @ subter3376206727240021173_a_nat
        @ ( collec9088346736735320338_a_nat
          @ ( produc4098677624135445996_nat_o
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) ) ) ).

% rpo_with_assms.supteq_subset_RPO_NS
thf(fact_620_rpo__with__assms_ORPO__NS__subst,axiom,
    ! [Prc: produc2248741730625028719ms_nat > produc2248741730625028719ms_nat > product_prod_o_o,Prl: produc2248741730625028719ms_nat > $o,S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,C: produc2248741730625028719ms_nat > order_tag,N2: nat,Sigma: sum_sum_a_nat > term_R1747997765655105994_a_nat] :
      ( ( rpo_wi5375545839774465844_FSyms @ Prc @ Prl )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 )
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( subst_4241131627535844496_a_nat @ S @ Sigma ) @ ( subst_4241131627535844496_a_nat @ T2 @ Sigma ) )
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_snd_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) ) ) ) ).

% rpo_with_assms.RPO_NS_subst
thf(fact_621_rpo__with__assms_ORPO__S__subst,axiom,
    ! [Prc: produc2248741730625028719ms_nat > produc2248741730625028719ms_nat > product_prod_o_o,Prl: produc2248741730625028719ms_nat > $o,S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,C: produc2248741730625028719ms_nat > order_tag,N2: nat,Sigma: sum_sum_a_nat > term_R1747997765655105994_a_nat] :
      ( ( rpo_wi5375545839774465844_FSyms @ Prc @ Prl )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 )
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( subst_4241131627535844496_a_nat @ S @ Sigma ) @ ( subst_4241131627535844496_a_nat @ T2 @ Sigma ) )
          @ ( collec9088346736735320338_a_nat
            @ ( produc4098677624135445996_nat_o
              @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( product_fst_o_o @ ( rpo_RP6885983473509681171_a_nat @ Prc @ Prl @ C @ N2 @ S4 @ T ) ) ) ) ) ) ) ).

% rpo_with_assms.RPO_S_subst
thf(fact_622_subterm_Obdd__above__Ioc,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit1868887373268119414_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_gr6446970317771492622_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Ioc
thf(fact_623_subterm_Obdd__above__Ico,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit1868887373268119414_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_at5605147574288828367_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Ico
thf(fact_624_subterm_Obdd__below__Ioc,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit8263634360281639266_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_gr6446970317771492622_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_below_Ioc
thf(fact_625_subterm_Obdd__below__Ico,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit8263634360281639266_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_at5605147574288828367_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_below_Ico
thf(fact_626_list__order__extension__impl_Olist__ext__ns,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,List_ext: ( sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o ) > list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o,S: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Ns: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
      ( ( list_l6805256357292310145on_nat @ S_list @ Ns_list @ List_ext )
     => ( ( Ns_list @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ S ) ) @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ Ns ) ) )
        = ( collec1293568243156533638on_nat
          @ ( produc7163990014243126832_nat_o
            @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] :
                ( product_snd_o_o
                @ ( List_ext
                  @ ^ [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_ns
thf(fact_627_list__order__extension__impl_Olist__ext__ns,axiom,
    ! [S_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,Ns_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,List_ext: ( list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o ) > list_l3889969465389992278_a_nat > list_l3889969465389992278_a_nat > product_prod_o_o,S: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Ns: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ( list_l2859941575254952092_a_nat @ S_list @ Ns_list @ List_ext )
     => ( ( Ns_list @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ S ) ) @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ Ns ) ) )
        = ( collec4515946210884203282_a_nat
          @ ( produc2556087688056060396_nat_o
            @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] :
                ( product_snd_o_o
                @ ( List_ext
                  @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_ns
thf(fact_628_list__order__extension__impl_Olist__ext__ns,axiom,
    ! [S_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,Ns_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,List_ext: ( term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o ) > list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o,S: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Ns: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ( list_l8247872550231653526_a_nat @ S_list @ Ns_list @ List_ext )
     => ( ( Ns_list @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ S ) ) @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ Ns ) ) )
        = ( collec8858057866656402194_a_nat
          @ ( produc4945827410475597548_nat_o
            @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] :
                ( product_snd_o_o
                @ ( List_ext
                  @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_ns
thf(fact_629_list__order__extension__impl_Olist__ext__ns,axiom,
    ! [S_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,Ns_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,List_ext: ( list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o ) > list_l8999537460041914605on_nat > list_l8999537460041914605on_nat > product_prod_o_o,S: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Ns: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ( list_l2120044010367530257on_nat @ S_list @ Ns_list @ List_ext )
     => ( ( Ns_list @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ S ) ) @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ Ns ) ) )
        = ( collec1004640060711064742on_nat
          @ ( produc9222835799945651472_nat_o
            @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] :
                ( product_snd_o_o
                @ ( List_ext
                  @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_ns
thf(fact_630_list__order__extension__impl_Olist__ext__ns,axiom,
    ! [S_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,Ns_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,List_ext: ( $o > $o > product_prod_o_o ) > list_o > list_o > product_prod_o_o,S: $o > $o > $o,Ns: $o > $o > $o] :
      ( ( list_l3256649549261819492impl_o @ S_list @ Ns_list @ List_ext )
     => ( ( Ns_list @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ S ) ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ Ns ) ) )
        = ( collec4435127810633735570list_o
          @ ( produc1052442579958709612st_o_o
            @ ^ [As: list_o,Bs: list_o] :
                ( product_snd_o_o
                @ ( List_ext
                  @ ^ [A3: $o,B3: $o] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_ns
thf(fact_631_list__order__extension__impl_Olist__ext__s,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,List_ext: ( sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o ) > list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o,S: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Ns: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
      ( ( list_l6805256357292310145on_nat @ S_list @ Ns_list @ List_ext )
     => ( ( S_list @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ S ) ) @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ Ns ) ) )
        = ( collec1293568243156533638on_nat
          @ ( produc7163990014243126832_nat_o
            @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] :
                ( product_fst_o_o
                @ ( List_ext
                  @ ^ [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_s
thf(fact_632_list__order__extension__impl_Olist__ext__s,axiom,
    ! [S_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,Ns_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,List_ext: ( list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o ) > list_l3889969465389992278_a_nat > list_l3889969465389992278_a_nat > product_prod_o_o,S: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Ns: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ( list_l2859941575254952092_a_nat @ S_list @ Ns_list @ List_ext )
     => ( ( S_list @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ S ) ) @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ Ns ) ) )
        = ( collec4515946210884203282_a_nat
          @ ( produc2556087688056060396_nat_o
            @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] :
                ( product_fst_o_o
                @ ( List_ext
                  @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_s
thf(fact_633_list__order__extension__impl_Olist__ext__s,axiom,
    ! [S_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,Ns_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,List_ext: ( term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o ) > list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o,S: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Ns: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ( list_l8247872550231653526_a_nat @ S_list @ Ns_list @ List_ext )
     => ( ( S_list @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ S ) ) @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ Ns ) ) )
        = ( collec8858057866656402194_a_nat
          @ ( produc4945827410475597548_nat_o
            @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] :
                ( product_fst_o_o
                @ ( List_ext
                  @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_s
thf(fact_634_list__order__extension__impl_Olist__ext__s,axiom,
    ! [S_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,Ns_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,List_ext: ( list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o ) > list_l8999537460041914605on_nat > list_l8999537460041914605on_nat > product_prod_o_o,S: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Ns: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ( list_l2120044010367530257on_nat @ S_list @ Ns_list @ List_ext )
     => ( ( S_list @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ S ) ) @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ Ns ) ) )
        = ( collec1004640060711064742on_nat
          @ ( produc9222835799945651472_nat_o
            @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] :
                ( product_fst_o_o
                @ ( List_ext
                  @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_s
thf(fact_635_list__order__extension__impl_Olist__ext__s,axiom,
    ! [S_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,Ns_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,List_ext: ( $o > $o > product_prod_o_o ) > list_o > list_o > product_prod_o_o,S: $o > $o > $o,Ns: $o > $o > $o] :
      ( ( list_l3256649549261819492impl_o @ S_list @ Ns_list @ List_ext )
     => ( ( S_list @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ S ) ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ Ns ) ) )
        = ( collec4435127810633735570list_o
          @ ( produc1052442579958709612st_o_o
            @ ^ [As: list_o,Bs: list_o] :
                ( product_fst_o_o
                @ ( List_ext
                  @ ^ [A3: $o,B3: $o] : ( product_Pair_o_o @ ( S @ A3 @ B3 ) @ ( Ns @ A3 @ B3 ) )
                  @ As
                  @ Bs ) ) ) ) ) ) ).

% list_order_extension_impl.list_ext_s
thf(fact_636_subterm_Obdd__belowI,axiom,
    ! [A5: set_te1179867837402157482_a_nat,M2: term_R1747997765655105994_a_nat] :
      ( ! [X: term_R1747997765655105994_a_nat] :
          ( ( member7260319590701762419_a_nat @ X @ A5 )
         => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X @ M2 ) @ subter3376206727240021173_a_nat ) )
     => ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 ) ) ).

% subterm.bdd_belowI
thf(fact_637_subterm_Obdd__below_OI,axiom,
    ! [A5: set_te1179867837402157482_a_nat,M3: term_R1747997765655105994_a_nat] :
      ( ! [X: term_R1747997765655105994_a_nat] :
          ( ( member7260319590701762419_a_nat @ X @ A5 )
         => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X @ M3 ) @ subter3376206727240021173_a_nat ) )
     => ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 ) ) ).

% subterm.bdd_below.I
thf(fact_638_subterm_Obdd__above_OI,axiom,
    ! [A5: set_te1179867837402157482_a_nat,M3: term_R1747997765655105994_a_nat] :
      ( ! [X: term_R1747997765655105994_a_nat] :
          ( ( member7260319590701762419_a_nat @ X @ A5 )
         => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ M3 @ X ) @ subter3376206727240021173_a_nat ) )
     => ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 ) ) ).

% subterm.bdd_above.I
thf(fact_639_subterm_Obdd__below_OE,axiom,
    ! [A5: set_te1179867837402157482_a_nat] :
      ( ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 )
     => ~ ! [M4: term_R1747997765655105994_a_nat] :
            ~ ! [X5: term_R1747997765655105994_a_nat] :
                ( ( member7260319590701762419_a_nat @ X5 @ A5 )
               => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X5 @ M4 ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.bdd_below.E
thf(fact_640_subterm_Obdd__below__mono,axiom,
    ! [B5: set_te1179867837402157482_a_nat,A5: set_te1179867837402157482_a_nat] :
      ( ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ B5 )
     => ( ( ord_le6135133001831709002_a_nat @ A5 @ B5 )
       => ( condit8263634360281639266_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A5 ) ) ) ).

% subterm.bdd_below_mono
thf(fact_641_subterm_Obdd__below_Ounfold,axiom,
    ! [A5: set_te1179867837402157482_a_nat] :
      ( ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 )
      = ( ? [M5: term_R1747997765655105994_a_nat] :
          ! [X2: term_R1747997765655105994_a_nat] :
            ( ( member7260319590701762419_a_nat @ X2 @ A5 )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ M5 ) @ subter3376206727240021173_a_nat ) ) ) ) ).

% subterm.bdd_below.unfold
thf(fact_642_subterm_Obdd__above_OE,axiom,
    ! [A5: set_te1179867837402157482_a_nat] :
      ( ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 )
     => ~ ! [M4: term_R1747997765655105994_a_nat] :
            ~ ! [X5: term_R1747997765655105994_a_nat] :
                ( ( member7260319590701762419_a_nat @ X5 @ A5 )
               => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ M4 @ X5 ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.bdd_above.E
thf(fact_643_subterm_Obdd__above__mono,axiom,
    ! [B5: set_te1179867837402157482_a_nat,A5: set_te1179867837402157482_a_nat] :
      ( ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ B5 )
     => ( ( ord_le6135133001831709002_a_nat @ A5 @ B5 )
       => ( condit1868887373268119414_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A5 ) ) ) ).

% subterm.bdd_above_mono
thf(fact_644_subterm_Obdd__above_Ounfold,axiom,
    ! [A5: set_te1179867837402157482_a_nat] :
      ( ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 )
      = ( ? [M5: term_R1747997765655105994_a_nat] :
          ! [X2: term_R1747997765655105994_a_nat] :
            ( ( member7260319590701762419_a_nat @ X2 @ A5 )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ M5 @ X2 ) @ subter3376206727240021173_a_nat ) ) ) ) ).

% subterm.bdd_above.unfold
thf(fact_645_subterm_Obdd__above__Ioo,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit1868887373268119414_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_gr1098500637439456618_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Ioo
thf(fact_646_subterm_Obdd__below__Ioo,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit8263634360281639266_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_gr1098500637439456618_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_below_Ioo
thf(fact_647_subterm_Obdd__above__Iio,axiom,
    ! [B: term_R1747997765655105994_a_nat] :
      ( condit1868887373268119414_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_le6356817768498327033_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ B ) ) ).

% subterm.bdd_above_Iio
thf(fact_648_subterm_Obdd__below__Ioi,axiom,
    ! [A: term_R1747997765655105994_a_nat] :
      ( condit8263634360281639266_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_gr4226962830520313764_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A ) ) ).

% subterm.bdd_below_Ioi
thf(fact_649_subterm_Obdd__above__Iic,axiom,
    ! [B: term_R1747997765655105994_a_nat] :
      ( condit1868887373268119414_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_at2477807202489097501_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ B ) ) ).

% subterm.bdd_above_Iic
thf(fact_650_subterm_OatMost__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,K: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_at2477807202489097501_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ K ) )
      = ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ K @ I ) @ subter3376206727240021173_a_nat ) ) ).

% subterm.atMost_iff
thf(fact_651_subterm_OgreaterThan__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,K: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_gr4226962830520313764_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ K ) )
      = ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ I @ K ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.greaterThan_iff
thf(fact_652_subterm_OlessThan__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,K: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_le6356817768498327033_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ K ) )
      = ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ K @ I ) @ subter8519896202621964801_a_nat ) ) ).

% subterm.lessThan_iff
thf(fact_653_subterm_OgreaterThanLessThan__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_gr1098500637439456618_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ L
          @ U ) )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ I @ L ) @ subter8519896202621964801_a_nat )
        & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ U @ I ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.greaterThanLessThan_iff
thf(fact_654_subterm_OgreaterThan__def,axiom,
    ! [L: term_R1747997765655105994_a_nat] :
      ( ( set_gr4226962830520313764_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ L )
      = ( collec1489781475602551989_a_nat
        @ ^ [X2: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ L ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.greaterThan_def
thf(fact_655_subterm_OlessThan__def,axiom,
    ! [U: term_R1747997765655105994_a_nat] :
      ( ( set_le6356817768498327033_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ U )
      = ( collec1489781475602551989_a_nat
        @ ^ [X2: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ U @ X2 ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.lessThan_def
thf(fact_656_subterm_OatMost__def,axiom,
    ! [U: term_R1747997765655105994_a_nat] :
      ( ( set_at2477807202489097501_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ U )
      = ( collec1489781475602551989_a_nat
        @ ^ [X2: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ U @ X2 ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.atMost_def
thf(fact_657_subterm_OIoi__le__Ico,axiom,
    ! [A: term_R1747997765655105994_a_nat] :
      ( ord_le6135133001831709002_a_nat
      @ ( set_gr4226962830520313764_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A )
      @ ( set_at3383287042032256009_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A ) ) ).

% subterm.Ioi_le_Ico
thf(fact_658_subterm_OIcc__subset__Iic__iff,axiom,
    ! [L: term_R1747997765655105994_a_nat,H: term_R1747997765655105994_a_nat,H2: term_R1747997765655105994_a_nat] :
      ( ( ord_le6135133001831709002_a_nat
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L
          @ H )
        @ ( set_at2477807202489097501_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ H2 ) )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ H @ L ) @ subter3376206727240021173_a_nat )
        | ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ H2 @ H ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.Icc_subset_Iic_iff
thf(fact_659_subterm_OgreaterThanAtMost__def,axiom,
    ! [L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( set_gr6446970317771492622_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ L
        @ U )
      = ( inf_in7710483629710275224_a_nat
        @ ( set_gr4226962830520313764_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ L )
        @ ( set_at2477807202489097501_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ U ) ) ) ).

% subterm.greaterThanAtMost_def
thf(fact_660_subterm_OatLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat,D2: term_R1747997765655105994_a_nat] :
      ( ( ord_le6135133001831709002_a_nat
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        @ ( set_at5605147574288828367_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ C
          @ D2 ) )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
       => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ C ) @ subter3376206727240021173_a_nat )
          & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ D2 @ B ) @ subter8519896202621964801_a_nat ) ) ) ) ).

% subterm.atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_661_subterm_Obdd__below__Ici,axiom,
    ! [A: term_R1747997765655105994_a_nat] :
      ( condit8263634360281639266_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_at3383287042032256009_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A ) ) ).

% subterm.bdd_below_Ici
thf(fact_662_Int__iff,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) )
      = ( ( member4231271695026834640_a_nat @ C @ A5 )
        & ( member4231271695026834640_a_nat @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_663_Int__iff,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) )
      = ( ( member6928026473249828560_a_nat @ C @ A5 )
        & ( member6928026473249828560_a_nat @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_664_Int__iff,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( inf_in1188219735355578729on_nat @ A5 @ B5 ) )
      = ( ( member4278285197320822468on_nat @ C @ A5 )
        & ( member4278285197320822468on_nat @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_665_Int__iff,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( inf_in505647112809072361on_nat @ A5 @ B5 ) )
      = ( ( member4221009242412147780on_nat @ C @ A5 )
        & ( member4221009242412147780on_nat @ C @ B5 ) ) ) ).

% Int_iff
thf(fact_666_IntI,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ A5 )
     => ( ( member4231271695026834640_a_nat @ C @ B5 )
       => ( member4231271695026834640_a_nat @ C @ ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) ) ) ) ).

% IntI
thf(fact_667_IntI,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ A5 )
     => ( ( member6928026473249828560_a_nat @ C @ B5 )
       => ( member6928026473249828560_a_nat @ C @ ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) ) ) ) ).

% IntI
thf(fact_668_IntI,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ A5 )
     => ( ( member4278285197320822468on_nat @ C @ B5 )
       => ( member4278285197320822468on_nat @ C @ ( inf_in1188219735355578729on_nat @ A5 @ B5 ) ) ) ) ).

% IntI
thf(fact_669_IntI,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ A5 )
     => ( ( member4221009242412147780on_nat @ C @ B5 )
       => ( member4221009242412147780on_nat @ C @ ( inf_in505647112809072361on_nat @ A5 @ B5 ) ) ) ) ).

% IntI
thf(fact_670_subterm_OIcc__eq__Icc,axiom,
    ! [L: term_R1747997765655105994_a_nat,H: term_R1747997765655105994_a_nat,L2: term_R1747997765655105994_a_nat,H2: term_R1747997765655105994_a_nat] :
      ( ( ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L
          @ H )
        = ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L2
          @ H2 ) )
      = ( ( ( L = L2 )
          & ( H = H2 ) )
        | ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ H @ L ) @ subter3376206727240021173_a_nat )
          & ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ H2 @ L2 ) @ subter3376206727240021173_a_nat ) ) ) ) ).

% subterm.Icc_eq_Icc
thf(fact_671_subterm_OatLeastAtMost__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L
          @ U ) )
      = ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ I @ L ) @ subter3376206727240021173_a_nat )
        & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ U @ I ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.atLeastAtMost_iff
thf(fact_672_subterm_OatLeast__iff,axiom,
    ! [I: term_R1747997765655105994_a_nat,K: term_R1747997765655105994_a_nat] :
      ( ( member7260319590701762419_a_nat @ I
        @ ( set_at3383287042032256009_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ K ) )
      = ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ I @ K ) @ subter3376206727240021173_a_nat ) ) ).

% subterm.atLeast_iff
thf(fact_673_subterm_Obdd__below__Icc,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit8263634360281639266_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_at1360634933218525683_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_below_Icc
thf(fact_674_subterm_Obdd__above__Icc,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( condit1868887373268119414_a_nat
      @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
      @ ( set_at1360634933218525683_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A
        @ B ) ) ).

% subterm.bdd_above_Icc
thf(fact_675_subterm_OatLeastatMost__subset__iff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat,D2: term_R1747997765655105994_a_nat] :
      ( ( ord_le6135133001831709002_a_nat
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ C
          @ D2 ) )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
        | ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ C ) @ subter3376206727240021173_a_nat )
          & ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ D2 @ B ) @ subter3376206727240021173_a_nat ) ) ) ) ).

% subterm.atLeastatMost_subset_iff
thf(fact_676_subterm_OIcc__subset__Ici__iff,axiom,
    ! [L: term_R1747997765655105994_a_nat,H: term_R1747997765655105994_a_nat,L2: term_R1747997765655105994_a_nat] :
      ( ( ord_le6135133001831709002_a_nat
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L
          @ H )
        @ ( set_at3383287042032256009_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L2 ) )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ H @ L ) @ subter3376206727240021173_a_nat )
        | ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ L @ L2 ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.Icc_subset_Ici_iff
thf(fact_677_Collect__conj__eq,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,Q: produc394670695091157671_a_nat > $o] :
      ( ( collec8858057866656402194_a_nat
        @ ^ [X2: produc394670695091157671_a_nat] :
            ( ( P2 @ X2 )
            & ( Q @ X2 ) ) )
      = ( inf_in3705958062578247285_a_nat @ ( collec8858057866656402194_a_nat @ P2 ) @ ( collec8858057866656402194_a_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_678_Collect__conj__eq,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ( collec9088346736735320338_a_nat
        @ ^ [X2: produc3043957382837978535_a_nat] :
            ( ( P2 @ X2 )
            & ( Q @ X2 ) ) )
      = ( inf_in5381721427322260597_a_nat @ ( collec9088346736735320338_a_nat @ P2 ) @ ( collec9088346736735320338_a_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_679_Collect__conj__eq,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,Q: produc5580075136493545115on_nat > $o] :
      ( ( collec1293568243156533638on_nat
        @ ^ [X2: produc5580075136493545115on_nat] :
            ( ( P2 @ X2 )
            & ( Q @ X2 ) ) )
      = ( inf_in505647112809072361on_nat @ ( collec1293568243156533638on_nat @ P2 ) @ ( collec1293568243156533638on_nat @ Q ) ) ) ).

% Collect_conj_eq
thf(fact_680_Int__Collect,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,P2: produc5238312538367656103_a_nat > $o] :
      ( ( member6928026473249828560_a_nat @ X3 @ ( inf_in7903224342872612981_a_nat @ A5 @ ( collec6671004892620940562_a_nat @ P2 ) ) )
      = ( ( member6928026473249828560_a_nat @ X3 @ A5 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_681_Int__Collect,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,P2: produc9176241080893687067on_nat > $o] :
      ( ( member4278285197320822468on_nat @ X3 @ ( inf_in1188219735355578729on_nat @ A5 @ ( collec1886838643678311430on_nat @ P2 ) ) )
      = ( ( member4278285197320822468on_nat @ X3 @ A5 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_682_Int__Collect,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,P2: produc394670695091157671_a_nat > $o] :
      ( ( member4231271695026834640_a_nat @ X3 @ ( inf_in3705958062578247285_a_nat @ A5 @ ( collec8858057866656402194_a_nat @ P2 ) ) )
      = ( ( member4231271695026834640_a_nat @ X3 @ A5 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_683_Int__Collect,axiom,
    ! [X3: produc3043957382837978535_a_nat,A5: set_Pr630484882382513799_a_nat,P2: produc3043957382837978535_a_nat > $o] :
      ( ( member6937586146271652048_a_nat @ X3 @ ( inf_in5381721427322260597_a_nat @ A5 @ ( collec9088346736735320338_a_nat @ P2 ) ) )
      = ( ( member6937586146271652048_a_nat @ X3 @ A5 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_684_Int__Collect,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,P2: produc5580075136493545115on_nat > $o] :
      ( ( member4221009242412147780on_nat @ X3 @ ( inf_in505647112809072361on_nat @ A5 @ ( collec1293568243156533638on_nat @ P2 ) ) )
      = ( ( member4221009242412147780on_nat @ X3 @ A5 )
        & ( P2 @ X3 ) ) ) ).

% Int_Collect
thf(fact_685_Int__def,axiom,
    ( inf_in7903224342872612981_a_nat
    = ( ^ [A6: set_Pr8529752790658994311_a_nat,B6: set_Pr8529752790658994311_a_nat] :
          ( collec6671004892620940562_a_nat
          @ ^ [X2: produc5238312538367656103_a_nat] :
              ( ( member6928026473249828560_a_nat @ X2 @ A6 )
              & ( member6928026473249828560_a_nat @ X2 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_686_Int__def,axiom,
    ( inf_in1188219735355578729on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B6: set_Pr3573155178515677307on_nat] :
          ( collec1886838643678311430on_nat
          @ ^ [X2: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X2 @ A6 )
              & ( member4278285197320822468on_nat @ X2 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_687_Int__def,axiom,
    ( inf_in3705958062578247285_a_nat
    = ( ^ [A6: set_Pr4991467605864603271_a_nat,B6: set_Pr4991467605864603271_a_nat] :
          ( collec8858057866656402194_a_nat
          @ ^ [X2: produc394670695091157671_a_nat] :
              ( ( member4231271695026834640_a_nat @ X2 @ A6 )
              & ( member4231271695026834640_a_nat @ X2 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_688_Int__def,axiom,
    ( inf_in5381721427322260597_a_nat
    = ( ^ [A6: set_Pr630484882382513799_a_nat,B6: set_Pr630484882382513799_a_nat] :
          ( collec9088346736735320338_a_nat
          @ ^ [X2: produc3043957382837978535_a_nat] :
              ( ( member6937586146271652048_a_nat @ X2 @ A6 )
              & ( member6937586146271652048_a_nat @ X2 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_689_Int__def,axiom,
    ( inf_in505647112809072361on_nat
    = ( ^ [A6: set_Pr7828957961435280891on_nat,B6: set_Pr7828957961435280891on_nat] :
          ( collec1293568243156533638on_nat
          @ ^ [X2: produc5580075136493545115on_nat] :
              ( ( member4221009242412147780on_nat @ X2 @ A6 )
              & ( member4221009242412147780on_nat @ X2 @ B6 ) ) ) ) ) ).

% Int_def
thf(fact_690_IntD2,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) )
     => ( member4231271695026834640_a_nat @ C @ B5 ) ) ).

% IntD2
thf(fact_691_IntD2,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) )
     => ( member6928026473249828560_a_nat @ C @ B5 ) ) ).

% IntD2
thf(fact_692_IntD2,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( inf_in1188219735355578729on_nat @ A5 @ B5 ) )
     => ( member4278285197320822468on_nat @ C @ B5 ) ) ).

% IntD2
thf(fact_693_IntD2,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( inf_in505647112809072361on_nat @ A5 @ B5 ) )
     => ( member4221009242412147780on_nat @ C @ B5 ) ) ).

% IntD2
thf(fact_694_IntD1,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) )
     => ( member4231271695026834640_a_nat @ C @ A5 ) ) ).

% IntD1
thf(fact_695_IntD1,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) )
     => ( member6928026473249828560_a_nat @ C @ A5 ) ) ).

% IntD1
thf(fact_696_IntD1,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( inf_in1188219735355578729on_nat @ A5 @ B5 ) )
     => ( member4278285197320822468on_nat @ C @ A5 ) ) ).

% IntD1
thf(fact_697_IntD1,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( inf_in505647112809072361on_nat @ A5 @ B5 ) )
     => ( member4221009242412147780on_nat @ C @ A5 ) ) ).

% IntD1
thf(fact_698_IntE,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) )
     => ~ ( ( member4231271695026834640_a_nat @ C @ A5 )
         => ~ ( member4231271695026834640_a_nat @ C @ B5 ) ) ) ).

% IntE
thf(fact_699_IntE,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) )
     => ~ ( ( member6928026473249828560_a_nat @ C @ A5 )
         => ~ ( member6928026473249828560_a_nat @ C @ B5 ) ) ) ).

% IntE
thf(fact_700_IntE,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( inf_in1188219735355578729on_nat @ A5 @ B5 ) )
     => ~ ( ( member4278285197320822468on_nat @ C @ A5 )
         => ~ ( member4278285197320822468on_nat @ C @ B5 ) ) ) ).

% IntE
thf(fact_701_IntE,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( inf_in505647112809072361on_nat @ A5 @ B5 ) )
     => ~ ( ( member4221009242412147780on_nat @ C @ A5 )
         => ~ ( member4221009242412147780on_nat @ C @ B5 ) ) ) ).

% IntE
thf(fact_702_Int__Collect__mono,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat,P2: produc5238312538367656103_a_nat > $o,Q: produc5238312538367656103_a_nat > $o] :
      ( ( ord_le7248185767425706023_a_nat @ A5 @ B5 )
     => ( ! [X: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X @ A5 )
           => ( ( P2 @ X )
             => ( Q @ X ) ) )
       => ( ord_le7248185767425706023_a_nat @ ( inf_in7903224342872612981_a_nat @ A5 @ ( collec6671004892620940562_a_nat @ P2 ) ) @ ( inf_in7903224342872612981_a_nat @ B5 @ ( collec6671004892620940562_a_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_703_Int__Collect__mono,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat,P2: produc9176241080893687067on_nat > $o,Q: produc9176241080893687067on_nat > $o] :
      ( ( ord_le927452247459997723on_nat @ A5 @ B5 )
     => ( ! [X: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X @ A5 )
           => ( ( P2 @ X )
             => ( Q @ X ) ) )
       => ( ord_le927452247459997723on_nat @ ( inf_in1188219735355578729on_nat @ A5 @ ( collec1886838643678311430on_nat @ P2 ) ) @ ( inf_in1188219735355578729on_nat @ B5 @ ( collec1886838643678311430on_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_704_Int__Collect__mono,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat,P2: produc394670695091157671_a_nat > $o,Q: produc394670695091157671_a_nat > $o] :
      ( ( ord_le998509814859404839_a_nat @ A5 @ B5 )
     => ( ! [X: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X @ A5 )
           => ( ( P2 @ X )
             => ( Q @ X ) ) )
       => ( ord_le998509814859404839_a_nat @ ( inf_in3705958062578247285_a_nat @ A5 @ ( collec8858057866656402194_a_nat @ P2 ) ) @ ( inf_in3705958062578247285_a_nat @ B5 @ ( collec8858057866656402194_a_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_705_Int__Collect__mono,axiom,
    ! [A5: set_Pr630484882382513799_a_nat,B5: set_Pr630484882382513799_a_nat,P2: produc3043957382837978535_a_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ( ord_le251666825002368551_a_nat @ A5 @ B5 )
     => ( ! [X: produc3043957382837978535_a_nat] :
            ( ( member6937586146271652048_a_nat @ X @ A5 )
           => ( ( P2 @ X )
             => ( Q @ X ) ) )
       => ( ord_le251666825002368551_a_nat @ ( inf_in5381721427322260597_a_nat @ A5 @ ( collec9088346736735320338_a_nat @ P2 ) ) @ ( inf_in5381721427322260597_a_nat @ B5 @ ( collec9088346736735320338_a_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_706_Int__Collect__mono,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat,P2: produc5580075136493545115on_nat > $o,Q: produc5580075136493545115on_nat > $o] :
      ( ( ord_le6423710186836778395on_nat @ A5 @ B5 )
     => ( ! [X: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X @ A5 )
           => ( ( P2 @ X )
             => ( Q @ X ) ) )
       => ( ord_le6423710186836778395on_nat @ ( inf_in505647112809072361on_nat @ A5 @ ( collec1293568243156533638on_nat @ P2 ) ) @ ( inf_in505647112809072361on_nat @ B5 @ ( collec1293568243156533638on_nat @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_707_subterm_OatLeastAtMost__def,axiom,
    ! [L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( set_at1360634933218525683_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ L
        @ U )
      = ( inf_in7710483629710275224_a_nat
        @ ( set_at3383287042032256009_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L )
        @ ( set_at2477807202489097501_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ U ) ) ) ).

% subterm.atLeastAtMost_def
thf(fact_708_subterm_OatLeastLessThan__def,axiom,
    ! [L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( set_at5605147574288828367_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ L
        @ U )
      = ( inf_in7710483629710275224_a_nat
        @ ( set_at3383287042032256009_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ L )
        @ ( set_le6356817768498327033_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ U ) ) ) ).

% subterm.atLeastLessThan_def
thf(fact_709_subterm_Obdd__below__Int2,axiom,
    ! [B5: set_te1179867837402157482_a_nat,A5: set_te1179867837402157482_a_nat] :
      ( ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ B5 )
     => ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ( inf_in7710483629710275224_a_nat @ A5 @ B5 ) ) ) ).

% subterm.bdd_below_Int2
thf(fact_710_subterm_Obdd__below__Int1,axiom,
    ! [A5: set_te1179867837402157482_a_nat,B5: set_te1179867837402157482_a_nat] :
      ( ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 )
     => ( condit8263634360281639266_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ( inf_in7710483629710275224_a_nat @ A5 @ B5 ) ) ) ).

% subterm.bdd_below_Int1
thf(fact_711_subterm_Obdd__above__Int2,axiom,
    ! [B5: set_te1179867837402157482_a_nat,A5: set_te1179867837402157482_a_nat] :
      ( ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ B5 )
     => ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ( inf_in7710483629710275224_a_nat @ A5 @ B5 ) ) ) ).

% subterm.bdd_above_Int2
thf(fact_712_subterm_Obdd__above__Int1,axiom,
    ! [A5: set_te1179867837402157482_a_nat,B5: set_te1179867837402157482_a_nat] :
      ( ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A5 )
     => ( condit1868887373268119414_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ( inf_in7710483629710275224_a_nat @ A5 @ B5 ) ) ) ).

% subterm.bdd_above_Int1
thf(fact_713_subterm_OatLeast__def,axiom,
    ! [L: term_R1747997765655105994_a_nat] :
      ( ( set_at3383287042032256009_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ L )
      = ( collec1489781475602551989_a_nat
        @ ^ [X2: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ L ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.atLeast_def
thf(fact_714_subterm_OgreaterThanLessThan__eq,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( set_gr1098500637439456618_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B )
      = ( inf_in7710483629710275224_a_nat
        @ ( set_gr4226962830520313764_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ A )
        @ ( set_le6356817768498327033_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ B ) ) ) ).

% subterm.greaterThanLessThan_eq
thf(fact_715_subterm_OgreaterThanLessThan__def,axiom,
    ! [L: term_R1747997765655105994_a_nat,U: term_R1747997765655105994_a_nat] :
      ( ( set_gr1098500637439456618_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ L
        @ U )
      = ( inf_in7710483629710275224_a_nat
        @ ( set_gr4226962830520313764_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ L )
        @ ( set_le6356817768498327033_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ U ) ) ) ).

% subterm.greaterThanLessThan_def
thf(fact_716_le__inf__iff,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z ) )
      = ( ( ord_less_eq_nat @ X3 @ Y3 )
        & ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% le_inf_iff
thf(fact_717_inf_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
      = ( ( ord_less_eq_nat @ A @ B )
        & ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.bounded_iff
thf(fact_718_inf_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI2
thf(fact_719_inf_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ C ) ) ).

% inf.coboundedI1
thf(fact_720_inf__Int__eq2,axiom,
    ! [R: set_Pr8529752790658994311_a_nat,S6: set_Pr8529752790658994311_a_nat] :
      ( ( inf_in7499259990892343772_nat_o
        @ ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ ( inf_in7903224342872612981_a_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_721_inf__Int__eq2,axiom,
    ! [R: set_Pr3573155178515677307on_nat,S6: set_Pr3573155178515677307on_nat] :
      ( ( inf_in2510263837201576250_nat_o
        @ ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ ( inf_in1188219735355578729on_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_722_inf__Int__eq2,axiom,
    ! [R: set_Pr737724919871395061on_nat,S6: set_Pr737724919871395061on_nat] :
      ( ( inf_in1805279390179427940_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ ( inf_in587691179868169863on_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_723_inf__Int__eq2,axiom,
    ! [R: set_Pr6244131468446592861on_nat,S6: set_Pr6244131468446592861on_nat] :
      ( ( inf_in5355765883627203288_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ ( inf_in4911864082805573707on_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_724_inf__Int__eq2,axiom,
    ! [R: set_Pr7828957961435280891on_nat,S6: set_Pr7828957961435280891on_nat] :
      ( ( inf_in9003242535668113850_nat_o
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ ( inf_in505647112809072361on_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_725_inf__Int__eq2,axiom,
    ! [R: set_Pr4991467605864603271_a_nat,S6: set_Pr4991467605864603271_a_nat] :
      ( ( inf_in9033586014723130844_nat_o
        @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ ( inf_in3705958062578247285_a_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_726_inf__Int__eq2,axiom,
    ! [R: set_Product_prod_o_o,S6: set_Product_prod_o_o] :
      ( ( inf_inf_o_o_o
        @ ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ R )
        @ ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ ( inf_in4898592226082374645od_o_o @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_727_inf__Int__eq2,axiom,
    ! [R: set_Pr630484882382513799_a_nat,S6: set_Pr630484882382513799_a_nat] :
      ( ( inf_in1647444548604681052_nat_o
        @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ R )
        @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ S6 ) )
      = ( ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ ( inf_in5381721427322260597_a_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq2
thf(fact_728_inf__set__def,axiom,
    ( inf_in7903224342872612981_a_nat
    = ( ^ [A6: set_Pr8529752790658994311_a_nat,B6: set_Pr8529752790658994311_a_nat] :
          ( collec6671004892620940562_a_nat
          @ ( inf_in7473963557951668392_nat_o
            @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ A6 )
            @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_729_inf__set__def,axiom,
    ( inf_in1188219735355578729on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B6: set_Pr3573155178515677307on_nat] :
          ( collec1886838643678311430on_nat
          @ ( inf_in2973073432369849012_nat_o
            @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ A6 )
            @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_730_inf__set__def,axiom,
    ( inf_in3705958062578247285_a_nat
    = ( ^ [A6: set_Pr4991467605864603271_a_nat,B6: set_Pr4991467605864603271_a_nat] :
          ( collec8858057866656402194_a_nat
          @ ( inf_in3918759502588701352_nat_o
            @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ A6 )
            @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_731_inf__set__def,axiom,
    ( inf_in5381721427322260597_a_nat
    = ( ^ [A6: set_Pr630484882382513799_a_nat,B6: set_Pr630484882382513799_a_nat] :
          ( collec9088346736735320338_a_nat
          @ ( inf_in4418476747659956392_nat_o
            @ ^ [X2: produc3043957382837978535_a_nat] : ( member6937586146271652048_a_nat @ X2 @ A6 )
            @ ^ [X2: produc3043957382837978535_a_nat] : ( member6937586146271652048_a_nat @ X2 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_732_inf__set__def,axiom,
    ( inf_in505647112809072361on_nat
    = ( ^ [A6: set_Pr7828957961435280891on_nat,B6: set_Pr7828957961435280891on_nat] :
          ( collec1293568243156533638on_nat
          @ ( inf_in3828304446256219444_nat_o
            @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ A6 )
            @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ B6 ) ) ) ) ) ).

% inf_set_def
thf(fact_733_inf__Int__eq,axiom,
    ! [R: set_Pr4991467605864603271_a_nat,S6: set_Pr4991467605864603271_a_nat] :
      ( ( inf_in3918759502588701352_nat_o
        @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ R )
        @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ S6 ) )
      = ( ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ ( inf_in3705958062578247285_a_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq
thf(fact_734_inf__Int__eq,axiom,
    ! [R: set_Pr8529752790658994311_a_nat,S6: set_Pr8529752790658994311_a_nat] :
      ( ( inf_in7473963557951668392_nat_o
        @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ R )
        @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ S6 ) )
      = ( ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ ( inf_in7903224342872612981_a_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq
thf(fact_735_inf__Int__eq,axiom,
    ! [R: set_Pr3573155178515677307on_nat,S6: set_Pr3573155178515677307on_nat] :
      ( ( inf_in2973073432369849012_nat_o
        @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ R )
        @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ S6 ) )
      = ( ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ ( inf_in1188219735355578729on_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq
thf(fact_736_inf__Int__eq,axiom,
    ! [R: set_Pr7828957961435280891on_nat,S6: set_Pr7828957961435280891on_nat] :
      ( ( inf_in3828304446256219444_nat_o
        @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ R )
        @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ S6 ) )
      = ( ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ ( inf_in505647112809072361on_nat @ R @ S6 ) ) ) ) ).

% inf_Int_eq
thf(fact_737_relation__inter,axiom,
    ! [P2: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Q: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
      ( ( inf_in3705958062578247285_a_nat @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ P2 ) ) @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ Q ) ) )
      = ( collec8858057866656402194_a_nat
        @ ( produc4945827410475597548_nat_o
          @ ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] :
              ( ( P2 @ X2 @ Y5 )
              & ( Q @ X2 @ Y5 ) ) ) ) ) ).

% relation_inter
thf(fact_738_relation__inter,axiom,
    ! [P2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Q: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
      ( ( inf_in5381721427322260597_a_nat @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ P2 ) ) @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ Q ) ) )
      = ( collec9088346736735320338_a_nat
        @ ( produc4098677624135445996_nat_o
          @ ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] :
              ( ( P2 @ X2 @ Y5 )
              & ( Q @ X2 @ Y5 ) ) ) ) ) ).

% relation_inter
thf(fact_739_relation__inter,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Q: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
      ( ( inf_in505647112809072361on_nat @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ P2 ) ) @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ Q ) ) )
      = ( collec1293568243156533638on_nat
        @ ( produc7163990014243126832_nat_o
          @ ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] :
              ( ( P2 @ X2 @ Y5 )
              & ( Q @ X2 @ Y5 ) ) ) ) ) ).

% relation_inter
thf(fact_740_relation__inter,axiom,
    ! [P2: $o > $o > $o,Q: $o > $o > $o] :
      ( ( inf_in4898592226082374645od_o_o @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ P2 ) ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ Q ) ) )
      = ( collec3167064739498627218od_o_o
        @ ( produc6197397395684419436_o_o_o
          @ ^ [X2: $o,Y5: $o] :
              ( ( P2 @ X2 @ Y5 )
              & ( Q @ X2 @ Y5 ) ) ) ) ) ).

% relation_inter
thf(fact_741_relation__inter,axiom,
    ! [P2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,Q: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ( inf_in4911864082805573707on_nat @ ( collec624303909867598824on_nat @ ( produc7728289706344822350_nat_o @ P2 ) ) @ ( collec624303909867598824on_nat @ ( produc7728289706344822350_nat_o @ Q ) ) )
      = ( collec624303909867598824on_nat
        @ ( produc7728289706344822350_nat_o
          @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] :
              ( ( P2 @ X2 @ Y5 )
              & ( Q @ X2 @ Y5 ) ) ) ) ) ).

% relation_inter
thf(fact_742_relation__inter,axiom,
    ! [P2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,Q: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ( inf_in587691179868169863on_nat @ ( collec8524684687135612820on_nat @ ( produc1360909077854731642_nat_o @ P2 ) ) @ ( collec8524684687135612820on_nat @ ( produc1360909077854731642_nat_o @ Q ) ) )
      = ( collec8524684687135612820on_nat
        @ ( produc1360909077854731642_nat_o
          @ ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] :
              ( ( P2 @ X2 @ Y5 )
              & ( Q @ X2 @ Y5 ) ) ) ) ) ).

% relation_inter
thf(fact_743_bounded__Max__nat,axiom,
    ! [P2: nat > $o,X3: nat,M3: nat] :
      ( ( P2 @ X3 )
     => ( ! [X: nat] :
            ( ( P2 @ X )
           => ( ord_less_eq_nat @ X @ M3 ) )
       => ~ ! [M6: nat] :
              ( ( P2 @ M6 )
             => ~ ! [X5: nat] :
                    ( ( P2 @ X5 )
                   => ( ord_less_eq_nat @ X5 @ M6 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_744_ord_OatLeastLessThan__iff,axiom,
    ! [I: produc394670695091157671_a_nat,Less_eq: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,Less: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,L: produc394670695091157671_a_nat,U: produc394670695091157671_a_nat] :
      ( ( member4231271695026834640_a_nat @ I @ ( set_at3786910554017667884_a_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less_eq @ L @ I )
        & ( Less @ I @ U ) ) ) ).

% ord.atLeastLessThan_iff
thf(fact_745_ord_OatLeastLessThan__iff,axiom,
    ! [I: produc5238312538367656103_a_nat,Less_eq: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > $o,Less: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > $o,L: produc5238312538367656103_a_nat,U: produc5238312538367656103_a_nat] :
      ( ( member6928026473249828560_a_nat @ I @ ( set_at8479263003327384876_a_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less_eq @ L @ I )
        & ( Less @ I @ U ) ) ) ).

% ord.atLeastLessThan_iff
thf(fact_746_ord_OatLeastLessThan__iff,axiom,
    ! [I: produc9176241080893687067on_nat,Less_eq: produc9176241080893687067on_nat > produc9176241080893687067on_nat > $o,Less: produc9176241080893687067on_nat > produc9176241080893687067on_nat > $o,L: produc9176241080893687067on_nat,U: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ I @ ( set_at4811824280310468384on_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less_eq @ L @ I )
        & ( Less @ I @ U ) ) ) ).

% ord.atLeastLessThan_iff
thf(fact_747_ord_OatLeastLessThan__iff,axiom,
    ! [I: produc5580075136493545115on_nat,Less_eq: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,Less: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,L: produc5580075136493545115on_nat,U: produc5580075136493545115on_nat] :
      ( ( member4221009242412147780on_nat @ I @ ( set_at6948628715166237856on_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less_eq @ L @ I )
        & ( Less @ I @ U ) ) ) ).

% ord.atLeastLessThan_iff
thf(fact_748_ord_OgreaterThanAtMost__iff,axiom,
    ! [I: produc394670695091157671_a_nat,Less_eq: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,Less: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,L: produc394670695091157671_a_nat,U: produc394670695091157671_a_nat] :
      ( ( member4231271695026834640_a_nat @ I @ ( set_gr3136671174511769579_a_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less @ L @ I )
        & ( Less_eq @ I @ U ) ) ) ).

% ord.greaterThanAtMost_iff
thf(fact_749_ord_OgreaterThanAtMost__iff,axiom,
    ! [I: produc5238312538367656103_a_nat,Less_eq: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > $o,Less: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > $o,L: produc5238312538367656103_a_nat,U: produc5238312538367656103_a_nat] :
      ( ( member6928026473249828560_a_nat @ I @ ( set_gr3193239122004864491_a_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less @ L @ I )
        & ( Less_eq @ I @ U ) ) ) ).

% ord.greaterThanAtMost_iff
thf(fact_750_ord_OgreaterThanAtMost__iff,axiom,
    ! [I: produc9176241080893687067on_nat,Less_eq: produc9176241080893687067on_nat > produc9176241080893687067on_nat > $o,Less: produc9176241080893687067on_nat > produc9176241080893687067on_nat > $o,L: produc9176241080893687067on_nat,U: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ I @ ( set_gr8546024018979262559on_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less @ L @ I )
        & ( Less_eq @ I @ U ) ) ) ).

% ord.greaterThanAtMost_iff
thf(fact_751_ord_OgreaterThanAtMost__iff,axiom,
    ! [I: produc5580075136493545115on_nat,Less_eq: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,Less: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,L: produc5580075136493545115on_nat,U: produc5580075136493545115on_nat] :
      ( ( member4221009242412147780on_nat @ I @ ( set_gr7287495252507771359on_nat @ Less_eq @ Less @ L @ U ) )
      = ( ( Less @ L @ I )
        & ( Less_eq @ I @ U ) ) ) ).

% ord.greaterThanAtMost_iff
thf(fact_752_ord_OatLeast__def,axiom,
    ( set_at7577166204269462630_a_nat
    = ( ^ [Less_eq2: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,L3: produc394670695091157671_a_nat] : ( collec8858057866656402194_a_nat @ ( Less_eq2 @ L3 ) ) ) ) ).

% ord.atLeast_def
thf(fact_753_ord_OatLeast__def,axiom,
    ( set_at6702604556921448038_a_nat
    = ( ^ [Less_eq2: produc3043957382837978535_a_nat > produc3043957382837978535_a_nat > $o,L3: produc3043957382837978535_a_nat] : ( collec9088346736735320338_a_nat @ ( Less_eq2 @ L3 ) ) ) ) ).

% ord.atLeast_def
thf(fact_754_ord_OatLeast__def,axiom,
    ( set_at2371042145679200474on_nat
    = ( ^ [Less_eq2: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,L3: produc5580075136493545115on_nat] : ( collec1293568243156533638on_nat @ ( Less_eq2 @ L3 ) ) ) ) ).

% ord.atLeast_def
thf(fact_755_ord_OgreaterThan__def,axiom,
    ( set_gr9152156306490183041_a_nat
    = ( ^ [Less2: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,L3: produc394670695091157671_a_nat] : ( collec8858057866656402194_a_nat @ ( Less2 @ L3 ) ) ) ) ).

% ord.greaterThan_def
thf(fact_756_ord_OgreaterThan__def,axiom,
    ( set_gr647247908036985473_a_nat
    = ( ^ [Less2: produc3043957382837978535_a_nat > produc3043957382837978535_a_nat > $o,L3: produc3043957382837978535_a_nat] : ( collec9088346736735320338_a_nat @ ( Less2 @ L3 ) ) ) ) ).

% ord.greaterThan_def
thf(fact_757_ord_OgreaterThan__def,axiom,
    ( set_gr2330129122582793333on_nat
    = ( ^ [Less2: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,L3: produc5580075136493545115on_nat] : ( collec1293568243156533638on_nat @ ( Less2 @ L3 ) ) ) ) ).

% ord.greaterThan_def
thf(fact_758_ord_OlessThan__def,axiom,
    ( set_le4328816260233045590_a_nat
    = ( ^ [Less2: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,U2: produc394670695091157671_a_nat] :
          ( collec8858057866656402194_a_nat
          @ ^ [X2: produc394670695091157671_a_nat] : ( Less2 @ X2 @ U2 ) ) ) ) ).

% ord.lessThan_def
thf(fact_759_ord_OlessThan__def,axiom,
    ( set_le9152149561825540182_a_nat
    = ( ^ [Less2: produc3043957382837978535_a_nat > produc3043957382837978535_a_nat > $o,U2: produc3043957382837978535_a_nat] :
          ( collec9088346736735320338_a_nat
          @ ^ [X2: produc3043957382837978535_a_nat] : ( Less2 @ X2 @ U2 ) ) ) ) ).

% ord.lessThan_def
thf(fact_760_ord_OlessThan__def,axiom,
    ( set_le3917444702330944202on_nat
    = ( ^ [Less2: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,U2: produc5580075136493545115on_nat] :
          ( collec1293568243156533638on_nat
          @ ^ [X2: produc5580075136493545115on_nat] : ( Less2 @ X2 @ U2 ) ) ) ) ).

% ord.lessThan_def
thf(fact_761_ord_OatMost__def,axiom,
    ( set_at3024382190904231802_a_nat
    = ( ^ [Less_eq2: produc394670695091157671_a_nat > produc394670695091157671_a_nat > $o,U2: produc394670695091157671_a_nat] :
          ( collec8858057866656402194_a_nat
          @ ^ [X2: produc394670695091157671_a_nat] : ( Less_eq2 @ X2 @ U2 ) ) ) ) ).

% ord.atMost_def
thf(fact_762_ord_OatMost__def,axiom,
    ( set_at1866530483518414202_a_nat
    = ( ^ [Less_eq2: produc3043957382837978535_a_nat > produc3043957382837978535_a_nat > $o,U2: produc3043957382837978535_a_nat] :
          ( collec9088346736735320338_a_nat
          @ ^ [X2: produc3043957382837978535_a_nat] : ( Less_eq2 @ X2 @ U2 ) ) ) ) ).

% ord.atMost_def
thf(fact_763_ord_OatMost__def,axiom,
    ( set_at7788539869816981998on_nat
    = ( ^ [Less_eq2: produc5580075136493545115on_nat > produc5580075136493545115on_nat > $o,U2: produc5580075136493545115on_nat] :
          ( collec1293568243156533638on_nat
          @ ^ [X2: produc5580075136493545115on_nat] : ( Less_eq2 @ X2 @ U2 ) ) ) ) ).

% ord.atMost_def
thf(fact_764_inf__sup__ord_I2_J,axiom,
    ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).

% inf_sup_ord(2)
thf(fact_765_inf__sup__ord_I1_J,axiom,
    ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).

% inf_sup_ord(1)
thf(fact_766_inf__le1,axiom,
    ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ X3 ) ).

% inf_le1
thf(fact_767_inf__le2,axiom,
    ! [X3: nat,Y3: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ X3 @ Y3 ) @ Y3 ) ).

% inf_le2
thf(fact_768_le__infE,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) )
     => ~ ( ( ord_less_eq_nat @ X3 @ A )
         => ~ ( ord_less_eq_nat @ X3 @ B ) ) ) ).

% le_infE
thf(fact_769_le__infI,axiom,
    ! [X3: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ X3 @ A )
     => ( ( ord_less_eq_nat @ X3 @ B )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ A @ B ) ) ) ) ).

% le_infI
thf(fact_770_inf__mono,axiom,
    ! [A: nat,C: nat,B: nat,D2: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ D2 )
       => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ ( inf_inf_nat @ C @ D2 ) ) ) ) ).

% inf_mono
thf(fact_771_le__infI1,axiom,
    ! [A: nat,X3: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).

% le_infI1
thf(fact_772_le__infI2,axiom,
    ! [B: nat,X3: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ X3 )
     => ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ X3 ) ) ).

% le_infI2
thf(fact_773_inf_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( A
        = ( inf_inf_nat @ A @ B ) ) ) ).

% inf.orderE
thf(fact_774_inf_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( inf_inf_nat @ A @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% inf.orderI
thf(fact_775_inf__unique,axiom,
    ! [F: nat > nat > nat,X3: nat,Y3: nat] :
      ( ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X @ Y ) @ X )
     => ( ! [X: nat,Y: nat] : ( ord_less_eq_nat @ ( F @ X @ Y ) @ Y )
       => ( ! [X: nat,Y: nat,Z3: nat] :
              ( ( ord_less_eq_nat @ X @ Y )
             => ( ( ord_less_eq_nat @ X @ Z3 )
               => ( ord_less_eq_nat @ X @ ( F @ Y @ Z3 ) ) ) )
         => ( ( inf_inf_nat @ X3 @ Y3 )
            = ( F @ X3 @ Y3 ) ) ) ) ) ).

% inf_unique
thf(fact_776_le__iff__inf,axiom,
    ( ord_less_eq_nat
    = ( ^ [X2: nat,Y5: nat] :
          ( ( inf_inf_nat @ X2 @ Y5 )
          = X2 ) ) ) ).

% le_iff_inf
thf(fact_777_inf_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( inf_inf_nat @ A @ B )
        = A ) ) ).

% inf.absorb1
thf(fact_778_inf_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( inf_inf_nat @ A @ B )
        = B ) ) ).

% inf.absorb2
thf(fact_779_inf__absorb1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( inf_inf_nat @ X3 @ Y3 )
        = X3 ) ) ).

% inf_absorb1
thf(fact_780_inf__absorb2,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( inf_inf_nat @ X3 @ Y3 )
        = Y3 ) ) ).

% inf_absorb2
thf(fact_781_inf_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) )
     => ~ ( ( ord_less_eq_nat @ A @ B )
         => ~ ( ord_less_eq_nat @ A @ C ) ) ) ).

% inf.boundedE
thf(fact_782_inf_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ C )
       => ( ord_less_eq_nat @ A @ ( inf_inf_nat @ B @ C ) ) ) ) ).

% inf.boundedI
thf(fact_783_inf__greatest,axiom,
    ! [X3: nat,Y3: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ X3 @ Z )
       => ( ord_less_eq_nat @ X3 @ ( inf_inf_nat @ Y3 @ Z ) ) ) ) ).

% inf_greatest
thf(fact_784_inf_Oorder__iff,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( A3
          = ( inf_inf_nat @ A3 @ B3 ) ) ) ) ).

% inf.order_iff
thf(fact_785_inf_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ A ) ).

% inf.cobounded1
thf(fact_786_inf_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ ( inf_inf_nat @ A @ B ) @ B ) ).

% inf.cobounded2
thf(fact_787_inf_Oabsorb__iff1,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( inf_inf_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% inf.absorb_iff1
thf(fact_788_inf_Oabsorb__iff2,axiom,
    ( ord_less_eq_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( inf_inf_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% inf.absorb_iff2
thf(fact_789_subterm_OgreaterThanLessThan__empty,axiom,
    ! [K: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ K @ L ) @ subter3376206727240021173_a_nat )
     => ( ( set_gr1098500637439456618_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ K
          @ L )
        = bot_bo6214183520602931734_a_nat ) ) ).

% subterm.greaterThanLessThan_empty
thf(fact_790_supt__def,axiom,
    ( subter8519896202621964801_a_nat
    = ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ subter5000939398133148941_a_nat ) ) ) ).

% supt_def
thf(fact_791_subterm_OgreaterThanAtMost__empty__iff2,axiom,
    ! [K: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat] :
      ( ( bot_bo6214183520602931734_a_nat
        = ( set_gr6446970317771492622_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ K
          @ L ) )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ L @ K ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.greaterThanAtMost_empty_iff2
thf(fact_792_subterm_OgreaterThanAtMost__empty__iff,axiom,
    ! [K: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat] :
      ( ( ( set_gr6446970317771492622_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ K
          @ L )
        = bot_bo6214183520602931734_a_nat )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ L @ K ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.greaterThanAtMost_empty_iff
thf(fact_793_subterm_OgreaterThanAtMost__empty,axiom,
    ! [K: term_R1747997765655105994_a_nat,L: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ K @ L ) @ subter3376206727240021173_a_nat )
     => ( ( set_gr6446970317771492622_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ K
          @ L )
        = bot_bo6214183520602931734_a_nat ) ) ).

% subterm.greaterThanAtMost_empty
thf(fact_794_empty__Collect__eq,axiom,
    ! [P2: produc394670695091157671_a_nat > $o] :
      ( ( bot_bo7892652260360037619_a_nat
        = ( collec8858057866656402194_a_nat @ P2 ) )
      = ( ! [X2: produc394670695091157671_a_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_795_empty__Collect__eq,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o] :
      ( ( bot_bo2698581962261985523_a_nat
        = ( collec9088346736735320338_a_nat @ P2 ) )
      = ( ! [X2: produc3043957382837978535_a_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_796_empty__Collect__eq,axiom,
    ! [P2: produc5580075136493545115on_nat > $o] :
      ( ( bot_bo3119703073163937383on_nat
        = ( collec1293568243156533638on_nat @ P2 ) )
      = ( ! [X2: produc5580075136493545115on_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% empty_Collect_eq
thf(fact_797_Collect__empty__eq,axiom,
    ! [P2: produc394670695091157671_a_nat > $o] :
      ( ( ( collec8858057866656402194_a_nat @ P2 )
        = bot_bo7892652260360037619_a_nat )
      = ( ! [X2: produc394670695091157671_a_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_798_Collect__empty__eq,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o] :
      ( ( ( collec9088346736735320338_a_nat @ P2 )
        = bot_bo2698581962261985523_a_nat )
      = ( ! [X2: produc3043957382837978535_a_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_799_Collect__empty__eq,axiom,
    ! [P2: produc5580075136493545115on_nat > $o] :
      ( ( ( collec1293568243156533638on_nat @ P2 )
        = bot_bo3119703073163937383on_nat )
      = ( ! [X2: produc5580075136493545115on_nat] :
            ~ ( P2 @ X2 ) ) ) ).

% Collect_empty_eq
thf(fact_800_all__not__in__conv,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat] :
      ( ( ! [X2: produc394670695091157671_a_nat] :
            ~ ( member4231271695026834640_a_nat @ X2 @ A5 ) )
      = ( A5 = bot_bo7892652260360037619_a_nat ) ) ).

% all_not_in_conv
thf(fact_801_all__not__in__conv,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat] :
      ( ( ! [X2: produc5238312538367656103_a_nat] :
            ~ ( member6928026473249828560_a_nat @ X2 @ A5 ) )
      = ( A5 = bot_bo330137521859356403_a_nat ) ) ).

% all_not_in_conv
thf(fact_802_all__not__in__conv,axiom,
    ! [A5: set_Pr3573155178515677307on_nat] :
      ( ( ! [X2: produc9176241080893687067on_nat] :
            ~ ( member4278285197320822468on_nat @ X2 @ A5 ) )
      = ( A5 = bot_bo2515438816433479911on_nat ) ) ).

% all_not_in_conv
thf(fact_803_all__not__in__conv,axiom,
    ! [A5: set_Pr7828957961435280891on_nat] :
      ( ( ! [X2: produc5580075136493545115on_nat] :
            ~ ( member4221009242412147780on_nat @ X2 @ A5 ) )
      = ( A5 = bot_bo3119703073163937383on_nat ) ) ).

% all_not_in_conv
thf(fact_804_empty__iff,axiom,
    ! [C: produc394670695091157671_a_nat] :
      ~ ( member4231271695026834640_a_nat @ C @ bot_bo7892652260360037619_a_nat ) ).

% empty_iff
thf(fact_805_empty__iff,axiom,
    ! [C: produc5238312538367656103_a_nat] :
      ~ ( member6928026473249828560_a_nat @ C @ bot_bo330137521859356403_a_nat ) ).

% empty_iff
thf(fact_806_empty__iff,axiom,
    ! [C: produc9176241080893687067on_nat] :
      ~ ( member4278285197320822468on_nat @ C @ bot_bo2515438816433479911on_nat ) ).

% empty_iff
thf(fact_807_empty__iff,axiom,
    ! [C: produc5580075136493545115on_nat] :
      ~ ( member4221009242412147780on_nat @ C @ bot_bo3119703073163937383on_nat ) ).

% empty_iff
thf(fact_808_subterm_OatLeastatMost__empty__iff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        = bot_bo6214183520602931734_a_nat )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.atLeastatMost_empty_iff
thf(fact_809_subterm_OatLeastatMost__empty__iff2,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( bot_bo6214183520602931734_a_nat
        = ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B ) )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat ) ) ) ).

% subterm.atLeastatMost_empty_iff2
thf(fact_810_subterm_Obdd__below__empty,axiom,
    ( condit8263634360281639266_a_nat
    @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
    @ bot_bo6214183520602931734_a_nat ) ).

% subterm.bdd_below_empty
thf(fact_811_subterm_Obdd__above__empty,axiom,
    ( condit1868887373268119414_a_nat
    @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
    @ bot_bo6214183520602931734_a_nat ) ).

% subterm.bdd_above_empty
thf(fact_812_subterm_OatLeastatMost__empty,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter8519896202621964801_a_nat )
     => ( ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        = bot_bo6214183520602931734_a_nat ) ) ).

% subterm.atLeastatMost_empty
thf(fact_813_subterm_OatLeastLessThan__empty,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ subter3376206727240021173_a_nat )
     => ( ( set_at5605147574288828367_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ A
          @ B )
        = bot_bo6214183520602931734_a_nat ) ) ).

% subterm.atLeastLessThan_empty
thf(fact_814_subterm_OatLeastLessThan__empty__iff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( ( set_at5605147574288828367_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ A
          @ B )
        = bot_bo6214183520602931734_a_nat )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.atLeastLessThan_empty_iff
thf(fact_815_subterm_OatLeastLessThan__empty__iff2,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( bot_bo6214183520602931734_a_nat
        = ( set_at5605147574288828367_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
          @ A
          @ B ) )
      = ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter8519896202621964801_a_nat ) ) ) ).

% subterm.atLeastLessThan_empty_iff2
thf(fact_816_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_817_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_818_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_819_empty__def,axiom,
    ( bot_bo7892652260360037619_a_nat
    = ( collec8858057866656402194_a_nat
      @ ^ [X2: produc394670695091157671_a_nat] : $false ) ) ).

% empty_def
thf(fact_820_empty__def,axiom,
    ( bot_bo2698581962261985523_a_nat
    = ( collec9088346736735320338_a_nat
      @ ^ [X2: produc3043957382837978535_a_nat] : $false ) ) ).

% empty_def
thf(fact_821_empty__def,axiom,
    ( bot_bo3119703073163937383on_nat
    = ( collec1293568243156533638on_nat
      @ ^ [X2: produc5580075136493545115on_nat] : $false ) ) ).

% empty_def
thf(fact_822_ex__in__conv,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat] :
      ( ( ? [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ A5 ) )
      = ( A5 != bot_bo7892652260360037619_a_nat ) ) ).

% ex_in_conv
thf(fact_823_ex__in__conv,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat] :
      ( ( ? [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ A5 ) )
      = ( A5 != bot_bo330137521859356403_a_nat ) ) ).

% ex_in_conv
thf(fact_824_ex__in__conv,axiom,
    ! [A5: set_Pr3573155178515677307on_nat] :
      ( ( ? [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ A5 ) )
      = ( A5 != bot_bo2515438816433479911on_nat ) ) ).

% ex_in_conv
thf(fact_825_ex__in__conv,axiom,
    ! [A5: set_Pr7828957961435280891on_nat] :
      ( ( ? [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ A5 ) )
      = ( A5 != bot_bo3119703073163937383on_nat ) ) ).

% ex_in_conv
thf(fact_826_equals0I,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat] :
      ( ! [Y: produc394670695091157671_a_nat] :
          ~ ( member4231271695026834640_a_nat @ Y @ A5 )
     => ( A5 = bot_bo7892652260360037619_a_nat ) ) ).

% equals0I
thf(fact_827_equals0I,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat] :
      ( ! [Y: produc5238312538367656103_a_nat] :
          ~ ( member6928026473249828560_a_nat @ Y @ A5 )
     => ( A5 = bot_bo330137521859356403_a_nat ) ) ).

% equals0I
thf(fact_828_equals0I,axiom,
    ! [A5: set_Pr3573155178515677307on_nat] :
      ( ! [Y: produc9176241080893687067on_nat] :
          ~ ( member4278285197320822468on_nat @ Y @ A5 )
     => ( A5 = bot_bo2515438816433479911on_nat ) ) ).

% equals0I
thf(fact_829_equals0I,axiom,
    ! [A5: set_Pr7828957961435280891on_nat] :
      ( ! [Y: produc5580075136493545115on_nat] :
          ~ ( member4221009242412147780on_nat @ Y @ A5 )
     => ( A5 = bot_bo3119703073163937383on_nat ) ) ).

% equals0I
thf(fact_830_equals0D,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,A: produc394670695091157671_a_nat] :
      ( ( A5 = bot_bo7892652260360037619_a_nat )
     => ~ ( member4231271695026834640_a_nat @ A @ A5 ) ) ).

% equals0D
thf(fact_831_equals0D,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,A: produc5238312538367656103_a_nat] :
      ( ( A5 = bot_bo330137521859356403_a_nat )
     => ~ ( member6928026473249828560_a_nat @ A @ A5 ) ) ).

% equals0D
thf(fact_832_equals0D,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,A: produc9176241080893687067on_nat] :
      ( ( A5 = bot_bo2515438816433479911on_nat )
     => ~ ( member4278285197320822468on_nat @ A @ A5 ) ) ).

% equals0D
thf(fact_833_equals0D,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,A: produc5580075136493545115on_nat] :
      ( ( A5 = bot_bo3119703073163937383on_nat )
     => ~ ( member4221009242412147780on_nat @ A @ A5 ) ) ).

% equals0D
thf(fact_834_emptyE,axiom,
    ! [A: produc394670695091157671_a_nat] :
      ~ ( member4231271695026834640_a_nat @ A @ bot_bo7892652260360037619_a_nat ) ).

% emptyE
thf(fact_835_emptyE,axiom,
    ! [A: produc5238312538367656103_a_nat] :
      ~ ( member6928026473249828560_a_nat @ A @ bot_bo330137521859356403_a_nat ) ).

% emptyE
thf(fact_836_emptyE,axiom,
    ! [A: produc9176241080893687067on_nat] :
      ~ ( member4278285197320822468on_nat @ A @ bot_bo2515438816433479911on_nat ) ).

% emptyE
thf(fact_837_emptyE,axiom,
    ! [A: produc5580075136493545115on_nat] :
      ~ ( member4221009242412147780on_nat @ A @ bot_bo3119703073163937383on_nat ) ).

% emptyE
thf(fact_838_disjoint__iff,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( ( inf_in3705958062578247285_a_nat @ A5 @ B5 )
        = bot_bo7892652260360037619_a_nat )
      = ( ! [X2: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X2 @ A5 )
           => ~ ( member4231271695026834640_a_nat @ X2 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_839_disjoint__iff,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( ( inf_in7903224342872612981_a_nat @ A5 @ B5 )
        = bot_bo330137521859356403_a_nat )
      = ( ! [X2: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X2 @ A5 )
           => ~ ( member6928026473249828560_a_nat @ X2 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_840_disjoint__iff,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( ( inf_in1188219735355578729on_nat @ A5 @ B5 )
        = bot_bo2515438816433479911on_nat )
      = ( ! [X2: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X2 @ A5 )
           => ~ ( member4278285197320822468on_nat @ X2 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_841_disjoint__iff,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( ( inf_in505647112809072361on_nat @ A5 @ B5 )
        = bot_bo3119703073163937383on_nat )
      = ( ! [X2: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X2 @ A5 )
           => ~ ( member4221009242412147780on_nat @ X2 @ B5 ) ) ) ) ).

% disjoint_iff
thf(fact_842_Int__emptyI,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ! [X: produc394670695091157671_a_nat] :
          ( ( member4231271695026834640_a_nat @ X @ A5 )
         => ~ ( member4231271695026834640_a_nat @ X @ B5 ) )
     => ( ( inf_in3705958062578247285_a_nat @ A5 @ B5 )
        = bot_bo7892652260360037619_a_nat ) ) ).

% Int_emptyI
thf(fact_843_Int__emptyI,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ! [X: produc5238312538367656103_a_nat] :
          ( ( member6928026473249828560_a_nat @ X @ A5 )
         => ~ ( member6928026473249828560_a_nat @ X @ B5 ) )
     => ( ( inf_in7903224342872612981_a_nat @ A5 @ B5 )
        = bot_bo330137521859356403_a_nat ) ) ).

% Int_emptyI
thf(fact_844_Int__emptyI,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ! [X: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X @ A5 )
         => ~ ( member4278285197320822468on_nat @ X @ B5 ) )
     => ( ( inf_in1188219735355578729on_nat @ A5 @ B5 )
        = bot_bo2515438816433479911on_nat ) ) ).

% Int_emptyI
thf(fact_845_Int__emptyI,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ! [X: produc5580075136493545115on_nat] :
          ( ( member4221009242412147780on_nat @ X @ A5 )
         => ~ ( member4221009242412147780on_nat @ X @ B5 ) )
     => ( ( inf_in505647112809072361on_nat @ A5 @ B5 )
        = bot_bo3119703073163937383on_nat ) ) ).

% Int_emptyI
thf(fact_846_subterm_Onot__empty__eq__Ici__eq__empty,axiom,
    ! [L: term_R1747997765655105994_a_nat] :
      ( bot_bo6214183520602931734_a_nat
     != ( set_at3383287042032256009_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ L ) ) ).

% subterm.not_empty_eq_Ici_eq_empty
thf(fact_847_subterm_Onot__empty__eq__Iic__eq__empty,axiom,
    ! [H: term_R1747997765655105994_a_nat] :
      ( bot_bo6214183520602931734_a_nat
     != ( set_at2477807202489097501_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ H ) ) ).

% subterm.not_empty_eq_Iic_eq_empty
thf(fact_848_subset__emptyI,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat] :
      ( ! [X: produc394670695091157671_a_nat] :
          ~ ( member4231271695026834640_a_nat @ X @ A5 )
     => ( ord_le998509814859404839_a_nat @ A5 @ bot_bo7892652260360037619_a_nat ) ) ).

% subset_emptyI
thf(fact_849_subset__emptyI,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat] :
      ( ! [X: produc5238312538367656103_a_nat] :
          ~ ( member6928026473249828560_a_nat @ X @ A5 )
     => ( ord_le7248185767425706023_a_nat @ A5 @ bot_bo330137521859356403_a_nat ) ) ).

% subset_emptyI
thf(fact_850_subset__emptyI,axiom,
    ! [A5: set_Pr3573155178515677307on_nat] :
      ( ! [X: produc9176241080893687067on_nat] :
          ~ ( member4278285197320822468on_nat @ X @ A5 )
     => ( ord_le927452247459997723on_nat @ A5 @ bot_bo2515438816433479911on_nat ) ) ).

% subset_emptyI
thf(fact_851_subset__emptyI,axiom,
    ! [A5: set_Pr7828957961435280891on_nat] :
      ( ! [X: produc5580075136493545115on_nat] :
          ~ ( member4221009242412147780on_nat @ X @ A5 )
     => ( ord_le6423710186836778395on_nat @ A5 @ bot_bo3119703073163937383on_nat ) ) ).

% subset_emptyI
thf(fact_852_subterm_OIio__Int__singleton,axiom,
    ! [K: term_R1747997765655105994_a_nat,X3: term_R1747997765655105994_a_nat] :
      ( ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ K @ X3 ) @ subter8519896202621964801_a_nat )
       => ( ( inf_in7710483629710275224_a_nat
            @ ( set_le6356817768498327033_a_nat
              @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
              @ K )
            @ ( insert8400145511831611290_a_nat @ X3 @ bot_bo6214183520602931734_a_nat ) )
          = ( insert8400145511831611290_a_nat @ X3 @ bot_bo6214183520602931734_a_nat ) ) )
      & ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ K @ X3 ) @ subter8519896202621964801_a_nat )
       => ( ( inf_in7710483629710275224_a_nat
            @ ( set_le6356817768498327033_a_nat
              @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
              @ K )
            @ ( insert8400145511831611290_a_nat @ X3 @ bot_bo6214183520602931734_a_nat ) )
          = bot_bo6214183520602931734_a_nat ) ) ) ).

% subterm.Iio_Int_singleton
thf(fact_853_supteq__def,axiom,
    ( subter3376206727240021173_a_nat
    = ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ subter2734268076276479833_a_nat ) ) ) ).

% supteq_def
thf(fact_854_insertCI,axiom,
    ! [A: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat,B: produc394670695091157671_a_nat] :
      ( ( ~ ( member4231271695026834640_a_nat @ A @ B5 )
       => ( A = B ) )
     => ( member4231271695026834640_a_nat @ A @ ( insert8772053866485626999_a_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_855_insertCI,axiom,
    ! [A: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat,B: produc5238312538367656103_a_nat] :
      ( ( ~ ( member6928026473249828560_a_nat @ A @ B5 )
       => ( A = B ) )
     => ( member6928026473249828560_a_nat @ A @ ( insert2316125241028329079_a_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_856_insertCI,axiom,
    ! [A: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat,B: produc9176241080893687067on_nat] :
      ( ( ~ ( member4278285197320822468on_nat @ A @ B5 )
       => ( A = B ) )
     => ( member4278285197320822468on_nat @ A @ ( insert210981557334383339on_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_857_insertCI,axiom,
    ! [A: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat,B: produc5580075136493545115on_nat] :
      ( ( ~ ( member4221009242412147780on_nat @ A @ B5 )
       => ( A = B ) )
     => ( member4221009242412147780on_nat @ A @ ( insert2664702451092716651on_nat @ B @ B5 ) ) ) ).

% insertCI
thf(fact_858_insert__iff,axiom,
    ! [A: produc394670695091157671_a_nat,B: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ ( insert8772053866485626999_a_nat @ B @ A5 ) )
      = ( ( A = B )
        | ( member4231271695026834640_a_nat @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_859_insert__iff,axiom,
    ! [A: produc5238312538367656103_a_nat,B: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ ( insert2316125241028329079_a_nat @ B @ A5 ) )
      = ( ( A = B )
        | ( member6928026473249828560_a_nat @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_860_insert__iff,axiom,
    ! [A: produc9176241080893687067on_nat,B: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( insert210981557334383339on_nat @ B @ A5 ) )
      = ( ( A = B )
        | ( member4278285197320822468on_nat @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_861_insert__iff,axiom,
    ! [A: produc5580075136493545115on_nat,B: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ A @ ( insert2664702451092716651on_nat @ B @ A5 ) )
      = ( ( A = B )
        | ( member4221009242412147780on_nat @ A @ A5 ) ) ) ).

% insert_iff
thf(fact_862_singletonI,axiom,
    ! [A: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ A @ ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) ) ).

% singletonI
thf(fact_863_singletonI,axiom,
    ! [A: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ A @ ( insert2316125241028329079_a_nat @ A @ bot_bo330137521859356403_a_nat ) ) ).

% singletonI
thf(fact_864_singletonI,axiom,
    ! [A: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ A @ ( insert210981557334383339on_nat @ A @ bot_bo2515438816433479911on_nat ) ) ).

% singletonI
thf(fact_865_singletonI,axiom,
    ! [A: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ A @ ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) ) ).

% singletonI
thf(fact_866_insert__subset,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( ord_le998509814859404839_a_nat @ ( insert8772053866485626999_a_nat @ X3 @ A5 ) @ B5 )
      = ( ( member4231271695026834640_a_nat @ X3 @ B5 )
        & ( ord_le998509814859404839_a_nat @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_867_insert__subset,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( ord_le7248185767425706023_a_nat @ ( insert2316125241028329079_a_nat @ X3 @ A5 ) @ B5 )
      = ( ( member6928026473249828560_a_nat @ X3 @ B5 )
        & ( ord_le7248185767425706023_a_nat @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_868_insert__subset,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( ord_le927452247459997723on_nat @ ( insert210981557334383339on_nat @ X3 @ A5 ) @ B5 )
      = ( ( member4278285197320822468on_nat @ X3 @ B5 )
        & ( ord_le927452247459997723on_nat @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_869_insert__subset,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( ord_le6423710186836778395on_nat @ ( insert2664702451092716651on_nat @ X3 @ A5 ) @ B5 )
      = ( ( member4221009242412147780on_nat @ X3 @ B5 )
        & ( ord_le6423710186836778395on_nat @ A5 @ B5 ) ) ) ).

% insert_subset
thf(fact_870_Int__insert__left__if0,axiom,
    ! [A: produc394670695091157671_a_nat,C4: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ~ ( member4231271695026834640_a_nat @ A @ C4 )
     => ( ( inf_in3705958062578247285_a_nat @ ( insert8772053866485626999_a_nat @ A @ B5 ) @ C4 )
        = ( inf_in3705958062578247285_a_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_871_Int__insert__left__if0,axiom,
    ! [A: produc5238312538367656103_a_nat,C4: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ~ ( member6928026473249828560_a_nat @ A @ C4 )
     => ( ( inf_in7903224342872612981_a_nat @ ( insert2316125241028329079_a_nat @ A @ B5 ) @ C4 )
        = ( inf_in7903224342872612981_a_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_872_Int__insert__left__if0,axiom,
    ! [A: produc9176241080893687067on_nat,C4: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ~ ( member4278285197320822468on_nat @ A @ C4 )
     => ( ( inf_in1188219735355578729on_nat @ ( insert210981557334383339on_nat @ A @ B5 ) @ C4 )
        = ( inf_in1188219735355578729on_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_873_Int__insert__left__if0,axiom,
    ! [A: produc5580075136493545115on_nat,C4: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ~ ( member4221009242412147780on_nat @ A @ C4 )
     => ( ( inf_in505647112809072361on_nat @ ( insert2664702451092716651on_nat @ A @ B5 ) @ C4 )
        = ( inf_in505647112809072361on_nat @ B5 @ C4 ) ) ) ).

% Int_insert_left_if0
thf(fact_874_Int__insert__left__if1,axiom,
    ! [A: produc394670695091157671_a_nat,C4: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ C4 )
     => ( ( inf_in3705958062578247285_a_nat @ ( insert8772053866485626999_a_nat @ A @ B5 ) @ C4 )
        = ( insert8772053866485626999_a_nat @ A @ ( inf_in3705958062578247285_a_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_875_Int__insert__left__if1,axiom,
    ! [A: produc5238312538367656103_a_nat,C4: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ C4 )
     => ( ( inf_in7903224342872612981_a_nat @ ( insert2316125241028329079_a_nat @ A @ B5 ) @ C4 )
        = ( insert2316125241028329079_a_nat @ A @ ( inf_in7903224342872612981_a_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_876_Int__insert__left__if1,axiom,
    ! [A: produc9176241080893687067on_nat,C4: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ A @ C4 )
     => ( ( inf_in1188219735355578729on_nat @ ( insert210981557334383339on_nat @ A @ B5 ) @ C4 )
        = ( insert210981557334383339on_nat @ A @ ( inf_in1188219735355578729on_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_877_Int__insert__left__if1,axiom,
    ! [A: produc5580075136493545115on_nat,C4: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ A @ C4 )
     => ( ( inf_in505647112809072361on_nat @ ( insert2664702451092716651on_nat @ A @ B5 ) @ C4 )
        = ( insert2664702451092716651on_nat @ A @ ( inf_in505647112809072361on_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_878_Int__insert__right__if0,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ~ ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( inf_in3705958062578247285_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ A @ B5 ) )
        = ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_879_Int__insert__right__if0,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ~ ( member6928026473249828560_a_nat @ A @ A5 )
     => ( ( inf_in7903224342872612981_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ A @ B5 ) )
        = ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_880_Int__insert__right__if0,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ~ ( member4278285197320822468on_nat @ A @ A5 )
     => ( ( inf_in1188219735355578729on_nat @ A5 @ ( insert210981557334383339on_nat @ A @ B5 ) )
        = ( inf_in1188219735355578729on_nat @ A5 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_881_Int__insert__right__if0,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ~ ( member4221009242412147780on_nat @ A @ A5 )
     => ( ( inf_in505647112809072361on_nat @ A5 @ ( insert2664702451092716651on_nat @ A @ B5 ) )
        = ( inf_in505647112809072361on_nat @ A5 @ B5 ) ) ) ).

% Int_insert_right_if0
thf(fact_882_Int__insert__right__if1,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( inf_in3705958062578247285_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ A @ B5 ) )
        = ( insert8772053866485626999_a_nat @ A @ ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_883_Int__insert__right__if1,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ A5 )
     => ( ( inf_in7903224342872612981_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ A @ B5 ) )
        = ( insert2316125241028329079_a_nat @ A @ ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_884_Int__insert__right__if1,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ A @ A5 )
     => ( ( inf_in1188219735355578729on_nat @ A5 @ ( insert210981557334383339on_nat @ A @ B5 ) )
        = ( insert210981557334383339on_nat @ A @ ( inf_in1188219735355578729on_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_885_Int__insert__right__if1,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ A @ A5 )
     => ( ( inf_in505647112809072361on_nat @ A5 @ ( insert2664702451092716651on_nat @ A @ B5 ) )
        = ( insert2664702451092716651on_nat @ A @ ( inf_in505647112809072361on_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_886_singleton__conv,axiom,
    ! [A: produc394670695091157671_a_nat] :
      ( ( collec8858057866656402194_a_nat
        @ ^ [X2: produc394670695091157671_a_nat] : ( X2 = A ) )
      = ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) ) ).

% singleton_conv
thf(fact_887_singleton__conv,axiom,
    ! [A: produc3043957382837978535_a_nat] :
      ( ( collec9088346736735320338_a_nat
        @ ^ [X2: produc3043957382837978535_a_nat] : ( X2 = A ) )
      = ( insert2097847225457054583_a_nat @ A @ bot_bo2698581962261985523_a_nat ) ) ).

% singleton_conv
thf(fact_888_singleton__conv,axiom,
    ! [A: produc5580075136493545115on_nat] :
      ( ( collec1293568243156533638on_nat
        @ ^ [X2: produc5580075136493545115on_nat] : ( X2 = A ) )
      = ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) ) ).

% singleton_conv
thf(fact_889_singleton__conv2,axiom,
    ! [A: produc394670695091157671_a_nat] :
      ( ( collec8858057866656402194_a_nat
        @ ( ^ [Y4: produc394670695091157671_a_nat,Z2: produc394670695091157671_a_nat] : ( Y4 = Z2 )
          @ A ) )
      = ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) ) ).

% singleton_conv2
thf(fact_890_singleton__conv2,axiom,
    ! [A: produc3043957382837978535_a_nat] :
      ( ( collec9088346736735320338_a_nat
        @ ( ^ [Y4: produc3043957382837978535_a_nat,Z2: produc3043957382837978535_a_nat] : ( Y4 = Z2 )
          @ A ) )
      = ( insert2097847225457054583_a_nat @ A @ bot_bo2698581962261985523_a_nat ) ) ).

% singleton_conv2
thf(fact_891_singleton__conv2,axiom,
    ! [A: produc5580075136493545115on_nat] :
      ( ( collec1293568243156533638on_nat
        @ ( ^ [Y4: produc5580075136493545115on_nat,Z2: produc5580075136493545115on_nat] : ( Y4 = Z2 )
          @ A ) )
      = ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) ) ).

% singleton_conv2
thf(fact_892_disjoint__insert_I2_J,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( bot_bo7892652260360037619_a_nat
        = ( inf_in3705958062578247285_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ B @ B5 ) ) )
      = ( ~ ( member4231271695026834640_a_nat @ B @ A5 )
        & ( bot_bo7892652260360037619_a_nat
          = ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_893_disjoint__insert_I2_J,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( bot_bo330137521859356403_a_nat
        = ( inf_in7903224342872612981_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ B @ B5 ) ) )
      = ( ~ ( member6928026473249828560_a_nat @ B @ A5 )
        & ( bot_bo330137521859356403_a_nat
          = ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_894_disjoint__insert_I2_J,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( bot_bo2515438816433479911on_nat
        = ( inf_in1188219735355578729on_nat @ A5 @ ( insert210981557334383339on_nat @ B @ B5 ) ) )
      = ( ~ ( member4278285197320822468on_nat @ B @ A5 )
        & ( bot_bo2515438816433479911on_nat
          = ( inf_in1188219735355578729on_nat @ A5 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_895_disjoint__insert_I2_J,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( bot_bo3119703073163937383on_nat
        = ( inf_in505647112809072361on_nat @ A5 @ ( insert2664702451092716651on_nat @ B @ B5 ) ) )
      = ( ~ ( member4221009242412147780on_nat @ B @ A5 )
        & ( bot_bo3119703073163937383on_nat
          = ( inf_in505647112809072361on_nat @ A5 @ B5 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_896_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr4991467605864603271_a_nat,A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( ( inf_in3705958062578247285_a_nat @ B5 @ ( insert8772053866485626999_a_nat @ A @ A5 ) )
        = bot_bo7892652260360037619_a_nat )
      = ( ~ ( member4231271695026834640_a_nat @ A @ B5 )
        & ( ( inf_in3705958062578247285_a_nat @ B5 @ A5 )
          = bot_bo7892652260360037619_a_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_897_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr8529752790658994311_a_nat,A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( ( inf_in7903224342872612981_a_nat @ B5 @ ( insert2316125241028329079_a_nat @ A @ A5 ) )
        = bot_bo330137521859356403_a_nat )
      = ( ~ ( member6928026473249828560_a_nat @ A @ B5 )
        & ( ( inf_in7903224342872612981_a_nat @ B5 @ A5 )
          = bot_bo330137521859356403_a_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_898_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr3573155178515677307on_nat,A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( ( inf_in1188219735355578729on_nat @ B5 @ ( insert210981557334383339on_nat @ A @ A5 ) )
        = bot_bo2515438816433479911on_nat )
      = ( ~ ( member4278285197320822468on_nat @ A @ B5 )
        & ( ( inf_in1188219735355578729on_nat @ B5 @ A5 )
          = bot_bo2515438816433479911on_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_899_disjoint__insert_I1_J,axiom,
    ! [B5: set_Pr7828957961435280891on_nat,A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( ( inf_in505647112809072361on_nat @ B5 @ ( insert2664702451092716651on_nat @ A @ A5 ) )
        = bot_bo3119703073163937383on_nat )
      = ( ~ ( member4221009242412147780on_nat @ A @ B5 )
        & ( ( inf_in505647112809072361on_nat @ B5 @ A5 )
          = bot_bo3119703073163937383on_nat ) ) ) ).

% disjoint_insert(1)
thf(fact_900_insert__disjoint_I2_J,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( bot_bo7892652260360037619_a_nat
        = ( inf_in3705958062578247285_a_nat @ ( insert8772053866485626999_a_nat @ A @ A5 ) @ B5 ) )
      = ( ~ ( member4231271695026834640_a_nat @ A @ B5 )
        & ( bot_bo7892652260360037619_a_nat
          = ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_901_insert__disjoint_I2_J,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( bot_bo330137521859356403_a_nat
        = ( inf_in7903224342872612981_a_nat @ ( insert2316125241028329079_a_nat @ A @ A5 ) @ B5 ) )
      = ( ~ ( member6928026473249828560_a_nat @ A @ B5 )
        & ( bot_bo330137521859356403_a_nat
          = ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_902_insert__disjoint_I2_J,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( bot_bo2515438816433479911on_nat
        = ( inf_in1188219735355578729on_nat @ ( insert210981557334383339on_nat @ A @ A5 ) @ B5 ) )
      = ( ~ ( member4278285197320822468on_nat @ A @ B5 )
        & ( bot_bo2515438816433479911on_nat
          = ( inf_in1188219735355578729on_nat @ A5 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_903_insert__disjoint_I2_J,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( bot_bo3119703073163937383on_nat
        = ( inf_in505647112809072361on_nat @ ( insert2664702451092716651on_nat @ A @ A5 ) @ B5 ) )
      = ( ~ ( member4221009242412147780on_nat @ A @ B5 )
        & ( bot_bo3119703073163937383on_nat
          = ( inf_in505647112809072361on_nat @ A5 @ B5 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_904_insert__disjoint_I1_J,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( ( inf_in3705958062578247285_a_nat @ ( insert8772053866485626999_a_nat @ A @ A5 ) @ B5 )
        = bot_bo7892652260360037619_a_nat )
      = ( ~ ( member4231271695026834640_a_nat @ A @ B5 )
        & ( ( inf_in3705958062578247285_a_nat @ A5 @ B5 )
          = bot_bo7892652260360037619_a_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_905_insert__disjoint_I1_J,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( ( inf_in7903224342872612981_a_nat @ ( insert2316125241028329079_a_nat @ A @ A5 ) @ B5 )
        = bot_bo330137521859356403_a_nat )
      = ( ~ ( member6928026473249828560_a_nat @ A @ B5 )
        & ( ( inf_in7903224342872612981_a_nat @ A5 @ B5 )
          = bot_bo330137521859356403_a_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_906_insert__disjoint_I1_J,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( ( inf_in1188219735355578729on_nat @ ( insert210981557334383339on_nat @ A @ A5 ) @ B5 )
        = bot_bo2515438816433479911on_nat )
      = ( ~ ( member4278285197320822468on_nat @ A @ B5 )
        & ( ( inf_in1188219735355578729on_nat @ A5 @ B5 )
          = bot_bo2515438816433479911on_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_907_insert__disjoint_I1_J,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( ( inf_in505647112809072361on_nat @ ( insert2664702451092716651on_nat @ A @ A5 ) @ B5 )
        = bot_bo3119703073163937383on_nat )
      = ( ~ ( member4221009242412147780on_nat @ A @ B5 )
        & ( ( inf_in505647112809072361on_nat @ A5 @ B5 )
          = bot_bo3119703073163937383on_nat ) ) ) ).

% insert_disjoint(1)
thf(fact_908_subterm_OatLeastAtMost__singleton__iff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,C: term_R1747997765655105994_a_nat] :
      ( ( ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        = ( insert8400145511831611290_a_nat @ C @ bot_bo6214183520602931734_a_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% subterm.atLeastAtMost_singleton_iff
thf(fact_909_subterm_OatLeastAtMost__singleton,axiom,
    ! [A: term_R1747997765655105994_a_nat] :
      ( ( set_at1360634933218525683_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ A
        @ A )
      = ( insert8400145511831611290_a_nat @ A @ bot_bo6214183520602931734_a_nat ) ) ).

% subterm.atLeastAtMost_singleton
thf(fact_910_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_911_bot__empty__eq,axiom,
    ( bot_bo6797488618402958762_nat_o
    = ( ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ bot_bo7892652260360037619_a_nat ) ) ) ).

% bot_empty_eq
thf(fact_912_bot__empty__eq,axiom,
    ( bot_bo2441825641851411370_nat_o
    = ( ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ bot_bo330137521859356403_a_nat ) ) ) ).

% bot_empty_eq
thf(fact_913_bot__empty__eq,axiom,
    ( bot_bo2409773297793940662_nat_o
    = ( ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ bot_bo2515438816433479911on_nat ) ) ) ).

% bot_empty_eq
thf(fact_914_bot__empty__eq,axiom,
    ( bot_bo8997232410071131958_nat_o
    = ( ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ bot_bo3119703073163937383on_nat ) ) ) ).

% bot_empty_eq
thf(fact_915_bot__empty__eq2,axiom,
    ( bot_bo7909424473289354330_nat_o
    = ( ^ [X2: multis6046679608167549072_a_nat,Y5: multis6046679608167549072_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ X2 @ Y5 ) @ bot_bo330137521859356403_a_nat ) ) ) ).

% bot_empty_eq2
thf(fact_916_bot__empty__eq2,axiom,
    ( bot_bo4377304567363464376_nat_o
    = ( ^ [X2: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X2 @ Y5 ) @ bot_bo2515438816433479911on_nat ) ) ) ).

% bot_empty_eq2
thf(fact_917_bot__empty__eq2,axiom,
    ( bot_bo9008160929533878630_nat_o
    = ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y5 ) @ bot_bo1745525042678953353on_nat ) ) ) ).

% bot_empty_eq2
thf(fact_918_bot__empty__eq2,axiom,
    ( bot_bo2987674557801227862_nat_o
    = ( ^ [X2: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y5 ) @ bot_bo6889268709867540425on_nat ) ) ) ).

% bot_empty_eq2
thf(fact_919_bot__empty__eq2,axiom,
    ( bot_bo2292562699024373048_nat_o
    = ( ^ [X2: list_S5253404628753938141on_nat,Y5: list_S5253404628753938141on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X2 @ Y5 ) @ bot_bo3119703073163937383on_nat ) ) ) ).

% bot_empty_eq2
thf(fact_920_bot__empty__eq2,axiom,
    ( bot_bo8706249483093850202_nat_o
    = ( ^ [X2: list_t5668669291736003536_a_nat,Y5: list_t5668669291736003536_a_nat] : ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ X2 @ Y5 ) @ bot_bo7892652260360037619_a_nat ) ) ) ).

% bot_empty_eq2
thf(fact_921_bot__empty__eq2,axiom,
    ( bot_bot_o_o_o
    = ( ^ [X2: $o,Y5: $o] : ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X2 @ Y5 ) @ bot_bo7073875226086086771od_o_o ) ) ) ).

% bot_empty_eq2
thf(fact_922_bot__empty__eq2,axiom,
    ( bot_bo2646606810212072410_nat_o
    = ( ^ [X2: term_R1747997765655105994_a_nat,Y5: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X2 @ Y5 ) @ bot_bo2698581962261985523_a_nat ) ) ) ).

% bot_empty_eq2
thf(fact_923_singleton__iff,axiom,
    ! [B: produc394670695091157671_a_nat,A: produc394670695091157671_a_nat] :
      ( ( member4231271695026834640_a_nat @ B @ ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_924_singleton__iff,axiom,
    ! [B: produc5238312538367656103_a_nat,A: produc5238312538367656103_a_nat] :
      ( ( member6928026473249828560_a_nat @ B @ ( insert2316125241028329079_a_nat @ A @ bot_bo330137521859356403_a_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_925_singleton__iff,axiom,
    ! [B: produc9176241080893687067on_nat,A: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ B @ ( insert210981557334383339on_nat @ A @ bot_bo2515438816433479911on_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_926_singleton__iff,axiom,
    ! [B: produc5580075136493545115on_nat,A: produc5580075136493545115on_nat] :
      ( ( member4221009242412147780on_nat @ B @ ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_927_bot__set__def,axiom,
    ( bot_bo7892652260360037619_a_nat
    = ( collec8858057866656402194_a_nat @ bot_bo6797488618402958762_nat_o ) ) ).

% bot_set_def
thf(fact_928_bot__set__def,axiom,
    ( bot_bo2698581962261985523_a_nat
    = ( collec9088346736735320338_a_nat @ bot_bo121387161548064170_nat_o ) ) ).

% bot_set_def
thf(fact_929_bot__set__def,axiom,
    ( bot_bo3119703073163937383on_nat
    = ( collec1293568243156533638on_nat @ bot_bo8997232410071131958_nat_o ) ) ).

% bot_set_def
thf(fact_930_singletonD,axiom,
    ! [B: produc394670695091157671_a_nat,A: produc394670695091157671_a_nat] :
      ( ( member4231271695026834640_a_nat @ B @ ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_931_singletonD,axiom,
    ! [B: produc5238312538367656103_a_nat,A: produc5238312538367656103_a_nat] :
      ( ( member6928026473249828560_a_nat @ B @ ( insert2316125241028329079_a_nat @ A @ bot_bo330137521859356403_a_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_932_singletonD,axiom,
    ! [B: produc9176241080893687067on_nat,A: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ B @ ( insert210981557334383339on_nat @ A @ bot_bo2515438816433479911on_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_933_singletonD,axiom,
    ! [B: produc5580075136493545115on_nat,A: produc5580075136493545115on_nat] :
      ( ( member4221009242412147780on_nat @ B @ ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_934_subset__insert,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ~ ( member4231271695026834640_a_nat @ X3 @ A5 )
     => ( ( ord_le998509814859404839_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ X3 @ B5 ) )
        = ( ord_le998509814859404839_a_nat @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_935_subset__insert,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ~ ( member6928026473249828560_a_nat @ X3 @ A5 )
     => ( ( ord_le7248185767425706023_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ X3 @ B5 ) )
        = ( ord_le7248185767425706023_a_nat @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_936_subset__insert,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X3 @ A5 )
     => ( ( ord_le927452247459997723on_nat @ A5 @ ( insert210981557334383339on_nat @ X3 @ B5 ) )
        = ( ord_le927452247459997723on_nat @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_937_subset__insert,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ~ ( member4221009242412147780on_nat @ X3 @ A5 )
     => ( ( ord_le6423710186836778395on_nat @ A5 @ ( insert2664702451092716651on_nat @ X3 @ B5 ) )
        = ( ord_le6423710186836778395on_nat @ A5 @ B5 ) ) ) ).

% subset_insert
thf(fact_938_Int__insert__right,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( ( member4231271695026834640_a_nat @ A @ A5 )
       => ( ( inf_in3705958062578247285_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ A @ B5 ) )
          = ( insert8772053866485626999_a_nat @ A @ ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) ) ) )
      & ( ~ ( member4231271695026834640_a_nat @ A @ A5 )
       => ( ( inf_in3705958062578247285_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ A @ B5 ) )
          = ( inf_in3705958062578247285_a_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_939_Int__insert__right,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( ( member6928026473249828560_a_nat @ A @ A5 )
       => ( ( inf_in7903224342872612981_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ A @ B5 ) )
          = ( insert2316125241028329079_a_nat @ A @ ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) ) ) )
      & ( ~ ( member6928026473249828560_a_nat @ A @ A5 )
       => ( ( inf_in7903224342872612981_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ A @ B5 ) )
          = ( inf_in7903224342872612981_a_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_940_Int__insert__right,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( ( member4278285197320822468on_nat @ A @ A5 )
       => ( ( inf_in1188219735355578729on_nat @ A5 @ ( insert210981557334383339on_nat @ A @ B5 ) )
          = ( insert210981557334383339on_nat @ A @ ( inf_in1188219735355578729on_nat @ A5 @ B5 ) ) ) )
      & ( ~ ( member4278285197320822468on_nat @ A @ A5 )
       => ( ( inf_in1188219735355578729on_nat @ A5 @ ( insert210981557334383339on_nat @ A @ B5 ) )
          = ( inf_in1188219735355578729on_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_941_Int__insert__right,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( ( member4221009242412147780on_nat @ A @ A5 )
       => ( ( inf_in505647112809072361on_nat @ A5 @ ( insert2664702451092716651on_nat @ A @ B5 ) )
          = ( insert2664702451092716651on_nat @ A @ ( inf_in505647112809072361on_nat @ A5 @ B5 ) ) ) )
      & ( ~ ( member4221009242412147780on_nat @ A @ A5 )
       => ( ( inf_in505647112809072361on_nat @ A5 @ ( insert2664702451092716651on_nat @ A @ B5 ) )
          = ( inf_in505647112809072361on_nat @ A5 @ B5 ) ) ) ) ).

% Int_insert_right
thf(fact_942_Int__insert__left,axiom,
    ! [A: produc394670695091157671_a_nat,C4: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( ( member4231271695026834640_a_nat @ A @ C4 )
       => ( ( inf_in3705958062578247285_a_nat @ ( insert8772053866485626999_a_nat @ A @ B5 ) @ C4 )
          = ( insert8772053866485626999_a_nat @ A @ ( inf_in3705958062578247285_a_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member4231271695026834640_a_nat @ A @ C4 )
       => ( ( inf_in3705958062578247285_a_nat @ ( insert8772053866485626999_a_nat @ A @ B5 ) @ C4 )
          = ( inf_in3705958062578247285_a_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_943_Int__insert__left,axiom,
    ! [A: produc5238312538367656103_a_nat,C4: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( ( member6928026473249828560_a_nat @ A @ C4 )
       => ( ( inf_in7903224342872612981_a_nat @ ( insert2316125241028329079_a_nat @ A @ B5 ) @ C4 )
          = ( insert2316125241028329079_a_nat @ A @ ( inf_in7903224342872612981_a_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member6928026473249828560_a_nat @ A @ C4 )
       => ( ( inf_in7903224342872612981_a_nat @ ( insert2316125241028329079_a_nat @ A @ B5 ) @ C4 )
          = ( inf_in7903224342872612981_a_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_944_Int__insert__left,axiom,
    ! [A: produc9176241080893687067on_nat,C4: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( ( member4278285197320822468on_nat @ A @ C4 )
       => ( ( inf_in1188219735355578729on_nat @ ( insert210981557334383339on_nat @ A @ B5 ) @ C4 )
          = ( insert210981557334383339on_nat @ A @ ( inf_in1188219735355578729on_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member4278285197320822468on_nat @ A @ C4 )
       => ( ( inf_in1188219735355578729on_nat @ ( insert210981557334383339on_nat @ A @ B5 ) @ C4 )
          = ( inf_in1188219735355578729on_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_945_Int__insert__left,axiom,
    ! [A: produc5580075136493545115on_nat,C4: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( ( member4221009242412147780on_nat @ A @ C4 )
       => ( ( inf_in505647112809072361on_nat @ ( insert2664702451092716651on_nat @ A @ B5 ) @ C4 )
          = ( insert2664702451092716651on_nat @ A @ ( inf_in505647112809072361on_nat @ B5 @ C4 ) ) ) )
      & ( ~ ( member4221009242412147780on_nat @ A @ C4 )
       => ( ( inf_in505647112809072361on_nat @ ( insert2664702451092716651on_nat @ A @ B5 ) @ C4 )
          = ( inf_in505647112809072361on_nat @ B5 @ C4 ) ) ) ) ).

% Int_insert_left
thf(fact_946_insertE,axiom,
    ! [A: produc394670695091157671_a_nat,B: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ ( insert8772053866485626999_a_nat @ B @ A5 ) )
     => ( ( A != B )
       => ( member4231271695026834640_a_nat @ A @ A5 ) ) ) ).

% insertE
thf(fact_947_insertE,axiom,
    ! [A: produc5238312538367656103_a_nat,B: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ ( insert2316125241028329079_a_nat @ B @ A5 ) )
     => ( ( A != B )
       => ( member6928026473249828560_a_nat @ A @ A5 ) ) ) ).

% insertE
thf(fact_948_insertE,axiom,
    ! [A: produc9176241080893687067on_nat,B: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( insert210981557334383339on_nat @ B @ A5 ) )
     => ( ( A != B )
       => ( member4278285197320822468on_nat @ A @ A5 ) ) ) ).

% insertE
thf(fact_949_insertE,axiom,
    ! [A: produc5580075136493545115on_nat,B: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ A @ ( insert2664702451092716651on_nat @ B @ A5 ) )
     => ( ( A != B )
       => ( member4221009242412147780on_nat @ A @ A5 ) ) ) ).

% insertE
thf(fact_950_insertI1,axiom,
    ! [A: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat] : ( member4231271695026834640_a_nat @ A @ ( insert8772053866485626999_a_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_951_insertI1,axiom,
    ! [A: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat] : ( member6928026473249828560_a_nat @ A @ ( insert2316125241028329079_a_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_952_insertI1,axiom,
    ! [A: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat] : ( member4278285197320822468on_nat @ A @ ( insert210981557334383339on_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_953_insertI1,axiom,
    ! [A: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat] : ( member4221009242412147780on_nat @ A @ ( insert2664702451092716651on_nat @ A @ B5 ) ) ).

% insertI1
thf(fact_954_insertI2,axiom,
    ! [A: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat,B: produc394670695091157671_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ B5 )
     => ( member4231271695026834640_a_nat @ A @ ( insert8772053866485626999_a_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_955_insertI2,axiom,
    ! [A: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat,B: produc5238312538367656103_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ B5 )
     => ( member6928026473249828560_a_nat @ A @ ( insert2316125241028329079_a_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_956_insertI2,axiom,
    ! [A: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat,B: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ A @ B5 )
     => ( member4278285197320822468on_nat @ A @ ( insert210981557334383339on_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_957_insertI2,axiom,
    ! [A: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat,B: produc5580075136493545115on_nat] :
      ( ( member4221009242412147780on_nat @ A @ B5 )
     => ( member4221009242412147780on_nat @ A @ ( insert2664702451092716651on_nat @ B @ B5 ) ) ) ).

% insertI2
thf(fact_958_Set_Oset__insert,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ X3 @ A5 )
     => ~ ! [B7: set_Pr4991467605864603271_a_nat] :
            ( ( A5
              = ( insert8772053866485626999_a_nat @ X3 @ B7 ) )
           => ( member4231271695026834640_a_nat @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_959_Set_Oset__insert,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ X3 @ A5 )
     => ~ ! [B7: set_Pr8529752790658994311_a_nat] :
            ( ( A5
              = ( insert2316125241028329079_a_nat @ X3 @ B7 ) )
           => ( member6928026473249828560_a_nat @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_960_Set_Oset__insert,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ X3 @ A5 )
     => ~ ! [B7: set_Pr3573155178515677307on_nat] :
            ( ( A5
              = ( insert210981557334383339on_nat @ X3 @ B7 ) )
           => ( member4278285197320822468on_nat @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_961_Set_Oset__insert,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ X3 @ A5 )
     => ~ ! [B7: set_Pr7828957961435280891on_nat] :
            ( ( A5
              = ( insert2664702451092716651on_nat @ X3 @ B7 ) )
           => ( member4221009242412147780on_nat @ X3 @ B7 ) ) ) ).

% Set.set_insert
thf(fact_962_insert__ident,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ~ ( member4231271695026834640_a_nat @ X3 @ A5 )
     => ( ~ ( member4231271695026834640_a_nat @ X3 @ B5 )
       => ( ( ( insert8772053866485626999_a_nat @ X3 @ A5 )
            = ( insert8772053866485626999_a_nat @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_963_insert__ident,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ~ ( member6928026473249828560_a_nat @ X3 @ A5 )
     => ( ~ ( member6928026473249828560_a_nat @ X3 @ B5 )
       => ( ( ( insert2316125241028329079_a_nat @ X3 @ A5 )
            = ( insert2316125241028329079_a_nat @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_964_insert__ident,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X3 @ A5 )
     => ( ~ ( member4278285197320822468on_nat @ X3 @ B5 )
       => ( ( ( insert210981557334383339on_nat @ X3 @ A5 )
            = ( insert210981557334383339on_nat @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_965_insert__ident,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ~ ( member4221009242412147780on_nat @ X3 @ A5 )
     => ( ~ ( member4221009242412147780on_nat @ X3 @ B5 )
       => ( ( ( insert2664702451092716651on_nat @ X3 @ A5 )
            = ( insert2664702451092716651on_nat @ X3 @ B5 ) )
          = ( A5 = B5 ) ) ) ) ).

% insert_ident
thf(fact_966_insert__absorb,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( insert8772053866485626999_a_nat @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_967_insert__absorb,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ A5 )
     => ( ( insert2316125241028329079_a_nat @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_968_insert__absorb,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ A @ A5 )
     => ( ( insert210981557334383339on_nat @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_969_insert__absorb,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ A @ A5 )
     => ( ( insert2664702451092716651on_nat @ A @ A5 )
        = A5 ) ) ).

% insert_absorb
thf(fact_970_insert__eq__iff,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ~ ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ~ ( member4231271695026834640_a_nat @ B @ B5 )
       => ( ( ( insert8772053866485626999_a_nat @ A @ A5 )
            = ( insert8772053866485626999_a_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr4991467605864603271_a_nat] :
                  ( ( A5
                    = ( insert8772053866485626999_a_nat @ B @ C5 ) )
                  & ~ ( member4231271695026834640_a_nat @ B @ C5 )
                  & ( B5
                    = ( insert8772053866485626999_a_nat @ A @ C5 ) )
                  & ~ ( member4231271695026834640_a_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_971_insert__eq__iff,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ~ ( member6928026473249828560_a_nat @ A @ A5 )
     => ( ~ ( member6928026473249828560_a_nat @ B @ B5 )
       => ( ( ( insert2316125241028329079_a_nat @ A @ A5 )
            = ( insert2316125241028329079_a_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr8529752790658994311_a_nat] :
                  ( ( A5
                    = ( insert2316125241028329079_a_nat @ B @ C5 ) )
                  & ~ ( member6928026473249828560_a_nat @ B @ C5 )
                  & ( B5
                    = ( insert2316125241028329079_a_nat @ A @ C5 ) )
                  & ~ ( member6928026473249828560_a_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_972_insert__eq__iff,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ~ ( member4278285197320822468on_nat @ A @ A5 )
     => ( ~ ( member4278285197320822468on_nat @ B @ B5 )
       => ( ( ( insert210981557334383339on_nat @ A @ A5 )
            = ( insert210981557334383339on_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr3573155178515677307on_nat] :
                  ( ( A5
                    = ( insert210981557334383339on_nat @ B @ C5 ) )
                  & ~ ( member4278285197320822468on_nat @ B @ C5 )
                  & ( B5
                    = ( insert210981557334383339on_nat @ A @ C5 ) )
                  & ~ ( member4278285197320822468on_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_973_insert__eq__iff,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ~ ( member4221009242412147780on_nat @ A @ A5 )
     => ( ~ ( member4221009242412147780on_nat @ B @ B5 )
       => ( ( ( insert2664702451092716651on_nat @ A @ A5 )
            = ( insert2664702451092716651on_nat @ B @ B5 ) )
          = ( ( ( A = B )
             => ( A5 = B5 ) )
            & ( ( A != B )
             => ? [C5: set_Pr7828957961435280891on_nat] :
                  ( ( A5
                    = ( insert2664702451092716651on_nat @ B @ C5 ) )
                  & ~ ( member4221009242412147780on_nat @ B @ C5 )
                  & ( B5
                    = ( insert2664702451092716651on_nat @ A @ C5 ) )
                  & ~ ( member4221009242412147780on_nat @ A @ C5 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_974_mk__disjoint__insert,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ? [B7: set_Pr4991467605864603271_a_nat] :
          ( ( A5
            = ( insert8772053866485626999_a_nat @ A @ B7 ) )
          & ~ ( member4231271695026834640_a_nat @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_975_mk__disjoint__insert,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ A5 )
     => ? [B7: set_Pr8529752790658994311_a_nat] :
          ( ( A5
            = ( insert2316125241028329079_a_nat @ A @ B7 ) )
          & ~ ( member6928026473249828560_a_nat @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_976_mk__disjoint__insert,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ A @ A5 )
     => ? [B7: set_Pr3573155178515677307on_nat] :
          ( ( A5
            = ( insert210981557334383339on_nat @ A @ B7 ) )
          & ~ ( member4278285197320822468on_nat @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_977_mk__disjoint__insert,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ A @ A5 )
     => ? [B7: set_Pr7828957961435280891on_nat] :
          ( ( A5
            = ( insert2664702451092716651on_nat @ A @ B7 ) )
          & ~ ( member4221009242412147780on_nat @ A @ B7 ) ) ) ).

% mk_disjoint_insert
thf(fact_978_insert__compr,axiom,
    ( insert2316125241028329079_a_nat
    = ( ^ [A3: produc5238312538367656103_a_nat,B6: set_Pr8529752790658994311_a_nat] :
          ( collec6671004892620940562_a_nat
          @ ^ [X2: produc5238312538367656103_a_nat] :
              ( ( X2 = A3 )
              | ( member6928026473249828560_a_nat @ X2 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_979_insert__compr,axiom,
    ( insert210981557334383339on_nat
    = ( ^ [A3: produc9176241080893687067on_nat,B6: set_Pr3573155178515677307on_nat] :
          ( collec1886838643678311430on_nat
          @ ^ [X2: produc9176241080893687067on_nat] :
              ( ( X2 = A3 )
              | ( member4278285197320822468on_nat @ X2 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_980_insert__compr,axiom,
    ( insert8772053866485626999_a_nat
    = ( ^ [A3: produc394670695091157671_a_nat,B6: set_Pr4991467605864603271_a_nat] :
          ( collec8858057866656402194_a_nat
          @ ^ [X2: produc394670695091157671_a_nat] :
              ( ( X2 = A3 )
              | ( member4231271695026834640_a_nat @ X2 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_981_insert__compr,axiom,
    ( insert2097847225457054583_a_nat
    = ( ^ [A3: produc3043957382837978535_a_nat,B6: set_Pr630484882382513799_a_nat] :
          ( collec9088346736735320338_a_nat
          @ ^ [X2: produc3043957382837978535_a_nat] :
              ( ( X2 = A3 )
              | ( member6937586146271652048_a_nat @ X2 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_982_insert__compr,axiom,
    ( insert2664702451092716651on_nat
    = ( ^ [A3: produc5580075136493545115on_nat,B6: set_Pr7828957961435280891on_nat] :
          ( collec1293568243156533638on_nat
          @ ^ [X2: produc5580075136493545115on_nat] :
              ( ( X2 = A3 )
              | ( member4221009242412147780on_nat @ X2 @ B6 ) ) ) ) ) ).

% insert_compr
thf(fact_983_insert__Collect,axiom,
    ! [A: produc394670695091157671_a_nat,P2: produc394670695091157671_a_nat > $o] :
      ( ( insert8772053866485626999_a_nat @ A @ ( collec8858057866656402194_a_nat @ P2 ) )
      = ( collec8858057866656402194_a_nat
        @ ^ [U2: produc394670695091157671_a_nat] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_984_insert__Collect,axiom,
    ! [A: produc3043957382837978535_a_nat,P2: produc3043957382837978535_a_nat > $o] :
      ( ( insert2097847225457054583_a_nat @ A @ ( collec9088346736735320338_a_nat @ P2 ) )
      = ( collec9088346736735320338_a_nat
        @ ^ [U2: produc3043957382837978535_a_nat] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_985_insert__Collect,axiom,
    ! [A: produc5580075136493545115on_nat,P2: produc5580075136493545115on_nat > $o] :
      ( ( insert2664702451092716651on_nat @ A @ ( collec1293568243156533638on_nat @ P2 ) )
      = ( collec1293568243156533638on_nat
        @ ^ [U2: produc5580075136493545115on_nat] :
            ( ( U2 != A )
           => ( P2 @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_986_insert__subsetI,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,X4: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ X3 @ A5 )
     => ( ( ord_le998509814859404839_a_nat @ X4 @ A5 )
       => ( ord_le998509814859404839_a_nat @ ( insert8772053866485626999_a_nat @ X3 @ X4 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_987_insert__subsetI,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,X4: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ X3 @ A5 )
     => ( ( ord_le7248185767425706023_a_nat @ X4 @ A5 )
       => ( ord_le7248185767425706023_a_nat @ ( insert2316125241028329079_a_nat @ X3 @ X4 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_988_insert__subsetI,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,X4: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ X3 @ A5 )
     => ( ( ord_le927452247459997723on_nat @ X4 @ A5 )
       => ( ord_le927452247459997723on_nat @ ( insert210981557334383339on_nat @ X3 @ X4 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_989_insert__subsetI,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,X4: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ X3 @ A5 )
     => ( ( ord_le6423710186836778395on_nat @ X4 @ A5 )
       => ( ord_le6423710186836778395on_nat @ ( insert2664702451092716651on_nat @ X3 @ X4 ) @ A5 ) ) ) ).

% insert_subsetI
thf(fact_990_trivial__rpo_Otrans__S__point,axiom,
    ! [X3: term_R1747997765655105994_a_nat,Y3: term_R1747997765655105994_a_nat,Z: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y3 ) @ bot_bo2698581962261985523_a_nat )
     => ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ Y3 @ Z ) @ bot_bo2698581962261985523_a_nat )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Z ) @ bot_bo2698581962261985523_a_nat ) ) ) ).

% trivial_rpo.trans_S_point
thf(fact_991_Collect__conv__if,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,A: produc394670695091157671_a_nat] :
      ( ( ( P2 @ A )
       => ( ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bo7892652260360037619_a_nat ) ) ) ).

% Collect_conv_if
thf(fact_992_Collect__conv__if,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,A: produc3043957382837978535_a_nat] :
      ( ( ( P2 @ A )
       => ( ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert2097847225457054583_a_nat @ A @ bot_bo2698581962261985523_a_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bo2698581962261985523_a_nat ) ) ) ).

% Collect_conv_if
thf(fact_993_Collect__conv__if,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,A: produc5580075136493545115on_nat] :
      ( ( ( P2 @ A )
       => ( ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( X2 = A )
                & ( P2 @ X2 ) ) )
          = bot_bo3119703073163937383on_nat ) ) ) ).

% Collect_conv_if
thf(fact_994_Collect__conv__if2,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,A: produc394670695091157671_a_nat] :
      ( ( ( P2 @ A )
       => ( ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bo7892652260360037619_a_nat ) ) ) ).

% Collect_conv_if2
thf(fact_995_Collect__conv__if2,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,A: produc3043957382837978535_a_nat] :
      ( ( ( P2 @ A )
       => ( ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert2097847225457054583_a_nat @ A @ bot_bo2698581962261985523_a_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bo2698581962261985523_a_nat ) ) ) ).

% Collect_conv_if2
thf(fact_996_Collect__conv__if2,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,A: produc5580075136493545115on_nat] :
      ( ( ( P2 @ A )
       => ( ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) ) )
      & ( ~ ( P2 @ A )
       => ( ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( A = X2 )
                & ( P2 @ X2 ) ) )
          = bot_bo3119703073163937383on_nat ) ) ) ).

% Collect_conv_if2
thf(fact_997_trivial__rpo_Osubst__S,axiom,
    ! [S: term_R1747997765655105994_a_nat,T2: term_R1747997765655105994_a_nat,Sigma: sum_sum_a_nat > term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ S @ T2 ) @ bot_bo2698581962261985523_a_nat )
     => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( subst_4241131627535844496_a_nat @ S @ Sigma ) @ ( subst_4241131627535844496_a_nat @ T2 @ Sigma ) ) @ bot_bo2698581962261985523_a_nat ) ) ).

% trivial_rpo.subst_S
thf(fact_998_ssubst__Pair__rhs,axiom,
    ! [R2: multis6046679608167549072_a_nat,S: multis6046679608167549072_a_nat,R: set_Pr8529752790658994311_a_nat,S5: multis6046679608167549072_a_nat] :
      ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_999_ssubst__Pair__rhs,axiom,
    ! [R2: multis1856558993800614429on_nat,S: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,S5: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1000_ssubst__Pair__rhs,axiom,
    ! [R2: list_S5253404628753938141on_nat,S: produc1714117091004681831on_nat,R: set_Pr737724919871395061on_nat,S5: produc1714117091004681831on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1001_ssubst__Pair__rhs,axiom,
    ! [R2: list_S5253404628753938141on_nat,S: produc118098540238352959on_nat,R: set_Pr6244131468446592861on_nat,S5: produc118098540238352959on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1002_ssubst__Pair__rhs,axiom,
    ! [R2: list_S5253404628753938141on_nat,S: list_S5253404628753938141on_nat,R: set_Pr7828957961435280891on_nat,S5: list_S5253404628753938141on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1003_ssubst__Pair__rhs,axiom,
    ! [R2: list_t5668669291736003536_a_nat,S: list_t5668669291736003536_a_nat,R: set_Pr4991467605864603271_a_nat,S5: list_t5668669291736003536_a_nat] :
      ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1004_ssubst__Pair__rhs,axiom,
    ! [R2: $o,S: $o,R: set_Product_prod_o_o,S5: $o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1005_ssubst__Pair__rhs,axiom,
    ! [R2: term_R1747997765655105994_a_nat,S: term_R1747997765655105994_a_nat,R: set_Pr630484882382513799_a_nat,S5: term_R1747997765655105994_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ R2 @ S ) @ R )
     => ( ( S5 = S )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ R2 @ S5 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1006_Collect__restrict,axiom,
    ! [X4: set_Pr8529752790658994311_a_nat,P2: produc5238312538367656103_a_nat > $o] :
      ( ord_le7248185767425706023_a_nat
      @ ( collec6671004892620940562_a_nat
        @ ^ [X2: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X2 @ X4 )
            & ( P2 @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_1007_Collect__restrict,axiom,
    ! [X4: set_Pr3573155178515677307on_nat,P2: produc9176241080893687067on_nat > $o] :
      ( ord_le927452247459997723on_nat
      @ ( collec1886838643678311430on_nat
        @ ^ [X2: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X2 @ X4 )
            & ( P2 @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_1008_Collect__restrict,axiom,
    ! [X4: set_Pr4991467605864603271_a_nat,P2: produc394670695091157671_a_nat > $o] :
      ( ord_le998509814859404839_a_nat
      @ ( collec8858057866656402194_a_nat
        @ ^ [X2: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X2 @ X4 )
            & ( P2 @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_1009_Collect__restrict,axiom,
    ! [X4: set_Pr630484882382513799_a_nat,P2: produc3043957382837978535_a_nat > $o] :
      ( ord_le251666825002368551_a_nat
      @ ( collec9088346736735320338_a_nat
        @ ^ [X2: produc3043957382837978535_a_nat] :
            ( ( member6937586146271652048_a_nat @ X2 @ X4 )
            & ( P2 @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_1010_Collect__restrict,axiom,
    ! [X4: set_Pr7828957961435280891on_nat,P2: produc5580075136493545115on_nat > $o] :
      ( ord_le6423710186836778395on_nat
      @ ( collec1293568243156533638on_nat
        @ ^ [X2: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X2 @ X4 )
            & ( P2 @ X2 ) ) )
      @ X4 ) ).

% Collect_restrict
thf(fact_1011_prop__restrict,axiom,
    ! [X3: produc5238312538367656103_a_nat,Z4: set_Pr8529752790658994311_a_nat,X4: set_Pr8529752790658994311_a_nat,P2: produc5238312538367656103_a_nat > $o] :
      ( ( member6928026473249828560_a_nat @ X3 @ Z4 )
     => ( ( ord_le7248185767425706023_a_nat @ Z4
          @ ( collec6671004892620940562_a_nat
            @ ^ [X2: produc5238312538367656103_a_nat] :
                ( ( member6928026473249828560_a_nat @ X2 @ X4 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_1012_prop__restrict,axiom,
    ! [X3: produc9176241080893687067on_nat,Z4: set_Pr3573155178515677307on_nat,X4: set_Pr3573155178515677307on_nat,P2: produc9176241080893687067on_nat > $o] :
      ( ( member4278285197320822468on_nat @ X3 @ Z4 )
     => ( ( ord_le927452247459997723on_nat @ Z4
          @ ( collec1886838643678311430on_nat
            @ ^ [X2: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X2 @ X4 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_1013_prop__restrict,axiom,
    ! [X3: produc394670695091157671_a_nat,Z4: set_Pr4991467605864603271_a_nat,X4: set_Pr4991467605864603271_a_nat,P2: produc394670695091157671_a_nat > $o] :
      ( ( member4231271695026834640_a_nat @ X3 @ Z4 )
     => ( ( ord_le998509814859404839_a_nat @ Z4
          @ ( collec8858057866656402194_a_nat
            @ ^ [X2: produc394670695091157671_a_nat] :
                ( ( member4231271695026834640_a_nat @ X2 @ X4 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_1014_prop__restrict,axiom,
    ! [X3: produc3043957382837978535_a_nat,Z4: set_Pr630484882382513799_a_nat,X4: set_Pr630484882382513799_a_nat,P2: produc3043957382837978535_a_nat > $o] :
      ( ( member6937586146271652048_a_nat @ X3 @ Z4 )
     => ( ( ord_le251666825002368551_a_nat @ Z4
          @ ( collec9088346736735320338_a_nat
            @ ^ [X2: produc3043957382837978535_a_nat] :
                ( ( member6937586146271652048_a_nat @ X2 @ X4 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_1015_prop__restrict,axiom,
    ! [X3: produc5580075136493545115on_nat,Z4: set_Pr7828957961435280891on_nat,X4: set_Pr7828957961435280891on_nat,P2: produc5580075136493545115on_nat > $o] :
      ( ( member4221009242412147780on_nat @ X3 @ Z4 )
     => ( ( ord_le6423710186836778395on_nat @ Z4
          @ ( collec1293568243156533638on_nat
            @ ^ [X2: produc5580075136493545115on_nat] :
                ( ( member4221009242412147780on_nat @ X2 @ X4 )
                & ( P2 @ X2 ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_1016_subterm_OatLeastAtMost__singleton_H,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( A = B )
     => ( ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        = ( insert8400145511831611290_a_nat @ A @ bot_bo6214183520602931734_a_nat ) ) ) ).

% subterm.atLeastAtMost_singleton'
thf(fact_1017_subterm_OgreaterThanAtMost__eq__atLeastAtMost__diff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( set_gr6446970317771492622_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B )
      = ( minus_1985411213817754769_a_nat
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        @ ( insert8400145511831611290_a_nat @ A @ bot_bo6214183520602931734_a_nat ) ) ) ).

% subterm.greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_1018_subterm_OatLeastLessThan__eq__atLeastAtMost__diff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat] :
      ( ( set_at5605147574288828367_a_nat
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
        @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter8519896202621964801_a_nat )
        @ A
        @ B )
      = ( minus_1985411213817754769_a_nat
        @ ( set_at1360634933218525683_a_nat
          @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
          @ A
          @ B )
        @ ( insert8400145511831611290_a_nat @ B @ bot_bo6214183520602931734_a_nat ) ) ) ).

% subterm.atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_1019_mul__ext__list_Os__map,axiom,
    ! [S6: set_Pr8529752790658994311_a_nat,F: multis6046679608167549072_a_nat > multis6046679608167549072_a_nat,NS2: set_Pr8529752790658994311_a_nat,As2: list_m6030409455938750486_a_nat,Bs2: list_m6030409455938750486_a_nat] :
      ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
          ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ S6 )
         => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
            ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ NS2 )
           => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ As2 @ Bs2 )
            @ ( collec521898700636174610_a_nat
              @ ( produc780312926067043308_nat_o
                @ ^ [As: list_m6030409455938750486_a_nat,Bs: list_m6030409455938750486_a_nat] : ( member348394700049801936_a_nat @ ( produc209738390134441623_a_nat @ ( mset_m7884049535971348773_a_nat @ As ) @ ( mset_m7884049535971348773_a_nat @ Bs ) ) @ ( multis5138272992776766887_a_nat @ NS2 @ S6 ) ) ) ) )
         => ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ ( map_mu6434090555345914948_a_nat @ F @ As2 ) @ ( map_mu6434090555345914948_a_nat @ F @ Bs2 ) )
            @ ( collec521898700636174610_a_nat
              @ ( produc780312926067043308_nat_o
                @ ^ [As: list_m6030409455938750486_a_nat,Bs: list_m6030409455938750486_a_nat] : ( member348394700049801936_a_nat @ ( produc209738390134441623_a_nat @ ( mset_m7884049535971348773_a_nat @ As ) @ ( mset_m7884049535971348773_a_nat @ Bs ) ) @ ( multis5138272992776766887_a_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_1020_mul__ext__list_Os__map,axiom,
    ! [S6: set_Pr3573155178515677307on_nat,F: multis1856558993800614429on_nat > multis1856558993800614429on_nat,NS2: set_Pr3573155178515677307on_nat,As2: list_m351779702951725613on_nat,Bs2: list_m351779702951725613on_nat] :
      ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
          ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ S6 )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ NS2 )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ As2 @ Bs2 )
            @ ( collec3327918327021630246on_nat
              @ ( produc5594322704402362000_nat_o
                @ ^ [As: list_m351779702951725613on_nat,Bs: list_m351779702951725613on_nat] : ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( mset_m1293634003157817032on_nat @ As ) @ ( mset_m1293634003157817032on_nat @ Bs ) ) @ ( multis5680658968000484038on_nat @ NS2 @ S6 ) ) ) ) )
         => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( map_mu5965434534149014016on_nat @ F @ As2 ) @ ( map_mu5965434534149014016on_nat @ F @ Bs2 ) )
            @ ( collec3327918327021630246on_nat
              @ ( produc5594322704402362000_nat_o
                @ ^ [As: list_m351779702951725613on_nat,Bs: list_m351779702951725613on_nat] : ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( mset_m1293634003157817032on_nat @ As ) @ ( mset_m1293634003157817032on_nat @ Bs ) ) @ ( multis5680658968000484038on_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_1021_mul__ext__list_Os__map,axiom,
    ! [S6: set_Pr7828957961435280891on_nat,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat,NS2: set_Pr7828957961435280891on_nat,As2: list_l8999537460041914605on_nat,Bs2: list_l8999537460041914605on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
          ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ S6 )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
            ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ NS2 )
           => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As2 @ Bs2 )
            @ ( collec1004640060711064742on_nat
              @ ( produc9222835799945651472_nat_o
                @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] : ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ As ) @ ( mset_l1352240747471131016on_nat @ Bs ) ) @ ( multis2514206365584361862on_nat @ NS2 @ S6 ) ) ) ) )
         => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( map_li4295621887133137792on_nat @ F @ As2 ) @ ( map_li4295621887133137792on_nat @ F @ Bs2 ) )
            @ ( collec1004640060711064742on_nat
              @ ( produc9222835799945651472_nat_o
                @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] : ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ As ) @ ( mset_l1352240747471131016on_nat @ Bs ) ) @ ( multis2514206365584361862on_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_1022_mul__ext__list_Os__map,axiom,
    ! [S6: set_Pr4991467605864603271_a_nat,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat,NS2: set_Pr4991467605864603271_a_nat,As2: list_l3889969465389992278_a_nat,Bs2: list_l3889969465389992278_a_nat] :
      ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
          ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ S6 )
         => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
            ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ NS2 )
           => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As2 @ Bs2 )
            @ ( collec4515946210884203282_a_nat
              @ ( produc2556087688056060396_nat_o
                @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] : ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ As ) @ ( mset_l7063064786757117541_a_nat @ Bs ) ) @ ( multis3722385149336313575_a_nat @ NS2 @ S6 ) ) ) ) )
         => ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ ( map_li7790450734011354692_a_nat @ F @ As2 ) @ ( map_li7790450734011354692_a_nat @ F @ Bs2 ) )
            @ ( collec4515946210884203282_a_nat
              @ ( produc2556087688056060396_nat_o
                @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] : ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ As ) @ ( mset_l7063064786757117541_a_nat @ Bs ) ) @ ( multis3722385149336313575_a_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_1023_mul__ext__list_Os__map,axiom,
    ! [S6: set_Product_prod_o_o,F: $o > $o,NS2: set_Product_prod_o_o,As2: list_o,Bs2: list_o] :
      ( ! [A2: $o,B2: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ S6 )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: $o,B2: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ NS2 )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As2 @ Bs2 )
            @ ( collec4435127810633735570list_o
              @ ( produc1052442579958709612st_o_o
                @ ^ [As: list_o,Bs: list_o] : ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ As ) @ ( mset_o @ Bs ) ) @ ( multiset_s_mul_ext_o @ NS2 @ S6 ) ) ) ) )
         => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( map_o_o @ F @ As2 ) @ ( map_o_o @ F @ Bs2 ) )
            @ ( collec4435127810633735570list_o
              @ ( produc1052442579958709612st_o_o
                @ ^ [As: list_o,Bs: list_o] : ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ As ) @ ( mset_o @ Bs ) ) @ ( multiset_s_mul_ext_o @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_1024_mul__ext__list_Os__map,axiom,
    ! [S6: set_Pr630484882382513799_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat,NS2: set_Pr630484882382513799_a_nat,As2: list_t5668669291736003536_a_nat,Bs2: list_t5668669291736003536_a_nat] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ S6 )
         => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
            ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ NS2 )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As2 @ Bs2 )
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis3666123592011920481_a_nat @ NS2 @ S6 ) ) ) ) )
         => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( map_te6398774066958647364_a_nat @ F @ As2 ) @ ( map_te6398774066958647364_a_nat @ F @ Bs2 ) )
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis3666123592011920481_a_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_1025_mul__ext__list_Os__map,axiom,
    ! [S6: set_Pr2535564044409765851on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS2: set_Pr2535564044409765851on_nat,As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat] :
      ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
          ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ S6 )
         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ NS2 )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs2 )
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis1620213760974180726on_nat @ NS2 @ S6 ) ) ) ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F @ As2 ) @ ( map_Su241112079427911264on_nat @ F @ Bs2 ) )
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis1620213760974180726on_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_1026_Diff__iff,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( minus_343231261423250670_a_nat @ A5 @ B5 ) )
      = ( ( member4231271695026834640_a_nat @ C @ A5 )
        & ~ ( member4231271695026834640_a_nat @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_1027_Diff__iff,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( minus_49980249911231214_a_nat @ A5 @ B5 ) )
      = ( ( member6928026473249828560_a_nat @ C @ A5 )
        & ~ ( member6928026473249828560_a_nat @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_1028_Diff__iff,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( minus_1132796316471476066on_nat @ A5 @ B5 ) )
      = ( ( member4278285197320822468on_nat @ C @ A5 )
        & ~ ( member4278285197320822468on_nat @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_1029_Diff__iff,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( minus_1670933346047824098on_nat @ A5 @ B5 ) )
      = ( ( member4221009242412147780on_nat @ C @ A5 )
        & ~ ( member4221009242412147780on_nat @ C @ B5 ) ) ) ).

% Diff_iff
thf(fact_1030_DiffI,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ A5 )
     => ( ~ ( member4231271695026834640_a_nat @ C @ B5 )
       => ( member4231271695026834640_a_nat @ C @ ( minus_343231261423250670_a_nat @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_1031_DiffI,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ A5 )
     => ( ~ ( member6928026473249828560_a_nat @ C @ B5 )
       => ( member6928026473249828560_a_nat @ C @ ( minus_49980249911231214_a_nat @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_1032_DiffI,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ A5 )
     => ( ~ ( member4278285197320822468on_nat @ C @ B5 )
       => ( member4278285197320822468on_nat @ C @ ( minus_1132796316471476066on_nat @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_1033_DiffI,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ A5 )
     => ( ~ ( member4221009242412147780on_nat @ C @ B5 )
       => ( member4221009242412147780on_nat @ C @ ( minus_1670933346047824098on_nat @ A5 @ B5 ) ) ) ) ).

% DiffI
thf(fact_1034_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ A )
      = zero_zero_nat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1035_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_1036_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_1037_insert__Diff1,axiom,
    ! [X3: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ X3 @ B5 )
     => ( ( minus_343231261423250670_a_nat @ ( insert8772053866485626999_a_nat @ X3 @ A5 ) @ B5 )
        = ( minus_343231261423250670_a_nat @ A5 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_1038_insert__Diff1,axiom,
    ! [X3: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ X3 @ B5 )
     => ( ( minus_49980249911231214_a_nat @ ( insert2316125241028329079_a_nat @ X3 @ A5 ) @ B5 )
        = ( minus_49980249911231214_a_nat @ A5 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_1039_insert__Diff1,axiom,
    ! [X3: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ X3 @ B5 )
     => ( ( minus_1132796316471476066on_nat @ ( insert210981557334383339on_nat @ X3 @ A5 ) @ B5 )
        = ( minus_1132796316471476066on_nat @ A5 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_1040_insert__Diff1,axiom,
    ! [X3: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ X3 @ B5 )
     => ( ( minus_1670933346047824098on_nat @ ( insert2664702451092716651on_nat @ X3 @ A5 ) @ B5 )
        = ( minus_1670933346047824098on_nat @ A5 @ B5 ) ) ) ).

% insert_Diff1
thf(fact_1041_Diff__insert0,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ~ ( member4231271695026834640_a_nat @ X3 @ A5 )
     => ( ( minus_343231261423250670_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ X3 @ B5 ) )
        = ( minus_343231261423250670_a_nat @ A5 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_1042_Diff__insert0,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ~ ( member6928026473249828560_a_nat @ X3 @ A5 )
     => ( ( minus_49980249911231214_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ X3 @ B5 ) )
        = ( minus_49980249911231214_a_nat @ A5 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_1043_Diff__insert0,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X3 @ A5 )
     => ( ( minus_1132796316471476066on_nat @ A5 @ ( insert210981557334383339on_nat @ X3 @ B5 ) )
        = ( minus_1132796316471476066on_nat @ A5 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_1044_Diff__insert0,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ~ ( member4221009242412147780on_nat @ X3 @ A5 )
     => ( ( minus_1670933346047824098on_nat @ A5 @ ( insert2664702451092716651on_nat @ X3 @ B5 ) )
        = ( minus_1670933346047824098on_nat @ A5 @ B5 ) ) ) ).

% Diff_insert0
thf(fact_1045_insert__Diff__if,axiom,
    ! [X3: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( ( member4231271695026834640_a_nat @ X3 @ B5 )
       => ( ( minus_343231261423250670_a_nat @ ( insert8772053866485626999_a_nat @ X3 @ A5 ) @ B5 )
          = ( minus_343231261423250670_a_nat @ A5 @ B5 ) ) )
      & ( ~ ( member4231271695026834640_a_nat @ X3 @ B5 )
       => ( ( minus_343231261423250670_a_nat @ ( insert8772053866485626999_a_nat @ X3 @ A5 ) @ B5 )
          = ( insert8772053866485626999_a_nat @ X3 @ ( minus_343231261423250670_a_nat @ A5 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1046_insert__Diff__if,axiom,
    ! [X3: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( ( member6928026473249828560_a_nat @ X3 @ B5 )
       => ( ( minus_49980249911231214_a_nat @ ( insert2316125241028329079_a_nat @ X3 @ A5 ) @ B5 )
          = ( minus_49980249911231214_a_nat @ A5 @ B5 ) ) )
      & ( ~ ( member6928026473249828560_a_nat @ X3 @ B5 )
       => ( ( minus_49980249911231214_a_nat @ ( insert2316125241028329079_a_nat @ X3 @ A5 ) @ B5 )
          = ( insert2316125241028329079_a_nat @ X3 @ ( minus_49980249911231214_a_nat @ A5 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1047_insert__Diff__if,axiom,
    ! [X3: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( ( member4278285197320822468on_nat @ X3 @ B5 )
       => ( ( minus_1132796316471476066on_nat @ ( insert210981557334383339on_nat @ X3 @ A5 ) @ B5 )
          = ( minus_1132796316471476066on_nat @ A5 @ B5 ) ) )
      & ( ~ ( member4278285197320822468on_nat @ X3 @ B5 )
       => ( ( minus_1132796316471476066on_nat @ ( insert210981557334383339on_nat @ X3 @ A5 ) @ B5 )
          = ( insert210981557334383339on_nat @ X3 @ ( minus_1132796316471476066on_nat @ A5 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1048_insert__Diff__if,axiom,
    ! [X3: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( ( member4221009242412147780on_nat @ X3 @ B5 )
       => ( ( minus_1670933346047824098on_nat @ ( insert2664702451092716651on_nat @ X3 @ A5 ) @ B5 )
          = ( minus_1670933346047824098on_nat @ A5 @ B5 ) ) )
      & ( ~ ( member4221009242412147780on_nat @ X3 @ B5 )
       => ( ( minus_1670933346047824098on_nat @ ( insert2664702451092716651on_nat @ X3 @ A5 ) @ B5 )
          = ( insert2664702451092716651on_nat @ X3 @ ( minus_1670933346047824098on_nat @ A5 @ B5 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_1049_set__diff__eq,axiom,
    ( minus_49980249911231214_a_nat
    = ( ^ [A6: set_Pr8529752790658994311_a_nat,B6: set_Pr8529752790658994311_a_nat] :
          ( collec6671004892620940562_a_nat
          @ ^ [X2: produc5238312538367656103_a_nat] :
              ( ( member6928026473249828560_a_nat @ X2 @ A6 )
              & ~ ( member6928026473249828560_a_nat @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1050_set__diff__eq,axiom,
    ( minus_1132796316471476066on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B6: set_Pr3573155178515677307on_nat] :
          ( collec1886838643678311430on_nat
          @ ^ [X2: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X2 @ A6 )
              & ~ ( member4278285197320822468on_nat @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1051_set__diff__eq,axiom,
    ( minus_343231261423250670_a_nat
    = ( ^ [A6: set_Pr4991467605864603271_a_nat,B6: set_Pr4991467605864603271_a_nat] :
          ( collec8858057866656402194_a_nat
          @ ^ [X2: produc394670695091157671_a_nat] :
              ( ( member4231271695026834640_a_nat @ X2 @ A6 )
              & ~ ( member4231271695026834640_a_nat @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1052_set__diff__eq,axiom,
    ( minus_6926475844497585134_a_nat
    = ( ^ [A6: set_Pr630484882382513799_a_nat,B6: set_Pr630484882382513799_a_nat] :
          ( collec9088346736735320338_a_nat
          @ ^ [X2: produc3043957382837978535_a_nat] :
              ( ( member6937586146271652048_a_nat @ X2 @ A6 )
              & ~ ( member6937586146271652048_a_nat @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1053_set__diff__eq,axiom,
    ( minus_1670933346047824098on_nat
    = ( ^ [A6: set_Pr7828957961435280891on_nat,B6: set_Pr7828957961435280891on_nat] :
          ( collec1293568243156533638on_nat
          @ ^ [X2: produc5580075136493545115on_nat] :
              ( ( member4221009242412147780on_nat @ X2 @ A6 )
              & ~ ( member4221009242412147780on_nat @ X2 @ B6 ) ) ) ) ) ).

% set_diff_eq
thf(fact_1054_cancel__ab__semigroup__add__class_Odiff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% cancel_ab_semigroup_add_class.diff_right_commute
thf(fact_1055_DiffD2,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( minus_343231261423250670_a_nat @ A5 @ B5 ) )
     => ~ ( member4231271695026834640_a_nat @ C @ B5 ) ) ).

% DiffD2
thf(fact_1056_DiffD2,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( minus_49980249911231214_a_nat @ A5 @ B5 ) )
     => ~ ( member6928026473249828560_a_nat @ C @ B5 ) ) ).

% DiffD2
thf(fact_1057_DiffD2,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( minus_1132796316471476066on_nat @ A5 @ B5 ) )
     => ~ ( member4278285197320822468on_nat @ C @ B5 ) ) ).

% DiffD2
thf(fact_1058_DiffD2,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( minus_1670933346047824098on_nat @ A5 @ B5 ) )
     => ~ ( member4221009242412147780on_nat @ C @ B5 ) ) ).

% DiffD2
thf(fact_1059_DiffD1,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( minus_343231261423250670_a_nat @ A5 @ B5 ) )
     => ( member4231271695026834640_a_nat @ C @ A5 ) ) ).

% DiffD1
thf(fact_1060_DiffD1,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( minus_49980249911231214_a_nat @ A5 @ B5 ) )
     => ( member6928026473249828560_a_nat @ C @ A5 ) ) ).

% DiffD1
thf(fact_1061_DiffD1,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( minus_1132796316471476066on_nat @ A5 @ B5 ) )
     => ( member4278285197320822468on_nat @ C @ A5 ) ) ).

% DiffD1
thf(fact_1062_DiffD1,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( minus_1670933346047824098on_nat @ A5 @ B5 ) )
     => ( member4221009242412147780on_nat @ C @ A5 ) ) ).

% DiffD1
thf(fact_1063_DiffE,axiom,
    ! [C: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ C @ ( minus_343231261423250670_a_nat @ A5 @ B5 ) )
     => ~ ( ( member4231271695026834640_a_nat @ C @ A5 )
         => ( member4231271695026834640_a_nat @ C @ B5 ) ) ) ).

% DiffE
thf(fact_1064_DiffE,axiom,
    ! [C: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ C @ ( minus_49980249911231214_a_nat @ A5 @ B5 ) )
     => ~ ( ( member6928026473249828560_a_nat @ C @ A5 )
         => ( member6928026473249828560_a_nat @ C @ B5 ) ) ) ).

% DiffE
thf(fact_1065_DiffE,axiom,
    ! [C: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C @ ( minus_1132796316471476066on_nat @ A5 @ B5 ) )
     => ~ ( ( member4278285197320822468on_nat @ C @ A5 )
         => ( member4278285197320822468on_nat @ C @ B5 ) ) ) ).

% DiffE
thf(fact_1066_DiffE,axiom,
    ! [C: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ C @ ( minus_1670933346047824098on_nat @ A5 @ B5 ) )
     => ~ ( ( member4221009242412147780on_nat @ C @ A5 )
         => ( member4221009242412147780on_nat @ C @ B5 ) ) ) ).

% DiffE
thf(fact_1067_insert__Diff,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( insert8772053866485626999_a_nat @ A @ ( minus_343231261423250670_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ A @ bot_bo7892652260360037619_a_nat ) ) )
        = A5 ) ) ).

% insert_Diff
thf(fact_1068_insert__Diff,axiom,
    ! [A: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ A @ A5 )
     => ( ( insert2316125241028329079_a_nat @ A @ ( minus_49980249911231214_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ A @ bot_bo330137521859356403_a_nat ) ) )
        = A5 ) ) ).

% insert_Diff
thf(fact_1069_insert__Diff,axiom,
    ! [A: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ A @ A5 )
     => ( ( insert210981557334383339on_nat @ A @ ( minus_1132796316471476066on_nat @ A5 @ ( insert210981557334383339on_nat @ A @ bot_bo2515438816433479911on_nat ) ) )
        = A5 ) ) ).

% insert_Diff
thf(fact_1070_insert__Diff,axiom,
    ! [A: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ A @ A5 )
     => ( ( insert2664702451092716651on_nat @ A @ ( minus_1670933346047824098on_nat @ A5 @ ( insert2664702451092716651on_nat @ A @ bot_bo3119703073163937383on_nat ) ) )
        = A5 ) ) ).

% insert_Diff
thf(fact_1071_Diff__insert__absorb,axiom,
    ! [X3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ~ ( member4231271695026834640_a_nat @ X3 @ A5 )
     => ( ( minus_343231261423250670_a_nat @ ( insert8772053866485626999_a_nat @ X3 @ A5 ) @ ( insert8772053866485626999_a_nat @ X3 @ bot_bo7892652260360037619_a_nat ) )
        = A5 ) ) ).

% Diff_insert_absorb
thf(fact_1072_Diff__insert__absorb,axiom,
    ! [X3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ~ ( member6928026473249828560_a_nat @ X3 @ A5 )
     => ( ( minus_49980249911231214_a_nat @ ( insert2316125241028329079_a_nat @ X3 @ A5 ) @ ( insert2316125241028329079_a_nat @ X3 @ bot_bo330137521859356403_a_nat ) )
        = A5 ) ) ).

% Diff_insert_absorb
thf(fact_1073_Diff__insert__absorb,axiom,
    ! [X3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X3 @ A5 )
     => ( ( minus_1132796316471476066on_nat @ ( insert210981557334383339on_nat @ X3 @ A5 ) @ ( insert210981557334383339on_nat @ X3 @ bot_bo2515438816433479911on_nat ) )
        = A5 ) ) ).

% Diff_insert_absorb
thf(fact_1074_Diff__insert__absorb,axiom,
    ! [X3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ~ ( member4221009242412147780on_nat @ X3 @ A5 )
     => ( ( minus_1670933346047824098on_nat @ ( insert2664702451092716651on_nat @ X3 @ A5 ) @ ( insert2664702451092716651on_nat @ X3 @ bot_bo3119703073163937383on_nat ) )
        = A5 ) ) ).

% Diff_insert_absorb
thf(fact_1075_subset__Diff__insert,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,B5: set_Pr4991467605864603271_a_nat,X3: produc394670695091157671_a_nat,C4: set_Pr4991467605864603271_a_nat] :
      ( ( ord_le998509814859404839_a_nat @ A5 @ ( minus_343231261423250670_a_nat @ B5 @ ( insert8772053866485626999_a_nat @ X3 @ C4 ) ) )
      = ( ( ord_le998509814859404839_a_nat @ A5 @ ( minus_343231261423250670_a_nat @ B5 @ C4 ) )
        & ~ ( member4231271695026834640_a_nat @ X3 @ A5 ) ) ) ).

% subset_Diff_insert
thf(fact_1076_subset__Diff__insert,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,B5: set_Pr8529752790658994311_a_nat,X3: produc5238312538367656103_a_nat,C4: set_Pr8529752790658994311_a_nat] :
      ( ( ord_le7248185767425706023_a_nat @ A5 @ ( minus_49980249911231214_a_nat @ B5 @ ( insert2316125241028329079_a_nat @ X3 @ C4 ) ) )
      = ( ( ord_le7248185767425706023_a_nat @ A5 @ ( minus_49980249911231214_a_nat @ B5 @ C4 ) )
        & ~ ( member6928026473249828560_a_nat @ X3 @ A5 ) ) ) ).

% subset_Diff_insert
thf(fact_1077_subset__Diff__insert,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,B5: set_Pr3573155178515677307on_nat,X3: produc9176241080893687067on_nat,C4: set_Pr3573155178515677307on_nat] :
      ( ( ord_le927452247459997723on_nat @ A5 @ ( minus_1132796316471476066on_nat @ B5 @ ( insert210981557334383339on_nat @ X3 @ C4 ) ) )
      = ( ( ord_le927452247459997723on_nat @ A5 @ ( minus_1132796316471476066on_nat @ B5 @ C4 ) )
        & ~ ( member4278285197320822468on_nat @ X3 @ A5 ) ) ) ).

% subset_Diff_insert
thf(fact_1078_subset__Diff__insert,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,B5: set_Pr7828957961435280891on_nat,X3: produc5580075136493545115on_nat,C4: set_Pr7828957961435280891on_nat] :
      ( ( ord_le6423710186836778395on_nat @ A5 @ ( minus_1670933346047824098on_nat @ B5 @ ( insert2664702451092716651on_nat @ X3 @ C4 ) ) )
      = ( ( ord_le6423710186836778395on_nat @ A5 @ ( minus_1670933346047824098on_nat @ B5 @ C4 ) )
        & ~ ( member4221009242412147780on_nat @ X3 @ A5 ) ) ) ).

% subset_Diff_insert
thf(fact_1079_is__singletonI_H,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat] :
      ( ( A5 != bot_bo7892652260360037619_a_nat )
     => ( ! [X: produc394670695091157671_a_nat,Y: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X @ A5 )
           => ( ( member4231271695026834640_a_nat @ Y @ A5 )
             => ( X = Y ) ) )
       => ( is_sin9624638647294363_a_nat @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1080_is__singletonI_H,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat] :
      ( ( A5 != bot_bo330137521859356403_a_nat )
     => ( ! [X: produc5238312538367656103_a_nat,Y: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X @ A5 )
           => ( ( member6928026473249828560_a_nat @ Y @ A5 )
             => ( X = Y ) ) )
       => ( is_sin3578329982680354715_a_nat @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1081_is__singletonI_H,axiom,
    ! [A5: set_Pr3573155178515677307on_nat] :
      ( ( A5 != bot_bo2515438816433479911on_nat )
     => ( ! [X: produc9176241080893687067on_nat,Y: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X @ A5 )
           => ( ( member4278285197320822468on_nat @ Y @ A5 )
             => ( X = Y ) ) )
       => ( is_sin2608175007680305679on_nat @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1082_is__singletonI_H,axiom,
    ! [A5: set_Pr7828957961435280891on_nat] :
      ( ( A5 != bot_bo3119703073163937383on_nat )
     => ( ! [X: produc5580075136493545115on_nat,Y: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X @ A5 )
           => ( ( member4221009242412147780on_nat @ Y @ A5 )
             => ( X = Y ) ) )
       => ( is_sin8499083445026163599on_nat @ A5 ) ) ) ).

% is_singletonI'
thf(fact_1083_subset__insert__iff,axiom,
    ! [A5: set_Pr4991467605864603271_a_nat,X3: produc394670695091157671_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( ord_le998509814859404839_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ X3 @ B5 ) )
      = ( ( ( member4231271695026834640_a_nat @ X3 @ A5 )
         => ( ord_le998509814859404839_a_nat @ ( minus_343231261423250670_a_nat @ A5 @ ( insert8772053866485626999_a_nat @ X3 @ bot_bo7892652260360037619_a_nat ) ) @ B5 ) )
        & ( ~ ( member4231271695026834640_a_nat @ X3 @ A5 )
         => ( ord_le998509814859404839_a_nat @ A5 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_1084_subset__insert__iff,axiom,
    ! [A5: set_Pr8529752790658994311_a_nat,X3: produc5238312538367656103_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( ord_le7248185767425706023_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ X3 @ B5 ) )
      = ( ( ( member6928026473249828560_a_nat @ X3 @ A5 )
         => ( ord_le7248185767425706023_a_nat @ ( minus_49980249911231214_a_nat @ A5 @ ( insert2316125241028329079_a_nat @ X3 @ bot_bo330137521859356403_a_nat ) ) @ B5 ) )
        & ( ~ ( member6928026473249828560_a_nat @ X3 @ A5 )
         => ( ord_le7248185767425706023_a_nat @ A5 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_1085_subset__insert__iff,axiom,
    ! [A5: set_Pr3573155178515677307on_nat,X3: produc9176241080893687067on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( ord_le927452247459997723on_nat @ A5 @ ( insert210981557334383339on_nat @ X3 @ B5 ) )
      = ( ( ( member4278285197320822468on_nat @ X3 @ A5 )
         => ( ord_le927452247459997723on_nat @ ( minus_1132796316471476066on_nat @ A5 @ ( insert210981557334383339on_nat @ X3 @ bot_bo2515438816433479911on_nat ) ) @ B5 ) )
        & ( ~ ( member4278285197320822468on_nat @ X3 @ A5 )
         => ( ord_le927452247459997723on_nat @ A5 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_1086_subset__insert__iff,axiom,
    ! [A5: set_Pr7828957961435280891on_nat,X3: produc5580075136493545115on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( ord_le6423710186836778395on_nat @ A5 @ ( insert2664702451092716651on_nat @ X3 @ B5 ) )
      = ( ( ( member4221009242412147780on_nat @ X3 @ A5 )
         => ( ord_le6423710186836778395on_nat @ ( minus_1670933346047824098on_nat @ A5 @ ( insert2664702451092716651on_nat @ X3 @ bot_bo3119703073163937383on_nat ) ) @ B5 ) )
        & ( ~ ( member4221009242412147780on_nat @ X3 @ A5 )
         => ( ord_le6423710186836778395on_nat @ A5 @ B5 ) ) ) ) ).

% subset_insert_iff
thf(fact_1087_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr8529752790658994311_a_nat > set_Pr8529752790658994311_a_nat > set_Pr4493201157488167559_a_nat,Ns_list: set_Pr8529752790658994311_a_nat > set_Pr8529752790658994311_a_nat > set_Pr4493201157488167559_a_nat,S6: set_Pr8529752790658994311_a_nat,F: multis6046679608167549072_a_nat > multis6046679608167549072_a_nat,NS2: set_Pr8529752790658994311_a_nat,As2: list_m6030409455938750486_a_nat,Bs2: list_m6030409455938750486_a_nat] :
      ( ( list_l6244125296607167708_a_nat @ S_list @ Ns_list )
     => ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
            ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ S6 )
           => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
              ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ NS2 )
             => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ As2 @ Bs2 ) @ ( Ns_list @ S6 @ NS2 ) )
           => ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ ( map_mu6434090555345914948_a_nat @ F @ As2 ) @ ( map_mu6434090555345914948_a_nat @ F @ Bs2 ) ) @ ( Ns_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_1088_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr3573155178515677307on_nat > set_Pr3573155178515677307on_nat > set_Pr5729023540515198363on_nat,Ns_list: set_Pr3573155178515677307on_nat > set_Pr3573155178515677307on_nat > set_Pr5729023540515198363on_nat,S6: set_Pr3573155178515677307on_nat,F: multis1856558993800614429on_nat > multis1856558993800614429on_nat,NS2: set_Pr3573155178515677307on_nat,As2: list_m351779702951725613on_nat,Bs2: list_m351779702951725613on_nat] :
      ( ( list_l2061016739618752721on_nat @ S_list @ Ns_list )
     => ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ S6 )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
              ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ NS2 )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ As2 @ Bs2 ) @ ( Ns_list @ S6 @ NS2 ) )
           => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( map_mu5965434534149014016on_nat @ F @ As2 ) @ ( map_mu5965434534149014016on_nat @ F @ Bs2 ) ) @ ( Ns_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_1089_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,S6: set_Pr2535564044409765851on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS2: set_Pr2535564044409765851on_nat,As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat] :
      ( ( list_l4998409290742832897on_nat @ S_list @ Ns_list )
     => ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ S6 )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
              ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ NS2 )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs2 ) @ ( Ns_list @ S6 @ NS2 ) )
           => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F @ As2 ) @ ( map_Su241112079427911264on_nat @ F @ Bs2 ) ) @ ( Ns_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_1090_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,Ns_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,S6: set_Pr7828957961435280891on_nat,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat,NS2: set_Pr7828957961435280891on_nat,As2: list_l8999537460041914605on_nat,Bs2: list_l8999537460041914605on_nat] :
      ( ( list_l1107520425274447249on_nat @ S_list @ Ns_list )
     => ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
            ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ S6 )
           => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
              ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ NS2 )
             => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As2 @ Bs2 ) @ ( Ns_list @ S6 @ NS2 ) )
           => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( map_li4295621887133137792on_nat @ F @ As2 ) @ ( map_li4295621887133137792on_nat @ F @ Bs2 ) ) @ ( Ns_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_1091_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,Ns_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,S6: set_Pr4991467605864603271_a_nat,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat,NS2: set_Pr4991467605864603271_a_nat,As2: list_l3889969465389992278_a_nat,Bs2: list_l3889969465389992278_a_nat] :
      ( ( list_l3040561122055473180_a_nat @ S_list @ Ns_list )
     => ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
            ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ S6 )
           => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
              ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ NS2 )
             => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As2 @ Bs2 ) @ ( Ns_list @ S6 @ NS2 ) )
           => ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ ( map_li7790450734011354692_a_nat @ F @ As2 ) @ ( map_li7790450734011354692_a_nat @ F @ Bs2 ) ) @ ( Ns_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_1092_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,Ns_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,S6: set_Product_prod_o_o,F: $o > $o,NS2: set_Product_prod_o_o,As2: list_o,Bs2: list_o] :
      ( ( list_l8919389732076785124sion_o @ S_list @ Ns_list )
     => ( ! [A2: $o,B2: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ S6 )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: $o,B2: $o] :
              ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ NS2 )
             => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As2 @ Bs2 ) @ ( Ns_list @ S6 @ NS2 ) )
           => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( map_o_o @ F @ As2 ) @ ( map_o_o @ F @ Bs2 ) ) @ ( Ns_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_1093_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,Ns_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,S6: set_Pr630484882382513799_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat,NS2: set_Pr630484882382513799_a_nat,As2: list_t5668669291736003536_a_nat,Bs2: list_t5668669291736003536_a_nat] :
      ( ( list_l9000741177501154326_a_nat @ S_list @ Ns_list )
     => ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
            ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ S6 )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
              ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ NS2 )
             => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As2 @ Bs2 ) @ ( Ns_list @ S6 @ NS2 ) )
           => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( map_te6398774066958647364_a_nat @ F @ As2 ) @ ( map_te6398774066958647364_a_nat @ F @ Bs2 ) ) @ ( Ns_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_1094_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr8529752790658994311_a_nat > set_Pr8529752790658994311_a_nat > set_Pr4493201157488167559_a_nat,Ns_list: set_Pr8529752790658994311_a_nat > set_Pr8529752790658994311_a_nat > set_Pr4493201157488167559_a_nat,S6: set_Pr8529752790658994311_a_nat,F: multis6046679608167549072_a_nat > multis6046679608167549072_a_nat,NS2: set_Pr8529752790658994311_a_nat,As2: list_m6030409455938750486_a_nat,Bs2: list_m6030409455938750486_a_nat] :
      ( ( list_l6244125296607167708_a_nat @ S_list @ Ns_list )
     => ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
            ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ S6 )
           => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
              ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ NS2 )
             => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ As2 @ Bs2 ) @ ( S_list @ S6 @ NS2 ) )
           => ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ ( map_mu6434090555345914948_a_nat @ F @ As2 ) @ ( map_mu6434090555345914948_a_nat @ F @ Bs2 ) ) @ ( S_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_1095_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr3573155178515677307on_nat > set_Pr3573155178515677307on_nat > set_Pr5729023540515198363on_nat,Ns_list: set_Pr3573155178515677307on_nat > set_Pr3573155178515677307on_nat > set_Pr5729023540515198363on_nat,S6: set_Pr3573155178515677307on_nat,F: multis1856558993800614429on_nat > multis1856558993800614429on_nat,NS2: set_Pr3573155178515677307on_nat,As2: list_m351779702951725613on_nat,Bs2: list_m351779702951725613on_nat] :
      ( ( list_l2061016739618752721on_nat @ S_list @ Ns_list )
     => ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ S6 )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
              ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ NS2 )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ As2 @ Bs2 ) @ ( S_list @ S6 @ NS2 ) )
           => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( map_mu5965434534149014016on_nat @ F @ As2 ) @ ( map_mu5965434534149014016on_nat @ F @ Bs2 ) ) @ ( S_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_1096_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,S6: set_Pr2535564044409765851on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS2: set_Pr2535564044409765851on_nat,As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat] :
      ( ( list_l4998409290742832897on_nat @ S_list @ Ns_list )
     => ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ S6 )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
              ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ NS2 )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs2 ) @ ( S_list @ S6 @ NS2 ) )
           => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F @ As2 ) @ ( map_Su241112079427911264on_nat @ F @ Bs2 ) ) @ ( S_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_1097_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,Ns_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,S6: set_Pr7828957961435280891on_nat,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat,NS2: set_Pr7828957961435280891on_nat,As2: list_l8999537460041914605on_nat,Bs2: list_l8999537460041914605on_nat] :
      ( ( list_l1107520425274447249on_nat @ S_list @ Ns_list )
     => ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
            ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ S6 )
           => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
              ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ NS2 )
             => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As2 @ Bs2 ) @ ( S_list @ S6 @ NS2 ) )
           => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( map_li4295621887133137792on_nat @ F @ As2 ) @ ( map_li4295621887133137792on_nat @ F @ Bs2 ) ) @ ( S_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_1098_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,Ns_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,S6: set_Pr4991467605864603271_a_nat,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat,NS2: set_Pr4991467605864603271_a_nat,As2: list_l3889969465389992278_a_nat,Bs2: list_l3889969465389992278_a_nat] :
      ( ( list_l3040561122055473180_a_nat @ S_list @ Ns_list )
     => ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
            ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ S6 )
           => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
              ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ NS2 )
             => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As2 @ Bs2 ) @ ( S_list @ S6 @ NS2 ) )
           => ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ ( map_li7790450734011354692_a_nat @ F @ As2 ) @ ( map_li7790450734011354692_a_nat @ F @ Bs2 ) ) @ ( S_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_1099_list__order__extension_Os__map,axiom,
    ! [S_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,Ns_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,S6: set_Product_prod_o_o,F: $o > $o,NS2: set_Product_prod_o_o,As2: list_o,Bs2: list_o] :
      ( ( list_l8919389732076785124sion_o @ S_list @ Ns_list )
     => ( ! [A2: $o,B2: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ S6 )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: $o,B2: $o] :
              ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ NS2 )
             => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As2 @ Bs2 ) @ ( S_list @ S6 @ NS2 ) )
           => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( map_o_o @ F @ As2 ) @ ( map_o_o @ F @ Bs2 ) ) @ ( S_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_1100_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,Ns_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,S6: set_Pr630484882382513799_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat,NS2: set_Pr630484882382513799_a_nat,As2: list_t5668669291736003536_a_nat,Bs2: list_t5668669291736003536_a_nat] :
      ( ( list_l9000741177501154326_a_nat @ S_list @ Ns_list )
     => ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
            ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ S6 )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
       => ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
              ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ NS2 )
             => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
         => ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As2 @ Bs2 ) @ ( S_list @ S6 @ NS2 ) )
           => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( map_te6398774066958647364_a_nat @ F @ As2 ) @ ( map_te6398774066958647364_a_nat @ F @ Bs2 ) ) @ ( S_list @ S6 @ NS2 ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_1101_mul__ext__list_Ons__map,axiom,
    ! [S6: set_Pr8529752790658994311_a_nat,F: multis6046679608167549072_a_nat > multis6046679608167549072_a_nat,NS2: set_Pr8529752790658994311_a_nat,As2: list_m6030409455938750486_a_nat,Bs2: list_m6030409455938750486_a_nat] :
      ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
          ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ S6 )
         => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: multis6046679608167549072_a_nat,B2: multis6046679608167549072_a_nat] :
            ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A2 @ B2 ) @ NS2 )
           => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ As2 @ Bs2 )
            @ ( collec521898700636174610_a_nat
              @ ( produc780312926067043308_nat_o
                @ ^ [As: list_m6030409455938750486_a_nat,Bs: list_m6030409455938750486_a_nat] : ( member348394700049801936_a_nat @ ( produc209738390134441623_a_nat @ ( mset_m7884049535971348773_a_nat @ As ) @ ( mset_m7884049535971348773_a_nat @ Bs ) ) @ ( multis3117391973379327429_a_nat @ NS2 @ S6 ) ) ) ) )
         => ( member7738011454448620752_a_nat @ ( produc8967729177179744407_a_nat @ ( map_mu6434090555345914948_a_nat @ F @ As2 ) @ ( map_mu6434090555345914948_a_nat @ F @ Bs2 ) )
            @ ( collec521898700636174610_a_nat
              @ ( produc780312926067043308_nat_o
                @ ^ [As: list_m6030409455938750486_a_nat,Bs: list_m6030409455938750486_a_nat] : ( member348394700049801936_a_nat @ ( produc209738390134441623_a_nat @ ( mset_m7884049535971348773_a_nat @ As ) @ ( mset_m7884049535971348773_a_nat @ Bs ) ) @ ( multis3117391973379327429_a_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_1102_mul__ext__list_Ons__map,axiom,
    ! [S6: set_Pr3573155178515677307on_nat,F: multis1856558993800614429on_nat > multis1856558993800614429on_nat,NS2: set_Pr3573155178515677307on_nat,As2: list_m351779702951725613on_nat,Bs2: list_m351779702951725613on_nat] :
      ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
          ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ S6 )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A2 @ B2 ) @ NS2 )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ As2 @ Bs2 )
            @ ( collec3327918327021630246on_nat
              @ ( produc5594322704402362000_nat_o
                @ ^ [As: list_m351779702951725613on_nat,Bs: list_m351779702951725613on_nat] : ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( mset_m1293634003157817032on_nat @ As ) @ ( mset_m1293634003157817032on_nat @ Bs ) ) @ ( multis3835232632530192936on_nat @ NS2 @ S6 ) ) ) ) )
         => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( map_mu5965434534149014016on_nat @ F @ As2 ) @ ( map_mu5965434534149014016on_nat @ F @ Bs2 ) )
            @ ( collec3327918327021630246on_nat
              @ ( produc5594322704402362000_nat_o
                @ ^ [As: list_m351779702951725613on_nat,Bs: list_m351779702951725613on_nat] : ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( mset_m1293634003157817032on_nat @ As ) @ ( mset_m1293634003157817032on_nat @ Bs ) ) @ ( multis3835232632530192936on_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_1103_mul__ext__list_Ons__map,axiom,
    ! [S6: set_Pr7828957961435280891on_nat,F: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat,NS2: set_Pr7828957961435280891on_nat,As2: list_l8999537460041914605on_nat,Bs2: list_l8999537460041914605on_nat] :
      ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
          ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ S6 )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat] :
            ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A2 @ B2 ) @ NS2 )
           => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As2 @ Bs2 )
            @ ( collec1004640060711064742on_nat
              @ ( produc9222835799945651472_nat_o
                @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] : ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ As ) @ ( mset_l1352240747471131016on_nat @ Bs ) ) @ ( multis1816195516039316712on_nat @ NS2 @ S6 ) ) ) ) )
         => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( map_li4295621887133137792on_nat @ F @ As2 ) @ ( map_li4295621887133137792on_nat @ F @ Bs2 ) )
            @ ( collec1004640060711064742on_nat
              @ ( produc9222835799945651472_nat_o
                @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] : ( member2553511916804391908on_nat @ ( produc5318233365966591219on_nat @ ( mset_l1352240747471131016on_nat @ As ) @ ( mset_l1352240747471131016on_nat @ Bs ) ) @ ( multis1816195516039316712on_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_1104_mul__ext__list_Ons__map,axiom,
    ! [S6: set_Pr4991467605864603271_a_nat,F: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat,NS2: set_Pr4991467605864603271_a_nat,As2: list_l3889969465389992278_a_nat,Bs2: list_l3889969465389992278_a_nat] :
      ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
          ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ S6 )
         => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: list_t5668669291736003536_a_nat,B2: list_t5668669291736003536_a_nat] :
            ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A2 @ B2 ) @ NS2 )
           => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As2 @ Bs2 )
            @ ( collec4515946210884203282_a_nat
              @ ( produc2556087688056060396_nat_o
                @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] : ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ As ) @ ( mset_l7063064786757117541_a_nat @ Bs ) ) @ ( multis6753421577935756037_a_nat @ NS2 @ S6 ) ) ) ) )
         => ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ ( map_li7790450734011354692_a_nat @ F @ As2 ) @ ( map_li7790450734011354692_a_nat @ F @ Bs2 ) )
            @ ( collec4515946210884203282_a_nat
              @ ( produc2556087688056060396_nat_o
                @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] : ( member2831970296344990928_a_nat @ ( produc4995849910639716503_a_nat @ ( mset_l7063064786757117541_a_nat @ As ) @ ( mset_l7063064786757117541_a_nat @ Bs ) ) @ ( multis6753421577935756037_a_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_1105_mul__ext__list_Ons__map,axiom,
    ! [S6: set_Product_prod_o_o,F: $o > $o,NS2: set_Product_prod_o_o,As2: list_o,Bs2: list_o] :
      ( ! [A2: $o,B2: $o] :
          ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ S6 )
         => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: $o,B2: $o] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A2 @ B2 ) @ NS2 )
           => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As2 @ Bs2 )
            @ ( collec4435127810633735570list_o
              @ ( produc1052442579958709612st_o_o
                @ ^ [As: list_o,Bs: list_o] : ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ As ) @ ( mset_o @ Bs ) ) @ ( multis1436808229551206221_ext_o @ NS2 @ S6 ) ) ) ) )
         => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ ( map_o_o @ F @ As2 ) @ ( map_o_o @ F @ Bs2 ) )
            @ ( collec4435127810633735570list_o
              @ ( produc1052442579958709612st_o_o
                @ ^ [As: list_o,Bs: list_o] : ( member6734461554573772112iset_o @ ( produc4035283259145089303iset_o @ ( mset_o @ As ) @ ( mset_o @ Bs ) ) @ ( multis1436808229551206221_ext_o @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_1106_mul__ext__list_Ons__map,axiom,
    ! [S6: set_Pr630484882382513799_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat,NS2: set_Pr630484882382513799_a_nat,As2: list_t5668669291736003536_a_nat,Bs2: list_t5668669291736003536_a_nat] :
      ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
          ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ S6 )
         => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: term_R1747997765655105994_a_nat,B2: term_R1747997765655105994_a_nat] :
            ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A2 @ B2 ) @ NS2 )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As2 @ Bs2 )
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis6945725571414890367_a_nat @ NS2 @ S6 ) ) ) ) )
         => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ ( map_te6398774066958647364_a_nat @ F @ As2 ) @ ( map_te6398774066958647364_a_nat @ F @ Bs2 ) )
            @ ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] : ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) ) @ ( multis6945725571414890367_a_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_1107_mul__ext__list_Ons__map,axiom,
    ! [S6: set_Pr2535564044409765851on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS2: set_Pr2535564044409765851on_nat,As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat] :
      ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
          ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ S6 )
         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ S6 ) )
     => ( ! [A2: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A2 @ B2 ) @ NS2 )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A2 ) @ ( F @ B2 ) ) @ NS2 ) )
       => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs2 )
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis4183089776494585304on_nat @ NS2 @ S6 ) ) ) ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F @ As2 ) @ ( map_Su241112079427911264on_nat @ F @ Bs2 ) )
            @ ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis4183089776494585304on_nat @ NS2 @ S6 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_1108_map__is__Nil__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
      ( ( ( map_li880244354692327940od_a_o @ F @ Xs2 )
        = nil_li1372968520647675725od_a_o )
      = ( Xs2 = nil_li1372968520647675725od_a_o ) ) ).

% map_is_Nil_conv
thf(fact_1109_map__is__Nil__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > term_R1747997765655105994_a_nat,Xs2: list_S5253404628753938141on_nat] :
      ( ( ( map_Su925086260876352759_a_nat @ F @ Xs2 )
        = nil_te8332778836088660176_a_nat )
      = ( Xs2 = nil_Su3078906344068544711on_nat ) ) ).

% map_is_Nil_conv
thf(fact_1110_Nil__is__map__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
      ( ( nil_li1372968520647675725od_a_o
        = ( map_li880244354692327940od_a_o @ F @ Xs2 ) )
      = ( Xs2 = nil_li1372968520647675725od_a_o ) ) ).

% Nil_is_map_conv
thf(fact_1111_Nil__is__map__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > term_R1747997765655105994_a_nat,Xs2: list_S5253404628753938141on_nat] :
      ( ( nil_te8332778836088660176_a_nat
        = ( map_Su925086260876352759_a_nat @ F @ Xs2 ) )
      = ( Xs2 = nil_Su3078906344068544711on_nat ) ) ).

% Nil_is_map_conv
thf(fact_1112_list_Omap__disc__iff,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,A: list_l8336264072933094989od_a_o] :
      ( ( ( map_li880244354692327940od_a_o @ F @ A )
        = nil_li1372968520647675725od_a_o )
      = ( A = nil_li1372968520647675725od_a_o ) ) ).

% list.map_disc_iff
thf(fact_1113_list_Omap__disc__iff,axiom,
    ! [F: sum_su4907927656612468301on_nat > term_R1747997765655105994_a_nat,A: list_S5253404628753938141on_nat] :
      ( ( ( map_Su925086260876352759_a_nat @ F @ A )
        = nil_te8332778836088660176_a_nat )
      = ( A = nil_Su3078906344068544711on_nat ) ) ).

% list.map_disc_iff
thf(fact_1114_diff__self__eq__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ M2 )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_1115_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_1116_diff__diff__cancel,axiom,
    ! [I: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I ) )
        = I ) ) ).

% diff_diff_cancel
thf(fact_1117_diff__is__0__eq,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% diff_is_0_eq
thf(fact_1118_diff__is__0__eq_H,axiom,
    ! [M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ( minus_minus_nat @ M2 @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_1119_minus__set__def,axiom,
    ( minus_49980249911231214_a_nat
    = ( ^ [A6: set_Pr8529752790658994311_a_nat,B6: set_Pr8529752790658994311_a_nat] :
          ( collec6671004892620940562_a_nat
          @ ( minus_9132118258730148719_nat_o
            @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ A6 )
            @ ^ [X2: produc5238312538367656103_a_nat] : ( member6928026473249828560_a_nat @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1120_minus__set__def,axiom,
    ( minus_1132796316471476066on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B6: set_Pr3573155178515677307on_nat] :
          ( collec1886838643678311430on_nat
          @ ( minus_3053068724314874363_nat_o
            @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ A6 )
            @ ^ [X2: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1121_minus__set__def,axiom,
    ( minus_343231261423250670_a_nat
    = ( ^ [A6: set_Pr4991467605864603271_a_nat,B6: set_Pr4991467605864603271_a_nat] :
          ( collec8858057866656402194_a_nat
          @ ( minus_5200695888155945327_nat_o
            @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ A6 )
            @ ^ [X2: produc394670695091157671_a_nat] : ( member4231271695026834640_a_nat @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1122_minus__set__def,axiom,
    ( minus_6926475844497585134_a_nat
    = ( ^ [A6: set_Pr630484882382513799_a_nat,B6: set_Pr630484882382513799_a_nat] :
          ( collec9088346736735320338_a_nat
          @ ( minus_1073715461003603567_nat_o
            @ ^ [X2: produc3043957382837978535_a_nat] : ( member6937586146271652048_a_nat @ X2 @ A6 )
            @ ^ [X2: produc3043957382837978535_a_nat] : ( member6937586146271652048_a_nat @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1123_minus__set__def,axiom,
    ( minus_1670933346047824098on_nat
    = ( ^ [A6: set_Pr7828957961435280891on_nat,B6: set_Pr7828957961435280891on_nat] :
          ( collec1293568243156533638on_nat
          @ ( minus_1284645405677936763_nat_o
            @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ A6 )
            @ ^ [X2: produc5580075136493545115on_nat] : ( member4221009242412147780on_nat @ X2 @ B6 ) ) ) ) ) ).

% minus_set_def
thf(fact_1124_diff__le__mono2,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M2 ) ) ) ).

% diff_le_mono2
thf(fact_1125_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_1126_diff__le__self,axiom,
    ! [M2: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ N2 ) @ M2 ) ).

% diff_le_self
thf(fact_1127_diff__le__mono,axiom,
    ! [M2: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M2 @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_1128_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M2 @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_1129_le__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M2 @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M2 @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_1130_eq__diff__iff,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M2 )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M2 @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M2 = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_1131_minus__nat_Odiff__0,axiom,
    ! [M2: nat] :
      ( ( minus_minus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% minus_nat.diff_0
thf(fact_1132_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_1133_list_Osimps_I8_J,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o] :
      ( ( map_li880244354692327940od_a_o @ F @ nil_li1372968520647675725od_a_o )
      = nil_li1372968520647675725od_a_o ) ).

% list.simps(8)
thf(fact_1134_list_Osimps_I8_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > term_R1747997765655105994_a_nat] :
      ( ( map_Su925086260876352759_a_nat @ F @ nil_Su3078906344068544711on_nat )
      = nil_te8332778836088660176_a_nat ) ).

% list.simps(8)
thf(fact_1135_pair__list__eqI,axiom,
    ! [Xs2: list_P4002435161011370285od_o_o,Ys2: list_P4002435161011370285od_o_o] :
      ( ( ( map_Pr7541730621154948341_o_o_o @ product_fst_o_o @ Xs2 )
        = ( map_Pr7541730621154948341_o_o_o @ product_fst_o_o @ Ys2 ) )
     => ( ( ( map_Pr7541730621154948341_o_o_o @ product_snd_o_o @ Xs2 )
          = ( map_Pr7541730621154948341_o_o_o @ product_snd_o_o @ Ys2 ) )
       => ( Xs2 = Ys2 ) ) ) ).

% pair_list_eqI
thf(fact_1136_pair__list__eqI,axiom,
    ! [Xs2: list_P5550462785134625709_a_nat,Ys2: list_P5550462785134625709_a_nat] :
      ( ( ( map_Pr5474893768019284525_a_nat @ produc3471587812718774147_a_nat @ Xs2 )
        = ( map_Pr5474893768019284525_a_nat @ produc3471587812718774147_a_nat @ Ys2 ) )
     => ( ( ( map_Pr5474893768019284525_a_nat @ produc4727165237793840837_a_nat @ Xs2 )
          = ( map_Pr5474893768019284525_a_nat @ produc4727165237793840837_a_nat @ Ys2 ) )
       => ( Xs2 = Ys2 ) ) ) ).

% pair_list_eqI
thf(fact_1137_pair__list__eqI,axiom,
    ! [Xs2: list_P2317284437984938017on_nat,Ys2: list_P2317284437984938017on_nat] :
      ( ( ( map_Pr8853982795285048372on_nat @ produc1822290298201838911on_nat @ Xs2 )
        = ( map_Pr8853982795285048372on_nat @ produc1822290298201838911on_nat @ Ys2 ) )
     => ( ( ( map_Pr8853982795285048372on_nat @ produc2374478376588790657on_nat @ Xs2 )
          = ( map_Pr8853982795285048372on_nat @ produc2374478376588790657on_nat @ Ys2 ) )
       => ( Xs2 = Ys2 ) ) ) ).

% pair_list_eqI
thf(fact_1138_pair__list__eqI,axiom,
    ! [Xs2: list_P6215481986061871491on_nat,Ys2: list_P6215481986061871491on_nat] :
      ( ( ( map_Pr7135454201604834710on_nat @ produc311592285738829985on_nat @ Xs2 )
        = ( map_Pr7135454201604834710on_nat @ produc311592285738829985on_nat @ Ys2 ) )
     => ( ( ( map_Pr6770859801276792056on_nat @ produc3934153714877691619on_nat @ Xs2 )
          = ( map_Pr6770859801276792056on_nat @ produc3934153714877691619on_nat @ Ys2 ) )
       => ( Xs2 = Ys2 ) ) ) ).

% pair_list_eqI
thf(fact_1139_pair__list__eqI,axiom,
    ! [Xs2: list_P5508476027986415439on_nat,Ys2: list_P5508476027986415439on_nat] :
      ( ( ( map_Pr473085249075997154on_nat @ produc6765150182969198981on_nat @ Xs2 )
        = ( map_Pr473085249075997154on_nat @ produc6765150182969198981on_nat @ Ys2 ) )
     => ( ( ( map_Pr5908406365050423074on_nat @ produc5312734412660135107on_nat @ Xs2 )
          = ( map_Pr5908406365050423074on_nat @ produc5312734412660135107on_nat @ Ys2 ) )
       => ( Xs2 = Ys2 ) ) ) ).

% pair_list_eqI
thf(fact_1140_term__lists__of__cnf__def,axiom,
    ( rPO_NP4415194297383320304_cnf_a
    = ( ^ [Cnf: list_l8336264072933094989od_a_o] :
          ( produc7585491754245536861_a_nat
          @ ^ [As: list_S5253404628753938141on_nat] :
              ( produc845213924535103881_a_nat
              @ ^ [Bs: list_S5253404628753938141on_nat] :
                  ( produc4711161665551991227_a_nat
                  @ ^ [S9: list_P9127870862658512897on_nat,NS3: list_P9127870862658512897on_nat] : ( produc5685097156399411607_a_nat @ ( map_Su925086260876352759_a_nat @ ( rPO_NP8124319294247464911term_a @ Cnf ) @ As ) @ ( map_Su925086260876352759_a_nat @ ( rPO_NP8124319294247464911term_a @ Cnf ) @ Bs ) ) ) )
          @ ( multis4221440410396274297_cnf_a @ Cnf ) ) ) ) ).

% term_lists_of_cnf_def
thf(fact_1141_abs_I2_J,axiom,
    ( bs
    = ( map_Su925086260876352759_a_nat @ ( rPO_NP8124319294247464911term_a @ cnf ) @ ys ) ) ).

% abs(2)
thf(fact_1142_abs_I1_J,axiom,
    ( as
    = ( map_Su925086260876352759_a_nat @ ( rPO_NP8124319294247464911term_a @ cnf ) @ xs ) ) ).

% abs(1)
thf(fact_1143__092_060open_062_Imap_A_Ims__elem__to__term_Acnf_J_Axs_M_Amap_A_Ims__elem__to__term_Acnf_J_Ays_J_A_061_A_Ias_M_Abs_J_092_060close_062,axiom,
    ( ( produc5685097156399411607_a_nat @ ( map_Su925086260876352759_a_nat @ ( rPO_NP8124319294247464911term_a @ cnf ) @ xs ) @ ( map_Su925086260876352759_a_nat @ ( rPO_NP8124319294247464911term_a @ cnf ) @ ys ) )
    = ( produc5685097156399411607_a_nat @ as @ bs ) ) ).

% \<open>(map (ms_elem_to_term cnf) xs, map (ms_elem_to_term cnf) ys) = (as, bs)\<close>
thf(fact_1144_diff__commute,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I @ K ) @ J ) ) ).

% diff_commute
thf(fact_1145_local_Omp,axiom,
    ( ( multis4221440410396274297_cnf_a @ cnf )
    = ( produc1545487600272464053on_nat @ xs @ ( produc4176415733022414961on_nat @ ys @ ( produc7152764266822961111on_nat @ s @ ns ) ) ) ) ).

% local.mp
thf(fact_1146_subterm_Omax__def,axiom,
    ! [B: term_R1747997765655105994_a_nat,A: term_R1747997765655105994_a_nat] :
      ( ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
       => ( ( max_te7465207290039160035_a_nat
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
            @ A
            @ B )
          = B ) )
      & ( ~ ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ B @ A ) @ subter3376206727240021173_a_nat )
       => ( ( max_te7465207290039160035_a_nat
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
            @ A
            @ B )
          = A ) ) ) ).

% subterm.max_def
thf(fact_1147_member__remove,axiom,
    ! [X3: produc394670695091157671_a_nat,Y3: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ X3 @ ( remove3877709235828608140_a_nat @ Y3 @ A5 ) )
      = ( ( member4231271695026834640_a_nat @ X3 @ A5 )
        & ( X3 != Y3 ) ) ) ).

% member_remove
thf(fact_1148_member__remove,axiom,
    ! [X3: produc5238312538367656103_a_nat,Y3: produc5238312538367656103_a_nat,A5: set_Pr8529752790658994311_a_nat] :
      ( ( member6928026473249828560_a_nat @ X3 @ ( remove120371333147491980_a_nat @ Y3 @ A5 ) )
      = ( ( member6928026473249828560_a_nat @ X3 @ A5 )
        & ( X3 != Y3 ) ) ) ).

% member_remove
thf(fact_1149_member__remove,axiom,
    ! [X3: produc9176241080893687067on_nat,Y3: produc9176241080893687067on_nat,A5: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ X3 @ ( remove1220511266357727360on_nat @ Y3 @ A5 ) )
      = ( ( member4278285197320822468on_nat @ X3 @ A5 )
        & ( X3 != Y3 ) ) ) ).

% member_remove
thf(fact_1150_member__remove,axiom,
    ! [X3: produc5580075136493545115on_nat,Y3: produc5580075136493545115on_nat,A5: set_Pr7828957961435280891on_nat] :
      ( ( member4221009242412147780on_nat @ X3 @ ( remove3904578265542657536on_nat @ Y3 @ A5 ) )
      = ( ( member4221009242412147780on_nat @ X3 @ A5 )
        & ( X3 != Y3 ) ) ) ).

% member_remove
thf(fact_1151__092_060open_062_Imset_Axs_M_Amset_Ays_J_A_092_060in_062_As__mul__ext_A_Iset_ANS_J_A_Iset_AS_J_092_060close_062,axiom,
    member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ xs ) @ ( mset_S7907735292211298936on_nat @ ys ) ) @ ( multis1620213760974180726on_nat @ ( set_Pr937530902923597456on_nat @ ns ) @ ( set_Pr937530902923597456on_nat @ s ) ) ).

% \<open>(mset xs, mset ys) \<in> s_mul_ext (set NS) (set S)\<close>
thf(fact_1152_mem,axiom,
    ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ xs @ ys )
    @ ( collec1293568243156533638on_nat
      @ ( produc7163990014243126832_nat_o
        @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis1620213760974180726on_nat @ ( set_Pr937530902923597456on_nat @ ns ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ) ) ).

% mem
thf(fact_1153__092_060open_062_I_092_060exists_062_092_060beta_062_O_Aeval__cnf_A_092_060beta_062_Acnf_J_A_061_A_I_Imset_Axs_M_Amset_Ays_J_A_092_060in_062_As__mul__ext_A_Iset_ANS_J_A_Iset_AS_J_J_092_060close_062,axiom,
    ( ( ? [Beta: a > $o] : ( propos4091448439431948687_cnf_a @ Beta @ cnf ) )
    = ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ xs ) @ ( mset_S7907735292211298936on_nat @ ys ) ) @ ( multis1620213760974180726on_nat @ ( set_Pr937530902923597456on_nat @ ns ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ).

% \<open>(\<exists>\<beta>. eval_cnf \<beta> cnf) = ((mset xs, mset ys) \<in> s_mul_ext (set NS) (set S))\<close>
thf(fact_1154_subterm_OGreatest__equality,axiom,
    ! [P2: term_R1747997765655105994_a_nat > $o,X3: term_R1747997765655105994_a_nat] :
      ( ( P2 @ X3 )
     => ( ! [Y: term_R1747997765655105994_a_nat] :
            ( ( P2 @ Y )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y ) @ subter3376206727240021173_a_nat ) )
       => ( ( greate439585953166210053_a_nat
            @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
            @ P2 )
          = X3 ) ) ) ).

% subterm.Greatest_equality
thf(fact_1155_set__empty,axiom,
    ! [Xs2: list_l8336264072933094989od_a_o] :
      ( ( ( set_li2629784761229999580od_a_o @ Xs2 )
        = bot_bo53931977818011539od_a_o )
      = ( Xs2 = nil_li1372968520647675725od_a_o ) ) ).

% set_empty
thf(fact_1156_set__empty,axiom,
    ! [Xs2: list_P9127870862658512897on_nat] :
      ( ( ( set_Pr937530902923597456on_nat @ Xs2 )
        = bot_bo1833791518489435207on_nat )
      = ( Xs2 = nil_Pr8614093721167239297on_nat ) ) ).

% set_empty
thf(fact_1157_set__empty2,axiom,
    ! [Xs2: list_l8336264072933094989od_a_o] :
      ( ( bot_bo53931977818011539od_a_o
        = ( set_li2629784761229999580od_a_o @ Xs2 ) )
      = ( Xs2 = nil_li1372968520647675725od_a_o ) ) ).

% set_empty2
thf(fact_1158_set__empty2,axiom,
    ! [Xs2: list_P9127870862658512897on_nat] :
      ( ( bot_bo1833791518489435207on_nat
        = ( set_Pr937530902923597456on_nat @ Xs2 ) )
      = ( Xs2 = nil_Pr8614093721167239297on_nat ) ) ).

% set_empty2
thf(fact_1159_subset__code_I1_J,axiom,
    ! [Xs2: list_P5550462785134625709_a_nat,B5: set_Pr4991467605864603271_a_nat] :
      ( ( ord_le998509814859404839_a_nat @ ( set_Pr7412993203964320572_a_nat @ Xs2 ) @ B5 )
      = ( ! [X2: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X2 @ ( set_Pr7412993203964320572_a_nat @ Xs2 ) )
           => ( member4231271695026834640_a_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1160_subset__code_I1_J,axiom,
    ! [Xs2: list_P5842215428982857133_a_nat,B5: set_Pr8529752790658994311_a_nat] :
      ( ( ord_le7248185767425706023_a_nat @ ( set_Pr7117250867156192572_a_nat @ Xs2 ) @ B5 )
      = ( ! [X2: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X2 @ ( set_Pr7117250867156192572_a_nat @ Xs2 ) )
           => ( member6928026473249828560_a_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1161_subset__code_I1_J,axiom,
    ! [Xs2: list_P1915136757390359201on_nat,B5: set_Pr3573155178515677307on_nat] :
      ( ( ord_le927452247459997723on_nat @ ( set_Pr8272476097931642672on_nat @ Xs2 ) @ B5 )
      = ( ! [X2: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X2 @ ( set_Pr8272476097931642672on_nat @ Xs2 ) )
           => ( member4278285197320822468on_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1162_subset__code_I1_J,axiom,
    ! [Xs2: list_P2317284437984938017on_nat,B5: set_Pr7828957961435280891on_nat] :
      ( ( ord_le6423710186836778395on_nat @ ( set_Pr4166446493430381744on_nat @ Xs2 ) @ B5 )
      = ( ! [X2: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X2 @ ( set_Pr4166446493430381744on_nat @ Xs2 ) )
           => ( member4221009242412147780on_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1163_subset__code_I1_J,axiom,
    ! [Xs2: list_P9127870862658512897on_nat,B5: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ Xs2 ) @ B5 )
      = ( ! [X2: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
           => ( member5906418179908956708on_nat @ X2 @ B5 ) ) ) ) ).

% subset_code(1)
thf(fact_1164_mset__eq__setD,axiom,
    ! [Xs2: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( ( mset_P6964650582947932560on_nat @ Xs2 )
        = ( mset_P6964650582947932560on_nat @ Ys2 ) )
     => ( ( set_Pr937530902923597456on_nat @ Xs2 )
        = ( set_Pr937530902923597456on_nat @ Ys2 ) ) ) ).

% mset_eq_setD
thf(fact_1165_mset__eq__setD,axiom,
    ! [Xs2: list_t5668669291736003536_a_nat,Ys2: list_t5668669291736003536_a_nat] :
      ( ( ( mset_t7141044528732312287_a_nat @ Xs2 )
        = ( mset_t7141044528732312287_a_nat @ Ys2 ) )
     => ( ( set_te3761484132649746399_a_nat @ Xs2 )
        = ( set_te3761484132649746399_a_nat @ Ys2 ) ) ) ).

% mset_eq_setD
thf(fact_1166_mset__eq__setD,axiom,
    ! [Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs2 )
        = ( mset_S7907735292211298936on_nat @ Ys2 ) )
     => ( ( set_Su4848624856859743608on_nat @ Xs2 )
        = ( set_Su4848624856859743608on_nat @ Ys2 ) ) ) ).

% mset_eq_setD
thf(fact_1167_empty__set,axiom,
    ( bot_bo53931977818011539od_a_o
    = ( set_li2629784761229999580od_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% empty_set
thf(fact_1168_empty__set,axiom,
    ( bot_bo1833791518489435207on_nat
    = ( set_Pr937530902923597456on_nat @ nil_Pr8614093721167239297on_nat ) ) ).

% empty_set
thf(fact_1169_mul__ext__cong,axiom,
    ! [Xs1: list_P5550462785134625709_a_nat,Ys1: list_P5550462785134625709_a_nat,Xs22: list_P5550462785134625709_a_nat,Ys22: list_P5550462785134625709_a_nat,F: produc394670695091157671_a_nat > produc394670695091157671_a_nat > product_prod_o_o,G: produc394670695091157671_a_nat > produc394670695091157671_a_nat > product_prod_o_o] :
      ( ( ( mset_P2640931962506309180_a_nat @ Xs1 )
        = ( mset_P2640931962506309180_a_nat @ Ys1 ) )
     => ( ( ( mset_P2640931962506309180_a_nat @ Xs22 )
          = ( mset_P2640931962506309180_a_nat @ Ys22 ) )
       => ( ! [X: produc394670695091157671_a_nat,X6: produc394670695091157671_a_nat] :
              ( ( member4231271695026834640_a_nat @ X @ ( set_Pr7412993203964320572_a_nat @ Ys1 ) )
             => ( ( member4231271695026834640_a_nat @ X6 @ ( set_Pr7412993203964320572_a_nat @ Ys22 ) )
               => ( ( F @ X @ X6 )
                  = ( G @ X @ X6 ) ) ) )
         => ( ( multis7464683662112179553_a_nat @ F @ Xs1 @ Xs22 )
            = ( multis7464683662112179553_a_nat @ G @ Ys1 @ Ys22 ) ) ) ) ) ).

% mul_ext_cong
thf(fact_1170_mul__ext__cong,axiom,
    ! [Xs1: list_P5842215428982857133_a_nat,Ys1: list_P5842215428982857133_a_nat,Xs22: list_P5842215428982857133_a_nat,Ys22: list_P5842215428982857133_a_nat,F: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > product_prod_o_o,G: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > product_prod_o_o] :
      ( ( ( mset_P3117112961179574332_a_nat @ Xs1 )
        = ( mset_P3117112961179574332_a_nat @ Ys1 ) )
     => ( ( ( mset_P3117112961179574332_a_nat @ Xs22 )
          = ( mset_P3117112961179574332_a_nat @ Ys22 ) )
       => ( ! [X: produc5238312538367656103_a_nat,X6: produc5238312538367656103_a_nat] :
              ( ( member6928026473249828560_a_nat @ X @ ( set_Pr7117250867156192572_a_nat @ Ys1 ) )
             => ( ( member6928026473249828560_a_nat @ X6 @ ( set_Pr7117250867156192572_a_nat @ Ys22 ) )
               => ( ( F @ X @ X6 )
                  = ( G @ X @ X6 ) ) ) )
         => ( ( multis2430777439941206881_a_nat @ F @ Xs1 @ Xs22 )
            = ( multis2430777439941206881_a_nat @ G @ Ys1 @ Ys22 ) ) ) ) ) ).

% mul_ext_cong
thf(fact_1171_mul__ext__cong,axiom,
    ! [Xs1: list_P1915136757390359201on_nat,Ys1: list_P1915136757390359201on_nat,Xs22: list_P1915136757390359201on_nat,Ys22: list_P1915136757390359201on_nat,F: produc9176241080893687067on_nat > produc9176241080893687067on_nat > product_prod_o_o,G: produc9176241080893687067on_nat > produc9176241080893687067on_nat > product_prod_o_o] :
      ( ( ( mset_P1678654427565054512on_nat @ Xs1 )
        = ( mset_P1678654427565054512on_nat @ Ys1 ) )
     => ( ( ( mset_P1678654427565054512on_nat @ Xs22 )
          = ( mset_P1678654427565054512on_nat @ Ys22 ) )
       => ( ! [X: produc9176241080893687067on_nat,X6: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X @ ( set_Pr8272476097931642672on_nat @ Ys1 ) )
             => ( ( member4278285197320822468on_nat @ X6 @ ( set_Pr8272476097931642672on_nat @ Ys22 ) )
               => ( ( F @ X @ X6 )
                  = ( G @ X @ X6 ) ) ) )
         => ( ( multis7516503858109221589on_nat @ F @ Xs1 @ Xs22 )
            = ( multis7516503858109221589on_nat @ G @ Ys1 @ Ys22 ) ) ) ) ) ).

% mul_ext_cong
thf(fact_1172_mul__ext__cong,axiom,
    ! [Xs1: list_P2317284437984938017on_nat,Ys1: list_P2317284437984938017on_nat,Xs22: list_P2317284437984938017on_nat,Ys22: list_P2317284437984938017on_nat,F: produc5580075136493545115on_nat > produc5580075136493545115on_nat > product_prod_o_o,G: produc5580075136493545115on_nat > produc5580075136493545115on_nat > product_prod_o_o] :
      ( ( ( mset_P2373404333907234736on_nat @ Xs1 )
        = ( mset_P2373404333907234736on_nat @ Ys1 ) )
     => ( ( ( mset_P2373404333907234736on_nat @ Xs22 )
          = ( mset_P2373404333907234736on_nat @ Ys22 ) )
       => ( ! [X: produc5580075136493545115on_nat,X6: produc5580075136493545115on_nat] :
              ( ( member4221009242412147780on_nat @ X @ ( set_Pr4166446493430381744on_nat @ Ys1 ) )
             => ( ( member4221009242412147780on_nat @ X6 @ ( set_Pr4166446493430381744on_nat @ Ys22 ) )
               => ( ( F @ X @ X6 )
                  = ( G @ X @ X6 ) ) ) )
         => ( ( multis4887289194134901845on_nat @ F @ Xs1 @ Xs22 )
            = ( multis4887289194134901845on_nat @ G @ Ys1 @ Ys22 ) ) ) ) ) ).

% mul_ext_cong
thf(fact_1173_mul__ext__cong,axiom,
    ! [Xs1: list_P9127870862658512897on_nat,Ys1: list_P9127870862658512897on_nat,Xs22: list_P9127870862658512897on_nat,Ys22: list_P9127870862658512897on_nat,F: produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o,G: produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o] :
      ( ( ( mset_P6964650582947932560on_nat @ Xs1 )
        = ( mset_P6964650582947932560on_nat @ Ys1 ) )
     => ( ( ( mset_P6964650582947932560on_nat @ Xs22 )
          = ( mset_P6964650582947932560on_nat @ Ys22 ) )
       => ( ! [X: produc1313295348606329723on_nat,X6: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ Ys1 ) )
             => ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Ys22 ) )
               => ( ( F @ X @ X6 )
                  = ( G @ X @ X6 ) ) ) )
         => ( ( multis1593711116464184117on_nat @ F @ Xs1 @ Xs22 )
            = ( multis1593711116464184117on_nat @ G @ Ys1 @ Ys22 ) ) ) ) ) ).

% mul_ext_cong
thf(fact_1174_mul__ext__cong,axiom,
    ! [Xs1: list_t5668669291736003536_a_nat,Ys1: list_t5668669291736003536_a_nat,Xs22: list_t5668669291736003536_a_nat,Ys22: list_t5668669291736003536_a_nat,F: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,G: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o] :
      ( ( ( mset_t7141044528732312287_a_nat @ Xs1 )
        = ( mset_t7141044528732312287_a_nat @ Ys1 ) )
     => ( ( ( mset_t7141044528732312287_a_nat @ Xs22 )
          = ( mset_t7141044528732312287_a_nat @ Ys22 ) )
       => ( ! [X: term_R1747997765655105994_a_nat,X6: term_R1747997765655105994_a_nat] :
              ( ( member7260319590701762419_a_nat @ X @ ( set_te3761484132649746399_a_nat @ Ys1 ) )
             => ( ( member7260319590701762419_a_nat @ X6 @ ( set_te3761484132649746399_a_nat @ Ys22 ) )
               => ( ( F @ X @ X6 )
                  = ( G @ X @ X6 ) ) ) )
         => ( ( multis2940008709989298052_a_nat @ F @ Xs1 @ Xs22 )
            = ( multis2940008709989298052_a_nat @ G @ Ys1 @ Ys22 ) ) ) ) ) ).

% mul_ext_cong
thf(fact_1175_mul__ext__cong,axiom,
    ! [Xs1: list_S5253404628753938141on_nat,Ys1: list_S5253404628753938141on_nat,Xs22: list_S5253404628753938141on_nat,Ys22: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,G: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs1 )
        = ( mset_S7907735292211298936on_nat @ Ys1 ) )
     => ( ( ( mset_S7907735292211298936on_nat @ Xs22 )
          = ( mset_S7907735292211298936on_nat @ Ys22 ) )
       => ( ! [X: sum_su4907927656612468301on_nat,X6: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Ys1 ) )
             => ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
               => ( ( F @ X @ X6 )
                  = ( G @ X @ X6 ) ) ) )
         => ( ( multis6586140943272769299on_nat @ F @ Xs1 @ Xs22 )
            = ( multis6586140943272769299on_nat @ G @ Ys1 @ Ys22 ) ) ) ) ) ).

% mul_ext_cong
thf(fact_1176_mul__ext__compat,axiom,
    ! [Ss: list_P5550462785134625709_a_nat,Ts: list_P5550462785134625709_a_nat,Us: list_P5550462785134625709_a_nat,F: produc394670695091157671_a_nat > produc394670695091157671_a_nat > product_prod_o_o] :
      ( ! [S7: produc394670695091157671_a_nat,T3: produc394670695091157671_a_nat,U3: produc394670695091157671_a_nat] :
          ( ( member4231271695026834640_a_nat @ S7 @ ( set_Pr7412993203964320572_a_nat @ Ss ) )
         => ( ( member4231271695026834640_a_nat @ T3 @ ( set_Pr7412993203964320572_a_nat @ Ts ) )
           => ( ( member4231271695026834640_a_nat @ U3 @ ( set_Pr7412993203964320572_a_nat @ Us ) )
             => ( ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_snd_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) ) ) ) ) )
     => ( ( ( ( product_snd_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis7464683662112179553_a_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_snd_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis7464683662112179553_a_nat @ F @ Ts @ Us ) ) )
         => ( product_snd_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ F @ Ss @ Us ) ) ) ) ) ).

% mul_ext_compat
thf(fact_1177_mul__ext__compat,axiom,
    ! [Ss: list_P5842215428982857133_a_nat,Ts: list_P5842215428982857133_a_nat,Us: list_P5842215428982857133_a_nat,F: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > product_prod_o_o] :
      ( ! [S7: produc5238312538367656103_a_nat,T3: produc5238312538367656103_a_nat,U3: produc5238312538367656103_a_nat] :
          ( ( member6928026473249828560_a_nat @ S7 @ ( set_Pr7117250867156192572_a_nat @ Ss ) )
         => ( ( member6928026473249828560_a_nat @ T3 @ ( set_Pr7117250867156192572_a_nat @ Ts ) )
           => ( ( member6928026473249828560_a_nat @ U3 @ ( set_Pr7117250867156192572_a_nat @ Us ) )
             => ( ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_snd_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) ) ) ) ) )
     => ( ( ( ( product_snd_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis2430777439941206881_a_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_snd_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis2430777439941206881_a_nat @ F @ Ts @ Us ) ) )
         => ( product_snd_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ F @ Ss @ Us ) ) ) ) ) ).

% mul_ext_compat
thf(fact_1178_mul__ext__compat,axiom,
    ! [Ss: list_P1915136757390359201on_nat,Ts: list_P1915136757390359201on_nat,Us: list_P1915136757390359201on_nat,F: produc9176241080893687067on_nat > produc9176241080893687067on_nat > product_prod_o_o] :
      ( ! [S7: produc9176241080893687067on_nat,T3: produc9176241080893687067on_nat,U3: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ S7 @ ( set_Pr8272476097931642672on_nat @ Ss ) )
         => ( ( member4278285197320822468on_nat @ T3 @ ( set_Pr8272476097931642672on_nat @ Ts ) )
           => ( ( member4278285197320822468on_nat @ U3 @ ( set_Pr8272476097931642672on_nat @ Us ) )
             => ( ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_snd_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) ) ) ) ) )
     => ( ( ( ( product_snd_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis7516503858109221589on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis7516503858109221589on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_snd_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis7516503858109221589on_nat @ F @ Ts @ Us ) ) )
         => ( product_snd_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis7516503858109221589on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis7516503858109221589on_nat @ F @ Ss @ Us ) ) ) ) ) ).

% mul_ext_compat
thf(fact_1179_mul__ext__compat,axiom,
    ! [Ss: list_P2317284437984938017on_nat,Ts: list_P2317284437984938017on_nat,Us: list_P2317284437984938017on_nat,F: produc5580075136493545115on_nat > produc5580075136493545115on_nat > product_prod_o_o] :
      ( ! [S7: produc5580075136493545115on_nat,T3: produc5580075136493545115on_nat,U3: produc5580075136493545115on_nat] :
          ( ( member4221009242412147780on_nat @ S7 @ ( set_Pr4166446493430381744on_nat @ Ss ) )
         => ( ( member4221009242412147780on_nat @ T3 @ ( set_Pr4166446493430381744on_nat @ Ts ) )
           => ( ( member4221009242412147780on_nat @ U3 @ ( set_Pr4166446493430381744on_nat @ Us ) )
             => ( ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_snd_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) ) ) ) ) )
     => ( ( ( ( product_snd_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis4887289194134901845on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis4887289194134901845on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_snd_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis4887289194134901845on_nat @ F @ Ts @ Us ) ) )
         => ( product_snd_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis4887289194134901845on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis4887289194134901845on_nat @ F @ Ss @ Us ) ) ) ) ) ).

% mul_ext_compat
thf(fact_1180_mul__ext__compat,axiom,
    ! [Ss: list_P9127870862658512897on_nat,Ts: list_P9127870862658512897on_nat,Us: list_P9127870862658512897on_nat,F: produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o] :
      ( ! [S7: produc1313295348606329723on_nat,T3: produc1313295348606329723on_nat,U3: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ S7 @ ( set_Pr937530902923597456on_nat @ Ss ) )
         => ( ( member5906418179908956708on_nat @ T3 @ ( set_Pr937530902923597456on_nat @ Ts ) )
           => ( ( member5906418179908956708on_nat @ U3 @ ( set_Pr937530902923597456on_nat @ Us ) )
             => ( ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_snd_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_snd_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_snd_o_o @ ( F @ S7 @ U3 ) ) )
                & ( ( ( product_fst_o_o @ ( F @ S7 @ T3 ) )
                    & ( product_fst_o_o @ ( F @ T3 @ U3 ) ) )
                 => ( product_fst_o_o @ ( F @ S7 @ U3 ) ) ) ) ) ) )
     => ( ( ( ( product_snd_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis1593711116464184117on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis1593711116464184117on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_snd_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Ts ) )
            & ( product_snd_o_o @ ( multis1593711116464184117on_nat @ F @ Ts @ Us ) ) )
         => ( product_snd_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Us ) ) )
        & ( ( ( product_fst_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Ts ) )
            & ( product_fst_o_o @ ( multis1593711116464184117on_nat @ F @ Ts @ Us ) ) )
         => ( product_fst_o_o @ ( multis1593711116464184117on_nat @ F @ Ss @ Us ) ) ) ) ) ).

% mul_ext_compat
thf(fact_1181_mul__ext__mono_I1_J,axiom,
    ! [Xs2: list_P5550462785134625709_a_nat,Ys2: list_P5550462785134625709_a_nat,P2: produc394670695091157671_a_nat > produc394670695091157671_a_nat > product_prod_o_o,P4: produc394670695091157671_a_nat > produc394670695091157671_a_nat > product_prod_o_o] :
      ( ! [X: produc394670695091157671_a_nat,Y: produc394670695091157671_a_nat] :
          ( ( member4231271695026834640_a_nat @ X @ ( set_Pr7412993203964320572_a_nat @ Xs2 ) )
         => ( ( member4231271695026834640_a_nat @ Y @ ( set_Pr7412993203964320572_a_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc394670695091157671_a_nat,Y: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X @ ( set_Pr7412993203964320572_a_nat @ Xs2 ) )
           => ( ( member4231271695026834640_a_nat @ Y @ ( set_Pr7412993203964320572_a_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_fst_o_o @ ( multis7464683662112179553_a_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(1)
thf(fact_1182_mul__ext__mono_I1_J,axiom,
    ! [Xs2: list_P5842215428982857133_a_nat,Ys2: list_P5842215428982857133_a_nat,P2: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > product_prod_o_o,P4: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > product_prod_o_o] :
      ( ! [X: produc5238312538367656103_a_nat,Y: produc5238312538367656103_a_nat] :
          ( ( member6928026473249828560_a_nat @ X @ ( set_Pr7117250867156192572_a_nat @ Xs2 ) )
         => ( ( member6928026473249828560_a_nat @ Y @ ( set_Pr7117250867156192572_a_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc5238312538367656103_a_nat,Y: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X @ ( set_Pr7117250867156192572_a_nat @ Xs2 ) )
           => ( ( member6928026473249828560_a_nat @ Y @ ( set_Pr7117250867156192572_a_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_fst_o_o @ ( multis2430777439941206881_a_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(1)
thf(fact_1183_mul__ext__mono_I1_J,axiom,
    ! [Xs2: list_P1915136757390359201on_nat,Ys2: list_P1915136757390359201on_nat,P2: produc9176241080893687067on_nat > produc9176241080893687067on_nat > product_prod_o_o,P4: produc9176241080893687067on_nat > produc9176241080893687067on_nat > product_prod_o_o] :
      ( ! [X: produc9176241080893687067on_nat,Y: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X @ ( set_Pr8272476097931642672on_nat @ Xs2 ) )
         => ( ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc9176241080893687067on_nat,Y: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X @ ( set_Pr8272476097931642672on_nat @ Xs2 ) )
           => ( ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_fst_o_o @ ( multis7516503858109221589on_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_fst_o_o @ ( multis7516503858109221589on_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(1)
thf(fact_1184_mul__ext__mono_I1_J,axiom,
    ! [Xs2: list_P2317284437984938017on_nat,Ys2: list_P2317284437984938017on_nat,P2: produc5580075136493545115on_nat > produc5580075136493545115on_nat > product_prod_o_o,P4: produc5580075136493545115on_nat > produc5580075136493545115on_nat > product_prod_o_o] :
      ( ! [X: produc5580075136493545115on_nat,Y: produc5580075136493545115on_nat] :
          ( ( member4221009242412147780on_nat @ X @ ( set_Pr4166446493430381744on_nat @ Xs2 ) )
         => ( ( member4221009242412147780on_nat @ Y @ ( set_Pr4166446493430381744on_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc5580075136493545115on_nat,Y: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X @ ( set_Pr4166446493430381744on_nat @ Xs2 ) )
           => ( ( member4221009242412147780on_nat @ Y @ ( set_Pr4166446493430381744on_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_fst_o_o @ ( multis4887289194134901845on_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_fst_o_o @ ( multis4887289194134901845on_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(1)
thf(fact_1185_mul__ext__mono_I1_J,axiom,
    ! [Xs2: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o,P4: produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o] :
      ( ! [X: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
         => ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
           => ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_fst_o_o @ ( multis1593711116464184117on_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_fst_o_o @ ( multis1593711116464184117on_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(1)
thf(fact_1186_mul__ext__mono_I2_J,axiom,
    ! [Xs2: list_P5550462785134625709_a_nat,Ys2: list_P5550462785134625709_a_nat,P2: produc394670695091157671_a_nat > produc394670695091157671_a_nat > product_prod_o_o,P4: produc394670695091157671_a_nat > produc394670695091157671_a_nat > product_prod_o_o] :
      ( ! [X: produc394670695091157671_a_nat,Y: produc394670695091157671_a_nat] :
          ( ( member4231271695026834640_a_nat @ X @ ( set_Pr7412993203964320572_a_nat @ Xs2 ) )
         => ( ( member4231271695026834640_a_nat @ Y @ ( set_Pr7412993203964320572_a_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc394670695091157671_a_nat,Y: produc394670695091157671_a_nat] :
            ( ( member4231271695026834640_a_nat @ X @ ( set_Pr7412993203964320572_a_nat @ Xs2 ) )
           => ( ( member4231271695026834640_a_nat @ Y @ ( set_Pr7412993203964320572_a_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_snd_o_o @ ( multis7464683662112179553_a_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_snd_o_o @ ( multis7464683662112179553_a_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(2)
thf(fact_1187_mul__ext__mono_I2_J,axiom,
    ! [Xs2: list_P5842215428982857133_a_nat,Ys2: list_P5842215428982857133_a_nat,P2: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > product_prod_o_o,P4: produc5238312538367656103_a_nat > produc5238312538367656103_a_nat > product_prod_o_o] :
      ( ! [X: produc5238312538367656103_a_nat,Y: produc5238312538367656103_a_nat] :
          ( ( member6928026473249828560_a_nat @ X @ ( set_Pr7117250867156192572_a_nat @ Xs2 ) )
         => ( ( member6928026473249828560_a_nat @ Y @ ( set_Pr7117250867156192572_a_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc5238312538367656103_a_nat,Y: produc5238312538367656103_a_nat] :
            ( ( member6928026473249828560_a_nat @ X @ ( set_Pr7117250867156192572_a_nat @ Xs2 ) )
           => ( ( member6928026473249828560_a_nat @ Y @ ( set_Pr7117250867156192572_a_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_snd_o_o @ ( multis2430777439941206881_a_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_snd_o_o @ ( multis2430777439941206881_a_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(2)
thf(fact_1188_mul__ext__mono_I2_J,axiom,
    ! [Xs2: list_P1915136757390359201on_nat,Ys2: list_P1915136757390359201on_nat,P2: produc9176241080893687067on_nat > produc9176241080893687067on_nat > product_prod_o_o,P4: produc9176241080893687067on_nat > produc9176241080893687067on_nat > product_prod_o_o] :
      ( ! [X: produc9176241080893687067on_nat,Y: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X @ ( set_Pr8272476097931642672on_nat @ Xs2 ) )
         => ( ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc9176241080893687067on_nat,Y: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X @ ( set_Pr8272476097931642672on_nat @ Xs2 ) )
           => ( ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_snd_o_o @ ( multis7516503858109221589on_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_snd_o_o @ ( multis7516503858109221589on_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(2)
thf(fact_1189_mul__ext__mono_I2_J,axiom,
    ! [Xs2: list_P2317284437984938017on_nat,Ys2: list_P2317284437984938017on_nat,P2: produc5580075136493545115on_nat > produc5580075136493545115on_nat > product_prod_o_o,P4: produc5580075136493545115on_nat > produc5580075136493545115on_nat > product_prod_o_o] :
      ( ! [X: produc5580075136493545115on_nat,Y: produc5580075136493545115on_nat] :
          ( ( member4221009242412147780on_nat @ X @ ( set_Pr4166446493430381744on_nat @ Xs2 ) )
         => ( ( member4221009242412147780on_nat @ Y @ ( set_Pr4166446493430381744on_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc5580075136493545115on_nat,Y: produc5580075136493545115on_nat] :
            ( ( member4221009242412147780on_nat @ X @ ( set_Pr4166446493430381744on_nat @ Xs2 ) )
           => ( ( member4221009242412147780on_nat @ Y @ ( set_Pr4166446493430381744on_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_snd_o_o @ ( multis4887289194134901845on_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_snd_o_o @ ( multis4887289194134901845on_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(2)
thf(fact_1190_mul__ext__mono_I2_J,axiom,
    ! [Xs2: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o,P4: produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o] :
      ( ! [X: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
         => ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Ys2 ) )
           => ( ( product_fst_o_o @ ( P2 @ X @ Y ) )
             => ( product_fst_o_o @ ( P4 @ X @ Y ) ) ) ) )
     => ( ! [X: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ Xs2 ) )
           => ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Ys2 ) )
             => ( ( product_snd_o_o @ ( P2 @ X @ Y ) )
               => ( product_snd_o_o @ ( P4 @ X @ Y ) ) ) ) )
       => ( ( product_snd_o_o @ ( multis1593711116464184117on_nat @ P2 @ Xs2 @ Ys2 ) )
         => ( product_snd_o_o @ ( multis1593711116464184117on_nat @ P4 @ Xs2 @ Ys2 ) ) ) ) ) ).

% mul_ext_mono(2)
thf(fact_1191_nstri__mul__ext__map,axiom,
    ! [Ss: list_S5253404628753938141on_nat,Ts: list_S5253404628753938141on_nat,Order: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,Order2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,F: sum_su4907927656612468301on_nat > term_R1747997765655105994_a_nat] :
      ( ! [S7: sum_su4907927656612468301on_nat,T3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ S7 @ ( set_Su4848624856859743608on_nat @ Ss ) )
         => ( ( member8476201367316484836on_nat @ T3 @ ( set_Su4848624856859743608on_nat @ Ts ) )
           => ( ( product_fst_o_o @ ( Order @ S7 @ T3 ) )
             => ( product_fst_o_o @ ( Order2 @ ( F @ S7 ) @ ( F @ T3 ) ) ) ) ) )
     => ( ! [S7: sum_su4907927656612468301on_nat,T3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ S7 @ ( set_Su4848624856859743608on_nat @ Ss ) )
           => ( ( member8476201367316484836on_nat @ T3 @ ( set_Su4848624856859743608on_nat @ Ts ) )
             => ( ( product_snd_o_o @ ( Order @ S7 @ T3 ) )
               => ( product_snd_o_o @ ( Order2 @ ( F @ S7 ) @ ( F @ T3 ) ) ) ) ) )
       => ( ( product_snd_o_o @ ( multis6586140943272769299on_nat @ Order @ Ss @ Ts ) )
         => ( product_snd_o_o @ ( multis2940008709989298052_a_nat @ Order2 @ ( map_Su925086260876352759_a_nat @ F @ Ss ) @ ( map_Su925086260876352759_a_nat @ F @ Ts ) ) ) ) ) ) ).

% nstri_mul_ext_map
thf(fact_1192_stri__mul__ext__map,axiom,
    ! [Ss: list_S5253404628753938141on_nat,Ts: list_S5253404628753938141on_nat,Order: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o,Order2: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o,F: sum_su4907927656612468301on_nat > term_R1747997765655105994_a_nat] :
      ( ! [S7: sum_su4907927656612468301on_nat,T3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ S7 @ ( set_Su4848624856859743608on_nat @ Ss ) )
         => ( ( member8476201367316484836on_nat @ T3 @ ( set_Su4848624856859743608on_nat @ Ts ) )
           => ( ( product_fst_o_o @ ( Order @ S7 @ T3 ) )
             => ( product_fst_o_o @ ( Order2 @ ( F @ S7 ) @ ( F @ T3 ) ) ) ) ) )
     => ( ! [S7: sum_su4907927656612468301on_nat,T3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ S7 @ ( set_Su4848624856859743608on_nat @ Ss ) )
           => ( ( member8476201367316484836on_nat @ T3 @ ( set_Su4848624856859743608on_nat @ Ts ) )
             => ( ( product_snd_o_o @ ( Order @ S7 @ T3 ) )
               => ( product_snd_o_o @ ( Order2 @ ( F @ S7 ) @ ( F @ T3 ) ) ) ) ) )
       => ( ( product_fst_o_o @ ( multis6586140943272769299on_nat @ Order @ Ss @ Ts ) )
         => ( product_fst_o_o @ ( multis2940008709989298052_a_nat @ Order2 @ ( map_Su925086260876352759_a_nat @ F @ Ss ) @ ( map_Su925086260876352759_a_nat @ F @ Ts ) ) ) ) ) ) ).

% stri_mul_ext_map
thf(fact_1193_subterm_OGreatestI2__order,axiom,
    ! [P2: term_R1747997765655105994_a_nat > $o,X3: term_R1747997765655105994_a_nat,Q: term_R1747997765655105994_a_nat > $o] :
      ( ( P2 @ X3 )
     => ( ! [Y: term_R1747997765655105994_a_nat] :
            ( ( P2 @ Y )
           => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X3 @ Y ) @ subter3376206727240021173_a_nat ) )
       => ( ! [X: term_R1747997765655105994_a_nat] :
              ( ( P2 @ X )
             => ( ! [Y6: term_R1747997765655105994_a_nat] :
                    ( ( P2 @ Y6 )
                   => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ X @ Y6 ) @ subter3376206727240021173_a_nat ) )
               => ( Q @ X ) ) )
         => ( Q
            @ ( greate439585953166210053_a_nat
              @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] : ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ T @ S4 ) @ subter3376206727240021173_a_nat )
              @ P2 ) ) ) ) ) ).

% subterm.GreatestI2_order
thf(fact_1194_multiset__problem__of__cnf_I2_J,axiom,
    ! [Cnf2: list_l8336264072933094989od_a_o,Left: list_S5253404628753938141on_nat,Right: list_S5253404628753938141on_nat,S6: list_P9127870862658512897on_nat,NSS: list_P9127870862658512897on_nat] :
      ( ( ( multis4221440410396274297_cnf_a @ Cnf2 )
        = ( produc1545487600272464053on_nat @ Left @ ( produc4176415733022414961on_nat @ Right @ ( produc7152764266822961111on_nat @ S6 @ NSS ) ) ) )
     => ( ( Cnf2 != nil_li1372968520647675725od_a_o )
       => ( ( ? [Beta: a > $o] : ( propos4091448439431948687_cnf_a @ Beta @ Cnf2 ) )
          = ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ Left ) @ ( mset_S7907735292211298936on_nat @ Right ) ) @ ( multis1620213760974180726on_nat @ ( set_Pr937530902923597456on_nat @ NSS ) @ ( set_Pr937530902923597456on_nat @ S6 ) ) ) ) ) ) ).

% multiset_problem_of_cnf(2)
thf(fact_1195_multiset__problem__of__cnf__mul__ext,axiom,
    ! [Cnf2: list_l8336264072933094989od_a_o,Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat,S6: list_P9127870862658512897on_nat,NS2: list_P9127870862658512897on_nat] :
      ( ( ( multis4221440410396274297_cnf_a @ Cnf2 )
        = ( produc1545487600272464053on_nat @ Xs2 @ ( produc4176415733022414961on_nat @ Ys2 @ ( produc7152764266822961111on_nat @ S6 @ NS2 ) ) ) )
     => ( ( Cnf2 != nil_li1372968520647675725od_a_o )
       => ( ( ? [Beta: a > $o] : ( propos4091448439431948687_cnf_a @ Beta @ Cnf2 ) )
          = ( ( multis6586140943272769299on_nat
              @ ^ [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( product_Pair_o_o @ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A3 @ B3 ) @ ( set_Pr937530902923597456on_nat @ S6 ) ) @ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A3 @ B3 ) @ ( set_Pr937530902923597456on_nat @ NS2 ) ) )
              @ Xs2
              @ Ys2 )
            = ( product_Pair_o_o @ $true @ $true ) ) ) ) ) ).

% multiset_problem_of_cnf_mul_ext
thf(fact_1196_multiset__problem__of__cnf_I1_J,axiom,
    ! [Cnf2: list_l8336264072933094989od_a_o,Left: list_S5253404628753938141on_nat,Right: list_S5253404628753938141on_nat,S6: list_P9127870862658512897on_nat,NSS: list_P9127870862658512897on_nat] :
      ( ( ( multis4221440410396274297_cnf_a @ Cnf2 )
        = ( produc1545487600272464053on_nat @ Left @ ( produc4176415733022414961on_nat @ Right @ ( produc7152764266822961111on_nat @ S6 @ NSS ) ) ) )
     => ( ( ? [Beta: a > $o] : ( propos4091448439431948687_cnf_a @ Beta @ Cnf2 ) )
        = ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ Left ) @ ( mset_S7907735292211298936on_nat @ Right ) ) @ ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ NSS ) @ ( set_Pr937530902923597456on_nat @ S6 ) ) ) ) ) ).

% multiset_problem_of_cnf(1)
thf(fact_1197_ms__elem__of__lit_Ocases,axiom,
    ! [X3: product_prod_o_o] :
      ( ! [X: $o] :
          ( X3
         != ( product_Pair_o_o @ X @ $true ) )
     => ~ ! [X: $o] :
            ( X3
           != ( product_Pair_o_o @ X @ $false ) ) ) ).

% ms_elem_of_lit.cases
thf(fact_1198_eval__cnf__alt__def,axiom,
    ( propos1613540924048857769_cnf_o
    = ( ^ [Alpha: $o > $o,Cnf: list_l2060169558395104435od_o_o] :
        ! [X2: list_P4002435161011370285od_o_o] :
          ( ( member1154424878073987030od_o_o @ X2 @ ( set_li7585892930703410242od_o_o @ Cnf ) )
         => ? [Y5: product_prod_o_o] :
              ( ( member7466972457876170832od_o_o @ Y5 @ ( set_Product_prod_o_o2 @ X2 ) )
              & ( ( Alpha @ ( product_fst_o_o @ Y5 ) )
                = ( product_snd_o_o @ Y5 ) ) ) ) ) ) ).

% eval_cnf_alt_def
thf(fact_1199_eval__cnf__alt__def,axiom,
    ( propos4091448439431948687_cnf_a
    = ( ^ [Alpha: a > $o,Cnf: list_l8336264072933094989od_a_o] :
        ! [X2: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Cnf ) )
         => ? [Y5: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ Y5 @ ( set_Product_prod_a_o2 @ X2 ) )
              & ( ( Alpha @ ( product_fst_a_o @ Y5 ) )
                = ( product_snd_a_o @ Y5 ) ) ) ) ) ) ).

% eval_cnf_alt_def
thf(fact_1200_list__order__extension__impl__axioms_Ointro,axiom,
    ! [S_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat,List_ext: ( produc1313295348606329723on_nat > produc1313295348606329723on_nat > product_prod_o_o ) > list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > product_prod_o_o,Ns_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat] :
      ( ! [S7: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,Ns5: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o] :
          ( ( S_list @ ( collec8365270904964013138on_nat @ ( produc3964550740795057708_nat_o @ S7 ) ) @ ( collec8365270904964013138on_nat @ ( produc3964550740795057708_nat_o @ Ns5 ) ) )
          = ( collec5627915860394808530on_nat
            @ ( produc4129062015429275820_nat_o
              @ ^ [As: list_P9127870862658512897on_nat,Bs: list_P9127870862658512897on_nat] :
                  ( product_fst_o_o
                  @ ( List_ext
                    @ ^ [A3: produc1313295348606329723on_nat,B3: produc1313295348606329723on_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                    @ As
                    @ Bs ) ) ) ) )
     => ( ! [S7: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,Ns5: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o] :
            ( ( Ns_list @ ( collec8365270904964013138on_nat @ ( produc3964550740795057708_nat_o @ S7 ) ) @ ( collec8365270904964013138on_nat @ ( produc3964550740795057708_nat_o @ Ns5 ) ) )
            = ( collec5627915860394808530on_nat
              @ ( produc4129062015429275820_nat_o
                @ ^ [As: list_P9127870862658512897on_nat,Bs: list_P9127870862658512897on_nat] :
                    ( product_snd_o_o
                    @ ( List_ext
                      @ ^ [A3: produc1313295348606329723on_nat,B3: produc1313295348606329723on_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                      @ As
                      @ Bs ) ) ) ) )
       => ( ! [S7: set_Pr8202701763255893447on_nat,Ns5: set_Pr8202701763255893447on_nat,S10: set_Pr8202701763255893447on_nat,Ns6: set_Pr8202701763255893447on_nat,As3: list_P9127870862658512897on_nat,Bs3: list_P9127870862658512897on_nat] :
              ( ( ord_le6981717135557662055on_nat
                @ ( inf_in5721575873212415925on_nat
                  @ ( produc4044997224725718792on_nat @ ( set_Pr937530902923597456on_nat @ As3 )
                    @ ^ [Uu: produc1313295348606329723on_nat] : ( set_Pr937530902923597456on_nat @ Bs3 ) )
                  @ Ns5 )
                @ Ns6 )
             => ( ( ord_le6981717135557662055on_nat
                  @ ( inf_in5721575873212415925on_nat
                    @ ( produc4044997224725718792on_nat @ ( set_Pr937530902923597456on_nat @ As3 )
                      @ ^ [Uu: produc1313295348606329723on_nat] : ( set_Pr937530902923597456on_nat @ Bs3 ) )
                    @ S7 )
                  @ S10 )
               => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As3 @ Bs3 ) @ ( S_list @ Ns5 @ S7 ) )
                 => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As3 @ Bs3 ) @ ( S_list @ Ns6 @ S10 ) ) ) ) )
         => ( ! [S7: set_Pr8202701763255893447on_nat,Ns5: set_Pr8202701763255893447on_nat,S10: set_Pr8202701763255893447on_nat,Ns6: set_Pr8202701763255893447on_nat,As3: list_P9127870862658512897on_nat,Bs3: list_P9127870862658512897on_nat] :
                ( ( ord_le6981717135557662055on_nat
                  @ ( inf_in5721575873212415925on_nat
                    @ ( produc4044997224725718792on_nat @ ( set_Pr937530902923597456on_nat @ As3 )
                      @ ^ [Uu: produc1313295348606329723on_nat] : ( set_Pr937530902923597456on_nat @ Bs3 ) )
                    @ Ns5 )
                  @ Ns6 )
               => ( ( ord_le6981717135557662055on_nat
                    @ ( inf_in5721575873212415925on_nat
                      @ ( produc4044997224725718792on_nat @ ( set_Pr937530902923597456on_nat @ As3 )
                        @ ^ [Uu: produc1313295348606329723on_nat] : ( set_Pr937530902923597456on_nat @ Bs3 ) )
                      @ S7 )
                    @ S10 )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns5 @ S7 ) )
                   => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns6 @ S10 ) ) ) ) )
           => ( list_l9085172305638076644on_nat @ S_list @ Ns_list @ List_ext ) ) ) ) ) ).

% list_order_extension_impl_axioms.intro
thf(fact_1201_list__order__extension__impl__axioms_Ointro,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,List_ext: ( sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > product_prod_o_o ) > list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat] :
      ( ! [S7: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Ns5: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
          ( ( S_list @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ S7 ) ) @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ Ns5 ) ) )
          = ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] :
                  ( product_fst_o_o
                  @ ( List_ext
                    @ ^ [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                    @ As
                    @ Bs ) ) ) ) )
     => ( ! [S7: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Ns5: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
            ( ( Ns_list @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ S7 ) ) @ ( collec5585564048161825126on_nat @ ( produc208325271403635792_nat_o @ Ns5 ) ) )
            = ( collec1293568243156533638on_nat
              @ ( produc7163990014243126832_nat_o
                @ ^ [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] :
                    ( product_snd_o_o
                    @ ( List_ext
                      @ ^ [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                      @ As
                      @ Bs ) ) ) ) )
       => ( ! [S7: set_Pr2535564044409765851on_nat,Ns5: set_Pr2535564044409765851on_nat,S10: set_Pr2535564044409765851on_nat,Ns6: set_Pr2535564044409765851on_nat,As3: list_S5253404628753938141on_nat,Bs3: list_S5253404628753938141on_nat] :
              ( ( ord_le6961793561435735931on_nat
                @ ( inf_in1157766134151114441on_nat
                  @ ( produc2390926880413004132on_nat @ ( set_Su4848624856859743608on_nat @ As3 )
                    @ ^ [Uu: sum_su4907927656612468301on_nat] : ( set_Su4848624856859743608on_nat @ Bs3 ) )
                  @ Ns5 )
                @ Ns6 )
             => ( ( ord_le6961793561435735931on_nat
                  @ ( inf_in1157766134151114441on_nat
                    @ ( produc2390926880413004132on_nat @ ( set_Su4848624856859743608on_nat @ As3 )
                      @ ^ [Uu: sum_su4907927656612468301on_nat] : ( set_Su4848624856859743608on_nat @ Bs3 ) )
                    @ S7 )
                  @ S10 )
               => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As3 @ Bs3 ) @ ( S_list @ Ns5 @ S7 ) )
                 => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As3 @ Bs3 ) @ ( S_list @ Ns6 @ S10 ) ) ) ) )
         => ( ! [S7: set_Pr2535564044409765851on_nat,Ns5: set_Pr2535564044409765851on_nat,S10: set_Pr2535564044409765851on_nat,Ns6: set_Pr2535564044409765851on_nat,As3: list_S5253404628753938141on_nat,Bs3: list_S5253404628753938141on_nat] :
                ( ( ord_le6961793561435735931on_nat
                  @ ( inf_in1157766134151114441on_nat
                    @ ( produc2390926880413004132on_nat @ ( set_Su4848624856859743608on_nat @ As3 )
                      @ ^ [Uu: sum_su4907927656612468301on_nat] : ( set_Su4848624856859743608on_nat @ Bs3 ) )
                    @ Ns5 )
                  @ Ns6 )
               => ( ( ord_le6961793561435735931on_nat
                    @ ( inf_in1157766134151114441on_nat
                      @ ( produc2390926880413004132on_nat @ ( set_Su4848624856859743608on_nat @ As3 )
                        @ ^ [Uu: sum_su4907927656612468301on_nat] : ( set_Su4848624856859743608on_nat @ Bs3 ) )
                      @ S7 )
                    @ S10 )
                 => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns5 @ S7 ) )
                   => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns6 @ S10 ) ) ) ) )
           => ( list_l7123588524199674916on_nat @ S_list @ Ns_list @ List_ext ) ) ) ) ) ).

% list_order_extension_impl_axioms.intro
thf(fact_1202_list__order__extension__impl__axioms_Ointro,axiom,
    ! [S_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat,List_ext: ( list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o ) > list_l3889969465389992278_a_nat > list_l3889969465389992278_a_nat > product_prod_o_o,Ns_list: set_Pr4991467605864603271_a_nat > set_Pr4991467605864603271_a_nat > set_Pr4022536337903447175_a_nat] :
      ( ! [S7: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Ns5: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
          ( ( S_list @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ S7 ) ) @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ Ns5 ) ) )
          = ( collec4515946210884203282_a_nat
            @ ( produc2556087688056060396_nat_o
              @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] :
                  ( product_fst_o_o
                  @ ( List_ext
                    @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                    @ As
                    @ Bs ) ) ) ) )
     => ( ! [S7: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o,Ns5: list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > $o] :
            ( ( Ns_list @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ S7 ) ) @ ( collec8858057866656402194_a_nat @ ( produc4945827410475597548_nat_o @ Ns5 ) ) )
            = ( collec4515946210884203282_a_nat
              @ ( produc2556087688056060396_nat_o
                @ ^ [As: list_l3889969465389992278_a_nat,Bs: list_l3889969465389992278_a_nat] :
                    ( product_snd_o_o
                    @ ( List_ext
                      @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                      @ As
                      @ Bs ) ) ) ) )
       => ( ! [S7: set_Pr4991467605864603271_a_nat,Ns5: set_Pr4991467605864603271_a_nat,S10: set_Pr4991467605864603271_a_nat,Ns6: set_Pr4991467605864603271_a_nat,As3: list_l3889969465389992278_a_nat,Bs3: list_l3889969465389992278_a_nat] :
              ( ( ord_le998509814859404839_a_nat
                @ ( inf_in3705958062578247285_a_nat
                  @ ( produc2363723359784558408_a_nat @ ( set_li4920527676005225317_a_nat @ As3 )
                    @ ^ [Uu: list_t5668669291736003536_a_nat] : ( set_li4920527676005225317_a_nat @ Bs3 ) )
                  @ Ns5 )
                @ Ns6 )
             => ( ( ord_le998509814859404839_a_nat
                  @ ( inf_in3705958062578247285_a_nat
                    @ ( produc2363723359784558408_a_nat @ ( set_li4920527676005225317_a_nat @ As3 )
                      @ ^ [Uu: list_t5668669291736003536_a_nat] : ( set_li4920527676005225317_a_nat @ Bs3 ) )
                    @ S7 )
                  @ S10 )
               => ( ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As3 @ Bs3 ) @ ( S_list @ Ns5 @ S7 ) )
                 => ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As3 @ Bs3 ) @ ( S_list @ Ns6 @ S10 ) ) ) ) )
         => ( ! [S7: set_Pr4991467605864603271_a_nat,Ns5: set_Pr4991467605864603271_a_nat,S10: set_Pr4991467605864603271_a_nat,Ns6: set_Pr4991467605864603271_a_nat,As3: list_l3889969465389992278_a_nat,Bs3: list_l3889969465389992278_a_nat] :
                ( ( ord_le998509814859404839_a_nat
                  @ ( inf_in3705958062578247285_a_nat
                    @ ( produc2363723359784558408_a_nat @ ( set_li4920527676005225317_a_nat @ As3 )
                      @ ^ [Uu: list_t5668669291736003536_a_nat] : ( set_li4920527676005225317_a_nat @ Bs3 ) )
                    @ Ns5 )
                  @ Ns6 )
               => ( ( ord_le998509814859404839_a_nat
                    @ ( inf_in3705958062578247285_a_nat
                      @ ( produc2363723359784558408_a_nat @ ( set_li4920527676005225317_a_nat @ As3 )
                        @ ^ [Uu: list_t5668669291736003536_a_nat] : ( set_li4920527676005225317_a_nat @ Bs3 ) )
                      @ S7 )
                    @ S10 )
                 => ( ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns5 @ S7 ) )
                   => ( member7165818983852820176_a_nat @ ( produc2502319126687419031_a_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns6 @ S10 ) ) ) ) )
           => ( list_l4393306550459998137_a_nat @ S_list @ Ns_list @ List_ext ) ) ) ) ) ).

% list_order_extension_impl_axioms.intro
thf(fact_1203_list__order__extension__impl__axioms_Ointro,axiom,
    ! [S_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat,List_ext: ( term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > product_prod_o_o ) > list_t5668669291736003536_a_nat > list_t5668669291736003536_a_nat > product_prod_o_o,Ns_list: set_Pr630484882382513799_a_nat > set_Pr630484882382513799_a_nat > set_Pr4991467605864603271_a_nat] :
      ( ! [S7: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Ns5: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
          ( ( S_list @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ S7 ) ) @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ Ns5 ) ) )
          = ( collec8858057866656402194_a_nat
            @ ( produc4945827410475597548_nat_o
              @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] :
                  ( product_fst_o_o
                  @ ( List_ext
                    @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                    @ As
                    @ Bs ) ) ) ) )
     => ( ! [S7: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o,Ns5: term_R1747997765655105994_a_nat > term_R1747997765655105994_a_nat > $o] :
            ( ( Ns_list @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ S7 ) ) @ ( collec9088346736735320338_a_nat @ ( produc4098677624135445996_nat_o @ Ns5 ) ) )
            = ( collec8858057866656402194_a_nat
              @ ( produc4945827410475597548_nat_o
                @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] :
                    ( product_snd_o_o
                    @ ( List_ext
                      @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                      @ As
                      @ Bs ) ) ) ) )
       => ( ! [S7: set_Pr630484882382513799_a_nat,Ns5: set_Pr630484882382513799_a_nat,S10: set_Pr630484882382513799_a_nat,Ns6: set_Pr630484882382513799_a_nat,As3: list_t5668669291736003536_a_nat,Bs3: list_t5668669291736003536_a_nat] :
              ( ( ord_le251666825002368551_a_nat
                @ ( inf_in5381721427322260597_a_nat
                  @ ( produc8862320176311285064_a_nat @ ( set_te3761484132649746399_a_nat @ As3 )
                    @ ^ [Uu: term_R1747997765655105994_a_nat] : ( set_te3761484132649746399_a_nat @ Bs3 ) )
                  @ Ns5 )
                @ Ns6 )
             => ( ( ord_le251666825002368551_a_nat
                  @ ( inf_in5381721427322260597_a_nat
                    @ ( produc8862320176311285064_a_nat @ ( set_te3761484132649746399_a_nat @ As3 )
                      @ ^ [Uu: term_R1747997765655105994_a_nat] : ( set_te3761484132649746399_a_nat @ Bs3 ) )
                    @ S7 )
                  @ S10 )
               => ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As3 @ Bs3 ) @ ( S_list @ Ns5 @ S7 ) )
                 => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As3 @ Bs3 ) @ ( S_list @ Ns6 @ S10 ) ) ) ) )
         => ( ! [S7: set_Pr630484882382513799_a_nat,Ns5: set_Pr630484882382513799_a_nat,S10: set_Pr630484882382513799_a_nat,Ns6: set_Pr630484882382513799_a_nat,As3: list_t5668669291736003536_a_nat,Bs3: list_t5668669291736003536_a_nat] :
                ( ( ord_le251666825002368551_a_nat
                  @ ( inf_in5381721427322260597_a_nat
                    @ ( produc8862320176311285064_a_nat @ ( set_te3761484132649746399_a_nat @ As3 )
                      @ ^ [Uu: term_R1747997765655105994_a_nat] : ( set_te3761484132649746399_a_nat @ Bs3 ) )
                    @ Ns5 )
                  @ Ns6 )
               => ( ( ord_le251666825002368551_a_nat
                    @ ( inf_in5381721427322260597_a_nat
                      @ ( produc8862320176311285064_a_nat @ ( set_te3761484132649746399_a_nat @ As3 )
                        @ ^ [Uu: term_R1747997765655105994_a_nat] : ( set_te3761484132649746399_a_nat @ Bs3 ) )
                      @ S7 )
                    @ S10 )
                 => ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns5 @ S7 ) )
                   => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns6 @ S10 ) ) ) ) )
           => ( list_l7372445390329087539_a_nat @ S_list @ Ns_list @ List_ext ) ) ) ) ) ).

% list_order_extension_impl_axioms.intro
thf(fact_1204_list__order__extension__impl__axioms_Ointro,axiom,
    ! [S_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat,List_ext: ( list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > product_prod_o_o ) > list_l8999537460041914605on_nat > list_l8999537460041914605on_nat > product_prod_o_o,Ns_list: set_Pr7828957961435280891on_nat > set_Pr7828957961435280891on_nat > set_Pr1573952985545973531on_nat] :
      ( ! [S7: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Ns5: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
          ( ( S_list @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ S7 ) ) @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ Ns5 ) ) )
          = ( collec1004640060711064742on_nat
            @ ( produc9222835799945651472_nat_o
              @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] :
                  ( product_fst_o_o
                  @ ( List_ext
                    @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                    @ As
                    @ Bs ) ) ) ) )
     => ( ! [S7: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Ns5: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o] :
            ( ( Ns_list @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ S7 ) ) @ ( collec1293568243156533638on_nat @ ( produc7163990014243126832_nat_o @ Ns5 ) ) )
            = ( collec1004640060711064742on_nat
              @ ( produc9222835799945651472_nat_o
                @ ^ [As: list_l8999537460041914605on_nat,Bs: list_l8999537460041914605on_nat] :
                    ( product_snd_o_o
                    @ ( List_ext
                      @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                      @ As
                      @ Bs ) ) ) ) )
       => ( ! [S7: set_Pr7828957961435280891on_nat,Ns5: set_Pr7828957961435280891on_nat,S10: set_Pr7828957961435280891on_nat,Ns6: set_Pr7828957961435280891on_nat,As3: list_l8999537460041914605on_nat,Bs3: list_l8999537460041914605on_nat] :
              ( ( ord_le6423710186836778395on_nat
                @ ( inf_in505647112809072361on_nat
                  @ ( produc4944392799965481604on_nat @ ( set_li4371160873219908744on_nat @ As3 )
                    @ ^ [Uu: list_S5253404628753938141on_nat] : ( set_li4371160873219908744on_nat @ Bs3 ) )
                  @ Ns5 )
                @ Ns6 )
             => ( ( ord_le6423710186836778395on_nat
                  @ ( inf_in505647112809072361on_nat
                    @ ( produc4944392799965481604on_nat @ ( set_li4371160873219908744on_nat @ As3 )
                      @ ^ [Uu: list_S5253404628753938141on_nat] : ( set_li4371160873219908744on_nat @ Bs3 ) )
                    @ S7 )
                  @ S10 )
               => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As3 @ Bs3 ) @ ( S_list @ Ns5 @ S7 ) )
                 => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As3 @ Bs3 ) @ ( S_list @ Ns6 @ S10 ) ) ) ) )
         => ( ! [S7: set_Pr7828957961435280891on_nat,Ns5: set_Pr7828957961435280891on_nat,S10: set_Pr7828957961435280891on_nat,Ns6: set_Pr7828957961435280891on_nat,As3: list_l8999537460041914605on_nat,Bs3: list_l8999537460041914605on_nat] :
                ( ( ord_le6423710186836778395on_nat
                  @ ( inf_in505647112809072361on_nat
                    @ ( produc4944392799965481604on_nat @ ( set_li4371160873219908744on_nat @ As3 )
                      @ ^ [Uu: list_S5253404628753938141on_nat] : ( set_li4371160873219908744on_nat @ Bs3 ) )
                    @ Ns5 )
                  @ Ns6 )
               => ( ( ord_le6423710186836778395on_nat
                    @ ( inf_in505647112809072361on_nat
                      @ ( produc4944392799965481604on_nat @ ( set_li4371160873219908744on_nat @ As3 )
                        @ ^ [Uu: list_S5253404628753938141on_nat] : ( set_li4371160873219908744on_nat @ Bs3 ) )
                      @ S7 )
                    @ S10 )
                 => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns5 @ S7 ) )
                   => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ As3 @ Bs3 ) @ ( Ns_list @ Ns6 @ S10 ) ) ) ) )
           => ( list_l8897637855595466036on_nat @ S_list @ Ns_list @ List_ext ) ) ) ) ) ).

% list_order_extension_impl_axioms.intro
thf(fact_1205_list__order__extension__impl__axioms_Ointro,axiom,
    ! [S_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o,List_ext: ( $o > $o > product_prod_o_o ) > list_o > list_o > product_prod_o_o,Ns_list: set_Product_prod_o_o > set_Product_prod_o_o > set_Pr6227168374412355847list_o] :
      ( ! [S7: $o > $o > $o,Ns5: $o > $o > $o] :
          ( ( S_list @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ S7 ) ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ Ns5 ) ) )
          = ( collec4435127810633735570list_o
            @ ( produc1052442579958709612st_o_o
              @ ^ [As: list_o,Bs: list_o] :
                  ( product_fst_o_o
                  @ ( List_ext
                    @ ^ [A3: $o,B3: $o] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                    @ As
                    @ Bs ) ) ) ) )
     => ( ! [S7: $o > $o > $o,Ns5: $o > $o > $o] :
            ( ( Ns_list @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ S7 ) ) @ ( collec3167064739498627218od_o_o @ ( produc6197397395684419436_o_o_o @ Ns5 ) ) )
            = ( collec4435127810633735570list_o
              @ ( produc1052442579958709612st_o_o
                @ ^ [As: list_o,Bs: list_o] :
                    ( product_snd_o_o
                    @ ( List_ext
                      @ ^ [A3: $o,B3: $o] : ( product_Pair_o_o @ ( S7 @ A3 @ B3 ) @ ( Ns5 @ A3 @ B3 ) )
                      @ As
                      @ Bs ) ) ) ) )
       => ( ! [S7: set_Product_prod_o_o,Ns5: set_Product_prod_o_o,S10: set_Product_prod_o_o,Ns6: set_Product_prod_o_o,As3: list_o,Bs3: list_o] :
              ( ( ord_le192195778054999463od_o_o
                @ ( inf_in4898592226082374645od_o_o
                  @ ( product_Sigma_o_o @ ( set_o2 @ As3 )
                    @ ^ [Uu: $o] : ( set_o2 @ Bs3 ) )
                  @ Ns5 )
                @ Ns6 )
             => ( ( ord_le192195778054999463od_o_o
                  @ ( inf_in4898592226082374645od_o_o
                    @ ( product_Sigma_o_o @ ( set_o2 @ As3 )
                      @ ^ [Uu: $o] : ( set_o2 @ Bs3 ) )
                    @ S7 )
                  @ S10 )
               => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As3 @ Bs3 ) @ ( S_list @ Ns5 @ S7 ) )
                 => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As3 @ Bs3 ) @ ( S_list @ Ns6 @ S10 ) ) ) ) )
         => ( ! [S7: set_Product_prod_o_o,Ns5: set_Product_prod_o_o,S10: set_Product_prod_o_o,Ns6: set_Product_prod_o_o,As3: list_o,Bs3: list_o] :
                ( ( ord_le192195778054999463od_o_o
                  @ ( inf_in4898592226082374645od_o_o
                    @ ( product_Sigma_o_o @ ( set_o2 @ As3 )
                      @ ^ [Uu: $o] : ( set_o2 @ Bs3 ) )
                    @ Ns5 )
                  @ Ns6 )
               => ( ( ord_le192195778054999463od_o_o
                    @ ( inf_in4898592226082374645od_o_o
                      @ ( product_Sigma_o_o @ ( set_o2 @ As3 )
                        @ ^ [Uu: $o] : ( set_o2 @ Bs3 ) )
                      @ S7 )
                    @ S10 )
                 => ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As3 @ Bs3 ) @ ( Ns_list @ Ns5 @ S7 ) )
                   => ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ As3 @ Bs3 ) @ ( Ns_list @ Ns6 @ S10 ) ) ) ) )
           => ( list_l2181461889072780801ioms_o @ S_list @ Ns_list @ List_ext ) ) ) ) ) ).

% list_order_extension_impl_axioms.intro
thf(fact_1206_mem__Sigma__iff,axiom,
    ! [A: $o,B: $o,A5: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A5 @ B5 ) )
      = ( ( member_o @ A @ A5 )
        & ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1207_mem__Sigma__iff,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,A5: set_te1179867837402157482_a_nat,B5: term_R1747997765655105994_a_nat > set_te1179867837402157482_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ ( produc8862320176311285064_a_nat @ A5 @ B5 ) )
      = ( ( member7260319590701762419_a_nat @ A @ A5 )
        & ( member7260319590701762419_a_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1208_mem__Sigma__iff,axiom,
    ! [A: multis6046679608167549072_a_nat,B: multis6046679608167549072_a_nat,A5: set_mu2955896406520724720_a_nat,B5: multis6046679608167549072_a_nat > set_mu2955896406520724720_a_nat] :
      ( ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A @ B ) @ ( produc591701207991684424_a_nat @ A5 @ B5 ) )
      = ( ( member2753119575031202745_a_nat @ A @ A5 )
        & ( member2753119575031202745_a_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1209_mem__Sigma__iff,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,A5: set_mu5400888710828039123on_nat,B5: multis1856558993800614429on_nat > set_mu5400888710828039123on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( produc3072496586818388228on_nat @ A5 @ B5 ) )
      = ( ( member2988082745645132084on_nat @ A @ A5 )
        & ( member2988082745645132084on_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1210_mem__Sigma__iff,axiom,
    ! [A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,A5: set_li1675222548550825619on_nat,B5: list_S5253404628753938141on_nat > set_li1675222548550825619on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A @ B ) @ ( produc4944392799965481604on_nat @ A5 @ B5 ) )
      = ( ( member86162308477975540on_nat @ A @ A5 )
        & ( member86162308477975540on_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1211_mem__Sigma__iff,axiom,
    ! [A: list_t5668669291736003536_a_nat,B: list_t5668669291736003536_a_nat,A5: set_li148843460256248368_a_nat,B5: list_t5668669291736003536_a_nat > set_li148843460256248368_a_nat] :
      ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A @ B ) @ ( produc2363723359784558408_a_nat @ A5 @ B5 ) )
      = ( ( member4440227334935380217_a_nat @ A @ A5 )
        & ( member4440227334935380217_a_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1212_mem__Sigma__iff,axiom,
    ! [A: produc394670695091157671_a_nat,B: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: produc394670695091157671_a_nat > set_Pr4991467605864603271_a_nat] :
      ( ( member110630379338885904_a_nat @ ( produc925431128169182039_a_nat @ A @ B ) @ ( produc7506687526141623048_a_nat @ A5 @ B5 ) )
      = ( ( member4231271695026834640_a_nat @ A @ A5 )
        & ( member4231271695026834640_a_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1213_mem__Sigma__iff,axiom,
    ! [A: produc394670695091157671_a_nat,B: produc5238312538367656103_a_nat,A5: set_Pr4991467605864603271_a_nat,B5: produc394670695091157671_a_nat > set_Pr8529752790658994311_a_nat] :
      ( ( member4535363888883972368_a_nat @ ( produc6823678566767065431_a_nat @ A @ B ) @ ( produc2994124225532640520_a_nat @ A5 @ B5 ) )
      = ( ( member4231271695026834640_a_nat @ A @ A5 )
        & ( member6928026473249828560_a_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1214_mem__Sigma__iff,axiom,
    ! [A: produc394670695091157671_a_nat,B: produc9176241080893687067on_nat,A5: set_Pr4991467605864603271_a_nat,B5: produc394670695091157671_a_nat > set_Pr3573155178515677307on_nat] :
      ( ( member2295143444729160836on_nat @ ( produc2923303845683488203on_nat @ A @ B ) @ ( produc3112540394484420860on_nat @ A5 @ B5 ) )
      = ( ( member4231271695026834640_a_nat @ A @ A5 )
        & ( member4278285197320822468on_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1215_mem__Sigma__iff,axiom,
    ! [A: produc394670695091157671_a_nat,B: produc5580075136493545115on_nat,A5: set_Pr4991467605864603271_a_nat,B5: produc394670695091157671_a_nat > set_Pr7828957961435280891on_nat] :
      ( ( member3390692376801263108on_nat @ ( produc7154530732722243403on_nat @ A @ B ) @ ( produc4848419089484379772on_nat @ A5 @ B5 ) )
      = ( ( member4231271695026834640_a_nat @ A @ A5 )
        & ( member4221009242412147780on_nat @ B @ ( B5 @ A ) ) ) ) ).

% mem_Sigma_iff
thf(fact_1216_SigmaI,axiom,
    ! [A: $o,A5: set_o,B: $o,B5: $o > set_o] :
      ( ( member_o @ A @ A5 )
     => ( ( member_o @ B @ ( B5 @ A ) )
       => ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1217_SigmaI,axiom,
    ! [A: term_R1747997765655105994_a_nat,A5: set_te1179867837402157482_a_nat,B: term_R1747997765655105994_a_nat,B5: term_R1747997765655105994_a_nat > set_te1179867837402157482_a_nat] :
      ( ( member7260319590701762419_a_nat @ A @ A5 )
     => ( ( member7260319590701762419_a_nat @ B @ ( B5 @ A ) )
       => ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ ( produc8862320176311285064_a_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1218_SigmaI,axiom,
    ! [A: multis6046679608167549072_a_nat,A5: set_mu2955896406520724720_a_nat,B: multis6046679608167549072_a_nat,B5: multis6046679608167549072_a_nat > set_mu2955896406520724720_a_nat] :
      ( ( member2753119575031202745_a_nat @ A @ A5 )
     => ( ( member2753119575031202745_a_nat @ B @ ( B5 @ A ) )
       => ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ A @ B ) @ ( produc591701207991684424_a_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1219_SigmaI,axiom,
    ! [A: multis1856558993800614429on_nat,A5: set_mu5400888710828039123on_nat,B: multis1856558993800614429on_nat,B5: multis1856558993800614429on_nat > set_mu5400888710828039123on_nat] :
      ( ( member2988082745645132084on_nat @ A @ A5 )
     => ( ( member2988082745645132084on_nat @ B @ ( B5 @ A ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( produc3072496586818388228on_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1220_SigmaI,axiom,
    ! [A: list_S5253404628753938141on_nat,A5: set_li1675222548550825619on_nat,B: list_S5253404628753938141on_nat,B5: list_S5253404628753938141on_nat > set_li1675222548550825619on_nat] :
      ( ( member86162308477975540on_nat @ A @ A5 )
     => ( ( member86162308477975540on_nat @ B @ ( B5 @ A ) )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A @ B ) @ ( produc4944392799965481604on_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1221_SigmaI,axiom,
    ! [A: list_t5668669291736003536_a_nat,A5: set_li148843460256248368_a_nat,B: list_t5668669291736003536_a_nat,B5: list_t5668669291736003536_a_nat > set_li148843460256248368_a_nat] :
      ( ( member4440227334935380217_a_nat @ A @ A5 )
     => ( ( member4440227334935380217_a_nat @ B @ ( B5 @ A ) )
       => ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A @ B ) @ ( produc2363723359784558408_a_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1222_SigmaI,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B: produc394670695091157671_a_nat,B5: produc394670695091157671_a_nat > set_Pr4991467605864603271_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( member4231271695026834640_a_nat @ B @ ( B5 @ A ) )
       => ( member110630379338885904_a_nat @ ( produc925431128169182039_a_nat @ A @ B ) @ ( produc7506687526141623048_a_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1223_SigmaI,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B: produc5238312538367656103_a_nat,B5: produc394670695091157671_a_nat > set_Pr8529752790658994311_a_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( member6928026473249828560_a_nat @ B @ ( B5 @ A ) )
       => ( member4535363888883972368_a_nat @ ( produc6823678566767065431_a_nat @ A @ B ) @ ( produc2994124225532640520_a_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1224_SigmaI,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B: produc9176241080893687067on_nat,B5: produc394670695091157671_a_nat > set_Pr3573155178515677307on_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( member4278285197320822468on_nat @ B @ ( B5 @ A ) )
       => ( member2295143444729160836on_nat @ ( produc2923303845683488203on_nat @ A @ B ) @ ( produc3112540394484420860on_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1225_SigmaI,axiom,
    ! [A: produc394670695091157671_a_nat,A5: set_Pr4991467605864603271_a_nat,B: produc5580075136493545115on_nat,B5: produc394670695091157671_a_nat > set_Pr7828957961435280891on_nat] :
      ( ( member4231271695026834640_a_nat @ A @ A5 )
     => ( ( member4221009242412147780on_nat @ B @ ( B5 @ A ) )
       => ( member3390692376801263108on_nat @ ( produc7154530732722243403on_nat @ A @ B ) @ ( produc4848419089484379772on_nat @ A5 @ B5 ) ) ) ) ).

% SigmaI
thf(fact_1226_Collect__case__prod,axiom,
    ! [P2: $o > $o,Q: $o > $o] :
      ( ( collec3167064739498627218od_o_o
        @ ( produc6197397395684419436_o_o_o
          @ ^ [A3: $o,B3: $o] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( product_Sigma_o_o @ ( collect_o @ P2 )
        @ ^ [Uu: $o] : ( collect_o @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1227_Collect__case__prod,axiom,
    ! [P2: term_R1747997765655105994_a_nat > $o,Q: term_R1747997765655105994_a_nat > $o] :
      ( ( collec9088346736735320338_a_nat
        @ ( produc4098677624135445996_nat_o
          @ ^ [A3: term_R1747997765655105994_a_nat,B3: term_R1747997765655105994_a_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc8862320176311285064_a_nat @ ( collec1489781475602551989_a_nat @ P2 )
        @ ^ [Uu: term_R1747997765655105994_a_nat] : ( collec1489781475602551989_a_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1228_Collect__case__prod,axiom,
    ! [P2: list_t5668669291736003536_a_nat > $o,Q: list_t5668669291736003536_a_nat > $o] :
      ( ( collec8858057866656402194_a_nat
        @ ( produc4945827410475597548_nat_o
          @ ^ [A3: list_t5668669291736003536_a_nat,B3: list_t5668669291736003536_a_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc2363723359784558408_a_nat @ ( collec1189993861229692219_a_nat @ P2 )
        @ ^ [Uu: list_t5668669291736003536_a_nat] : ( collec1189993861229692219_a_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1229_Collect__case__prod,axiom,
    ! [P2: list_S5253404628753938141on_nat > $o,Q: list_S5253404628753938141on_nat > $o] :
      ( ( collec1293568243156533638on_nat
        @ ( produc7163990014243126832_nat_o
          @ ^ [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc4944392799965481604on_nat @ ( collec7158715823234635826on_nat @ P2 )
        @ ^ [Uu: list_S5253404628753938141on_nat] : ( collec7158715823234635826on_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1230_Collect__case__prod,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ( collec1156221682988523858_a_nat
        @ ( produc6883322048168523052_nat_o
          @ ^ [A3: produc3043957382837978535_a_nat,B3: produc3043957382837978535_a_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc2756613766766351624_a_nat @ ( collec9088346736735320338_a_nat @ P2 )
        @ ^ [Uu: produc3043957382837978535_a_nat] : ( collec9088346736735320338_a_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1231_Collect__case__prod,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ( collec8422034144856598610_a_nat
        @ ( produc6383932149206365740_nat_o
          @ ^ [A3: produc394670695091157671_a_nat,B3: produc3043957382837978535_a_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc3150125322957360904_a_nat @ ( collec8858057866656402194_a_nat @ P2 )
        @ ^ [Uu: produc394670695091157671_a_nat] : ( collec9088346736735320338_a_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1232_Collect__case__prod,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,Q: produc394670695091157671_a_nat > $o] :
      ( ( collec2067242455984674386_a_nat
        @ ( produc1643750412922618924_nat_o
          @ ^ [A3: produc3043957382837978535_a_nat,B3: produc394670695091157671_a_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc731014684085993736_a_nat @ ( collec9088346736735320338_a_nat @ P2 )
        @ ^ [Uu: produc3043957382837978535_a_nat] : ( collec8858057866656402194_a_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1233_Collect__case__prod,axiom,
    ! [P2: produc3043957382837978535_a_nat > $o,Q: produc5580075136493545115on_nat > $o] :
      ( ( collec1534616496477947974on_nat
        @ ( produc5750995641172989240_nat_o
          @ ^ [A3: produc3043957382837978535_a_nat,B3: produc5580075136493545115on_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc4469673562123320444on_nat @ ( collec9088346736735320338_a_nat @ P2 )
        @ ^ [Uu: produc3043957382837978535_a_nat] : ( collec1293568243156533638on_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1234_Collect__case__prod,axiom,
    ! [P2: produc5580075136493545115on_nat > $o,Q: produc3043957382837978535_a_nat > $o] :
      ( ( collec1217699247902277470_a_nat
        @ ( produc9121615584184547360_nat_o
          @ ^ [A3: produc5580075136493545115on_nat,B3: produc3043957382837978535_a_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc914789730243609492_a_nat @ ( collec1293568243156533638on_nat @ P2 )
        @ ^ [Uu: produc5580075136493545115on_nat] : ( collec9088346736735320338_a_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1235_Collect__case__prod,axiom,
    ! [P2: produc394670695091157671_a_nat > $o,Q: produc394670695091157671_a_nat > $o] :
      ( ( collec8768692579076431186_a_nat
        @ ( produc8599003699881166124_nat_o
          @ ^ [A3: produc394670695091157671_a_nat,B3: produc394670695091157671_a_nat] :
              ( ( P2 @ A3 )
              & ( Q @ B3 ) ) ) )
      = ( produc7506687526141623048_a_nat @ ( collec8858057866656402194_a_nat @ P2 )
        @ ^ [Uu: produc394670695091157671_a_nat] : ( collec8858057866656402194_a_nat @ Q ) ) ) ).

% Collect_case_prod
thf(fact_1236_SigmaE2,axiom,
    ! [A: list_t5668669291736003536_a_nat,B: list_t5668669291736003536_a_nat,A5: set_li148843460256248368_a_nat,B5: list_t5668669291736003536_a_nat > set_li148843460256248368_a_nat] :
      ( ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ A @ B ) @ ( produc2363723359784558408_a_nat @ A5 @ B5 ) )
     => ~ ( ( member4440227334935380217_a_nat @ A @ A5 )
         => ~ ( member4440227334935380217_a_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1237_SigmaE2,axiom,
    ! [A: $o,B: $o,A5: set_o,B5: $o > set_o] :
      ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ A @ B ) @ ( product_Sigma_o_o @ A5 @ B5 ) )
     => ~ ( ( member_o @ A @ A5 )
         => ~ ( member_o @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1238_SigmaE2,axiom,
    ! [A: term_R1747997765655105994_a_nat,B: term_R1747997765655105994_a_nat,A5: set_te1179867837402157482_a_nat,B5: term_R1747997765655105994_a_nat > set_te1179867837402157482_a_nat] :
      ( ( member6937586146271652048_a_nat @ ( produc4880773030396931735_a_nat @ A @ B ) @ ( produc8862320176311285064_a_nat @ A5 @ B5 ) )
     => ~ ( ( member7260319590701762419_a_nat @ A @ A5 )
         => ~ ( member7260319590701762419_a_nat @ B @ ( B5 @ A ) ) ) ) ).

% SigmaE2
thf(fact_1239_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_1240_Nat_Oadd__0__right,axiom,
    ! [M2: nat] :
      ( ( plus_plus_nat @ M2 @ zero_zero_nat )
      = M2 ) ).

% Nat.add_0_right
thf(fact_1241_nat__add__left__cancel__le,axiom,
    ! [K: nat,M2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_1242_diff__diff__left,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I @ J ) @ K )
      = ( minus_minus_nat @ I @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_1243_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_1244_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_1245_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_1246_diff__add__inverse2,axiom,
    ! [M2: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M2 @ N2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse2
thf(fact_1247_diff__add__inverse,axiom,
    ! [N2: nat,M2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M2 ) @ N2 )
      = M2 ) ).

% diff_add_inverse
thf(fact_1248_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_1249_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_1250_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_1251_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_1252_add__leE,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M2 @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_1253_le__add1,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M2 ) ) ).

% le_add1
thf(fact_1254_le__add2,axiom,
    ! [N2: nat,M2: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M2 @ N2 ) ) ).

% le_add2
thf(fact_1255_add__leD1,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ M2 @ N2 ) ) ).

% add_leD1
thf(fact_1256_add__leD2,axiom,
    ! [M2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_1257_le__Suc__ex,axiom,
    ! [K: nat,L: nat] :
      ( ( ord_less_eq_nat @ K @ L )
     => ? [N3: nat] :
          ( L
          = ( plus_plus_nat @ K @ N3 ) ) ) ).

% le_Suc_ex
thf(fact_1258_add__le__mono,axiom,
    ! [I: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_1259_add__le__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_1260_trans__le__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_le_add1
thf(fact_1261_trans__le__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_le_add2
thf(fact_1262_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M: nat,N: nat] :
        ? [K2: nat] :
          ( N
          = ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% nat_le_iff_add
thf(fact_1263_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_1264_Nat_Ole__imp__diff__is__add,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( minus_minus_nat @ J @ I )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_1265_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_1266_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_1267_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_1268_le__diff__conv,axiom,
    ! [J: nat,K: nat,I: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I @ K ) ) ) ).

% le_diff_conv
thf(fact_1269_Euclid__induct,axiom,
    ! [P2: nat > nat > $o,A: nat,B: nat] :
      ( ! [A2: nat,B2: nat] :
          ( ( P2 @ A2 @ B2 )
          = ( P2 @ B2 @ A2 ) )
     => ( ! [A2: nat] : ( P2 @ A2 @ zero_zero_nat )
       => ( ! [A2: nat,B2: nat] :
              ( ( P2 @ A2 @ B2 )
             => ( P2 @ A2 @ ( plus_plus_nat @ A2 @ B2 ) ) )
         => ( P2 @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_1270_or2__def,axiom,
    ( multis3616315163204145547ir_or2
    = ( ^ [A3: product_prod_o_o,B3: product_prod_o_o] :
          ( product_Pair_o_o
          @ ( ( product_fst_o_o @ A3 )
            | ( product_fst_o_o @ B3 ) )
          @ ( ( product_snd_o_o @ A3 )
            | ( product_snd_o_o @ B3 ) ) ) ) ) ).

% or2_def

% Conjectures (1)
thf(conj_0,conjecture,
    ( member4231271695026834640_a_nat @ ( produc5685097156399411607_a_nat @ as @ bs )
    @ ( collec8858057866656402194_a_nat
      @ ( produc4945827410475597548_nat_o
        @ ^ [As: list_t5668669291736003536_a_nat,Bs: list_t5668669291736003536_a_nat] :
            ( member6928026473249828560_a_nat @ ( produc7112426491196973975_a_nat @ ( mset_t7141044528732312287_a_nat @ As ) @ ( mset_t7141044528732312287_a_nat @ Bs ) )
            @ ( multis3666123592011920481_a_nat
              @ ( collec9088346736735320338_a_nat
                @ ( produc4098677624135445996_nat_o
                  @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
                      ( product_snd_o_o
                      @ ( rpo_RP6885983473509681171_a_nat
                        @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
                        @ ^ [F2: produc2248741730625028719ms_nat] : $true
                        @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
                        @ zero_zero_nat
                        @ S4
                        @ T ) ) ) )
              @ ( collec9088346736735320338_a_nat
                @ ( produc4098677624135445996_nat_o
                  @ ^ [S4: term_R1747997765655105994_a_nat,T: term_R1747997765655105994_a_nat] :
                      ( product_fst_o_o
                      @ ( rpo_RP6885983473509681171_a_nat
                        @ ^ [F2: produc2248741730625028719ms_nat,G2: produc2248741730625028719ms_nat] : ( product_Pair_o_o @ $false @ $true )
                        @ ^ [F2: produc2248741730625028719ms_nat] : $true
                        @ ^ [Uu: produc2248741730625028719ms_nat] : order_Mul
                        @ zero_zero_nat
                        @ S4
                        @ T ) ) ) ) ) ) ) ) ) ).

%------------------------------------------------------------------------------