%------------------------------------------------------------------------------
% 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/0003_Multiset_Ordering_NP_Hard/prob_00222_010759__14165114_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1955 ( 500 unt; 682 typ;   0 def)
%            Number of atoms       : 4733 (2041 equ;   0 cnn)
%            Maximal formula atoms :   30 (   3 avg)
%            Number of connectives : 20075 ( 762   ~; 117   |; 483   &;16304   @)
%                                         (   0 <=>;2409  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   9 avg)
%            Number of types       :  159 ( 158 usr)
%            Number of type conns  : 1176 (1176   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  527 ( 524 usr;  39 con; 0-4 aty)
%            Number of variables   : 5422 (  25   ^;4993   !; 404   ?;5422   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:40.947
%------------------------------------------------------------------------------
% Could-be-implicit typings (158)
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% Explicit typings (524)
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thf(sy_c_AList_Oupdate_001_062_It__Nat__Onat_Mt__Nat__Onat_J_001t__Nat__Onat,type,
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    update6820902895451041720on_nat: multis1856558993800614429on_nat > multis1856558993800614429on_nat > list_P1915136757390359201on_nat > list_P1915136757390359201on_nat ).

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    member6036888723768207770on_nat: produc5752899724758015619on_nat > set_Pr5934048715616008505on_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__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_J,type,
    member8058359081245219834on_nat: produc5436805147852408291on_nat > set_Pr7248776618047545241on_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__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_v_NSS,type,
    nss: list_P9127870862658512897on_nat ).

thf(sy_v_NS____,type,
    ns: list_P9127870862658512897on_nat ).

thf(sy_v_S,type,
    s: list_P9127870862658512897on_nat ).

thf(sy_v_cnf,type,
    cnf: list_l8336264072933094989od_a_o ).

thf(sy_v_left,type,
    left: list_S5253404628753938141on_nat ).

thf(sy_v_right,type,
    right: list_S5253404628753938141on_nat ).

thf(sy_v_xs____,type,
    xs: list_a ).

% Relevant facts (1272)
thf(fact_0_one__direction_I1_J,axiom,
    ( ? [Beta: a > $o] : ( propos4091448439431948687_cnf_a @ Beta @ cnf )
   => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ left ) @ ( mset_S7907735292211298936on_nat @ right ) ) @ ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ nss ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ).

% one_direction(1)
thf(fact_1_other__direction,axiom,
    ( ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ left ) @ ( mset_S7907735292211298936on_nat @ right ) ) @ ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ nss ) @ ( set_Pr937530902923597456on_nat @ s ) ) )
      | ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ left ) @ ( mset_S7907735292211298936on_nat @ right ) ) @ ( multis1620213760974180726on_nat @ ( set_Pr937530902923597456on_nat @ nss ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) )
   => ? [Beta2: a > $o] : ( propos4091448439431948687_cnf_a @ Beta2 @ cnf ) ) ).

% other_direction
thf(fact_2_one__direction_I2_J,axiom,
    ( ? [Beta: a > $o] : ( propos4091448439431948687_cnf_a @ Beta @ cnf )
   => ( ( cnf != nil_li1372968520647675725od_a_o )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ left ) @ ( mset_S7907735292211298936on_nat @ right ) ) @ ( multis1620213760974180726on_nat @ ( set_Pr937530902923597456on_nat @ nss ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ) ).

% one_direction(2)
thf(fact_3_prod_Oinject,axiom,
    ! [X1: list_l8999537460041914605on_nat,X2: list_P5508476027986415439on_nat,Y1: list_l8999537460041914605on_nat,Y2: list_P5508476027986415439on_nat] :
      ( ( ( produc3605646969090963669on_nat @ X1 @ X2 )
        = ( produc3605646969090963669on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_4_prod_Oinject,axiom,
    ! [X1: list_nat_nat,X2: list_nat,Y1: list_nat_nat,Y2: list_nat] :
      ( ( ( produc7978589510830832328st_nat @ X1 @ X2 )
        = ( produc7978589510830832328st_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_5_prod_Oinject,axiom,
    ! [X1: produc1313295348606329723on_nat > produc1313295348606329723on_nat,X2: produc1714117091004681831on_nat,Y1: produc1313295348606329723on_nat > produc1313295348606329723on_nat,Y2: produc1714117091004681831on_nat] :
      ( ( ( produc5429469637799208745on_nat @ X1 @ X2 )
        = ( produc5429469637799208745on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_6_prod_Oinject,axiom,
    ! [X1: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,X2: list_l8336264072933094989od_a_o,Y1: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Y2: list_l8336264072933094989od_a_o] :
      ( ( ( produc6784936493919748534od_a_o @ X1 @ X2 )
        = ( produc6784936493919748534od_a_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_7_prod_Oinject,axiom,
    ! [X1: nat > nat,X2: list_P5366131564312172150at_nat,Y1: nat > nat,Y2: list_P5366131564312172150at_nat] :
      ( ( ( produc17425388850148510at_nat @ X1 @ X2 )
        = ( produc17425388850148510at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_8_prod_Oinject,axiom,
    ! [X1: multis1856558993800614429on_nat,X2: multis1856558993800614429on_nat,Y1: multis1856558993800614429on_nat,Y2: multis1856558993800614429on_nat] :
      ( ( ( produc5200047356169809875on_nat @ X1 @ X2 )
        = ( produc5200047356169809875on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_9_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X2: produc118098540238352959on_nat,Y1: list_S5253404628753938141on_nat,Y2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ X1 @ X2 )
        = ( produc1545487600272464053on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X2: produc1714117091004681831on_nat,Y1: list_S5253404628753938141on_nat,Y2: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ X1 @ X2 )
        = ( produc4176415733022414961on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: list_P9127870862658512897on_nat,X2: list_P9127870862658512897on_nat,Y1: list_P9127870862658512897on_nat,Y2: list_P9127870862658512897on_nat] :
      ( ( ( produc7152764266822961111on_nat @ X1 @ X2 )
        = ( produc7152764266822961111on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: nat > nat,X2: nat,Y1: nat > nat,Y2: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X2 )
        = ( produc72220940542539688at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
    ! [A: list_l8999537460041914605on_nat,B: list_P5508476027986415439on_nat,A2: list_l8999537460041914605on_nat,B2: list_P5508476027986415439on_nat] :
      ( ( ( produc3605646969090963669on_nat @ A @ B )
        = ( produc3605646969090963669on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: list_nat_nat,B: list_nat,A2: list_nat_nat,B2: list_nat] :
      ( ( ( produc7978589510830832328st_nat @ A @ B )
        = ( produc7978589510830832328st_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B: produc1714117091004681831on_nat,A2: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B2: produc1714117091004681831on_nat] :
      ( ( ( produc5429469637799208745on_nat @ A @ B )
        = ( produc5429469637799208745on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
    ! [A: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,B: list_l8336264072933094989od_a_o,A2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,B2: list_l8336264072933094989od_a_o] :
      ( ( ( produc6784936493919748534od_a_o @ A @ B )
        = ( produc6784936493919748534od_a_o @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: list_P5366131564312172150at_nat,A2: nat > nat,B2: list_P5366131564312172150at_nat] :
      ( ( ( produc17425388850148510at_nat @ A @ B )
        = ( produc17425388850148510at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
      ( ( ( produc5200047356169809875on_nat @ A @ B )
        = ( produc5200047356169809875on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_20_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat,A2: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B )
        = ( produc4176415733022414961on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_21_old_Oprod_Oinject,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,A2: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( ( produc7152764266822961111on_nat @ A @ B )
        = ( produc7152764266822961111on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_22_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_23_mset__eq__setD,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( ( mset_l1352240747471131016on_nat @ Xs )
        = ( mset_l1352240747471131016on_nat @ Ys ) )
     => ( ( set_li4371160873219908744on_nat @ Xs )
        = ( set_li4371160873219908744on_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_24_mset__eq__setD,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( set_nat2 @ Xs )
        = ( set_nat2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_25_mset__eq__setD,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat] :
      ( ( ( mset_nat_nat @ Xs )
        = ( mset_nat_nat @ Ys ) )
     => ( ( set_nat_nat2 @ Xs )
        = ( set_nat_nat2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_26_mset__eq__setD,axiom,
    ! [Xs: list_P6215481986061871491on_nat,Ys: list_P6215481986061871491on_nat] :
      ( ( ( mset_P43429913489005074on_nat @ Xs )
        = ( mset_P43429913489005074on_nat @ Ys ) )
     => ( ( set_Pr4584522749497345810on_nat @ Xs )
        = ( set_Pr4584522749497345810on_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_27_mset__eq__setD,axiom,
    ! [Xs: list_P5366131564312172150at_nat,Ys: list_P5366131564312172150at_nat] :
      ( ( ( mset_P2828777261068555141at_nat @ Xs )
        = ( mset_P2828777261068555141at_nat @ Ys ) )
     => ( ( set_Pr1941165898809491589at_nat @ Xs )
        = ( set_Pr1941165898809491589at_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_28_mset__eq__setD,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( ( mset_m1293634003157817032on_nat @ Xs )
        = ( mset_m1293634003157817032on_nat @ Ys ) )
     => ( ( set_mu1790088485793245640on_nat @ Xs )
        = ( set_mu1790088485793245640on_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_29_mset__eq__setD,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( ( mset_l2262431794812116886on_nat @ Xs )
        = ( mset_l2262431794812116886on_nat @ Ys ) )
     => ( ( set_li8154783781463198870on_nat @ Xs )
        = ( set_li8154783781463198870on_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_30_mset__eq__setD,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( mset_l8139550355095933148od_a_o @ Xs )
        = ( mset_l8139550355095933148od_a_o @ Ys ) )
     => ( ( set_li2629784761229999580od_a_o @ Xs )
        = ( set_li2629784761229999580od_a_o @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_31_mset__eq__setD,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( mset_P6964650582947932560on_nat @ Xs )
        = ( mset_P6964650582947932560on_nat @ Ys ) )
     => ( ( set_Pr937530902923597456on_nat @ Xs )
        = ( set_Pr937530902923597456on_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_32_mset__eq__setD,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs )
        = ( mset_S7907735292211298936on_nat @ Ys ) )
     => ( ( set_Su4848624856859743608on_nat @ Xs )
        = ( set_Su4848624856859743608on_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_33_NSS,axiom,
    ( nss
    = ( append2865821678206098582on_nat @ ns @ s ) ) ).

% NSS
thf(fact_34_xs__def,axiom,
    ( xs
    = ( multis4741186478440321083_cnf_a @ cnf ) ) ).

% xs_def
thf(fact_35_assms,axiom,
    ( ( multis4221440410396274297_cnf_a @ cnf )
    = ( produc1545487600272464053on_nat @ left @ ( produc4176415733022414961on_nat @ right @ ( produc7152764266822961111on_nat @ s @ nss ) ) ) ) ).

% assms
thf(fact_36_s__ns__mul__ext,axiom,
    ! [A3: multis1714066462791002893od_a_o,B3: multis1714066462791002893od_a_o,Ns: set_Pr7420778282784443591od_a_o,S: set_Pr7420778282784443591od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ A3 @ B3 ) @ ( multis7840163239927194462od_a_o @ Ns @ S ) )
     => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ A3 @ B3 ) @ ( multis5290879534621336956od_a_o @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_37_s__ns__mul__ext,axiom,
    ! [A3: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis3147760155352349208on_nat @ Ns @ S ) )
     => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_38_s__ns__mul__ext,axiom,
    ! [A3: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis5680658968000484038on_nat @ Ns @ S ) )
     => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis3835232632530192936on_nat @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_39_s__ns__mul__ext,axiom,
    ! [A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis1332936165424366514on_nat @ Ns @ S ) )
     => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis3046611884734373584on_nat @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_40_s__ns__mul__ext,axiom,
    ! [A3: multis4835999835052358849on_nat,B3: multis4835999835052358849on_nat,Ns: set_Pr8202701763255893447on_nat,S: set_Pr8202701763255893447on_nat] :
      ( ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ A3 @ B3 ) @ ( multis9203262125639806226on_nat @ Ns @ S ) )
     => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ A3 @ B3 ) @ ( multis762940313840963120on_nat @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_41_s__ns__mul__ext,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ).

% s_ns_mul_ext
thf(fact_42_ex__mset,axiom,
    ! [X: multis4286687964247850563on_nat] :
    ? [Xs2: list_P6215481986061871491on_nat] :
      ( ( mset_P43429913489005074on_nat @ Xs2 )
      = X ) ).

% ex_mset
thf(fact_43_ex__mset,axiom,
    ! [X: multis4792816872523369270at_nat] :
    ? [Xs2: list_P5366131564312172150at_nat] :
      ( ( mset_P2828777261068555141at_nat @ Xs2 )
      = X ) ).

% ex_mset
thf(fact_44_ex__mset,axiom,
    ! [X: multis6850599868085929325on_nat] :
    ? [Xs2: list_m351779702951725613on_nat] :
      ( ( mset_m1293634003157817032on_nat @ Xs2 )
      = X ) ).

% ex_mset
thf(fact_45_ex__mset,axiom,
    ! [X: multis5168333519641040071on_nat] :
    ? [Xs2: list_l1726306699111145479on_nat] :
      ( ( mset_l2262431794812116886on_nat @ Xs2 )
      = X ) ).

% ex_mset
thf(fact_46_ex__mset,axiom,
    ! [X: multis1714066462791002893od_a_o] :
    ? [Xs2: list_l8336264072933094989od_a_o] :
      ( ( mset_l8139550355095933148od_a_o @ Xs2 )
      = X ) ).

% ex_mset
thf(fact_47_ex__mset,axiom,
    ! [X: multis1856558993800614429on_nat] :
    ? [Xs2: list_S5253404628753938141on_nat] :
      ( ( mset_S7907735292211298936on_nat @ Xs2 )
      = X ) ).

% ex_mset
thf(fact_48_ex__mset,axiom,
    ! [X: multis4835999835052358849on_nat] :
    ? [Xs2: list_P9127870862658512897on_nat] :
      ( ( mset_P6964650582947932560on_nat @ Xs2 )
      = X ) ).

% ex_mset
thf(fact_49_old_Oprod_Oexhaust,axiom,
    ! [Y: produc6470441716360615453on_nat] :
      ~ ! [A4: list_l8999537460041914605on_nat,B4: list_P5508476027986415439on_nat] :
          ( Y
         != ( produc3605646969090963669on_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_50_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4155638208356863248st_nat] :
      ~ ! [A4: list_nat_nat,B4: list_nat] :
          ( Y
         != ( produc7978589510830832328st_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_51_old_Oprod_Oexhaust,axiom,
    ! [Y: produc5902793192644279159on_nat] :
      ~ ! [A4: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B4: produc1714117091004681831on_nat] :
          ( Y
         != ( produc5429469637799208745on_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_52_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8998563774250336198od_a_o] :
      ~ ! [A4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,B4: list_l8336264072933094989od_a_o] :
          ( Y
         != ( produc6784936493919748534od_a_o @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_53_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1458239053189343596at_nat] :
      ~ ! [A4: nat > nat,B4: list_P5366131564312172150at_nat] :
          ( Y
         != ( produc17425388850148510at_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_54_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9176241080893687067on_nat] :
      ~ ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat] :
          ( Y
         != ( produc5200047356169809875on_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_55_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A4: list_S5253404628753938141on_nat,B4: produc118098540238352959on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_56_old_Oprod_Oexhaust,axiom,
    ! [Y: produc118098540238352959on_nat] :
      ~ ! [A4: list_S5253404628753938141on_nat,B4: produc1714117091004681831on_nat] :
          ( Y
         != ( produc4176415733022414961on_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_57_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1714117091004681831on_nat] :
      ~ ! [A4: list_P9127870862658512897on_nat,B4: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc7152764266822961111on_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_58_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8199716216217303280at_nat] :
      ~ ! [A4: nat > nat,B4: nat] :
          ( Y
         != ( produc72220940542539688at_nat @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_59_surj__pair,axiom,
    ! [P: produc6470441716360615453on_nat] :
    ? [X3: list_l8999537460041914605on_nat,Y3: list_P5508476027986415439on_nat] :
      ( P
      = ( produc3605646969090963669on_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_60_surj__pair,axiom,
    ! [P: produc4155638208356863248st_nat] :
    ? [X3: list_nat_nat,Y3: list_nat] :
      ( P
      = ( produc7978589510830832328st_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_61_surj__pair,axiom,
    ! [P: produc5902793192644279159on_nat] :
    ? [X3: produc1313295348606329723on_nat > produc1313295348606329723on_nat,Y3: produc1714117091004681831on_nat] :
      ( P
      = ( produc5429469637799208745on_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_62_surj__pair,axiom,
    ! [P: produc8998563774250336198od_a_o] :
    ? [X3: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Y3: list_l8336264072933094989od_a_o] :
      ( P
      = ( produc6784936493919748534od_a_o @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_63_surj__pair,axiom,
    ! [P: produc1458239053189343596at_nat] :
    ? [X3: nat > nat,Y3: list_P5366131564312172150at_nat] :
      ( P
      = ( produc17425388850148510at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_64_surj__pair,axiom,
    ! [P: produc9176241080893687067on_nat] :
    ? [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat] :
      ( P
      = ( produc5200047356169809875on_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_65_surj__pair,axiom,
    ! [P: produc8325617224762090749on_nat] :
    ? [X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat] :
      ( P
      = ( produc1545487600272464053on_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_66_surj__pair,axiom,
    ! [P: produc118098540238352959on_nat] :
    ? [X3: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat] :
      ( P
      = ( produc4176415733022414961on_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_67_surj__pair,axiom,
    ! [P: produc1714117091004681831on_nat] :
    ? [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
      ( P
      = ( produc7152764266822961111on_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_68_surj__pair,axiom,
    ! [P: produc8199716216217303280at_nat] :
    ? [X3: nat > nat,Y3: nat] :
      ( P
      = ( produc72220940542539688at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_69_prod__cases,axiom,
    ! [P2: produc6470441716360615453on_nat > $o,P: produc6470441716360615453on_nat] :
      ( ! [A4: list_l8999537460041914605on_nat,B4: list_P5508476027986415439on_nat] : ( P2 @ ( produc3605646969090963669on_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_70_prod__cases,axiom,
    ! [P2: produc4155638208356863248st_nat > $o,P: produc4155638208356863248st_nat] :
      ( ! [A4: list_nat_nat,B4: list_nat] : ( P2 @ ( produc7978589510830832328st_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_71_prod__cases,axiom,
    ! [P2: produc5902793192644279159on_nat > $o,P: produc5902793192644279159on_nat] :
      ( ! [A4: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B4: produc1714117091004681831on_nat] : ( P2 @ ( produc5429469637799208745on_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_72_prod__cases,axiom,
    ! [P2: produc8998563774250336198od_a_o > $o,P: produc8998563774250336198od_a_o] :
      ( ! [A4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,B4: list_l8336264072933094989od_a_o] : ( P2 @ ( produc6784936493919748534od_a_o @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_73_prod__cases,axiom,
    ! [P2: produc1458239053189343596at_nat > $o,P: produc1458239053189343596at_nat] :
      ( ! [A4: nat > nat,B4: list_P5366131564312172150at_nat] : ( P2 @ ( produc17425388850148510at_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_74_prod__cases,axiom,
    ! [P2: produc9176241080893687067on_nat > $o,P: produc9176241080893687067on_nat] :
      ( ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat] : ( P2 @ ( produc5200047356169809875on_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_75_prod__cases,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ! [A4: list_S5253404628753938141on_nat,B4: produc118098540238352959on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_76_prod__cases,axiom,
    ! [P2: produc118098540238352959on_nat > $o,P: produc118098540238352959on_nat] :
      ( ! [A4: list_S5253404628753938141on_nat,B4: produc1714117091004681831on_nat] : ( P2 @ ( produc4176415733022414961on_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_77_prod__cases,axiom,
    ! [P2: produc1714117091004681831on_nat > $o,P: produc1714117091004681831on_nat] :
      ( ! [A4: list_P9127870862658512897on_nat,B4: list_P9127870862658512897on_nat] : ( P2 @ ( produc7152764266822961111on_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_78_prod__cases,axiom,
    ! [P2: produc8199716216217303280at_nat > $o,P: produc8199716216217303280at_nat] :
      ( ! [A4: nat > nat,B4: nat] : ( P2 @ ( produc72220940542539688at_nat @ A4 @ B4 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_79_prod__induct4,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X4: produc8325617224762090749on_nat] :
      ( ! [A4: list_S5253404628753938141on_nat,B4: list_S5253404628753938141on_nat,C: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A4 @ ( produc4176415733022414961on_nat @ B4 @ ( produc7152764266822961111on_nat @ C @ D ) ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct4
thf(fact_80_prod__induct3,axiom,
    ! [P2: produc5902793192644279159on_nat > $o,X4: produc5902793192644279159on_nat] :
      ( ! [A4: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B4: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] : ( P2 @ ( produc5429469637799208745on_nat @ A4 @ ( produc7152764266822961111on_nat @ B4 @ C ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct3
thf(fact_81_prod__induct3,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X4: produc8325617224762090749on_nat] :
      ( ! [A4: list_S5253404628753938141on_nat,B4: list_S5253404628753938141on_nat,C: produc1714117091004681831on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A4 @ ( produc4176415733022414961on_nat @ B4 @ C ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct3
thf(fact_82_prod__induct3,axiom,
    ! [P2: produc118098540238352959on_nat > $o,X4: produc118098540238352959on_nat] :
      ( ! [A4: list_S5253404628753938141on_nat,B4: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] : ( P2 @ ( produc4176415733022414961on_nat @ A4 @ ( produc7152764266822961111on_nat @ B4 @ C ) ) )
     => ( P2 @ X4 ) ) ).

% prod_induct3
thf(fact_83_prod__cases4,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A4: list_S5253404628753938141on_nat,B4: list_S5253404628753938141on_nat,C: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A4 @ ( produc4176415733022414961on_nat @ B4 @ ( produc7152764266822961111on_nat @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_84_prod__cases3,axiom,
    ! [Y: produc5902793192644279159on_nat] :
      ~ ! [A4: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B4: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc5429469637799208745on_nat @ A4 @ ( produc7152764266822961111on_nat @ B4 @ C ) ) ) ).

% prod_cases3
thf(fact_85_prod__cases3,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A4: list_S5253404628753938141on_nat,B4: list_S5253404628753938141on_nat,C: produc1714117091004681831on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A4 @ ( produc4176415733022414961on_nat @ B4 @ C ) ) ) ).

% prod_cases3
thf(fact_86_prod__cases3,axiom,
    ! [Y: produc118098540238352959on_nat] :
      ~ ! [A4: list_S5253404628753938141on_nat,B4: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc4176415733022414961on_nat @ A4 @ ( produc7152764266822961111on_nat @ B4 @ C ) ) ) ).

% prod_cases3
thf(fact_87_Pair__inject,axiom,
    ! [A: list_l8999537460041914605on_nat,B: list_P5508476027986415439on_nat,A2: list_l8999537460041914605on_nat,B2: list_P5508476027986415439on_nat] :
      ( ( ( produc3605646969090963669on_nat @ A @ B )
        = ( produc3605646969090963669on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_88_Pair__inject,axiom,
    ! [A: list_nat_nat,B: list_nat,A2: list_nat_nat,B2: list_nat] :
      ( ( ( produc7978589510830832328st_nat @ A @ B )
        = ( produc7978589510830832328st_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_89_Pair__inject,axiom,
    ! [A: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B: produc1714117091004681831on_nat,A2: produc1313295348606329723on_nat > produc1313295348606329723on_nat,B2: produc1714117091004681831on_nat] :
      ( ( ( produc5429469637799208745on_nat @ A @ B )
        = ( produc5429469637799208745on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_90_Pair__inject,axiom,
    ! [A: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,B: list_l8336264072933094989od_a_o,A2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,B2: list_l8336264072933094989od_a_o] :
      ( ( ( produc6784936493919748534od_a_o @ A @ B )
        = ( produc6784936493919748534od_a_o @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_91_Pair__inject,axiom,
    ! [A: nat > nat,B: list_P5366131564312172150at_nat,A2: nat > nat,B2: list_P5366131564312172150at_nat] :
      ( ( ( produc17425388850148510at_nat @ A @ B )
        = ( produc17425388850148510at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_92_Pair__inject,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,A2: multis1856558993800614429on_nat,B2: multis1856558993800614429on_nat] :
      ( ( ( produc5200047356169809875on_nat @ A @ B )
        = ( produc5200047356169809875on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_93_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_94_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat,A2: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B )
        = ( produc4176415733022414961on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_95_Pair__inject,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,A2: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( ( produc7152764266822961111on_nat @ A @ B )
        = ( produc7152764266822961111on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_96_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_97_append_Oright__neutral,axiom,
    ! [A: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ A @ nil_li2682182165152866823on_nat )
      = A ) ).

% append.right_neutral
thf(fact_98_append_Oright__neutral,axiom,
    ! [A: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ A @ nil_li5563851662744376407on_nat )
      = A ) ).

% append.right_neutral
thf(fact_99_append_Oright__neutral,axiom,
    ! [A: list_l7431997615882138323od_a_o] :
      ( ( append6363893464373751912od_a_o @ A @ nil_li6902169125192129875od_a_o )
      = A ) ).

% append.right_neutral
thf(fact_100_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_101_append_Oright__neutral,axiom,
    ! [A: list_m3172874511867307804at_nat] :
      ( ( append64277455992063665at_nat @ A @ nil_mu4464041163730640006at_nat )
      = A ) ).

% append.right_neutral
thf(fact_102_append_Oright__neutral,axiom,
    ! [A: list_l4071084838796328136at_nat] :
      ( ( append3017019745180361949at_nat @ A @ nil_li268076387995464392at_nat )
      = A ) ).

% append.right_neutral
thf(fact_103_append_Oright__neutral,axiom,
    ! [A: list_nat_nat] :
      ( ( append_nat_nat @ A @ nil_nat_nat )
      = A ) ).

% append.right_neutral
thf(fact_104_append_Oright__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ A @ nil_Pr8614093721167239297on_nat )
      = A ) ).

% append.right_neutral
thf(fact_105_append_Oright__neutral,axiom,
    ! [A: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ A @ nil_li1372968520647675725od_a_o )
      = A ) ).

% append.right_neutral
thf(fact_106_append__Nil2,axiom,
    ! [Xs: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ Xs @ nil_li2682182165152866823on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_107_append__Nil2,axiom,
    ! [Xs: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ Xs @ nil_li5563851662744376407on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_108_append__Nil2,axiom,
    ! [Xs: list_l7431997615882138323od_a_o] :
      ( ( append6363893464373751912od_a_o @ Xs @ nil_li6902169125192129875od_a_o )
      = Xs ) ).

% append_Nil2
thf(fact_109_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_110_append__Nil2,axiom,
    ! [Xs: list_m3172874511867307804at_nat] :
      ( ( append64277455992063665at_nat @ Xs @ nil_mu4464041163730640006at_nat )
      = Xs ) ).

% append_Nil2
thf(fact_111_append__Nil2,axiom,
    ! [Xs: list_l4071084838796328136at_nat] :
      ( ( append3017019745180361949at_nat @ Xs @ nil_li268076387995464392at_nat )
      = Xs ) ).

% append_Nil2
thf(fact_112_append__Nil2,axiom,
    ! [Xs: list_nat_nat] :
      ( ( append_nat_nat @ Xs @ nil_nat_nat )
      = Xs ) ).

% append_Nil2
thf(fact_113_append__Nil2,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ Xs @ nil_Pr8614093721167239297on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_114_append__Nil2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ Xs @ nil_li1372968520647675725od_a_o )
      = Xs ) ).

% append_Nil2
thf(fact_115_append__self__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li2682182165152866823on_nat ) ) ).

% append_self_conv
thf(fact_116_append__self__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li5563851662744376407on_nat ) ) ).

% append_self_conv
thf(fact_117_append__self__conv,axiom,
    ! [Xs: list_l7431997615882138323od_a_o,Ys: list_l7431997615882138323od_a_o] :
      ( ( ( append6363893464373751912od_a_o @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li6902169125192129875od_a_o ) ) ).

% append_self_conv
thf(fact_118_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_119_append__self__conv,axiom,
    ! [Xs: list_m3172874511867307804at_nat,Ys: list_m3172874511867307804at_nat] :
      ( ( ( append64277455992063665at_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_mu4464041163730640006at_nat ) ) ).

% append_self_conv
thf(fact_120_append__self__conv,axiom,
    ! [Xs: list_l4071084838796328136at_nat,Ys: list_l4071084838796328136at_nat] :
      ( ( ( append3017019745180361949at_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li268076387995464392at_nat ) ) ).

% append_self_conv
thf(fact_121_append__self__conv,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat] :
      ( ( ( append_nat_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat_nat ) ) ).

% append_self_conv
thf(fact_122_append__self__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Pr8614093721167239297on_nat ) ) ).

% append_self_conv
thf(fact_123_append__self__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_li1372968520647675725od_a_o ) ) ).

% append_self_conv
thf(fact_124_self__append__conv,axiom,
    ! [Y: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( Y
        = ( append3839213209832495004on_nat @ Y @ Ys ) )
      = ( Ys = nil_li2682182165152866823on_nat ) ) ).

% self_append_conv
thf(fact_125_self__append__conv,axiom,
    ! [Y: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( Y
        = ( append6174841615030841602on_nat @ Y @ Ys ) )
      = ( Ys = nil_li5563851662744376407on_nat ) ) ).

% self_append_conv
thf(fact_126_self__append__conv,axiom,
    ! [Y: list_l7431997615882138323od_a_o,Ys: list_l7431997615882138323od_a_o] :
      ( ( Y
        = ( append6363893464373751912od_a_o @ Y @ Ys ) )
      = ( Ys = nil_li6902169125192129875od_a_o ) ) ).

% self_append_conv
thf(fact_127_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_128_self__append__conv,axiom,
    ! [Y: list_m3172874511867307804at_nat,Ys: list_m3172874511867307804at_nat] :
      ( ( Y
        = ( append64277455992063665at_nat @ Y @ Ys ) )
      = ( Ys = nil_mu4464041163730640006at_nat ) ) ).

% self_append_conv
thf(fact_129_self__append__conv,axiom,
    ! [Y: list_l4071084838796328136at_nat,Ys: list_l4071084838796328136at_nat] :
      ( ( Y
        = ( append3017019745180361949at_nat @ Y @ Ys ) )
      = ( Ys = nil_li268076387995464392at_nat ) ) ).

% self_append_conv
thf(fact_130_self__append__conv,axiom,
    ! [Y: list_nat_nat,Ys: list_nat_nat] :
      ( ( Y
        = ( append_nat_nat @ Y @ Ys ) )
      = ( Ys = nil_nat_nat ) ) ).

% self_append_conv
thf(fact_131_self__append__conv,axiom,
    ! [Y: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( Y
        = ( append2865821678206098582on_nat @ Y @ Ys ) )
      = ( Ys = nil_Pr8614093721167239297on_nat ) ) ).

% self_append_conv
thf(fact_132_self__append__conv,axiom,
    ! [Y: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( Y
        = ( append2159120467191820514od_a_o @ Y @ Ys ) )
      = ( Ys = nil_li1372968520647675725od_a_o ) ) ).

% self_append_conv
thf(fact_133_append__self__conv2,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li2682182165152866823on_nat ) ) ).

% append_self_conv2
thf(fact_134_append__self__conv2,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li5563851662744376407on_nat ) ) ).

% append_self_conv2
thf(fact_135_append__self__conv2,axiom,
    ! [Xs: list_l7431997615882138323od_a_o,Ys: list_l7431997615882138323od_a_o] :
      ( ( ( append6363893464373751912od_a_o @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li6902169125192129875od_a_o ) ) ).

% append_self_conv2
thf(fact_136_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_137_append__self__conv2,axiom,
    ! [Xs: list_m3172874511867307804at_nat,Ys: list_m3172874511867307804at_nat] :
      ( ( ( append64277455992063665at_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_mu4464041163730640006at_nat ) ) ).

% append_self_conv2
thf(fact_138_append__self__conv2,axiom,
    ! [Xs: list_l4071084838796328136at_nat,Ys: list_l4071084838796328136at_nat] :
      ( ( ( append3017019745180361949at_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li268076387995464392at_nat ) ) ).

% append_self_conv2
thf(fact_139_append__self__conv2,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat] :
      ( ( ( append_nat_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat_nat ) ) ).

% append_self_conv2
thf(fact_140_append__self__conv2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% append_self_conv2
thf(fact_141_append__self__conv2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% append_self_conv2
thf(fact_142_self__append__conv2,axiom,
    ! [Y: list_l1726306699111145479on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( Y
        = ( append3839213209832495004on_nat @ Xs @ Y ) )
      = ( Xs = nil_li2682182165152866823on_nat ) ) ).

% self_append_conv2
thf(fact_143_self__append__conv2,axiom,
    ! [Y: list_l8999537460041914605on_nat,Xs: list_l8999537460041914605on_nat] :
      ( ( Y
        = ( append6174841615030841602on_nat @ Xs @ Y ) )
      = ( Xs = nil_li5563851662744376407on_nat ) ) ).

% self_append_conv2
thf(fact_144_self__append__conv2,axiom,
    ! [Y: list_l7431997615882138323od_a_o,Xs: list_l7431997615882138323od_a_o] :
      ( ( Y
        = ( append6363893464373751912od_a_o @ Xs @ Y ) )
      = ( Xs = nil_li6902169125192129875od_a_o ) ) ).

% self_append_conv2
thf(fact_145_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_146_self__append__conv2,axiom,
    ! [Y: list_m3172874511867307804at_nat,Xs: list_m3172874511867307804at_nat] :
      ( ( Y
        = ( append64277455992063665at_nat @ Xs @ Y ) )
      = ( Xs = nil_mu4464041163730640006at_nat ) ) ).

% self_append_conv2
thf(fact_147_self__append__conv2,axiom,
    ! [Y: list_l4071084838796328136at_nat,Xs: list_l4071084838796328136at_nat] :
      ( ( Y
        = ( append3017019745180361949at_nat @ Xs @ Y ) )
      = ( Xs = nil_li268076387995464392at_nat ) ) ).

% self_append_conv2
thf(fact_148_self__append__conv2,axiom,
    ! [Y: list_nat_nat,Xs: list_nat_nat] :
      ( ( Y
        = ( append_nat_nat @ Xs @ Y ) )
      = ( Xs = nil_nat_nat ) ) ).

% self_append_conv2
thf(fact_149_self__append__conv2,axiom,
    ! [Y: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( Y
        = ( append2865821678206098582on_nat @ Xs @ Y ) )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% self_append_conv2
thf(fact_150_self__append__conv2,axiom,
    ! [Y: list_l8336264072933094989od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( Y
        = ( append2159120467191820514od_a_o @ Xs @ Y ) )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% self_append_conv2
thf(fact_151_Nil__is__append__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( nil_li2682182165152866823on_nat
        = ( append3839213209832495004on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li2682182165152866823on_nat )
        & ( Ys = nil_li2682182165152866823on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_152_Nil__is__append__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( nil_li5563851662744376407on_nat
        = ( append6174841615030841602on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li5563851662744376407on_nat )
        & ( Ys = nil_li5563851662744376407on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_153_Nil__is__append__conv,axiom,
    ! [Xs: list_l7431997615882138323od_a_o,Ys: list_l7431997615882138323od_a_o] :
      ( ( nil_li6902169125192129875od_a_o
        = ( append6363893464373751912od_a_o @ Xs @ Ys ) )
      = ( ( Xs = nil_li6902169125192129875od_a_o )
        & ( Ys = nil_li6902169125192129875od_a_o ) ) ) ).

% Nil_is_append_conv
thf(fact_154_Nil__is__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_nat
        = ( append_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_155_Nil__is__append__conv,axiom,
    ! [Xs: list_m3172874511867307804at_nat,Ys: list_m3172874511867307804at_nat] :
      ( ( nil_mu4464041163730640006at_nat
        = ( append64277455992063665at_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_mu4464041163730640006at_nat )
        & ( Ys = nil_mu4464041163730640006at_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_156_Nil__is__append__conv,axiom,
    ! [Xs: list_l4071084838796328136at_nat,Ys: list_l4071084838796328136at_nat] :
      ( ( nil_li268076387995464392at_nat
        = ( append3017019745180361949at_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li268076387995464392at_nat )
        & ( Ys = nil_li268076387995464392at_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_157_Nil__is__append__conv,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat] :
      ( ( nil_nat_nat
        = ( append_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat_nat )
        & ( Ys = nil_nat_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_158_Nil__is__append__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( nil_Pr8614093721167239297on_nat
        = ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Pr8614093721167239297on_nat )
        & ( Ys = nil_Pr8614093721167239297on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_159_Nil__is__append__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( nil_li1372968520647675725od_a_o
        = ( append2159120467191820514od_a_o @ Xs @ Ys ) )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        & ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% Nil_is_append_conv
thf(fact_160_append__is__Nil__conv,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys )
        = nil_li2682182165152866823on_nat )
      = ( ( Xs = nil_li2682182165152866823on_nat )
        & ( Ys = nil_li2682182165152866823on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_161_append__is__Nil__conv,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys )
        = nil_li5563851662744376407on_nat )
      = ( ( Xs = nil_li5563851662744376407on_nat )
        & ( Ys = nil_li5563851662744376407on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_162_append__is__Nil__conv,axiom,
    ! [Xs: list_l7431997615882138323od_a_o,Ys: list_l7431997615882138323od_a_o] :
      ( ( ( append6363893464373751912od_a_o @ Xs @ Ys )
        = nil_li6902169125192129875od_a_o )
      = ( ( Xs = nil_li6902169125192129875od_a_o )
        & ( Ys = nil_li6902169125192129875od_a_o ) ) ) ).

% append_is_Nil_conv
thf(fact_163_append__is__Nil__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = nil_nat )
      = ( ( Xs = nil_nat )
        & ( Ys = nil_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_164_append__is__Nil__conv,axiom,
    ! [Xs: list_m3172874511867307804at_nat,Ys: list_m3172874511867307804at_nat] :
      ( ( ( append64277455992063665at_nat @ Xs @ Ys )
        = nil_mu4464041163730640006at_nat )
      = ( ( Xs = nil_mu4464041163730640006at_nat )
        & ( Ys = nil_mu4464041163730640006at_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_165_append__is__Nil__conv,axiom,
    ! [Xs: list_l4071084838796328136at_nat,Ys: list_l4071084838796328136at_nat] :
      ( ( ( append3017019745180361949at_nat @ Xs @ Ys )
        = nil_li268076387995464392at_nat )
      = ( ( Xs = nil_li268076387995464392at_nat )
        & ( Ys = nil_li268076387995464392at_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_166_append__is__Nil__conv,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat] :
      ( ( ( append_nat_nat @ Xs @ Ys )
        = nil_nat_nat )
      = ( ( Xs = nil_nat_nat )
        & ( Ys = nil_nat_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_167_append__is__Nil__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = nil_Pr8614093721167239297on_nat )
      = ( ( Xs = nil_Pr8614093721167239297on_nat )
        & ( Ys = nil_Pr8614093721167239297on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_168_append__is__Nil__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ Ys )
        = nil_li1372968520647675725od_a_o )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        & ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% append_is_Nil_conv
thf(fact_169_append_Oassoc,axiom,
    ! [A: list_l1726306699111145479on_nat,B: list_l1726306699111145479on_nat,C2: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ ( append3839213209832495004on_nat @ A @ B ) @ C2 )
      = ( append3839213209832495004on_nat @ A @ ( append3839213209832495004on_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_170_append_Oassoc,axiom,
    ! [A: list_l8336264072933094989od_a_o,B: list_l8336264072933094989od_a_o,C2: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ ( append2159120467191820514od_a_o @ A @ B ) @ C2 )
      = ( append2159120467191820514od_a_o @ A @ ( append2159120467191820514od_a_o @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_171_append_Oassoc,axiom,
    ! [A: list_l8999537460041914605on_nat,B: list_l8999537460041914605on_nat,C2: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ ( append6174841615030841602on_nat @ A @ B ) @ C2 )
      = ( append6174841615030841602on_nat @ A @ ( append6174841615030841602on_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_172_append_Oassoc,axiom,
    ! [A: list_nat,B: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B ) @ C2 )
      = ( append_nat @ A @ ( append_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_173_append_Oassoc,axiom,
    ! [A: list_nat_nat,B: list_nat_nat,C2: list_nat_nat] :
      ( ( append_nat_nat @ ( append_nat_nat @ A @ B ) @ C2 )
      = ( append_nat_nat @ A @ ( append_nat_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_174_append_Oassoc,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,C2: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ A @ B ) @ C2 )
      = ( append2865821678206098582on_nat @ A @ ( append2865821678206098582on_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_175_append__assoc,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) @ Zs )
      = ( append3839213209832495004on_nat @ Xs @ ( append3839213209832495004on_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_176_append__assoc,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ Zs )
      = ( append2159120467191820514od_a_o @ Xs @ ( append2159120467191820514od_a_o @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_177_append__assoc,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) @ Zs )
      = ( append6174841615030841602on_nat @ Xs @ ( append6174841615030841602on_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_178_append__assoc,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( append_nat @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat @ Xs @ ( append_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_179_append__assoc,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat,Zs: list_nat_nat] :
      ( ( append_nat_nat @ ( append_nat_nat @ Xs @ Ys ) @ Zs )
      = ( append_nat_nat @ Xs @ ( append_nat_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_180_append__assoc,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ Zs )
      = ( append2865821678206098582on_nat @ Xs @ ( append2865821678206098582on_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_181_append__same__eq,axiom,
    ! [Ys: list_l1726306699111145479on_nat,Xs: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Ys @ Xs )
        = ( append3839213209832495004on_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_182_append__same__eq,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,Xs: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Ys @ Xs )
        = ( append2159120467191820514od_a_o @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_183_append__same__eq,axiom,
    ! [Ys: list_l8999537460041914605on_nat,Xs: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Ys @ Xs )
        = ( append6174841615030841602on_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_184_append__same__eq,axiom,
    ! [Ys: list_nat,Xs: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Ys @ Xs )
        = ( append_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_185_append__same__eq,axiom,
    ! [Ys: list_nat_nat,Xs: list_nat_nat,Zs: list_nat_nat] :
      ( ( ( append_nat_nat @ Ys @ Xs )
        = ( append_nat_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_186_append__same__eq,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys @ Xs )
        = ( append2865821678206098582on_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_187_same__append__eq,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys )
        = ( append3839213209832495004on_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_188_same__append__eq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ Ys )
        = ( append2159120467191820514od_a_o @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_189_same__append__eq,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys )
        = ( append6174841615030841602on_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_190_same__append__eq,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_191_same__append__eq,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat,Zs: list_nat_nat] :
      ( ( ( append_nat_nat @ Xs @ Ys )
        = ( append_nat_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_192_same__append__eq,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = ( append2865821678206098582on_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_193_append__eq__append__conv2,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat,Ts: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Ys )
        = ( append3839213209832495004on_nat @ Zs @ Ts ) )
      = ( ? [Us: list_l1726306699111145479on_nat] :
            ( ( ( Xs
                = ( append3839213209832495004on_nat @ Zs @ Us ) )
              & ( ( append3839213209832495004on_nat @ Us @ Ys )
                = Ts ) )
            | ( ( ( append3839213209832495004on_nat @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append3839213209832495004on_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_194_append__eq__append__conv2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o,Ts: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ Ys )
        = ( append2159120467191820514od_a_o @ Zs @ Ts ) )
      = ( ? [Us: list_l8336264072933094989od_a_o] :
            ( ( ( Xs
                = ( append2159120467191820514od_a_o @ Zs @ Us ) )
              & ( ( append2159120467191820514od_a_o @ Us @ Ys )
                = Ts ) )
            | ( ( ( append2159120467191820514od_a_o @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append2159120467191820514od_a_o @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_195_append__eq__append__conv2,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat,Ts: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Ys )
        = ( append6174841615030841602on_nat @ Zs @ Ts ) )
      = ( ? [Us: list_l8999537460041914605on_nat] :
            ( ( ( Xs
                = ( append6174841615030841602on_nat @ Zs @ Us ) )
              & ( ( append6174841615030841602on_nat @ Us @ Ys )
                = Ts ) )
            | ( ( ( append6174841615030841602on_nat @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append6174841615030841602on_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_196_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us ) )
              & ( ( append_nat @ Us @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_197_append__eq__append__conv2,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat,Zs: list_nat_nat,Ts: list_nat_nat] :
      ( ( ( append_nat_nat @ Xs @ Ys )
        = ( append_nat_nat @ Zs @ Ts ) )
      = ( ? [Us: list_nat_nat] :
            ( ( ( Xs
                = ( append_nat_nat @ Zs @ Us ) )
              & ( ( append_nat_nat @ Us @ Ys )
                = Ts ) )
            | ( ( ( append_nat_nat @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append_nat_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_198_append__eq__append__conv2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,Ts: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = ( append2865821678206098582on_nat @ Zs @ Ts ) )
      = ( ? [Us: list_P9127870862658512897on_nat] :
            ( ( ( Xs
                = ( append2865821678206098582on_nat @ Zs @ Us ) )
              & ( ( append2865821678206098582on_nat @ Us @ Ys )
                = Ts ) )
            | ( ( ( append2865821678206098582on_nat @ Xs @ Us )
                = Zs )
              & ( Ys
                = ( append2865821678206098582on_nat @ Us @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_199_append__eq__appendI,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Xs1: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Us2: list_l1726306699111145479on_nat] :
      ( ( ( append3839213209832495004on_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append3839213209832495004on_nat @ Xs1 @ Us2 ) )
       => ( ( append3839213209832495004on_nat @ Xs @ Ys )
          = ( append3839213209832495004on_nat @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_200_append__eq__appendI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Xs1: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Us2: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append2159120467191820514od_a_o @ Xs1 @ Us2 ) )
       => ( ( append2159120467191820514od_a_o @ Xs @ Ys )
          = ( append2159120467191820514od_a_o @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_201_append__eq__appendI,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Xs1: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Us2: list_l8999537460041914605on_nat] :
      ( ( ( append6174841615030841602on_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append6174841615030841602on_nat @ Xs1 @ Us2 ) )
       => ( ( append6174841615030841602on_nat @ Xs @ Ys )
          = ( append6174841615030841602on_nat @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_202_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us2: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us2 ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_203_append__eq__appendI,axiom,
    ! [Xs: list_nat_nat,Xs1: list_nat_nat,Zs: list_nat_nat,Ys: list_nat_nat,Us2: list_nat_nat] :
      ( ( ( append_nat_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat_nat @ Xs1 @ Us2 ) )
       => ( ( append_nat_nat @ Xs @ Ys )
          = ( append_nat_nat @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_204_append__eq__appendI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Xs1: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Us2: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append2865821678206098582on_nat @ Xs1 @ Us2 ) )
       => ( ( append2865821678206098582on_nat @ Xs @ Ys )
          = ( append2865821678206098582on_nat @ Zs @ Us2 ) ) ) ) ).

% append_eq_appendI
thf(fact_205_eq__Nil__appendI,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append3839213209832495004on_nat @ nil_li2682182165152866823on_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_206_eq__Nil__appendI,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append6174841615030841602on_nat @ nil_li5563851662744376407on_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_207_eq__Nil__appendI,axiom,
    ! [Xs: list_l7431997615882138323od_a_o,Ys: list_l7431997615882138323od_a_o] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append6363893464373751912od_a_o @ nil_li6902169125192129875od_a_o @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_208_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_209_eq__Nil__appendI,axiom,
    ! [Xs: list_m3172874511867307804at_nat,Ys: list_m3172874511867307804at_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append64277455992063665at_nat @ nil_mu4464041163730640006at_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_210_eq__Nil__appendI,axiom,
    ! [Xs: list_l4071084838796328136at_nat,Ys: list_l4071084838796328136at_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append3017019745180361949at_nat @ nil_li268076387995464392at_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_211_eq__Nil__appendI,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat_nat @ nil_nat_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_212_eq__Nil__appendI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_213_eq__Nil__appendI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append2159120467191820514od_a_o @ nil_li1372968520647675725od_a_o @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_214_append_Oleft__neutral,axiom,
    ! [A: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ nil_li2682182165152866823on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_215_append_Oleft__neutral,axiom,
    ! [A: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ nil_li5563851662744376407on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_216_append_Oleft__neutral,axiom,
    ! [A: list_l7431997615882138323od_a_o] :
      ( ( append6363893464373751912od_a_o @ nil_li6902169125192129875od_a_o @ A )
      = A ) ).

% append.left_neutral
thf(fact_217_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_218_append_Oleft__neutral,axiom,
    ! [A: list_m3172874511867307804at_nat] :
      ( ( append64277455992063665at_nat @ nil_mu4464041163730640006at_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_219_append_Oleft__neutral,axiom,
    ! [A: list_l4071084838796328136at_nat] :
      ( ( append3017019745180361949at_nat @ nil_li268076387995464392at_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_220_append_Oleft__neutral,axiom,
    ! [A: list_nat_nat] :
      ( ( append_nat_nat @ nil_nat_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_221_append_Oleft__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_222_append_Oleft__neutral,axiom,
    ! [A: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ nil_li1372968520647675725od_a_o @ A )
      = A ) ).

% append.left_neutral
thf(fact_223_append__Nil,axiom,
    ! [Ys: list_l1726306699111145479on_nat] :
      ( ( append3839213209832495004on_nat @ nil_li2682182165152866823on_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_224_append__Nil,axiom,
    ! [Ys: list_l8999537460041914605on_nat] :
      ( ( append6174841615030841602on_nat @ nil_li5563851662744376407on_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_225_append__Nil,axiom,
    ! [Ys: list_l7431997615882138323od_a_o] :
      ( ( append6363893464373751912od_a_o @ nil_li6902169125192129875od_a_o @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_226_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_227_append__Nil,axiom,
    ! [Ys: list_m3172874511867307804at_nat] :
      ( ( append64277455992063665at_nat @ nil_mu4464041163730640006at_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_228_append__Nil,axiom,
    ! [Ys: list_l4071084838796328136at_nat] :
      ( ( append3017019745180361949at_nat @ nil_li268076387995464392at_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_229_append__Nil,axiom,
    ! [Ys: list_nat_nat] :
      ( ( append_nat_nat @ nil_nat_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_230_append__Nil,axiom,
    ! [Ys: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_231_append__Nil,axiom,
    ! [Ys: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ nil_li1372968520647675725od_a_o @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_232_mem__Collect__eq,axiom,
    ! [A: list_P3196869577824546119od_a_o,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( member5261019209154752368od_a_o @ A @ ( collec5117750660557753778od_a_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_233_mem__Collect__eq,axiom,
    ! [A: list_S5253404628753938141on_nat,P2: list_S5253404628753938141on_nat > $o] :
      ( ( member86162308477975540on_nat @ A @ ( collec7158715823234635826on_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_234_mem__Collect__eq,axiom,
    ! [A: list_l8336264072933094989od_a_o,P2: list_l8336264072933094989od_a_o > $o] :
      ( ( member8885487059802017782od_a_o @ A @ ( collec3723402900366632248od_a_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_235_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_236_mem__Collect__eq,axiom,
    ! [A: nat > nat,P2: ( nat > nat ) > $o] :
      ( ( member_nat_nat @ A @ ( collect_nat_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_237_mem__Collect__eq,axiom,
    ! [A: produc9176241080893687067on_nat,P2: produc9176241080893687067on_nat > $o] :
      ( ( member4278285197320822468on_nat @ A @ ( collec1886838643678311430on_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_238_Collect__mem__eq,axiom,
    ! [A3: set_li5011860890973139239od_a_o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X5: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_239_Collect__mem__eq,axiom,
    ! [A3: set_li1675222548550825619on_nat] :
      ( ( collec7158715823234635826on_nat
        @ ^ [X5: list_S5253404628753938141on_nat] : ( member86162308477975540on_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_240_Collect__mem__eq,axiom,
    ! [A3: set_li7049945101328808109od_a_o] :
      ( ( collec3723402900366632248od_a_o
        @ ^ [X5: list_l8336264072933094989od_a_o] : ( member8885487059802017782od_a_o @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_241_Collect__mem__eq,axiom,
    ! [A3: set_nat] :
      ( ( collect_nat
        @ ^ [X5: nat] : ( member_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_242_Collect__mem__eq,axiom,
    ! [A3: set_nat_nat] :
      ( ( collect_nat_nat
        @ ^ [X5: nat > nat] : ( member_nat_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_243_Collect__mem__eq,axiom,
    ! [A3: set_Pr3573155178515677307on_nat] :
      ( ( collec1886838643678311430on_nat
        @ ^ [X5: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X5 @ A3 ) )
      = A3 ) ).

% Collect_mem_eq
thf(fact_244_Collect__cong,axiom,
    ! [P2: produc9176241080893687067on_nat > $o,Q: produc9176241080893687067on_nat > $o] :
      ( ! [X3: produc9176241080893687067on_nat] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec1886838643678311430on_nat @ P2 )
        = ( collec1886838643678311430on_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_245_lexord__same__pref__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append2159120467191820514od_a_o @ Xs @ Zs ) ) @ ( lexord3862817758065375290od_a_o @ R ) )
      = ( ? [X5: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X5 @ ( set_li2629784761229999580od_a_o @ Xs ) )
            & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X5 @ X5 ) @ R ) )
        | ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Ys @ Zs ) @ ( lexord3862817758065375290od_a_o @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_246_lexord__same__pref__iff,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat,R: set_Pr7828957961435280891on_nat] :
      ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) @ ( append6174841615030841602on_nat @ Xs @ Zs ) ) @ ( lexord5052122733320026026on_nat @ R ) )
      = ( ? [X5: list_S5253404628753938141on_nat] :
            ( ( member86162308477975540on_nat @ X5 @ ( set_li4371160873219908744on_nat @ Xs ) )
            & ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X5 @ X5 ) @ R ) )
        | ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ Ys @ Zs ) @ ( lexord5052122733320026026on_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_247_lexord__same__pref__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lexord_nat @ R ) )
      = ( ? [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X5 @ X5 ) @ R ) )
        | ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_248_lexord__same__pref__iff,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat_nat,Zs: list_nat_nat,R: set_Pr7682762132356531903at_nat] :
      ( ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ ( append_nat_nat @ Xs @ Ys ) @ ( append_nat_nat @ Xs @ Zs ) ) @ ( lexord_nat_nat @ R ) )
      = ( ? [X5: nat > nat] :
            ( ( member_nat_nat @ X5 @ ( set_nat_nat2 @ Xs ) )
            & ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X5 @ X5 ) @ R ) )
        | ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ Ys @ Zs ) @ ( lexord_nat_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_249_lexord__same__pref__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( append3523812819975775986on_nat @ Xs @ Zs ) ) @ ( lexord4030218662367584666on_nat @ R ) )
      = ( ? [X5: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X5 @ ( set_Su4848624856859743608on_nat @ Xs ) )
            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X5 @ X5 ) @ R ) )
        | ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Ys @ Zs ) @ ( lexord4030218662367584666on_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_250_lexord__same__pref__iff,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,Zs: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( append8234005794073363010on_nat @ Xs @ Ys ) @ ( append8234005794073363010on_nat @ Xs @ Zs ) ) @ ( lexord5514055260552411370on_nat @ R ) )
      = ( ? [X5: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X5 @ ( set_mu1790088485793245640on_nat @ Xs ) )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X5 @ X5 ) @ R ) )
        | ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Ys @ Zs ) @ ( lexord5514055260552411370on_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_251_lexord__same__pref__iff,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) @ ( append3839213209832495004on_nat @ Xs @ Zs ) ) @ ( lexord4827596522773364468on_nat @ R ) )
      = ( ? [X5: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X5 @ ( set_li8154783781463198870on_nat @ Xs ) )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X5 @ X5 ) @ R ) )
        | ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Ys @ Zs ) @ ( lexord4827596522773364468on_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_252_lexord__same__pref__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( append2865821678206098582on_nat @ Xs @ Zs ) ) @ ( lexord3493870746117591022on_nat @ R ) )
      = ( ? [X5: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X5 @ ( set_Pr937530902923597456on_nat @ Xs ) )
            & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X5 @ X5 ) @ R ) )
        | ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ys @ Zs ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_253_list__ex1__simps_I1_J,axiom,
    ! [P2: list_l8336264072933094989od_a_o > $o] :
      ~ ( list_e9073588913345584662od_a_o @ P2 @ nil_li6902169125192129875od_a_o ) ).

% list_ex1_simps(1)
thf(fact_254_list__ex1__simps_I1_J,axiom,
    ! [P2: nat > $o] :
      ~ ( list_ex1_nat @ P2 @ nil_nat ) ).

% list_ex1_simps(1)
thf(fact_255_list__ex1__simps_I1_J,axiom,
    ! [P2: ( multis1856558993800614429on_nat > nat ) > $o] :
      ~ ( list_e1603148173240128643at_nat @ P2 @ nil_mu4464041163730640006at_nat ) ).

% list_ex1_simps(1)
thf(fact_256_list__ex1__simps_I1_J,axiom,
    ! [P2: ( list_P9127870862658512897on_nat > nat ) > $o] :
      ~ ( list_e705310933721537675at_nat @ P2 @ nil_li268076387995464392at_nat ) ).

% list_ex1_simps(1)
thf(fact_257_list__ex1__simps_I1_J,axiom,
    ! [P2: ( nat > nat ) > $o] :
      ~ ( list_ex1_nat_nat @ P2 @ nil_nat_nat ) ).

% list_ex1_simps(1)
thf(fact_258_list__ex1__simps_I1_J,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o] :
      ~ ( list_e4461624505202298052on_nat @ P2 @ nil_Su3078906344068544711on_nat ) ).

% list_ex1_simps(1)
thf(fact_259_list__ex1__simps_I1_J,axiom,
    ! [P2: produc1313295348606329723on_nat > $o] :
      ~ ( list_e3536954311976427588on_nat @ P2 @ nil_Pr8614093721167239297on_nat ) ).

% list_ex1_simps(1)
thf(fact_260_list__ex1__simps_I1_J,axiom,
    ! [P2: produc9176241080893687067on_nat > $o] :
      ~ ( list_e6763340909265628388on_nat @ P2 @ nil_Pr2652050626817158177on_nat ) ).

% list_ex1_simps(1)
thf(fact_261_list__ex1__simps_I1_J,axiom,
    ! [P2: list_P3196869577824546119od_a_o > $o] :
      ~ ( list_e8133552692542901648od_a_o @ P2 @ nil_li1372968520647675725od_a_o ) ).

% list_ex1_simps(1)
thf(fact_262_lex__append__left__iff,axiom,
    ! [R: set_Pr7420778282784443591od_a_o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ! [X3: list_P3196869577824546119od_a_o] :
          ~ ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ X3 ) @ R )
     => ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append2159120467191820514od_a_o @ Xs @ Zs ) ) @ ( lex_li6160749805527239089od_a_o @ R ) )
        = ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Ys @ Zs ) @ ( lex_li6160749805527239089od_a_o @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_263_lex__append__left__iff,axiom,
    ! [R: set_Pr7828957961435280891on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat] :
      ( ! [X3: list_S5253404628753938141on_nat] :
          ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X3 @ X3 ) @ R )
     => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) @ ( append6174841615030841602on_nat @ Xs @ Zs ) ) @ ( lex_li8469410665615468275on_nat @ R ) )
        = ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ Ys @ Zs ) @ ( lex_li8469410665615468275on_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_264_lex__append__left__iff,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
        = ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_265_lex__append__left__iff,axiom,
    ! [R: set_Pr7682762132356531903at_nat,Xs: list_nat_nat,Ys: list_nat_nat,Zs: list_nat_nat] :
      ( ! [X3: nat > nat] :
          ~ ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X3 @ X3 ) @ R )
     => ( ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ ( append_nat_nat @ Xs @ Ys ) @ ( append_nat_nat @ Xs @ Zs ) ) @ ( lex_nat_nat @ R ) )
        = ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ Ys @ Zs ) @ ( lex_nat_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_266_lex__append__left__iff,axiom,
    ! [R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ! [X3: produc1313295348606329723on_nat] :
          ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ X3 ) @ R )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( append2865821678206098582on_nat @ Xs @ Zs ) ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
        = ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ys @ Zs ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_267_lex__append__left__iff,axiom,
    ! [R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,Zs: list_m351779702951725613on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ X3 ) @ R )
     => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( append8234005794073363010on_nat @ Xs @ Ys ) @ ( append8234005794073363010on_nat @ Xs @ Zs ) ) @ ( lex_mu3406824806971463219on_nat @ R ) )
        = ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Ys @ Zs ) @ ( lex_mu3406824806971463219on_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_268_lex__append__left__iff,axiom,
    ! [R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ X3 ) @ R )
     => ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) @ ( append3839213209832495004on_nat @ Xs @ Zs ) ) @ ( lex_li6505813899493256299on_nat @ R ) )
        = ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Ys @ Zs ) @ ( lex_li6505813899493256299on_nat @ R ) ) ) ) ).

% lex_append_left_iff
thf(fact_269_lex__append__leftD,axiom,
    ! [R: set_Pr7420778282784443591od_a_o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ! [X3: list_P3196869577824546119od_a_o] :
          ~ ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ X3 ) @ R )
     => ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append2159120467191820514od_a_o @ Xs @ Zs ) ) @ ( lex_li6160749805527239089od_a_o @ R ) )
       => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Ys @ Zs ) @ ( lex_li6160749805527239089od_a_o @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_270_lex__append__leftD,axiom,
    ! [R: set_Pr7828957961435280891on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat] :
      ( ! [X3: list_S5253404628753938141on_nat] :
          ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X3 @ X3 ) @ R )
     => ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) @ ( append6174841615030841602on_nat @ Xs @ Zs ) ) @ ( lex_li8469410665615468275on_nat @ R ) )
       => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ Ys @ Zs ) @ ( lex_li8469410665615468275on_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_271_lex__append__leftD,axiom,
    ! [R: set_Pr1261947904930325089at_nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ! [X3: nat] :
          ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ X3 ) @ R )
     => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Xs @ Zs ) ) @ ( lex_nat @ R ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Ys @ Zs ) @ ( lex_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_272_lex__append__leftD,axiom,
    ! [R: set_Pr7682762132356531903at_nat,Xs: list_nat_nat,Ys: list_nat_nat,Zs: list_nat_nat] :
      ( ! [X3: nat > nat] :
          ~ ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ X3 @ X3 ) @ R )
     => ( ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ ( append_nat_nat @ Xs @ Ys ) @ ( append_nat_nat @ Xs @ Zs ) ) @ ( lex_nat_nat @ R ) )
       => ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ Ys @ Zs ) @ ( lex_nat_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_273_lex__append__leftD,axiom,
    ! [R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ! [X3: produc1313295348606329723on_nat] :
          ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ X3 ) @ R )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( append2865821678206098582on_nat @ Xs @ Zs ) ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ys @ Zs ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_274_lex__append__leftD,axiom,
    ! [R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,Zs: list_m351779702951725613on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ X3 ) @ R )
     => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( append8234005794073363010on_nat @ Xs @ Ys ) @ ( append8234005794073363010on_nat @ Xs @ Zs ) ) @ ( lex_mu3406824806971463219on_nat @ R ) )
       => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Ys @ Zs ) @ ( lex_mu3406824806971463219on_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_275_lex__append__leftD,axiom,
    ! [R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ X3 ) @ R )
     => ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) @ ( append3839213209832495004on_nat @ Xs @ Zs ) ) @ ( lex_li6505813899493256299on_nat @ R ) )
       => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Ys @ Zs ) @ ( lex_li6505813899493256299on_nat @ R ) ) ) ) ).

% lex_append_leftD
thf(fact_276_Nil__lenlex__iff1,axiom,
    ! [Ns: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ nil_li2682182165152866823on_nat @ Ns ) @ ( lenlex2876470432240847372on_nat @ R ) )
      = ( Ns != nil_li2682182165152866823on_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_277_Nil__lenlex__iff1,axiom,
    ! [Ns: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ nil_mu4837018292623784855on_nat @ Ns ) @ ( lenlex3982385085865065426on_nat @ R ) )
      = ( Ns != nil_mu4837018292623784855on_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_278_Nil__lenlex__iff1,axiom,
    ! [Ns: list_l7431997615882138323od_a_o,R: set_Pr6888711208267297863od_a_o] :
      ( ( member8826157602383945744od_a_o @ ( produc3489511922157336151od_a_o @ nil_li6902169125192129875od_a_o @ Ns ) @ ( lenlex5735267928069597912od_a_o @ R ) )
      = ( Ns != nil_li6902169125192129875od_a_o ) ) ).

% Nil_lenlex_iff1
thf(fact_279_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Ns ) @ ( lenlex_nat @ R ) )
      = ( Ns != nil_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_280_Nil__lenlex__iff1,axiom,
    ! [Ns: list_m3172874511867307804at_nat,R: set_Pr460121339593100505at_nat] :
      ( ( member4130738906983378370at_nat @ ( produc1914166117740277329at_nat @ nil_mu4464041163730640006at_nat @ Ns ) @ ( lenlex1687073198467196993at_nat @ R ) )
      = ( Ns != nil_mu4464041163730640006at_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_281_Nil__lenlex__iff1,axiom,
    ! [Ns: list_l4071084838796328136at_nat,R: set_Pr3988080509287995271at_nat] :
      ( ( member3277963254928738256at_nat @ ( produc5313904832467951767at_nat @ nil_li268076387995464392at_nat @ Ns ) @ ( lenlex7919362645870767949at_nat @ R ) )
      = ( Ns != nil_li268076387995464392at_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_282_Nil__lenlex__iff1,axiom,
    ! [Ns: list_nat_nat,R: set_Pr7682762132356531903at_nat] :
      ( ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ nil_nat_nat @ Ns ) @ ( lenlex_nat_nat @ R ) )
      = ( Ns != nil_nat_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_283_Nil__lenlex__iff1,axiom,
    ! [Ns: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Ns ) @ ( lenlex2563728411540087634od_a_o @ R ) )
      = ( Ns != nil_li1372968520647675725od_a_o ) ) ).

% Nil_lenlex_iff1
thf(fact_284_Nil__lenlex__iff1,axiom,
    ! [Ns: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ns ) @ ( lenlex6631009945862686470on_nat @ R ) )
      = ( Ns != nil_Pr8614093721167239297on_nat ) ) ).

% Nil_lenlex_iff1
thf(fact_285_ns__mul__ext__refl__local,axiom,
    ! [Ns: set_Pr8586234676504894535on_nat,A3: multis5168333519641040071on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( locall5924098968501045722on_nat @ Ns @ A3 )
     => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ A3 ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_refl_local
thf(fact_286_ns__mul__ext__refl__local,axiom,
    ! [Ns: set_Pr3573155178515677307on_nat,A3: multis6850599868085929325on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( locall5469519386152921476on_nat @ Ns @ A3 )
     => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ A3 ) @ ( multis3835232632530192936on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_refl_local
thf(fact_287_ns__mul__ext__refl__local,axiom,
    ! [Ns: set_Pr1446391457923625415on_nat,A3: multis2912115315702009697on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( locall1369730039502324340on_nat @ Ns @ A3 )
     => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ A3 ) @ ( multis3046611884734373584on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_refl_local
thf(fact_288_ns__mul__ext__refl__local,axiom,
    ! [Ns: set_Pr8202701763255893447on_nat,A3: multis4835999835052358849on_nat,S: set_Pr8202701763255893447on_nat] :
      ( ( locall2707793954520237524on_nat @ Ns @ A3 )
     => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ A3 @ A3 ) @ ( multis762940313840963120on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_refl_local
thf(fact_289_ns__mul__ext__refl__local,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat,A3: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( locall2418947207864277812on_nat @ Ns @ A3 )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ A3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_refl_local
thf(fact_290_partition__P,axiom,
    ! [P2: list_P9127870862658512897on_nat > $o,Xs: list_l1726306699111145479on_nat,Yes: list_l1726306699111145479on_nat,No: list_l1726306699111145479on_nat] :
      ( ( ( partit3938626929531101056on_nat @ P2 @ Xs )
        = ( produc4739740463051851607on_nat @ Yes @ No ) )
     => ( ! [X6: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X6 @ ( set_li8154783781463198870on_nat @ Yes ) )
           => ( P2 @ X6 ) )
        & ! [X6: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X6 @ ( set_li8154783781463198870on_nat @ No ) )
           => ~ ( P2 @ X6 ) ) ) ) ).

% partition_P
thf(fact_291_partition__P,axiom,
    ! [P2: list_S5253404628753938141on_nat > $o,Xs: list_l8999537460041914605on_nat,Yes: list_l8999537460041914605on_nat,No: list_l8999537460041914605on_nat] :
      ( ( ( partit2300071607570179614on_nat @ P2 @ Xs )
        = ( produc1341766448070154867on_nat @ Yes @ No ) )
     => ( ! [X6: list_S5253404628753938141on_nat] :
            ( ( member86162308477975540on_nat @ X6 @ ( set_li4371160873219908744on_nat @ Yes ) )
           => ( P2 @ X6 ) )
        & ! [X6: list_S5253404628753938141on_nat] :
            ( ( member86162308477975540on_nat @ X6 @ ( set_li4371160873219908744on_nat @ No ) )
           => ~ ( P2 @ X6 ) ) ) ) ).

% partition_P
thf(fact_292_partition__P,axiom,
    ! [P2: nat > $o,Xs: list_nat,Yes: list_nat,No: list_nat] :
      ( ( ( partition_nat @ P2 @ Xs )
        = ( produc2694037385005941721st_nat @ Yes @ No ) )
     => ( ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_nat2 @ Yes ) )
           => ( P2 @ X6 ) )
        & ! [X6: nat] :
            ( ( member_nat @ X6 @ ( set_nat2 @ No ) )
           => ~ ( P2 @ X6 ) ) ) ) ).

% partition_P
thf(fact_293_partition__P,axiom,
    ! [P2: ( nat > nat ) > $o,Xs: list_nat_nat,Yes: list_nat_nat,No: list_nat_nat] :
      ( ( ( partition_nat_nat @ P2 @ Xs )
        = ( produc4657003450915473719at_nat @ Yes @ No ) )
     => ( ! [X6: nat > nat] :
            ( ( member_nat_nat @ X6 @ ( set_nat_nat2 @ Yes ) )
           => ( P2 @ X6 ) )
        & ! [X6: nat > nat] :
            ( ( member_nat_nat @ X6 @ ( set_nat_nat2 @ No ) )
           => ~ ( P2 @ X6 ) ) ) ) ).

% partition_P
thf(fact_294_partition__P,axiom,
    ! [P2: list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Yes: list_l8336264072933094989od_a_o,No: list_l8336264072933094989od_a_o] :
      ( ( ( partit8903024691838806214od_a_o @ P2 @ Xs )
        = ( produc7858527432735436247od_a_o @ Yes @ No ) )
     => ( ! [X6: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Yes ) )
           => ( P2 @ X6 ) )
        & ! [X6: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ No ) )
           => ~ ( P2 @ X6 ) ) ) ) ).

% partition_P
thf(fact_295_partition__P,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Yes: list_S5253404628753938141on_nat,No: list_S5253404628753938141on_nat] :
      ( ( ( partit4951714822998333966on_nat @ P2 @ Xs )
        = ( produc2912653624306349395on_nat @ Yes @ No ) )
     => ( ! [X6: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Yes ) )
           => ( P2 @ X6 ) )
        & ! [X6: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ No ) )
           => ~ ( P2 @ X6 ) ) ) ) ).

% partition_P
thf(fact_296_partition__P,axiom,
    ! [P2: produc1313295348606329723on_nat > $o,Xs: list_P9127870862658512897on_nat,Yes: list_P9127870862658512897on_nat,No: list_P9127870862658512897on_nat] :
      ( ( ( partit4945910930363278970on_nat @ P2 @ Xs )
        = ( produc7152764266822961111on_nat @ Yes @ No ) )
     => ( ! [X6: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Yes ) )
           => ( P2 @ X6 ) )
        & ! [X6: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ No ) )
           => ~ ( P2 @ X6 ) ) ) ) ).

% partition_P
thf(fact_297_lexord__append__leftD,axiom,
    ! [X4: list_l8999537460041914605on_nat,U: list_l8999537460041914605on_nat,V: list_l8999537460041914605on_nat,R: set_Pr7828957961435280891on_nat] :
      ( ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ ( append6174841615030841602on_nat @ X4 @ U ) @ ( append6174841615030841602on_nat @ X4 @ V ) ) @ ( lexord5052122733320026026on_nat @ R ) )
     => ( ! [A4: list_S5253404628753938141on_nat] :
            ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ A4 @ A4 ) @ R )
       => ( member6090043136585992548on_nat @ ( produc1341766448070154867on_nat @ U @ V ) @ ( lexord5052122733320026026on_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_298_lexord__append__leftD,axiom,
    ! [X4: list_nat,U: list_nat,V: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ X4 @ U ) @ ( append_nat @ X4 @ V ) ) @ ( lexord_nat @ R ) )
     => ( ! [A4: nat] :
            ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A4 @ A4 ) @ R )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ U @ V ) @ ( lexord_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_299_lexord__append__leftD,axiom,
    ! [X4: list_nat_nat,U: list_nat_nat,V: list_nat_nat,R: set_Pr7682762132356531903at_nat] :
      ( ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ ( append_nat_nat @ X4 @ U ) @ ( append_nat_nat @ X4 @ V ) ) @ ( lexord_nat_nat @ R ) )
     => ( ! [A4: nat > nat] :
            ~ ( member1096249278104964872at_nat @ ( produc5770335208449155351at_nat @ A4 @ A4 ) @ R )
       => ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ U @ V ) @ ( lexord_nat_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_300_lexord__append__leftD,axiom,
    ! [X4: list_S5253404628753938141on_nat,U: list_S5253404628753938141on_nat,V: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ X4 @ U ) @ ( append3523812819975775986on_nat @ X4 @ V ) ) @ ( lexord4030218662367584666on_nat @ R ) )
     => ( ! [A4: sum_su4907927656612468301on_nat] :
            ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A4 @ A4 ) @ R )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ U @ V ) @ ( lexord4030218662367584666on_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_301_lexord__append__leftD,axiom,
    ! [X4: list_l8336264072933094989od_a_o,U: list_l8336264072933094989od_a_o,V: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( append2159120467191820514od_a_o @ X4 @ U ) @ ( append2159120467191820514od_a_o @ X4 @ V ) ) @ ( lexord3862817758065375290od_a_o @ R ) )
     => ( ! [A4: list_P3196869577824546119od_a_o] :
            ~ ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A4 @ A4 ) @ R )
       => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ U @ V ) @ ( lexord3862817758065375290od_a_o @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_302_lexord__append__leftD,axiom,
    ! [X4: list_m351779702951725613on_nat,U: list_m351779702951725613on_nat,V: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( append8234005794073363010on_nat @ X4 @ U ) @ ( append8234005794073363010on_nat @ X4 @ V ) ) @ ( lexord5514055260552411370on_nat @ R ) )
     => ( ! [A4: multis1856558993800614429on_nat] :
            ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ A4 ) @ R )
       => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ U @ V ) @ ( lexord5514055260552411370on_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_303_lexord__append__leftD,axiom,
    ! [X4: list_l1726306699111145479on_nat,U: list_l1726306699111145479on_nat,V: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( append3839213209832495004on_nat @ X4 @ U ) @ ( append3839213209832495004on_nat @ X4 @ V ) ) @ ( lexord4827596522773364468on_nat @ R ) )
     => ( ! [A4: list_P9127870862658512897on_nat] :
            ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A4 @ A4 ) @ R )
       => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ U @ V ) @ ( lexord4827596522773364468on_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_304_lexord__append__leftD,axiom,
    ! [X4: list_P9127870862658512897on_nat,U: list_P9127870862658512897on_nat,V: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ X4 @ U ) @ ( append2865821678206098582on_nat @ X4 @ V ) ) @ ( lexord3493870746117591022on_nat @ R ) )
     => ( ! [A4: produc1313295348606329723on_nat] :
            ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ A4 @ A4 ) @ R )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ U @ V ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ) ).

% lexord_append_leftD
thf(fact_305_lexord__irreflexive,axiom,
    ! [R: set_Pr2535564044409765851on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat] :
          ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ X3 ) @ R )
     => ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Xs ) @ ( lexord4030218662367584666on_nat @ R ) ) ) ).

% lexord_irreflexive
thf(fact_306_lexord__irreflexive,axiom,
    ! [R: set_Pr7420778282784443591od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ! [X3: list_P3196869577824546119od_a_o] :
          ~ ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ X3 ) @ R )
     => ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Xs ) @ ( lexord3862817758065375290od_a_o @ R ) ) ) ).

% lexord_irreflexive
thf(fact_307_lexord__irreflexive,axiom,
    ! [R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ! [X3: produc1313295348606329723on_nat] :
          ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ X3 ) @ R )
     => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Xs ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ).

% lexord_irreflexive
thf(fact_308_lexord__irreflexive,axiom,
    ! [R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ X3 ) @ R )
     => ~ ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Xs ) @ ( lexord5514055260552411370on_nat @ R ) ) ) ).

% lexord_irreflexive
thf(fact_309_lexord__irreflexive,axiom,
    ! [R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ X3 ) @ R )
     => ~ ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Xs ) @ ( lexord4827596522773364468on_nat @ R ) ) ) ).

% lexord_irreflexive
thf(fact_310_lexord__linear,axiom,
    ! [R: set_Pr2535564044409765851on_nat,X4: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
      ( ! [A4: sum_su4907927656612468301on_nat,B4: sum_su4907927656612468301on_nat] :
          ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A4 @ B4 ) @ R )
          | ( A4 = B4 )
          | ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ B4 @ A4 ) @ R ) )
     => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ X4 @ Y ) @ ( lexord4030218662367584666on_nat @ R ) )
        | ( X4 = Y )
        | ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Y @ X4 ) @ ( lexord4030218662367584666on_nat @ R ) ) ) ) ).

% lexord_linear
thf(fact_311_lexord__linear,axiom,
    ! [R: set_Pr7420778282784443591od_a_o,X4: list_l8336264072933094989od_a_o,Y: list_l8336264072933094989od_a_o] :
      ( ! [A4: list_P3196869577824546119od_a_o,B4: list_P3196869577824546119od_a_o] :
          ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A4 @ B4 ) @ R )
          | ( A4 = B4 )
          | ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ B4 @ A4 ) @ R ) )
     => ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ X4 @ Y ) @ ( lexord3862817758065375290od_a_o @ R ) )
        | ( X4 = Y )
        | ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Y @ X4 ) @ ( lexord3862817758065375290od_a_o @ R ) ) ) ) ).

% lexord_linear
thf(fact_312_lexord__linear,axiom,
    ! [R: set_Pr8202701763255893447on_nat,X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ( ! [A4: produc1313295348606329723on_nat,B4: produc1313295348606329723on_nat] :
          ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ A4 @ B4 ) @ R )
          | ( A4 = B4 )
          | ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ B4 @ A4 ) @ R ) )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( lexord3493870746117591022on_nat @ R ) )
        | ( X4 = Y )
        | ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y @ X4 ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ) ).

% lexord_linear
thf(fact_313_lexord__linear,axiom,
    ! [R: set_Pr3573155178515677307on_nat,X4: list_m351779702951725613on_nat,Y: list_m351779702951725613on_nat] :
      ( ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat] :
          ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ R )
          | ( A4 = B4 )
          | ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B4 @ A4 ) @ R ) )
     => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ X4 @ Y ) @ ( lexord5514055260552411370on_nat @ R ) )
        | ( X4 = Y )
        | ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Y @ X4 ) @ ( lexord5514055260552411370on_nat @ R ) ) ) ) ).

% lexord_linear
thf(fact_314_lexord__linear,axiom,
    ! [R: set_Pr8586234676504894535on_nat,X4: list_l1726306699111145479on_nat,Y: list_l1726306699111145479on_nat] :
      ( ! [A4: list_P9127870862658512897on_nat,B4: list_P9127870862658512897on_nat] :
          ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A4 @ B4 ) @ R )
          | ( A4 = B4 )
          | ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B4 @ A4 ) @ R ) )
     => ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ X4 @ Y ) @ ( lexord4827596522773364468on_nat @ R ) )
        | ( X4 = Y )
        | ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Y @ X4 ) @ ( lexord4827596522773364468on_nat @ R ) ) ) ) ).

% lexord_linear
thf(fact_315_lexord__Nil__right,axiom,
    ! [X4: list_l4071084838796328136at_nat,R: set_Pr3988080509287995271at_nat] :
      ~ ( member3277963254928738256at_nat @ ( produc5313904832467951767at_nat @ X4 @ nil_li268076387995464392at_nat ) @ ( lexord300201617405405237at_nat @ R ) ) ).

% lexord_Nil_right
thf(fact_316_lexord__Nil__right,axiom,
    ! [X4: list_nat_nat,R: set_Pr7682762132356531903at_nat] :
      ~ ( member6315332072434022184at_nat @ ( produc4657003450915473719at_nat @ X4 @ nil_nat_nat ) @ ( lexord_nat_nat @ R ) ) ).

% lexord_Nil_right
thf(fact_317_lexord__Nil__right,axiom,
    ! [X4: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) @ ( lexord3862817758065375290od_a_o @ R ) ) ).

% lexord_Nil_right
thf(fact_318_lexord__Nil__right,axiom,
    ! [X4: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) @ ( lexord3493870746117591022on_nat @ R ) ) ).

% lexord_Nil_right
thf(fact_319_lexord__append__leftI,axiom,
    ! [U: list_P9127870862658512897on_nat,V: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,X4: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ U @ V ) @ ( lexord3493870746117591022on_nat @ R ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ X4 @ U ) @ ( append2865821678206098582on_nat @ X4 @ V ) ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ).

% lexord_append_leftI
thf(fact_320_list__ex1__iff,axiom,
    ( list_e6763340909265628388on_nat
    = ( ^ [P3: produc9176241080893687067on_nat > $o,Xs3: list_P1915136757390359201on_nat] :
        ? [X5: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X5 @ ( set_Pr8272476097931642672on_nat @ Xs3 ) )
          & ( P3 @ X5 )
          & ! [Y4: produc9176241080893687067on_nat] :
              ( ( ( member4278285197320822468on_nat @ Y4 @ ( set_Pr8272476097931642672on_nat @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X5 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_321_list__ex1__iff,axiom,
    ( list_e3536954311976427588on_nat
    = ( ^ [P3: produc1313295348606329723on_nat > $o,Xs3: list_P9127870862658512897on_nat] :
        ? [X5: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X5 @ ( set_Pr937530902923597456on_nat @ Xs3 ) )
          & ( P3 @ X5 )
          & ! [Y4: produc1313295348606329723on_nat] :
              ( ( ( member5906418179908956708on_nat @ Y4 @ ( set_Pr937530902923597456on_nat @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X5 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_322_list__ex1__iff,axiom,
    ( list_e4461624505202298052on_nat
    = ( ^ [P3: sum_su4907927656612468301on_nat > $o,Xs3: list_S5253404628753938141on_nat] :
        ? [X5: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X5 @ ( set_Su4848624856859743608on_nat @ Xs3 ) )
          & ( P3 @ X5 )
          & ! [Y4: sum_su4907927656612468301on_nat] :
              ( ( ( member8476201367316484836on_nat @ Y4 @ ( set_Su4848624856859743608on_nat @ Xs3 ) )
                & ( P3 @ Y4 ) )
             => ( Y4 = X5 ) ) ) ) ) ).

% list_ex1_iff
thf(fact_323_Nil__notin__lex,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Ys ) @ ( lex_li6160749805527239089od_a_o @ R ) ) ).

% Nil_notin_lex
thf(fact_324_Nil__notin__lex,axiom,
    ! [Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ).

% Nil_notin_lex
thf(fact_325_Nil2__notin__lex,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ nil_li1372968520647675725od_a_o ) @ ( lex_li6160749805527239089od_a_o @ R ) ) ).

% Nil2_notin_lex
thf(fact_326_Nil2__notin__lex,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ nil_Pr8614093721167239297on_nat ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ).

% Nil2_notin_lex
thf(fact_327_lex__append__leftI,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ys @ Zs ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( append2865821678206098582on_nat @ Xs @ Zs ) ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ) ).

% lex_append_leftI
thf(fact_328_lenlex__irreflexive,axiom,
    ! [R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ! [X3: produc1313295348606329723on_nat] :
          ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ X3 ) @ R )
     => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Xs ) @ ( lenlex6631009945862686470on_nat @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_329_lenlex__irreflexive,axiom,
    ! [R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ X3 ) @ R )
     => ~ ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Xs ) @ ( lenlex3982385085865065426on_nat @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_330_lenlex__irreflexive,axiom,
    ! [R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ X3 ) @ R )
     => ~ ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Xs ) @ ( lenlex2876470432240847372on_nat @ R ) ) ) ).

% lenlex_irreflexive
thf(fact_331_Nil__lenlex__iff2,axiom,
    ! [Ns: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Ns @ nil_li1372968520647675725od_a_o ) @ ( lenlex2563728411540087634od_a_o @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_332_Nil__lenlex__iff2,axiom,
    ! [Ns: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ns @ nil_Pr8614093721167239297on_nat ) @ ( lenlex6631009945862686470on_nat @ R ) ) ).

% Nil_lenlex_iff2
thf(fact_333_lexord__partial__trans,axiom,
    ! [Xs: list_P1915136757390359201on_nat,R: set_Pr1446391457923625415on_nat,Ys: list_P1915136757390359201on_nat,Zs: list_P1915136757390359201on_nat] :
      ( ! [X3: produc9176241080893687067on_nat,Y3: produc9176241080893687067on_nat,Z: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X3 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
         => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ R )
           => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ R )
             => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ R ) ) ) )
     => ( ( member5896657130317552784on_nat @ ( produc3175127650907945431on_nat @ Xs @ Ys ) @ ( lexord6816048723392330382on_nat @ R ) )
       => ( ( member5896657130317552784on_nat @ ( produc3175127650907945431on_nat @ Ys @ Zs ) @ ( lexord6816048723392330382on_nat @ R ) )
         => ( member5896657130317552784on_nat @ ( produc3175127650907945431on_nat @ Xs @ Zs ) @ ( lexord6816048723392330382on_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_334_lexord__partial__trans,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Z: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
         => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ R )
           => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ R )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ R ) ) ) )
     => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( lexord4030218662367584666on_nat @ R ) )
       => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Ys @ Zs ) @ ( lexord4030218662367584666on_nat @ R ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Zs ) @ ( lexord4030218662367584666on_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_335_lexord__partial__trans,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ! [X3: produc1313295348606329723on_nat,Y3: produc1313295348606329723on_nat,Z: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Xs ) )
         => ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ Y3 ) @ R )
           => ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ Y3 @ Z ) @ R )
             => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ Z ) @ R ) ) ) )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lexord3493870746117591022on_nat @ R ) )
       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ys @ Zs ) @ ( lexord3493870746117591022on_nat @ R ) )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Zs ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_336_lexord__partial__trans,axiom,
    ! [Xs: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat,Ys: list_m351779702951725613on_nat,Zs: list_m351779702951725613on_nat] :
      ( ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat,Z: multis1856558993800614429on_nat] :
          ( ( member2988082745645132084on_nat @ X3 @ ( set_mu1790088485793245640on_nat @ Xs ) )
         => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ R )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ R )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ R ) ) ) )
     => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( lexord5514055260552411370on_nat @ R ) )
       => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Ys @ Zs ) @ ( lexord5514055260552411370on_nat @ R ) )
         => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Zs ) @ ( lexord5514055260552411370on_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_337_lexord__partial__trans,axiom,
    ! [Xs: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] :
          ( ( member783415501022416426on_nat @ X3 @ ( set_li8154783781463198870on_nat @ Xs ) )
         => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ R )
           => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ R )
             => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ R ) ) ) )
     => ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( lexord4827596522773364468on_nat @ R ) )
       => ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Ys @ Zs ) @ ( lexord4827596522773364468on_nat @ R ) )
         => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Zs ) @ ( lexord4827596522773364468on_nat @ R ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_338_partition_Osimps_I1_J,axiom,
    ! [P2: list_P3196869577824546119od_a_o > $o] :
      ( ( partit8903024691838806214od_a_o @ P2 @ nil_li1372968520647675725od_a_o )
      = ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% partition.simps(1)
thf(fact_339_partition_Osimps_I1_J,axiom,
    ! [P2: produc1313295348606329723on_nat > $o] :
      ( ( partit4945910930363278970on_nat @ P2 @ nil_Pr8614093721167239297on_nat )
      = ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ nil_Pr8614093721167239297on_nat ) ) ).

% partition.simps(1)
thf(fact_340_can__select__set__list__ex1,axiom,
    ! [P2: produc1313295348606329723on_nat > $o,A3: list_P9127870862658512897on_nat] :
      ( ( can_se7488105254466884472on_nat @ P2 @ ( set_Pr937530902923597456on_nat @ A3 ) )
      = ( list_e3536954311976427588on_nat @ P2 @ A3 ) ) ).

% can_select_set_list_ex1
thf(fact_341_can__select__set__list__ex1,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,A3: list_S5253404628753938141on_nat] :
      ( ( can_se5303938684849109648on_nat @ P2 @ ( set_Su4848624856859743608on_nat @ A3 ) )
      = ( list_e4461624505202298052on_nat @ P2 @ A3 ) ) ).

% can_select_set_list_ex1
thf(fact_342_lexord__append__left__rightI,axiom,
    ! [A: produc1313295348606329723on_nat,B: produc1313295348606329723on_nat,R: set_Pr8202701763255893447on_nat,U: list_P9127870862658512897on_nat,X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ A @ B ) @ R )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ U @ ( cons_P1582596238967110705on_nat @ A @ X4 ) ) @ ( append2865821678206098582on_nat @ U @ ( cons_P1582596238967110705on_nat @ B @ Y ) ) ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_343_lexord__append__left__rightI,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,U: list_m351779702951725613on_nat,X4: list_m351779702951725613on_nat,Y: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ R )
     => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( append8234005794073363010on_nat @ U @ ( cons_m1493500683067190247on_nat @ A @ X4 ) ) @ ( append8234005794073363010on_nat @ U @ ( cons_m1493500683067190247on_nat @ B @ Y ) ) ) @ ( lexord5514055260552411370on_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_344_lexord__append__left__rightI,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,U: list_l1726306699111145479on_nat,X4: list_l1726306699111145479on_nat,Y: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ R )
     => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( append3839213209832495004on_nat @ U @ ( cons_l3028799188195208631on_nat @ A @ X4 ) ) @ ( append3839213209832495004on_nat @ U @ ( cons_l3028799188195208631on_nat @ B @ Y ) ) ) @ ( lexord4827596522773364468on_nat @ R ) ) ) ).

% lexord_append_left_rightI
thf(fact_345_lexord__Nil__left,axiom,
    ! [Y: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Y ) @ ( lexord3862817758065375290od_a_o @ R ) )
      = ( ? [A5: list_P3196869577824546119od_a_o,X5: list_l8336264072933094989od_a_o] :
            ( Y
            = ( cons_l9179691953475787517od_a_o @ A5 @ X5 ) ) ) ) ).

% lexord_Nil_left
thf(fact_346_lexord__Nil__left,axiom,
    ! [Y: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Y ) @ ( lexord3493870746117591022on_nat @ R ) )
      = ( ? [A5: produc1313295348606329723on_nat,X5: list_P9127870862658512897on_nat] :
            ( Y
            = ( cons_P1582596238967110705on_nat @ A5 @ X5 ) ) ) ) ).

% lexord_Nil_left
thf(fact_347_lexord__cons__cons,axiom,
    ! [A: multis1856558993800614429on_nat,X4: list_m351779702951725613on_nat,B: multis1856558993800614429on_nat,Y: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ A @ X4 ) @ ( cons_m1493500683067190247on_nat @ B @ Y ) ) @ ( lexord5514055260552411370on_nat @ R ) )
      = ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ X4 @ Y ) @ ( lexord5514055260552411370on_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_348_lexord__cons__cons,axiom,
    ! [A: list_P9127870862658512897on_nat,X4: list_l1726306699111145479on_nat,B: list_P9127870862658512897on_nat,Y: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ A @ X4 ) @ ( cons_l3028799188195208631on_nat @ B @ Y ) ) @ ( lexord4827596522773364468on_nat @ R ) )
      = ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ X4 @ Y ) @ ( lexord4827596522773364468on_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_349_lexord__cons__cons,axiom,
    ! [A: produc1313295348606329723on_nat,X4: list_P9127870862658512897on_nat,B: produc1313295348606329723on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ A @ X4 ) @ ( cons_P1582596238967110705on_nat @ B @ Y ) ) @ ( lexord3493870746117591022on_nat @ R ) )
      = ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ A @ B ) @ R )
        | ( ( A = B )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_350_s__mul__ext__union__compat,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) )
     => ( ( locall2418947207864277812on_nat @ Ns @ C3 )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B3 @ C3 ) ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ) ).

% s_mul_ext_union_compat
thf(fact_351_ns__mul__ext__union__compat,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ( ( locall2418947207864277812on_nat @ Ns @ C3 )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B3 @ C3 ) ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ) ).

% ns_mul_ext_union_compat
thf(fact_352_bind__simps_I1_J,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_l8336264072933094989od_a_o] :
      ( ( bind_l3689935036704996679od_a_o @ nil_li1372968520647675725od_a_o @ F )
      = nil_li1372968520647675725od_a_o ) ).

% bind_simps(1)
thf(fact_353_lexord__lex,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
      = ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( lexord3493870746117591022on_nat @ R ) )
        & ( ( size_s7490768528479217773on_nat @ X4 )
          = ( size_s7490768528479217773on_nat @ Y ) ) ) ) ).

% lexord_lex
thf(fact_354_mult2__ex__dom_Ocases,axiom,
    ! [X4: produc140257497307482913od_a_o] :
      ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o] :
          ( X4
         != ( produc8132124342406683665od_a_o @ X3 @ ( produc4956319469386570225od_a_o @ Xs2 @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Ys2 ) ) ) )
     => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o] :
            ( X4
           != ( produc8132124342406683665od_a_o @ X3 @ ( produc4956319469386570225od_a_o @ Xs2 @ ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) @ Ys2 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_355_mult2__ex__dom_Ocases,axiom,
    ! [X4: produc7326124634926975417on_nat] :
      ( ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
          ( X4
         != ( produc2628373118283442345on_nat @ X3 @ ( produc1138821556964385341on_nat @ Xs2 @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys2 ) ) ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat,Y3: produc1313295348606329723on_nat,Ys3: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
            ( X4
           != ( produc2628373118283442345on_nat @ X3 @ ( produc1138821556964385341on_nat @ Xs2 @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ Y3 @ Ys3 ) @ Ys2 ) ) ) ) ) ).

% mult2_ex_dom.cases
thf(fact_356_s__ns__mul__ext__union__compat,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat,D2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ C3 @ D2 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B3 @ D2 ) ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ) ).

% s_ns_mul_ext_union_compat
thf(fact_357_partition__set,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Yes: list_S5253404628753938141on_nat,No: list_S5253404628753938141on_nat] :
      ( ( ( partit4951714822998333966on_nat @ P2 @ Xs )
        = ( produc2912653624306349395on_nat @ Yes @ No ) )
     => ( ( sup_su343302011326411951on_nat @ ( set_Su4848624856859743608on_nat @ Yes ) @ ( set_Su4848624856859743608on_nat @ No ) )
        = ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% partition_set
thf(fact_358_partition__set,axiom,
    ! [P2: produc1313295348606329723on_nat > $o,Xs: list_P9127870862658512897on_nat,Yes: list_P9127870862658512897on_nat,No: list_P9127870862658512897on_nat] :
      ( ( ( partit4945910930363278970on_nat @ P2 @ Xs )
        = ( produc7152764266822961111on_nat @ Yes @ No ) )
     => ( ( sup_su6884158799674119087on_nat @ ( set_Pr937530902923597456on_nat @ Yes ) @ ( set_Pr937530902923597456on_nat @ No ) )
        = ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% partition_set
thf(fact_359_append__eq__append__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Us2: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
      ( ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
        | ( ( size_s7490768528479217773on_nat @ Us2 )
          = ( size_s7490768528479217773on_nat @ Vs ) ) )
     => ( ( ( append2865821678206098582on_nat @ Xs @ Us2 )
          = ( append2865821678206098582on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us2 = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_360_append1__eq__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) )
        = ( append2865821678206098582on_nat @ Ys @ ( cons_P1582596238967110705on_nat @ Y @ nil_Pr8614093721167239297on_nat ) ) )
      = ( ( Xs = Ys )
        & ( X4 = Y ) ) ) ).

% append1_eq_conv
thf(fact_361_append1__eq__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) )
        = ( append2159120467191820514od_a_o @ Ys @ ( cons_l9179691953475787517od_a_o @ Y @ nil_li1372968520647675725od_a_o ) ) )
      = ( ( Xs = Ys )
        & ( X4 = Y ) ) ) ).

% append1_eq_conv
thf(fact_362_set__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( set_Pr937530902923597456on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( sup_su6884158799674119087on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ ( set_Pr937530902923597456on_nat @ Ys ) ) ) ).

% set_append
thf(fact_363_set__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( set_Su4848624856859743608on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( sup_su343302011326411951on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ).

% set_append
thf(fact_364_mset__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( mset_S7907735292211298936on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) @ ( mset_S7907735292211298936on_nat @ Ys ) ) ) ).

% mset_append
thf(fact_365_mset__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( mset_P6964650582947932560on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) @ ( mset_P6964650582947932560on_nat @ Ys ) ) ) ).

% mset_append
thf(fact_366_Cons__in__lex,axiom,
    ! [X4: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ X4 @ Xs ) @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) ) @ ( lex_mu3406824806971463219on_nat @ R ) )
      = ( ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ R )
          & ( ( size_s6603137509925400257on_nat @ Xs )
            = ( size_s6603137509925400257on_nat @ Ys ) ) )
        | ( ( X4 = Y )
          & ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( lex_mu3406824806971463219on_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_367_Cons__in__lex,axiom,
    ! [X4: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X4 @ Xs ) @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) ) @ ( lex_li6505813899493256299on_nat @ R ) )
      = ( ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ R )
          & ( ( size_s129639936549468787on_nat @ Xs )
            = ( size_s129639936549468787on_nat @ Ys ) ) )
        | ( ( X4 = Y )
          & ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( lex_li6505813899493256299on_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_368_Cons__in__lex,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
      = ( ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ R )
          & ( ( size_s7490768528479217773on_nat @ Xs )
            = ( size_s7490768528479217773on_nat @ Ys ) ) )
        | ( ( X4 = Y )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ) ) ) ).

% Cons_in_lex
thf(fact_369_list__induct2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,P2: list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( P2 @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o )
       => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] :
              ( ( ( size_s3379185846251495609od_a_o @ Xs2 )
                = ( size_s3379185846251495609od_a_o @ Ys3 ) )
             => ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) ) ) )
         => ( P2 @ Xs @ Ys ) ) ) ) ).

% list_induct2
thf(fact_370_list__induct3,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o,P2: list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( ( size_s3379185846251495609od_a_o @ Ys )
          = ( size_s3379185846251495609od_a_o @ Zs ) )
       => ( ( P2 @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o )
         => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o,Z: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o] :
                ( ( ( size_s3379185846251495609od_a_o @ Xs2 )
                  = ( size_s3379185846251495609od_a_o @ Ys3 ) )
               => ( ( ( size_s3379185846251495609od_a_o @ Ys3 )
                    = ( size_s3379185846251495609od_a_o @ Zs2 ) )
                 => ( ( P2 @ Xs2 @ Ys3 @ Zs2 )
                   => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) @ ( cons_l9179691953475787517od_a_o @ Z @ Zs2 ) ) ) ) )
           => ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_371_list__induct4,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o,Ws: list_l8336264072933094989od_a_o,P2: list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( ( size_s3379185846251495609od_a_o @ Ys )
          = ( size_s3379185846251495609od_a_o @ Zs ) )
       => ( ( ( size_s3379185846251495609od_a_o @ Zs )
            = ( size_s3379185846251495609od_a_o @ Ws ) )
         => ( ( P2 @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o )
           => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o,Z: list_P3196869577824546119od_a_o,Zs2: list_l8336264072933094989od_a_o,W: list_P3196869577824546119od_a_o,Ws2: list_l8336264072933094989od_a_o] :
                  ( ( ( size_s3379185846251495609od_a_o @ Xs2 )
                    = ( size_s3379185846251495609od_a_o @ Ys3 ) )
                 => ( ( ( size_s3379185846251495609od_a_o @ Ys3 )
                      = ( size_s3379185846251495609od_a_o @ Zs2 ) )
                   => ( ( ( size_s3379185846251495609od_a_o @ Zs2 )
                        = ( size_s3379185846251495609od_a_o @ Ws2 ) )
                     => ( ( P2 @ Xs2 @ Ys3 @ Zs2 @ Ws2 )
                       => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) @ ( cons_l9179691953475787517od_a_o @ Z @ Zs2 ) @ ( cons_l9179691953475787517od_a_o @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_372_mult2__ex__dom0_Ocases,axiom,
    ! [X4: produc2662161832495817990_a_o_o] :
      ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o,B4: $o] :
          ( X4
         != ( produc3663832065306206966_a_o_o @ X3 @ ( produc666131611034064342_a_o_o @ Xs2 @ ( produc6403904292155313084_a_o_o @ nil_li1372968520647675725od_a_o @ ( produc6433402059633706786_a_o_o @ Ys2 @ B4 ) ) ) ) )
     => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o,B4: $o] :
            ( X4
           != ( produc3663832065306206966_a_o_o @ X3 @ ( produc666131611034064342_a_o_o @ Xs2 @ ( produc6403904292155313084_a_o_o @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) @ ( produc6433402059633706786_a_o_o @ Ys2 @ B4 ) ) ) ) ) ) ).

% mult2_ex_dom0.cases
thf(fact_373_mult2__impl0_Ocases,axiom,
    ! [X4: produc2083629628261547084_a_o_o] :
      ( ! [B4: $o] :
          ( X4
         != ( produc6403904292155313084_a_o_o @ nil_li1372968520647675725od_a_o @ ( produc6433402059633706786_a_o_o @ nil_li1372968520647675725od_a_o @ B4 ) ) )
     => ( ! [V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o,B4: $o] :
            ( X4
           != ( produc6403904292155313084_a_o_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) @ ( produc6433402059633706786_a_o_o @ nil_li1372968520647675725od_a_o @ B4 ) ) )
       => ( ! [V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o,B4: $o] :
              ( X4
             != ( produc6403904292155313084_a_o_o @ nil_li1372968520647675725od_a_o @ ( produc6433402059633706786_a_o_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) @ B4 ) ) )
         => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o,B4: $o] :
                ( X4
               != ( produc6403904292155313084_a_o_o @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( produc6433402059633706786_a_o_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) @ B4 ) ) ) ) ) ) ).

% mult2_impl0.cases
thf(fact_374_same__length__different,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( Xs != Ys )
     => ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
       => ? [Pre: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat,Xs4: list_P9127870862658512897on_nat,Y3: produc1313295348606329723on_nat,Ys2: list_P9127870862658512897on_nat] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append2865821678206098582on_nat @ Pre @ ( append2865821678206098582on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) @ Xs4 ) ) )
            & ( Ys
              = ( append2865821678206098582on_nat @ Pre @ ( append2865821678206098582on_nat @ ( cons_P1582596238967110705on_nat @ Y3 @ nil_Pr8614093721167239297on_nat ) @ Ys2 ) ) ) ) ) ) ).

% same_length_different
thf(fact_375_same__length__different,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( Xs != Ys )
     => ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
       => ? [Pre: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o,Xs4: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys2: list_l8336264072933094989od_a_o] :
            ( ( X3 != Y3 )
            & ( Xs
              = ( append2159120467191820514od_a_o @ Pre @ ( append2159120467191820514od_a_o @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) @ Xs4 ) ) )
            & ( Ys
              = ( append2159120467191820514od_a_o @ Pre @ ( append2159120467191820514od_a_o @ ( cons_l9179691953475787517od_a_o @ Y3 @ nil_li1372968520647675725od_a_o ) @ Ys2 ) ) ) ) ) ) ).

% same_length_different
thf(fact_376_transpose_Ocases,axiom,
    ! [X4: list_l7431997615882138323od_a_o] :
      ( ( X4 != nil_li6902169125192129875od_a_o )
     => ( ! [Xss: list_l7431997615882138323od_a_o] :
            ( X4
           != ( cons_l8465095039735648515od_a_o @ nil_li1372968520647675725od_a_o @ Xss ) )
       => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Xss: list_l7431997615882138323od_a_o] :
              ( X4
             != ( cons_l8465095039735648515od_a_o @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_377_transpose_Ocases,axiom,
    ! [X4: list_l8336264072933094989od_a_o] :
      ( ( X4 != nil_li1372968520647675725od_a_o )
     => ( ! [Xss: list_l8336264072933094989od_a_o] :
            ( X4
           != ( cons_l9179691953475787517od_a_o @ nil_Product_prod_a_o @ Xss ) )
       => ~ ! [X3: product_prod_a_o,Xs2: list_P3196869577824546119od_a_o,Xss: list_l8336264072933094989od_a_o] :
              ( X4
             != ( cons_l9179691953475787517od_a_o @ ( cons_P8471407633269037175od_a_o @ X3 @ Xs2 ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_378_list__nonempty__induct,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P2: list_l8336264072933094989od_a_o > $o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o] : ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) )
       => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
              ( ( Xs2 != nil_li1372968520647675725od_a_o )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_379_list__induct2_H,axiom,
    ! [P2: list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( P2 @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] : ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ nil_li1372968520647675725od_a_o )
       => ( ! [Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] : ( P2 @ nil_li1372968520647675725od_a_o @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) )
         => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] :
                ( ( P2 @ Xs2 @ Ys3 )
               => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_380_neq__Nil__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
      = ( ? [Y4: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
            ( Xs
            = ( cons_l9179691953475787517od_a_o @ Y4 @ Ys4 ) ) ) ) ).

% neq_Nil_conv
thf(fact_381_successively_Ocases,axiom,
    ! [X4: produc8998563774250336198od_a_o] :
      ( ! [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o] :
          ( X4
         != ( produc6784936493919748534od_a_o @ P4 @ nil_li1372968520647675725od_a_o ) )
     => ( ! [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,X3: list_P3196869577824546119od_a_o] :
            ( X4
           != ( produc6784936493919748534od_a_o @ P4 @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) ) )
       => ~ ! [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,X3: list_P3196869577824546119od_a_o,Y3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
              ( X4
             != ( produc6784936493919748534od_a_o @ P4 @ ( cons_l9179691953475787517od_a_o @ X3 @ ( cons_l9179691953475787517od_a_o @ Y3 @ Xs2 ) ) ) ) ) ) ).

% successively.cases
thf(fact_382_remdups__adj_Ocases,axiom,
    ! [X4: list_l8336264072933094989od_a_o] :
      ( ( X4 != nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o] :
            ( X4
           != ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) )
       => ~ ! [X3: list_P3196869577824546119od_a_o,Y3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
              ( X4
             != ( cons_l9179691953475787517od_a_o @ X3 @ ( cons_l9179691953475787517od_a_o @ Y3 @ Xs2 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_383_sorted__wrt_Ocases,axiom,
    ! [X4: produc8998563774250336198od_a_o] :
      ( ! [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o] :
          ( X4
         != ( produc6784936493919748534od_a_o @ P4 @ nil_li1372968520647675725od_a_o ) )
     => ~ ! [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,X3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] :
            ( X4
           != ( produc6784936493919748534od_a_o @ P4 @ ( cons_l9179691953475787517od_a_o @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_384_list_Oexhaust,axiom,
    ! [Y: list_l8336264072933094989od_a_o] :
      ( ( Y != nil_li1372968520647675725od_a_o )
     => ~ ! [X21: list_P3196869577824546119od_a_o,X22: list_l8336264072933094989od_a_o] :
            ( Y
           != ( cons_l9179691953475787517od_a_o @ X21 @ X22 ) ) ) ).

% list.exhaust
thf(fact_385_list_OdiscI,axiom,
    ! [List: list_l8336264072933094989od_a_o,X212: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o] :
      ( ( List
        = ( cons_l9179691953475787517od_a_o @ X212 @ X222 ) )
     => ( List != nil_li1372968520647675725od_a_o ) ) ).

% list.discI
thf(fact_386_list_Odistinct_I1_J,axiom,
    ! [X212: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o] :
      ( nil_li1372968520647675725od_a_o
     != ( cons_l9179691953475787517od_a_o @ X212 @ X222 ) ) ).

% list.distinct(1)
thf(fact_387_list_Oset__intros_I2_J,axiom,
    ! [Y: produc9176241080893687067on_nat,X222: list_P1915136757390359201on_nat,X212: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ X222 ) )
     => ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ ( cons_P6897958070471385041on_nat @ X212 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_388_list_Oset__intros_I2_J,axiom,
    ! [Y: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat,X212: produc1313295348606329723on_nat] :
      ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ X222 ) )
     => ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X212 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_389_list_Oset__intros_I2_J,axiom,
    ! [Y: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat,X212: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ X222 ) )
     => ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X212 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_390_list_Oset__intros_I1_J,axiom,
    ! [X212: produc9176241080893687067on_nat,X222: list_P1915136757390359201on_nat] : ( member4278285197320822468on_nat @ X212 @ ( set_Pr8272476097931642672on_nat @ ( cons_P6897958070471385041on_nat @ X212 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_391_list_Oset__intros_I1_J,axiom,
    ! [X212: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat] : ( member5906418179908956708on_nat @ X212 @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X212 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_392_list_Oset__intros_I1_J,axiom,
    ! [X212: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] : ( member8476201367316484836on_nat @ X212 @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X212 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_393_list_Oset__cases,axiom,
    ! [E: produc9176241080893687067on_nat,A: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ E @ ( set_Pr8272476097931642672on_nat @ A ) )
     => ( ! [Z2: list_P1915136757390359201on_nat] :
            ( A
           != ( cons_P6897958070471385041on_nat @ E @ Z2 ) )
       => ~ ! [Z1: produc9176241080893687067on_nat,Z2: list_P1915136757390359201on_nat] :
              ( ( A
                = ( cons_P6897958070471385041on_nat @ Z1 @ Z2 ) )
             => ~ ( member4278285197320822468on_nat @ E @ ( set_Pr8272476097931642672on_nat @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_394_list_Oset__cases,axiom,
    ! [E: produc1313295348606329723on_nat,A: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ E @ ( set_Pr937530902923597456on_nat @ A ) )
     => ( ! [Z2: list_P9127870862658512897on_nat] :
            ( A
           != ( cons_P1582596238967110705on_nat @ E @ Z2 ) )
       => ~ ! [Z1: produc1313295348606329723on_nat,Z2: list_P9127870862658512897on_nat] :
              ( ( A
                = ( cons_P1582596238967110705on_nat @ Z1 @ Z2 ) )
             => ~ ( member5906418179908956708on_nat @ E @ ( set_Pr937530902923597456on_nat @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_395_list_Oset__cases,axiom,
    ! [E: sum_su4907927656612468301on_nat,A: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ E @ ( set_Su4848624856859743608on_nat @ A ) )
     => ( ! [Z2: list_S5253404628753938141on_nat] :
            ( A
           != ( cons_S6797538523646374167on_nat @ E @ Z2 ) )
       => ~ ! [Z1: sum_su4907927656612468301on_nat,Z2: list_S5253404628753938141on_nat] :
              ( ( A
                = ( cons_S6797538523646374167on_nat @ Z1 @ Z2 ) )
             => ~ ( member8476201367316484836on_nat @ E @ ( set_Su4848624856859743608on_nat @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_396_set__ConsD,axiom,
    ! [Y: produc9176241080893687067on_nat,X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ ( cons_P6897958070471385041on_nat @ X4 @ Xs ) ) )
     => ( ( Y = X4 )
        | ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_397_set__ConsD,axiom,
    ! [Y: produc1313295348606329723on_nat,X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) ) )
     => ( ( Y = X4 )
        | ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_398_set__ConsD,axiom,
    ! [Y: sum_su4907927656612468301on_nat,X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X4 @ Xs ) ) )
     => ( ( Y = X4 )
        | ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_399_can__select__def,axiom,
    ( can_se6095669637142871064on_nat
    = ( ^ [P3: produc9176241080893687067on_nat > $o,A6: set_Pr3573155178515677307on_nat] :
        ? [X5: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X5 @ A6 )
          & ( P3 @ X5 )
          & ! [Y4: produc9176241080893687067on_nat] :
              ( ( ( member4278285197320822468on_nat @ Y4 @ A6 )
                & ( P3 @ Y4 ) )
             => ( Y4 = X5 ) ) ) ) ) ).

% can_select_def
thf(fact_400_Cons__eq__appendI,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs1: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( cons_P1582596238967110705on_nat @ X4 @ Xs1 )
        = Ys )
     => ( ( Xs
          = ( append2865821678206098582on_nat @ Xs1 @ Zs ) )
       => ( ( cons_P1582596238967110705on_nat @ X4 @ Xs )
          = ( append2865821678206098582on_nat @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_401_append__Cons,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ Ys )
      = ( cons_P1582596238967110705on_nat @ X4 @ ( append2865821678206098582on_nat @ Xs @ Ys ) ) ) ).

% append_Cons
thf(fact_402_mset__eq__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs )
        = ( mset_S7907735292211298936on_nat @ Ys ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% mset_eq_length
thf(fact_403_mset__eq__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( mset_P6964650582947932560on_nat @ Xs )
        = ( mset_P6964650582947932560on_nat @ Ys ) )
     => ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) ) ) ).

% mset_eq_length
thf(fact_404_mult2__impl_Ocases,axiom,
    ! [X4: produc6649435091139611495od_a_o] :
      ( ( X4
       != ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o ) )
     => ( ! [V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o] :
            ( X4
           != ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) @ nil_li1372968520647675725od_a_o ) )
       => ( ! [V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o] :
              ( X4
             != ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) ) )
         => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o] :
                ( X4
               != ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_405_mult2__impl_Ocases,axiom,
    ! [X4: produc1714117091004681831on_nat] :
      ( ( X4
       != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ nil_Pr8614093721167239297on_nat ) )
     => ( ! [V2: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
            ( X4
           != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ V2 @ Va ) @ nil_Pr8614093721167239297on_nat ) )
       => ( ! [V2: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
              ( X4
             != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ ( cons_P1582596238967110705on_nat @ V2 @ Va ) ) )
         => ~ ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat,V2: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
                ( X4
               != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs2 ) @ ( cons_P1582596238967110705on_nat @ V2 @ Va ) ) ) ) ) ) ).

% mult2_impl.cases
thf(fact_406_rev__nonempty__induct,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: list_P9127870862658512897on_nat > $o] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat] : ( P2 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) )
       => ( ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat] :
              ( ( Xs2 != nil_Pr8614093721167239297on_nat )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( append2865821678206098582on_nat @ Xs2 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_407_rev__nonempty__induct,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P2: list_l8336264072933094989od_a_o > $o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o] : ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) )
       => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
              ( ( Xs2 != nil_li1372968520647675725od_a_o )
             => ( ( P2 @ Xs2 )
               => ( P2 @ ( append2159120467191820514od_a_o @ Xs2 @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% rev_nonempty_induct
thf(fact_408_append__eq__Cons__conv,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys @ Zs )
        = ( cons_P1582596238967110705on_nat @ X4 @ Xs ) )
      = ( ( ( Ys = nil_Pr8614093721167239297on_nat )
          & ( Zs
            = ( cons_P1582596238967110705on_nat @ X4 @ Xs ) ) )
        | ? [Ys5: list_P9127870862658512897on_nat] :
            ( ( Ys
              = ( cons_P1582596238967110705on_nat @ X4 @ Ys5 ) )
            & ( ( append2865821678206098582on_nat @ Ys5 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_409_append__eq__Cons__conv,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Ys @ Zs )
        = ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
      = ( ( ( Ys = nil_li1372968520647675725od_a_o )
          & ( Zs
            = ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) ) )
        | ? [Ys5: list_l8336264072933094989od_a_o] :
            ( ( Ys
              = ( cons_l9179691953475787517od_a_o @ X4 @ Ys5 ) )
            & ( ( append2159120467191820514od_a_o @ Ys5 @ Zs )
              = Xs ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_410_Cons__eq__append__conv,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( cons_P1582596238967110705on_nat @ X4 @ Xs )
        = ( append2865821678206098582on_nat @ Ys @ Zs ) )
      = ( ( ( Ys = nil_Pr8614093721167239297on_nat )
          & ( ( cons_P1582596238967110705on_nat @ X4 @ Xs )
            = Zs ) )
        | ? [Ys5: list_P9127870862658512897on_nat] :
            ( ( ( cons_P1582596238967110705on_nat @ X4 @ Ys5 )
              = Ys )
            & ( Xs
              = ( append2865821678206098582on_nat @ Ys5 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_411_Cons__eq__append__conv,axiom,
    ! [X4: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( ( cons_l9179691953475787517od_a_o @ X4 @ Xs )
        = ( append2159120467191820514od_a_o @ Ys @ Zs ) )
      = ( ( ( Ys = nil_li1372968520647675725od_a_o )
          & ( ( cons_l9179691953475787517od_a_o @ X4 @ Xs )
            = Zs ) )
        | ? [Ys5: list_l8336264072933094989od_a_o] :
            ( ( ( cons_l9179691953475787517od_a_o @ X4 @ Ys5 )
              = Ys )
            & ( Xs
              = ( append2159120467191820514od_a_o @ Ys5 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_412_rev__exhaust,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ~ ! [Ys3: list_P9127870862658512897on_nat,Y3: produc1313295348606329723on_nat] :
            ( Xs
           != ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ Y3 @ nil_Pr8614093721167239297on_nat ) ) ) ) ).

% rev_exhaust
thf(fact_413_rev__exhaust,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ~ ! [Ys3: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o] :
            ( Xs
           != ( append2159120467191820514od_a_o @ Ys3 @ ( cons_l9179691953475787517od_a_o @ Y3 @ nil_li1372968520647675725od_a_o ) ) ) ) ).

% rev_exhaust
thf(fact_414_rev__induct,axiom,
    ! [P2: list_P9127870862658512897on_nat > $o,Xs: list_P9127870862658512897on_nat] :
      ( ( P2 @ nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat] :
            ( ( P2 @ Xs2 )
           => ( P2 @ ( append2865821678206098582on_nat @ Xs2 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) ) ) )
       => ( P2 @ Xs ) ) ) ).

% rev_induct
thf(fact_415_rev__induct,axiom,
    ! [P2: list_l8336264072933094989od_a_o > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ( P2 @ nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
            ( ( P2 @ Xs2 )
           => ( P2 @ ( append2159120467191820514od_a_o @ Xs2 @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) ) ) )
       => ( P2 @ Xs ) ) ) ).

% rev_induct
thf(fact_416_split__list__first__prop__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ( ? [X5: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X5 @ ( set_Pr937530902923597456on_nat @ Xs ) )
            & ( P2 @ X5 ) ) )
      = ( ? [Ys4: list_P9127870862658512897on_nat,X5: produc1313295348606329723on_nat] :
            ( ? [Zs3: list_P9127870862658512897on_nat] :
                ( Xs
                = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X5 @ Zs3 ) ) )
            & ( P2 @ X5 )
            & ! [Y4: produc1313295348606329723on_nat] :
                ( ( member5906418179908956708on_nat @ Y4 @ ( set_Pr937530902923597456on_nat @ Ys4 ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_417_split__list__first__prop__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ? [X5: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X5 @ ( set_Su4848624856859743608on_nat @ Xs ) )
            & ( P2 @ X5 ) ) )
      = ( ? [Ys4: list_S5253404628753938141on_nat,X5: sum_su4907927656612468301on_nat] :
            ( ? [Zs3: list_S5253404628753938141on_nat] :
                ( Xs
                = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X5 @ Zs3 ) ) )
            & ( P2 @ X5 )
            & ! [Y4: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y4 @ ( set_Su4848624856859743608on_nat @ Ys4 ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_418_split__list__last__prop__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ( ? [X5: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X5 @ ( set_Pr937530902923597456on_nat @ Xs ) )
            & ( P2 @ X5 ) ) )
      = ( ? [Ys4: list_P9127870862658512897on_nat,X5: produc1313295348606329723on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X5 @ Zs3 ) ) )
            & ( P2 @ X5 )
            & ! [Y4: produc1313295348606329723on_nat] :
                ( ( member5906418179908956708on_nat @ Y4 @ ( set_Pr937530902923597456on_nat @ Zs3 ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_419_split__list__last__prop__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ? [X5: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X5 @ ( set_Su4848624856859743608on_nat @ Xs ) )
            & ( P2 @ X5 ) ) )
      = ( ? [Ys4: list_S5253404628753938141on_nat,X5: sum_su4907927656612468301on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X5 @ Zs3 ) ) )
            & ( P2 @ X5 )
            & ! [Y4: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y4 @ ( set_Su4848624856859743608on_nat @ Zs3 ) )
               => ~ ( P2 @ Y4 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_420_in__set__conv__decomp__first,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
      = ( ? [Ys4: list_P1915136757390359201on_nat,Zs3: list_P1915136757390359201on_nat] :
            ( ( Xs
              = ( append1010812867689791286on_nat @ Ys4 @ ( cons_P6897958070471385041on_nat @ X4 @ Zs3 ) ) )
            & ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_421_in__set__conv__decomp__first,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [Ys4: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X4 @ Zs3 ) ) )
            & ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_422_in__set__conv__decomp__first,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [Ys4: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X4 @ Zs3 ) ) )
            & ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Ys4 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_423_in__set__conv__decomp__last,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
      = ( ? [Ys4: list_P1915136757390359201on_nat,Zs3: list_P1915136757390359201on_nat] :
            ( ( Xs
              = ( append1010812867689791286on_nat @ Ys4 @ ( cons_P6897958070471385041on_nat @ X4 @ Zs3 ) ) )
            & ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_424_in__set__conv__decomp__last,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [Ys4: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X4 @ Zs3 ) ) )
            & ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_425_in__set__conv__decomp__last,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [Ys4: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X4 @ Zs3 ) ) )
            & ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_426_split__list__first__propE,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys3: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat] :
            ( ? [Zs2: list_P9127870862658512897on_nat] :
                ( Xs
                = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: produc1313295348606329723on_nat] :
                    ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_427_split__list__first__propE,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys3: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
            ( ? [Zs2: list_S5253404628753938141on_nat] :
                ( Xs
                = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_428_split__list__last__propE,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys3: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: produc1313295348606329723on_nat] :
                    ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Zs2 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_429_split__list__last__propE,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys3: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs2 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Zs2 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_430_split__list__first__prop,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys3: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat] :
          ( ? [Zs2: list_P9127870862658512897on_nat] :
              ( Xs
              = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_431_split__list__first__prop,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys3: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
          ( ? [Zs2: list_S5253404628753938141on_nat] :
              ( Xs
              = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_432_split__list__last__prop,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys3: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Zs2 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_433_split__list__last__prop,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys3: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Zs2 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_434_in__set__conv__decomp,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
      = ( ? [Ys4: list_P1915136757390359201on_nat,Zs3: list_P1915136757390359201on_nat] :
            ( Xs
            = ( append1010812867689791286on_nat @ Ys4 @ ( cons_P6897958070471385041on_nat @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_435_in__set__conv__decomp,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [Ys4: list_P9127870862658512897on_nat,Zs3: list_P9127870862658512897on_nat] :
            ( Xs
            = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_436_in__set__conv__decomp,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [Ys4: list_S5253404628753938141on_nat,Zs3: list_S5253404628753938141on_nat] :
            ( Xs
            = ( append3523812819975775986on_nat @ Ys4 @ ( cons_S6797538523646374167on_nat @ X4 @ Zs3 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_437_append__Cons__eq__iff,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat,Ys: list_P1915136757390359201on_nat,Xs5: list_P1915136757390359201on_nat,Ys6: list_P1915136757390359201on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Ys ) )
       => ( ( ( append1010812867689791286on_nat @ Xs @ ( cons_P6897958070471385041on_nat @ X4 @ Ys ) )
            = ( append1010812867689791286on_nat @ Xs5 @ ( cons_P6897958070471385041on_nat @ X4 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_438_append__Cons__eq__iff,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Xs5: list_P9127870862658512897on_nat,Ys6: list_P9127870862658512897on_nat] :
      ( ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Ys ) )
       => ( ( ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ Ys ) )
            = ( append2865821678206098582on_nat @ Xs5 @ ( cons_P1582596238967110705on_nat @ X4 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_439_append__Cons__eq__iff,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Xs5: list_S5253404628753938141on_nat,Ys6: list_S5253404628753938141on_nat] :
      ( ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Ys ) )
       => ( ( ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X4 @ Ys ) )
            = ( append3523812819975775986on_nat @ Xs5 @ ( cons_S6797538523646374167on_nat @ X4 @ Ys6 ) ) )
          = ( ( Xs = Xs5 )
            & ( Ys = Ys6 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_440_split__list__propE,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys3: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat] :
            ( ? [Zs2: list_P9127870862658512897on_nat] :
                ( Xs
                = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs2 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_441_split__list__propE,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ~ ! [Ys3: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
            ( ? [Zs2: list_S5253404628753938141on_nat] :
                ( Xs
                = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs2 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_442_split__list__first,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ? [Ys3: list_P1915136757390359201on_nat,Zs2: list_P1915136757390359201on_nat] :
          ( ( Xs
            = ( append1010812867689791286on_nat @ Ys3 @ ( cons_P6897958070471385041on_nat @ X4 @ Zs2 ) ) )
          & ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_443_split__list__first,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ? [Ys3: list_P9127870862658512897on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X4 @ Zs2 ) ) )
          & ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_444_split__list__first,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ? [Ys3: list_S5253404628753938141on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X4 @ Zs2 ) ) )
          & ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_445_split__list__prop,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X6: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys3: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat] :
          ( ? [Zs2: list_P9127870862658512897on_nat] :
              ( Xs
              = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_446_split__list__prop,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ? [X6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs ) )
          & ( P2 @ X6 ) )
     => ? [Ys3: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
          ( ? [Zs2: list_S5253404628753938141on_nat] :
              ( Xs
              = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X3 @ Zs2 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_447_split__list__last,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ? [Ys3: list_P1915136757390359201on_nat,Zs2: list_P1915136757390359201on_nat] :
          ( ( Xs
            = ( append1010812867689791286on_nat @ Ys3 @ ( cons_P6897958070471385041on_nat @ X4 @ Zs2 ) ) )
          & ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_448_split__list__last,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ? [Ys3: list_P9127870862658512897on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X4 @ Zs2 ) ) )
          & ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_449_split__list__last,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ? [Ys3: list_S5253404628753938141on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X4 @ Zs2 ) ) )
          & ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Zs2 ) ) ) ) ).

% split_list_last
thf(fact_450_split__list,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ? [Ys3: list_P1915136757390359201on_nat,Zs2: list_P1915136757390359201on_nat] :
          ( Xs
          = ( append1010812867689791286on_nat @ Ys3 @ ( cons_P6897958070471385041on_nat @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_451_split__list,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ? [Ys3: list_P9127870862658512897on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( Xs
          = ( append2865821678206098582on_nat @ Ys3 @ ( cons_P1582596238967110705on_nat @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_452_split__list,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ? [Ys3: list_S5253404628753938141on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( Xs
          = ( append3523812819975775986on_nat @ Ys3 @ ( cons_S6797538523646374167on_nat @ X4 @ Zs2 ) ) ) ) ).

% split_list
thf(fact_453_map__tailrec__rev_Ocases,axiom,
    ! [X4: produc5902793192644279159on_nat] :
      ( ! [F2: produc1313295348606329723on_nat > produc1313295348606329723on_nat,Bs: list_P9127870862658512897on_nat] :
          ( X4
         != ( produc5429469637799208745on_nat @ F2 @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Bs ) ) )
     => ~ ! [F2: produc1313295348606329723on_nat > produc1313295348606329723on_nat,A4: produc1313295348606329723on_nat,As: list_P9127870862658512897on_nat,Bs: list_P9127870862658512897on_nat] :
            ( X4
           != ( produc5429469637799208745on_nat @ F2 @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ A4 @ As ) @ Bs ) ) ) ) ).

% map_tailrec_rev.cases
thf(fact_454_shuffles_Ocases,axiom,
    ! [X4: produc6649435091139611495od_a_o] :
      ( ! [Ys3: list_l8336264072933094989od_a_o] :
          ( X4
         != ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Ys3 ) )
     => ( ! [Xs2: list_l8336264072933094989od_a_o] :
            ( X4
           != ( produc7858527432735436247od_a_o @ Xs2 @ nil_li1372968520647675725od_a_o ) )
       => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] :
              ( X4
             != ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_455_shuffles_Ocases,axiom,
    ! [X4: produc1714117091004681831on_nat] :
      ( ! [Ys3: list_P9127870862658512897on_nat] :
          ( X4
         != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys3 ) )
     => ( ! [Xs2: list_P9127870862658512897on_nat] :
            ( X4
           != ( produc7152764266822961111on_nat @ Xs2 @ nil_Pr8614093721167239297on_nat ) )
       => ~ ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat,Y3: produc1313295348606329723on_nat,Ys3: list_P9127870862658512897on_nat] :
              ( X4
             != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs2 ) @ ( cons_P1582596238967110705on_nat @ Y3 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_456_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc6649435091139611495od_a_o] :
      ( ! [Ys3: list_l8336264072933094989od_a_o] :
          ( X4
         != ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Ys3 ) )
     => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Ys3: list_l8336264072933094989od_a_o] :
            ( X4
           != ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_457_subset__eq__mset__impl_Ocases,axiom,
    ! [X4: produc1714117091004681831on_nat] :
      ( ! [Ys3: list_P9127870862658512897on_nat] :
          ( X4
         != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys3 ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat,Ys3: list_P9127870862658512897on_nat] :
            ( X4
           != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_458_ns__ns__mul__ext__union__compat,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat,D2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ C3 @ D2 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B3 @ D2 ) ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ) ).

% ns_ns_mul_ext_union_compat
thf(fact_459_s__mul__ext__extend__left,axiom,
    ! [B3: multis1856558993800614429on_nat,C3: multis1856558993800614429on_nat,W2: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,A3: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B3 @ C3 ) @ ( multis1620213760974180726on_nat @ W2 @ S2 ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ B3 ) @ C3 ) @ ( multis1620213760974180726on_nat @ W2 @ S2 ) ) ) ).

% s_mul_ext_extend_left
thf(fact_460_lexord__sufE,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Qs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Zs ) @ ( append2865821678206098582on_nat @ Ys @ Qs ) ) @ ( lexord3493870746117591022on_nat @ R ) )
     => ( ( Xs != Ys )
       => ( ( ( size_s7490768528479217773on_nat @ Xs )
            = ( size_s7490768528479217773on_nat @ Ys ) )
         => ( ( ( size_s7490768528479217773on_nat @ Zs )
              = ( size_s7490768528479217773on_nat @ Qs ) )
           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_461_lex__append__rightI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,Vs: list_P9127870862658512897on_nat,Us2: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
     => ( ( ( size_s7490768528479217773on_nat @ Vs )
          = ( size_s7490768528479217773on_nat @ Us2 ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Us2 ) @ ( append2865821678206098582on_nat @ Ys @ Vs ) ) @ ( lex_Pr6276626268559582437on_nat @ R ) ) ) ) ).

% lex_append_rightI
thf(fact_462_lexord__append__rightI,axiom,
    ! [Y: list_P9127870862658512897on_nat,X4: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ? [B5: produc1313295348606329723on_nat,Z3: list_P9127870862658512897on_nat] :
          ( Y
          = ( cons_P1582596238967110705on_nat @ B5 @ Z3 ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ ( append2865821678206098582on_nat @ X4 @ Y ) ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ).

% lexord_append_rightI
thf(fact_463_lenlex__append1,axiom,
    ! [Us2: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,R2: set_Pr8202701763255893447on_nat,Vs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Us2 @ Xs ) @ ( lenlex6631009945862686470on_nat @ R2 ) )
     => ( ( ( size_s7490768528479217773on_nat @ Vs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Us2 @ Vs ) @ ( append2865821678206098582on_nat @ Xs @ Ys ) ) @ ( lenlex6631009945862686470on_nat @ R2 ) ) ) ) ).

% lenlex_append1
thf(fact_464_mul__ex__dom_Ocases,axiom,
    ! [X4: produc7254190109447244321od_a_o] :
      ( ! [Xs4: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] :
          ( X4
         != ( produc7470031014590217233od_a_o @ nil_li1372968520647675725od_a_o @ ( produc4213510470271429111od_a_o @ Xs4 @ ( produc528540703176342621od_a_o @ Y3 @ Ys3 ) ) ) )
     => ~ ! [X3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o,Xs4: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] :
            ( X4
           != ( produc7470031014590217233od_a_o @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) @ ( produc4213510470271429111od_a_o @ Xs4 @ ( produc528540703176342621od_a_o @ Y3 @ Ys3 ) ) ) ) ) ).

% mul_ex_dom.cases
thf(fact_465_set__union,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( set_Pr937530902923597456on_nat @ ( union_6364243231197012597on_nat @ Xs @ Ys ) )
      = ( sup_su6884158799674119087on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ ( set_Pr937530902923597456on_nat @ Ys ) ) ) ).

% set_union
thf(fact_466_set__union,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( set_Su4848624856859743608on_nat @ ( union_82372077702528979on_nat @ Xs @ Ys ) )
      = ( sup_su343302011326411951on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ).

% set_union
thf(fact_467_mul__ext__impl_Ocases,axiom,
    ! [X4: produc6649435091139611495od_a_o] :
      ( ( X4
       != ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o ) )
     => ( ! [V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o] :
            ( X4
           != ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) ) )
       => ( ! [V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o] :
              ( X4
             != ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) @ nil_li1372968520647675725od_a_o ) )
         => ~ ! [V2: list_P3196869577824546119od_a_o,Va: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] :
                ( X4
               != ( produc7858527432735436247od_a_o @ ( cons_l9179691953475787517od_a_o @ V2 @ Va ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_468_mul__ext__impl_Ocases,axiom,
    ! [X4: produc1714117091004681831on_nat] :
      ( ( X4
       != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ nil_Pr8614093721167239297on_nat ) )
     => ( ! [V2: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
            ( X4
           != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ ( cons_P1582596238967110705on_nat @ V2 @ Va ) ) )
       => ( ! [V2: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat] :
              ( X4
             != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ V2 @ Va ) @ nil_Pr8614093721167239297on_nat ) )
         => ~ ! [V2: produc1313295348606329723on_nat,Va: list_P9127870862658512897on_nat,Y3: produc1313295348606329723on_nat,Ys3: list_P9127870862658512897on_nat] :
                ( X4
               != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ V2 @ Va ) @ ( cons_P1582596238967110705on_nat @ Y3 @ Ys3 ) ) ) ) ) ) ).

% mul_ext_impl.cases
thf(fact_469_Un__iff,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( sup_su588211329720815695on_nat @ A3 @ B3 ) )
      = ( ( member4278285197320822468on_nat @ C2 @ A3 )
        | ( member4278285197320822468on_nat @ C2 @ B3 ) ) ) ).

% Un_iff
thf(fact_470_UnCI,axiom,
    ! [C2: produc9176241080893687067on_nat,B3: set_Pr3573155178515677307on_nat,A3: set_Pr3573155178515677307on_nat] :
      ( ( ~ ( member4278285197320822468on_nat @ C2 @ B3 )
       => ( member4278285197320822468on_nat @ C2 @ A3 ) )
     => ( member4278285197320822468on_nat @ C2 @ ( sup_su588211329720815695on_nat @ A3 @ B3 ) ) ) ).

% UnCI
thf(fact_471_UnE,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( sup_su588211329720815695on_nat @ A3 @ B3 ) )
     => ( ~ ( member4278285197320822468on_nat @ C2 @ A3 )
       => ( member4278285197320822468on_nat @ C2 @ B3 ) ) ) ).

% UnE
thf(fact_472_UnI1,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ A3 )
     => ( member4278285197320822468on_nat @ C2 @ ( sup_su588211329720815695on_nat @ A3 @ B3 ) ) ) ).

% UnI1
thf(fact_473_UnI2,axiom,
    ! [C2: produc9176241080893687067on_nat,B3: set_Pr3573155178515677307on_nat,A3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ B3 )
     => ( member4278285197320822468on_nat @ C2 @ ( sup_su588211329720815695on_nat @ A3 @ B3 ) ) ) ).

% UnI2
thf(fact_474_the__elem__set,axiom,
    ! [X4: list_P3196869577824546119od_a_o] :
      ( ( the_el4730552915664895548od_a_o @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_475_the__elem__set,axiom,
    ! [X4: produc1313295348606329723on_nat] :
      ( ( the_el8283737788879144688on_nat @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_476_the__elem__set,axiom,
    ! [X4: sum_su4907927656612468301on_nat] :
      ( ( the_el7555745616574930200on_nat @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X4 @ nil_Su3078906344068544711on_nat ) ) )
      = X4 ) ).

% the_elem_set
thf(fact_477_delete__aux_Ocases,axiom,
    ! [X4: produc7374814426355550521on_nat] :
      ( ! [K: multis1856558993800614429on_nat] :
          ( X4
         != ( produc2731917574445325163on_nat @ K @ nil_Pr2652050626817158177on_nat ) )
     => ~ ! [K: multis1856558993800614429on_nat,K2: multis1856558993800614429on_nat,V2: multis1856558993800614429on_nat,Xs2: list_P1915136757390359201on_nat] :
            ( X4
           != ( produc2731917574445325163on_nat @ K @ ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_478_delete__aux_Ocases,axiom,
    ! [X4: produc5056679816642863195on_nat] :
      ( ! [K: list_S5253404628753938141on_nat] :
          ( X4
         != ( produc8252945450212337037on_nat @ K @ nil_Pr5986419849129872387on_nat ) )
     => ~ ! [K: list_S5253404628753938141on_nat,K2: list_S5253404628753938141on_nat,V2: produc118098540238352959on_nat,Xs2: list_P6215481986061871491on_nat] :
            ( X4
           != ( produc8252945450212337037on_nat @ K @ ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_479_delete__aux_Ocases,axiom,
    ! [X4: produc6588776972112385165on_nat] :
      ( ! [K: list_S5253404628753938141on_nat] :
          ( X4
         != ( produc9043615093183382213on_nat @ K @ nil_Pr2927124501823088569on_nat ) )
     => ~ ! [K: list_S5253404628753938141on_nat,K2: list_S5253404628753938141on_nat,V2: produc1714117091004681831on_nat,Xs2: list_P5508476027986415439on_nat] :
            ( X4
           != ( produc9043615093183382213on_nat @ K @ ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_480_delete__aux_Ocases,axiom,
    ! [X4: produc4504349007393439315on_nat] :
      ( ! [K: list_P9127870862658512897on_nat] :
          ( X4
         != ( produc2966718627504340547on_nat @ K @ nil_Pr707104663622130797on_nat ) )
     => ~ ! [K: list_P9127870862658512897on_nat,K2: list_P9127870862658512897on_nat,V2: list_P9127870862658512897on_nat,Xs2: list_P8734936519038225773on_nat] :
            ( X4
           != ( produc2966718627504340547on_nat @ K @ ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_481_delete__aux_Ocases,axiom,
    ! [X4: produc1458239053189343596at_nat] :
      ( ! [K: nat > nat] :
          ( X4
         != ( produc17425388850148510at_nat @ K @ nil_Pr2223394031645031670at_nat ) )
     => ~ ! [K: nat > nat,K2: nat > nat,V2: nat,Xs2: list_P5366131564312172150at_nat] :
            ( X4
           != ( produc17425388850148510at_nat @ K @ ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K2 @ V2 ) @ Xs2 ) ) ) ) ).

% delete_aux.cases
thf(fact_482_product__lists_Osimps_I1_J,axiom,
    ( ( produc6266365578930819638od_a_o @ nil_li6902169125192129875od_a_o )
    = ( cons_l8465095039735648515od_a_o @ nil_li1372968520647675725od_a_o @ nil_li6902169125192129875od_a_o ) ) ).

% product_lists.simps(1)
thf(fact_483_product__lists_Osimps_I1_J,axiom,
    ( ( produc3814538932794474544od_a_o @ nil_li1372968520647675725od_a_o )
    = ( cons_l9179691953475787517od_a_o @ nil_Product_prod_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% product_lists.simps(1)
thf(fact_484_subseqs_Osimps_I1_J,axiom,
    ( ( subseq3821054824087877782od_a_o @ nil_Product_prod_a_o )
    = ( cons_l9179691953475787517od_a_o @ nil_Product_prod_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% subseqs.simps(1)
thf(fact_485_subseqs_Osimps_I1_J,axiom,
    ( ( subseq4901587760977928092od_a_o @ nil_li1372968520647675725od_a_o )
    = ( cons_l8465095039735648515od_a_o @ nil_li1372968520647675725od_a_o @ nil_li6902169125192129875od_a_o ) ) ).

% subseqs.simps(1)
thf(fact_486_length__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( size_s7490768528479217773on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ ( size_s7490768528479217773on_nat @ Ys ) ) ) ).

% length_append
thf(fact_487_compose_Ocases,axiom,
    ! [X4: produc1714117091004681831on_nat] :
      ( ! [Ys3: list_P9127870862658512897on_nat] :
          ( X4
         != ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Ys3 ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat,Ys3: list_P9127870862658512897on_nat] :
            ( X4
           != ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs2 ) @ Ys3 ) ) ) ).

% compose.cases
thf(fact_488_update__with__aux_Osimps_I1_J,axiom,
    ! [V: multis1856558993800614429on_nat,K3: multis1856558993800614429on_nat,F: multis1856558993800614429on_nat > multis1856558993800614429on_nat] :
      ( ( update882873918760498622on_nat @ V @ K3 @ F @ nil_Pr2652050626817158177on_nat )
      = ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K3 @ ( F @ V ) ) @ nil_Pr2652050626817158177on_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_489_update__with__aux_Osimps_I1_J,axiom,
    ! [V: produc118098540238352959on_nat,K3: list_S5253404628753938141on_nat,F: produc118098540238352959on_nat > produc118098540238352959on_nat] :
      ( ( update1380325430104989088on_nat @ V @ K3 @ F @ nil_Pr5986419849129872387on_nat )
      = ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K3 @ ( F @ V ) ) @ nil_Pr5986419849129872387on_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_490_update__with__aux_Osimps_I1_J,axiom,
    ! [V: produc1714117091004681831on_nat,K3: list_S5253404628753938141on_nat,F: produc1714117091004681831on_nat > produc1714117091004681831on_nat] :
      ( ( update6077729823055279170on_nat @ V @ K3 @ F @ nil_Pr2927124501823088569on_nat )
      = ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K3 @ ( F @ V ) ) @ nil_Pr2927124501823088569on_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_491_update__with__aux_Osimps_I1_J,axiom,
    ! [V: list_P9127870862658512897on_nat,K3: list_P9127870862658512897on_nat,F: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat] :
      ( ( update2179041433245623618on_nat @ V @ K3 @ F @ nil_Pr707104663622130797on_nat )
      = ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K3 @ ( F @ V ) ) @ nil_Pr707104663622130797on_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_492_update__with__aux_Osimps_I1_J,axiom,
    ! [V: nat,K3: nat > nat,F: nat > nat] :
      ( ( update516555593404511891at_nat @ V @ K3 @ F @ nil_Pr2223394031645031670at_nat )
      = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K3 @ ( F @ V ) ) @ nil_Pr2223394031645031670at_nat ) ) ).

% update_with_aux.simps(1)
thf(fact_493_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o,Y: list_P3196869577824546119od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) ) @ ( append2159120467191820514od_a_o @ Ys @ ( cons_l9179691953475787517od_a_o @ Y @ nil_li1372968520647675725od_a_o ) ) ) @ ( listre4186574385322583590od_a_o @ R ) )
      = ( ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre4186574385322583590od_a_o @ R ) )
          & ( X4 = Y ) )
        | ( ( Xs = Ys )
          & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X4 @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_494_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_m351779702951725613on_nat,X4: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( append8234005794073363010on_nat @ Xs @ ( cons_m1493500683067190247on_nat @ X4 @ nil_mu4837018292623784855on_nat ) ) @ ( append8234005794073363010on_nat @ Ys @ ( cons_m1493500683067190247on_nat @ Y @ nil_mu4837018292623784855on_nat ) ) ) @ ( listre7145697762236742142on_nat @ R ) )
      = ( ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( listre7145697762236742142on_nat @ R ) )
          & ( X4 = Y ) )
        | ( ( Xs = Ys )
          & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_495_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_l1726306699111145479on_nat,X4: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( append3839213209832495004on_nat @ Xs @ ( cons_l3028799188195208631on_nat @ X4 @ nil_li2682182165152866823on_nat ) ) @ ( append3839213209832495004on_nat @ Ys @ ( cons_l3028799188195208631on_nat @ Y @ nil_li2682182165152866823on_nat ) ) ) @ ( listre2468548762442407136on_nat @ R ) )
      = ( ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( listre2468548762442407136on_nat @ R ) )
          & ( X4 = Y ) )
        | ( ( Xs = Ys )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_496_snoc__listrel1__snoc__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) ) @ ( append2865821678206098582on_nat @ Ys @ ( cons_P1582596238967110705on_nat @ Y @ nil_Pr8614093721167239297on_nat ) ) ) @ ( listre7129750038259667930on_nat @ R ) )
      = ( ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
          & ( X4 = Y ) )
        | ( ( Xs = Ys )
          & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ R ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_497_Cons__lenlex__iff,axiom,
    ! [M: multis1856558993800614429on_nat,Ms: list_m351779702951725613on_nat,N: multis1856558993800614429on_nat,Ns: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ M @ Ms ) @ ( cons_m1493500683067190247on_nat @ N @ Ns ) ) @ ( lenlex3982385085865065426on_nat @ R ) )
      = ( ( ord_less_nat @ ( size_s6603137509925400257on_nat @ Ms ) @ ( size_s6603137509925400257on_nat @ Ns ) )
        | ( ( ( size_s6603137509925400257on_nat @ Ms )
            = ( size_s6603137509925400257on_nat @ Ns ) )
          & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Ms @ Ns ) @ ( lenlex3982385085865065426on_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_498_Cons__lenlex__iff,axiom,
    ! [M: list_P9127870862658512897on_nat,Ms: list_l1726306699111145479on_nat,N: list_P9127870862658512897on_nat,Ns: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ M @ Ms ) @ ( cons_l3028799188195208631on_nat @ N @ Ns ) ) @ ( lenlex2876470432240847372on_nat @ R ) )
      = ( ( ord_less_nat @ ( size_s129639936549468787on_nat @ Ms ) @ ( size_s129639936549468787on_nat @ Ns ) )
        | ( ( ( size_s129639936549468787on_nat @ Ms )
            = ( size_s129639936549468787on_nat @ Ns ) )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Ms @ Ns ) @ ( lenlex2876470432240847372on_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_499_Cons__lenlex__iff,axiom,
    ! [M: produc1313295348606329723on_nat,Ms: list_P9127870862658512897on_nat,N: produc1313295348606329723on_nat,Ns: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ M @ Ms ) @ ( cons_P1582596238967110705on_nat @ N @ Ns ) ) @ ( lenlex6631009945862686470on_nat @ R ) )
      = ( ( ord_less_nat @ ( size_s7490768528479217773on_nat @ Ms ) @ ( size_s7490768528479217773on_nat @ Ns ) )
        | ( ( ( size_s7490768528479217773on_nat @ Ms )
            = ( size_s7490768528479217773on_nat @ Ns ) )
          & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ M @ N ) @ R ) )
        | ( ( M = N )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ms @ Ns ) @ ( lenlex6631009945862686470on_nat @ R ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_500_delete__aux_Oelims,axiom,
    ! [X4: multis1856558993800614429on_nat,Xa2: list_P1915136757390359201on_nat,Y: list_P1915136757390359201on_nat] :
      ( ( ( delete2329322700792945250on_nat @ X4 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr2652050626817158177on_nat )
         => ( Y != nil_Pr2652050626817158177on_nat ) )
       => ~ ! [K2: multis1856558993800614429on_nat,V2: multis1856558993800614429on_nat,Xs2: list_P1915136757390359201on_nat] :
              ( ( Xa2
                = ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K2 @ V2 ) @ Xs2 ) )
             => ~ ( ( ( X4 = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X4 != K2 )
                   => ( Y
                      = ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K2 @ V2 ) @ ( delete2329322700792945250on_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_501_delete__aux_Oelims,axiom,
    ! [X4: list_S5253404628753938141on_nat,Xa2: list_P6215481986061871491on_nat,Y: list_P6215481986061871491on_nat] :
      ( ( ( delete2560205632146498372on_nat @ X4 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr5986419849129872387on_nat )
         => ( Y != nil_Pr5986419849129872387on_nat ) )
       => ~ ! [K2: list_S5253404628753938141on_nat,V2: produc118098540238352959on_nat,Xs2: list_P6215481986061871491on_nat] :
              ( ( Xa2
                = ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K2 @ V2 ) @ Xs2 ) )
             => ~ ( ( ( X4 = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X4 != K2 )
                   => ( Y
                      = ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K2 @ V2 ) @ ( delete2560205632146498372on_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_502_delete__aux_Oelims,axiom,
    ! [X4: list_S5253404628753938141on_nat,Xa2: list_P5508476027986415439on_nat,Y: list_P5508476027986415439on_nat] :
      ( ( ( delete3393325587366434082on_nat @ X4 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr2927124501823088569on_nat )
         => ( Y != nil_Pr2927124501823088569on_nat ) )
       => ~ ! [K2: list_S5253404628753938141on_nat,V2: produc1714117091004681831on_nat,Xs2: list_P5508476027986415439on_nat] :
              ( ( Xa2
                = ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K2 @ V2 ) @ Xs2 ) )
             => ~ ( ( ( X4 = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X4 != K2 )
                   => ( Y
                      = ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K2 @ V2 ) @ ( delete3393325587366434082on_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_503_delete__aux_Oelims,axiom,
    ! [X4: list_P9127870862658512897on_nat,Xa2: list_P8734936519038225773on_nat,Y: list_P8734936519038225773on_nat] :
      ( ( ( delete4533364535300879334on_nat @ X4 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr707104663622130797on_nat )
         => ( Y != nil_Pr707104663622130797on_nat ) )
       => ~ ! [K2: list_P9127870862658512897on_nat,V2: list_P9127870862658512897on_nat,Xs2: list_P8734936519038225773on_nat] :
              ( ( Xa2
                = ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K2 @ V2 ) @ Xs2 ) )
             => ~ ( ( ( X4 = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X4 != K2 )
                   => ( Y
                      = ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K2 @ V2 ) @ ( delete4533364535300879334on_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_504_delete__aux_Oelims,axiom,
    ! [X4: nat > nat,Xa2: list_P5366131564312172150at_nat,Y: list_P5366131564312172150at_nat] :
      ( ( ( delete7917733657354291895at_nat @ X4 @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Pr2223394031645031670at_nat )
         => ( Y != nil_Pr2223394031645031670at_nat ) )
       => ~ ! [K2: nat > nat,V2: nat,Xs2: list_P5366131564312172150at_nat] :
              ( ( Xa2
                = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K2 @ V2 ) @ Xs2 ) )
             => ~ ( ( ( X4 = K2 )
                   => ( Y = Xs2 ) )
                  & ( ( X4 != K2 )
                   => ( Y
                      = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K2 @ V2 ) @ ( delete7917733657354291895at_nat @ X4 @ Xs2 ) ) ) ) ) ) ) ) ).

% delete_aux.elims
thf(fact_505_delete__aux__eq__Nil__conv,axiom,
    ! [K3: multis1856558993800614429on_nat,Ts: list_P1915136757390359201on_nat] :
      ( ( ( delete2329322700792945250on_nat @ K3 @ Ts )
        = nil_Pr2652050626817158177on_nat )
      = ( ( Ts = nil_Pr2652050626817158177on_nat )
        | ? [V3: multis1856558993800614429on_nat] :
            ( Ts
            = ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K3 @ V3 ) @ nil_Pr2652050626817158177on_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_506_delete__aux__eq__Nil__conv,axiom,
    ! [K3: list_S5253404628753938141on_nat,Ts: list_P6215481986061871491on_nat] :
      ( ( ( delete2560205632146498372on_nat @ K3 @ Ts )
        = nil_Pr5986419849129872387on_nat )
      = ( ( Ts = nil_Pr5986419849129872387on_nat )
        | ? [V3: produc118098540238352959on_nat] :
            ( Ts
            = ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K3 @ V3 ) @ nil_Pr5986419849129872387on_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_507_delete__aux__eq__Nil__conv,axiom,
    ! [K3: list_S5253404628753938141on_nat,Ts: list_P5508476027986415439on_nat] :
      ( ( ( delete3393325587366434082on_nat @ K3 @ Ts )
        = nil_Pr2927124501823088569on_nat )
      = ( ( Ts = nil_Pr2927124501823088569on_nat )
        | ? [V3: produc1714117091004681831on_nat] :
            ( Ts
            = ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K3 @ V3 ) @ nil_Pr2927124501823088569on_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_508_delete__aux__eq__Nil__conv,axiom,
    ! [K3: list_P9127870862658512897on_nat,Ts: list_P8734936519038225773on_nat] :
      ( ( ( delete4533364535300879334on_nat @ K3 @ Ts )
        = nil_Pr707104663622130797on_nat )
      = ( ( Ts = nil_Pr707104663622130797on_nat )
        | ? [V3: list_P9127870862658512897on_nat] :
            ( Ts
            = ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K3 @ V3 ) @ nil_Pr707104663622130797on_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_509_delete__aux__eq__Nil__conv,axiom,
    ! [K3: nat > nat,Ts: list_P5366131564312172150at_nat] :
      ( ( ( delete7917733657354291895at_nat @ K3 @ Ts )
        = nil_Pr2223394031645031670at_nat )
      = ( ( Ts = nil_Pr2223394031645031670at_nat )
        | ? [V3: nat] :
            ( Ts
            = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K3 @ V3 ) @ nil_Pr2223394031645031670at_nat ) ) ) ) ).

% delete_aux_eq_Nil_conv
thf(fact_510_s__mul__ext__self__extend__left,axiom,
    ! [A3: multis1856558993800614429on_nat,W2: set_Pr2535564044409765851on_nat,B3: multis1856558993800614429on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( A3 != zero_z8702567803361615526on_nat )
     => ( ( locall2418947207864277812on_nat @ W2 @ B3 )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ B3 ) @ B3 ) @ ( multis1620213760974180726on_nat @ W2 @ S2 ) ) ) ) ).

% s_mul_ext_self_extend_left
thf(fact_511_s__mul__ext__ne__extend__left,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,C3: multis1856558993800614429on_nat,W2: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( A3 != zero_z8702567803361615526on_nat )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B3 @ C3 ) @ ( multis4183089776494585304on_nat @ W2 @ S2 ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ B3 ) @ C3 ) @ ( multis1620213760974180726on_nat @ W2 @ S2 ) ) ) ) ).

% s_mul_ext_ne_extend_left
thf(fact_512_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_513_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_514_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_515_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_516_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_517_add__eq__0__iff__both__eq__0,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X4 @ Y )
        = zero_zero_nat )
      = ( ( X4 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_518_zero__eq__add__iff__both__eq__0,axiom,
    ! [X4: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X4 @ Y ) )
      = ( ( X4 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_519_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_520_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_521_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_522_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_523_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_524_mset__zero__iff__right,axiom,
    ! [X4: list_l8336264072933094989od_a_o] :
      ( ( zero_z8905637151584487364od_a_o
        = ( mset_l8139550355095933148od_a_o @ X4 ) )
      = ( X4 = nil_li1372968520647675725od_a_o ) ) ).

% mset_zero_iff_right
thf(fact_525_mset__zero__iff__right,axiom,
    ! [X4: list_S5253404628753938141on_nat] :
      ( ( zero_z8702567803361615526on_nat
        = ( mset_S7907735292211298936on_nat @ X4 ) )
      = ( X4 = nil_Su3078906344068544711on_nat ) ) ).

% mset_zero_iff_right
thf(fact_526_mset__zero__iff__right,axiom,
    ! [X4: list_P9127870862658512897on_nat] :
      ( ( zero_z2712726346186620152on_nat
        = ( mset_P6964650582947932560on_nat @ X4 ) )
      = ( X4 = nil_Pr8614093721167239297on_nat ) ) ).

% mset_zero_iff_right
thf(fact_527_mset__zero__iff,axiom,
    ! [X4: list_l8336264072933094989od_a_o] :
      ( ( ( mset_l8139550355095933148od_a_o @ X4 )
        = zero_z8905637151584487364od_a_o )
      = ( X4 = nil_li1372968520647675725od_a_o ) ) ).

% mset_zero_iff
thf(fact_528_mset__zero__iff,axiom,
    ! [X4: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ X4 )
        = zero_z8702567803361615526on_nat )
      = ( X4 = nil_Su3078906344068544711on_nat ) ) ).

% mset_zero_iff
thf(fact_529_mset__zero__iff,axiom,
    ! [X4: list_P9127870862658512897on_nat] :
      ( ( ( mset_P6964650582947932560on_nat @ X4 )
        = zero_z2712726346186620152on_nat )
      = ( X4 = nil_Pr8614093721167239297on_nat ) ) ).

% mset_zero_iff
thf(fact_530_Cons__listrel1__Cons,axiom,
    ! [X4: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ X4 @ Xs ) @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) ) @ ( listre7145697762236742142on_nat @ R ) )
      = ( ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X4 = Y )
          & ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( listre7145697762236742142on_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_531_Cons__listrel1__Cons,axiom,
    ! [X4: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X4 @ Xs ) @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) ) @ ( listre2468548762442407136on_nat @ R ) )
      = ( ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X4 = Y )
          & ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( listre2468548762442407136on_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_532_Cons__listrel1__Cons,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) @ ( listre7129750038259667930on_nat @ R ) )
      = ( ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X4 = Y )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_533_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_534_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_535_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_536_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_537_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_538_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_539_mset_Osimps_I1_J,axiom,
    ( ( mset_l8139550355095933148od_a_o @ nil_li1372968520647675725od_a_o )
    = zero_z8905637151584487364od_a_o ) ).

% mset.simps(1)
thf(fact_540_mset_Osimps_I1_J,axiom,
    ( ( mset_S7907735292211298936on_nat @ nil_Su3078906344068544711on_nat )
    = zero_z8702567803361615526on_nat ) ).

% mset.simps(1)
thf(fact_541_mset_Osimps_I1_J,axiom,
    ( ( mset_P6964650582947932560on_nat @ nil_Pr8614093721167239297on_nat )
    = zero_z2712726346186620152on_nat ) ).

% mset.simps(1)
thf(fact_542_listrel1I2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,X4: produc1313295348606329723on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ ( cons_P1582596238967110705on_nat @ X4 @ Ys ) ) @ ( listre7129750038259667930on_nat @ R ) ) ) ).

% listrel1I2
thf(fact_543_not__Nil__listrel1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Xs ) @ ( listre4186574385322583590od_a_o @ R ) ) ).

% not_Nil_listrel1
thf(fact_544_not__Nil__listrel1,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Xs ) @ ( listre7129750038259667930on_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_545_not__listrel1__Nil,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ nil_li1372968520647675725od_a_o ) @ ( listre4186574385322583590od_a_o @ R ) ) ).

% not_listrel1_Nil
thf(fact_546_not__listrel1__Nil,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ nil_Pr8614093721167239297on_nat ) @ ( listre7129750038259667930on_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_547_listrel1__eq__len,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
     => ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_548_append__listrel1I,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,Us2: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
      ( ( ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
          & ( Us2 = Vs ) )
        | ( ( Xs = Ys )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Us2 @ Vs ) @ ( listre7129750038259667930on_nat @ R ) ) ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Us2 ) @ ( append2865821678206098582on_nat @ Ys @ Vs ) ) @ ( listre7129750038259667930on_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_549_ns__mul__ext__bottom__uniqueness,axiom,
    ! [A3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ A3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ( A3 = zero_z8702567803361615526on_nat ) ) ).

% ns_mul_ext_bottom_uniqueness
thf(fact_550_ns__mul__ext__bottom2,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ( ( B3 != zero_z8702567803361615526on_nat )
       => ( A3 != zero_z8702567803361615526on_nat ) ) ) ).

% ns_mul_ext_bottom2
thf(fact_551_ns__mul__ext__bottom,axiom,
    ! [A3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ zero_z8702567803361615526on_nat ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ).

% ns_mul_ext_bottom
thf(fact_552_s__mul__ext__bottom__strict,axiom,
    ! [A3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ A3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ).

% s_mul_ext_bottom_strict
thf(fact_553_s__mul__ext__bottom,axiom,
    ! [A3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( A3 != zero_z8702567803361615526on_nat )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ zero_z8702567803361615526on_nat ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ).

% s_mul_ext_bottom
thf(fact_554_maps__simps_I2_J,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_l8336264072933094989od_a_o] :
      ( ( maps_l8079236806422966529od_a_o @ F @ nil_li1372968520647675725od_a_o )
      = nil_li1372968520647675725od_a_o ) ).

% maps_simps(2)
thf(fact_555_listrel1I1,axiom,
    ! [X4: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat,R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ R )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ ( cons_P1582596238967110705on_nat @ Y @ Xs ) ) @ ( listre7129750038259667930on_nat @ R ) ) ) ).

% listrel1I1
thf(fact_556_listrel1I1,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ R )
     => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ X4 @ Xs ) @ ( cons_m1493500683067190247on_nat @ Y @ Xs ) ) @ ( listre7145697762236742142on_nat @ R ) ) ) ).

% listrel1I1
thf(fact_557_listrel1I1,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ R )
     => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X4 @ Xs ) @ ( cons_l3028799188195208631on_nat @ Y @ Xs ) ) @ ( listre2468548762442407136on_nat @ R ) ) ) ).

% listrel1I1
thf(fact_558_Cons__listrel1E1,axiom,
    ! [X4: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ X4 @ Xs ) @ Ys ) @ ( listre7145697762236742142on_nat @ R ) )
     => ( ! [Y3: multis1856558993800614429on_nat] :
            ( ( Ys
              = ( cons_m1493500683067190247on_nat @ Y3 @ Xs ) )
           => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_m351779702951725613on_nat] :
              ( ( Ys
                = ( cons_m1493500683067190247on_nat @ X4 @ Zs2 ) )
             => ~ ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Zs2 ) @ ( listre7145697762236742142on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_559_Cons__listrel1E1,axiom,
    ! [X4: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X4 @ Xs ) @ Ys ) @ ( listre2468548762442407136on_nat @ R ) )
     => ( ! [Y3: list_P9127870862658512897on_nat] :
            ( ( Ys
              = ( cons_l3028799188195208631on_nat @ Y3 @ Xs ) )
           => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_l1726306699111145479on_nat] :
              ( ( Ys
                = ( cons_l3028799188195208631on_nat @ X4 @ Zs2 ) )
             => ~ ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Zs2 ) @ ( listre2468548762442407136on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_560_Cons__listrel1E1,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
     => ( ! [Y3: produc1313295348606329723on_nat] :
            ( ( Ys
              = ( cons_P1582596238967110705on_nat @ Y3 @ Xs ) )
           => ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y3 ) @ R ) )
       => ~ ! [Zs2: list_P9127870862658512897on_nat] :
              ( ( Ys
                = ( cons_P1582596238967110705on_nat @ X4 @ Zs2 ) )
             => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Zs2 ) @ ( listre7129750038259667930on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_561_Cons__listrel1E2,axiom,
    ! [Xs: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) ) @ ( listre7145697762236742142on_nat @ R ) )
     => ( ! [X3: multis1856558993800614429on_nat] :
            ( ( Xs
              = ( cons_m1493500683067190247on_nat @ X3 @ Ys ) )
           => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_m351779702951725613on_nat] :
              ( ( Xs
                = ( cons_m1493500683067190247on_nat @ Y @ Zs2 ) )
             => ~ ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Zs2 @ Ys ) @ ( listre7145697762236742142on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_562_Cons__listrel1E2,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) ) @ ( listre2468548762442407136on_nat @ R ) )
     => ( ! [X3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( cons_l3028799188195208631on_nat @ X3 @ Ys ) )
           => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_l1726306699111145479on_nat] :
              ( ( Xs
                = ( cons_l3028799188195208631on_nat @ Y @ Zs2 ) )
             => ~ ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Zs2 @ Ys ) @ ( listre2468548762442407136on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_563_Cons__listrel1E2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) @ ( listre7129750038259667930on_nat @ R ) )
     => ( ! [X3: produc1313295348606329723on_nat] :
            ( ( Xs
              = ( cons_P1582596238967110705on_nat @ X3 @ Ys ) )
           => ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ Y ) @ R ) )
       => ~ ! [Zs2: list_P9127870862658512897on_nat] :
              ( ( Xs
                = ( cons_P1582596238967110705on_nat @ Y @ Zs2 ) )
             => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Zs2 @ Ys ) @ ( listre7129750038259667930on_nat @ R ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_564_delete__aux_Osimps_I2_J,axiom,
    ! [K3: multis1856558993800614429on_nat,K4: multis1856558993800614429on_nat,V: multis1856558993800614429on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete2329322700792945250on_nat @ K3 @ ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete2329322700792945250on_nat @ K3 @ ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K4 @ V ) @ Xs ) )
          = ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K4 @ V ) @ ( delete2329322700792945250on_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_565_delete__aux_Osimps_I2_J,axiom,
    ! [K3: list_S5253404628753938141on_nat,K4: list_S5253404628753938141on_nat,V: produc118098540238352959on_nat,Xs: list_P6215481986061871491on_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete2560205632146498372on_nat @ K3 @ ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete2560205632146498372on_nat @ K3 @ ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K4 @ V ) @ Xs ) )
          = ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K4 @ V ) @ ( delete2560205632146498372on_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_566_delete__aux_Osimps_I2_J,axiom,
    ! [K3: list_S5253404628753938141on_nat,K4: list_S5253404628753938141on_nat,V: produc1714117091004681831on_nat,Xs: list_P5508476027986415439on_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete3393325587366434082on_nat @ K3 @ ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete3393325587366434082on_nat @ K3 @ ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K4 @ V ) @ Xs ) )
          = ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K4 @ V ) @ ( delete3393325587366434082on_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_567_delete__aux_Osimps_I2_J,axiom,
    ! [K3: list_P9127870862658512897on_nat,K4: list_P9127870862658512897on_nat,V: list_P9127870862658512897on_nat,Xs: list_P8734936519038225773on_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete4533364535300879334on_nat @ K3 @ ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete4533364535300879334on_nat @ K3 @ ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K4 @ V ) @ Xs ) )
          = ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K4 @ V ) @ ( delete4533364535300879334on_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_568_delete__aux_Osimps_I2_J,axiom,
    ! [K3: nat > nat,K4: nat > nat,V: nat,Xs: list_P5366131564312172150at_nat] :
      ( ( ( K3 = K4 )
       => ( ( delete7917733657354291895at_nat @ K3 @ ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K4 @ V ) @ Xs ) )
          = Xs ) )
      & ( ( K3 != K4 )
       => ( ( delete7917733657354291895at_nat @ K3 @ ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K4 @ V ) @ Xs ) )
          = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K4 @ V ) @ ( delete7917733657354291895at_nat @ K3 @ Xs ) ) ) ) ) ).

% delete_aux.simps(2)
thf(fact_569_listrel1E,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( listre7145697762236742142on_nat @ R ) )
     => ~ ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ R )
           => ! [Us3: list_m351779702951725613on_nat,Vs2: list_m351779702951725613on_nat] :
                ( ( Xs
                  = ( append8234005794073363010on_nat @ Us3 @ ( cons_m1493500683067190247on_nat @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append8234005794073363010on_nat @ Us3 @ ( cons_m1493500683067190247on_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_570_listrel1E,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( listre2468548762442407136on_nat @ R ) )
     => ~ ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
            ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ R )
           => ! [Us3: list_l1726306699111145479on_nat,Vs2: list_l1726306699111145479on_nat] :
                ( ( Xs
                  = ( append3839213209832495004on_nat @ Us3 @ ( cons_l3028799188195208631on_nat @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append3839213209832495004on_nat @ Us3 @ ( cons_l3028799188195208631on_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_571_listrel1E,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Y3: produc1313295348606329723on_nat] :
            ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ Y3 ) @ R )
           => ! [Us3: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
                ( ( Xs
                  = ( append2865821678206098582on_nat @ Us3 @ ( cons_P1582596238967110705on_nat @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append2865821678206098582on_nat @ Us3 @ ( cons_P1582596238967110705on_nat @ Y3 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_572_listrel1I,axiom,
    ! [X4: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat,R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat,Us2: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ R )
     => ( ( Xs
          = ( append2865821678206098582on_nat @ Us2 @ ( cons_P1582596238967110705on_nat @ X4 @ Vs ) ) )
       => ( ( Ys
            = ( append2865821678206098582on_nat @ Us2 @ ( cons_P1582596238967110705on_nat @ Y @ Vs ) ) )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_573_listrel1I,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat,Us2: list_m351779702951725613on_nat,Vs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ R )
     => ( ( Xs
          = ( append8234005794073363010on_nat @ Us2 @ ( cons_m1493500683067190247on_nat @ X4 @ Vs ) ) )
       => ( ( Ys
            = ( append8234005794073363010on_nat @ Us2 @ ( cons_m1493500683067190247on_nat @ Y @ Vs ) ) )
         => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( listre7145697762236742142on_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_574_listrel1I,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat,Us2: list_l1726306699111145479on_nat,Vs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ R )
     => ( ( Xs
          = ( append3839213209832495004on_nat @ Us2 @ ( cons_l3028799188195208631on_nat @ X4 @ Vs ) ) )
       => ( ( Ys
            = ( append3839213209832495004on_nat @ Us2 @ ( cons_l3028799188195208631on_nat @ Y @ Vs ) ) )
         => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( listre2468548762442407136on_nat @ R ) ) ) ) ) ).

% listrel1I
thf(fact_575_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_576_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_577_s__mul__ext__intro,axiom,
    ! [Xs: multis2912115315702009697on_nat,Xs1: list_P1915136757390359201on_nat,Xs22: list_P1915136757390359201on_nat,Ys: multis2912115315702009697on_nat,Ys1: list_P1915136757390359201on_nat,Ys22: list_P1915136757390359201on_nat,NS: set_Pr1446391457923625415on_nat,S2: set_Pr1446391457923625415on_nat] :
      ( ( Xs
        = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Xs1 ) @ ( mset_P1678654427565054512on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Ys1 ) @ ( mset_P1678654427565054512on_nat @ Ys22 ) ) )
       => ( ( ( size_s8191359103794446093on_nat @ Xs1 )
            = ( size_s8191359103794446093on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s8191359103794446093on_nat @ Ys1 ) )
               => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( nth_Pr6961551319198328098on_nat @ Xs1 @ I ) @ ( nth_Pr6961551319198328098on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ( Xs22 != nil_Pr2652050626817158177on_nat )
             => ( ! [Y3: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Y3 @ ( set_Pr8272476097931642672on_nat @ Ys22 ) )
                   => ? [X6: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ X6 @ ( set_Pr8272476097931642672on_nat @ Xs22 ) )
                        & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X6 @ Y3 ) @ S2 ) ) )
               => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ Xs @ Ys ) @ ( multis1332936165424366514on_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_578_s__mul__ext__intro,axiom,
    ! [Xs: multis1714066462791002893od_a_o,Xs1: list_l8336264072933094989od_a_o,Xs22: list_l8336264072933094989od_a_o,Ys: multis1714066462791002893od_a_o,Ys1: list_l8336264072933094989od_a_o,Ys22: list_l8336264072933094989od_a_o,NS: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o] :
      ( ( Xs
        = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs1 ) @ ( mset_l8139550355095933148od_a_o @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Ys1 ) @ ( mset_l8139550355095933148od_a_o @ Ys22 ) ) )
       => ( ( ( size_s3379185846251495609od_a_o @ Xs1 )
            = ( size_s3379185846251495609od_a_o @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Ys1 ) )
               => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs1 @ I ) @ ( nth_li3248255784598043854od_a_o @ Ys1 @ I ) ) @ NS ) )
           => ( ( Xs22 != nil_li1372968520647675725od_a_o )
             => ( ! [Y3: list_P3196869577824546119od_a_o] :
                    ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Ys22 ) )
                   => ? [X6: list_P3196869577824546119od_a_o] :
                        ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Xs22 ) )
                        & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X6 @ Y3 ) @ S2 ) ) )
               => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis7840163239927194462od_a_o @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_579_s__mul__ext__intro,axiom,
    ! [Xs: multis4835999835052358849on_nat,Xs1: list_P9127870862658512897on_nat,Xs22: list_P9127870862658512897on_nat,Ys: multis4835999835052358849on_nat,Ys1: list_P9127870862658512897on_nat,Ys22: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S2: set_Pr8202701763255893447on_nat] :
      ( ( Xs
        = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs1 ) @ ( mset_P6964650582947932560on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys1 ) @ ( mset_P6964650582947932560on_nat @ Ys22 ) ) )
       => ( ( ( size_s7490768528479217773on_nat @ Xs1 )
            = ( size_s7490768528479217773on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Ys1 ) )
               => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs1 @ I ) @ ( nth_Pr3336893728888831618on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ( Xs22 != nil_Pr8614093721167239297on_nat )
             => ( ! [Y3: produc1313295348606329723on_nat] :
                    ( ( member5906418179908956708on_nat @ Y3 @ ( set_Pr937530902923597456on_nat @ Ys22 ) )
                   => ? [X6: produc1313295348606329723on_nat] :
                        ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs22 ) )
                        & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X6 @ Y3 ) @ S2 ) ) )
               => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ Xs @ Ys ) @ ( multis9203262125639806226on_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_580_s__mul__ext__intro,axiom,
    ! [Xs: multis6850599868085929325on_nat,Xs1: list_m351779702951725613on_nat,Xs22: list_m351779702951725613on_nat,Ys: multis6850599868085929325on_nat,Ys1: list_m351779702951725613on_nat,Ys22: list_m351779702951725613on_nat,NS: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat] :
      ( ( Xs
        = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Xs1 ) @ ( mset_m1293634003157817032on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Ys1 ) @ ( mset_m1293634003157817032on_nat @ Ys22 ) ) )
       => ( ( ( size_s6603137509925400257on_nat @ Xs1 )
            = ( size_s6603137509925400257on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Ys1 ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs1 @ I ) @ ( nth_mu8151576519813129558on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ( Xs22 != nil_mu4837018292623784855on_nat )
             => ( ! [Y3: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Y3 @ ( set_mu1790088485793245640on_nat @ Ys22 ) )
                   => ? [X6: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ X6 @ ( set_mu1790088485793245640on_nat @ Xs22 ) )
                        & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X6 @ Y3 ) @ S2 ) ) )
               => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Xs @ Ys ) @ ( multis5680658968000484038on_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_581_s__mul__ext__intro,axiom,
    ! [Xs: multis5168333519641040071on_nat,Xs1: list_l1726306699111145479on_nat,Xs22: list_l1726306699111145479on_nat,Ys: multis5168333519641040071on_nat,Ys1: list_l1726306699111145479on_nat,Ys22: list_l1726306699111145479on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat] :
      ( ( Xs
        = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs1 ) @ ( mset_l2262431794812116886on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys1 ) @ ( mset_l2262431794812116886on_nat @ Ys22 ) ) )
       => ( ( ( size_s129639936549468787on_nat @ Xs1 )
            = ( size_s129639936549468787on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Ys1 ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs1 @ I ) @ ( nth_li1302981970230039432on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ( Xs22 != nil_li2682182165152866823on_nat )
             => ( ! [Y3: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Y3 @ ( set_li8154783781463198870on_nat @ Ys22 ) )
                   => ? [X6: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ X6 @ ( set_li8154783781463198870on_nat @ Xs22 ) )
                        & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X6 @ Y3 ) @ S2 ) ) )
               => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Xs @ Ys ) @ ( multis3147760155352349208on_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_582_s__mul__ext__intro,axiom,
    ! [Xs: multis1856558993800614429on_nat,Xs1: list_S5253404628753938141on_nat,Xs22: list_S5253404628753938141on_nat,Ys: multis1856558993800614429on_nat,Ys1: list_S5253404628753938141on_nat,Ys22: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( Xs
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs1 ) @ ( mset_S7907735292211298936on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys1 ) @ ( mset_S7907735292211298936on_nat @ Ys22 ) ) )
       => ( ( ( size_s5686234912572693873on_nat @ Xs1 )
            = ( size_s5686234912572693873on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys1 ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs1 @ I ) @ ( nth_Su8471564416988646406on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ( Xs22 != nil_Su3078906344068544711on_nat )
             => ( ! [Y3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
                   => ? [X6: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs22 ) )
                        & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X6 @ Y3 ) @ S2 ) ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis1620213760974180726on_nat @ NS @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_583_s__mul__ext__elim,axiom,
    ! [Xs: multis1714066462791002893od_a_o,Ys: multis1714066462791002893od_a_o,NS: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis7840163239927194462od_a_o @ NS @ S2 ) )
     => ? [Xs12: list_l8336264072933094989od_a_o,Xs23: list_l8336264072933094989od_a_o] :
          ( ( Xs
            = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs12 ) @ ( mset_l8139550355095933148od_a_o @ Xs23 ) ) )
          & ? [Ys12: list_l8336264072933094989od_a_o,Ys23: list_l8336264072933094989od_a_o] :
              ( ( Ys
                = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Ys12 ) @ ( mset_l8139550355095933148od_a_o @ Ys23 ) ) )
              & ( ( size_s3379185846251495609od_a_o @ Xs12 )
                = ( size_s3379185846251495609od_a_o @ Ys12 ) )
              & ( Xs23 != nil_li1372968520647675725od_a_o )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Ys12 ) )
                 => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs12 @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ X6 @ ( set_li2629784761229999580od_a_o @ Ys23 ) )
                 => ? [Xa3: list_P3196869577824546119od_a_o] :
                      ( ( member5261019209154752368od_a_o @ Xa3 @ ( set_li2629784761229999580od_a_o @ Xs23 ) )
                      & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_584_s__mul__ext__elim,axiom,
    ! [Xs: multis4835999835052358849on_nat,Ys: multis4835999835052358849on_nat,NS: set_Pr8202701763255893447on_nat,S2: set_Pr8202701763255893447on_nat] :
      ( ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ Xs @ Ys ) @ ( multis9203262125639806226on_nat @ NS @ S2 ) )
     => ? [Xs12: list_P9127870862658512897on_nat,Xs23: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs12 ) @ ( mset_P6964650582947932560on_nat @ Xs23 ) ) )
          & ? [Ys12: list_P9127870862658512897on_nat,Ys23: list_P9127870862658512897on_nat] :
              ( ( Ys
                = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys12 ) @ ( mset_P6964650582947932560on_nat @ Ys23 ) ) )
              & ( ( size_s7490768528479217773on_nat @ Xs12 )
                = ( size_s7490768528479217773on_nat @ Ys12 ) )
              & ( Xs23 != nil_Pr8614093721167239297on_nat )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Ys12 ) )
                 => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs12 @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: produc1313295348606329723on_nat] :
                  ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Ys23 ) )
                 => ? [Xa3: produc1313295348606329723on_nat] :
                      ( ( member5906418179908956708on_nat @ Xa3 @ ( set_Pr937530902923597456on_nat @ Xs23 ) )
                      & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_585_s__mul__ext__elim,axiom,
    ! [Xs: multis6850599868085929325on_nat,Ys: multis6850599868085929325on_nat,NS: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Xs @ Ys ) @ ( multis5680658968000484038on_nat @ NS @ S2 ) )
     => ? [Xs12: list_m351779702951725613on_nat,Xs23: list_m351779702951725613on_nat] :
          ( ( Xs
            = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Xs12 ) @ ( mset_m1293634003157817032on_nat @ Xs23 ) ) )
          & ? [Ys12: list_m351779702951725613on_nat,Ys23: list_m351779702951725613on_nat] :
              ( ( Ys
                = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Ys12 ) @ ( mset_m1293634003157817032on_nat @ Ys23 ) ) )
              & ( ( size_s6603137509925400257on_nat @ Xs12 )
                = ( size_s6603137509925400257on_nat @ Ys12 ) )
              & ( Xs23 != nil_mu4837018292623784855on_nat )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s6603137509925400257on_nat @ Ys12 ) )
                 => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs12 @ I2 ) @ ( nth_mu8151576519813129558on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ X6 @ ( set_mu1790088485793245640on_nat @ Ys23 ) )
                 => ? [Xa3: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Xa3 @ ( set_mu1790088485793245640on_nat @ Xs23 ) )
                      & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_586_s__mul__ext__elim,axiom,
    ! [Xs: multis5168333519641040071on_nat,Ys: multis5168333519641040071on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Xs @ Ys ) @ ( multis3147760155352349208on_nat @ NS @ S2 ) )
     => ? [Xs12: list_l1726306699111145479on_nat,Xs23: list_l1726306699111145479on_nat] :
          ( ( Xs
            = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs12 ) @ ( mset_l2262431794812116886on_nat @ Xs23 ) ) )
          & ? [Ys12: list_l1726306699111145479on_nat,Ys23: list_l1726306699111145479on_nat] :
              ( ( Ys
                = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys12 ) @ ( mset_l2262431794812116886on_nat @ Ys23 ) ) )
              & ( ( size_s129639936549468787on_nat @ Xs12 )
                = ( size_s129639936549468787on_nat @ Ys12 ) )
              & ( Xs23 != nil_li2682182165152866823on_nat )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Ys12 ) )
                 => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs12 @ I2 ) @ ( nth_li1302981970230039432on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ X6 @ ( set_li8154783781463198870on_nat @ Ys23 ) )
                 => ? [Xa3: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Xa3 @ ( set_li8154783781463198870on_nat @ Xs23 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_587_s__mul__ext__elim,axiom,
    ! [Xs: multis1856558993800614429on_nat,Ys: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis1620213760974180726on_nat @ NS @ S2 ) )
     => ? [Xs12: list_S5253404628753938141on_nat,Xs23: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs12 ) @ ( mset_S7907735292211298936on_nat @ Xs23 ) ) )
          & ? [Ys12: list_S5253404628753938141on_nat,Ys23: list_S5253404628753938141on_nat] :
              ( ( Ys
                = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys12 ) @ ( mset_S7907735292211298936on_nat @ Ys23 ) ) )
              & ( ( size_s5686234912572693873on_nat @ Xs12 )
                = ( size_s5686234912572693873on_nat @ Ys12 ) )
              & ( Xs23 != nil_Su3078906344068544711on_nat )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Ys12 ) )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs12 @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Ys23 ) )
                 => ? [Xa3: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Xa3 @ ( set_Su4848624856859743608on_nat @ Xs23 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_588_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat,C3: multis2912115315702009697on_nat,D2: multis2912115315702009697on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis3046611884734373584on_nat @ Ns @ S ) )
     => ( ( C3 != zero_z1502369238205801624on_nat )
       => ( ! [D: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ D @ ( set_ms7239842528327418072on_nat @ D2 ) )
             => ? [C4: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ C4 @ ( set_ms7239842528327418072on_nat @ C3 ) )
                  & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ C4 @ D ) @ S ) ) )
         => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ ( plus_p7340530524814416408on_nat @ A3 @ C3 ) @ ( plus_p7340530524814416408on_nat @ B3 @ D2 ) ) @ ( multis1332936165424366514on_nat @ Ns @ S ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_589_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat,C3: multis6850599868085929325on_nat,D2: multis6850599868085929325on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis3835232632530192936on_nat @ Ns @ S ) )
     => ( ( C3 != zero_z4144889538238539638on_nat )
       => ( ! [D: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ D @ ( set_ms2805207930803453472on_nat @ D2 ) )
             => ? [C4: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ C4 @ ( set_ms2805207930803453472on_nat @ C3 ) )
                  & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ C4 @ D ) @ S ) ) )
         => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( plus_p4342429221931815414on_nat @ A3 @ C3 ) @ ( plus_p4342429221931815414on_nat @ B3 @ D2 ) ) @ ( multis5680658968000484038on_nat @ Ns @ S ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_590_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat,C3: multis5168333519641040071on_nat,D2: multis5168333519641040071on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis664119935410451510on_nat @ Ns @ S ) )
     => ( ( C3 != zero_z8385993896842825086on_nat )
       => ( ! [D: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ D @ ( set_ms6185292591494683198on_nat @ D2 ) )
             => ? [C4: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ C4 @ ( set_ms6185292591494683198on_nat @ C3 ) )
                  & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ C4 @ D ) @ S ) ) )
         => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( plus_p7957816880051667710on_nat @ A3 @ C3 ) @ ( plus_p7957816880051667710on_nat @ B3 @ D2 ) ) @ ( multis3147760155352349208on_nat @ Ns @ S ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_591_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat,D2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ( ( C3 != zero_z8702567803361615526on_nat )
       => ( ! [D: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ D @ ( set_ms3922534532180125136on_nat @ D2 ) )
             => ? [C4: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ C4 @ ( set_ms3922534532180125136on_nat @ C3 ) )
                  & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ C4 @ D ) @ S ) ) )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A3 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B3 @ D2 ) ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_592_length__0__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% length_0_conv
thf(fact_593_length__greater__0__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ Xs ) )
      = ( Xs != nil_li1372968520647675725od_a_o ) ) ).

% length_greater_0_conv
thf(fact_594_set__mset__mset,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) )
      = ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% set_mset_mset
thf(fact_595_set__mset__mset,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( set_ms8042534003875616824on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) )
      = ( set_Pr937530902923597456on_nat @ Xs ) ) ).

% set_mset_mset
thf(fact_596_nth__append__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ Ys ) ) @ ( size_s7490768528479217773on_nat @ Xs ) )
      = X4 ) ).

% nth_append_length
thf(fact_597_nth__append__length__plus,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,N: nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ N ) )
      = ( nth_Pr3336893728888831618on_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_598_multiset__nonemptyE,axiom,
    ! [A3: multis2912115315702009697on_nat] :
      ( ( A3 != zero_z1502369238205801624on_nat )
     => ~ ! [X3: produc9176241080893687067on_nat] :
            ~ ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ A3 ) ) ) ).

% multiset_nonemptyE
thf(fact_599_union__iff,axiom,
    ! [A: produc9176241080893687067on_nat,A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ ( plus_p7340530524814416408on_nat @ A3 @ B3 ) ) )
      = ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ A3 ) )
        | ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ B3 ) ) ) ) ).

% union_iff
thf(fact_600_nth__mem__mset,axiom,
    ! [I3: nat,Ls: list_P1915136757390359201on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s8191359103794446093on_nat @ Ls ) )
     => ( member4278285197320822468on_nat @ ( nth_Pr6961551319198328098on_nat @ Ls @ I3 ) @ ( set_ms7239842528327418072on_nat @ ( mset_P1678654427565054512on_nat @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_601_nth__mem__mset,axiom,
    ! [I3: nat,Ls: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Ls ) )
     => ( member8476201367316484836on_nat @ ( nth_Su8471564416988646406on_nat @ Ls @ I3 ) @ ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_602_nth__mem__mset,axiom,
    ! [I3: nat,Ls: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Ls ) )
     => ( member5906418179908956708on_nat @ ( nth_Pr3336893728888831618on_nat @ Ls @ I3 ) @ ( set_ms8042534003875616824on_nat @ ( mset_P6964650582947932560on_nat @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_603_in__multiset__in__set,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ ( mset_P1678654427565054512on_nat @ Xs ) ) )
      = ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_604_in__multiset__in__set,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) ) )
      = ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_605_in__multiset__in__set,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_ms8042534003875616824on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) ) )
      = ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_606_ns__mul__ext__point,axiom,
    ! [As2: multis2912115315702009697on_nat,Bs2: multis2912115315702009697on_nat,NS: set_Pr1446391457923625415on_nat,S2: set_Pr1446391457923625415on_nat,B: produc9176241080893687067on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ As2 @ Bs2 ) @ ( multis3046611884734373584on_nat @ NS @ S2 ) )
     => ( ( member4278285197320822468on_nat @ B @ ( set_ms7239842528327418072on_nat @ Bs2 ) )
       => ? [X3: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ As2 ) )
            & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ B ) @ ( sup_su5655913927576555419on_nat @ NS @ S2 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_607_ns__mul__ext__point,axiom,
    ! [As2: multis6850599868085929325on_nat,Bs2: multis6850599868085929325on_nat,NS: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat,B: multis1856558993800614429on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ As2 @ Bs2 ) @ ( multis3835232632530192936on_nat @ NS @ S2 ) )
     => ( ( member2988082745645132084on_nat @ B @ ( set_ms2805207930803453472on_nat @ Bs2 ) )
       => ? [X3: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ As2 ) )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ B ) @ ( sup_su588211329720815695on_nat @ NS @ S2 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_608_ns__mul__ext__point,axiom,
    ! [As2: multis5168333519641040071on_nat,Bs2: multis5168333519641040071on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat,B: list_P9127870862658512897on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ As2 @ Bs2 ) @ ( multis664119935410451510on_nat @ NS @ S2 ) )
     => ( ( member783415501022416426on_nat @ B @ ( set_ms6185292591494683198on_nat @ Bs2 ) )
       => ? [X3: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ As2 ) )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ B ) @ ( sup_su5722365171377032731on_nat @ NS @ S2 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_609_ns__mul__ext__point,axiom,
    ! [As2: multis1856558993800614429on_nat,Bs2: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,B: sum_su4907927656612468301on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ As2 @ Bs2 ) @ ( multis4183089776494585304on_nat @ NS @ S2 ) )
     => ( ( member8476201367316484836on_nat @ B @ ( set_ms3922534532180125136on_nat @ Bs2 ) )
       => ? [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ As2 ) )
            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ B ) @ ( sup_su6884158799674119087on_nat @ NS @ S2 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_610_s__mul__ext__point,axiom,
    ! [As2: multis2912115315702009697on_nat,Bs2: multis2912115315702009697on_nat,NS: set_Pr1446391457923625415on_nat,S2: set_Pr1446391457923625415on_nat,B: produc9176241080893687067on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ As2 @ Bs2 ) @ ( multis1332936165424366514on_nat @ NS @ S2 ) )
     => ( ( member4278285197320822468on_nat @ B @ ( set_ms7239842528327418072on_nat @ Bs2 ) )
       => ? [X3: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ As2 ) )
            & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ B ) @ ( sup_su5655913927576555419on_nat @ NS @ S2 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_611_s__mul__ext__point,axiom,
    ! [As2: multis6850599868085929325on_nat,Bs2: multis6850599868085929325on_nat,NS: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat,B: multis1856558993800614429on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ As2 @ Bs2 ) @ ( multis5680658968000484038on_nat @ NS @ S2 ) )
     => ( ( member2988082745645132084on_nat @ B @ ( set_ms2805207930803453472on_nat @ Bs2 ) )
       => ? [X3: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ As2 ) )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ B ) @ ( sup_su588211329720815695on_nat @ NS @ S2 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_612_s__mul__ext__point,axiom,
    ! [As2: multis5168333519641040071on_nat,Bs2: multis5168333519641040071on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat,B: list_P9127870862658512897on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ As2 @ Bs2 ) @ ( multis3147760155352349208on_nat @ NS @ S2 ) )
     => ( ( member783415501022416426on_nat @ B @ ( set_ms6185292591494683198on_nat @ Bs2 ) )
       => ? [X3: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ As2 ) )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ B ) @ ( sup_su5722365171377032731on_nat @ NS @ S2 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_613_s__mul__ext__point,axiom,
    ! [As2: multis1856558993800614429on_nat,Bs2: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,B: sum_su4907927656612468301on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ As2 @ Bs2 ) @ ( multis1620213760974180726on_nat @ NS @ S2 ) )
     => ( ( member8476201367316484836on_nat @ B @ ( set_ms3922534532180125136on_nat @ Bs2 ) )
       => ? [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ As2 ) )
            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ B ) @ ( sup_su6884158799674119087on_nat @ NS @ S2 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_614_list_Osize_I3_J,axiom,
    ( ( size_s3379185846251495609od_a_o @ nil_li1372968520647675725od_a_o )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_615_locally__refl__def,axiom,
    ( locall1369730039502324340on_nat
    = ( ^ [R3: set_Pr1446391457923625415on_nat,A6: multis2912115315702009697on_nat] :
        ! [A5: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ A5 @ ( set_ms7239842528327418072on_nat @ A6 ) )
         => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A5 @ A5 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_616_locally__refl__def,axiom,
    ( locall5469519386152921476on_nat
    = ( ^ [R3: set_Pr3573155178515677307on_nat,A6: multis6850599868085929325on_nat] :
        ! [A5: multis1856558993800614429on_nat] :
          ( ( member2988082745645132084on_nat @ A5 @ ( set_ms2805207930803453472on_nat @ A6 ) )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A5 @ A5 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_617_locally__refl__def,axiom,
    ( locall5924098968501045722on_nat
    = ( ^ [R3: set_Pr8586234676504894535on_nat,A6: multis5168333519641040071on_nat] :
        ! [A5: list_P9127870862658512897on_nat] :
          ( ( member783415501022416426on_nat @ A5 @ ( set_ms6185292591494683198on_nat @ A6 ) )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A5 @ A5 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_618_all__set__conv__all__nth,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ( ! [X5: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X5 @ ( set_Pr937530902923597456on_nat @ Xs ) )
           => ( P2 @ X5 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s7490768528479217773on_nat @ Xs ) )
           => ( P2 @ ( nth_Pr3336893728888831618on_nat @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_619_all__set__conv__all__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ! [X5: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X5 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P2 @ X5 ) ) )
      = ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ I4 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_620_all__nth__imp__all__set,axiom,
    ! [Xs: list_P1915136757390359201on_nat,P2: produc9176241080893687067on_nat > $o,X4: produc9176241080893687067on_nat] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s8191359103794446093on_nat @ Xs ) )
         => ( P2 @ ( nth_Pr6961551319198328098on_nat @ Xs @ I ) ) )
     => ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
       => ( P2 @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_621_all__nth__imp__all__set,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o,X4: produc1313295348606329723on_nat] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs ) )
         => ( P2 @ ( nth_Pr3336893728888831618on_nat @ Xs @ I ) ) )
     => ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
       => ( P2 @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_622_all__nth__imp__all__set,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o,X4: sum_su4907927656612468301on_nat] :
      ( ! [I: nat] :
          ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
         => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) ) )
     => ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( P2 @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_623_in__set__conv__nth,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s8191359103794446093on_nat @ Xs ) )
            & ( ( nth_Pr6961551319198328098on_nat @ Xs @ I4 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_624_in__set__conv__nth,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s7490768528479217773on_nat @ Xs ) )
            & ( ( nth_Pr3336893728888831618on_nat @ Xs @ I4 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_625_in__set__conv__nth,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( nth_Su8471564416988646406on_nat @ Xs @ I4 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_626_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,P2: produc1313295348606329723on_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ! [X3: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X3 @ ( set_Pr937530902923597456on_nat @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_Pr3336893728888831618on_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_627_list__ball__nth,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_628_nth__mem,axiom,
    ! [N: nat,Xs: list_P1915136757390359201on_nat] :
      ( ( ord_less_nat @ N @ ( size_s8191359103794446093on_nat @ Xs ) )
     => ( member4278285197320822468on_nat @ ( nth_Pr6961551319198328098on_nat @ Xs @ N ) @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_629_nth__mem,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( member5906418179908956708on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ N ) @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_630_nth__mem,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( member8476201367316484836on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_631_ns__mul__extI__old,axiom,
    ! [A3: multis2912115315702009697on_nat,Xs: list_P1915136757390359201on_nat,A22: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,Ys: list_P1915136757390359201on_nat,B22: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( A3
        = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s8191359103794446093on_nat @ Xs )
            = ( size_s8191359103794446093on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s8191359103794446093on_nat @ Ys ) )
               => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( nth_Pr6961551319198328098on_nat @ Xs @ I ) @ ( nth_Pr6961551319198328098on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ! [B4: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ B4 @ ( set_ms7239842528327418072on_nat @ B22 ) )
                 => ? [A7: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ A7 @ ( set_ms7239842528327418072on_nat @ A22 ) )
                      & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis3046611884734373584on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_632_ns__mul__extI__old,axiom,
    ! [A3: multis4835999835052358849on_nat,Xs: list_P9127870862658512897on_nat,A22: multis4835999835052358849on_nat,B3: multis4835999835052358849on_nat,Ys: list_P9127870862658512897on_nat,B22: multis4835999835052358849on_nat,Ns: set_Pr8202701763255893447on_nat,S: set_Pr8202701763255893447on_nat] :
      ( ( A3
        = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s7490768528479217773on_nat @ Xs )
            = ( size_s7490768528479217773on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Ys ) )
               => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I ) @ ( nth_Pr3336893728888831618on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ! [B4: produc1313295348606329723on_nat] :
                  ( ( member5906418179908956708on_nat @ B4 @ ( set_ms8042534003875616824on_nat @ B22 ) )
                 => ? [A7: produc1313295348606329723on_nat] :
                      ( ( member5906418179908956708on_nat @ A7 @ ( set_ms8042534003875616824on_nat @ A22 ) )
                      & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ A3 @ B3 ) @ ( multis762940313840963120on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_633_ns__mul__extI__old,axiom,
    ! [A3: multis6850599868085929325on_nat,Xs: list_m351779702951725613on_nat,A22: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,Ys: list_m351779702951725613on_nat,B22: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( A3
        = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s6603137509925400257on_nat @ Xs )
            = ( size_s6603137509925400257on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Ys ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs @ I ) @ ( nth_mu8151576519813129558on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ! [B4: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ B4 @ ( set_ms2805207930803453472on_nat @ B22 ) )
                 => ? [A7: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ A7 @ ( set_ms2805207930803453472on_nat @ A22 ) )
                      & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis3835232632530192936on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_634_ns__mul__extI__old,axiom,
    ! [A3: multis5168333519641040071on_nat,Xs: list_l1726306699111145479on_nat,A22: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,Ys: list_l1726306699111145479on_nat,B22: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( A3
        = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s129639936549468787on_nat @ Xs )
            = ( size_s129639936549468787on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Ys ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ I ) @ ( nth_li1302981970230039432on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ! [B4: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ B4 @ ( set_ms6185292591494683198on_nat @ B22 ) )
                 => ? [A7: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ A7 @ ( set_ms6185292591494683198on_nat @ A22 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_635_ns__mul__extI__old,axiom,
    ! [A3: multis1856558993800614429on_nat,Xs: list_S5253404628753938141on_nat,A22: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ys: list_S5253404628753938141on_nat,B22: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( A3
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s5686234912572693873on_nat @ Xs )
            = ( size_s5686234912572693873on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ! [B4: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ B4 @ ( set_ms3922534532180125136on_nat @ B22 ) )
                 => ? [A7: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A22 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_636_length__pos__if__in__set,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s8191359103794446093on_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_637_length__pos__if__in__set,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s7490768528479217773on_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_638_length__pos__if__in__set,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5686234912572693873on_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_639_s__mul__extI__old,axiom,
    ! [A3: multis2912115315702009697on_nat,Xs: list_P1915136757390359201on_nat,A22: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,Ys: list_P1915136757390359201on_nat,B22: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( A3
        = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s8191359103794446093on_nat @ Xs )
            = ( size_s8191359103794446093on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s8191359103794446093on_nat @ Ys ) )
               => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( nth_Pr6961551319198328098on_nat @ Xs @ I ) @ ( nth_Pr6961551319198328098on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ( A22 != zero_z1502369238205801624on_nat )
             => ( ! [B4: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ B4 @ ( set_ms7239842528327418072on_nat @ B22 ) )
                   => ? [A7: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ A7 @ ( set_ms7239842528327418072on_nat @ A22 ) )
                        & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A7 @ B4 ) @ S ) ) )
               => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis1332936165424366514on_nat @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_640_s__mul__extI__old,axiom,
    ! [A3: multis4835999835052358849on_nat,Xs: list_P9127870862658512897on_nat,A22: multis4835999835052358849on_nat,B3: multis4835999835052358849on_nat,Ys: list_P9127870862658512897on_nat,B22: multis4835999835052358849on_nat,Ns: set_Pr8202701763255893447on_nat,S: set_Pr8202701763255893447on_nat] :
      ( ( A3
        = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s7490768528479217773on_nat @ Xs )
            = ( size_s7490768528479217773on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Ys ) )
               => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I ) @ ( nth_Pr3336893728888831618on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ( A22 != zero_z2712726346186620152on_nat )
             => ( ! [B4: produc1313295348606329723on_nat] :
                    ( ( member5906418179908956708on_nat @ B4 @ ( set_ms8042534003875616824on_nat @ B22 ) )
                   => ? [A7: produc1313295348606329723on_nat] :
                        ( ( member5906418179908956708on_nat @ A7 @ ( set_ms8042534003875616824on_nat @ A22 ) )
                        & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ A7 @ B4 ) @ S ) ) )
               => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ A3 @ B3 ) @ ( multis9203262125639806226on_nat @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_641_s__mul__extI__old,axiom,
    ! [A3: multis6850599868085929325on_nat,Xs: list_m351779702951725613on_nat,A22: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,Ys: list_m351779702951725613on_nat,B22: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( A3
        = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s6603137509925400257on_nat @ Xs )
            = ( size_s6603137509925400257on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Ys ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs @ I ) @ ( nth_mu8151576519813129558on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ( A22 != zero_z4144889538238539638on_nat )
             => ( ! [B4: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ B4 @ ( set_ms2805207930803453472on_nat @ B22 ) )
                   => ? [A7: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ A7 @ ( set_ms2805207930803453472on_nat @ A22 ) )
                        & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A7 @ B4 ) @ S ) ) )
               => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis5680658968000484038on_nat @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_642_s__mul__extI__old,axiom,
    ! [A3: multis5168333519641040071on_nat,Xs: list_l1726306699111145479on_nat,A22: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,Ys: list_l1726306699111145479on_nat,B22: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( A3
        = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s129639936549468787on_nat @ Xs )
            = ( size_s129639936549468787on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Ys ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ I ) @ ( nth_li1302981970230039432on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ( A22 != zero_z8385993896842825086on_nat )
             => ( ! [B4: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ B4 @ ( set_ms6185292591494683198on_nat @ B22 ) )
                   => ? [A7: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ A7 @ ( set_ms6185292591494683198on_nat @ A22 ) )
                        & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A7 @ B4 ) @ S ) ) )
               => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis3147760155352349208on_nat @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_643_s__mul__extI__old,axiom,
    ! [A3: multis1856558993800614429on_nat,Xs: list_S5253404628753938141on_nat,A22: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ys: list_S5253404628753938141on_nat,B22: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( A3
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) @ A22 ) )
     => ( ( B3
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s5686234912572693873on_nat @ Xs )
            = ( size_s5686234912572693873on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I ) ) @ Ns ) )
           => ( ( A22 != zero_z8702567803361615526on_nat )
             => ( ! [B4: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ B4 @ ( set_ms3922534532180125136on_nat @ B22 ) )
                   => ? [A7: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A22 ) )
                        & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B4 ) @ S ) ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_644_ns__mul__ext__NS__union__S,axiom,
    ! [NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( multis4183089776494585304on_nat @ ( sup_su6884158799674119087on_nat @ NS @ S2 ) @ S2 )
      = ( multis4183089776494585304on_nat @ NS @ S2 ) ) ).

% ns_mul_ext_NS_union_S
thf(fact_645_all__s__s__mul__ext,axiom,
    ! [A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,S: set_Pr1446391457923625415on_nat,Ns: set_Pr1446391457923625415on_nat] :
      ( ( A3 != zero_z1502369238205801624on_nat )
     => ( ! [B4: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ B4 @ ( set_ms7239842528327418072on_nat @ B3 ) )
           => ? [A7: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ A7 @ ( set_ms7239842528327418072on_nat @ A3 ) )
                & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A7 @ B4 ) @ S ) ) )
       => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis1332936165424366514on_nat @ Ns @ S ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_646_all__s__s__mul__ext,axiom,
    ! [A3: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( A3 != zero_z4144889538238539638on_nat )
     => ( ! [B4: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ B4 @ ( set_ms2805207930803453472on_nat @ B3 ) )
           => ? [A7: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ A7 @ ( set_ms2805207930803453472on_nat @ A3 ) )
                & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A7 @ B4 ) @ S ) ) )
       => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis5680658968000484038on_nat @ Ns @ S ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_647_all__s__s__mul__ext,axiom,
    ! [A3: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( A3 != zero_z8385993896842825086on_nat )
     => ( ! [B4: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ B4 @ ( set_ms6185292591494683198on_nat @ B3 ) )
           => ? [A7: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ A7 @ ( set_ms6185292591494683198on_nat @ A3 ) )
                & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A7 @ B4 ) @ S ) ) )
       => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis3147760155352349208on_nat @ Ns @ S ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_648_all__s__s__mul__ext,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( A3 != zero_z8702567803361615526on_nat )
     => ( ! [B4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ B4 @ ( set_ms3922534532180125136on_nat @ B3 ) )
           => ? [A7: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A3 ) )
                & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B4 ) @ S ) ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_649_ns__mul__ext__elim,axiom,
    ! [Xs: multis4835999835052358849on_nat,Ys: multis4835999835052358849on_nat,NS: set_Pr8202701763255893447on_nat,S2: set_Pr8202701763255893447on_nat] :
      ( ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ Xs @ Ys ) @ ( multis762940313840963120on_nat @ NS @ S2 ) )
     => ? [Xs12: list_P9127870862658512897on_nat,Xs23: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs12 ) @ ( mset_P6964650582947932560on_nat @ Xs23 ) ) )
          & ? [Ys12: list_P9127870862658512897on_nat,Ys23: list_P9127870862658512897on_nat] :
              ( ( Ys
                = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys12 ) @ ( mset_P6964650582947932560on_nat @ Ys23 ) ) )
              & ( ( size_s7490768528479217773on_nat @ Xs12 )
                = ( size_s7490768528479217773on_nat @ Ys12 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Ys12 ) )
                 => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs12 @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: produc1313295348606329723on_nat] :
                  ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Ys23 ) )
                 => ? [Xa3: produc1313295348606329723on_nat] :
                      ( ( member5906418179908956708on_nat @ Xa3 @ ( set_Pr937530902923597456on_nat @ Xs23 ) )
                      & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_650_ns__mul__ext__elim,axiom,
    ! [Xs: multis6850599868085929325on_nat,Ys: multis6850599868085929325on_nat,NS: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Xs @ Ys ) @ ( multis3835232632530192936on_nat @ NS @ S2 ) )
     => ? [Xs12: list_m351779702951725613on_nat,Xs23: list_m351779702951725613on_nat] :
          ( ( Xs
            = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Xs12 ) @ ( mset_m1293634003157817032on_nat @ Xs23 ) ) )
          & ? [Ys12: list_m351779702951725613on_nat,Ys23: list_m351779702951725613on_nat] :
              ( ( Ys
                = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Ys12 ) @ ( mset_m1293634003157817032on_nat @ Ys23 ) ) )
              & ( ( size_s6603137509925400257on_nat @ Xs12 )
                = ( size_s6603137509925400257on_nat @ Ys12 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s6603137509925400257on_nat @ Ys12 ) )
                 => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs12 @ I2 ) @ ( nth_mu8151576519813129558on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ X6 @ ( set_mu1790088485793245640on_nat @ Ys23 ) )
                 => ? [Xa3: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Xa3 @ ( set_mu1790088485793245640on_nat @ Xs23 ) )
                      & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_651_ns__mul__ext__elim,axiom,
    ! [Xs: multis5168333519641040071on_nat,Ys: multis5168333519641040071on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Xs @ Ys ) @ ( multis664119935410451510on_nat @ NS @ S2 ) )
     => ? [Xs12: list_l1726306699111145479on_nat,Xs23: list_l1726306699111145479on_nat] :
          ( ( Xs
            = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs12 ) @ ( mset_l2262431794812116886on_nat @ Xs23 ) ) )
          & ? [Ys12: list_l1726306699111145479on_nat,Ys23: list_l1726306699111145479on_nat] :
              ( ( Ys
                = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys12 ) @ ( mset_l2262431794812116886on_nat @ Ys23 ) ) )
              & ( ( size_s129639936549468787on_nat @ Xs12 )
                = ( size_s129639936549468787on_nat @ Ys12 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Ys12 ) )
                 => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs12 @ I2 ) @ ( nth_li1302981970230039432on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ X6 @ ( set_li8154783781463198870on_nat @ Ys23 ) )
                 => ? [Xa3: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Xa3 @ ( set_li8154783781463198870on_nat @ Xs23 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_652_ns__mul__ext__elim,axiom,
    ! [Xs: multis1856558993800614429on_nat,Ys: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis4183089776494585304on_nat @ NS @ S2 ) )
     => ? [Xs12: list_S5253404628753938141on_nat,Xs23: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs12 ) @ ( mset_S7907735292211298936on_nat @ Xs23 ) ) )
          & ? [Ys12: list_S5253404628753938141on_nat,Ys23: list_S5253404628753938141on_nat] :
              ( ( Ys
                = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys12 ) @ ( mset_S7907735292211298936on_nat @ Ys23 ) ) )
              & ( ( size_s5686234912572693873on_nat @ Xs12 )
                = ( size_s5686234912572693873on_nat @ Ys12 ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Ys12 ) )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs12 @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Ys12 @ I2 ) ) @ NS ) )
              & ! [X6: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Ys23 ) )
                 => ? [Xa3: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Xa3 @ ( set_Su4848624856859743608on_nat @ Xs23 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa3 @ X6 ) @ S2 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_653_ns__mul__ext__intro,axiom,
    ! [Xs: multis2912115315702009697on_nat,Xs1: list_P1915136757390359201on_nat,Xs22: list_P1915136757390359201on_nat,Ys: multis2912115315702009697on_nat,Ys1: list_P1915136757390359201on_nat,Ys22: list_P1915136757390359201on_nat,NS: set_Pr1446391457923625415on_nat,S2: set_Pr1446391457923625415on_nat] :
      ( ( Xs
        = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Xs1 ) @ ( mset_P1678654427565054512on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p7340530524814416408on_nat @ ( mset_P1678654427565054512on_nat @ Ys1 ) @ ( mset_P1678654427565054512on_nat @ Ys22 ) ) )
       => ( ( ( size_s8191359103794446093on_nat @ Xs1 )
            = ( size_s8191359103794446093on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s8191359103794446093on_nat @ Ys1 ) )
               => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( nth_Pr6961551319198328098on_nat @ Xs1 @ I ) @ ( nth_Pr6961551319198328098on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ! [Y3: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Y3 @ ( set_Pr8272476097931642672on_nat @ Ys22 ) )
                 => ? [X6: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ X6 @ ( set_Pr8272476097931642672on_nat @ Xs22 ) )
                      & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X6 @ Y3 ) @ S2 ) ) )
             => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ Xs @ Ys ) @ ( multis3046611884734373584on_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_654_ns__mul__ext__intro,axiom,
    ! [Xs: multis4835999835052358849on_nat,Xs1: list_P9127870862658512897on_nat,Xs22: list_P9127870862658512897on_nat,Ys: multis4835999835052358849on_nat,Ys1: list_P9127870862658512897on_nat,Ys22: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S2: set_Pr8202701763255893447on_nat] :
      ( ( Xs
        = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs1 ) @ ( mset_P6964650582947932560on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys1 ) @ ( mset_P6964650582947932560on_nat @ Ys22 ) ) )
       => ( ( ( size_s7490768528479217773on_nat @ Xs1 )
            = ( size_s7490768528479217773on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Ys1 ) )
               => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs1 @ I ) @ ( nth_Pr3336893728888831618on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ! [Y3: produc1313295348606329723on_nat] :
                  ( ( member5906418179908956708on_nat @ Y3 @ ( set_Pr937530902923597456on_nat @ Ys22 ) )
                 => ? [X6: produc1313295348606329723on_nat] :
                      ( ( member5906418179908956708on_nat @ X6 @ ( set_Pr937530902923597456on_nat @ Xs22 ) )
                      & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X6 @ Y3 ) @ S2 ) ) )
             => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ Xs @ Ys ) @ ( multis762940313840963120on_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_655_ns__mul__ext__intro,axiom,
    ! [Xs: multis6850599868085929325on_nat,Xs1: list_m351779702951725613on_nat,Xs22: list_m351779702951725613on_nat,Ys: multis6850599868085929325on_nat,Ys1: list_m351779702951725613on_nat,Ys22: list_m351779702951725613on_nat,NS: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat] :
      ( ( Xs
        = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Xs1 ) @ ( mset_m1293634003157817032on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p4342429221931815414on_nat @ ( mset_m1293634003157817032on_nat @ Ys1 ) @ ( mset_m1293634003157817032on_nat @ Ys22 ) ) )
       => ( ( ( size_s6603137509925400257on_nat @ Xs1 )
            = ( size_s6603137509925400257on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Ys1 ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs1 @ I ) @ ( nth_mu8151576519813129558on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ! [Y3: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Y3 @ ( set_mu1790088485793245640on_nat @ Ys22 ) )
                 => ? [X6: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ X6 @ ( set_mu1790088485793245640on_nat @ Xs22 ) )
                      & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X6 @ Y3 ) @ S2 ) ) )
             => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Xs @ Ys ) @ ( multis3835232632530192936on_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_656_ns__mul__ext__intro,axiom,
    ! [Xs: multis5168333519641040071on_nat,Xs1: list_l1726306699111145479on_nat,Xs22: list_l1726306699111145479on_nat,Ys: multis5168333519641040071on_nat,Ys1: list_l1726306699111145479on_nat,Ys22: list_l1726306699111145479on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat] :
      ( ( Xs
        = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs1 ) @ ( mset_l2262431794812116886on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys1 ) @ ( mset_l2262431794812116886on_nat @ Ys22 ) ) )
       => ( ( ( size_s129639936549468787on_nat @ Xs1 )
            = ( size_s129639936549468787on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Ys1 ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs1 @ I ) @ ( nth_li1302981970230039432on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ! [Y3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Y3 @ ( set_li8154783781463198870on_nat @ Ys22 ) )
                 => ? [X6: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ X6 @ ( set_li8154783781463198870on_nat @ Xs22 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X6 @ Y3 ) @ S2 ) ) )
             => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Xs @ Ys ) @ ( multis664119935410451510on_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_657_ns__mul__ext__intro,axiom,
    ! [Xs: multis1856558993800614429on_nat,Xs1: list_S5253404628753938141on_nat,Xs22: list_S5253404628753938141on_nat,Ys: multis1856558993800614429on_nat,Ys1: list_S5253404628753938141on_nat,Ys22: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( Xs
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs1 ) @ ( mset_S7907735292211298936on_nat @ Xs22 ) ) )
     => ( ( Ys
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys1 ) @ ( mset_S7907735292211298936on_nat @ Ys22 ) ) )
       => ( ( ( size_s5686234912572693873on_nat @ Xs1 )
            = ( size_s5686234912572693873on_nat @ Ys1 ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys1 ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs1 @ I ) @ ( nth_Su8471564416988646406on_nat @ Ys1 @ I ) ) @ NS ) )
           => ( ! [Y3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
                 => ? [X6: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ X6 @ ( set_Su4848624856859743608on_nat @ Xs22 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X6 @ Y3 ) @ S2 ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis4183089776494585304on_nat @ NS @ S2 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_658_all__ns__ns__mul__ext,axiom,
    ! [As2: list_P9127870862658512897on_nat,Bs2: list_P9127870862658512897on_nat,Ns: set_Pr8202701763255893447on_nat,S: set_Pr8202701763255893447on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ As2 )
        = ( size_s7490768528479217773on_nat @ Bs2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Bs2 ) )
           => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ As2 @ I ) @ ( nth_Pr3336893728888831618on_nat @ Bs2 @ I ) ) @ Ns ) )
       => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ ( mset_P6964650582947932560on_nat @ As2 ) @ ( mset_P6964650582947932560on_nat @ Bs2 ) ) @ ( multis762940313840963120on_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_659_all__ns__ns__mul__ext,axiom,
    ! [As2: list_m351779702951725613on_nat,Bs2: list_m351779702951725613on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( ( size_s6603137509925400257on_nat @ As2 )
        = ( size_s6603137509925400257on_nat @ Bs2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Bs2 ) )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ As2 @ I ) @ ( nth_mu8151576519813129558on_nat @ Bs2 @ I ) ) @ Ns ) )
       => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( mset_m1293634003157817032on_nat @ As2 ) @ ( mset_m1293634003157817032on_nat @ Bs2 ) ) @ ( multis3835232632530192936on_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_660_all__ns__ns__mul__ext,axiom,
    ! [As2: list_l1726306699111145479on_nat,Bs2: list_l1726306699111145479on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( ( size_s129639936549468787on_nat @ As2 )
        = ( size_s129639936549468787on_nat @ Bs2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Bs2 ) )
           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ As2 @ I ) @ ( nth_li1302981970230039432on_nat @ Bs2 @ I ) ) @ Ns ) )
       => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( mset_l2262431794812116886on_nat @ As2 ) @ ( mset_l2262431794812116886on_nat @ Bs2 ) ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_661_all__ns__ns__mul__ext,axiom,
    ! [As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ As2 )
        = ( size_s5686234912572693873on_nat @ Bs2 ) )
     => ( ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Bs2 ) )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As2 @ I ) @ ( nth_Su8471564416988646406on_nat @ Bs2 @ I ) ) @ Ns ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As2 ) @ ( mset_S7907735292211298936on_nat @ Bs2 ) ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_662_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list4838562191945764047od_a_o @ N @ nil_Product_prod_a_o )
          = ( cons_l9179691953475787517od_a_o @ nil_Product_prod_a_o @ nil_li1372968520647675725od_a_o ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list4838562191945764047od_a_o @ N @ nil_Product_prod_a_o )
          = nil_li1372968520647675725od_a_o ) ) ) ).

% n_lists_Nil
thf(fact_663_n__lists__Nil,axiom,
    ! [N: nat] :
      ( ( ( N = zero_zero_nat )
       => ( ( n_list931218472929919829od_a_o @ N @ nil_li1372968520647675725od_a_o )
          = ( cons_l8465095039735648515od_a_o @ nil_li1372968520647675725od_a_o @ nil_li6902169125192129875od_a_o ) ) )
      & ( ( N != zero_zero_nat )
       => ( ( n_list931218472929919829od_a_o @ N @ nil_li1372968520647675725od_a_o )
          = nil_li6902169125192129875od_a_o ) ) ) ).

% n_lists_Nil
thf(fact_664_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_P3196869577824546119od_a_o] :
      ( ( n_list4838562191945764047od_a_o @ zero_zero_nat @ Xs )
      = ( cons_l9179691953475787517od_a_o @ nil_Product_prod_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% n_lists.simps(1)
thf(fact_665_n__lists_Osimps_I1_J,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( n_list931218472929919829od_a_o @ zero_zero_nat @ Xs )
      = ( cons_l8465095039735648515od_a_o @ nil_li1372968520647675725od_a_o @ nil_li6902169125192129875od_a_o ) ) ).

% n_lists.simps(1)
thf(fact_666_listrel1__iff__update,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( listre7145697762236742142on_nat @ R ) )
      = ( ? [Y4: multis1856558993800614429on_nat,N2: nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs @ N2 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N2 @ ( size_s6603137509925400257on_nat @ Xs ) )
            & ( Ys
              = ( list_u4451328684322502447on_nat @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_667_listrel1__iff__update,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( listre2468548762442407136on_nat @ R ) )
      = ( ? [Y4: list_P9127870862658512897on_nat,N2: nat] :
            ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ N2 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N2 @ ( size_s129639936549468787on_nat @ Xs ) )
            & ( Ys
              = ( list_u184235112954345583on_nat @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_668_listrel1__iff__update,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
      = ( ? [Y4: produc1313295348606329723on_nat,N2: nat] :
            ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ N2 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N2 @ ( size_s7490768528479217773on_nat @ Xs ) )
            & ( Ys
              = ( list_u1840724836389618409on_nat @ Xs @ N2 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_669_s__mul__ext__IdI,axiom,
    ! [X: multis6850599868085929325on_nat,M2: multis6850599868085929325on_nat,Z4: multis6850599868085929325on_nat,N3: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,R2: set_Pr3573155178515677307on_nat] :
      ( ( X != zero_z4144889538238539638on_nat )
     => ( ( M2
          = ( plus_p4342429221931815414on_nat @ X @ Z4 ) )
       => ( ( N3
            = ( plus_p4342429221931815414on_nat @ Y5 @ Z4 ) )
         => ( ! [X3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
               => ? [Xa: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Xa @ ( set_ms2805207930803453472on_nat @ X ) )
                    & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xa @ X3 ) @ R2 ) ) )
           => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ M2 @ N3 ) @ ( multis5680658968000484038on_nat @ id_mul1802627190162231233on_nat @ R2 ) ) ) ) ) ) ).

% s_mul_ext_IdI
thf(fact_670_s__mul__ext__IdI,axiom,
    ! [X: multis5168333519641040071on_nat,M2: multis5168333519641040071on_nat,Z4: multis5168333519641040071on_nat,N3: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,R2: set_Pr8586234676504894535on_nat] :
      ( ( X != zero_z8385993896842825086on_nat )
     => ( ( M2
          = ( plus_p7957816880051667710on_nat @ X @ Z4 ) )
       => ( ( N3
            = ( plus_p7957816880051667710on_nat @ Y5 @ Z4 ) )
         => ( ! [X3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
               => ? [Xa: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Xa @ ( set_ms6185292591494683198on_nat @ X ) )
                    & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xa @ X3 ) @ R2 ) ) )
           => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ M2 @ N3 ) @ ( multis3147760155352349208on_nat @ id_lis4774886207970145629on_nat @ R2 ) ) ) ) ) ) ).

% s_mul_ext_IdI
thf(fact_671_s__mul__ext__IdI,axiom,
    ! [X: multis1856558993800614429on_nat,M2: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,N3: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,R2: set_Pr2535564044409765851on_nat] :
      ( ( X != zero_z8702567803361615526on_nat )
     => ( ( M2
          = ( plus_p2586647234311342374on_nat @ X @ Z4 ) )
       => ( ( N3
            = ( plus_p2586647234311342374on_nat @ Y5 @ Z4 ) )
         => ( ! [X3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
               => ? [Xa: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Xa @ ( set_ms3922534532180125136on_nat @ X ) )
                    & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa @ X3 ) @ R2 ) ) )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M2 @ N3 ) @ ( multis1620213760974180726on_nat @ id_Sum1147813627050573297on_nat @ R2 ) ) ) ) ) ) ).

% s_mul_ext_IdI
thf(fact_672_s__mul__ext__IdE,axiom,
    ! [M2: multis6850599868085929325on_nat,N3: multis6850599868085929325on_nat,R2: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ M2 @ N3 ) @ ( multis5680658968000484038on_nat @ id_mul1802627190162231233on_nat @ R2 ) )
     => ~ ! [X7: multis6850599868085929325on_nat] :
            ( ( X7 != zero_z4144889538238539638on_nat )
           => ! [Y6: multis6850599868085929325on_nat,Z5: multis6850599868085929325on_nat] :
                ( ( M2
                  = ( plus_p4342429221931815414on_nat @ X7 @ Z5 ) )
               => ( ( N3
                    = ( plus_p4342429221931815414on_nat @ Y6 @ Z5 ) )
                 => ~ ! [X6: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ X6 @ ( set_ms2805207930803453472on_nat @ Y6 ) )
                       => ? [Xa3: multis1856558993800614429on_nat] :
                            ( ( member2988082745645132084on_nat @ Xa3 @ ( set_ms2805207930803453472on_nat @ X7 ) )
                            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xa3 @ X6 ) @ R2 ) ) ) ) ) ) ) ).

% s_mul_ext_IdE
thf(fact_673_s__mul__ext__IdE,axiom,
    ! [M2: multis5168333519641040071on_nat,N3: multis5168333519641040071on_nat,R2: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ M2 @ N3 ) @ ( multis3147760155352349208on_nat @ id_lis4774886207970145629on_nat @ R2 ) )
     => ~ ! [X7: multis5168333519641040071on_nat] :
            ( ( X7 != zero_z8385993896842825086on_nat )
           => ! [Y6: multis5168333519641040071on_nat,Z5: multis5168333519641040071on_nat] :
                ( ( M2
                  = ( plus_p7957816880051667710on_nat @ X7 @ Z5 ) )
               => ( ( N3
                    = ( plus_p7957816880051667710on_nat @ Y6 @ Z5 ) )
                 => ~ ! [X6: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ X6 @ ( set_ms6185292591494683198on_nat @ Y6 ) )
                       => ? [Xa3: list_P9127870862658512897on_nat] :
                            ( ( member783415501022416426on_nat @ Xa3 @ ( set_ms6185292591494683198on_nat @ X7 ) )
                            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xa3 @ X6 ) @ R2 ) ) ) ) ) ) ) ).

% s_mul_ext_IdE
thf(fact_674_s__mul__ext__IdE,axiom,
    ! [M2: multis1856558993800614429on_nat,N3: multis1856558993800614429on_nat,R2: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M2 @ N3 ) @ ( multis1620213760974180726on_nat @ id_Sum1147813627050573297on_nat @ R2 ) )
     => ~ ! [X7: multis1856558993800614429on_nat] :
            ( ( X7 != zero_z8702567803361615526on_nat )
           => ! [Y6: multis1856558993800614429on_nat,Z5: multis1856558993800614429on_nat] :
                ( ( M2
                  = ( plus_p2586647234311342374on_nat @ X7 @ Z5 ) )
               => ( ( N3
                    = ( plus_p2586647234311342374on_nat @ Y6 @ Z5 ) )
                 => ~ ! [X6: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ X6 @ ( set_ms3922534532180125136on_nat @ Y6 ) )
                       => ? [Xa3: sum_su4907927656612468301on_nat] :
                            ( ( member8476201367316484836on_nat @ Xa3 @ ( set_ms3922534532180125136on_nat @ X7 ) )
                            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa3 @ X6 ) @ R2 ) ) ) ) ) ) ) ).

% s_mul_ext_IdE
thf(fact_675_list__update__nonempty,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,K3: nat,X4: list_P3196869577824546119od_a_o] :
      ( ( ( list_u329291161401623989od_a_o @ Xs @ K3 @ X4 )
        = nil_li1372968520647675725od_a_o )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% list_update_nonempty
thf(fact_676_list__update__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat] :
      ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ Ys ) ) @ ( size_s7490768528479217773on_nat @ Xs ) @ Y )
      = ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_677_set__swap,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( set_Pr937530902923597456on_nat @ ( list_u1840724836389618409on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ ( nth_Pr3336893728888831618on_nat @ Xs @ J ) ) @ J @ ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) ) )
          = ( set_Pr937530902923597456on_nat @ Xs ) ) ) ) ).

% set_swap
thf(fact_678_set__swap,axiom,
    ! [I3: nat,Xs: list_S5253404628753938141on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) @ J @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) ) )
          = ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% set_swap
thf(fact_679_ns__mul__ext__Id__eq,axiom,
    ! [R2: set_Pr2535564044409765851on_nat] :
      ( ( multis4183089776494585304on_nat @ id_Sum1147813627050573297on_nat @ R2 )
      = ( sup_su588211329720815695on_nat @ ( multis1620213760974180726on_nat @ id_Sum1147813627050573297on_nat @ R2 ) @ id_mul1802627190162231233on_nat ) ) ).

% ns_mul_ext_Id_eq
thf(fact_680_list__update__code_I1_J,axiom,
    ! [I3: nat,Y: list_P3196869577824546119od_a_o] :
      ( ( list_u329291161401623989od_a_o @ nil_li1372968520647675725od_a_o @ I3 @ Y )
      = nil_li1372968520647675725od_a_o ) ).

% list_update_code(1)
thf(fact_681_list__update_Osimps_I1_J,axiom,
    ! [I3: nat,V: list_P3196869577824546119od_a_o] :
      ( ( list_u329291161401623989od_a_o @ nil_li1372968520647675725od_a_o @ I3 @ V )
      = nil_li1372968520647675725od_a_o ) ).

% list_update.simps(1)
thf(fact_682_set__update__memI,axiom,
    ! [N: nat,Xs: list_P1915136757390359201on_nat,X4: produc9176241080893687067on_nat] :
      ( ( ord_less_nat @ N @ ( size_s8191359103794446093on_nat @ Xs ) )
     => ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( list_u8615448375230402185on_nat @ Xs @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_683_set__update__memI,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( list_u1840724836389618409on_nat @ Xs @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_684_set__update__memI,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,X4: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ Xs @ N @ X4 ) ) ) ) ).

% set_update_memI
thf(fact_685_list__update__append1,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ I3 @ X4 )
        = ( append2865821678206098582on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X4 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_686_mset__swap,axiom,
    ! [I3: nat,Ls: list_S5253404628753938141on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Ls ) )
     => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Ls ) )
       => ( ( mset_S7907735292211298936on_nat @ ( list_u8577817230714425439on_nat @ ( list_u8577817230714425439on_nat @ Ls @ J @ ( nth_Su8471564416988646406on_nat @ Ls @ I3 ) ) @ I3 @ ( nth_Su8471564416988646406on_nat @ Ls @ J ) ) )
          = ( mset_S7907735292211298936on_nat @ Ls ) ) ) ) ).

% mset_swap
thf(fact_687_mset__swap,axiom,
    ! [I3: nat,Ls: list_P9127870862658512897on_nat,J: nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Ls ) )
     => ( ( ord_less_nat @ J @ ( size_s7490768528479217773on_nat @ Ls ) )
       => ( ( mset_P6964650582947932560on_nat @ ( list_u1840724836389618409on_nat @ ( list_u1840724836389618409on_nat @ Ls @ J @ ( nth_Pr3336893728888831618on_nat @ Ls @ I3 ) ) @ I3 @ ( nth_Pr3336893728888831618on_nat @ Ls @ J ) ) )
          = ( mset_P6964650582947932560on_nat @ Ls ) ) ) ) ).

% mset_swap
thf(fact_688_ns__mul__ext__IdI,axiom,
    ! [M2: multis6850599868085929325on_nat,X: multis6850599868085929325on_nat,Z4: multis6850599868085929325on_nat,N3: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,R2: set_Pr3573155178515677307on_nat] :
      ( ( M2
        = ( plus_p4342429221931815414on_nat @ X @ Z4 ) )
     => ( ( N3
          = ( plus_p4342429221931815414on_nat @ Y5 @ Z4 ) )
       => ( ! [X3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
             => ? [Xa: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Xa @ ( set_ms2805207930803453472on_nat @ X ) )
                  & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xa @ X3 ) @ R2 ) ) )
         => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ M2 @ N3 ) @ ( multis3835232632530192936on_nat @ id_mul1802627190162231233on_nat @ R2 ) ) ) ) ) ).

% ns_mul_ext_IdI
thf(fact_689_ns__mul__ext__IdI,axiom,
    ! [M2: multis5168333519641040071on_nat,X: multis5168333519641040071on_nat,Z4: multis5168333519641040071on_nat,N3: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,R2: set_Pr8586234676504894535on_nat] :
      ( ( M2
        = ( plus_p7957816880051667710on_nat @ X @ Z4 ) )
     => ( ( N3
          = ( plus_p7957816880051667710on_nat @ Y5 @ Z4 ) )
       => ( ! [X3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
             => ? [Xa: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Xa @ ( set_ms6185292591494683198on_nat @ X ) )
                  & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xa @ X3 ) @ R2 ) ) )
         => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ M2 @ N3 ) @ ( multis664119935410451510on_nat @ id_lis4774886207970145629on_nat @ R2 ) ) ) ) ) ).

% ns_mul_ext_IdI
thf(fact_690_ns__mul__ext__IdI,axiom,
    ! [M2: multis1856558993800614429on_nat,X: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,N3: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,R2: set_Pr2535564044409765851on_nat] :
      ( ( M2
        = ( plus_p2586647234311342374on_nat @ X @ Z4 ) )
     => ( ( N3
          = ( plus_p2586647234311342374on_nat @ Y5 @ Z4 ) )
       => ( ! [X3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
             => ? [Xa: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Xa @ ( set_ms3922534532180125136on_nat @ X ) )
                  & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa @ X3 ) @ R2 ) ) )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M2 @ N3 ) @ ( multis4183089776494585304on_nat @ id_Sum1147813627050573297on_nat @ R2 ) ) ) ) ) ).

% ns_mul_ext_IdI
thf(fact_691_ns__mul__ext__IdE,axiom,
    ! [M2: multis6850599868085929325on_nat,N3: multis6850599868085929325on_nat,R2: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ M2 @ N3 ) @ ( multis3835232632530192936on_nat @ id_mul1802627190162231233on_nat @ R2 ) )
     => ~ ! [X7: multis6850599868085929325on_nat,Y6: multis6850599868085929325on_nat,Z5: multis6850599868085929325on_nat] :
            ( ( M2
              = ( plus_p4342429221931815414on_nat @ X7 @ Z5 ) )
           => ( ( N3
                = ( plus_p4342429221931815414on_nat @ Y6 @ Z5 ) )
             => ~ ! [X6: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ X6 @ ( set_ms2805207930803453472on_nat @ Y6 ) )
                   => ? [Xa3: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ Xa3 @ ( set_ms2805207930803453472on_nat @ X7 ) )
                        & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xa3 @ X6 ) @ R2 ) ) ) ) ) ) ).

% ns_mul_ext_IdE
thf(fact_692_ns__mul__ext__IdE,axiom,
    ! [M2: multis5168333519641040071on_nat,N3: multis5168333519641040071on_nat,R2: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ M2 @ N3 ) @ ( multis664119935410451510on_nat @ id_lis4774886207970145629on_nat @ R2 ) )
     => ~ ! [X7: multis5168333519641040071on_nat,Y6: multis5168333519641040071on_nat,Z5: multis5168333519641040071on_nat] :
            ( ( M2
              = ( plus_p7957816880051667710on_nat @ X7 @ Z5 ) )
           => ( ( N3
                = ( plus_p7957816880051667710on_nat @ Y6 @ Z5 ) )
             => ~ ! [X6: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ X6 @ ( set_ms6185292591494683198on_nat @ Y6 ) )
                   => ? [Xa3: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ Xa3 @ ( set_ms6185292591494683198on_nat @ X7 ) )
                        & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xa3 @ X6 ) @ R2 ) ) ) ) ) ) ).

% ns_mul_ext_IdE
thf(fact_693_ns__mul__ext__IdE,axiom,
    ! [M2: multis1856558993800614429on_nat,N3: multis1856558993800614429on_nat,R2: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M2 @ N3 ) @ ( multis4183089776494585304on_nat @ id_Sum1147813627050573297on_nat @ R2 ) )
     => ~ ! [X7: multis1856558993800614429on_nat,Y6: multis1856558993800614429on_nat,Z5: multis1856558993800614429on_nat] :
            ( ( M2
              = ( plus_p2586647234311342374on_nat @ X7 @ Z5 ) )
           => ( ( N3
                = ( plus_p2586647234311342374on_nat @ Y6 @ Z5 ) )
             => ~ ! [X6: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ X6 @ ( set_ms3922534532180125136on_nat @ Y6 ) )
                   => ? [Xa3: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ Xa3 @ ( set_ms3922534532180125136on_nat @ X7 ) )
                        & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa3 @ X6 ) @ R2 ) ) ) ) ) ) ).

% ns_mul_ext_IdE
thf(fact_694_IdI,axiom,
    ! [A: multis1856558993800614429on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ A ) @ id_mul1802627190162231233on_nat ) ).

% IdI
thf(fact_695_IdI,axiom,
    ! [A: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ A ) @ id_lis4774886207970145629on_nat ) ).

% IdI
thf(fact_696_pair__in__Id__conv,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ id_mul1802627190162231233on_nat )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_697_pair__in__Id__conv,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ id_lis4774886207970145629on_nat )
      = ( A = B ) ) ).

% pair_in_Id_conv
thf(fact_698_IdE,axiom,
    ! [P: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ P @ id_mul1802627190162231233on_nat )
     => ~ ! [X3: multis1856558993800614429on_nat] :
            ( P
           != ( produc5200047356169809875on_nat @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_699_IdE,axiom,
    ! [P: produc1714117091004681831on_nat] :
      ( ( member4629988367768146576on_nat @ P @ id_lis4774886207970145629on_nat )
     => ~ ! [X3: list_P9127870862658512897on_nat] :
            ( P
           != ( produc7152764266822961111on_nat @ X3 @ X3 ) ) ) ).

% IdE
thf(fact_700_IdD,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ id_mul1802627190162231233on_nat )
     => ( A = B ) ) ).

% IdD
thf(fact_701_IdD,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ id_lis4774886207970145629on_nat )
     => ( A = B ) ) ).

% IdD
thf(fact_702_in__mset__fold__plus__iff,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat,NN: multis1044950388305188647on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ ( fold_m9173186124463788235on_nat @ plus_p7340530524814416408on_nat @ M2 @ NN ) ) )
      = ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
        | ? [N4: multis2912115315702009697on_nat] :
            ( ( member3844701217656106634on_nat @ N4 @ ( set_ms722812799953292958on_nat @ NN ) )
            & ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ N4 ) ) ) ) ) ).

% in_mset_fold_plus_iff
thf(fact_703_enumerate__simps_I1_J,axiom,
    ! [N: nat] :
      ( ( enumer7178362783773462060od_a_o @ N @ nil_li1372968520647675725od_a_o )
      = nil_Pr1392216289592255974od_a_o ) ).

% enumerate_simps(1)
thf(fact_704_enumerate__append__eq,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( enumer158069492004818656on_nat @ N @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append7842736379566049733on_nat @ ( enumer158069492004818656on_nat @ N @ Xs ) @ ( enumer158069492004818656on_nat @ ( plus_plus_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) ) @ Ys ) ) ) ).

% enumerate_append_eq
thf(fact_705_SuccI,axiom,
    ! [Kl: list_P1915136757390359201on_nat,K3: produc9176241080893687067on_nat,Kl2: set_li8167635853205255297on_nat] :
      ( ( member6200664750067624906on_nat @ ( append1010812867689791286on_nat @ Kl @ ( cons_P6897958070471385041on_nat @ K3 @ nil_Pr2652050626817158177on_nat ) ) @ Kl2 )
     => ( member4278285197320822468on_nat @ K3 @ ( bNF_Gr7372635924212258486on_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_706_SuccI,axiom,
    ! [Kl: list_P9127870862658512897on_nat,K3: produc1313295348606329723on_nat,Kl2: set_li1748721617258726113on_nat] :
      ( ( member783415501022416426on_nat @ ( append2865821678206098582on_nat @ Kl @ ( cons_P1582596238967110705on_nat @ K3 @ nil_Pr8614093721167239297on_nat ) ) @ Kl2 )
     => ( member5906418179908956708on_nat @ K3 @ ( bNF_Gr8764099174735115286on_nat @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_707_SuccI,axiom,
    ! [Kl: list_l8336264072933094989od_a_o,K3: list_P3196869577824546119od_a_o,Kl2: set_li7049945101328808109od_a_o] :
      ( ( member8885487059802017782od_a_o @ ( append2159120467191820514od_a_o @ Kl @ ( cons_l9179691953475787517od_a_o @ K3 @ nil_li1372968520647675725od_a_o ) ) @ Kl2 )
     => ( member5261019209154752368od_a_o @ K3 @ ( bNF_Gr142345107972655202od_a_o @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_708_SuccD,axiom,
    ! [K3: produc9176241080893687067on_nat,Kl2: set_li8167635853205255297on_nat,Kl: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ K3 @ ( bNF_Gr7372635924212258486on_nat @ Kl2 @ Kl ) )
     => ( member6200664750067624906on_nat @ ( append1010812867689791286on_nat @ Kl @ ( cons_P6897958070471385041on_nat @ K3 @ nil_Pr2652050626817158177on_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_709_SuccD,axiom,
    ! [K3: produc1313295348606329723on_nat,Kl2: set_li1748721617258726113on_nat,Kl: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ K3 @ ( bNF_Gr8764099174735115286on_nat @ Kl2 @ Kl ) )
     => ( member783415501022416426on_nat @ ( append2865821678206098582on_nat @ Kl @ ( cons_P1582596238967110705on_nat @ K3 @ nil_Pr8614093721167239297on_nat ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_710_SuccD,axiom,
    ! [K3: list_P3196869577824546119od_a_o,Kl2: set_li7049945101328808109od_a_o,Kl: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ K3 @ ( bNF_Gr142345107972655202od_a_o @ Kl2 @ Kl ) )
     => ( member8885487059802017782od_a_o @ ( append2159120467191820514od_a_o @ Kl @ ( cons_l9179691953475787517od_a_o @ K3 @ nil_li1372968520647675725od_a_o ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_711_nth__zip,axiom,
    ! [I3: nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s6603137509925400257on_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s6603137509925400257on_nat @ Ys ) )
       => ( ( nth_Pr6961551319198328098on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) @ I3 )
          = ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs @ I3 ) @ ( nth_mu8151576519813129558on_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_712_nth__zip,axiom,
    ! [I3: nat,Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s650155532264187265on_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s2647645505745046499on_nat @ Ys ) )
       => ( ( nth_Pr2044732159338816772on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) @ I3 )
          = ( produc1545487600272464053on_nat @ ( nth_li4091121329102240790on_nat @ Xs @ I3 ) @ ( nth_Pr1046342761279222648on_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_713_nth__zip,axiom,
    ! [I3: nat,Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s650155532264187265on_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s4814362452863045593on_nat @ Ys ) )
       => ( ( nth_Pr1046342761279222648on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) @ I3 )
          = ( produc4176415733022414961on_nat @ ( nth_li4091121329102240790on_nat @ Xs @ I3 ) @ ( nth_Pr2007431530836016110on_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_714_nth__zip,axiom,
    ! [I3: nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s129639936549468787on_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_s129639936549468787on_nat @ Ys ) )
       => ( ( nth_Pr2007431530836016110on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) @ I3 )
          = ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ I3 ) @ ( nth_li1302981970230039432on_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_715_nth__zip,axiom,
    ! [I3: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s8208510060688613859at_nat @ Xs ) )
     => ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr4952906083868132727at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I3 )
          = ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) ) ) ) ).

% nth_zip
thf(fact_716_s__mul__extI,axiom,
    ! [A3: multis2912115315702009697on_nat,A1: multis2912115315702009697on_nat,A22: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,B1: multis2912115315702009697on_nat,B22: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( A3
        = ( plus_p7340530524814416408on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p7340530524814416408on_nat @ B1 @ B22 ) )
       => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A1 @ B1 ) @ ( multis5394615859312135327on_nat @ Ns ) )
         => ( ( A22 != zero_z1502369238205801624on_nat )
           => ( ! [B4: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ B4 @ ( set_ms7239842528327418072on_nat @ B22 ) )
                 => ? [A7: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ A7 @ ( set_ms7239842528327418072on_nat @ A22 ) )
                      & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis1332936165424366514on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_717_s__mul__extI,axiom,
    ! [A3: multis6850599868085929325on_nat,A1: multis6850599868085929325on_nat,A22: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,B1: multis6850599868085929325on_nat,B22: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( A3
        = ( plus_p4342429221931815414on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p4342429221931815414on_nat @ B1 @ B22 ) )
       => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A1 @ B1 ) @ ( multis1913027631269000601on_nat @ Ns ) )
         => ( ( A22 != zero_z4144889538238539638on_nat )
           => ( ! [B4: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ B4 @ ( set_ms2805207930803453472on_nat @ B22 ) )
                 => ? [A7: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ A7 @ ( set_ms2805207930803453472on_nat @ A22 ) )
                      & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis5680658968000484038on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_718_s__mul__extI,axiom,
    ! [A3: multis5168333519641040071on_nat,A1: multis5168333519641040071on_nat,A22: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,B1: multis5168333519641040071on_nat,B22: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( A3
        = ( plus_p7957816880051667710on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p7957816880051667710on_nat @ B1 @ B22 ) )
       => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A1 @ B1 ) @ ( multis4607348483316519301on_nat @ Ns ) )
         => ( ( A22 != zero_z8385993896842825086on_nat )
           => ( ! [B4: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ B4 @ ( set_ms6185292591494683198on_nat @ B22 ) )
                 => ? [A7: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ A7 @ ( set_ms6185292591494683198on_nat @ A22 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis3147760155352349208on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_719_s__mul__extI,axiom,
    ! [A3: multis1856558993800614429on_nat,A1: multis1856558993800614429on_nat,A22: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,B1: multis1856558993800614429on_nat,B22: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( A3
        = ( plus_p2586647234311342374on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p2586647234311342374on_nat @ B1 @ B22 ) )
       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A1 @ B1 ) @ ( multis636354481432518601on_nat @ Ns ) )
         => ( ( A22 != zero_z8702567803361615526on_nat )
           => ( ! [B4: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ B4 @ ( set_ms3922534532180125136on_nat @ B22 ) )
                 => ? [A7: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A22 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B4 ) @ S ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ) ) ) ) ).

% s_mul_extI
thf(fact_720_Nil__eq__zip__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( nil_Pr5733333942050161901od_a_o
        = ( zip_li3837685382659732087od_a_o @ Xs @ Ys ) )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        | ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% Nil_eq_zip_iff
thf(fact_721_zip__eq__Nil__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( zip_li3837685382659732087od_a_o @ Xs @ Ys )
        = nil_Pr5733333942050161901od_a_o )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        | ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% zip_eq_Nil_iff
thf(fact_722_zip__Cons__Cons,axiom,
    ! [X4: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( zip_mu5118375602418390771on_nat @ ( cons_m1493500683067190247on_nat @ X4 @ Xs ) @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) )
      = ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_723_zip__Cons__Cons,axiom,
    ! [X4: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,Y: produc118098540238352959on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( zip_li7388484927123455957on_nat @ ( cons_l5857885737047467175on_nat @ X4 @ Xs ) @ ( cons_P4518942764240009225on_nat @ Y @ Ys ) )
      = ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y ) @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_724_zip__Cons__Cons,axiom,
    ! [X4: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,Y: produc1714117091004681831on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( zip_li8749512630040955217on_nat @ ( cons_l5857885737047467175on_nat @ X4 @ Xs ) @ ( cons_P2382207818294396957on_nat @ Y @ Ys ) )
      = ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ X4 @ Y ) @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_725_zip__Cons__Cons,axiom,
    ! [X4: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( zip_li3935435632236560119on_nat @ ( cons_l3028799188195208631on_nat @ X4 @ Xs ) @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) )
      = ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_726_zip__Cons__Cons,axiom,
    ! [X4: nat > nat,Xs: list_nat_nat,Y: nat,Ys: list_nat] :
      ( ( zip_nat_nat_nat @ ( cons_nat_nat @ X4 @ Xs ) @ ( cons_nat @ Y @ Ys ) )
      = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X4 @ Y ) @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_727_zip__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Us2: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Us2 ) )
     => ( ( zip_Su8130003265398491731on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( append3523812819975775986on_nat @ Us2 @ Vs ) )
        = ( append2865821678206098582on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Us2 ) @ ( zip_Su8130003265398491731on_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_728_zip__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Us2: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Us2 ) )
     => ( ( zip_Pr4229164709604849015on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( append2865821678206098582on_nat @ Us2 @ Vs ) )
        = ( append3895018001099526018on_nat @ ( zip_Pr4229164709604849015on_nat @ Xs @ Us2 ) @ ( zip_Pr4229164709604849015on_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_729_ex__mset__zip__left,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Xs5: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( ( mset_S7907735292211298936on_nat @ Xs5 )
          = ( mset_S7907735292211298936on_nat @ Xs ) )
       => ? [Ys2: list_S5253404628753938141on_nat] :
            ( ( ( size_s5686234912572693873on_nat @ Ys2 )
              = ( size_s5686234912572693873on_nat @ Xs5 ) )
            & ( ( mset_P6964650582947932560on_nat @ ( zip_Su8130003265398491731on_nat @ Xs5 @ Ys2 ) )
              = ( mset_P6964650582947932560on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) ) ) ) ) ) ).

% ex_mset_zip_left
thf(fact_730_zip__update,axiom,
    ! [Xs: list_m351779702951725613on_nat,I3: nat,X4: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat] :
      ( ( zip_mu5118375602418390771on_nat @ ( list_u4451328684322502447on_nat @ Xs @ I3 @ X4 ) @ ( list_u4451328684322502447on_nat @ Ys @ I3 @ Y ) )
      = ( list_u8615448375230402185on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) @ I3 @ ( produc5200047356169809875on_nat @ X4 @ Y ) ) ) ).

% zip_update
thf(fact_731_zip__update,axiom,
    ! [Xs: list_l8999537460041914605on_nat,I3: nat,X4: list_S5253404628753938141on_nat,Ys: list_P5508476027986415439on_nat,Y: produc118098540238352959on_nat] :
      ( ( zip_li7388484927123455957on_nat @ ( list_u3873310082918383599on_nat @ Xs @ I3 @ X4 ) @ ( list_u3336313086739624785on_nat @ Ys @ I3 @ Y ) )
      = ( list_u550736303359258219on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) @ I3 @ ( produc1545487600272464053on_nat @ X4 @ Y ) ) ) ).

% zip_update
thf(fact_732_zip__update,axiom,
    ! [Xs: list_l8999537460041914605on_nat,I3: nat,X4: list_S5253404628753938141on_nat,Ys: list_P8734936519038225773on_nat,Y: produc1714117091004681831on_nat] :
      ( ( zip_li8749512630040955217on_nat @ ( list_u3873310082918383599on_nat @ Xs @ I3 @ X4 ) @ ( list_u3278723242984596181on_nat @ Ys @ I3 @ Y ) )
      = ( list_u3336313086739624785on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) @ I3 @ ( produc4176415733022414961on_nat @ X4 @ Y ) ) ) ).

% zip_update
thf(fact_733_zip__update,axiom,
    ! [Xs: list_l1726306699111145479on_nat,I3: nat,X4: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( zip_li3935435632236560119on_nat @ ( list_u184235112954345583on_nat @ Xs @ I3 @ X4 ) @ ( list_u184235112954345583on_nat @ Ys @ I3 @ Y ) )
      = ( list_u3278723242984596181on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) @ I3 @ ( produc7152764266822961111on_nat @ X4 @ Y ) ) ) ).

% zip_update
thf(fact_734_zip__update,axiom,
    ! [Xs: list_nat_nat,I3: nat,X4: nat > nat,Ys: list_nat,Y: nat] :
      ( ( zip_nat_nat_nat @ ( list_update_nat_nat @ Xs @ I3 @ X4 ) @ ( list_update_nat @ Ys @ I3 @ Y ) )
      = ( list_u5493879213508619102at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) @ I3 @ ( produc72220940542539688at_nat @ X4 @ Y ) ) ) ).

% zip_update
thf(fact_735_multpw__refl_H,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat,A3: multis1856558993800614429on_nat] :
      ( ( locall2418947207864277812on_nat @ Ns @ A3 )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ A3 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ).

% multpw_refl'
thf(fact_736_zip__eq__ConsE,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,Xy: produc9176241080893687067on_nat,Xys: list_P1915136757390359201on_nat] :
      ( ( ( zip_mu5118375602418390771on_nat @ Xs @ Ys )
        = ( cons_P6897958070471385041on_nat @ Xy @ Xys ) )
     => ~ ! [X3: multis1856558993800614429on_nat,Xs4: list_m351779702951725613on_nat] :
            ( ( Xs
              = ( cons_m1493500683067190247on_nat @ X3 @ Xs4 ) )
           => ! [Y3: multis1856558993800614429on_nat,Ys2: list_m351779702951725613on_nat] :
                ( ( Ys
                  = ( cons_m1493500683067190247on_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc5200047356169809875on_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_mu5118375602418390771on_nat @ Xs4 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_737_zip__eq__ConsE,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat,Xy: produc8325617224762090749on_nat,Xys: list_P6215481986061871491on_nat] :
      ( ( ( zip_li7388484927123455957on_nat @ Xs @ Ys )
        = ( cons_P8471348786669567923on_nat @ Xy @ Xys ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs4: list_l8999537460041914605on_nat] :
            ( ( Xs
              = ( cons_l5857885737047467175on_nat @ X3 @ Xs4 ) )
           => ! [Y3: produc118098540238352959on_nat,Ys2: list_P5508476027986415439on_nat] :
                ( ( Ys
                  = ( cons_P4518942764240009225on_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc1545487600272464053on_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_li7388484927123455957on_nat @ Xs4 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_738_zip__eq__ConsE,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat,Xy: produc118098540238352959on_nat,Xys: list_P5508476027986415439on_nat] :
      ( ( ( zip_li8749512630040955217on_nat @ Xs @ Ys )
        = ( cons_P4518942764240009225on_nat @ Xy @ Xys ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs4: list_l8999537460041914605on_nat] :
            ( ( Xs
              = ( cons_l5857885737047467175on_nat @ X3 @ Xs4 ) )
           => ! [Y3: produc1714117091004681831on_nat,Ys2: list_P8734936519038225773on_nat] :
                ( ( Ys
                  = ( cons_P2382207818294396957on_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc4176415733022414961on_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_li8749512630040955217on_nat @ Xs4 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_739_zip__eq__ConsE,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Xy: produc1714117091004681831on_nat,Xys: list_P8734936519038225773on_nat] :
      ( ( ( zip_li3935435632236560119on_nat @ Xs @ Ys )
        = ( cons_P2382207818294396957on_nat @ Xy @ Xys ) )
     => ~ ! [X3: list_P9127870862658512897on_nat,Xs4: list_l1726306699111145479on_nat] :
            ( ( Xs
              = ( cons_l3028799188195208631on_nat @ X3 @ Xs4 ) )
           => ! [Y3: list_P9127870862658512897on_nat,Ys2: list_l1726306699111145479on_nat] :
                ( ( Ys
                  = ( cons_l3028799188195208631on_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc7152764266822961111on_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_li3935435632236560119on_nat @ Xs4 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_740_zip__eq__ConsE,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,Xy: produc8199716216217303280at_nat,Xys: list_P5366131564312172150at_nat] :
      ( ( ( zip_nat_nat_nat @ Xs @ Ys )
        = ( cons_P4219629788700907686at_nat @ Xy @ Xys ) )
     => ~ ! [X3: nat > nat,Xs4: list_nat_nat] :
            ( ( Xs
              = ( cons_nat_nat @ X3 @ Xs4 ) )
           => ! [Y3: nat,Ys2: list_nat] :
                ( ( Ys
                  = ( cons_nat @ Y3 @ Ys2 ) )
               => ( ( Xy
                    = ( produc72220940542539688at_nat @ X3 @ Y3 ) )
                 => ( Xys
                   != ( zip_nat_nat_nat @ Xs4 @ Ys2 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_741_set__zip__rightD,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y ) @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) ) )
     => ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_742_set__zip__rightD,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( set_Pr8272476097931642672on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) ) )
     => ( member2988082745645132084on_nat @ Y @ ( set_mu1790088485793245640on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_743_set__zip__rightD,axiom,
    ! [X4: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y ) @ ( set_Pr4584522749497345810on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) )
     => ( member4585374397974051670on_nat @ Y @ ( set_Pr8183371313083159530on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_744_set__zip__rightD,axiom,
    ! [X4: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X4 @ Y ) @ ( set_Pr8183371313083159530on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) )
     => ( member4629988367768146576on_nat @ Y @ ( set_Pr6705596042943559932on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_745_set__zip__rightD,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) )
     => ( member783415501022416426on_nat @ Y @ ( set_li8154783781463198870on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_746_set__zip__rightD,axiom,
    ! [X4: nat > nat,Y: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_747_set__zip__leftD,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y ) @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) ) )
     => ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_748_set__zip__leftD,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( set_Pr8272476097931642672on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) ) )
     => ( member2988082745645132084on_nat @ X4 @ ( set_mu1790088485793245640on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_749_set__zip__leftD,axiom,
    ! [X4: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y ) @ ( set_Pr4584522749497345810on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) )
     => ( member86162308477975540on_nat @ X4 @ ( set_li4371160873219908744on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_750_set__zip__leftD,axiom,
    ! [X4: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X4 @ Y ) @ ( set_Pr8183371313083159530on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) )
     => ( member86162308477975540on_nat @ X4 @ ( set_li4371160873219908744on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_751_set__zip__leftD,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) )
     => ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_752_set__zip__leftD,axiom,
    ! [X4: nat > nat,Y: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat_nat @ X4 @ ( set_nat_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_753_in__set__zipE,axiom,
    ! [X4: nat > nat,Y: nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat_nat @ X4 @ ( set_nat_nat2 @ Xs ) )
         => ~ ( member_nat @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_754_in__set__zipE,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y ) @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
         => ~ ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_755_in__set__zipE,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( set_Pr8272476097931642672on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member2988082745645132084on_nat @ X4 @ ( set_mu1790088485793245640on_nat @ Xs ) )
         => ~ ( member2988082745645132084on_nat @ Y @ ( set_mu1790088485793245640on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_756_in__set__zipE,axiom,
    ! [X4: produc1313295348606329723on_nat,Y: sum_su4907927656612468301on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member4423147992870482318on_nat @ ( produc6127739654764711537on_nat @ X4 @ Y ) @ ( set_Pr4357461787844060706on_nat @ ( zip_Pr4381645364492640081on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
         => ~ ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_757_in__set__zipE,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Y: produc1313295348606329723on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member8058359081245219834on_nat @ ( produc5881223965920569109on_nat @ X4 @ Y ) @ ( set_Pr7992672876218798222on_nat @ ( zip_Su4135129675648497653on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
         => ~ ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_758_in__set__zipE,axiom,
    ! [X4: produc9176241080893687067on_nat,Y: sum_su4907927656612468301on_nat,Xs: list_P1915136757390359201on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member3190426813638135086on_nat @ ( produc5351261619647891473on_nat @ X4 @ Y ) @ ( set_Pr1257392981786754498on_nat @ ( zip_Pr2961279957876259057on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
         => ~ ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_759_in__set__zipE,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Y: produc9176241080893687067on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_P1915136757390359201on_nat] :
      ( ( member6036888723768207770on_nat @ ( produc4743540071926527669on_nat @ X4 @ Y ) @ ( set_Pr4103854891916827182on_nat @ ( zip_Su2353558410154895253on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
         => ~ ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_760_in__set__zipE,axiom,
    ! [X4: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ ( set_Pr3855996318391990908on_nat @ ( zip_Pr4229164709604849015on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
         => ~ ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_761_in__set__zipE,axiom,
    ! [X4: produc9176241080893687067on_nat,Y: produc1313295348606329723on_nat,Xs: list_P1915136757390359201on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member7531489447933518960on_nat @ ( produc1223934467720830135on_nat @ X4 @ Y ) @ ( set_Pr1289081361443199708on_nat @ ( zip_Pr4296189396232164311on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
         => ~ ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_762_in__set__zipE,axiom,
    ! [X4: produc1313295348606329723on_nat,Y: produc9176241080893687067on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P1915136757390359201on_nat] :
      ( ( member6971310113140963248on_nat @ ( produc1091298452295717879on_nat @ X4 @ Y ) @ ( set_Pr728902026650643996on_nat @ ( zip_Pr4163553380807052055on_nat @ Xs @ Ys ) ) )
     => ~ ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
         => ~ ( member4278285197320822468on_nat @ Y @ ( set_Pr8272476097931642672on_nat @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_763_zip__same,axiom,
    ! [A: produc9176241080893687067on_nat,B: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A @ B ) @ ( set_Pr4220369340323606140on_nat @ ( zip_Pr3749958079798905207on_nat @ Xs @ Xs ) ) )
      = ( ( member4278285197320822468on_nat @ A @ ( set_Pr8272476097931642672on_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_764_zip__same,axiom,
    ! [A: produc1313295348606329723on_nat,B: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ A @ B ) @ ( set_Pr3855996318391990908on_nat @ ( zip_Pr4229164709604849015on_nat @ Xs @ Xs ) ) )
      = ( ( member5906418179908956708on_nat @ A @ ( set_Pr937530902923597456on_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_765_zip__same,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ B ) @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Xs ) ) )
      = ( ( member8476201367316484836on_nat @ A @ ( set_Su4848624856859743608on_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_766_zip__same,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,Xs: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( set_Pr8272476097931642672on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Xs ) ) )
      = ( ( member2988082745645132084on_nat @ A @ ( set_mu1790088485793245640on_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_767_zip__same,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Xs ) ) )
      = ( ( member783415501022416426on_nat @ A @ ( set_li8154783781463198870on_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_768_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ Ys ) )
       => ~ ! [X3: sum_su4907927656612468301on_nat] :
              ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y ) @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_769_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat] :
      ( ( ( size_s6603137509925400257on_nat @ Xs )
        = ( size_s6603137509925400257on_nat @ Ys ) )
     => ( ( member2988082745645132084on_nat @ Y @ ( set_mu1790088485793245640on_nat @ Ys ) )
       => ~ ! [X3: multis1856558993800614429on_nat] :
              ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y ) @ ( set_Pr8272476097931642672on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_770_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat,Y: produc118098540238352959on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = ( size_s2647645505745046499on_nat @ Ys ) )
     => ( ( member4585374397974051670on_nat @ Y @ ( set_Pr8183371313083159530on_nat @ Ys ) )
       => ~ ! [X3: list_S5253404628753938141on_nat] :
              ~ ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y ) @ ( set_Pr4584522749497345810on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_771_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat,Y: produc1714117091004681831on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = ( size_s4814362452863045593on_nat @ Ys ) )
     => ( ( member4629988367768146576on_nat @ Y @ ( set_Pr6705596042943559932on_nat @ Ys ) )
       => ~ ! [X3: list_S5253404628753938141on_nat] :
              ~ ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y ) @ ( set_Pr8183371313083159530on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_772_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( ( size_s129639936549468787on_nat @ Xs )
        = ( size_s129639936549468787on_nat @ Ys ) )
     => ( ( member783415501022416426on_nat @ Y @ ( set_li8154783781463198870on_nat @ Ys ) )
       => ~ ! [X3: list_P9127870862658512897on_nat] :
              ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_773_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,Y: nat] :
      ( ( ( size_s8208510060688613859at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X3: nat > nat] :
              ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_774_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,X4: sum_su4907927656612468301on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ~ ! [Y3: sum_su4907927656612468301on_nat] :
              ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y3 ) @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_775_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,X4: multis1856558993800614429on_nat] :
      ( ( ( size_s6603137509925400257on_nat @ Xs )
        = ( size_s6603137509925400257on_nat @ Ys ) )
     => ( ( member2988082745645132084on_nat @ X4 @ ( set_mu1790088485793245640on_nat @ Xs ) )
       => ~ ! [Y3: multis1856558993800614429on_nat] :
              ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y3 ) @ ( set_Pr8272476097931642672on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_776_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat,X4: list_S5253404628753938141on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = ( size_s2647645505745046499on_nat @ Ys ) )
     => ( ( member86162308477975540on_nat @ X4 @ ( set_li4371160873219908744on_nat @ Xs ) )
       => ~ ! [Y3: produc118098540238352959on_nat] :
              ~ ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y3 ) @ ( set_Pr4584522749497345810on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_777_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat,X4: list_S5253404628753938141on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = ( size_s4814362452863045593on_nat @ Ys ) )
     => ( ( member86162308477975540on_nat @ X4 @ ( set_li4371160873219908744on_nat @ Xs ) )
       => ~ ! [Y3: produc1714117091004681831on_nat] :
              ~ ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X4 @ Y3 ) @ ( set_Pr8183371313083159530on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_778_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,X4: list_P9127870862658512897on_nat] :
      ( ( ( size_s129639936549468787on_nat @ Xs )
        = ( size_s129639936549468787on_nat @ Ys ) )
     => ( ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Xs ) )
       => ~ ! [Y3: list_P9127870862658512897on_nat] :
              ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y3 ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_779_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,X4: nat > nat] :
      ( ( ( size_s8208510060688613859at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat_nat @ X4 @ ( set_nat_nat2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y3 ) @ ( set_Pr1941165898809491589at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_780_ns__mul__extI,axiom,
    ! [A3: multis2912115315702009697on_nat,A1: multis2912115315702009697on_nat,A22: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,B1: multis2912115315702009697on_nat,B22: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( A3
        = ( plus_p7340530524814416408on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p7340530524814416408on_nat @ B1 @ B22 ) )
       => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A1 @ B1 ) @ ( multis5394615859312135327on_nat @ Ns ) )
         => ( ! [B4: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ B4 @ ( set_ms7239842528327418072on_nat @ B22 ) )
               => ? [A7: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ A7 @ ( set_ms7239842528327418072on_nat @ A22 ) )
                    & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A7 @ B4 ) @ S ) ) )
           => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis3046611884734373584on_nat @ Ns @ S ) ) ) ) ) ) ).

% ns_mul_extI
thf(fact_781_ns__mul__extI,axiom,
    ! [A3: multis6850599868085929325on_nat,A1: multis6850599868085929325on_nat,A22: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,B1: multis6850599868085929325on_nat,B22: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( A3
        = ( plus_p4342429221931815414on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p4342429221931815414on_nat @ B1 @ B22 ) )
       => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A1 @ B1 ) @ ( multis1913027631269000601on_nat @ Ns ) )
         => ( ! [B4: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ B4 @ ( set_ms2805207930803453472on_nat @ B22 ) )
               => ? [A7: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ A7 @ ( set_ms2805207930803453472on_nat @ A22 ) )
                    & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A7 @ B4 ) @ S ) ) )
           => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis3835232632530192936on_nat @ Ns @ S ) ) ) ) ) ) ).

% ns_mul_extI
thf(fact_782_ns__mul__extI,axiom,
    ! [A3: multis5168333519641040071on_nat,A1: multis5168333519641040071on_nat,A22: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,B1: multis5168333519641040071on_nat,B22: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( A3
        = ( plus_p7957816880051667710on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p7957816880051667710on_nat @ B1 @ B22 ) )
       => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A1 @ B1 ) @ ( multis4607348483316519301on_nat @ Ns ) )
         => ( ! [B4: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ B4 @ ( set_ms6185292591494683198on_nat @ B22 ) )
               => ? [A7: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ A7 @ ( set_ms6185292591494683198on_nat @ A22 ) )
                    & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A7 @ B4 ) @ S ) ) )
           => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ) ) ) ).

% ns_mul_extI
thf(fact_783_ns__mul__extI,axiom,
    ! [A3: multis1856558993800614429on_nat,A1: multis1856558993800614429on_nat,A22: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,B1: multis1856558993800614429on_nat,B22: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( A3
        = ( plus_p2586647234311342374on_nat @ A1 @ A22 ) )
     => ( ( B3
          = ( plus_p2586647234311342374on_nat @ B1 @ B22 ) )
       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A1 @ B1 ) @ ( multis636354481432518601on_nat @ Ns ) )
         => ( ! [B4: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ B4 @ ( set_ms3922534532180125136on_nat @ B22 ) )
               => ? [A7: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A22 ) )
                    & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B4 ) @ S ) ) )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ) ) ) ).

% ns_mul_extI
thf(fact_784_ns__mul__extE,axiom,
    ! [A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis3046611884734373584on_nat @ Ns @ S ) )
     => ~ ! [A12: multis2912115315702009697on_nat,A23: multis2912115315702009697on_nat] :
            ( ( A3
              = ( plus_p7340530524814416408on_nat @ A12 @ A23 ) )
           => ! [B12: multis2912115315702009697on_nat,B23: multis2912115315702009697on_nat] :
                ( ( B3
                  = ( plus_p7340530524814416408on_nat @ B12 @ B23 ) )
               => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A12 @ B12 ) @ ( multis5394615859312135327on_nat @ Ns ) )
                 => ~ ! [B5: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ B5 @ ( set_ms7239842528327418072on_nat @ B23 ) )
                       => ? [A4: produc9176241080893687067on_nat] :
                            ( ( member4278285197320822468on_nat @ A4 @ ( set_ms7239842528327418072on_nat @ A23 ) )
                            & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ).

% ns_mul_extE
thf(fact_785_ns__mul__extE,axiom,
    ! [A3: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis3835232632530192936on_nat @ Ns @ S ) )
     => ~ ! [A12: multis6850599868085929325on_nat,A23: multis6850599868085929325on_nat] :
            ( ( A3
              = ( plus_p4342429221931815414on_nat @ A12 @ A23 ) )
           => ! [B12: multis6850599868085929325on_nat,B23: multis6850599868085929325on_nat] :
                ( ( B3
                  = ( plus_p4342429221931815414on_nat @ B12 @ B23 ) )
               => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A12 @ B12 ) @ ( multis1913027631269000601on_nat @ Ns ) )
                 => ~ ! [B5: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ B5 @ ( set_ms2805207930803453472on_nat @ B23 ) )
                       => ? [A4: multis1856558993800614429on_nat] :
                            ( ( member2988082745645132084on_nat @ A4 @ ( set_ms2805207930803453472on_nat @ A23 ) )
                            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ).

% ns_mul_extE
thf(fact_786_ns__mul__extE,axiom,
    ! [A3: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis664119935410451510on_nat @ Ns @ S ) )
     => ~ ! [A12: multis5168333519641040071on_nat,A23: multis5168333519641040071on_nat] :
            ( ( A3
              = ( plus_p7957816880051667710on_nat @ A12 @ A23 ) )
           => ! [B12: multis5168333519641040071on_nat,B23: multis5168333519641040071on_nat] :
                ( ( B3
                  = ( plus_p7957816880051667710on_nat @ B12 @ B23 ) )
               => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A12 @ B12 ) @ ( multis4607348483316519301on_nat @ Ns ) )
                 => ~ ! [B5: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ B5 @ ( set_ms6185292591494683198on_nat @ B23 ) )
                       => ? [A4: list_P9127870862658512897on_nat] :
                            ( ( member783415501022416426on_nat @ A4 @ ( set_ms6185292591494683198on_nat @ A23 ) )
                            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ).

% ns_mul_extE
thf(fact_787_ns__mul__extE,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) )
     => ~ ! [A12: multis1856558993800614429on_nat,A23: multis1856558993800614429on_nat] :
            ( ( A3
              = ( plus_p2586647234311342374on_nat @ A12 @ A23 ) )
           => ! [B12: multis1856558993800614429on_nat,B23: multis1856558993800614429on_nat] :
                ( ( B3
                  = ( plus_p2586647234311342374on_nat @ B12 @ B23 ) )
               => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A12 @ B12 ) @ ( multis636354481432518601on_nat @ Ns ) )
                 => ~ ! [B5: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ B5 @ ( set_ms3922534532180125136on_nat @ B23 ) )
                       => ? [A4: sum_su4907927656612468301on_nat] :
                            ( ( member8476201367316484836on_nat @ A4 @ ( set_ms3922534532180125136on_nat @ A23 ) )
                            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ).

% ns_mul_extE
thf(fact_788_multpw__listI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X: multis4835999835052358849on_nat,Y5: multis4835999835052358849on_nat,Ns: set_Pr8202701763255893447on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) )
     => ( ( X
          = ( mset_P6964650582947932560on_nat @ Xs ) )
       => ( ( Y5
            = ( mset_P6964650582947932560on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Ys ) )
               => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I ) @ ( nth_Pr3336893728888831618on_nat @ Ys @ I ) ) @ Ns ) )
           => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ X @ Y5 ) @ ( multis308651003543936767on_nat @ Ns ) ) ) ) ) ) ).

% multpw_listI
thf(fact_789_multpw__listI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( X
          = ( mset_S7907735292211298936on_nat @ Xs ) )
       => ( ( Y5
            = ( mset_S7907735292211298936on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I ) ) @ Ns ) )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ) ) ).

% multpw_listI
thf(fact_790_multpw__listI,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( ( size_s6603137509925400257on_nat @ Xs )
        = ( size_s6603137509925400257on_nat @ Ys ) )
     => ( ( X
          = ( mset_m1293634003157817032on_nat @ Xs ) )
       => ( ( Y5
            = ( mset_m1293634003157817032on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Ys ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs @ I ) @ ( nth_mu8151576519813129558on_nat @ Ys @ I ) ) @ Ns ) )
           => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis1913027631269000601on_nat @ Ns ) ) ) ) ) ) ).

% multpw_listI
thf(fact_791_multpw__listI,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( ( size_s129639936549468787on_nat @ Xs )
        = ( size_s129639936549468787on_nat @ Ys ) )
     => ( ( X
          = ( mset_l2262431794812116886on_nat @ Xs ) )
       => ( ( Y5
            = ( mset_l2262431794812116886on_nat @ Ys ) )
         => ( ! [I: nat] :
                ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Ys ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ I ) @ ( nth_li1302981970230039432on_nat @ Ys @ I ) ) @ Ns ) )
           => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis4607348483316519301on_nat @ Ns ) ) ) ) ) ) ).

% multpw_listI
thf(fact_792_multpw__listE,axiom,
    ! [X: multis4835999835052358849on_nat,Y5: multis4835999835052358849on_nat,Ns: set_Pr8202701763255893447on_nat] :
      ( ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ X @ Y5 ) @ ( multis308651003543936767on_nat @ Ns ) )
     => ~ ! [Xs2: list_P9127870862658512897on_nat,Ys3: list_P9127870862658512897on_nat] :
            ( ( ( size_s7490768528479217773on_nat @ Xs2 )
              = ( size_s7490768528479217773on_nat @ Ys3 ) )
           => ( ( X
                = ( mset_P6964650582947932560on_nat @ Xs2 ) )
             => ( ( Y5
                  = ( mset_P6964650582947932560on_nat @ Ys3 ) )
               => ~ ! [I2: nat] :
                      ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Ys3 ) )
                     => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs2 @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Ys3 @ I2 ) ) @ Ns ) ) ) ) ) ) ).

% multpw_listE
thf(fact_793_multpw__listE,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis1913027631269000601on_nat @ Ns ) )
     => ~ ! [Xs2: list_m351779702951725613on_nat,Ys3: list_m351779702951725613on_nat] :
            ( ( ( size_s6603137509925400257on_nat @ Xs2 )
              = ( size_s6603137509925400257on_nat @ Ys3 ) )
           => ( ( X
                = ( mset_m1293634003157817032on_nat @ Xs2 ) )
             => ( ( Y5
                  = ( mset_m1293634003157817032on_nat @ Ys3 ) )
               => ~ ! [I2: nat] :
                      ( ( ord_less_nat @ I2 @ ( size_s6603137509925400257on_nat @ Ys3 ) )
                     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs2 @ I2 ) @ ( nth_mu8151576519813129558on_nat @ Ys3 @ I2 ) ) @ Ns ) ) ) ) ) ) ).

% multpw_listE
thf(fact_794_multpw__listE,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis4607348483316519301on_nat @ Ns ) )
     => ~ ! [Xs2: list_l1726306699111145479on_nat,Ys3: list_l1726306699111145479on_nat] :
            ( ( ( size_s129639936549468787on_nat @ Xs2 )
              = ( size_s129639936549468787on_nat @ Ys3 ) )
           => ( ( X
                = ( mset_l2262431794812116886on_nat @ Xs2 ) )
             => ( ( Y5
                  = ( mset_l2262431794812116886on_nat @ Ys3 ) )
               => ~ ! [I2: nat] :
                      ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Ys3 ) )
                     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs2 @ I2 ) @ ( nth_li1302981970230039432on_nat @ Ys3 @ I2 ) ) @ Ns ) ) ) ) ) ) ).

% multpw_listE
thf(fact_795_multpw__listE,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ~ ! [Xs2: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
            ( ( ( size_s5686234912572693873on_nat @ Xs2 )
              = ( size_s5686234912572693873on_nat @ Ys3 ) )
           => ( ( X
                = ( mset_S7907735292211298936on_nat @ Xs2 ) )
             => ( ( Y5
                  = ( mset_S7907735292211298936on_nat @ Ys3 ) )
               => ~ ! [I2: nat] :
                      ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Ys3 ) )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs2 @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Ys3 @ I2 ) ) @ Ns ) ) ) ) ) ) ).

% multpw_listE
thf(fact_796_s__mul__extE,axiom,
    ! [A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A3 @ B3 ) @ ( multis1332936165424366514on_nat @ Ns @ S ) )
     => ~ ! [A12: multis2912115315702009697on_nat,A23: multis2912115315702009697on_nat] :
            ( ( A3
              = ( plus_p7340530524814416408on_nat @ A12 @ A23 ) )
           => ! [B12: multis2912115315702009697on_nat,B23: multis2912115315702009697on_nat] :
                ( ( B3
                  = ( plus_p7340530524814416408on_nat @ B12 @ B23 ) )
               => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ A12 @ B12 ) @ ( multis5394615859312135327on_nat @ Ns ) )
                 => ( ( A23 != zero_z1502369238205801624on_nat )
                   => ~ ! [B5: produc9176241080893687067on_nat] :
                          ( ( member4278285197320822468on_nat @ B5 @ ( set_ms7239842528327418072on_nat @ B23 ) )
                         => ? [A4: produc9176241080893687067on_nat] :
                              ( ( member4278285197320822468on_nat @ A4 @ ( set_ms7239842528327418072on_nat @ A23 ) )
                              & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ) ).

% s_mul_extE
thf(fact_797_s__mul__extE,axiom,
    ! [A3: multis6850599868085929325on_nat,B3: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A3 @ B3 ) @ ( multis5680658968000484038on_nat @ Ns @ S ) )
     => ~ ! [A12: multis6850599868085929325on_nat,A23: multis6850599868085929325on_nat] :
            ( ( A3
              = ( plus_p4342429221931815414on_nat @ A12 @ A23 ) )
           => ! [B12: multis6850599868085929325on_nat,B23: multis6850599868085929325on_nat] :
                ( ( B3
                  = ( plus_p4342429221931815414on_nat @ B12 @ B23 ) )
               => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A12 @ B12 ) @ ( multis1913027631269000601on_nat @ Ns ) )
                 => ( ( A23 != zero_z4144889538238539638on_nat )
                   => ~ ! [B5: multis1856558993800614429on_nat] :
                          ( ( member2988082745645132084on_nat @ B5 @ ( set_ms2805207930803453472on_nat @ B23 ) )
                         => ? [A4: multis1856558993800614429on_nat] :
                              ( ( member2988082745645132084on_nat @ A4 @ ( set_ms2805207930803453472on_nat @ A23 ) )
                              & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ) ).

% s_mul_extE
thf(fact_798_s__mul__extE,axiom,
    ! [A3: multis5168333519641040071on_nat,B3: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A3 @ B3 ) @ ( multis3147760155352349208on_nat @ Ns @ S ) )
     => ~ ! [A12: multis5168333519641040071on_nat,A23: multis5168333519641040071on_nat] :
            ( ( A3
              = ( plus_p7957816880051667710on_nat @ A12 @ A23 ) )
           => ! [B12: multis5168333519641040071on_nat,B23: multis5168333519641040071on_nat] :
                ( ( B3
                  = ( plus_p7957816880051667710on_nat @ B12 @ B23 ) )
               => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A12 @ B12 ) @ ( multis4607348483316519301on_nat @ Ns ) )
                 => ( ( A23 != zero_z8385993896842825086on_nat )
                   => ~ ! [B5: list_P9127870862658512897on_nat] :
                          ( ( member783415501022416426on_nat @ B5 @ ( set_ms6185292591494683198on_nat @ B23 ) )
                         => ? [A4: list_P9127870862658512897on_nat] :
                              ( ( member783415501022416426on_nat @ A4 @ ( set_ms6185292591494683198on_nat @ A23 ) )
                              & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ) ).

% s_mul_extE
thf(fact_799_s__mul__extE,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A3 @ B3 ) @ ( multis1620213760974180726on_nat @ Ns @ S ) )
     => ~ ! [A12: multis1856558993800614429on_nat,A23: multis1856558993800614429on_nat] :
            ( ( A3
              = ( plus_p2586647234311342374on_nat @ A12 @ A23 ) )
           => ! [B12: multis1856558993800614429on_nat,B23: multis1856558993800614429on_nat] :
                ( ( B3
                  = ( plus_p2586647234311342374on_nat @ B12 @ B23 ) )
               => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A12 @ B12 ) @ ( multis636354481432518601on_nat @ Ns ) )
                 => ( ( A23 != zero_z8702567803361615526on_nat )
                   => ~ ! [B5: sum_su4907927656612468301on_nat] :
                          ( ( member8476201367316484836on_nat @ B5 @ ( set_ms3922534532180125136on_nat @ B23 ) )
                         => ? [A4: sum_su4907927656612468301on_nat] :
                              ( ( member8476201367316484836on_nat @ A4 @ ( set_ms3922534532180125136on_nat @ A23 ) )
                              & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A4 @ B5 ) @ S ) ) ) ) ) ) ) ) ).

% s_mul_extE
thf(fact_800_multpw__emptyL,axiom,
    ! [X: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ X ) @ ( multis636354481432518601on_nat @ Ns ) )
      = ( X = zero_z8702567803361615526on_nat ) ) ).

% multpw_emptyL
thf(fact_801_multpw__emptyR,axiom,
    ! [X: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ zero_z8702567803361615526on_nat ) @ ( multis636354481432518601on_nat @ Ns ) )
      = ( X = zero_z8702567803361615526on_nat ) ) ).

% multpw_emptyR
thf(fact_802_multpw__splitR,axiom,
    ! [X12: multis1856558993800614429on_nat,X23: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ X12 @ X23 ) @ Y5 ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ~ ! [Y12: multis1856558993800614429on_nat,Y22: multis1856558993800614429on_nat] :
            ( ( Y5
              = ( plus_p2586647234311342374on_nat @ Y12 @ Y22 ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y12 ) @ ( multis636354481432518601on_nat @ Ns ) )
             => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X23 @ Y22 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ) ).

% multpw_splitR
thf(fact_803_multpw__splitL,axiom,
    ! [X: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ~ ! [X13: multis1856558993800614429on_nat,X24: multis1856558993800614429on_nat] :
            ( ( X
              = ( plus_p2586647234311342374on_nat @ X13 @ X24 ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X13 @ Y13 ) @ ( multis636354481432518601on_nat @ Ns ) )
             => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X24 @ Y23 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ) ).

% multpw_splitL
thf(fact_804_multpw__add,axiom,
    ! [X12: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,X23: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X23 @ Y23 ) @ ( multis636354481432518601on_nat @ Ns ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ X12 @ X23 ) @ ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ).

% multpw_add
thf(fact_805_multpw_Oempty,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ zero_z8702567803361615526on_nat ) @ ( multis636354481432518601on_nat @ Ns ) ) ).

% multpw.empty
thf(fact_806_empty__Shift,axiom,
    ! [Kl2: set_li8167635853205255297on_nat,K3: produc9176241080893687067on_nat] :
      ( ( member6200664750067624906on_nat @ nil_Pr2652050626817158177on_nat @ Kl2 )
     => ( ( member4278285197320822468on_nat @ K3 @ ( bNF_Gr7372635924212258486on_nat @ Kl2 @ nil_Pr2652050626817158177on_nat ) )
       => ( member6200664750067624906on_nat @ nil_Pr2652050626817158177on_nat @ ( bNF_Gr2284405008283128634on_nat @ Kl2 @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_807_empty__Shift,axiom,
    ! [Kl2: set_li7049945101328808109od_a_o,K3: list_P3196869577824546119od_a_o] :
      ( ( member8885487059802017782od_a_o @ nil_li1372968520647675725od_a_o @ Kl2 )
     => ( ( member5261019209154752368od_a_o @ K3 @ ( bNF_Gr142345107972655202od_a_o @ Kl2 @ nil_li1372968520647675725od_a_o ) )
       => ( member8885487059802017782od_a_o @ nil_li1372968520647675725od_a_o @ ( bNF_Gr198862587184010982od_a_o @ Kl2 @ K3 ) ) ) ) ).

% empty_Shift
thf(fact_808_mult2__altE,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,B: $o,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ B @ Ns @ S ) )
     => ~ ! [X13: multis2912115315702009697on_nat,X24: multis2912115315702009697on_nat] :
            ( ( X
              = ( plus_p7340530524814416408on_nat @ X13 @ X24 ) )
           => ! [Y12: multis2912115315702009697on_nat,Y22: multis2912115315702009697on_nat] :
                ( ( Y5
                  = ( plus_p7340530524814416408on_nat @ Y12 @ Y22 ) )
               => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X13 @ Y12 ) @ ( multis5394615859312135327on_nat @ Ns ) )
                 => ( ( B
                      | ( Y22 != zero_z1502369238205801624on_nat ) )
                   => ~ ! [X6: produc9176241080893687067on_nat] :
                          ( ( member4278285197320822468on_nat @ X6 @ ( set_ms7239842528327418072on_nat @ X24 ) )
                         => ? [Y3: produc9176241080893687067on_nat] :
                              ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y22 ) )
                              & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_altE
thf(fact_809_mult2__altE,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,B: $o,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ B @ Ns @ S ) )
     => ~ ! [X13: multis6850599868085929325on_nat,X24: multis6850599868085929325on_nat] :
            ( ( X
              = ( plus_p4342429221931815414on_nat @ X13 @ X24 ) )
           => ! [Y12: multis6850599868085929325on_nat,Y22: multis6850599868085929325on_nat] :
                ( ( Y5
                  = ( plus_p4342429221931815414on_nat @ Y12 @ Y22 ) )
               => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X13 @ Y12 ) @ ( multis1913027631269000601on_nat @ Ns ) )
                 => ( ( B
                      | ( Y22 != zero_z4144889538238539638on_nat ) )
                   => ~ ! [X6: multis1856558993800614429on_nat] :
                          ( ( member2988082745645132084on_nat @ X6 @ ( set_ms2805207930803453472on_nat @ X24 ) )
                         => ? [Y3: multis1856558993800614429on_nat] :
                              ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y22 ) )
                              & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_altE
thf(fact_810_mult2__altE,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,B: $o,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ B @ Ns @ S ) )
     => ~ ! [X13: multis5168333519641040071on_nat,X24: multis5168333519641040071on_nat] :
            ( ( X
              = ( plus_p7957816880051667710on_nat @ X13 @ X24 ) )
           => ! [Y12: multis5168333519641040071on_nat,Y22: multis5168333519641040071on_nat] :
                ( ( Y5
                  = ( plus_p7957816880051667710on_nat @ Y12 @ Y22 ) )
               => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X13 @ Y12 ) @ ( multis4607348483316519301on_nat @ Ns ) )
                 => ( ( B
                      | ( Y22 != zero_z8385993896842825086on_nat ) )
                   => ~ ! [X6: list_P9127870862658512897on_nat] :
                          ( ( member783415501022416426on_nat @ X6 @ ( set_ms6185292591494683198on_nat @ X24 ) )
                         => ? [Y3: list_P9127870862658512897on_nat] :
                              ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y22 ) )
                              & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_altE
thf(fact_811_mult2__altE,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,B: $o,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ B @ Ns @ S ) )
     => ~ ! [X13: multis1856558993800614429on_nat,X24: multis1856558993800614429on_nat] :
            ( ( X
              = ( plus_p2586647234311342374on_nat @ X13 @ X24 ) )
           => ! [Y12: multis1856558993800614429on_nat,Y22: multis1856558993800614429on_nat] :
                ( ( Y5
                  = ( plus_p2586647234311342374on_nat @ Y12 @ Y22 ) )
               => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X13 @ Y12 ) @ ( multis636354481432518601on_nat @ Ns ) )
                 => ( ( B
                      | ( Y22 != zero_z8702567803361615526on_nat ) )
                   => ~ ! [X6: sum_su4907927656612468301on_nat] :
                          ( ( member8476201367316484836on_nat @ X6 @ ( set_ms3922534532180125136on_nat @ X24 ) )
                         => ? [Y3: sum_su4907927656612468301on_nat] :
                              ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y22 ) )
                              & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_altE
thf(fact_812_mult2__altI,axiom,
    ! [X: multis2912115315702009697on_nat,X12: multis2912115315702009697on_nat,X23: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Y13: multis2912115315702009697on_nat,Y23: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,B: $o,S: set_Pr1446391457923625415on_nat] :
      ( ( X
        = ( plus_p7340530524814416408on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p7340530524814416408on_nat @ Y13 @ Y23 ) )
       => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X12 @ Y13 ) @ ( multis5394615859312135327on_nat @ Ns ) )
         => ( ( B
              | ( Y23 != zero_z1502369238205801624on_nat ) )
           => ( ! [X3: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X23 ) )
                 => ? [Y7: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ Y7 @ ( set_ms7239842528327418072on_nat @ Y23 ) )
                      & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ B @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_altI
thf(fact_813_mult2__altI,axiom,
    ! [X: multis6850599868085929325on_nat,X12: multis6850599868085929325on_nat,X23: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Y13: multis6850599868085929325on_nat,Y23: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,B: $o,S: set_Pr3573155178515677307on_nat] :
      ( ( X
        = ( plus_p4342429221931815414on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p4342429221931815414on_nat @ Y13 @ Y23 ) )
       => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X12 @ Y13 ) @ ( multis1913027631269000601on_nat @ Ns ) )
         => ( ( B
              | ( Y23 != zero_z4144889538238539638on_nat ) )
           => ( ! [X3: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X23 ) )
                 => ? [Y7: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Y7 @ ( set_ms2805207930803453472on_nat @ Y23 ) )
                      & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ B @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_altI
thf(fact_814_mult2__altI,axiom,
    ! [X: multis5168333519641040071on_nat,X12: multis5168333519641040071on_nat,X23: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Y13: multis5168333519641040071on_nat,Y23: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,B: $o,S: set_Pr8586234676504894535on_nat] :
      ( ( X
        = ( plus_p7957816880051667710on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p7957816880051667710on_nat @ Y13 @ Y23 ) )
       => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X12 @ Y13 ) @ ( multis4607348483316519301on_nat @ Ns ) )
         => ( ( B
              | ( Y23 != zero_z8385993896842825086on_nat ) )
           => ( ! [X3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X23 ) )
                 => ? [Y7: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Y7 @ ( set_ms6185292591494683198on_nat @ Y23 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ B @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_altI
thf(fact_815_mult2__altI,axiom,
    ! [X: multis1856558993800614429on_nat,X12: multis1856558993800614429on_nat,X23: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,B: $o,S: set_Pr2535564044409765851on_nat] :
      ( ( X
        = ( plus_p2586647234311342374on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) )
       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis636354481432518601on_nat @ Ns ) )
         => ( ( B
              | ( Y23 != zero_z8702567803361615526on_nat ) )
           => ( ! [X3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X23 ) )
                 => ? [Y7: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Y7 @ ( set_ms3922534532180125136on_nat @ Y23 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ B @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_altI
thf(fact_816_mult2__alt__sE,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) )
     => ~ ! [X13: multis2912115315702009697on_nat,X24: multis2912115315702009697on_nat] :
            ( ( X
              = ( plus_p7340530524814416408on_nat @ X13 @ X24 ) )
           => ! [Y12: multis2912115315702009697on_nat,Y22: multis2912115315702009697on_nat] :
                ( ( Y5
                  = ( plus_p7340530524814416408on_nat @ Y12 @ Y22 ) )
               => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X13 @ Y12 ) @ ( multis5394615859312135327on_nat @ Ns ) )
                 => ( ( Y22 != zero_z1502369238205801624on_nat )
                   => ~ ! [X6: produc9176241080893687067on_nat] :
                          ( ( member4278285197320822468on_nat @ X6 @ ( set_ms7239842528327418072on_nat @ X24 ) )
                         => ? [Y3: produc9176241080893687067on_nat] :
                              ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y22 ) )
                              & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_alt_sE
thf(fact_817_mult2__alt__sE,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) )
     => ~ ! [X13: multis6850599868085929325on_nat,X24: multis6850599868085929325on_nat] :
            ( ( X
              = ( plus_p4342429221931815414on_nat @ X13 @ X24 ) )
           => ! [Y12: multis6850599868085929325on_nat,Y22: multis6850599868085929325on_nat] :
                ( ( Y5
                  = ( plus_p4342429221931815414on_nat @ Y12 @ Y22 ) )
               => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X13 @ Y12 ) @ ( multis1913027631269000601on_nat @ Ns ) )
                 => ( ( Y22 != zero_z4144889538238539638on_nat )
                   => ~ ! [X6: multis1856558993800614429on_nat] :
                          ( ( member2988082745645132084on_nat @ X6 @ ( set_ms2805207930803453472on_nat @ X24 ) )
                         => ? [Y3: multis1856558993800614429on_nat] :
                              ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y22 ) )
                              & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_alt_sE
thf(fact_818_mult2__alt__sE,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) )
     => ~ ! [X13: multis5168333519641040071on_nat,X24: multis5168333519641040071on_nat] :
            ( ( X
              = ( plus_p7957816880051667710on_nat @ X13 @ X24 ) )
           => ! [Y12: multis5168333519641040071on_nat,Y22: multis5168333519641040071on_nat] :
                ( ( Y5
                  = ( plus_p7957816880051667710on_nat @ Y12 @ Y22 ) )
               => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X13 @ Y12 ) @ ( multis4607348483316519301on_nat @ Ns ) )
                 => ( ( Y22 != zero_z8385993896842825086on_nat )
                   => ~ ! [X6: list_P9127870862658512897on_nat] :
                          ( ( member783415501022416426on_nat @ X6 @ ( set_ms6185292591494683198on_nat @ X24 ) )
                         => ? [Y3: list_P9127870862658512897on_nat] :
                              ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y22 ) )
                              & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_alt_sE
thf(fact_819_mult2__alt__sE,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
     => ~ ! [X13: multis1856558993800614429on_nat,X24: multis1856558993800614429on_nat] :
            ( ( X
              = ( plus_p2586647234311342374on_nat @ X13 @ X24 ) )
           => ! [Y12: multis1856558993800614429on_nat,Y22: multis1856558993800614429on_nat] :
                ( ( Y5
                  = ( plus_p2586647234311342374on_nat @ Y12 @ Y22 ) )
               => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X13 @ Y12 ) @ ( multis636354481432518601on_nat @ Ns ) )
                 => ( ( Y22 != zero_z8702567803361615526on_nat )
                   => ~ ! [X6: sum_su4907927656612468301on_nat] :
                          ( ( member8476201367316484836on_nat @ X6 @ ( set_ms3922534532180125136on_nat @ X24 ) )
                         => ? [Y3: sum_su4907927656612468301on_nat] :
                              ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y22 ) )
                              & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ) ).

% mult2_alt_sE
thf(fact_820_mult2__alt__s__implies__mult2__alt__ns,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) ) ) ).

% mult2_alt_s_implies_mult2_alt_ns
thf(fact_821_mult2__alt__emptyL,axiom,
    ! [Y5: multis1856558993800614429on_nat,B: $o,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ Y5 ) @ ( multis2224597371354980665on_nat @ B @ Ns @ S ) )
      = ( B
        | ( Y5 != zero_z8702567803361615526on_nat ) ) ) ).

% mult2_alt_emptyL
thf(fact_822_mult2__alt__emptyR,axiom,
    ! [X: multis1856558993800614429on_nat,B: $o,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ zero_z8702567803361615526on_nat ) @ ( multis2224597371354980665on_nat @ B @ Ns @ S ) )
      = ( B
        & ( X = zero_z8702567803361615526on_nat ) ) ) ).

% mult2_alt_emptyR
thf(fact_823_mult2__alt__ns__ns__add,axiom,
    ! [X12: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,X23: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X23 @ Y23 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ X12 @ X23 ) @ ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) ) ) ) ).

% mult2_alt_ns_ns_add
thf(fact_824_mult2__alt__s__ns__add,axiom,
    ! [X12: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,X23: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X23 @ Y23 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ X12 @ X23 ) @ ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ) ).

% mult2_alt_s_ns_add
thf(fact_825_mult2__alt__ns__s__add,axiom,
    ! [X12: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,X23: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X23 @ Y23 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ X12 @ X23 ) @ ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ) ).

% mult2_alt_ns_s_add
thf(fact_826_mult2__alt__s__s__add,axiom,
    ! [X12: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,X23: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X23 @ Y23 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ X12 @ X23 ) @ ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ) ).

% mult2_alt_s_s_add
thf(fact_827_mult2__alt__add,axiom,
    ! [X12: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,B13: $o,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,X23: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat,B24: $o] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis2224597371354980665on_nat @ B13 @ Ns @ S ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X23 @ Y23 ) @ ( multis2224597371354980665on_nat @ B24 @ Ns @ S ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ X12 @ X23 ) @ ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) )
          @ ( multis2224597371354980665on_nat
            @ ( B13
              & B24 )
            @ Ns
            @ S ) ) ) ) ).

% mult2_alt_add
thf(fact_828_multpw__implies__mult2__alt__ns,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) ) ) ).

% multpw_implies_mult2_alt_ns
thf(fact_829_trans__mult2__alt__local,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Z4: multis2912115315702009697on_nat,S: set_Pr1446391457923625415on_nat,Ns: set_Pr1446391457923625415on_nat,B13: $o,B24: $o] :
      ( ! [X3: produc9176241080893687067on_nat,Y3: produc9176241080893687067on_nat,Z: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
         => ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
           => ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
             => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
               => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                 => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: produc9176241080893687067on_nat,Y3: produc9176241080893687067on_nat,Z: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
           => ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
             => ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
               => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                 => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                   => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: produc9176241080893687067on_nat,Y3: produc9176241080893687067on_nat,Z: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
             => ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
               => ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                 => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                     => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: produc9176241080893687067on_nat,Y3: produc9176241080893687067on_nat,Z: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
               => ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                 => ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                       => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ B13 @ Ns @ S ) )
             => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ Y5 @ Z4 ) @ ( multis5731881090987732783on_nat @ B24 @ Ns @ S ) )
               => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Z4 )
                  @ ( multis5731881090987732783on_nat
                    @ ( B13
                      & B24 )
                    @ Ns
                    @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_local
thf(fact_830_trans__mult2__alt__local,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat,B13: $o,B24: $o] :
      ( ! [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Z: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
         => ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
           => ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
             => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
               => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Z: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
           => ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
             => ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
               => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                   => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Z: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
             => ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
               => ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Z: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
               => ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                 => ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                       => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ B13 @ Ns @ S ) )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y5 @ Z4 ) @ ( multis2224597371354980665on_nat @ B24 @ Ns @ S ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Z4 )
                  @ ( multis2224597371354980665on_nat
                    @ ( B13
                      & B24 )
                    @ Ns
                    @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_local
thf(fact_831_trans__mult2__alt__local,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Z4: multis6850599868085929325on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat,B13: $o,B24: $o] :
      ( ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat,Z: multis1856558993800614429on_nat] :
          ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
         => ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
           => ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
               => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                 => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat,Z: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
           => ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
             => ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
               => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                   => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat,Z: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
             => ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
               => ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat,Z: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
               => ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                 => ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ B13 @ Ns @ S ) )
             => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Y5 @ Z4 ) @ ( multis394508389068283657on_nat @ B24 @ Ns @ S ) )
               => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Z4 )
                  @ ( multis394508389068283657on_nat
                    @ ( B13
                      & B24 )
                    @ Ns
                    @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_local
thf(fact_832_trans__mult2__alt__local,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Z4: multis5168333519641040071on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat,B13: $o,B24: $o] :
      ( ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] :
          ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
         => ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
           => ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
             => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
               => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                 => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
           => ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
             => ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
               => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                   => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
             => ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
               => ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
               => ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                 => ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ B13 @ Ns @ S ) )
             => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Y5 @ Z4 ) @ ( multis1733271916442156053on_nat @ B24 @ Ns @ S ) )
               => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Z4 )
                  @ ( multis1733271916442156053on_nat
                    @ ( B13
                      & B24 )
                    @ Ns
                    @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_local
thf(fact_833_trans__mult2__alt__s__s__local,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Z4: multis2912115315702009697on_nat,S: set_Pr1446391457923625415on_nat,Ns: set_Pr1446391457923625415on_nat] :
      ( ! [X3: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
         => ! [Y3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
             => ! [Z: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                 => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                     => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
           => ! [Y3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
               => ! [Z: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                       => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
             => ! [Y3: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                 => ! [Z: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                         => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
               => ! [Y3: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                   => ! [Z: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                           => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) )
             => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ Y5 @ Z4 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) )
               => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Z4 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_s_local
thf(fact_834_trans__mult2__alt__s__s__local,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
         => ! [Y3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
             => ! [Z: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
           => ! [Y3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
               => ! [Z: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                       => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
             => ! [Y3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                 => ! [Z: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
               => ! [Y3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                   => ! [Z: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y5 @ Z4 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Z4 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_s_local
thf(fact_835_trans__mult2__alt__s__s__local,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Z4: multis6850599868085929325on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
         => ! [Y3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
             => ! [Z: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
           => ! [Y3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
               => ! [Z: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
             => ! [Y3: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                 => ! [Z: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
               => ! [Y3: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                   => ! [Z: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) )
             => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Y5 @ Z4 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) )
               => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Z4 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_s_local
thf(fact_836_trans__mult2__alt__s__s__local,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Z4: multis5168333519641040071on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
         => ! [Y3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
             => ! [Z: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
           => ! [Y3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
               => ! [Z: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
             => ! [Y3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                 => ! [Z: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
               => ! [Y3: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                   => ! [Z: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) )
             => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Y5 @ Z4 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) )
               => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Z4 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_s_local
thf(fact_837_trans__mult2__alt__ns__s__local,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Z4: multis2912115315702009697on_nat,S: set_Pr1446391457923625415on_nat,Ns: set_Pr1446391457923625415on_nat] :
      ( ! [X3: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
         => ! [Y3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
             => ! [Z: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                 => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                     => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
           => ! [Y3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
               => ! [Z: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                       => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
             => ! [Y3: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                 => ! [Z: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                         => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
               => ! [Y3: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                   => ! [Z: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                           => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $true @ Ns @ S ) )
             => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ Y5 @ Z4 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) )
               => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Z4 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_s_local
thf(fact_838_trans__mult2__alt__ns__s__local,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
         => ! [Y3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
             => ! [Z: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
           => ! [Y3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
               => ! [Z: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                       => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
             => ! [Y3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                 => ! [Z: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
               => ! [Y3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                   => ! [Z: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y5 @ Z4 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Z4 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_s_local
thf(fact_839_trans__mult2__alt__ns__s__local,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Z4: multis6850599868085929325on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
         => ! [Y3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
             => ! [Z: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
           => ! [Y3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
               => ! [Z: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
             => ! [Y3: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                 => ! [Z: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
               => ! [Y3: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                   => ! [Z: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $true @ Ns @ S ) )
             => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Y5 @ Z4 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) )
               => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Z4 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_s_local
thf(fact_840_trans__mult2__alt__ns__s__local,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Z4: multis5168333519641040071on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
         => ! [Y3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
             => ! [Z: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
           => ! [Y3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
               => ! [Z: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
             => ! [Y3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                 => ! [Z: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
               => ! [Y3: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                   => ! [Z: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $true @ Ns @ S ) )
             => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Y5 @ Z4 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) )
               => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Z4 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_s_local
thf(fact_841_trans__mult2__alt__s__ns__local,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Z4: multis2912115315702009697on_nat,S: set_Pr1446391457923625415on_nat,Ns: set_Pr1446391457923625415on_nat] :
      ( ! [X3: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
         => ! [Y3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
             => ! [Z: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                 => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                     => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
           => ! [Y3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
               => ! [Z: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                       => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
             => ! [Y3: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                 => ! [Z: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                         => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
               => ! [Y3: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                   => ! [Z: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                           => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) )
             => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ Y5 @ Z4 ) @ ( multis5731881090987732783on_nat @ $true @ Ns @ S ) )
               => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Z4 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_ns_local
thf(fact_842_trans__mult2__alt__s__ns__local,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
         => ! [Y3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
             => ! [Z: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
           => ! [Y3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
               => ! [Z: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                       => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
             => ! [Y3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                 => ! [Z: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
               => ! [Y3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                   => ! [Z: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y5 @ Z4 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Z4 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_ns_local
thf(fact_843_trans__mult2__alt__s__ns__local,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Z4: multis6850599868085929325on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
         => ! [Y3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
             => ! [Z: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
           => ! [Y3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
               => ! [Z: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
             => ! [Y3: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                 => ! [Z: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
               => ! [Y3: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                   => ! [Z: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) )
             => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Y5 @ Z4 ) @ ( multis394508389068283657on_nat @ $true @ Ns @ S ) )
               => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Z4 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_ns_local
thf(fact_844_trans__mult2__alt__s__ns__local,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Z4: multis5168333519641040071on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
         => ! [Y3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
             => ! [Z: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
           => ! [Y3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
               => ! [Z: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
             => ! [Y3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                 => ! [Z: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
               => ! [Y3: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                   => ! [Z: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) )
             => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Y5 @ Z4 ) @ ( multis1733271916442156053on_nat @ $true @ Ns @ S ) )
               => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Z4 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_s_ns_local
thf(fact_845_trans__mult2__alt__ns__ns__local,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Z4: multis2912115315702009697on_nat,S: set_Pr1446391457923625415on_nat,Ns: set_Pr1446391457923625415on_nat] :
      ( ! [X3: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
         => ! [Y3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
             => ! [Z: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                 => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                     => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
           => ! [Y3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
               => ! [Z: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                   => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ S )
                       => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
             => ! [Y3: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                 => ! [Z: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                     => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                         => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X ) )
               => ! [Y3: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y5 ) )
                   => ! [Z: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ Z @ ( set_ms7239842528327418072on_nat @ Z4 ) )
                       => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ Y3 @ Z ) @ Ns )
                           => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $true @ Ns @ S ) )
             => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ Y5 @ Z4 ) @ ( multis5731881090987732783on_nat @ $true @ Ns @ S ) )
               => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Z4 ) @ ( multis5731881090987732783on_nat @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_ns_local
thf(fact_846_trans__mult2__alt__ns__ns__local,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
         => ! [Y3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
             => ! [Z: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
           => ! [Y3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
               => ! [Z: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                   => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ S )
                       => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
             => ! [Y3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                 => ! [Z: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X ) )
               => ! [Y3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y5 ) )
                   => ! [Z: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ Z @ ( set_ms3922534532180125136on_nat @ Z4 ) )
                       => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Y3 @ Z ) @ Ns )
                           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y5 @ Z4 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Z4 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_ns_local
thf(fact_847_trans__mult2__alt__ns__ns__local,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Z4: multis6850599868085929325on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ! [X3: multis1856558993800614429on_nat] :
          ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
         => ! [Y3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
             => ! [Z: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
           => ! [Y3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
               => ! [Z: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                   => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ S )
                       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
             => ! [Y3: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                 => ! [Z: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X ) )
               => ! [Y3: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y5 ) )
                   => ! [Z: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ Z @ ( set_ms2805207930803453472on_nat @ Z4 ) )
                       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $true @ Ns @ S ) )
             => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Y5 @ Z4 ) @ ( multis394508389068283657on_nat @ $true @ Ns @ S ) )
               => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Z4 ) @ ( multis394508389068283657on_nat @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_ns_local
thf(fact_848_trans__mult2__alt__ns__ns__local,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Z4: multis5168333519641040071on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat] :
          ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
         => ! [Y3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
             => ! [Z: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
     => ( ! [X3: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
           => ! [Y3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
               => ! [Z: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                   => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ S )
                       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
       => ( ! [X3: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
             => ! [Y3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                 => ! [Z: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ S ) ) ) ) ) )
         => ( ! [X3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X ) )
               => ! [Y3: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y5 ) )
                   => ! [Z: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ Z @ ( set_ms6185292591494683198on_nat @ Z4 ) )
                       => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                         => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ Ns )
                           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Z ) @ Ns ) ) ) ) ) )
           => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $true @ Ns @ S ) )
             => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Y5 @ Z4 ) @ ( multis1733271916442156053on_nat @ $true @ Ns @ S ) )
               => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Z4 ) @ ( multis1733271916442156053on_nat @ $true @ Ns @ S ) ) ) ) ) ) ) ) ).

% trans_mult2_alt_ns_ns_local
thf(fact_849_mult2__alt__nsI,axiom,
    ! [X: multis2912115315702009697on_nat,X12: multis2912115315702009697on_nat,X23: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Y13: multis2912115315702009697on_nat,Y23: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( X
        = ( plus_p7340530524814416408on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p7340530524814416408on_nat @ Y13 @ Y23 ) )
       => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X12 @ Y13 ) @ ( multis5394615859312135327on_nat @ Ns ) )
         => ( ! [X3: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X23 ) )
               => ? [Y7: produc9176241080893687067on_nat] :
                    ( ( member4278285197320822468on_nat @ Y7 @ ( set_ms7239842528327418072on_nat @ Y23 ) )
                    & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y7 ) @ S ) ) )
           => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $true @ Ns @ S ) ) ) ) ) ) ).

% mult2_alt_nsI
thf(fact_850_mult2__alt__nsI,axiom,
    ! [X: multis6850599868085929325on_nat,X12: multis6850599868085929325on_nat,X23: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Y13: multis6850599868085929325on_nat,Y23: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( X
        = ( plus_p4342429221931815414on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p4342429221931815414on_nat @ Y13 @ Y23 ) )
       => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X12 @ Y13 ) @ ( multis1913027631269000601on_nat @ Ns ) )
         => ( ! [X3: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X23 ) )
               => ? [Y7: multis1856558993800614429on_nat] :
                    ( ( member2988082745645132084on_nat @ Y7 @ ( set_ms2805207930803453472on_nat @ Y23 ) )
                    & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y7 ) @ S ) ) )
           => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $true @ Ns @ S ) ) ) ) ) ) ).

% mult2_alt_nsI
thf(fact_851_mult2__alt__nsI,axiom,
    ! [X: multis5168333519641040071on_nat,X12: multis5168333519641040071on_nat,X23: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Y13: multis5168333519641040071on_nat,Y23: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( X
        = ( plus_p7957816880051667710on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p7957816880051667710on_nat @ Y13 @ Y23 ) )
       => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X12 @ Y13 ) @ ( multis4607348483316519301on_nat @ Ns ) )
         => ( ! [X3: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X23 ) )
               => ? [Y7: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Y7 @ ( set_ms6185292591494683198on_nat @ Y23 ) )
                    & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y7 ) @ S ) ) )
           => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $true @ Ns @ S ) ) ) ) ) ) ).

% mult2_alt_nsI
thf(fact_852_mult2__alt__nsI,axiom,
    ! [X: multis1856558993800614429on_nat,X12: multis1856558993800614429on_nat,X23: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( X
        = ( plus_p2586647234311342374on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) )
       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis636354481432518601on_nat @ Ns ) )
         => ( ! [X3: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X23 ) )
               => ? [Y7: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y7 @ ( set_ms3922534532180125136on_nat @ Y23 ) )
                    & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y7 ) @ S ) ) )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) ) ) ) ) ) ).

% mult2_alt_nsI
thf(fact_853_mult2__alt__nsE,axiom,
    ! [X: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $true @ Ns @ S ) )
     => ~ ! [X13: multis2912115315702009697on_nat,X24: multis2912115315702009697on_nat] :
            ( ( X
              = ( plus_p7340530524814416408on_nat @ X13 @ X24 ) )
           => ! [Y12: multis2912115315702009697on_nat,Y22: multis2912115315702009697on_nat] :
                ( ( Y5
                  = ( plus_p7340530524814416408on_nat @ Y12 @ Y22 ) )
               => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X13 @ Y12 ) @ ( multis5394615859312135327on_nat @ Ns ) )
                 => ~ ! [X6: produc9176241080893687067on_nat] :
                        ( ( member4278285197320822468on_nat @ X6 @ ( set_ms7239842528327418072on_nat @ X24 ) )
                       => ? [Y3: produc9176241080893687067on_nat] :
                            ( ( member4278285197320822468on_nat @ Y3 @ ( set_ms7239842528327418072on_nat @ Y22 ) )
                            & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).

% mult2_alt_nsE
thf(fact_854_mult2__alt__nsE,axiom,
    ! [X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $true @ Ns @ S ) )
     => ~ ! [X13: multis6850599868085929325on_nat,X24: multis6850599868085929325on_nat] :
            ( ( X
              = ( plus_p4342429221931815414on_nat @ X13 @ X24 ) )
           => ! [Y12: multis6850599868085929325on_nat,Y22: multis6850599868085929325on_nat] :
                ( ( Y5
                  = ( plus_p4342429221931815414on_nat @ Y12 @ Y22 ) )
               => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X13 @ Y12 ) @ ( multis1913027631269000601on_nat @ Ns ) )
                 => ~ ! [X6: multis1856558993800614429on_nat] :
                        ( ( member2988082745645132084on_nat @ X6 @ ( set_ms2805207930803453472on_nat @ X24 ) )
                       => ? [Y3: multis1856558993800614429on_nat] :
                            ( ( member2988082745645132084on_nat @ Y3 @ ( set_ms2805207930803453472on_nat @ Y22 ) )
                            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).

% mult2_alt_nsE
thf(fact_855_mult2__alt__nsE,axiom,
    ! [X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $true @ Ns @ S ) )
     => ~ ! [X13: multis5168333519641040071on_nat,X24: multis5168333519641040071on_nat] :
            ( ( X
              = ( plus_p7957816880051667710on_nat @ X13 @ X24 ) )
           => ! [Y12: multis5168333519641040071on_nat,Y22: multis5168333519641040071on_nat] :
                ( ( Y5
                  = ( plus_p7957816880051667710on_nat @ Y12 @ Y22 ) )
               => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X13 @ Y12 ) @ ( multis4607348483316519301on_nat @ Ns ) )
                 => ~ ! [X6: list_P9127870862658512897on_nat] :
                        ( ( member783415501022416426on_nat @ X6 @ ( set_ms6185292591494683198on_nat @ X24 ) )
                       => ? [Y3: list_P9127870862658512897on_nat] :
                            ( ( member783415501022416426on_nat @ Y3 @ ( set_ms6185292591494683198on_nat @ Y22 ) )
                            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).

% mult2_alt_nsE
thf(fact_856_mult2__alt__nsE,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $true @ Ns @ S ) )
     => ~ ! [X13: multis1856558993800614429on_nat,X24: multis1856558993800614429on_nat] :
            ( ( X
              = ( plus_p2586647234311342374on_nat @ X13 @ X24 ) )
           => ! [Y12: multis1856558993800614429on_nat,Y22: multis1856558993800614429on_nat] :
                ( ( Y5
                  = ( plus_p2586647234311342374on_nat @ Y12 @ Y22 ) )
               => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X13 @ Y12 ) @ ( multis636354481432518601on_nat @ Ns ) )
                 => ~ ! [X6: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ X6 @ ( set_ms3922534532180125136on_nat @ X24 ) )
                       => ? [Y3: sum_su4907927656612468301on_nat] :
                            ( ( member8476201367316484836on_nat @ Y3 @ ( set_ms3922534532180125136on_nat @ Y22 ) )
                            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X6 @ Y3 ) @ S ) ) ) ) ) ) ) ).

% mult2_alt_nsE
thf(fact_857_mult2__alt__sI,axiom,
    ! [X: multis2912115315702009697on_nat,X12: multis2912115315702009697on_nat,X23: multis2912115315702009697on_nat,Y5: multis2912115315702009697on_nat,Y13: multis2912115315702009697on_nat,Y23: multis2912115315702009697on_nat,Ns: set_Pr1446391457923625415on_nat,S: set_Pr1446391457923625415on_nat] :
      ( ( X
        = ( plus_p7340530524814416408on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p7340530524814416408on_nat @ Y13 @ Y23 ) )
       => ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X12 @ Y13 ) @ ( multis5394615859312135327on_nat @ Ns ) )
         => ( ( Y23 != zero_z1502369238205801624on_nat )
           => ( ! [X3: produc9176241080893687067on_nat] :
                  ( ( member4278285197320822468on_nat @ X3 @ ( set_ms7239842528327418072on_nat @ X23 ) )
                 => ? [Y7: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ Y7 @ ( set_ms7239842528327418072on_nat @ Y23 ) )
                      & ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ X @ Y5 ) @ ( multis5731881090987732783on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_alt_sI
thf(fact_858_mult2__alt__sI,axiom,
    ! [X: multis6850599868085929325on_nat,X12: multis6850599868085929325on_nat,X23: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Y13: multis6850599868085929325on_nat,Y23: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( X
        = ( plus_p4342429221931815414on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p4342429221931815414on_nat @ Y13 @ Y23 ) )
       => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X12 @ Y13 ) @ ( multis1913027631269000601on_nat @ Ns ) )
         => ( ( Y23 != zero_z4144889538238539638on_nat )
           => ( ! [X3: multis1856558993800614429on_nat] :
                  ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ X23 ) )
                 => ? [Y7: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ Y7 @ ( set_ms2805207930803453472on_nat @ Y23 ) )
                      & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_alt_sI
thf(fact_859_mult2__alt__sI,axiom,
    ! [X: multis5168333519641040071on_nat,X12: multis5168333519641040071on_nat,X23: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Y13: multis5168333519641040071on_nat,Y23: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( X
        = ( plus_p7957816880051667710on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p7957816880051667710on_nat @ Y13 @ Y23 ) )
       => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X12 @ Y13 ) @ ( multis4607348483316519301on_nat @ Ns ) )
         => ( ( Y23 != zero_z8385993896842825086on_nat )
           => ( ! [X3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ X23 ) )
                 => ? [Y7: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Y7 @ ( set_ms6185292591494683198on_nat @ Y23 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_alt_sI
thf(fact_860_mult2__alt__sI,axiom,
    ! [X: multis1856558993800614429on_nat,X12: multis1856558993800614429on_nat,X23: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Y13: multis1856558993800614429on_nat,Y23: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( X
        = ( plus_p2586647234311342374on_nat @ X12 @ X23 ) )
     => ( ( Y5
          = ( plus_p2586647234311342374on_nat @ Y13 @ Y23 ) )
       => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X12 @ Y13 ) @ ( multis636354481432518601on_nat @ Ns ) )
         => ( ( Y23 != zero_z8702567803361615526on_nat )
           => ( ! [X3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ X23 ) )
                 => ? [Y7: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Y7 @ ( set_ms3922534532180125136on_nat @ Y23 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y7 ) @ S ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ) ) ) ) ).

% mult2_alt_sI
thf(fact_861_listrel__iff__nth,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( listre2938678831095920429on_nat @ R ) )
      = ( ( ( size_s6603137509925400257on_nat @ Xs )
          = ( size_s6603137509925400257on_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s6603137509925400257on_nat @ Xs ) )
           => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs @ N2 ) @ ( nth_mu8151576519813129558on_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_862_listrel__iff__nth,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat,R: set_Pr6244131468446592861on_nat] :
      ( ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ Xs @ Ys ) @ ( listre1567628990915239951on_nat @ R ) )
      = ( ( ( size_s650155532264187265on_nat @ Xs )
          = ( size_s2647645505745046499on_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s650155532264187265on_nat @ Xs ) )
           => ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ ( nth_li4091121329102240790on_nat @ Xs @ N2 ) @ ( nth_Pr1046342761279222648on_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_863_listrel__iff__nth,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat,R: set_Pr737724919871395061on_nat] :
      ( ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ Xs @ Ys ) @ ( listre1486203369474292887on_nat @ R ) )
      = ( ( ( size_s650155532264187265on_nat @ Xs )
          = ( size_s4814362452863045593on_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s650155532264187265on_nat @ Xs ) )
           => ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ ( nth_li4091121329102240790on_nat @ Xs @ N2 ) @ ( nth_Pr2007431530836016110on_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_864_listrel__iff__nth,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( listre5026495727120537649on_nat @ R ) )
      = ( ( ( size_s129639936549468787on_nat @ Xs )
          = ( size_s129639936549468787on_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s129639936549468787on_nat @ Xs ) )
           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ N2 ) @ ( nth_li1302981970230039432on_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_865_listrel__iff__nth,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R ) )
      = ( ( ( size_s8208510060688613859at_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s8208510060688613859at_nat @ Xs ) )
           => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_866_listrel__iff__nth,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre6301251474747857073on_nat @ R ) )
      = ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s7490768528479217773on_nat @ Xs ) )
           => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ N2 ) @ ( nth_Pr3336893728888831618on_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_867_one__step__implies__mult,axiom,
    ! [J2: multis1856558993800614429on_nat,K5: multis1856558993800614429on_nat,R: set_Pr2535564044409765851on_nat,I5: multis1856558993800614429on_nat] :
      ( ( J2 != zero_z8702567803361615526on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ K5 ) )
           => ? [Xa: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ Xa @ ( set_ms3922534532180125136on_nat @ J2 ) )
                & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Xa ) @ R ) ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ I5 @ K5 ) @ ( plus_p2586647234311342374on_nat @ I5 @ J2 ) ) @ ( mult_S8265718338682012467on_nat @ R ) ) ) ) ).

% one_step_implies_mult
thf(fact_868_one__step__implies__mult,axiom,
    ! [J2: multis6850599868085929325on_nat,K5: multis6850599868085929325on_nat,R: set_Pr3573155178515677307on_nat,I5: multis6850599868085929325on_nat] :
      ( ( J2 != zero_z4144889538238539638on_nat )
     => ( ! [X3: multis1856558993800614429on_nat] :
            ( ( member2988082745645132084on_nat @ X3 @ ( set_ms2805207930803453472on_nat @ K5 ) )
           => ? [Xa: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ Xa @ ( set_ms2805207930803453472on_nat @ J2 ) )
                & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Xa ) @ R ) ) )
       => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( plus_p4342429221931815414on_nat @ I5 @ K5 ) @ ( plus_p4342429221931815414on_nat @ I5 @ J2 ) ) @ ( mult_m3495053697227556867on_nat @ R ) ) ) ) ).

% one_step_implies_mult
thf(fact_869_one__step__implies__mult,axiom,
    ! [J2: multis5168333519641040071on_nat,K5: multis5168333519641040071on_nat,R: set_Pr8586234676504894535on_nat,I5: multis5168333519641040071on_nat] :
      ( ( J2 != zero_z8385993896842825086on_nat )
     => ( ! [X3: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X3 @ ( set_ms6185292591494683198on_nat @ K5 ) )
           => ? [Xa: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ Xa @ ( set_ms6185292591494683198on_nat @ J2 ) )
                & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Xa ) @ R ) ) )
       => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( plus_p7957816880051667710on_nat @ I5 @ K5 ) @ ( plus_p7957816880051667710on_nat @ I5 @ J2 ) ) @ ( mult_l6788743291290228379on_nat @ R ) ) ) ) ).

% one_step_implies_mult
thf(fact_870_lex__take__index,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( lex_mu3406824806971463219on_nat @ R ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Ys ) )
             => ( ( ( take_m329436973745802671on_nat @ I @ Xs )
                  = ( take_m329436973745802671on_nat @ I @ Ys ) )
               => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ Xs @ I ) @ ( nth_mu8151576519813129558on_nat @ Ys @ I ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_871_lex__take__index,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( lex_li6505813899493256299on_nat @ R ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Ys ) )
             => ( ( ( take_l5410376112614949359on_nat @ I @ Xs )
                  = ( take_l5410376112614949359on_nat @ I @ Ys ) )
               => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ I ) @ ( nth_li1302981970230039432on_nat @ Ys @ I ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_872_lex__take__index,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( lex_Pr6276626268559582437on_nat @ R ) )
     => ~ ! [I: nat] :
            ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs ) )
           => ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Ys ) )
             => ( ( ( take_P4655611586102217833on_nat @ I @ Xs )
                  = ( take_P4655611586102217833on_nat @ I @ Ys ) )
               => ~ ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I ) @ ( nth_Pr3336893728888831618on_nat @ Ys @ I ) ) @ R ) ) ) ) ) ).

% lex_take_index
thf(fact_873_take__eq__Nil2,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( nil_li1372968520647675725od_a_o
        = ( take_l5436743365630562101od_a_o @ N @ Xs ) )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_li1372968520647675725od_a_o ) ) ) ).

% take_eq_Nil2
thf(fact_874_take__eq__Nil,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( take_l5436743365630562101od_a_o @ N @ Xs )
        = nil_li1372968520647675725od_a_o )
      = ( ( N = zero_zero_nat )
        | ( Xs = nil_li1372968520647675725od_a_o ) ) ) ).

% take_eq_Nil
thf(fact_875_take0,axiom,
    ( ( take_l5436743365630562101od_a_o @ zero_zero_nat )
    = ( ^ [Xs3: list_l8336264072933094989od_a_o] : nil_li1372968520647675725od_a_o ) ) ).

% take0
thf(fact_876_take__Nil,axiom,
    ! [N: nat] :
      ( ( take_l5436743365630562101od_a_o @ N @ nil_li1372968520647675725od_a_o )
      = nil_li1372968520647675725od_a_o ) ).

% take_Nil
thf(fact_877_in__set__takeD,axiom,
    ! [X4: produc9176241080893687067on_nat,N: nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( take_P8771938209668683785on_nat @ N @ Xs ) ) )
     => ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_878_in__set__takeD,axiom,
    ! [X4: produc1313295348606329723on_nat,N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) ) )
     => ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_879_in__set__takeD,axiom,
    ! [X4: sum_su4907927656612468301on_nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ N @ Xs ) ) )
     => ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% in_set_takeD
thf(fact_880_take__0,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( take_l5436743365630562101od_a_o @ zero_zero_nat @ Xs )
      = nil_li1372968520647675725od_a_o ) ).

% take_0
thf(fact_881_listrel__Nil2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ nil_li1372968520647675725od_a_o ) @ ( listre9037316752283237297od_a_o @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil2
thf(fact_882_listrel__Nil2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ nil_Pr8614093721167239297on_nat ) @ ( listre6301251474747857073on_nat @ R ) )
     => ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% listrel_Nil2
thf(fact_883_listrel__Nil1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Xs ) @ ( listre9037316752283237297od_a_o @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil1
thf(fact_884_listrel__Nil1,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ Xs ) @ ( listre6301251474747857073on_nat @ R ) )
     => ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% listrel_Nil1
thf(fact_885_listrel_ONil,axiom,
    ! [R: set_Pr7420778282784443591od_a_o] : ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o ) @ ( listre9037316752283237297od_a_o @ R ) ) ).

% listrel.Nil
thf(fact_886_listrel_ONil,axiom,
    ! [R: set_Pr8202701763255893447on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ nil_Pr8614093721167239297on_nat ) @ ( listre6301251474747857073on_nat @ R ) ) ).

% listrel.Nil
thf(fact_887_listrel__eq__len,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre6301251474747857073on_nat @ R ) )
     => ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_888_listrel__Cons2,axiom,
    ! [Xs: list_m351779702951725613on_nat,Y: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) ) @ ( listre2938678831095920429on_nat @ R ) )
     => ~ ! [X3: multis1856558993800614429on_nat,Xs2: list_m351779702951725613on_nat] :
            ( ( Xs
              = ( cons_m1493500683067190247on_nat @ X3 @ Xs2 ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y ) @ R )
             => ~ ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs2 @ Ys ) @ ( listre2938678831095920429on_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_889_listrel__Cons2,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Y: produc118098540238352959on_nat,Ys: list_P5508476027986415439on_nat,R: set_Pr6244131468446592861on_nat] :
      ( ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ Xs @ ( cons_P4518942764240009225on_nat @ Y @ Ys ) ) @ ( listre1567628990915239951on_nat @ R ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs2: list_l8999537460041914605on_nat] :
            ( ( Xs
              = ( cons_l5857885737047467175on_nat @ X3 @ Xs2 ) )
           => ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y ) @ R )
             => ~ ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ Xs2 @ Ys ) @ ( listre1567628990915239951on_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_890_listrel__Cons2,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Y: produc1714117091004681831on_nat,Ys: list_P8734936519038225773on_nat,R: set_Pr737724919871395061on_nat] :
      ( ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ Xs @ ( cons_P2382207818294396957on_nat @ Y @ Ys ) ) @ ( listre1486203369474292887on_nat @ R ) )
     => ~ ! [X3: list_S5253404628753938141on_nat,Xs2: list_l8999537460041914605on_nat] :
            ( ( Xs
              = ( cons_l5857885737047467175on_nat @ X3 @ Xs2 ) )
           => ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y ) @ R )
             => ~ ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ Xs2 @ Ys ) @ ( listre1486203369474292887on_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_891_listrel__Cons2,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) ) @ ( listre5026495727120537649on_nat @ R ) )
     => ~ ! [X3: list_P9127870862658512897on_nat,Xs2: list_l1726306699111145479on_nat] :
            ( ( Xs
              = ( cons_l3028799188195208631on_nat @ X3 @ Xs2 ) )
           => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ R )
             => ~ ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs2 @ Ys ) @ ( listre5026495727120537649on_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_892_listrel__Cons2,axiom,
    ! [Xs: list_nat_nat,Y: nat,Ys: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R ) )
     => ~ ! [X3: nat > nat,Xs2: list_nat_nat] :
            ( ( Xs
              = ( cons_nat_nat @ X3 @ Xs2 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y ) @ R )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs2 @ Ys ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_893_listrel__Cons2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) @ ( listre6301251474747857073on_nat @ R ) )
     => ~ ! [X3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( cons_P1582596238967110705on_nat @ X3 @ Xs2 ) )
           => ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ Y ) @ R )
             => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs2 @ Ys ) @ ( listre6301251474747857073on_nat @ R ) ) ) ) ) ).

% listrel_Cons2
thf(fact_894_listrel__Cons1,axiom,
    ! [Y: multis1856558993800614429on_nat,Ys: list_m351779702951725613on_nat,Xs: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) @ Xs ) @ ( listre2938678831095920429on_nat @ R ) )
     => ~ ! [Y3: multis1856558993800614429on_nat,Ys3: list_m351779702951725613on_nat] :
            ( ( Xs
              = ( cons_m1493500683067190247on_nat @ Y3 @ Ys3 ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y @ Y3 ) @ R )
             => ~ ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Ys @ Ys3 ) @ ( listre2938678831095920429on_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_895_listrel__Cons1,axiom,
    ! [Y: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat,Xs: list_P5508476027986415439on_nat,R: set_Pr6244131468446592861on_nat] :
      ( ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ ( cons_l5857885737047467175on_nat @ Y @ Ys ) @ Xs ) @ ( listre1567628990915239951on_nat @ R ) )
     => ~ ! [Y3: produc118098540238352959on_nat,Ys3: list_P5508476027986415439on_nat] :
            ( ( Xs
              = ( cons_P4518942764240009225on_nat @ Y3 @ Ys3 ) )
           => ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ Y @ Y3 ) @ R )
             => ~ ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ Ys @ Ys3 ) @ ( listre1567628990915239951on_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_896_listrel__Cons1,axiom,
    ! [Y: list_S5253404628753938141on_nat,Ys: list_l8999537460041914605on_nat,Xs: list_P8734936519038225773on_nat,R: set_Pr737724919871395061on_nat] :
      ( ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ ( cons_l5857885737047467175on_nat @ Y @ Ys ) @ Xs ) @ ( listre1486203369474292887on_nat @ R ) )
     => ~ ! [Y3: produc1714117091004681831on_nat,Ys3: list_P8734936519038225773on_nat] :
            ( ( Xs
              = ( cons_P2382207818294396957on_nat @ Y3 @ Ys3 ) )
           => ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ Y @ Y3 ) @ R )
             => ~ ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ Ys @ Ys3 ) @ ( listre1486203369474292887on_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_897_listrel__Cons1,axiom,
    ! [Y: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,Xs: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) @ Xs ) @ ( listre5026495727120537649on_nat @ R ) )
     => ~ ! [Y3: list_P9127870862658512897on_nat,Ys3: list_l1726306699111145479on_nat] :
            ( ( Xs
              = ( cons_l3028799188195208631on_nat @ Y3 @ Ys3 ) )
           => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y @ Y3 ) @ R )
             => ~ ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Ys @ Ys3 ) @ ( listre5026495727120537649on_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_898_listrel__Cons1,axiom,
    ! [Y: nat > nat,Ys: list_nat_nat,Xs: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ Y @ Ys ) @ Xs ) @ ( listrel_nat_nat_nat @ R ) )
     => ~ ! [Y3: nat,Ys3: list_nat] :
            ( ( Xs
              = ( cons_nat @ Y3 @ Ys3 ) )
           => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ Y @ Y3 ) @ R )
             => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Ys @ Ys3 ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_899_listrel__Cons1,axiom,
    ! [Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) @ Xs ) @ ( listre6301251474747857073on_nat @ R ) )
     => ~ ! [Y3: produc1313295348606329723on_nat,Ys3: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( cons_P1582596238967110705on_nat @ Y3 @ Ys3 ) )
           => ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ Y @ Y3 ) @ R )
             => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ys @ Ys3 ) @ ( listre6301251474747857073on_nat @ R ) ) ) ) ) ).

% listrel_Cons1
thf(fact_900_listrel_OCons,axiom,
    ! [X4: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat,R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ R )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre6301251474747857073on_nat @ R ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) @ ( listre6301251474747857073on_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_901_listrel_OCons,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ R )
     => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( listre2938678831095920429on_nat @ R ) )
       => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ X4 @ Xs ) @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) ) @ ( listre2938678831095920429on_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_902_listrel_OCons,axiom,
    ! [X4: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat,R: set_Pr6244131468446592861on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y ) @ R )
     => ( ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ Xs @ Ys ) @ ( listre1567628990915239951on_nat @ R ) )
       => ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ ( cons_l5857885737047467175on_nat @ X4 @ Xs ) @ ( cons_P4518942764240009225on_nat @ Y @ Ys ) ) @ ( listre1567628990915239951on_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_903_listrel_OCons,axiom,
    ! [X4: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat,R: set_Pr737724919871395061on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X4 @ Y ) @ R )
     => ( ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ Xs @ Ys ) @ ( listre1486203369474292887on_nat @ R ) )
       => ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ ( cons_l5857885737047467175on_nat @ X4 @ Xs ) @ ( cons_P2382207818294396957on_nat @ Y @ Ys ) ) @ ( listre1486203369474292887on_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_904_listrel_OCons,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ R )
     => ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( listre5026495727120537649on_nat @ R ) )
       => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X4 @ Xs ) @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) ) @ ( listre5026495727120537649on_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_905_listrel_OCons,axiom,
    ! [X4: nat > nat,Y: nat,R: set_Pr9093778441882193744at_nat,Xs: list_nat_nat,Ys: list_nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y ) @ R )
     => ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R ) )
       => ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ ( cons_nat_nat @ X4 @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ).

% listrel.Cons
thf(fact_906_listrel_Osimps,axiom,
    ! [A13: list_l8336264072933094989od_a_o,A24: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ A13 @ A24 ) @ ( listre9037316752283237297od_a_o @ R ) )
      = ( ( ( A13 = nil_li1372968520647675725od_a_o )
          & ( A24 = nil_li1372968520647675725od_a_o ) )
        | ? [X5: list_P3196869577824546119od_a_o,Y4: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Ys4: list_l8336264072933094989od_a_o] :
            ( ( A13
              = ( cons_l9179691953475787517od_a_o @ X5 @ Xs3 ) )
            & ( A24
              = ( cons_l9179691953475787517od_a_o @ Y4 @ Ys4 ) )
            & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X5 @ Y4 ) @ R )
            & ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs3 @ Ys4 ) @ ( listre9037316752283237297od_a_o @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_907_listrel_Osimps,axiom,
    ! [A13: list_m351779702951725613on_nat,A24: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ A13 @ A24 ) @ ( listre2938678831095920429on_nat @ R ) )
      = ( ( ( A13 = nil_mu4837018292623784855on_nat )
          & ( A24 = nil_mu4837018292623784855on_nat ) )
        | ? [X5: multis1856558993800614429on_nat,Y4: multis1856558993800614429on_nat,Xs3: list_m351779702951725613on_nat,Ys4: list_m351779702951725613on_nat] :
            ( ( A13
              = ( cons_m1493500683067190247on_nat @ X5 @ Xs3 ) )
            & ( A24
              = ( cons_m1493500683067190247on_nat @ Y4 @ Ys4 ) )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X5 @ Y4 ) @ R )
            & ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs3 @ Ys4 ) @ ( listre2938678831095920429on_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_908_listrel_Osimps,axiom,
    ! [A13: list_l8999537460041914605on_nat,A24: list_P5508476027986415439on_nat,R: set_Pr6244131468446592861on_nat] :
      ( ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ A13 @ A24 ) @ ( listre1567628990915239951on_nat @ R ) )
      = ( ( ( A13 = nil_li5563851662744376407on_nat )
          & ( A24 = nil_Pr2927124501823088569on_nat ) )
        | ? [X5: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat,Xs3: list_l8999537460041914605on_nat,Ys4: list_P5508476027986415439on_nat] :
            ( ( A13
              = ( cons_l5857885737047467175on_nat @ X5 @ Xs3 ) )
            & ( A24
              = ( cons_P4518942764240009225on_nat @ Y4 @ Ys4 ) )
            & ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X5 @ Y4 ) @ R )
            & ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ Xs3 @ Ys4 ) @ ( listre1567628990915239951on_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_909_listrel_Osimps,axiom,
    ! [A13: list_l8999537460041914605on_nat,A24: list_P8734936519038225773on_nat,R: set_Pr737724919871395061on_nat] :
      ( ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ A13 @ A24 ) @ ( listre1486203369474292887on_nat @ R ) )
      = ( ( ( A13 = nil_li5563851662744376407on_nat )
          & ( A24 = nil_Pr707104663622130797on_nat ) )
        | ? [X5: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat,Xs3: list_l8999537460041914605on_nat,Ys4: list_P8734936519038225773on_nat] :
            ( ( A13
              = ( cons_l5857885737047467175on_nat @ X5 @ Xs3 ) )
            & ( A24
              = ( cons_P2382207818294396957on_nat @ Y4 @ Ys4 ) )
            & ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X5 @ Y4 ) @ R )
            & ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ Xs3 @ Ys4 ) @ ( listre1486203369474292887on_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_910_listrel_Osimps,axiom,
    ! [A13: list_l1726306699111145479on_nat,A24: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ A13 @ A24 ) @ ( listre5026495727120537649on_nat @ R ) )
      = ( ( ( A13 = nil_li2682182165152866823on_nat )
          & ( A24 = nil_li2682182165152866823on_nat ) )
        | ? [X5: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat,Xs3: list_l1726306699111145479on_nat,Ys4: list_l1726306699111145479on_nat] :
            ( ( A13
              = ( cons_l3028799188195208631on_nat @ X5 @ Xs3 ) )
            & ( A24
              = ( cons_l3028799188195208631on_nat @ Y4 @ Ys4 ) )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X5 @ Y4 ) @ R )
            & ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs3 @ Ys4 ) @ ( listre5026495727120537649on_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_911_listrel_Osimps,axiom,
    ! [A13: list_nat_nat,A24: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A13 @ A24 ) @ ( listrel_nat_nat_nat @ R ) )
      = ( ( ( A13 = nil_nat_nat )
          & ( A24 = nil_nat ) )
        | ? [X5: nat > nat,Y4: nat,Xs3: list_nat_nat,Ys4: list_nat] :
            ( ( A13
              = ( cons_nat_nat @ X5 @ Xs3 ) )
            & ( A24
              = ( cons_nat @ Y4 @ Ys4 ) )
            & ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X5 @ Y4 ) @ R )
            & ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs3 @ Ys4 ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_912_listrel_Osimps,axiom,
    ! [A13: list_P9127870862658512897on_nat,A24: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A13 @ A24 ) @ ( listre6301251474747857073on_nat @ R ) )
      = ( ( ( A13 = nil_Pr8614093721167239297on_nat )
          & ( A24 = nil_Pr8614093721167239297on_nat ) )
        | ? [X5: produc1313295348606329723on_nat,Y4: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Ys4: list_P9127870862658512897on_nat] :
            ( ( A13
              = ( cons_P1582596238967110705on_nat @ X5 @ Xs3 ) )
            & ( A24
              = ( cons_P1582596238967110705on_nat @ Y4 @ Ys4 ) )
            & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X5 @ Y4 ) @ R )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs3 @ Ys4 ) @ ( listre6301251474747857073on_nat @ R ) ) ) ) ) ).

% listrel.simps
thf(fact_913_listrel_Ocases,axiom,
    ! [A13: list_l8336264072933094989od_a_o,A24: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ A13 @ A24 ) @ ( listre9037316752283237297od_a_o @ R ) )
     => ( ( ( A13 = nil_li1372968520647675725od_a_o )
         => ( A24 != nil_li1372968520647675725od_a_o ) )
       => ~ ! [X3: list_P3196869577824546119od_a_o,Y3: list_P3196869577824546119od_a_o,Xs2: list_l8336264072933094989od_a_o] :
              ( ( A13
                = ( cons_l9179691953475787517od_a_o @ X3 @ Xs2 ) )
             => ! [Ys3: list_l8336264072933094989od_a_o] :
                  ( ( A24
                    = ( cons_l9179691953475787517od_a_o @ Y3 @ Ys3 ) )
                 => ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ Y3 ) @ R )
                   => ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs2 @ Ys3 ) @ ( listre9037316752283237297od_a_o @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_914_listrel_Ocases,axiom,
    ! [A13: list_m351779702951725613on_nat,A24: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ A13 @ A24 ) @ ( listre2938678831095920429on_nat @ R ) )
     => ( ( ( A13 = nil_mu4837018292623784855on_nat )
         => ( A24 != nil_mu4837018292623784855on_nat ) )
       => ~ ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat,Xs2: list_m351779702951725613on_nat] :
              ( ( A13
                = ( cons_m1493500683067190247on_nat @ X3 @ Xs2 ) )
             => ! [Ys3: list_m351779702951725613on_nat] :
                  ( ( A24
                    = ( cons_m1493500683067190247on_nat @ Y3 @ Ys3 ) )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs2 @ Ys3 ) @ ( listre2938678831095920429on_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_915_listrel_Ocases,axiom,
    ! [A13: list_l8999537460041914605on_nat,A24: list_P5508476027986415439on_nat,R: set_Pr6244131468446592861on_nat] :
      ( ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ A13 @ A24 ) @ ( listre1567628990915239951on_nat @ R ) )
     => ( ( ( A13 = nil_li5563851662744376407on_nat )
         => ( A24 != nil_Pr2927124501823088569on_nat ) )
       => ~ ! [X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat,Xs2: list_l8999537460041914605on_nat] :
              ( ( A13
                = ( cons_l5857885737047467175on_nat @ X3 @ Xs2 ) )
             => ! [Ys3: list_P5508476027986415439on_nat] :
                  ( ( A24
                    = ( cons_P4518942764240009225on_nat @ Y3 @ Ys3 ) )
                 => ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ Xs2 @ Ys3 ) @ ( listre1567628990915239951on_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_916_listrel_Ocases,axiom,
    ! [A13: list_l8999537460041914605on_nat,A24: list_P8734936519038225773on_nat,R: set_Pr737724919871395061on_nat] :
      ( ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ A13 @ A24 ) @ ( listre1486203369474292887on_nat @ R ) )
     => ( ( ( A13 = nil_li5563851662744376407on_nat )
         => ( A24 != nil_Pr707104663622130797on_nat ) )
       => ~ ! [X3: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat,Xs2: list_l8999537460041914605on_nat] :
              ( ( A13
                = ( cons_l5857885737047467175on_nat @ X3 @ Xs2 ) )
             => ! [Ys3: list_P8734936519038225773on_nat] :
                  ( ( A24
                    = ( cons_P2382207818294396957on_nat @ Y3 @ Ys3 ) )
                 => ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ Xs2 @ Ys3 ) @ ( listre1486203369474292887on_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_917_listrel_Ocases,axiom,
    ! [A13: list_l1726306699111145479on_nat,A24: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ A13 @ A24 ) @ ( listre5026495727120537649on_nat @ R ) )
     => ( ( ( A13 = nil_li2682182165152866823on_nat )
         => ( A24 != nil_li2682182165152866823on_nat ) )
       => ~ ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat,Xs2: list_l1726306699111145479on_nat] :
              ( ( A13
                = ( cons_l3028799188195208631on_nat @ X3 @ Xs2 ) )
             => ! [Ys3: list_l1726306699111145479on_nat] :
                  ( ( A24
                    = ( cons_l3028799188195208631on_nat @ Y3 @ Ys3 ) )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs2 @ Ys3 ) @ ( listre5026495727120537649on_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_918_listrel_Ocases,axiom,
    ! [A13: list_nat_nat,A24: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ A13 @ A24 ) @ ( listrel_nat_nat_nat @ R ) )
     => ( ( ( A13 = nil_nat_nat )
         => ( A24 != nil_nat ) )
       => ~ ! [X3: nat > nat,Y3: nat,Xs2: list_nat_nat] :
              ( ( A13
                = ( cons_nat_nat @ X3 @ Xs2 ) )
             => ! [Ys3: list_nat] :
                  ( ( A24
                    = ( cons_nat @ Y3 @ Ys3 ) )
                 => ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs2 @ Ys3 ) @ ( listrel_nat_nat_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_919_listrel_Ocases,axiom,
    ! [A13: list_P9127870862658512897on_nat,A24: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A13 @ A24 ) @ ( listre6301251474747857073on_nat @ R ) )
     => ( ( ( A13 = nil_Pr8614093721167239297on_nat )
         => ( A24 != nil_Pr8614093721167239297on_nat ) )
       => ~ ! [X3: produc1313295348606329723on_nat,Y3: produc1313295348606329723on_nat,Xs2: list_P9127870862658512897on_nat] :
              ( ( A13
                = ( cons_P1582596238967110705on_nat @ X3 @ Xs2 ) )
             => ! [Ys3: list_P9127870862658512897on_nat] :
                  ( ( A24
                    = ( cons_P1582596238967110705on_nat @ Y3 @ Ys3 ) )
                 => ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X3 @ Y3 ) @ R )
                   => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs2 @ Ys3 ) @ ( listre6301251474747857073on_nat @ R ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_920_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( take_P4655611586102217833on_nat @ ( suc @ I3 ) @ Xs )
        = ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ I3 @ Xs ) @ ( cons_P1582596238967110705on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) @ nil_Pr8614093721167239297on_nat ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_921_take__Suc__conv__app__nth,axiom,
    ! [I3: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( take_l5436743365630562101od_a_o @ ( suc @ I3 ) @ Xs )
        = ( append2159120467191820514od_a_o @ ( take_l5436743365630562101od_a_o @ I3 @ Xs ) @ ( cons_l9179691953475787517od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) @ nil_li1372968520647675725od_a_o ) ) ) ) ).

% take_Suc_conv_app_nth
thf(fact_922_lexord__take__index__conv,axiom,
    ! [X4: list_m351779702951725613on_nat,Y: list_m351779702951725613on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ X4 @ Y ) @ ( lexord5514055260552411370on_nat @ R ) )
      = ( ( ( ord_less_nat @ ( size_s6603137509925400257on_nat @ X4 ) @ ( size_s6603137509925400257on_nat @ Y ) )
          & ( ( take_m329436973745802671on_nat @ ( size_s6603137509925400257on_nat @ X4 ) @ Y )
            = X4 ) )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( ord_min_nat @ ( size_s6603137509925400257on_nat @ X4 ) @ ( size_s6603137509925400257on_nat @ Y ) ) )
            & ( ( take_m329436973745802671on_nat @ I4 @ X4 )
              = ( take_m329436973745802671on_nat @ I4 @ Y ) )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ X4 @ I4 ) @ ( nth_mu8151576519813129558on_nat @ Y @ I4 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_923_lexord__take__index__conv,axiom,
    ! [X4: list_l1726306699111145479on_nat,Y: list_l1726306699111145479on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ X4 @ Y ) @ ( lexord4827596522773364468on_nat @ R ) )
      = ( ( ( ord_less_nat @ ( size_s129639936549468787on_nat @ X4 ) @ ( size_s129639936549468787on_nat @ Y ) )
          & ( ( take_l5410376112614949359on_nat @ ( size_s129639936549468787on_nat @ X4 ) @ Y )
            = X4 ) )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( ord_min_nat @ ( size_s129639936549468787on_nat @ X4 ) @ ( size_s129639936549468787on_nat @ Y ) ) )
            & ( ( take_l5410376112614949359on_nat @ I4 @ X4 )
              = ( take_l5410376112614949359on_nat @ I4 @ Y ) )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ X4 @ I4 ) @ ( nth_li1302981970230039432on_nat @ Y @ I4 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_924_lexord__take__index__conv,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( lexord3493870746117591022on_nat @ R ) )
      = ( ( ( ord_less_nat @ ( size_s7490768528479217773on_nat @ X4 ) @ ( size_s7490768528479217773on_nat @ Y ) )
          & ( ( take_P4655611586102217833on_nat @ ( size_s7490768528479217773on_nat @ X4 ) @ Y )
            = X4 ) )
        | ? [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( ord_min_nat @ ( size_s7490768528479217773on_nat @ X4 ) @ ( size_s7490768528479217773on_nat @ Y ) ) )
            & ( ( take_P4655611586102217833on_nat @ I4 @ X4 )
              = ( take_P4655611586102217833on_nat @ I4 @ Y ) )
            & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ X4 @ I4 ) @ ( nth_Pr3336893728888831618on_nat @ Y @ I4 ) ) @ R ) ) ) ) ).

% lexord_take_index_conv
thf(fact_925_in__measures_I2_J,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,F: multis1856558993800614429on_nat > nat,Fs: list_m3172874511867307804at_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( measur7411324770174205213on_nat @ ( cons_m8717053863055185110at_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) )
        | ( ( ( F @ X4 )
            = ( F @ Y ) )
          & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( measur7411324770174205213on_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_926_in__measures_I2_J,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,F: list_P9127870862658512897on_nat > nat,Fs: list_l4071084838796328136at_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( measur2781751509508144385on_nat @ ( cons_l971637515602310776at_nat @ F @ Fs ) ) )
      = ( ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) )
        | ( ( ( F @ X4 )
            = ( F @ Y ) )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( measur2781751509508144385on_nat @ Fs ) ) ) ) ) ).

% in_measures(2)
thf(fact_927_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat,Ns_list: set_Pr8202701763255893447on_nat > set_Pr8202701763255893447on_nat > set_Pr8586234676504894535on_nat,As2: list_P9127870862658512897on_nat,Bs2: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S2: set_Pr8202701763255893447on_nat] :
      ( ( list_l5400313399564392903on_nat @ S_list @ Ns_list )
     => ( ( ( size_s7490768528479217773on_nat @ As2 )
          = ( size_s7490768528479217773on_nat @ Bs2 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Bs2 ) )
             => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ As2 @ I ) @ ( nth_Pr3336893728888831618on_nat @ Bs2 @ I ) ) @ NS ) )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As2 @ Bs2 ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_928_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr3573155178515677307on_nat > set_Pr3573155178515677307on_nat > set_Pr5729023540515198363on_nat,Ns_list: set_Pr3573155178515677307on_nat > set_Pr3573155178515677307on_nat > set_Pr5729023540515198363on_nat,As2: list_m351779702951725613on_nat,Bs2: list_m351779702951725613on_nat,NS: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat] :
      ( ( list_l2061016739618752721on_nat @ S_list @ Ns_list )
     => ( ( ( size_s6603137509925400257on_nat @ As2 )
          = ( size_s6603137509925400257on_nat @ Bs2 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s6603137509925400257on_nat @ Bs2 ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ As2 @ I ) @ ( nth_mu8151576519813129558on_nat @ Bs2 @ I ) ) @ NS ) )
         => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ As2 @ Bs2 ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_929_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr8586234676504894535on_nat > set_Pr8586234676504894535on_nat > set_Pr9215882266573312199on_nat,Ns_list: set_Pr8586234676504894535on_nat > set_Pr8586234676504894535on_nat > set_Pr9215882266573312199on_nat,As2: list_l1726306699111145479on_nat,Bs2: list_l1726306699111145479on_nat,NS: set_Pr8586234676504894535on_nat,S2: set_Pr8586234676504894535on_nat] :
      ( ( list_l2763215806797416525on_nat @ S_list @ Ns_list )
     => ( ( ( size_s129639936549468787on_nat @ As2 )
          = ( size_s129639936549468787on_nat @ Bs2 ) )
       => ( ! [I: nat] :
              ( ( ord_less_nat @ I @ ( size_s129639936549468787on_nat @ Bs2 ) )
             => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ As2 @ I ) @ ( nth_li1302981970230039432on_nat @ Bs2 @ I ) ) @ NS ) )
         => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ As2 @ Bs2 ) @ ( Ns_list @ S2 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_930_zip__append1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( zip_Su8130003265398491731on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ Zs )
      = ( append2865821678206098582on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ ( take_S1147720861222069983on_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ Zs ) ) @ ( zip_Su8130003265398491731on_nat @ Ys @ ( drop_S4725550819094782855on_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ Zs ) ) ) ) ).

% zip_append1
thf(fact_931_in__measures_I1_J,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat] :
      ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( measur7411324770174205213on_nat @ nil_mu4464041163730640006at_nat ) ) ).

% in_measures(1)
thf(fact_932_in__measures_I1_J,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( measur2781751509508144385on_nat @ nil_li268076387995464392at_nat ) ) ).

% in_measures(1)
thf(fact_933_append__take__drop__id,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) @ ( drop_P7041600534018571713on_nat @ N @ Xs ) )
      = Xs ) ).

% append_take_drop_id
thf(fact_934_in__set__dropD,axiom,
    ! [X4: produc9176241080893687067on_nat,N: nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( drop_P1029249954889453921on_nat @ N @ Xs ) ) )
     => ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ).

% in_set_dropD
thf(fact_935_in__set__dropD,axiom,
    ! [X4: produc1313295348606329723on_nat,N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( drop_P7041600534018571713on_nat @ N @ Xs ) ) )
     => ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% in_set_dropD
thf(fact_936_in__set__dropD,axiom,
    ! [X4: sum_su4907927656612468301on_nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( drop_S4725550819094782855on_nat @ N @ Xs ) ) )
     => ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% in_set_dropD
thf(fact_937_drop__Nil,axiom,
    ! [N: nat] :
      ( ( drop_l2897732520037520525od_a_o @ N @ nil_li1372968520647675725od_a_o )
      = nil_li1372968520647675725od_a_o ) ).

% drop_Nil
thf(fact_938_id__take__nth__drop,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( Xs
        = ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ I3 @ Xs ) @ ( cons_P1582596238967110705on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ I3 ) @ ( drop_P7041600534018571713on_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).

% id_take_nth_drop
thf(fact_939_upd__conv__take__nth__drop,axiom,
    ! [I3: nat,Xs: list_P9127870862658512897on_nat,A: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( list_u1840724836389618409on_nat @ Xs @ I3 @ A )
        = ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ I3 @ Xs ) @ ( cons_P1582596238967110705on_nat @ A @ ( drop_P7041600534018571713on_nat @ ( suc @ I3 ) @ Xs ) ) ) ) ) ).

% upd_conv_take_nth_drop
thf(fact_940_append__eq__conv__conj,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Ys )
        = Zs )
      = ( ( Xs
          = ( take_P4655611586102217833on_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ Zs ) )
        & ( Ys
          = ( drop_P7041600534018571713on_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ Zs ) ) ) ) ).

% append_eq_conv_conj
thf(fact_941_take__add,axiom,
    ! [I3: nat,J: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( take_P4655611586102217833on_nat @ ( plus_plus_nat @ I3 @ J ) @ Xs )
      = ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ I3 @ Xs ) @ ( take_P4655611586102217833on_nat @ J @ ( drop_P7041600534018571713on_nat @ I3 @ Xs ) ) ) ) ).

% take_add
thf(fact_942_measures__less,axiom,
    ! [F: multis1856558993800614429on_nat > nat,X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,Fs: list_m3172874511867307804at_nat] :
      ( ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( measur7411324770174205213on_nat @ ( cons_m8717053863055185110at_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_943_measures__less,axiom,
    ! [F: list_P9127870862658512897on_nat > nat,X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Fs: list_l4071084838796328136at_nat] :
      ( ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( measur2781751509508144385on_nat @ ( cons_l971637515602310776at_nat @ F @ Fs ) ) ) ) ).

% measures_less
thf(fact_944_length__Suc__conv__rev,axiom,
    ! [Xs: list_P9127870862658512897on_nat,N: nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: produc1313295348606329723on_nat,Ys4: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Ys4 @ ( cons_P1582596238967110705on_nat @ Y4 @ nil_Pr8614093721167239297on_nat ) ) )
            & ( ( size_s7490768528479217773on_nat @ Ys4 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_945_length__Suc__conv__rev,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,N: nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( suc @ N ) )
      = ( ? [Y4: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Ys4 @ ( cons_l9179691953475787517od_a_o @ Y4 @ nil_li1372968520647675725od_a_o ) ) )
            & ( ( size_s3379185846251495609od_a_o @ Ys4 )
              = N ) ) ) ) ).

% length_Suc_conv_rev
thf(fact_946_length__append__singleton,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] :
      ( ( size_s7490768528479217773on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) ) )
      = ( suc @ ( size_s7490768528479217773on_nat @ Xs ) ) ) ).

% length_append_singleton
thf(fact_947_length__append__singleton,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) ) )
      = ( suc @ ( size_s3379185846251495609od_a_o @ Xs ) ) ) ).

% length_append_singleton
thf(fact_948_zip__append2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( zip_Su8130003265398491731on_nat @ Xs @ ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( append2865821678206098582on_nat @ ( zip_Su8130003265398491731on_nat @ ( take_S1147720861222069983on_nat @ ( size_s5686234912572693873on_nat @ Ys ) @ Xs ) @ Ys ) @ ( zip_Su8130003265398491731on_nat @ ( drop_S4725550819094782855on_nat @ ( size_s5686234912572693873on_nat @ Ys ) @ Xs ) @ Zs ) ) ) ).

% zip_append2
thf(fact_949_take__hd__drop,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) @ ( cons_P1582596238967110705on_nat @ ( hd_Pro2908541307909527270on_nat @ ( drop_P7041600534018571713on_nat @ N @ Xs ) ) @ nil_Pr8614093721167239297on_nat ) )
        = ( take_P4655611586102217833on_nat @ ( suc @ N ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_950_take__hd__drop,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( append2159120467191820514od_a_o @ ( take_l5436743365630562101od_a_o @ N @ Xs ) @ ( cons_l9179691953475787517od_a_o @ ( hd_lis660529642729407282od_a_o @ ( drop_l2897732520037520525od_a_o @ N @ Xs ) ) @ nil_li1372968520647675725od_a_o ) )
        = ( take_l5436743365630562101od_a_o @ ( suc @ N ) @ Xs ) ) ) ).

% take_hd_drop
thf(fact_951_inf__concat__simple_Ocases,axiom,
    ! [X4: produc8199716216217303280at_nat] :
      ( ! [F2: nat > nat] :
          ( X4
         != ( produc72220940542539688at_nat @ F2 @ zero_zero_nat ) )
     => ~ ! [F2: nat > nat,N5: nat] :
            ( X4
           != ( produc72220940542539688at_nat @ F2 @ ( suc @ N5 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_952_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_953_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_954_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_955_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_956_hd__append2,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ( ( hd_Pro2908541307909527270on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
        = ( hd_Pro2908541307909527270on_nat @ Xs ) ) ) ).

% hd_append2
thf(fact_957_hd__append2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( hd_lis660529642729407282od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
        = ( hd_lis660529642729407282od_a_o @ Xs ) ) ) ).

% hd_append2
thf(fact_958_drop__eq__Nil2,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( nil_li1372968520647675725od_a_o
        = ( drop_l2897732520037520525od_a_o @ N @ Xs ) )
      = ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ N ) ) ).

% drop_eq_Nil2
thf(fact_959_drop__eq__Nil,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( drop_l2897732520037520525od_a_o @ N @ Xs )
        = nil_li1372968520647675725od_a_o )
      = ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ N ) ) ).

% drop_eq_Nil
thf(fact_960_drop__all,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,N: nat] :
      ( ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ N )
     => ( ( drop_l2897732520037520525od_a_o @ N @ Xs )
        = nil_li1372968520647675725od_a_o ) ) ).

% drop_all
thf(fact_961_ass__list__to__single__list_Ocases,axiom,
    ! [X4: list_P5366131564312172150at_nat] :
      ( ( X4 != nil_Pr2223394031645031670at_nat )
     => ~ ! [X3: nat > nat,N5: nat,Xs2: list_P5366131564312172150at_nat] :
            ( X4
           != ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ X3 @ N5 ) @ Xs2 ) ) ) ).

% ass_list_to_single_list.cases
thf(fact_962_hd__zip,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( Ys != nil_li1372968520647675725od_a_o )
       => ( ( hd_Pro5665698334477552338od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs @ Ys ) )
          = ( produc8077463541153823575od_a_o @ ( hd_lis660529642729407282od_a_o @ Xs ) @ ( hd_lis660529642729407282od_a_o @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_963_hd__zip,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( Xs != nil_mu4837018292623784855on_nat )
     => ( ( Ys != nil_mu4837018292623784855on_nat )
       => ( ( hd_Pro7900403694626400646on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) )
          = ( produc5200047356169809875on_nat @ ( hd_mul4361412637287379954on_nat @ Xs ) @ ( hd_mul4361412637287379954on_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_964_hd__zip,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( Xs != nil_li5563851662744376407on_nat )
     => ( ( Ys != nil_Pr2927124501823088569on_nat )
       => ( ( hd_Pro4663940589246654824on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) )
          = ( produc1545487600272464053on_nat @ ( hd_lis8677980075046437554on_nat @ Xs ) @ ( hd_Pro6238629103468366356on_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_965_hd__zip,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( Xs != nil_li5563851662744376407on_nat )
     => ( ( Ys != nil_Pr707104663622130797on_nat )
       => ( ( hd_Pro6238629103468366356on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) )
          = ( produc4176415733022414961on_nat @ ( hd_lis8677980075046437554on_nat @ Xs ) @ ( hd_Pro4877181668135357010on_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_966_hd__zip,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( Xs != nil_li2682182165152866823on_nat )
     => ( ( Ys != nil_li2682182165152866823on_nat )
       => ( ( hd_Pro4877181668135357010on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) )
          = ( produc7152764266822961111on_nat @ ( hd_lis7205458727479863788on_nat @ Xs ) @ ( hd_lis7205458727479863788on_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_967_hd__zip,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat] :
      ( ( Xs != nil_nat_nat )
     => ( ( Ys != nil_nat )
       => ( ( hd_Pro5468458281938048987at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) )
          = ( produc72220940542539688at_nat @ ( hd_nat_nat @ Xs ) @ ( hd_nat @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_968_list_Oset__sel_I1_J,axiom,
    ! [A: list_P1915136757390359201on_nat] :
      ( ( A != nil_Pr2652050626817158177on_nat )
     => ( member4278285197320822468on_nat @ ( hd_Pro7900403694626400646on_nat @ A ) @ ( set_Pr8272476097931642672on_nat @ A ) ) ) ).

% list.set_sel(1)
thf(fact_969_list_Oset__sel_I1_J,axiom,
    ! [A: list_l8336264072933094989od_a_o] :
      ( ( A != nil_li1372968520647675725od_a_o )
     => ( member5261019209154752368od_a_o @ ( hd_lis660529642729407282od_a_o @ A ) @ ( set_li2629784761229999580od_a_o @ A ) ) ) ).

% list.set_sel(1)
thf(fact_970_list_Oset__sel_I1_J,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( A != nil_Pr8614093721167239297on_nat )
     => ( member5906418179908956708on_nat @ ( hd_Pro2908541307909527270on_nat @ A ) @ ( set_Pr937530902923597456on_nat @ A ) ) ) ).

% list.set_sel(1)
thf(fact_971_list_Oset__sel_I1_J,axiom,
    ! [A: list_S5253404628753938141on_nat] :
      ( ( A != nil_Su3078906344068544711on_nat )
     => ( member8476201367316484836on_nat @ ( hd_Sum7036853328632959138on_nat @ A ) @ ( set_Su4848624856859743608on_nat @ A ) ) ) ).

% list.set_sel(1)
thf(fact_972_hd__in__set,axiom,
    ! [Xs: list_P1915136757390359201on_nat] :
      ( ( Xs != nil_Pr2652050626817158177on_nat )
     => ( member4278285197320822468on_nat @ ( hd_Pro7900403694626400646on_nat @ Xs ) @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ).

% hd_in_set
thf(fact_973_hd__in__set,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( member5261019209154752368od_a_o @ ( hd_lis660529642729407282od_a_o @ Xs ) @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ).

% hd_in_set
thf(fact_974_hd__in__set,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ( member5906418179908956708on_nat @ ( hd_Pro2908541307909527270on_nat @ Xs ) @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% hd_in_set
thf(fact_975_hd__in__set,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( Xs != nil_Su3078906344068544711on_nat )
     => ( member8476201367316484836on_nat @ ( hd_Sum7036853328632959138on_nat @ Xs ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% hd_in_set
thf(fact_976_longest__common__prefix,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
    ? [Ps: list_P9127870862658512897on_nat,Xs4: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( Xs
        = ( append2865821678206098582on_nat @ Ps @ Xs4 ) )
      & ( Ys
        = ( append2865821678206098582on_nat @ Ps @ Ys2 ) )
      & ( ( Xs4 = nil_Pr8614093721167239297on_nat )
        | ( Ys2 = nil_Pr8614093721167239297on_nat )
        | ( ( hd_Pro2908541307909527270on_nat @ Xs4 )
         != ( hd_Pro2908541307909527270on_nat @ Ys2 ) ) ) ) ).

% longest_common_prefix
thf(fact_977_longest__common__prefix,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
    ? [Ps: list_l8336264072933094989od_a_o,Xs4: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o] :
      ( ( Xs
        = ( append2159120467191820514od_a_o @ Ps @ Xs4 ) )
      & ( Ys
        = ( append2159120467191820514od_a_o @ Ps @ Ys2 ) )
      & ( ( Xs4 = nil_li1372968520647675725od_a_o )
        | ( Ys2 = nil_li1372968520647675725od_a_o )
        | ( ( hd_lis660529642729407282od_a_o @ Xs4 )
         != ( hd_lis660529642729407282od_a_o @ Ys2 ) ) ) ) ).

% longest_common_prefix
thf(fact_978_hd__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( Xs = nil_Pr8614093721167239297on_nat )
       => ( ( hd_Pro2908541307909527270on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( hd_Pro2908541307909527270on_nat @ Ys ) ) )
      & ( ( Xs != nil_Pr8614093721167239297on_nat )
       => ( ( hd_Pro2908541307909527270on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( hd_Pro2908541307909527270on_nat @ Xs ) ) ) ) ).

% hd_append
thf(fact_979_hd__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( Xs = nil_li1372968520647675725od_a_o )
       => ( ( hd_lis660529642729407282od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
          = ( hd_lis660529642729407282od_a_o @ Ys ) ) )
      & ( ( Xs != nil_li1372968520647675725od_a_o )
       => ( ( hd_lis660529642729407282od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
          = ( hd_lis660529642729407282od_a_o @ Xs ) ) ) ) ).

% hd_append
thf(fact_980_add__nonpos__eq__0__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X4 @ Y )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_981_add__nonneg__eq__0__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X4 @ Y )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_982_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_983_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_984_add__increasing2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C2 )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing2
thf(fact_985_add__decreasing2,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C2 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing2
thf(fact_986_add__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_increasing
thf(fact_987_add__decreasing,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C2 @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C2 ) @ B ) ) ) ).

% add_decreasing
thf(fact_988_hd__conv__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( hd_lis660529642729407282od_a_o @ Xs )
        = ( nth_li3248255784598043854od_a_o @ Xs @ zero_zero_nat ) ) ) ).

% hd_conv_nth
thf(fact_989_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing2
thf(fact_990_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% add_strict_increasing
thf(fact_991_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_992_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_993_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_994_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_995_lenlex__length,axiom,
    ! [Ms: list_P9127870862658512897on_nat,Ns: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ms @ Ns ) @ ( lenlex6631009945862686470on_nat @ R ) )
     => ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Ms ) @ ( size_s7490768528479217773on_nat @ Ns ) ) ) ).

% lenlex_length
thf(fact_996_append__eq__append__conv__if,axiom,
    ! [Xs_1: list_P9127870862658512897on_nat,Xs_2: list_P9127870862658512897on_nat,Ys_1: list_P9127870862658512897on_nat,Ys_2: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs_1 @ Xs_2 )
        = ( append2865821678206098582on_nat @ Ys_1 @ Ys_2 ) )
      = ( ( ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Xs_1 ) @ ( size_s7490768528479217773on_nat @ Ys_1 ) )
         => ( ( Xs_1
              = ( take_P4655611586102217833on_nat @ ( size_s7490768528479217773on_nat @ Xs_1 ) @ Ys_1 ) )
            & ( Xs_2
              = ( append2865821678206098582on_nat @ ( drop_P7041600534018571713on_nat @ ( size_s7490768528479217773on_nat @ Xs_1 ) @ Ys_1 ) @ Ys_2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ Xs_1 ) @ ( size_s7490768528479217773on_nat @ Ys_1 ) )
         => ( ( ( take_P4655611586102217833on_nat @ ( size_s7490768528479217773on_nat @ Ys_1 ) @ Xs_1 )
              = Ys_1 )
            & ( ( append2865821678206098582on_nat @ ( drop_P7041600534018571713on_nat @ ( size_s7490768528479217773on_nat @ Ys_1 ) @ Xs_1 ) @ Xs_2 )
              = Ys_2 ) ) ) ) ) ).

% append_eq_append_conv_if
thf(fact_997_lexord__sufI,axiom,
    ! [U: list_P9127870862658512897on_nat,W3: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,V: list_P9127870862658512897on_nat,Z6: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ U @ W3 ) @ ( lexord3493870746117591022on_nat @ R ) )
     => ( ( ord_less_eq_nat @ ( size_s7490768528479217773on_nat @ W3 ) @ ( size_s7490768528479217773on_nat @ U ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ U @ V ) @ ( append2865821678206098582on_nat @ W3 @ Z6 ) ) @ ( lexord3493870746117591022on_nat @ R ) ) ) ) ).

% lexord_sufI
thf(fact_998_measures__lesseq,axiom,
    ! [F: multis1856558993800614429on_nat > nat,X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,Fs: list_m3172874511867307804at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( measur7411324770174205213on_nat @ Fs ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( measur7411324770174205213on_nat @ ( cons_m8717053863055185110at_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_999_measures__lesseq,axiom,
    ! [F: list_P9127870862658512897on_nat > nat,X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Fs: list_l4071084838796328136at_nat] :
      ( ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y ) )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( measur2781751509508144385on_nat @ Fs ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( measur2781751509508144385on_nat @ ( cons_l971637515602310776at_nat @ F @ Fs ) ) ) ) ) ).

% measures_lesseq
thf(fact_1000_nth__equal__first__eq,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat,N: nat] :
      ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s8191359103794446093on_nat @ Xs ) )
       => ( ( ( nth_Pr6961551319198328098on_nat @ ( cons_P6897958070471385041on_nat @ X4 @ Xs ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1001_nth__equal__first__eq,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,N: nat] :
      ( ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( ( nth_Pr3336893728888831618on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1002_nth__equal__first__eq,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,N: nat] :
      ( ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ( ord_less_eq_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( ( nth_Su8471564416988646406on_nat @ ( cons_S6797538523646374167on_nat @ X4 @ Xs ) @ N )
            = X4 )
          = ( N = zero_zero_nat ) ) ) ) ).

% nth_equal_first_eq
thf(fact_1003_mset__zip__take__Cons__drop__twice,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,J: nat,X4: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_s7490768528479217773on_nat @ Ys ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( mset_P3992433801195507068on_nat @ ( zip_Pr4229164709604849015on_nat @ ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ J @ Xs ) @ ( cons_P1582596238967110705on_nat @ X4 @ ( drop_P7041600534018571713on_nat @ J @ Xs ) ) ) @ ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ J @ Ys ) @ ( cons_P1582596238967110705on_nat @ Y @ ( drop_P7041600534018571713on_nat @ J @ Ys ) ) ) ) )
          = ( add_ms662851516662172101on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ ( mset_P3992433801195507068on_nat @ ( zip_Pr4229164709604849015on_nat @ Xs @ Ys ) ) ) ) ) ) ).

% mset_zip_take_Cons_drop_twice
thf(fact_1004_mset__zip__take__Cons__drop__twice,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,J: nat,X4: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( mset_P6964650582947932560on_nat @ ( zip_Su8130003265398491731on_nat @ ( append3523812819975775986on_nat @ ( take_S1147720861222069983on_nat @ J @ Xs ) @ ( cons_S6797538523646374167on_nat @ X4 @ ( drop_S4725550819094782855on_nat @ J @ Xs ) ) ) @ ( append3523812819975775986on_nat @ ( take_S1147720861222069983on_nat @ J @ Ys ) @ ( cons_S6797538523646374167on_nat @ Y @ ( drop_S4725550819094782855on_nat @ J @ Ys ) ) ) ) )
          = ( add_ms8088439773566242649on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y ) @ ( mset_P6964650582947932560on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) ) ) ) ) ) ).

% mset_zip_take_Cons_drop_twice
thf(fact_1005_mset__zip__take__Cons__drop__twice,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat,J: nat,X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat] :
      ( ( ( size_s6603137509925400257on_nat @ Xs )
        = ( size_s6603137509925400257on_nat @ Ys ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s6603137509925400257on_nat @ Xs ) )
       => ( ( mset_P1678654427565054512on_nat @ ( zip_mu5118375602418390771on_nat @ ( append8234005794073363010on_nat @ ( take_m329436973745802671on_nat @ J @ Xs ) @ ( cons_m1493500683067190247on_nat @ X4 @ ( drop_m9104577059763177047on_nat @ J @ Xs ) ) ) @ ( append8234005794073363010on_nat @ ( take_m329436973745802671on_nat @ J @ Ys ) @ ( cons_m1493500683067190247on_nat @ Y @ ( drop_m9104577059763177047on_nat @ J @ Ys ) ) ) ) )
          = ( add_ms5516047754188811001on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( mset_P1678654427565054512on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) ) ) ) ) ) ).

% mset_zip_take_Cons_drop_twice
thf(fact_1006_mset__zip__take__Cons__drop__twice,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat,J: nat,X4: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = ( size_s2647645505745046499on_nat @ Ys ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s650155532264187265on_nat @ Xs ) )
       => ( ( mset_P43429913489005074on_nat @ ( zip_li7388484927123455957on_nat @ ( append6174841615030841602on_nat @ ( take_l6840013448454723183on_nat @ J @ Xs ) @ ( cons_l5857885737047467175on_nat @ X4 @ ( drop_l3393446327006289687on_nat @ J @ Xs ) ) ) @ ( append858208364616284260on_nat @ ( take_P8490018312761117137on_nat @ J @ Ys ) @ ( cons_P4518942764240009225on_nat @ Y @ ( drop_P5036690033322947193on_nat @ J @ Ys ) ) ) ) )
          = ( add_ms6606606094511819483on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y ) @ ( mset_P43429913489005074on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) ) ) ) ) ).

% mset_zip_take_Cons_drop_twice
thf(fact_1007_mset__zip__take__Cons__drop__twice,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat,J: nat,X4: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
      ( ( ( size_s650155532264187265on_nat @ Xs )
        = ( size_s4814362452863045593on_nat @ Ys ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s650155532264187265on_nat @ Xs ) )
       => ( ( mset_P9133150700452511978on_nat @ ( zip_li8749512630040955217on_nat @ ( append6174841615030841602on_nat @ ( take_l6840013448454723183on_nat @ J @ Xs ) @ ( cons_l5857885737047467175on_nat @ X4 @ ( drop_l3393446327006289687on_nat @ J @ Xs ) ) ) @ ( append2981160587596371970on_nat @ ( take_P3706611629360606293on_nat @ J @ Ys ) @ ( cons_P2382207818294396957on_nat @ Y @ ( drop_P6127876734035304877on_nat @ J @ Ys ) ) ) ) )
          = ( add_ms1143978036601055969on_nat @ ( produc4176415733022414961on_nat @ X4 @ Y ) @ ( mset_P9133150700452511978on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) ) ) ) ) ).

% mset_zip_take_Cons_drop_twice
thf(fact_1008_mset__zip__take__Cons__drop__twice,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat,J: nat,X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( ( size_s129639936549468787on_nat @ Xs )
        = ( size_s129639936549468787on_nat @ Ys ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s129639936549468787on_nat @ Xs ) )
       => ( ( mset_P1379524417292127228on_nat @ ( zip_li3935435632236560119on_nat @ ( append3839213209832495004on_nat @ ( take_l5410376112614949359on_nat @ J @ Xs ) @ ( cons_l3028799188195208631on_nat @ X4 @ ( drop_l138928210601027399on_nat @ J @ Xs ) ) ) @ ( append3839213209832495004on_nat @ ( take_l5410376112614949359on_nat @ J @ Ys ) @ ( cons_l3028799188195208631on_nat @ Y @ ( drop_l138928210601027399on_nat @ J @ Ys ) ) ) ) )
          = ( add_ms8490070595691190085on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( mset_P1379524417292127228on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) ) ) ) ) ).

% mset_zip_take_Cons_drop_twice
thf(fact_1009_mset__zip__take__Cons__drop__twice,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,J: nat,X4: nat > nat,Y: nat] :
      ( ( ( size_s8208510060688613859at_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( ord_less_eq_nat @ J @ ( size_s8208510060688613859at_nat @ Xs ) )
       => ( ( mset_P2828777261068555141at_nat @ ( zip_nat_nat_nat @ ( append_nat_nat @ ( take_nat_nat @ J @ Xs ) @ ( cons_nat_nat @ X4 @ ( drop_nat_nat @ J @ Xs ) ) ) @ ( append_nat @ ( take_nat @ J @ Ys ) @ ( cons_nat @ Y @ ( drop_nat @ J @ Ys ) ) ) ) )
          = ( add_ms4680508776633798606at_nat @ ( produc72220940542539688at_nat @ X4 @ Y ) @ ( mset_P2828777261068555141at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) ) ) ) ) ) ).

% mset_zip_take_Cons_drop_twice
thf(fact_1010_in__replicate__mset,axiom,
    ! [X4: produc9176241080893687067on_nat,N: nat,Y: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ ( replic561267910315885681on_nat @ N @ Y ) ) )
      = ( ( ord_less_nat @ zero_zero_nat @ N )
        & ( X4 = Y ) ) ) ).

% in_replicate_mset
thf(fact_1011_mset__single__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( ( mset_l8139550355095933148od_a_o @ Xs )
        = ( add_ms6706211442996992549od_a_o @ X4 @ zero_z8905637151584487364od_a_o ) )
      = ( Xs
        = ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) ) ) ).

% mset_single_iff
thf(fact_1012_mset__single__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X4: sum_su4907927656612468301on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs )
        = ( add_ms8575691854974865391on_nat @ X4 @ zero_z8702567803361615526on_nat ) )
      = ( Xs
        = ( cons_S6797538523646374167on_nat @ X4 @ nil_Su3078906344068544711on_nat ) ) ) ).

% mset_single_iff
thf(fact_1013_mset__single__iff,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] :
      ( ( ( mset_P6964650582947932560on_nat @ Xs )
        = ( add_ms8088439773566242649on_nat @ X4 @ zero_z2712726346186620152on_nat ) )
      = ( Xs
        = ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) ) ) ).

% mset_single_iff
thf(fact_1014_mset__single__iff__right,axiom,
    ! [X4: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( add_ms6706211442996992549od_a_o @ X4 @ zero_z8905637151584487364od_a_o )
        = ( mset_l8139550355095933148od_a_o @ Xs ) )
      = ( Xs
        = ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) ) ) ).

% mset_single_iff_right
thf(fact_1015_mset__single__iff__right,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( add_ms8575691854974865391on_nat @ X4 @ zero_z8702567803361615526on_nat )
        = ( mset_S7907735292211298936on_nat @ Xs ) )
      = ( Xs
        = ( cons_S6797538523646374167on_nat @ X4 @ nil_Su3078906344068544711on_nat ) ) ) ).

% mset_single_iff_right
thf(fact_1016_mset__single__iff__right,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( add_ms8088439773566242649on_nat @ X4 @ zero_z2712726346186620152on_nat )
        = ( mset_P6964650582947932560on_nat @ Xs ) )
      = ( Xs
        = ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) ) ) ).

% mset_single_iff_right
thf(fact_1017_ns__mul__ext__singleton,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,Ns: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ Ns )
     => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( add_ms8432739108091334335on_nat @ A @ zero_z4144889538238539638on_nat ) @ ( add_ms8432739108091334335on_nat @ B @ zero_z4144889538238539638on_nat ) ) @ ( multis3835232632530192936on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton
thf(fact_1018_ns__mul__ext__singleton,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ Ns )
     => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( add_ms213608174324010207on_nat @ A @ zero_z8385993896842825086on_nat ) @ ( add_ms213608174324010207on_nat @ B @ zero_z8385993896842825086on_nat ) ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton
thf(fact_1019_ns__mul__ext__singleton,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ B ) @ Ns )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( add_ms8575691854974865391on_nat @ A @ zero_z8702567803361615526on_nat ) @ ( add_ms8575691854974865391on_nat @ B @ zero_z8702567803361615526on_nat ) ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton
thf(fact_1020_s__mul__ext__singleton,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ S )
     => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( add_ms8432739108091334335on_nat @ A @ zero_z4144889538238539638on_nat ) @ ( add_ms8432739108091334335on_nat @ B @ zero_z4144889538238539638on_nat ) ) @ ( multis5680658968000484038on_nat @ Ns @ S ) ) ) ).

% s_mul_ext_singleton
thf(fact_1021_s__mul__ext__singleton,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ S )
     => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( add_ms213608174324010207on_nat @ A @ zero_z8385993896842825086on_nat ) @ ( add_ms213608174324010207on_nat @ B @ zero_z8385993896842825086on_nat ) ) @ ( multis3147760155352349208on_nat @ Ns @ S ) ) ) ).

% s_mul_ext_singleton
thf(fact_1022_s__mul__ext__singleton,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ B ) @ S )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( add_ms8575691854974865391on_nat @ A @ zero_z8702567803361615526on_nat ) @ ( add_ms8575691854974865391on_nat @ B @ zero_z8702567803361615526on_nat ) ) @ ( multis1620213760974180726on_nat @ Ns @ S ) ) ) ).

% s_mul_ext_singleton
thf(fact_1023_subset__code_I1_J,axiom,
    ! [Xs: list_P1915136757390359201on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( ord_le927452247459997723on_nat @ ( set_Pr8272476097931642672on_nat @ Xs ) @ B3 )
      = ( ! [X5: produc9176241080893687067on_nat] :
            ( ( member4278285197320822468on_nat @ X5 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
           => ( member4278285197320822468on_nat @ X5 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1024_subset__code_I1_J,axiom,
    ! [Xs: list_P9127870862658512897on_nat,B3: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ B3 )
      = ( ! [X5: produc1313295348606329723on_nat] :
            ( ( member5906418179908956708on_nat @ X5 @ ( set_Pr937530902923597456on_nat @ Xs ) )
           => ( member5906418179908956708on_nat @ X5 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1025_subset__code_I1_J,axiom,
    ! [Xs: list_S5253404628753938141on_nat,B3: set_Su266273576555847555on_nat] :
      ( ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ B3 )
      = ( ! [X5: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X5 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( member8476201367316484836on_nat @ X5 @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_1026_ns__mul__ext__mono,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat,Ns2: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ Ns @ Ns2 )
     => ( ( ord_le6961793561435735931on_nat @ S @ S3 )
       => ( ord_le927452247459997723on_nat @ ( multis4183089776494585304on_nat @ Ns @ S ) @ ( multis4183089776494585304on_nat @ Ns2 @ S3 ) ) ) ) ).

% ns_mul_ext_mono
thf(fact_1027_s__mul__ext__mono,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat,Ns2: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ Ns @ Ns2 )
     => ( ( ord_le6961793561435735931on_nat @ S @ S3 )
       => ( ord_le927452247459997723on_nat @ ( multis1620213760974180726on_nat @ Ns @ S ) @ ( multis1620213760974180726on_nat @ Ns2 @ S3 ) ) ) ) ).

% s_mul_ext_mono
thf(fact_1028_subrelI,axiom,
    ! [R: set_Pr3573155178515677307on_nat,S: set_Pr3573155178515677307on_nat] :
      ( ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat] :
          ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ R )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le927452247459997723on_nat @ R @ S ) ) ).

% subrelI
thf(fact_1029_subrelI,axiom,
    ! [R: set_Pr6244131468446592861on_nat,S: set_Pr6244131468446592861on_nat] :
      ( ! [X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat] :
          ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y3 ) @ R )
         => ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3399424890108663549on_nat @ R @ S ) ) ).

% subrelI
thf(fact_1030_subrelI,axiom,
    ! [R: set_Pr737724919871395061on_nat,S: set_Pr737724919871395061on_nat] :
      ( ! [X3: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat] :
          ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y3 ) @ R )
         => ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le248320493815342421on_nat @ R @ S ) ) ).

% subrelI
thf(fact_1031_subrelI,axiom,
    ! [R: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
          ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ R )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le2174946625535087591on_nat @ R @ S ) ) ).

% subrelI
thf(fact_1032_subrelI,axiom,
    ! [R: set_Pr9093778441882193744at_nat,S: set_Pr9093778441882193744at_nat] :
      ( ! [X3: nat > nat,Y3: nat] :
          ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) @ R )
         => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X3 @ Y3 ) @ S ) )
     => ( ord_le3678578370064672496at_nat @ R @ S ) ) ).

% subrelI
thf(fact_1033_mset__add,axiom,
    ! [A: produc9176241080893687067on_nat,A3: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ A3 ) )
     => ~ ! [B6: multis2912115315702009697on_nat] :
            ( A3
           != ( add_ms5516047754188811001on_nat @ A @ B6 ) ) ) ).

% mset_add
thf(fact_1034_multi__member__split,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ? [A8: multis2912115315702009697on_nat] :
          ( M2
          = ( add_ms5516047754188811001on_nat @ X4 @ A8 ) ) ) ).

% multi_member_split
thf(fact_1035_insert__noteq__member,axiom,
    ! [B: produc9176241080893687067on_nat,B3: multis2912115315702009697on_nat,C2: produc9176241080893687067on_nat,C3: multis2912115315702009697on_nat] :
      ( ( ( add_ms5516047754188811001on_nat @ B @ B3 )
        = ( add_ms5516047754188811001on_nat @ C2 @ C3 ) )
     => ( ( B != C2 )
       => ( member4278285197320822468on_nat @ C2 @ ( set_ms7239842528327418072on_nat @ B3 ) ) ) ) ).

% insert_noteq_member
thf(fact_1036_union__single__eq__member,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat,N3: multis2912115315702009697on_nat] :
      ( ( ( add_ms5516047754188811001on_nat @ X4 @ M2 )
        = N3 )
     => ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ N3 ) ) ) ).

% union_single_eq_member
thf(fact_1037_multi__member__last,axiom,
    ! [X4: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) ) ).

% multi_member_last
thf(fact_1038_mset_Osimps_I2_J,axiom,
    ! [A: sum_su4907927656612468301on_nat,X4: list_S5253404628753938141on_nat] :
      ( ( mset_S7907735292211298936on_nat @ ( cons_S6797538523646374167on_nat @ A @ X4 ) )
      = ( add_ms8575691854974865391on_nat @ A @ ( mset_S7907735292211298936on_nat @ X4 ) ) ) ).

% mset.simps(2)
thf(fact_1039_mset_Osimps_I2_J,axiom,
    ! [A: produc1313295348606329723on_nat,X4: list_P9127870862658512897on_nat] :
      ( ( mset_P6964650582947932560on_nat @ ( cons_P1582596238967110705on_nat @ A @ X4 ) )
      = ( add_ms8088439773566242649on_nat @ A @ ( mset_P6964650582947932560on_nat @ X4 ) ) ) ).

% mset.simps(2)
thf(fact_1040_set__subset__Cons,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] : ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_1041_set__subset__Cons,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X4: sum_su4907927656612468301on_nat] : ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X4 @ Xs ) ) ) ).

% set_subset_Cons
thf(fact_1042_set__take__subset,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] : ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) ) @ ( set_Pr937530902923597456on_nat @ Xs ) ) ).

% set_take_subset
thf(fact_1043_set__take__subset,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] : ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ N @ Xs ) ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% set_take_subset
thf(fact_1044_set__drop__subset,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat] : ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( drop_P7041600534018571713on_nat @ N @ Xs ) ) @ ( set_Pr937530902923597456on_nat @ Xs ) ) ).

% set_drop_subset
thf(fact_1045_set__drop__subset,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] : ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( drop_S4725550819094782855on_nat @ N @ Xs ) ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% set_drop_subset
thf(fact_1046_set__update__subsetI,axiom,
    ! [Xs: list_P1915136757390359201on_nat,A3: set_Pr3573155178515677307on_nat,X4: produc9176241080893687067on_nat,I3: nat] :
      ( ( ord_le927452247459997723on_nat @ ( set_Pr8272476097931642672on_nat @ Xs ) @ A3 )
     => ( ( member4278285197320822468on_nat @ X4 @ A3 )
       => ( ord_le927452247459997723on_nat @ ( set_Pr8272476097931642672on_nat @ ( list_u8615448375230402185on_nat @ Xs @ I3 @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_1047_set__update__subsetI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,A3: set_Pr2535564044409765851on_nat,X4: produc1313295348606329723on_nat,I3: nat] :
      ( ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ A3 )
     => ( ( member5906418179908956708on_nat @ X4 @ A3 )
       => ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I3 @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_1048_set__update__subsetI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,A3: set_Su266273576555847555on_nat,X4: sum_su4907927656612468301on_nat,I3: nat] :
      ( ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ A3 )
     => ( ( member8476201367316484836on_nat @ X4 @ A3 )
       => ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I3 @ X4 ) ) @ A3 ) ) ) ).

% set_update_subsetI
thf(fact_1049_ns__mul__ext__some__S__in__NS,axiom,
    ! [S4: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,NS: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ S4 @ S2 )
     => ( ( multis4183089776494585304on_nat @ ( sup_su6884158799674119087on_nat @ NS @ S4 ) @ S2 )
        = ( multis4183089776494585304on_nat @ NS @ S2 ) ) ) ).

% ns_mul_ext_some_S_in_NS
thf(fact_1050_multi__member__this,axiom,
    ! [X4: produc9176241080893687067on_nat,XS: multis2912115315702009697on_nat] : ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ ( plus_p7340530524814416408on_nat @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) @ XS ) ) ) ).

% multi_member_this
thf(fact_1051_multi__member__skip,axiom,
    ! [X4: produc9176241080893687067on_nat,XS: multis2912115315702009697on_nat,Y: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ XS ) )
     => ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ ( plus_p7340530524814416408on_nat @ ( add_ms5516047754188811001on_nat @ Y @ zero_z1502369238205801624on_nat ) @ XS ) ) ) ) ).

% multi_member_skip
thf(fact_1052_multpw__split1R,axiom,
    ! [X4: multis1856558993800614429on_nat,X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( add_ms8432739108091334335on_nat @ X4 @ X ) @ Y5 ) @ ( multis1913027631269000601on_nat @ Ns ) )
     => ~ ! [Z: multis1856558993800614429on_nat,Z5: multis6850599868085929325on_nat] :
            ( ( Y5
              = ( add_ms8432739108091334335on_nat @ Z @ Z5 ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Z ) @ Ns )
             => ~ ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Z5 ) @ ( multis1913027631269000601on_nat @ Ns ) ) ) ) ) ).

% multpw_split1R
thf(fact_1053_multpw__split1R,axiom,
    ! [X4: list_P9127870862658512897on_nat,X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( add_ms213608174324010207on_nat @ X4 @ X ) @ Y5 ) @ ( multis4607348483316519301on_nat @ Ns ) )
     => ~ ! [Z: list_P9127870862658512897on_nat,Z5: multis5168333519641040071on_nat] :
            ( ( Y5
              = ( add_ms213608174324010207on_nat @ Z @ Z5 ) )
           => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Z ) @ Ns )
             => ~ ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Z5 ) @ ( multis4607348483316519301on_nat @ Ns ) ) ) ) ) ).

% multpw_split1R
thf(fact_1054_multpw__split1R,axiom,
    ! [X4: sum_su4907927656612468301on_nat,X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( add_ms8575691854974865391on_nat @ X4 @ X ) @ Y5 ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ~ ! [Z: sum_su4907927656612468301on_nat,Z5: multis1856558993800614429on_nat] :
            ( ( Y5
              = ( add_ms8575691854974865391on_nat @ Z @ Z5 ) )
           => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Z ) @ Ns )
             => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Z5 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ) ).

% multpw_split1R
thf(fact_1055_multpw__split1L,axiom,
    ! [X: multis6850599868085929325on_nat,Y: multis1856558993800614429on_nat,Y5: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ ( add_ms8432739108091334335on_nat @ Y @ Y5 ) ) @ ( multis1913027631269000601on_nat @ Ns ) )
     => ~ ! [Z: multis1856558993800614429on_nat,Z5: multis6850599868085929325on_nat] :
            ( ( X
              = ( add_ms8432739108091334335on_nat @ Z @ Z5 ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Z @ Y ) @ Ns )
             => ~ ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ Z5 @ Y5 ) @ ( multis1913027631269000601on_nat @ Ns ) ) ) ) ) ).

% multpw_split1L
thf(fact_1056_multpw__split1L,axiom,
    ! [X: multis5168333519641040071on_nat,Y: list_P9127870862658512897on_nat,Y5: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ ( add_ms213608174324010207on_nat @ Y @ Y5 ) ) @ ( multis4607348483316519301on_nat @ Ns ) )
     => ~ ! [Z: list_P9127870862658512897on_nat,Z5: multis5168333519641040071on_nat] :
            ( ( X
              = ( add_ms213608174324010207on_nat @ Z @ Z5 ) )
           => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Z @ Y ) @ Ns )
             => ~ ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Z5 @ Y5 ) @ ( multis4607348483316519301on_nat @ Ns ) ) ) ) ) ).

% multpw_split1L
thf(fact_1057_multpw__split1L,axiom,
    ! [X: multis1856558993800614429on_nat,Y: sum_su4907927656612468301on_nat,Y5: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ ( add_ms8575691854974865391on_nat @ Y @ Y5 ) ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ~ ! [Z: sum_su4907927656612468301on_nat,Z5: multis1856558993800614429on_nat] :
            ( ( X
              = ( add_ms8575691854974865391on_nat @ Z @ Z5 ) )
           => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Z @ Y ) @ Ns )
             => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Z5 @ Y5 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ) ).

% multpw_split1L
thf(fact_1058_multpw_Oadd,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat,Ns: set_Pr2535564044409765851on_nat,X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y ) @ Ns )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Y5 ) @ ( multis636354481432518601on_nat @ Ns ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( add_ms8575691854974865391on_nat @ X4 @ X ) @ ( add_ms8575691854974865391on_nat @ Y @ Y5 ) ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ).

% multpw.add
thf(fact_1059_multpw_Oadd,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,Ns: set_Pr3573155178515677307on_nat,X: multis6850599868085929325on_nat,Y5: multis6850599868085929325on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ Ns )
     => ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X @ Y5 ) @ ( multis1913027631269000601on_nat @ Ns ) )
       => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( add_ms8432739108091334335on_nat @ X4 @ X ) @ ( add_ms8432739108091334335on_nat @ Y @ Y5 ) ) @ ( multis1913027631269000601on_nat @ Ns ) ) ) ) ).

% multpw.add
thf(fact_1060_multpw_Oadd,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Ns: set_Pr8586234676504894535on_nat,X: multis5168333519641040071on_nat,Y5: multis5168333519641040071on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ Ns )
     => ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X @ Y5 ) @ ( multis4607348483316519301on_nat @ Ns ) )
       => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( add_ms213608174324010207on_nat @ X4 @ X ) @ ( add_ms213608174324010207on_nat @ Y @ Y5 ) ) @ ( multis4607348483316519301on_nat @ Ns ) ) ) ) ).

% multpw.add
thf(fact_1061_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ M @ Xs ) ) @ ( set_Pr937530902923597456on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1062_set__take__subset__set__take,axiom,
    ! [M: nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_eq_nat @ M @ N )
     => ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ M @ Xs ) ) @ ( set_Su4848624856859743608on_nat @ ( take_S1147720861222069983on_nat @ N @ Xs ) ) ) ) ).

% set_take_subset_set_take
thf(fact_1063_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( drop_P7041600534018571713on_nat @ M @ Xs ) ) @ ( set_Pr937530902923597456on_nat @ ( drop_P7041600534018571713on_nat @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_1064_set__drop__subset__set__drop,axiom,
    ! [N: nat,M: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_eq_nat @ N @ M )
     => ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( drop_S4725550819094782855on_nat @ M @ Xs ) ) @ ( set_Su4848624856859743608on_nat @ ( drop_S4725550819094782855on_nat @ N @ Xs ) ) ) ) ).

% set_drop_subset_set_drop
thf(fact_1065_multpw__single,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y ) @ Ns )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( add_ms8575691854974865391on_nat @ X4 @ zero_z8702567803361615526on_nat ) @ ( add_ms8575691854974865391on_nat @ Y @ zero_z8702567803361615526on_nat ) ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ).

% multpw_single
thf(fact_1066_multpw__single,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ Ns )
     => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( add_ms8432739108091334335on_nat @ X4 @ zero_z4144889538238539638on_nat ) @ ( add_ms8432739108091334335on_nat @ Y @ zero_z4144889538238539638on_nat ) ) @ ( multis1913027631269000601on_nat @ Ns ) ) ) ).

% multpw_single
thf(fact_1067_multpw__single,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ Ns )
     => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( add_ms213608174324010207on_nat @ X4 @ zero_z8385993896842825086on_nat ) @ ( add_ms213608174324010207on_nat @ Y @ zero_z8385993896842825086on_nat ) ) @ ( multis4607348483316519301on_nat @ Ns ) ) ) ).

% multpw_single
thf(fact_1068_multpw_Osimps,axiom,
    ! [A13: multis6850599868085929325on_nat,A24: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A13 @ A24 ) @ ( multis1913027631269000601on_nat @ Ns ) )
      = ( ( ( A13 = zero_z4144889538238539638on_nat )
          & ( A24 = zero_z4144889538238539638on_nat ) )
        | ? [X5: multis1856558993800614429on_nat,Y4: multis1856558993800614429on_nat,X8: multis6850599868085929325on_nat,Y8: multis6850599868085929325on_nat] :
            ( ( A13
              = ( add_ms8432739108091334335on_nat @ X5 @ X8 ) )
            & ( A24
              = ( add_ms8432739108091334335on_nat @ Y4 @ Y8 ) )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X5 @ Y4 ) @ Ns )
            & ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X8 @ Y8 ) @ ( multis1913027631269000601on_nat @ Ns ) ) ) ) ) ).

% multpw.simps
thf(fact_1069_multpw_Osimps,axiom,
    ! [A13: multis5168333519641040071on_nat,A24: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A13 @ A24 ) @ ( multis4607348483316519301on_nat @ Ns ) )
      = ( ( ( A13 = zero_z8385993896842825086on_nat )
          & ( A24 = zero_z8385993896842825086on_nat ) )
        | ? [X5: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat,X8: multis5168333519641040071on_nat,Y8: multis5168333519641040071on_nat] :
            ( ( A13
              = ( add_ms213608174324010207on_nat @ X5 @ X8 ) )
            & ( A24
              = ( add_ms213608174324010207on_nat @ Y4 @ Y8 ) )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X5 @ Y4 ) @ Ns )
            & ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X8 @ Y8 ) @ ( multis4607348483316519301on_nat @ Ns ) ) ) ) ) ).

% multpw.simps
thf(fact_1070_multpw_Osimps,axiom,
    ! [A13: multis1856558993800614429on_nat,A24: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A13 @ A24 ) @ ( multis636354481432518601on_nat @ Ns ) )
      = ( ( ( A13 = zero_z8702567803361615526on_nat )
          & ( A24 = zero_z8702567803361615526on_nat ) )
        | ? [X5: sum_su4907927656612468301on_nat,Y4: sum_su4907927656612468301on_nat,X8: multis1856558993800614429on_nat,Y8: multis1856558993800614429on_nat] :
            ( ( A13
              = ( add_ms8575691854974865391on_nat @ X5 @ X8 ) )
            & ( A24
              = ( add_ms8575691854974865391on_nat @ Y4 @ Y8 ) )
            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X5 @ Y4 ) @ Ns )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X8 @ Y8 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ) ).

% multpw.simps
thf(fact_1071_multpw_Ocases,axiom,
    ! [A13: multis6850599868085929325on_nat,A24: multis6850599868085929325on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ A13 @ A24 ) @ ( multis1913027631269000601on_nat @ Ns ) )
     => ( ( ( A13 = zero_z4144889538238539638on_nat )
         => ( A24 != zero_z4144889538238539638on_nat ) )
       => ~ ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat,X7: multis6850599868085929325on_nat] :
              ( ( A13
                = ( add_ms8432739108091334335on_nat @ X3 @ X7 ) )
             => ! [Y6: multis6850599868085929325on_nat] :
                  ( ( A24
                    = ( add_ms8432739108091334335on_nat @ Y3 @ Y6 ) )
                 => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ Y3 ) @ Ns )
                   => ~ ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ X7 @ Y6 ) @ ( multis1913027631269000601on_nat @ Ns ) ) ) ) ) ) ) ).

% multpw.cases
thf(fact_1072_multpw_Ocases,axiom,
    ! [A13: multis5168333519641040071on_nat,A24: multis5168333519641040071on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ A13 @ A24 ) @ ( multis4607348483316519301on_nat @ Ns ) )
     => ( ( ( A13 = zero_z8385993896842825086on_nat )
         => ( A24 != zero_z8385993896842825086on_nat ) )
       => ~ ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat,X7: multis5168333519641040071on_nat] :
              ( ( A13
                = ( add_ms213608174324010207on_nat @ X3 @ X7 ) )
             => ! [Y6: multis5168333519641040071on_nat] :
                  ( ( A24
                    = ( add_ms213608174324010207on_nat @ Y3 @ Y6 ) )
                 => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y3 ) @ Ns )
                   => ~ ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ X7 @ Y6 ) @ ( multis4607348483316519301on_nat @ Ns ) ) ) ) ) ) ) ).

% multpw.cases
thf(fact_1073_multpw_Ocases,axiom,
    ! [A13: multis1856558993800614429on_nat,A24: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A13 @ A24 ) @ ( multis636354481432518601on_nat @ Ns ) )
     => ( ( ( A13 = zero_z8702567803361615526on_nat )
         => ( A24 != zero_z8702567803361615526on_nat ) )
       => ~ ! [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,X7: multis1856558993800614429on_nat] :
              ( ( A13
                = ( add_ms8575691854974865391on_nat @ X3 @ X7 ) )
             => ! [Y6: multis1856558993800614429on_nat] :
                  ( ( A24
                    = ( add_ms8575691854974865391on_nat @ Y3 @ Y6 ) )
                 => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y3 ) @ Ns )
                   => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X7 @ Y6 ) @ ( multis636354481432518601on_nat @ Ns ) ) ) ) ) ) ) ).

% multpw.cases
thf(fact_1074_mult2__alt__s__single,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ B ) @ S )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( add_ms8575691854974865391on_nat @ A @ zero_z8702567803361615526on_nat ) @ ( add_ms8575691854974865391on_nat @ B @ zero_z8702567803361615526on_nat ) ) @ ( multis2224597371354980665on_nat @ $false @ Ns @ S ) ) ) ).

% mult2_alt_s_single
thf(fact_1075_mult2__alt__s__single,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ S )
     => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( add_ms8432739108091334335on_nat @ A @ zero_z4144889538238539638on_nat ) @ ( add_ms8432739108091334335on_nat @ B @ zero_z4144889538238539638on_nat ) ) @ ( multis394508389068283657on_nat @ $false @ Ns @ S ) ) ) ).

% mult2_alt_s_single
thf(fact_1076_mult2__alt__s__single,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ S )
     => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( add_ms213608174324010207on_nat @ A @ zero_z8385993896842825086on_nat ) @ ( add_ms213608174324010207on_nat @ B @ zero_z8385993896842825086on_nat ) ) @ ( multis1733271916442156053on_nat @ $false @ Ns @ S ) ) ) ).

% mult2_alt_s_single
thf(fact_1077_ns__mul__ext__singleton2,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,S: set_Pr3573155178515677307on_nat,Ns: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ S )
     => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ ( add_ms8432739108091334335on_nat @ A @ zero_z4144889538238539638on_nat ) @ ( add_ms8432739108091334335on_nat @ B @ zero_z4144889538238539638on_nat ) ) @ ( multis3835232632530192936on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton2
thf(fact_1078_ns__mul__ext__singleton2,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,S: set_Pr8586234676504894535on_nat,Ns: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ S )
     => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( add_ms213608174324010207on_nat @ A @ zero_z8385993896842825086on_nat ) @ ( add_ms213608174324010207on_nat @ B @ zero_z8385993896842825086on_nat ) ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton2
thf(fact_1079_ns__mul__ext__singleton2,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,S: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ B ) @ S )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( add_ms8575691854974865391on_nat @ A @ zero_z8702567803361615526on_nat ) @ ( add_ms8575691854974865391on_nat @ B @ zero_z8702567803361615526on_nat ) ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_singleton2
thf(fact_1080_subsetI,axiom,
    ! [A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ! [X3: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X3 @ A3 )
         => ( member4278285197320822468on_nat @ X3 @ B3 ) )
     => ( ord_le927452247459997723on_nat @ A3 @ B3 ) ) ).

% subsetI
thf(fact_1081_in__mono,axiom,
    ! [A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat,X4: produc9176241080893687067on_nat] :
      ( ( ord_le927452247459997723on_nat @ A3 @ B3 )
     => ( ( member4278285197320822468on_nat @ X4 @ A3 )
       => ( member4278285197320822468on_nat @ X4 @ B3 ) ) ) ).

% in_mono
thf(fact_1082_subsetD,axiom,
    ! [A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat,C2: produc9176241080893687067on_nat] :
      ( ( ord_le927452247459997723on_nat @ A3 @ B3 )
     => ( ( member4278285197320822468on_nat @ C2 @ A3 )
       => ( member4278285197320822468on_nat @ C2 @ B3 ) ) ) ).

% subsetD
thf(fact_1083_subset__eq,axiom,
    ( ord_le927452247459997723on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B7: set_Pr3573155178515677307on_nat] :
        ! [X5: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ X5 @ A6 )
         => ( member4278285197320822468on_nat @ X5 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_1084_subset__iff,axiom,
    ( ord_le927452247459997723on_nat
    = ( ^ [A6: set_Pr3573155178515677307on_nat,B7: set_Pr3573155178515677307on_nat] :
        ! [T: produc9176241080893687067on_nat] :
          ( ( member4278285197320822468on_nat @ T @ A6 )
         => ( member4278285197320822468on_nat @ T @ B7 ) ) ) ) ).

% subset_iff
thf(fact_1085_update_Osimps_I1_J,axiom,
    ! [K3: multis1856558993800614429on_nat,V: multis1856558993800614429on_nat] :
      ( ( update6820902895451041720on_nat @ K3 @ V @ nil_Pr2652050626817158177on_nat )
      = ( cons_P6897958070471385041on_nat @ ( produc5200047356169809875on_nat @ K3 @ V ) @ nil_Pr2652050626817158177on_nat ) ) ).

% update.simps(1)
thf(fact_1086_update_Osimps_I1_J,axiom,
    ! [K3: list_S5253404628753938141on_nat,V: produc118098540238352959on_nat] :
      ( ( update4833328981212444826on_nat @ K3 @ V @ nil_Pr5986419849129872387on_nat )
      = ( cons_P8471348786669567923on_nat @ ( produc1545487600272464053on_nat @ K3 @ V ) @ nil_Pr5986419849129872387on_nat ) ) ).

% update.simps(1)
thf(fact_1087_update_Osimps_I1_J,axiom,
    ! [K3: list_S5253404628753938141on_nat,V: produc1714117091004681831on_nat] :
      ( ( update2224492077062527052on_nat @ K3 @ V @ nil_Pr2927124501823088569on_nat )
      = ( cons_P4518942764240009225on_nat @ ( produc4176415733022414961on_nat @ K3 @ V ) @ nil_Pr2927124501823088569on_nat ) ) ).

% update.simps(1)
thf(fact_1088_update_Osimps_I1_J,axiom,
    ! [K3: list_P9127870862658512897on_nat,V: list_P9127870862658512897on_nat] :
      ( ( update4684173787400992828on_nat @ K3 @ V @ nil_Pr707104663622130797on_nat )
      = ( cons_P2382207818294396957on_nat @ ( produc7152764266822961111on_nat @ K3 @ V ) @ nil_Pr707104663622130797on_nat ) ) ).

% update.simps(1)
thf(fact_1089_update_Osimps_I1_J,axiom,
    ! [K3: nat > nat,V: nat] :
      ( ( update_nat_nat_nat @ K3 @ V @ nil_Pr2223394031645031670at_nat )
      = ( cons_P4219629788700907686at_nat @ ( produc72220940542539688at_nat @ K3 @ V ) @ nil_Pr2223394031645031670at_nat ) ) ).

% update.simps(1)
thf(fact_1090_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R4: set_Pr3573155178515677307on_nat,R2: set_Pr3573155178515677307on_nat,Seq: nat > multis1856558993800614429on_nat] :
      ( ( ord_le927452247459997723on_nat @ R4 @ R2 )
     => ( ! [I: nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R4 )
       => ! [I2: nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R2 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_1091_Abstract__Rewriting_Ochain__mono,axiom,
    ! [R4: set_Pr8586234676504894535on_nat,R2: set_Pr8586234676504894535on_nat,Seq: nat > list_P9127870862658512897on_nat] :
      ( ( ord_le2174946625535087591on_nat @ R4 @ R2 )
     => ( ! [I: nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( Seq @ I ) @ ( Seq @ ( suc @ I ) ) ) @ R4 )
       => ! [I2: nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( Seq @ I2 ) @ ( Seq @ ( suc @ I2 ) ) ) @ R2 ) ) ) ).

% Abstract_Rewriting.chain_mono
thf(fact_1092_mult1E,axiom,
    ! [N3: multis2912115315702009697on_nat,M2: multis2912115315702009697on_nat,R: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ N3 @ M2 ) @ ( mult1_5496832715746181134on_nat @ R ) )
     => ~ ! [A4: produc9176241080893687067on_nat,M0: multis2912115315702009697on_nat] :
            ( ( M2
              = ( add_ms5516047754188811001on_nat @ A4 @ M0 ) )
           => ! [K6: multis2912115315702009697on_nat] :
                ( ( N3
                  = ( plus_p7340530524814416408on_nat @ M0 @ K6 ) )
               => ~ ! [B5: produc9176241080893687067on_nat] :
                      ( ( member4278285197320822468on_nat @ B5 @ ( set_ms7239842528327418072on_nat @ K6 ) )
                     => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ B5 @ A4 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_1093_mult1E,axiom,
    ! [N3: multis6850599868085929325on_nat,M2: multis6850599868085929325on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ N3 @ M2 ) @ ( mult1_6031627987982144362on_nat @ R ) )
     => ~ ! [A4: multis1856558993800614429on_nat,M0: multis6850599868085929325on_nat] :
            ( ( M2
              = ( add_ms8432739108091334335on_nat @ A4 @ M0 ) )
           => ! [K6: multis6850599868085929325on_nat] :
                ( ( N3
                  = ( plus_p4342429221931815414on_nat @ M0 @ K6 ) )
               => ~ ! [B5: multis1856558993800614429on_nat] :
                      ( ( member2988082745645132084on_nat @ B5 @ ( set_ms2805207930803453472on_nat @ K6 ) )
                     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B5 @ A4 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_1094_mult1E,axiom,
    ! [N3: multis5168333519641040071on_nat,M2: multis5168333519641040071on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ N3 @ M2 ) @ ( mult1_616571009830880372on_nat @ R ) )
     => ~ ! [A4: list_P9127870862658512897on_nat,M0: multis5168333519641040071on_nat] :
            ( ( M2
              = ( add_ms213608174324010207on_nat @ A4 @ M0 ) )
           => ! [K6: multis5168333519641040071on_nat] :
                ( ( N3
                  = ( plus_p7957816880051667710on_nat @ M0 @ K6 ) )
               => ~ ! [B5: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ B5 @ ( set_ms6185292591494683198on_nat @ K6 ) )
                     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B5 @ A4 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_1095_mult1E,axiom,
    ! [N3: multis1856558993800614429on_nat,M2: multis1856558993800614429on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ N3 @ M2 ) @ ( mult1_354936085848472602on_nat @ R ) )
     => ~ ! [A4: sum_su4907927656612468301on_nat,M0: multis1856558993800614429on_nat] :
            ( ( M2
              = ( add_ms8575691854974865391on_nat @ A4 @ M0 ) )
           => ! [K6: multis1856558993800614429on_nat] :
                ( ( N3
                  = ( plus_p2586647234311342374on_nat @ M0 @ K6 ) )
               => ~ ! [B5: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ B5 @ ( set_ms3922534532180125136on_nat @ K6 ) )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ B5 @ A4 ) @ R ) ) ) ) ) ).

% mult1E
thf(fact_1096_mult1I,axiom,
    ! [M2: multis2912115315702009697on_nat,A: produc9176241080893687067on_nat,M02: multis2912115315702009697on_nat,N3: multis2912115315702009697on_nat,K5: multis2912115315702009697on_nat,R: set_Pr1446391457923625415on_nat] :
      ( ( M2
        = ( add_ms5516047754188811001on_nat @ A @ M02 ) )
     => ( ( N3
          = ( plus_p7340530524814416408on_nat @ M02 @ K5 ) )
       => ( ! [B4: produc9176241080893687067on_nat] :
              ( ( member4278285197320822468on_nat @ B4 @ ( set_ms7239842528327418072on_nat @ K5 ) )
             => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ B4 @ A ) @ R ) )
         => ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ N3 @ M2 ) @ ( mult1_5496832715746181134on_nat @ R ) ) ) ) ) ).

% mult1I
thf(fact_1097_mult1I,axiom,
    ! [M2: multis1856558993800614429on_nat,A: sum_su4907927656612468301on_nat,M02: multis1856558993800614429on_nat,N3: multis1856558993800614429on_nat,K5: multis1856558993800614429on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( M2
        = ( add_ms8575691854974865391on_nat @ A @ M02 ) )
     => ( ( N3
          = ( plus_p2586647234311342374on_nat @ M02 @ K5 ) )
       => ( ! [B4: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ B4 @ ( set_ms3922534532180125136on_nat @ K5 ) )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ B4 @ A ) @ R ) )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ N3 @ M2 ) @ ( mult1_354936085848472602on_nat @ R ) ) ) ) ) ).

% mult1I
thf(fact_1098_mult1I,axiom,
    ! [M2: multis6850599868085929325on_nat,A: multis1856558993800614429on_nat,M02: multis6850599868085929325on_nat,N3: multis6850599868085929325on_nat,K5: multis6850599868085929325on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( M2
        = ( add_ms8432739108091334335on_nat @ A @ M02 ) )
     => ( ( N3
          = ( plus_p4342429221931815414on_nat @ M02 @ K5 ) )
       => ( ! [B4: multis1856558993800614429on_nat] :
              ( ( member2988082745645132084on_nat @ B4 @ ( set_ms2805207930803453472on_nat @ K5 ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B4 @ A ) @ R ) )
         => ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ N3 @ M2 ) @ ( mult1_6031627987982144362on_nat @ R ) ) ) ) ) ).

% mult1I
thf(fact_1099_mult1I,axiom,
    ! [M2: multis5168333519641040071on_nat,A: list_P9127870862658512897on_nat,M02: multis5168333519641040071on_nat,N3: multis5168333519641040071on_nat,K5: multis5168333519641040071on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( M2
        = ( add_ms213608174324010207on_nat @ A @ M02 ) )
     => ( ( N3
          = ( plus_p7957816880051667710on_nat @ M02 @ K5 ) )
       => ( ! [B4: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ B4 @ ( set_ms6185292591494683198on_nat @ K5 ) )
             => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B4 @ A ) @ R ) )
         => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ N3 @ M2 ) @ ( mult1_616571009830880372on_nat @ R ) ) ) ) ) ).

% mult1I
thf(fact_1100_psubsetD,axiom,
    ! [A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat,C2: produc9176241080893687067on_nat] :
      ( ( ord_le2556122027443587367on_nat @ A3 @ B3 )
     => ( ( member4278285197320822468on_nat @ C2 @ A3 )
       => ( member4278285197320822468on_nat @ C2 @ B3 ) ) ) ).

% psubsetD
thf(fact_1101_not__less__empty,axiom,
    ! [M2: multis1856558993800614429on_nat,R: set_Pr2535564044409765851on_nat] :
      ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M2 @ zero_z8702567803361615526on_nat ) @ ( mult1_354936085848472602on_nat @ R ) ) ).

% not_less_empty
thf(fact_1102_mult1__union,axiom,
    ! [B3: multis1856558993800614429on_nat,D2: multis1856558993800614429on_nat,R: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B3 @ D2 ) @ ( mult1_354936085848472602on_nat @ R ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ C3 @ B3 ) @ ( plus_p2586647234311342374on_nat @ C3 @ D2 ) ) @ ( mult1_354936085848472602on_nat @ R ) ) ) ).

% mult1_union
thf(fact_1103_shift_Ocases,axiom,
    ! [X4: produc8199716216217303280at_nat] :
      ~ ! [F2: nat > nat,J3: nat] :
          ( X4
         != ( produc72220940542539688at_nat @ F2 @ J3 ) ) ).

% shift.cases
thf(fact_1104_less__add,axiom,
    ! [N3: multis2912115315702009697on_nat,A: produc9176241080893687067on_nat,M02: multis2912115315702009697on_nat,R: set_Pr1446391457923625415on_nat] :
      ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ N3 @ ( add_ms5516047754188811001on_nat @ A @ M02 ) ) @ ( mult1_5496832715746181134on_nat @ R ) )
     => ( ? [M3: multis2912115315702009697on_nat] :
            ( ( member10543167634940048on_nat @ ( produc6755973333635976663on_nat @ M3 @ M02 ) @ ( mult1_5496832715746181134on_nat @ R ) )
            & ( N3
              = ( add_ms5516047754188811001on_nat @ A @ M3 ) ) )
        | ? [K6: multis2912115315702009697on_nat] :
            ( ! [B5: produc9176241080893687067on_nat] :
                ( ( member4278285197320822468on_nat @ B5 @ ( set_ms7239842528327418072on_nat @ K6 ) )
               => ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ B5 @ A ) @ R ) )
            & ( N3
              = ( plus_p7340530524814416408on_nat @ M02 @ K6 ) ) ) ) ) ).

% less_add
thf(fact_1105_less__add,axiom,
    ! [N3: multis6850599868085929325on_nat,A: multis1856558993800614429on_nat,M02: multis6850599868085929325on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ N3 @ ( add_ms8432739108091334335on_nat @ A @ M02 ) ) @ ( mult1_6031627987982144362on_nat @ R ) )
     => ( ? [M3: multis6850599868085929325on_nat] :
            ( ( member460288039655959140on_nat @ ( produc2379226833201183603on_nat @ M3 @ M02 ) @ ( mult1_6031627987982144362on_nat @ R ) )
            & ( N3
              = ( add_ms8432739108091334335on_nat @ A @ M3 ) ) )
        | ? [K6: multis6850599868085929325on_nat] :
            ( ! [B5: multis1856558993800614429on_nat] :
                ( ( member2988082745645132084on_nat @ B5 @ ( set_ms2805207930803453472on_nat @ K6 ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B5 @ A ) @ R ) )
            & ( N3
              = ( plus_p4342429221931815414on_nat @ M02 @ K6 ) ) ) ) ) ).

% less_add
thf(fact_1106_less__add,axiom,
    ! [N3: multis5168333519641040071on_nat,A: list_P9127870862658512897on_nat,M02: multis5168333519641040071on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ N3 @ ( add_ms213608174324010207on_nat @ A @ M02 ) ) @ ( mult1_616571009830880372on_nat @ R ) )
     => ( ? [M3: multis5168333519641040071on_nat] :
            ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ M3 @ M02 ) @ ( mult1_616571009830880372on_nat @ R ) )
            & ( N3
              = ( add_ms213608174324010207on_nat @ A @ M3 ) ) )
        | ? [K6: multis5168333519641040071on_nat] :
            ( ! [B5: list_P9127870862658512897on_nat] :
                ( ( member783415501022416426on_nat @ B5 @ ( set_ms6185292591494683198on_nat @ K6 ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B5 @ A ) @ R ) )
            & ( N3
              = ( plus_p7957816880051667710on_nat @ M02 @ K6 ) ) ) ) ) ).

% less_add
thf(fact_1107_less__add,axiom,
    ! [N3: multis1856558993800614429on_nat,A: sum_su4907927656612468301on_nat,M02: multis1856558993800614429on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ N3 @ ( add_ms8575691854974865391on_nat @ A @ M02 ) ) @ ( mult1_354936085848472602on_nat @ R ) )
     => ( ? [M3: multis1856558993800614429on_nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ M3 @ M02 ) @ ( mult1_354936085848472602on_nat @ R ) )
            & ( N3
              = ( add_ms8575691854974865391on_nat @ A @ M3 ) ) )
        | ? [K6: multis1856558993800614429on_nat] :
            ( ! [B5: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ B5 @ ( set_ms3922534532180125136on_nat @ K6 ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ B5 @ A ) @ R ) )
            & ( N3
              = ( plus_p2586647234311342374on_nat @ M02 @ K6 ) ) ) ) ) ).

% less_add
thf(fact_1108_mset__update,axiom,
    ! [I3: nat,Ls: list_S5253404628753938141on_nat,V: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Ls ) )
     => ( ( mset_S7907735292211298936on_nat @ ( list_u8577817230714425439on_nat @ Ls @ I3 @ V ) )
        = ( add_ms8575691854974865391on_nat @ V @ ( minus_6645206844887421558on_nat @ ( mset_S7907735292211298936on_nat @ Ls ) @ ( add_ms8575691854974865391on_nat @ ( nth_Su8471564416988646406on_nat @ Ls @ I3 ) @ zero_z8702567803361615526on_nat ) ) ) ) ) ).

% mset_update
thf(fact_1109_mset__update,axiom,
    ! [I3: nat,Ls: list_P9127870862658512897on_nat,V: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Ls ) )
     => ( ( mset_P6964650582947932560on_nat @ ( list_u1840724836389618409on_nat @ Ls @ I3 @ V ) )
        = ( add_ms8088439773566242649on_nat @ V @ ( minus_2641405189499471144on_nat @ ( mset_P6964650582947932560on_nat @ Ls ) @ ( add_ms8088439773566242649on_nat @ ( nth_Pr3336893728888831618on_nat @ Ls @ I3 ) @ zero_z2712726346186620152on_nat ) ) ) ) ) ).

% mset_update
thf(fact_1110_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_1111_diff__add__mset__swap,axiom,
    ! [B: produc9176241080893687067on_nat,A3: multis2912115315702009697on_nat,M2: multis2912115315702009697on_nat] :
      ( ~ ( member4278285197320822468on_nat @ B @ ( set_ms7239842528327418072on_nat @ A3 ) )
     => ( ( minus_802841720353717448on_nat @ ( add_ms5516047754188811001on_nat @ B @ M2 ) @ A3 )
        = ( add_ms5516047754188811001on_nat @ B @ ( minus_802841720353717448on_nat @ M2 @ A3 ) ) ) ) ).

% diff_add_mset_swap
thf(fact_1112_insert__DiffM,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( add_ms5516047754188811001on_nat @ X4 @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) )
        = M2 ) ) ).

% insert_DiffM
thf(fact_1113_diff__union__swap2,axiom,
    ! [Y: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat,X4: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ Y @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( minus_802841720353717448on_nat @ ( add_ms5516047754188811001on_nat @ X4 @ M2 ) @ ( add_ms5516047754188811001on_nat @ Y @ zero_z1502369238205801624on_nat ) )
        = ( add_ms5516047754188811001on_nat @ X4 @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ Y @ zero_z1502369238205801624on_nat ) ) ) ) ) ).

% diff_union_swap2
thf(fact_1114_in__diffD,axiom,
    ! [A: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat,N3: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ ( minus_802841720353717448on_nat @ M2 @ N3 ) ) )
     => ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ M2 ) ) ) ).

% in_diffD
thf(fact_1115_diff__single__trivial,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) )
        = M2 ) ) ).

% diff_single_trivial
thf(fact_1116_diff__single__eq__union,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat,N3: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) )
          = N3 )
        = ( M2
          = ( add_ms5516047754188811001on_nat @ X4 @ N3 ) ) ) ) ).

% diff_single_eq_union
thf(fact_1117_multi__drop__mem__not__eq,axiom,
    ! [C2: produc9176241080893687067on_nat,B3: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( set_ms7239842528327418072on_nat @ B3 ) )
     => ( ( minus_802841720353717448on_nat @ B3 @ ( add_ms5516047754188811001on_nat @ C2 @ zero_z1502369238205801624on_nat ) )
       != B3 ) ) ).

% multi_drop_mem_not_eq
thf(fact_1118_add__mset__remove__trivial__If,axiom,
    ! [A: produc9176241080893687067on_nat,N3: multis2912115315702009697on_nat] :
      ( ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ N3 ) )
       => ( ( add_ms5516047754188811001on_nat @ A @ ( minus_802841720353717448on_nat @ N3 @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) ) )
          = N3 ) )
      & ( ~ ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ N3 ) )
       => ( ( add_ms5516047754188811001on_nat @ A @ ( minus_802841720353717448on_nat @ N3 @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) ) )
          = ( add_ms5516047754188811001on_nat @ A @ N3 ) ) ) ) ).

% add_mset_remove_trivial_If
thf(fact_1119_add__mset__remove__trivial__eq,axiom,
    ! [N3: multis2912115315702009697on_nat,A: produc9176241080893687067on_nat] :
      ( ( N3
        = ( add_ms5516047754188811001on_nat @ A @ ( minus_802841720353717448on_nat @ N3 @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) ) ) )
      = ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ N3 ) ) ) ).

% add_mset_remove_trivial_eq
thf(fact_1120_multiset__add__sub__el__shuffle,axiom,
    ! [C2: produc9176241080893687067on_nat,B3: multis2912115315702009697on_nat,B: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( set_ms7239842528327418072on_nat @ B3 ) )
     => ( ( B != C2 )
       => ( ( add_ms5516047754188811001on_nat @ B @ ( minus_802841720353717448on_nat @ B3 @ ( add_ms5516047754188811001on_nat @ C2 @ zero_z1502369238205801624on_nat ) ) )
          = ( minus_802841720353717448on_nat @ ( add_ms5516047754188811001on_nat @ B @ B3 ) @ ( add_ms5516047754188811001on_nat @ C2 @ zero_z1502369238205801624on_nat ) ) ) ) ) ).

% multiset_add_sub_el_shuffle
thf(fact_1121_more__than__one__mset__mset__diff,axiom,
    ! [A: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) ) ) )
     => ( ( set_ms7239842528327418072on_nat @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) ) )
        = ( set_ms7239842528327418072on_nat @ M2 ) ) ) ).

% more_than_one_mset_mset_diff
thf(fact_1122_mem__multiset__diff,axiom,
    ! [X4: produc9176241080893687067on_nat,A3: multis2912115315702009697on_nat,Y: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ A3 ) )
     => ( ( X4 != Y )
       => ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ ( minus_802841720353717448on_nat @ A3 @ ( add_ms5516047754188811001on_nat @ Y @ zero_z1502369238205801624on_nat ) ) ) ) ) ) ).

% mem_multiset_diff
thf(fact_1123_insert__DiffM2,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( plus_p7340530524814416408on_nat @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) )
        = M2 ) ) ).

% insert_DiffM2
thf(fact_1124_diff__union__single__conv,axiom,
    ! [A: produc9176241080893687067on_nat,J2: multis2912115315702009697on_nat,I5: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ J2 ) )
     => ( ( minus_802841720353717448on_nat @ ( plus_p7340530524814416408on_nat @ I5 @ J2 ) @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) )
        = ( plus_p7340530524814416408on_nat @ I5 @ ( minus_802841720353717448on_nat @ J2 @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) ) ) ) ) ).

% diff_union_single_conv
thf(fact_1125_take__append,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( take_P4655611586102217833on_nat @ N @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( take_P4655611586102217833on_nat @ N @ Xs ) @ ( take_P4655611586102217833on_nat @ ( minus_minus_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) ) @ Ys ) ) ) ).

% take_append
thf(fact_1126_drop__append,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( drop_P7041600534018571713on_nat @ N @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( drop_P7041600534018571713on_nat @ N @ Xs ) @ ( drop_P7041600534018571713on_nat @ ( minus_minus_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) ) @ Ys ) ) ) ).

% drop_append
thf(fact_1127_psubset__imp__ex__mem,axiom,
    ! [A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( ord_le2556122027443587367on_nat @ A3 @ B3 )
     => ? [B4: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ B4 @ ( minus_1132796316471476066on_nat @ B3 @ A3 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_1128_nth__append,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ N )
          = ( nth_Pr3336893728888831618on_nat @ Xs @ N ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ N )
          = ( nth_Pr3336893728888831618on_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) ) ) ) ) ) ).

% nth_append
thf(fact_1129_list__update__append,axiom,
    ! [N: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] :
      ( ( ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ N @ X4 )
          = ( append2865821678206098582on_nat @ ( list_u1840724836389618409on_nat @ Xs @ N @ X4 ) @ Ys ) ) )
      & ( ~ ( ord_less_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) )
       => ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ N @ X4 )
          = ( append2865821678206098582on_nat @ Xs @ ( list_u1840724836389618409on_nat @ Ys @ ( minus_minus_nat @ N @ ( size_s7490768528479217773on_nat @ Xs ) ) @ X4 ) ) ) ) ) ).

% list_update_append
thf(fact_1130_DiffI,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ A3 )
     => ( ~ ( member4278285197320822468on_nat @ C2 @ B3 )
       => ( member4278285197320822468on_nat @ C2 @ ( minus_1132796316471476066on_nat @ A3 @ B3 ) ) ) ) ).

% DiffI
thf(fact_1131_Diff__iff,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( minus_1132796316471476066on_nat @ A3 @ B3 ) )
      = ( ( member4278285197320822468on_nat @ C2 @ A3 )
        & ~ ( member4278285197320822468on_nat @ C2 @ B3 ) ) ) ).

% Diff_iff
thf(fact_1132_single__subset__iff,axiom,
    ! [A: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( subset6465164242270104578on_nat @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) @ M2 )
      = ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ M2 ) ) ) ).

% single_subset_iff
thf(fact_1133_DiffE,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( minus_1132796316471476066on_nat @ A3 @ B3 ) )
     => ~ ( ( member4278285197320822468on_nat @ C2 @ A3 )
         => ( member4278285197320822468on_nat @ C2 @ B3 ) ) ) ).

% DiffE
thf(fact_1134_DiffD1,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( minus_1132796316471476066on_nat @ A3 @ B3 ) )
     => ( member4278285197320822468on_nat @ C2 @ A3 ) ) ).

% DiffD1
thf(fact_1135_DiffD2,axiom,
    ! [C2: produc9176241080893687067on_nat,A3: set_Pr3573155178515677307on_nat,B3: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ C2 @ ( minus_1132796316471476066on_nat @ A3 @ B3 ) )
     => ~ ( member4278285197320822468on_nat @ C2 @ B3 ) ) ).

% DiffD2
thf(fact_1136_subset__mult2__alt,axiom,
    ! [X: multis1856558993800614429on_nat,Y5: multis1856558993800614429on_nat,Z4: multis1856558993800614429on_nat,B: $o,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat,B2: $o] :
      ( ( subset6916030864335142182on_nat @ X @ Y5 )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y5 @ Z4 ) @ ( multis2224597371354980665on_nat @ B @ Ns @ S ) )
       => ( ( B
           => B2 )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X @ Z4 ) @ ( multis2224597371354980665on_nat @ B2 @ Ns @ S ) ) ) ) ) ).

% subset_mult2_alt
thf(fact_1137_mset__subset__eqD,axiom,
    ! [A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat,X4: produc9176241080893687067on_nat] :
      ( ( subset6465164242270104578on_nat @ A3 @ B3 )
     => ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ A3 ) )
       => ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ B3 ) ) ) ) ).

% mset_subset_eqD
thf(fact_1138_multi__subset__induct,axiom,
    ! [F3: multis2912115315702009697on_nat,A3: multis2912115315702009697on_nat,P2: multis2912115315702009697on_nat > $o] :
      ( ( subset6465164242270104578on_nat @ F3 @ A3 )
     => ( ( P2 @ zero_z1502369238205801624on_nat )
       => ( ! [A4: produc9176241080893687067on_nat,F4: multis2912115315702009697on_nat] :
              ( ( member4278285197320822468on_nat @ A4 @ ( set_ms7239842528327418072on_nat @ A3 ) )
             => ( ( P2 @ F4 )
               => ( P2 @ ( add_ms5516047754188811001on_nat @ A4 @ F4 ) ) ) )
         => ( P2 @ F3 ) ) ) ) ).

% multi_subset_induct
thf(fact_1139_mset__subset__eq__single,axiom,
    ! [A: produc9176241080893687067on_nat,B3: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ B3 ) )
     => ( subset6465164242270104578on_nat @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) @ B3 ) ) ).

% mset_subset_eq_single
thf(fact_1140_subseteq__mset__imp__ns__mul__ext__Id,axiom,
    ! [A3: multis1856558993800614429on_nat,B3: multis1856558993800614429on_nat,R2: set_Pr2535564044409765851on_nat] :
      ( ( subset6916030864335142182on_nat @ A3 @ B3 )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B3 @ A3 ) @ ( multis4183089776494585304on_nat @ id_Sum1147813627050573297on_nat @ R2 ) ) ) ).

% subseteq_mset_imp_ns_mul_ext_Id
thf(fact_1141_insert__subset__eq__iff,axiom,
    ! [A: produc9176241080893687067on_nat,A3: multis2912115315702009697on_nat,B3: multis2912115315702009697on_nat] :
      ( ( subset6465164242270104578on_nat @ ( add_ms5516047754188811001on_nat @ A @ A3 ) @ B3 )
      = ( ( member4278285197320822468on_nat @ A @ ( set_ms7239842528327418072on_nat @ B3 ) )
        & ( subset6465164242270104578on_nat @ A3 @ ( minus_802841720353717448on_nat @ B3 @ ( add_ms5516047754188811001on_nat @ A @ zero_z1502369238205801624on_nat ) ) ) ) ) ).

% insert_subset_eq_iff
thf(fact_1142_mset__remove1,axiom,
    ! [A: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( mset_S7907735292211298936on_nat @ ( remove1288255167417399217on_nat @ A @ Xs ) )
      = ( minus_6645206844887421558on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) @ ( add_ms8575691854974865391on_nat @ A @ zero_z8702567803361615526on_nat ) ) ) ).

% mset_remove1
thf(fact_1143_mset__remove1,axiom,
    ! [A: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( mset_P6964650582947932560on_nat @ ( remove3942973658249323799on_nat @ A @ Xs ) )
      = ( minus_2641405189499471144on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) @ ( add_ms8088439773566242649on_nat @ A @ zero_z2712726346186620152on_nat ) ) ) ).

% mset_remove1
thf(fact_1144_in__set__remove1,axiom,
    ! [A: produc9176241080893687067on_nat,B: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( A != B )
     => ( ( member4278285197320822468on_nat @ A @ ( set_Pr8272476097931642672on_nat @ ( remove8774177389924145847on_nat @ B @ Xs ) ) )
        = ( member4278285197320822468on_nat @ A @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1145_in__set__remove1,axiom,
    ! [A: produc1313295348606329723on_nat,B: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( A != B )
     => ( ( member5906418179908956708on_nat @ A @ ( set_Pr937530902923597456on_nat @ ( remove3942973658249323799on_nat @ B @ Xs ) ) )
        = ( member5906418179908956708on_nat @ A @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1146_in__set__remove1,axiom,
    ! [A: sum_su4907927656612468301on_nat,B: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( A != B )
     => ( ( member8476201367316484836on_nat @ A @ ( set_Su4848624856859743608on_nat @ ( remove1288255167417399217on_nat @ B @ Xs ) ) )
        = ( member8476201367316484836on_nat @ A @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% in_set_remove1
thf(fact_1147_length__remove1,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
       => ( ( size_s8191359103794446093on_nat @ ( remove8774177389924145847on_nat @ X4 @ Xs ) )
          = ( minus_minus_nat @ ( size_s8191359103794446093on_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
       => ( ( size_s8191359103794446093on_nat @ ( remove8774177389924145847on_nat @ X4 @ Xs ) )
          = ( size_s8191359103794446093on_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1148_length__remove1,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
       => ( ( size_s7490768528479217773on_nat @ ( remove3942973658249323799on_nat @ X4 @ Xs ) )
          = ( minus_minus_nat @ ( size_s7490768528479217773on_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
       => ( ( size_s7490768528479217773on_nat @ ( remove3942973658249323799on_nat @ X4 @ Xs ) )
          = ( size_s7490768528479217773on_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1149_length__remove1,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( ( size_s5686234912572693873on_nat @ ( remove1288255167417399217on_nat @ X4 @ Xs ) )
          = ( minus_minus_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ one_one_nat ) ) )
      & ( ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( ( size_s5686234912572693873on_nat @ ( remove1288255167417399217on_nat @ X4 @ Xs ) )
          = ( size_s5686234912572693873on_nat @ Xs ) ) ) ) ).

% length_remove1
thf(fact_1150_set__remove1__subset,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] : ( ord_le6961793561435735931on_nat @ ( set_Pr937530902923597456on_nat @ ( remove3942973658249323799on_nat @ X4 @ Xs ) ) @ ( set_Pr937530902923597456on_nat @ Xs ) ) ).

% set_remove1_subset
thf(fact_1151_set__remove1__subset,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] : ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( remove1288255167417399217on_nat @ X4 @ Xs ) ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% set_remove1_subset
thf(fact_1152_remove1__append,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat,Ys: list_P1915136757390359201on_nat] :
      ( ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
       => ( ( remove8774177389924145847on_nat @ X4 @ ( append1010812867689791286on_nat @ Xs @ Ys ) )
          = ( append1010812867689791286on_nat @ ( remove8774177389924145847on_nat @ X4 @ Xs ) @ Ys ) ) )
      & ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
       => ( ( remove8774177389924145847on_nat @ X4 @ ( append1010812867689791286on_nat @ Xs @ Ys ) )
          = ( append1010812867689791286on_nat @ Xs @ ( remove8774177389924145847on_nat @ X4 @ Ys ) ) ) ) ) ).

% remove1_append
thf(fact_1153_remove1__append,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
       => ( ( remove3942973658249323799on_nat @ X4 @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( append2865821678206098582on_nat @ ( remove3942973658249323799on_nat @ X4 @ Xs ) @ Ys ) ) )
      & ( ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
       => ( ( remove3942973658249323799on_nat @ X4 @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( append2865821678206098582on_nat @ Xs @ ( remove3942973658249323799on_nat @ X4 @ Ys ) ) ) ) ) ).

% remove1_append
thf(fact_1154_remove1__append,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( ( remove1288255167417399217on_nat @ X4 @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
          = ( append3523812819975775986on_nat @ ( remove1288255167417399217on_nat @ X4 @ Xs ) @ Ys ) ) )
      & ( ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( ( remove1288255167417399217on_nat @ X4 @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
          = ( append3523812819975775986on_nat @ Xs @ ( remove1288255167417399217on_nat @ X4 @ Ys ) ) ) ) ) ).

% remove1_append
thf(fact_1155_notin__set__remove1,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat,Y: produc9176241080893687067on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( remove8774177389924145847on_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1156_notin__set__remove1,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat] :
      ( ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( remove3942973658249323799on_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1157_notin__set__remove1,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( remove1288255167417399217on_nat @ Y @ Xs ) ) ) ) ).

% notin_set_remove1
thf(fact_1158_remove1__idem,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ( ( remove8774177389924145847on_nat @ X4 @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1159_remove1__idem,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ~ ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ( remove3942973658249323799on_nat @ X4 @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1160_remove1__idem,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ~ ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ( remove1288255167417399217on_nat @ X4 @ Xs )
        = Xs ) ) ).

% remove1_idem
thf(fact_1161_remove1_Osimps_I1_J,axiom,
    ! [X4: list_P3196869577824546119od_a_o] :
      ( ( remove4626611885917946083od_a_o @ X4 @ nil_li1372968520647675725od_a_o )
      = nil_li1372968520647675725od_a_o ) ).

% remove1.simps(1)
thf(fact_1162_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_1163_remove1__split,axiom,
    ! [A: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat,Ys: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ A @ ( set_Pr8272476097931642672on_nat @ Xs ) )
     => ( ( ( remove8774177389924145847on_nat @ A @ Xs )
          = Ys )
        = ( ? [Ls2: list_P1915136757390359201on_nat,Rs: list_P1915136757390359201on_nat] :
              ( ( Xs
                = ( append1010812867689791286on_nat @ Ls2 @ ( cons_P6897958070471385041on_nat @ A @ Rs ) ) )
              & ~ ( member4278285197320822468on_nat @ A @ ( set_Pr8272476097931642672on_nat @ Ls2 ) )
              & ( Ys
                = ( append1010812867689791286on_nat @ Ls2 @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1164_remove1__split,axiom,
    ! [A: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ A @ ( set_Pr937530902923597456on_nat @ Xs ) )
     => ( ( ( remove3942973658249323799on_nat @ A @ Xs )
          = Ys )
        = ( ? [Ls2: list_P9127870862658512897on_nat,Rs: list_P9127870862658512897on_nat] :
              ( ( Xs
                = ( append2865821678206098582on_nat @ Ls2 @ ( cons_P1582596238967110705on_nat @ A @ Rs ) ) )
              & ~ ( member5906418179908956708on_nat @ A @ ( set_Pr937530902923597456on_nat @ Ls2 ) )
              & ( Ys
                = ( append2865821678206098582on_nat @ Ls2 @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1165_remove1__split,axiom,
    ! [A: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ A @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ( ( remove1288255167417399217on_nat @ A @ Xs )
          = Ys )
        = ( ? [Ls2: list_S5253404628753938141on_nat,Rs: list_S5253404628753938141on_nat] :
              ( ( Xs
                = ( append3523812819975775986on_nat @ Ls2 @ ( cons_S6797538523646374167on_nat @ A @ Rs ) ) )
              & ~ ( member8476201367316484836on_nat @ A @ ( set_Su4848624856859743608on_nat @ Ls2 ) )
              & ( Ys
                = ( append3523812819975775986on_nat @ Ls2 @ Rs ) ) ) ) ) ) ).

% remove1_split
thf(fact_1166_take__Cons_H,axiom,
    ! [N: nat,X4: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( N = zero_zero_nat )
       => ( ( take_l5436743365630562101od_a_o @ N @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
          = nil_li1372968520647675725od_a_o ) )
      & ( ( N != zero_zero_nat )
       => ( ( take_l5436743365630562101od_a_o @ N @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
          = ( cons_l9179691953475787517od_a_o @ X4 @ ( take_l5436743365630562101od_a_o @ ( minus_minus_nat @ N @ one_one_nat ) @ Xs ) ) ) ) ) ).

% take_Cons'
thf(fact_1167_size__Diff__singleton,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( size_s4491293915487501261on_nat @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) )
        = ( minus_minus_nat @ ( size_s4491293915487501261on_nat @ M2 ) @ one_one_nat ) ) ) ).

% size_Diff_singleton
thf(fact_1168_size__mset,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( size_s708034413772738737on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% size_mset
thf(fact_1169_size__mset,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( size_s5414930716960531757on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) )
      = ( size_s7490768528479217773on_nat @ Xs ) ) ).

% size_mset
thf(fact_1170_butlast__snoc,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] :
      ( ( butlas3749561085738797005on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1171_butlast__snoc,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( butlas7945703636940585113od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) ) )
      = Xs ) ).

% butlast_snoc
thf(fact_1172_size__eq__Suc__imp__elem,axiom,
    ! [M2: multis2912115315702009697on_nat,N: nat] :
      ( ( ( size_s4491293915487501261on_nat @ M2 )
        = ( suc @ N ) )
     => ? [A4: produc9176241080893687067on_nat] : ( member4278285197320822468on_nat @ A4 @ ( set_ms7239842528327418072on_nat @ M2 ) ) ) ).

% size_eq_Suc_imp_elem
thf(fact_1173_butlast_Osimps_I2_J,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( ( Xs = nil_li1372968520647675725od_a_o )
       => ( ( butlas7945703636940585113od_a_o @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
          = nil_li1372968520647675725od_a_o ) )
      & ( ( Xs != nil_li1372968520647675725od_a_o )
       => ( ( butlas7945703636940585113od_a_o @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
          = ( cons_l9179691953475787517od_a_o @ X4 @ ( butlas7945703636940585113od_a_o @ Xs ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_1174_butlast__append,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( Ys = nil_Pr8614093721167239297on_nat )
       => ( ( butlas3749561085738797005on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( butlas3749561085738797005on_nat @ Xs ) ) )
      & ( ( Ys != nil_Pr8614093721167239297on_nat )
       => ( ( butlas3749561085738797005on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( append2865821678206098582on_nat @ Xs @ ( butlas3749561085738797005on_nat @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1175_butlast__append,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( Ys = nil_li1372968520647675725od_a_o )
       => ( ( butlas7945703636940585113od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
          = ( butlas7945703636940585113od_a_o @ Xs ) ) )
      & ( ( Ys != nil_li1372968520647675725od_a_o )
       => ( ( butlas7945703636940585113od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
          = ( append2159120467191820514od_a_o @ Xs @ ( butlas7945703636940585113od_a_o @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_1176_butlast_Osimps_I1_J,axiom,
    ( ( butlas7945703636940585113od_a_o @ nil_li1372968520647675725od_a_o )
    = nil_li1372968520647675725od_a_o ) ).

% butlast.simps(1)
thf(fact_1177_in__set__butlastD,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( butlas4897968515323772781on_nat @ Xs ) ) )
     => ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_1178_in__set__butlastD,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( butlas3749561085738797005on_nat @ Xs ) ) )
     => ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_1179_in__set__butlastD,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( butlas6846127204869900411on_nat @ Xs ) ) )
     => ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% in_set_butlastD
thf(fact_1180_in__set__butlast__appendI,axiom,
    ! [X4: produc9176241080893687067on_nat,Xs: list_P1915136757390359201on_nat,Ys: list_P1915136757390359201on_nat] :
      ( ( ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( butlas4897968515323772781on_nat @ Xs ) ) )
        | ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( butlas4897968515323772781on_nat @ Ys ) ) ) )
     => ( member4278285197320822468on_nat @ X4 @ ( set_Pr8272476097931642672on_nat @ ( butlas4897968515323772781on_nat @ ( append1010812867689791286on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1181_in__set__butlast__appendI,axiom,
    ! [X4: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( butlas3749561085738797005on_nat @ Xs ) ) )
        | ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( butlas3749561085738797005on_nat @ Ys ) ) ) )
     => ( member5906418179908956708on_nat @ X4 @ ( set_Pr937530902923597456on_nat @ ( butlas3749561085738797005on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1182_in__set__butlast__appendI,axiom,
    ! [X4: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( butlas6846127204869900411on_nat @ Xs ) ) )
        | ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( butlas6846127204869900411on_nat @ Ys ) ) ) )
     => ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ ( butlas6846127204869900411on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_1183_size__Suc__Diff1,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( suc @ ( size_s4491293915487501261on_nat @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) ) )
        = ( size_s4491293915487501261on_nat @ M2 ) ) ) ).

% size_Suc_Diff1
thf(fact_1184_size__Diff1__less,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ord_less_nat @ ( size_s4491293915487501261on_nat @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) ) @ ( size_s4491293915487501261on_nat @ M2 ) ) ) ).

% size_Diff1_less
thf(fact_1185_size__Diff2__less,axiom,
    ! [X4: produc9176241080893687067on_nat,M2: multis2912115315702009697on_nat,Y: produc9176241080893687067on_nat] :
      ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ M2 ) )
     => ( ( member4278285197320822468on_nat @ Y @ ( set_ms7239842528327418072on_nat @ M2 ) )
       => ( ord_less_nat @ ( size_s4491293915487501261on_nat @ ( minus_802841720353717448on_nat @ ( minus_802841720353717448on_nat @ M2 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) @ ( add_ms5516047754188811001on_nat @ Y @ zero_z1502369238205801624on_nat ) ) ) @ ( size_s4491293915487501261on_nat @ M2 ) ) ) ) ).

% size_Diff2_less
thf(fact_1186_size__Diff__singleton__if,axiom,
    ! [X4: produc9176241080893687067on_nat,A3: multis2912115315702009697on_nat] :
      ( ( ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ A3 ) )
       => ( ( size_s4491293915487501261on_nat @ ( minus_802841720353717448on_nat @ A3 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) )
          = ( minus_minus_nat @ ( size_s4491293915487501261on_nat @ A3 ) @ one_one_nat ) ) )
      & ( ~ ( member4278285197320822468on_nat @ X4 @ ( set_ms7239842528327418072on_nat @ A3 ) )
       => ( ( size_s4491293915487501261on_nat @ ( minus_802841720353717448on_nat @ A3 @ ( add_ms5516047754188811001on_nat @ X4 @ zero_z1502369238205801624on_nat ) ) )
          = ( size_s4491293915487501261on_nat @ A3 ) ) ) ) ).

% size_Diff_singleton_if
thf(fact_1187_append__butlast__last__id,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ( ( append2865821678206098582on_nat @ ( butlas3749561085738797005on_nat @ Xs ) @ ( cons_P1582596238967110705on_nat @ ( last_P4395001986731763098on_nat @ Xs ) @ nil_Pr8614093721167239297on_nat ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1188_append__butlast__last__id,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( append2159120467191820514od_a_o @ ( butlas7945703636940585113od_a_o @ Xs ) @ ( cons_l9179691953475787517od_a_o @ ( last_l256304712951322598od_a_o @ Xs ) @ nil_li1372968520647675725od_a_o ) )
        = Xs ) ) ).

% append_butlast_last_id
thf(fact_1189_last__list__update,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,K3: nat,X4: list_P3196869577824546119od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( ( K3
            = ( minus_minus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ one_one_nat ) )
         => ( ( last_l256304712951322598od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ K3 @ X4 ) )
            = X4 ) )
        & ( ( K3
           != ( minus_minus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ one_one_nat ) )
         => ( ( last_l256304712951322598od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ K3 @ X4 ) )
            = ( last_l256304712951322598od_a_o @ Xs ) ) ) ) ) ).

% last_list_update
thf(fact_1190_last__appendR,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( Ys != nil_Pr8614093721167239297on_nat )
     => ( ( last_P4395001986731763098on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
        = ( last_P4395001986731763098on_nat @ Ys ) ) ) ).

% last_appendR
thf(fact_1191_last__appendR,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( Ys != nil_li1372968520647675725od_a_o )
     => ( ( last_l256304712951322598od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
        = ( last_l256304712951322598od_a_o @ Ys ) ) ) ).

% last_appendR
thf(fact_1192_last__appendL,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( Ys = nil_Pr8614093721167239297on_nat )
     => ( ( last_P4395001986731763098on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
        = ( last_P4395001986731763098on_nat @ Xs ) ) ) ).

% last_appendL
thf(fact_1193_last__appendL,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( Ys = nil_li1372968520647675725od_a_o )
     => ( ( last_l256304712951322598od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
        = ( last_l256304712951322598od_a_o @ Xs ) ) ) ).

% last_appendL
thf(fact_1194_last__snoc,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat] :
      ( ( last_P4395001986731763098on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) ) )
      = X4 ) ).

% last_snoc
thf(fact_1195_last__snoc,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( last_l256304712951322598od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) ) )
      = X4 ) ).

% last_snoc
thf(fact_1196_last__zip,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( Ys != nil_li1372968520647675725od_a_o )
       => ( ( ( size_s3379185846251495609od_a_o @ Xs )
            = ( size_s3379185846251495609od_a_o @ Ys ) )
         => ( ( last_P1244636564506044294od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs @ Ys ) )
            = ( produc8077463541153823575od_a_o @ ( last_l256304712951322598od_a_o @ Xs ) @ ( last_l256304712951322598od_a_o @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1197_last__zip,axiom,
    ! [Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( Xs != nil_mu4837018292623784855on_nat )
     => ( ( Ys != nil_mu4837018292623784855on_nat )
       => ( ( ( size_s6603137509925400257on_nat @ Xs )
            = ( size_s6603137509925400257on_nat @ Ys ) )
         => ( ( last_P79704018437493818on_nat @ ( zip_mu5118375602418390771on_nat @ Xs @ Ys ) )
            = ( produc5200047356169809875on_nat @ ( last_m71750120343304254on_nat @ Xs ) @ ( last_m71750120343304254on_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1198_last__zip,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( Xs != nil_li5563851662744376407on_nat )
     => ( ( Ys != nil_Pr2927124501823088569on_nat )
       => ( ( ( size_s650155532264187265on_nat @ Xs )
            = ( size_s2647645505745046499on_nat @ Ys ) )
         => ( ( last_P5034829980338590748on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) )
            = ( produc1545487600272464053on_nat @ ( last_l2060209757957955326on_nat @ Xs ) @ ( last_P3805086993262556768on_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1199_last__zip,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( Xs != nil_li5563851662744376407on_nat )
     => ( ( Ys != nil_Pr707104663622130797on_nat )
       => ( ( ( size_s650155532264187265on_nat @ Xs )
            = ( size_s4814362452863045593on_nat @ Ys ) )
         => ( ( last_P3805086993262556768on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) )
            = ( produc4176415733022414961on_nat @ ( last_l2060209757957955326on_nat @ Xs ) @ ( last_P2091710660802692870on_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1200_last__zip,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( Xs != nil_li2682182165152866823on_nat )
     => ( ( Ys != nil_li2682182165152866823on_nat )
       => ( ( ( size_s129639936549468787on_nat @ Xs )
            = ( size_s129639936549468787on_nat @ Ys ) )
         => ( ( last_P2091710660802692870on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) )
            = ( produc7152764266822961111on_nat @ ( last_l1792177154342325920on_nat @ Xs ) @ ( last_l1792177154342325920on_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1201_last__zip,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat] :
      ( ( Xs != nil_nat_nat )
     => ( ( Ys != nil_nat )
       => ( ( ( size_s8208510060688613859at_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ( last_P1651921108130080911at_nat @ ( zip_nat_nat_nat @ Xs @ Ys ) )
            = ( produc72220940542539688at_nat @ ( last_nat_nat @ Xs ) @ ( last_nat @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1202_last__in__set,axiom,
    ! [As2: list_P1915136757390359201on_nat] :
      ( ( As2 != nil_Pr2652050626817158177on_nat )
     => ( member4278285197320822468on_nat @ ( last_P79704018437493818on_nat @ As2 ) @ ( set_Pr8272476097931642672on_nat @ As2 ) ) ) ).

% last_in_set
thf(fact_1203_last__in__set,axiom,
    ! [As2: list_l8336264072933094989od_a_o] :
      ( ( As2 != nil_li1372968520647675725od_a_o )
     => ( member5261019209154752368od_a_o @ ( last_l256304712951322598od_a_o @ As2 ) @ ( set_li2629784761229999580od_a_o @ As2 ) ) ) ).

% last_in_set
thf(fact_1204_last__in__set,axiom,
    ! [As2: list_P9127870862658512897on_nat] :
      ( ( As2 != nil_Pr8614093721167239297on_nat )
     => ( member5906418179908956708on_nat @ ( last_P4395001986731763098on_nat @ As2 ) @ ( set_Pr937530902923597456on_nat @ As2 ) ) ) ).

% last_in_set
thf(fact_1205_last__in__set,axiom,
    ! [As2: list_S5253404628753938141on_nat] :
      ( ( As2 != nil_Su3078906344068544711on_nat )
     => ( member8476201367316484836on_nat @ ( last_S444411360336806638on_nat @ As2 ) @ ( set_Su4848624856859743608on_nat @ As2 ) ) ) ).

% last_in_set
thf(fact_1206_hd__Nil__eq__last,axiom,
    ( ( hd_lis660529642729407282od_a_o @ nil_li1372968520647675725od_a_o )
    = ( last_l256304712951322598od_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% hd_Nil_eq_last
thf(fact_1207_longest__common__suffix,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
    ? [Ss: list_P9127870862658512897on_nat,Xs4: list_P9127870862658512897on_nat,Ys2: list_P9127870862658512897on_nat] :
      ( ( Xs
        = ( append2865821678206098582on_nat @ Xs4 @ Ss ) )
      & ( Ys
        = ( append2865821678206098582on_nat @ Ys2 @ Ss ) )
      & ( ( Xs4 = nil_Pr8614093721167239297on_nat )
        | ( Ys2 = nil_Pr8614093721167239297on_nat )
        | ( ( last_P4395001986731763098on_nat @ Xs4 )
         != ( last_P4395001986731763098on_nat @ Ys2 ) ) ) ) ).

% longest_common_suffix
thf(fact_1208_longest__common__suffix,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
    ? [Ss: list_l8336264072933094989od_a_o,Xs4: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o] :
      ( ( Xs
        = ( append2159120467191820514od_a_o @ Xs4 @ Ss ) )
      & ( Ys
        = ( append2159120467191820514od_a_o @ Ys2 @ Ss ) )
      & ( ( Xs4 = nil_li1372968520647675725od_a_o )
        | ( Ys2 = nil_li1372968520647675725od_a_o )
        | ( ( last_l256304712951322598od_a_o @ Xs4 )
         != ( last_l256304712951322598od_a_o @ Ys2 ) ) ) ) ).

% longest_common_suffix
thf(fact_1209_last__append,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( Ys = nil_Pr8614093721167239297on_nat )
       => ( ( last_P4395001986731763098on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( last_P4395001986731763098on_nat @ Xs ) ) )
      & ( ( Ys != nil_Pr8614093721167239297on_nat )
       => ( ( last_P4395001986731763098on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
          = ( last_P4395001986731763098on_nat @ Ys ) ) ) ) ).

% last_append
thf(fact_1210_last__append,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( Ys = nil_li1372968520647675725od_a_o )
       => ( ( last_l256304712951322598od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
          = ( last_l256304712951322598od_a_o @ Xs ) ) )
      & ( ( Ys != nil_li1372968520647675725od_a_o )
       => ( ( last_l256304712951322598od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
          = ( last_l256304712951322598od_a_o @ Ys ) ) ) ) ).

% last_append
thf(fact_1211_last_Osimps,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( ( Xs = nil_li1372968520647675725od_a_o )
       => ( ( last_l256304712951322598od_a_o @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
          = X4 ) )
      & ( ( Xs != nil_li1372968520647675725od_a_o )
       => ( ( last_l256304712951322598od_a_o @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
          = ( last_l256304712951322598od_a_o @ Xs ) ) ) ) ).

% last.simps
thf(fact_1212_last__ConsL,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( Xs = nil_li1372968520647675725od_a_o )
     => ( ( last_l256304712951322598od_a_o @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
        = X4 ) ) ).

% last_ConsL
thf(fact_1213_last__ConsR,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( last_l256304712951322598od_a_o @ ( cons_l9179691953475787517od_a_o @ X4 @ Xs ) )
        = ( last_l256304712951322598od_a_o @ Xs ) ) ) ).

% last_ConsR
thf(fact_1214_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X4: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X4 @ nil_Pr8614093721167239297on_nat ) )
        = Ys )
      = ( ( Ys != nil_Pr8614093721167239297on_nat )
        & ( ( butlas3749561085738797005on_nat @ Ys )
          = Xs )
        & ( ( last_P4395001986731763098on_nat @ Ys )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1215_snoc__eq__iff__butlast,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X4: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X4 @ nil_li1372968520647675725od_a_o ) )
        = Ys )
      = ( ( Ys != nil_li1372968520647675725od_a_o )
        & ( ( butlas7945703636940585113od_a_o @ Ys )
          = Xs )
        & ( ( last_l256304712951322598od_a_o @ Ys )
          = X4 ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_1216_last__conv__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ( last_l256304712951322598od_a_o @ Xs )
        = ( nth_li3248255784598043854od_a_o @ Xs @ ( minus_minus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ one_one_nat ) ) ) ) ).

% last_conv_nth
thf(fact_1217_Relative__Rewriting_Ortrancl__list__conv,axiom,
    ! [S: multis1856558993800614429on_nat,T2: multis1856558993800614429on_nat,R2: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ S @ T2 ) @ ( transi3597390680068855002on_nat @ R2 ) )
      = ( ? [List2: list_m351779702951725613on_nat] :
            ( ( ( last_m71750120343304254on_nat @ ( cons_m1493500683067190247on_nat @ S @ List2 ) )
              = T2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s6603137509925400257on_nat @ List2 ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( nth_mu8151576519813129558on_nat @ ( cons_m1493500683067190247on_nat @ S @ List2 ) @ I4 ) @ ( nth_mu8151576519813129558on_nat @ ( cons_m1493500683067190247on_nat @ S @ List2 ) @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ) ).

% Relative_Rewriting.rtrancl_list_conv
thf(fact_1218_Relative__Rewriting_Ortrancl__list__conv,axiom,
    ! [S: list_P9127870862658512897on_nat,T2: list_P9127870862658512897on_nat,R2: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ S @ T2 ) @ ( transi2210373407710287620on_nat @ R2 ) )
      = ( ? [List2: list_l1726306699111145479on_nat] :
            ( ( ( last_l1792177154342325920on_nat @ ( cons_l3028799188195208631on_nat @ S @ List2 ) )
              = T2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ ( size_s129639936549468787on_nat @ List2 ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ ( cons_l3028799188195208631on_nat @ S @ List2 ) @ I4 ) @ ( nth_li1302981970230039432on_nat @ ( cons_l3028799188195208631on_nat @ S @ List2 ) @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ) ).

% Relative_Rewriting.rtrancl_list_conv
thf(fact_1219_listrel__rtrancl__refl,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Xs ) @ ( listre6301251474747857073on_nat @ ( transi2539530154838355966on_nat @ R ) ) ) ).

% listrel_rtrancl_refl
thf(fact_1220_rtrancl__listrel1__ConsI1,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,X4: produc1313295348606329723on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( transi2210373407710287620on_nat @ ( listre7129750038259667930on_nat @ R ) ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ ( cons_P1582596238967110705on_nat @ X4 @ Ys ) ) @ ( transi2210373407710287620on_nat @ ( listre7129750038259667930on_nat @ R ) ) ) ) ).

% rtrancl_listrel1_ConsI1
thf(fact_1221_rtrancl__listrel1__eq__len,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( transi2210373407710287620on_nat @ ( listre7129750038259667930on_nat @ R ) ) )
     => ( ( size_s7490768528479217773on_nat @ X4 )
        = ( size_s7490768528479217773on_nat @ Y ) ) ) ).

% rtrancl_listrel1_eq_len
thf(fact_1222_chain__imp__rtrancl,axiom,
    ! [S2: nat > multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,I3: nat,J: nat] :
      ( ! [I: nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( S2 @ I ) @ ( S2 @ ( suc @ I ) ) ) @ R )
     => ( ( ord_less_eq_nat @ I3 @ J )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( S2 @ I3 ) @ ( S2 @ J ) ) @ ( transi3597390680068855002on_nat @ R ) ) ) ) ).

% chain_imp_rtrancl
thf(fact_1223_chain__imp__rtrancl,axiom,
    ! [S2: nat > list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,I3: nat,J: nat] :
      ( ! [I: nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( S2 @ I ) @ ( S2 @ ( suc @ I ) ) ) @ R )
     => ( ( ord_less_eq_nat @ I3 @ J )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( S2 @ I3 ) @ ( S2 @ J ) ) @ ( transi2210373407710287620on_nat @ R ) ) ) ) ).

% chain_imp_rtrancl
thf(fact_1224_listrel__rtrancl__trans,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre6301251474747857073on_nat @ ( transi2539530154838355966on_nat @ R ) ) )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Ys @ Zs ) @ ( listre6301251474747857073on_nat @ ( transi2539530154838355966on_nat @ R ) ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Zs ) @ ( listre6301251474747857073on_nat @ ( transi2539530154838355966on_nat @ R ) ) ) ) ) ).

% listrel_rtrancl_trans
thf(fact_1225_rtrancl__listrel1__ConsI2,axiom,
    ! [X4: produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat,R: set_Pr8202701763255893447on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X4 @ Y ) @ ( transi2539530154838355966on_nat @ R ) )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( transi2210373407710287620on_nat @ ( listre7129750038259667930on_nat @ R ) ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X4 @ Xs ) @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) @ ( transi2210373407710287620on_nat @ ( listre7129750038259667930on_nat @ R ) ) ) ) ) ).

% rtrancl_listrel1_ConsI2
thf(fact_1226_rtrancl__listrel1__ConsI2,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,Xs: list_m351779702951725613on_nat,Ys: list_m351779702951725613on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ Xs @ Ys ) @ ( transi87064574546512490on_nat @ ( listre7145697762236742142on_nat @ R ) ) )
       => ( member51483655959556068on_nat @ ( produc96799490136572147on_nat @ ( cons_m1493500683067190247on_nat @ X4 @ Xs ) @ ( cons_m1493500683067190247on_nat @ Y @ Ys ) ) @ ( transi87064574546512490on_nat @ ( listre7145697762236742142on_nat @ R ) ) ) ) ) ).

% rtrancl_listrel1_ConsI2
thf(fact_1227_rtrancl__listrel1__ConsI2,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ Xs @ Ys ) @ ( transi7002813000506487178on_nat @ ( listre2468548762442407136on_nat @ R ) ) )
       => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ ( cons_l3028799188195208631on_nat @ X4 @ Xs ) @ ( cons_l3028799188195208631on_nat @ Y @ Ys ) ) @ ( transi7002813000506487178on_nat @ ( listre2468548762442407136on_nat @ R ) ) ) ) ) ).

% rtrancl_listrel1_ConsI2
thf(fact_1228_listrel__reflcl__if__listrel1,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre6301251474747857073on_nat @ ( transi2539530154838355966on_nat @ R ) ) ) ) ).

% listrel_reflcl_if_listrel1
thf(fact_1229_rtrancl__listrel1__if__listrel,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre6301251474747857073on_nat @ R ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( transi2210373407710287620on_nat @ ( listre7129750038259667930on_nat @ R ) ) ) ) ).

% rtrancl_listrel1_if_listrel
thf(fact_1230_rtrancl__imp__seq,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( transi3597390680068855002on_nat @ R ) )
     => ? [F2: nat > multis1856558993800614429on_nat,N5: nat] :
          ( ( ( F2 @ zero_zero_nat )
            = X4 )
          & ( ( F2 @ N5 )
            = Y )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N5 )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R ) ) ) ) ).

% rtrancl_imp_seq
thf(fact_1231_rtrancl__imp__seq,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( transi2210373407710287620on_nat @ R ) )
     => ? [F2: nat > list_P9127870862658512897on_nat,N5: nat] :
          ( ( ( F2 @ zero_zero_nat )
            = X4 )
          & ( ( F2 @ N5 )
            = Y )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ N5 )
             => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) ) @ R ) ) ) ) ).

% rtrancl_imp_seq
thf(fact_1232_rtrancl__fun__conv,axiom,
    ! [S: multis1856558993800614429on_nat,T2: multis1856558993800614429on_nat,R2: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ S @ T2 ) @ ( transi3597390680068855002on_nat @ R2 ) )
      = ( ? [F5: nat > multis1856558993800614429on_nat,N2: nat] :
            ( ( ( F5 @ zero_zero_nat )
              = S )
            & ( ( F5 @ N2 )
              = T2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ N2 )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ) ).

% rtrancl_fun_conv
thf(fact_1233_rtrancl__fun__conv,axiom,
    ! [S: list_P9127870862658512897on_nat,T2: list_P9127870862658512897on_nat,R2: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ S @ T2 ) @ ( transi2210373407710287620on_nat @ R2 ) )
      = ( ? [F5: nat > list_P9127870862658512897on_nat,N2: nat] :
            ( ( ( F5 @ zero_zero_nat )
              = S )
            & ( ( F5 @ N2 )
              = T2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ N2 )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( F5 @ I4 ) @ ( F5 @ ( suc @ I4 ) ) ) @ R2 ) ) ) ) ) ).

% rtrancl_fun_conv
thf(fact_1234_rtrancl__induct2,axiom,
    ! [Ax: multis1856558993800614429on_nat,Ay: multis1856558993800614429on_nat,Bx: multis1856558993800614429on_nat,By: multis1856558993800614429on_nat,R: set_Pr1446391457923625415on_nat,P2: multis1856558993800614429on_nat > multis1856558993800614429on_nat > $o] :
      ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ Ax @ Ay ) @ ( produc5200047356169809875on_nat @ Bx @ By ) ) @ ( transi8183264230337399454on_nat @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Aa: multis1856558993800614429on_nat,Ba: multis1856558993800614429on_nat] :
              ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ Ax @ Ay ) @ ( produc5200047356169809875on_nat @ A4 @ B4 ) ) @ ( transi8183264230337399454on_nat @ R ) )
             => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( produc5200047356169809875on_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B4 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1235_rtrancl__induct2,axiom,
    ! [Ax: list_S5253404628753938141on_nat,Ay: produc118098540238352959on_nat,Bx: list_S5253404628753938141on_nat,By: produc118098540238352959on_nat,R: set_Pr226151479150194759on_nat,P2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ Ax @ Ay ) @ ( produc1545487600272464053on_nat @ Bx @ By ) ) @ ( transi2063249972354845312on_nat @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: list_S5253404628753938141on_nat,B4: produc118098540238352959on_nat,Aa: list_S5253404628753938141on_nat,Ba: produc118098540238352959on_nat] :
              ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ Ax @ Ay ) @ ( produc1545487600272464053on_nat @ A4 @ B4 ) ) @ ( transi2063249972354845312on_nat @ R ) )
             => ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ A4 @ B4 ) @ ( produc1545487600272464053on_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B4 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1236_rtrancl__induct2,axiom,
    ! [Ax: list_S5253404628753938141on_nat,Ay: produc1714117091004681831on_nat,Bx: list_S5253404628753938141on_nat,By: produc1714117091004681831on_nat,R: set_Pr1367747515129921727on_nat,P2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ Ax @ Ay ) @ ( produc4176415733022414961on_nat @ Bx @ By ) ) @ ( transi4207821869261036284on_nat @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: list_S5253404628753938141on_nat,B4: produc1714117091004681831on_nat,Aa: list_S5253404628753938141on_nat,Ba: produc1714117091004681831on_nat] :
              ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ Ax @ Ay ) @ ( produc4176415733022414961on_nat @ A4 @ B4 ) ) @ ( transi4207821869261036284on_nat @ R ) )
             => ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ A4 @ B4 ) @ ( produc4176415733022414961on_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B4 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1237_rtrancl__induct2,axiom,
    ! [Ax: list_P9127870862658512897on_nat,Ay: list_P9127870862658512897on_nat,Bx: list_P9127870862658512897on_nat,By: list_P9127870862658512897on_nat,R: set_Pr3667176616129746119on_nat,P2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o] :
      ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ Ax @ Ay ) @ ( produc7152764266822961111on_nat @ Bx @ By ) ) @ ( transi7059048071873120106on_nat @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: list_P9127870862658512897on_nat,B4: list_P9127870862658512897on_nat,Aa: list_P9127870862658512897on_nat,Ba: list_P9127870862658512897on_nat] :
              ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ Ax @ Ay ) @ ( produc7152764266822961111on_nat @ A4 @ B4 ) ) @ ( transi7059048071873120106on_nat @ R ) )
             => ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ A4 @ B4 ) @ ( produc7152764266822961111on_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B4 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1238_rtrancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R: set_Pr7047708467701517959at_nat,P2: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R ) )
     => ( ( P2 @ Ax @ Ay )
       => ( ! [A4: nat > nat,B4: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ A4 @ B4 ) ) @ ( transi3422749762518297843at_nat @ R ) )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B4 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R )
               => ( ( P2 @ A4 @ B4 )
                 => ( P2 @ Aa @ Ba ) ) ) )
         => ( P2 @ Bx @ By ) ) ) ) ).

% rtrancl_induct2
thf(fact_1239_converse__rtranclE2,axiom,
    ! [Xa2: multis1856558993800614429on_nat,Xb: multis1856558993800614429on_nat,Za: multis1856558993800614429on_nat,Zb: multis1856558993800614429on_nat,R: set_Pr1446391457923625415on_nat] :
      ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ Xa2 @ Xb ) @ ( produc5200047356169809875on_nat @ Za @ Zb ) ) @ ( transi8183264230337399454on_nat @ R ) )
     => ( ( ( produc5200047356169809875on_nat @ Xa2 @ Xb )
         != ( produc5200047356169809875on_nat @ Za @ Zb ) )
       => ~ ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat] :
              ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ Xa2 @ Xb ) @ ( produc5200047356169809875on_nat @ A4 @ B4 ) ) @ R )
             => ~ ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( produc5200047356169809875on_nat @ Za @ Zb ) ) @ ( transi8183264230337399454on_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1240_converse__rtranclE2,axiom,
    ! [Xa2: list_S5253404628753938141on_nat,Xb: produc118098540238352959on_nat,Za: list_S5253404628753938141on_nat,Zb: produc118098540238352959on_nat,R: set_Pr226151479150194759on_nat] :
      ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ Xa2 @ Xb ) @ ( produc1545487600272464053on_nat @ Za @ Zb ) ) @ ( transi2063249972354845312on_nat @ R ) )
     => ( ( ( produc1545487600272464053on_nat @ Xa2 @ Xb )
         != ( produc1545487600272464053on_nat @ Za @ Zb ) )
       => ~ ! [A4: list_S5253404628753938141on_nat,B4: produc118098540238352959on_nat] :
              ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ Xa2 @ Xb ) @ ( produc1545487600272464053on_nat @ A4 @ B4 ) ) @ R )
             => ~ ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ A4 @ B4 ) @ ( produc1545487600272464053on_nat @ Za @ Zb ) ) @ ( transi2063249972354845312on_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1241_converse__rtranclE2,axiom,
    ! [Xa2: list_S5253404628753938141on_nat,Xb: produc1714117091004681831on_nat,Za: list_S5253404628753938141on_nat,Zb: produc1714117091004681831on_nat,R: set_Pr1367747515129921727on_nat] :
      ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ Xa2 @ Xb ) @ ( produc4176415733022414961on_nat @ Za @ Zb ) ) @ ( transi4207821869261036284on_nat @ R ) )
     => ( ( ( produc4176415733022414961on_nat @ Xa2 @ Xb )
         != ( produc4176415733022414961on_nat @ Za @ Zb ) )
       => ~ ! [A4: list_S5253404628753938141on_nat,B4: produc1714117091004681831on_nat] :
              ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ Xa2 @ Xb ) @ ( produc4176415733022414961on_nat @ A4 @ B4 ) ) @ R )
             => ~ ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ A4 @ B4 ) @ ( produc4176415733022414961on_nat @ Za @ Zb ) ) @ ( transi4207821869261036284on_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1242_converse__rtranclE2,axiom,
    ! [Xa2: list_P9127870862658512897on_nat,Xb: list_P9127870862658512897on_nat,Za: list_P9127870862658512897on_nat,Zb: list_P9127870862658512897on_nat,R: set_Pr3667176616129746119on_nat] :
      ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ Xa2 @ Xb ) @ ( produc7152764266822961111on_nat @ Za @ Zb ) ) @ ( transi7059048071873120106on_nat @ R ) )
     => ( ( ( produc7152764266822961111on_nat @ Xa2 @ Xb )
         != ( produc7152764266822961111on_nat @ Za @ Zb ) )
       => ~ ! [A4: list_P9127870862658512897on_nat,B4: list_P9127870862658512897on_nat] :
              ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ Xa2 @ Xb ) @ ( produc7152764266822961111on_nat @ A4 @ B4 ) ) @ R )
             => ~ ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ A4 @ B4 ) @ ( produc7152764266822961111on_nat @ Za @ Zb ) ) @ ( transi7059048071873120106on_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1243_converse__rtranclE2,axiom,
    ! [Xa2: nat > nat,Xb: nat,Za: nat > nat,Zb: nat,R: set_Pr7047708467701517959at_nat] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa2 @ Xb ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R ) )
     => ( ( ( produc72220940542539688at_nat @ Xa2 @ Xb )
         != ( produc72220940542539688at_nat @ Za @ Zb ) )
       => ~ ! [A4: nat > nat,B4: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Xa2 @ Xb ) @ ( produc72220940542539688at_nat @ A4 @ B4 ) ) @ R )
             => ~ ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B4 ) @ ( produc72220940542539688at_nat @ Za @ Zb ) ) @ ( transi3422749762518297843at_nat @ R ) ) ) ) ) ).

% converse_rtranclE2
thf(fact_1244_converse__rtrancl__induct2,axiom,
    ! [Ax: multis1856558993800614429on_nat,Ay: multis1856558993800614429on_nat,Bx: multis1856558993800614429on_nat,By: multis1856558993800614429on_nat,R: set_Pr1446391457923625415on_nat,P2: multis1856558993800614429on_nat > multis1856558993800614429on_nat > $o] :
      ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ Ax @ Ay ) @ ( produc5200047356169809875on_nat @ Bx @ By ) ) @ ( transi8183264230337399454on_nat @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Aa: multis1856558993800614429on_nat,Ba: multis1856558993800614429on_nat] :
              ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( produc5200047356169809875on_nat @ Aa @ Ba ) ) @ R )
             => ( ( member238560603788358672on_nat @ ( produc1755979922506592855on_nat @ ( produc5200047356169809875on_nat @ Aa @ Ba ) @ ( produc5200047356169809875on_nat @ Bx @ By ) ) @ ( transi8183264230337399454on_nat @ R ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B4 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1245_converse__rtrancl__induct2,axiom,
    ! [Ax: list_S5253404628753938141on_nat,Ay: produc118098540238352959on_nat,Bx: list_S5253404628753938141on_nat,By: produc118098540238352959on_nat,R: set_Pr226151479150194759on_nat,P2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ Ax @ Ay ) @ ( produc1545487600272464053on_nat @ Bx @ By ) ) @ ( transi2063249972354845312on_nat @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: list_S5253404628753938141on_nat,B4: produc118098540238352959on_nat,Aa: list_S5253404628753938141on_nat,Ba: produc118098540238352959on_nat] :
              ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ A4 @ B4 ) @ ( produc1545487600272464053on_nat @ Aa @ Ba ) ) @ R )
             => ( ( member5431087842157015696on_nat @ ( produc1374777139333890519on_nat @ ( produc1545487600272464053on_nat @ Aa @ Ba ) @ ( produc1545487600272464053on_nat @ Bx @ By ) ) @ ( transi2063249972354845312on_nat @ R ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B4 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1246_converse__rtrancl__induct2,axiom,
    ! [Ax: list_S5253404628753938141on_nat,Ay: produc1714117091004681831on_nat,Bx: list_S5253404628753938141on_nat,By: produc1714117091004681831on_nat,R: set_Pr1367747515129921727on_nat,P2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ Ax @ Ay ) @ ( produc4176415733022414961on_nat @ Bx @ By ) ) @ ( transi4207821869261036284on_nat @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: list_S5253404628753938141on_nat,B4: produc1714117091004681831on_nat,Aa: list_S5253404628753938141on_nat,Ba: produc1714117091004681831on_nat] :
              ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ A4 @ B4 ) @ ( produc4176415733022414961on_nat @ Aa @ Ba ) ) @ R )
             => ( ( member8718785170798833928on_nat @ ( produc5289854005164020503on_nat @ ( produc4176415733022414961on_nat @ Aa @ Ba ) @ ( produc4176415733022414961on_nat @ Bx @ By ) ) @ ( transi4207821869261036284on_nat @ R ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B4 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1247_converse__rtrancl__induct2,axiom,
    ! [Ax: list_P9127870862658512897on_nat,Ay: list_P9127870862658512897on_nat,Bx: list_P9127870862658512897on_nat,By: list_P9127870862658512897on_nat,R: set_Pr3667176616129746119on_nat,P2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o] :
      ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ Ax @ Ay ) @ ( produc7152764266822961111on_nat @ Bx @ By ) ) @ ( transi7059048071873120106on_nat @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: list_P9127870862658512897on_nat,B4: list_P9127870862658512897on_nat,Aa: list_P9127870862658512897on_nat,Ba: list_P9127870862658512897on_nat] :
              ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ A4 @ B4 ) @ ( produc7152764266822961111on_nat @ Aa @ Ba ) ) @ R )
             => ( ( member5584747866559478544on_nat @ ( produc914474163118732119on_nat @ ( produc7152764266822961111on_nat @ Aa @ Ba ) @ ( produc7152764266822961111on_nat @ Bx @ By ) ) @ ( transi7059048071873120106on_nat @ R ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B4 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1248_converse__rtrancl__induct2,axiom,
    ! [Ax: nat > nat,Ay: nat,Bx: nat > nat,By: nat,R: set_Pr7047708467701517959at_nat,P2: ( nat > nat ) > nat > $o] :
      ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Ax @ Ay ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R ) )
     => ( ( P2 @ Bx @ By )
       => ( ! [A4: nat > nat,B4: nat,Aa: nat > nat,Ba: nat] :
              ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ A4 @ B4 ) @ ( produc72220940542539688at_nat @ Aa @ Ba ) ) @ R )
             => ( ( member3795392313164512464at_nat @ ( produc2768974569381949847at_nat @ ( produc72220940542539688at_nat @ Aa @ Ba ) @ ( produc72220940542539688at_nat @ Bx @ By ) ) @ ( transi3422749762518297843at_nat @ R ) )
               => ( ( P2 @ Aa @ Ba )
                 => ( P2 @ A4 @ B4 ) ) ) )
         => ( P2 @ Ax @ Ay ) ) ) ) ).

% converse_rtrancl_induct2
thf(fact_1249_converse__rtrancl__into__rtrancl,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,C2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ R )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B @ C2 ) @ ( transi3597390680068855002on_nat @ R ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ C2 ) @ ( transi3597390680068855002on_nat @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_1250_converse__rtrancl__into__rtrancl,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,C2: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ R )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B @ C2 ) @ ( transi2210373407710287620on_nat @ R ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ C2 ) @ ( transi2210373407710287620on_nat @ R ) ) ) ) ).

% converse_rtrancl_into_rtrancl
thf(fact_1251_converse__rtrancl__induct,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,P2: multis1856558993800614429on_nat > $o] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( P2 @ B )
       => ( ! [Y3: multis1856558993800614429on_nat,Z: multis1856558993800614429on_nat] :
              ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ R )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Z @ B ) @ ( transi3597390680068855002on_nat @ R ) )
               => ( ( P2 @ Z )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_1252_converse__rtrancl__induct,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,P2: list_P9127870862658512897on_nat > $o] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( P2 @ B )
       => ( ! [Y3: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] :
              ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ R )
             => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Z @ B ) @ ( transi2210373407710287620on_nat @ R ) )
               => ( ( P2 @ Z )
                 => ( P2 @ Y3 ) ) ) )
         => ( P2 @ A ) ) ) ) ).

% converse_rtrancl_induct
thf(fact_1253_converse__rtranclE,axiom,
    ! [X4: multis1856558993800614429on_nat,Z6: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Z6 ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( X4 != Z6 )
       => ~ ! [Y3: multis1856558993800614429on_nat] :
              ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y3 ) @ R )
             => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z6 ) @ ( transi3597390680068855002on_nat @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_1254_converse__rtranclE,axiom,
    ! [X4: list_P9127870862658512897on_nat,Z6: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Z6 ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( X4 != Z6 )
       => ~ ! [Y3: list_P9127870862658512897on_nat] :
              ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y3 ) @ R )
             => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z6 ) @ ( transi2210373407710287620on_nat @ R ) ) ) ) ) ).

% converse_rtranclE
thf(fact_1255_rtrancl__induct,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,P2: multis1856558993800614429on_nat > $o] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( P2 @ A )
       => ( ! [Y3: multis1856558993800614429on_nat,Z: multis1856558993800614429on_nat] :
              ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ Y3 ) @ ( transi3597390680068855002on_nat @ R ) )
             => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ Z ) @ R )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_1256_rtrancl__induct,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,P2: list_P9127870862658512897on_nat > $o] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( P2 @ A )
       => ( ! [Y3: list_P9127870862658512897on_nat,Z: list_P9127870862658512897on_nat] :
              ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ Y3 ) @ ( transi2210373407710287620on_nat @ R ) )
             => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ Z ) @ R )
               => ( ( P2 @ Y3 )
                 => ( P2 @ Z ) ) ) )
         => ( P2 @ B ) ) ) ) ).

% rtrancl_induct
thf(fact_1257_rtrancl__trans,axiom,
    ! [X4: multis1856558993800614429on_nat,Y: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,Z6: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Y ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y @ Z6 ) @ ( transi3597390680068855002on_nat @ R ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X4 @ Z6 ) @ ( transi3597390680068855002on_nat @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_1258_rtrancl__trans,axiom,
    ! [X4: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,Z6: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y @ Z6 ) @ ( transi2210373407710287620on_nat @ R ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Z6 ) @ ( transi2210373407710287620on_nat @ R ) ) ) ) ).

% rtrancl_trans
thf(fact_1259_rtranclE,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( A != B )
       => ~ ! [Y3: multis1856558993800614429on_nat] :
              ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ Y3 ) @ ( transi3597390680068855002on_nat @ R ) )
             => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_1260_rtranclE,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( A != B )
       => ~ ! [Y3: list_P9127870862658512897on_nat] :
              ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ Y3 ) @ ( transi2210373407710287620on_nat @ R ) )
             => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ B ) @ R ) ) ) ) ).

% rtranclE
thf(fact_1261_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat,C2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B @ C2 ) @ R )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ C2 ) @ ( transi3597390680068855002on_nat @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_1262_rtrancl_Ortrancl__into__rtrancl,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat,C2: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B @ C2 ) @ R )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ C2 ) @ ( transi2210373407710287620on_nat @ R ) ) ) ) ).

% rtrancl.rtrancl_into_rtrancl
thf(fact_1263_rtrancl_Ortrancl__refl,axiom,
    ! [A: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ A ) @ ( transi3597390680068855002on_nat @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_1264_rtrancl_Ortrancl__refl,axiom,
    ! [A: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ A ) @ ( transi2210373407710287620on_nat @ R ) ) ).

% rtrancl.rtrancl_refl
thf(fact_1265_rtrancl_Osimps,axiom,
    ! [A13: multis1856558993800614429on_nat,A24: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A13 @ A24 ) @ ( transi3597390680068855002on_nat @ R ) )
      = ( ? [A5: multis1856558993800614429on_nat] :
            ( ( A13 = A5 )
            & ( A24 = A5 ) )
        | ? [A5: multis1856558993800614429on_nat,B8: multis1856558993800614429on_nat,C5: multis1856558993800614429on_nat] :
            ( ( A13 = A5 )
            & ( A24 = C5 )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A5 @ B8 ) @ ( transi3597390680068855002on_nat @ R ) )
            & ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B8 @ C5 ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_1266_rtrancl_Osimps,axiom,
    ! [A13: list_P9127870862658512897on_nat,A24: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A13 @ A24 ) @ ( transi2210373407710287620on_nat @ R ) )
      = ( ? [A5: list_P9127870862658512897on_nat] :
            ( ( A13 = A5 )
            & ( A24 = A5 ) )
        | ? [A5: list_P9127870862658512897on_nat,B8: list_P9127870862658512897on_nat,C5: list_P9127870862658512897on_nat] :
            ( ( A13 = A5 )
            & ( A24 = C5 )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A5 @ B8 ) @ ( transi2210373407710287620on_nat @ R ) )
            & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B8 @ C5 ) @ R ) ) ) ) ).

% rtrancl.simps
thf(fact_1267_rtrancl_Ocases,axiom,
    ! [A13: multis1856558993800614429on_nat,A24: multis1856558993800614429on_nat,R: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A13 @ A24 ) @ ( transi3597390680068855002on_nat @ R ) )
     => ( ( A24 != A13 )
       => ~ ! [B4: multis1856558993800614429on_nat] :
              ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A13 @ B4 ) @ ( transi3597390680068855002on_nat @ R ) )
             => ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B4 @ A24 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_1268_rtrancl_Ocases,axiom,
    ! [A13: list_P9127870862658512897on_nat,A24: list_P9127870862658512897on_nat,R: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A13 @ A24 ) @ ( transi2210373407710287620on_nat @ R ) )
     => ( ( A24 != A13 )
       => ~ ! [B4: list_P9127870862658512897on_nat] :
              ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A13 @ B4 ) @ ( transi2210373407710287620on_nat @ R ) )
             => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ B4 @ A24 ) @ R ) ) ) ) ).

% rtrancl.cases
thf(fact_1269_in__rtrancl__UnI,axiom,
    ! [X4: produc9176241080893687067on_nat,R2: set_Pr3573155178515677307on_nat,S2: set_Pr3573155178515677307on_nat] :
      ( ( ( member4278285197320822468on_nat @ X4 @ ( transi3597390680068855002on_nat @ R2 ) )
        | ( member4278285197320822468on_nat @ X4 @ ( transi3597390680068855002on_nat @ S2 ) ) )
     => ( member4278285197320822468on_nat @ X4 @ ( transi3597390680068855002on_nat @ ( sup_su588211329720815695on_nat @ R2 @ S2 ) ) ) ) ).

% in_rtrancl_UnI
thf(fact_1270_rtrancl__Un__separator__converseE,axiom,
    ! [A: multis1856558993800614429on_nat,B: multis1856558993800614429on_nat,P2: set_Pr3573155178515677307on_nat,Q: set_Pr3573155178515677307on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( transi3597390680068855002on_nat @ ( sup_su588211329720815695on_nat @ P2 @ Q ) ) )
     => ( ! [X3: multis1856558993800614429on_nat,Y3: multis1856558993800614429on_nat] :
            ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ X3 @ B ) @ ( transi3597390680068855002on_nat @ P2 ) )
           => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Y3 @ X3 ) @ Q )
             => ( Y3 = X3 ) ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A @ B ) @ ( transi3597390680068855002on_nat @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE
thf(fact_1271_rtrancl__Un__separator__converseE,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,P2: set_Pr8586234676504894535on_nat,Q: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ ( transi2210373407710287620on_nat @ ( sup_su5722365171377032731on_nat @ P2 @ Q ) ) )
     => ( ! [X3: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
            ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ B ) @ ( transi2210373407710287620on_nat @ P2 ) )
           => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y3 @ X3 ) @ Q )
             => ( Y3 = X3 ) ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B ) @ ( transi2210373407710287620on_nat @ P2 ) ) ) ) ).

% rtrancl_Un_separator_converseE

% Conjectures (1)
thf(conj_0,conjecture,
    ( ( ? [Beta3: a > $o] : ( propos4091448439431948687_cnf_a @ Beta3 @ cnf ) )
    = ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ left ) @ ( mset_S7907735292211298936on_nat @ right ) ) @ ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ nss ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ).

%------------------------------------------------------------------------------