TPTP Problem File: SLH0406^1.p

%------------------------------------------------------------------------------
% File     : SLH0000^1 : TPTP v8.2.0. Released v8.2.0.
% Domain   : Archive of Formal Proofs
% Problem  :
% Version  : Especial.
% English  :

% Refs     : [Des23] Desharnais (2023), Email to Geoff Sutcliffe
% Source   : [Des23]
% Names    : Multiset_Ordering_NPC/0003_Multiset_Ordering_NP_Hard/prob_00194_009084__14143314_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 2091 ( 583 unt; 823 typ;   0 def)
%            Number of atoms       : 3783 (1479 equ;   0 cnn)
%            Maximal formula atoms :   12 (   2 avg)
%            Number of connectives : 12896 ( 309   ~;  60   |; 359   &;10678   @)
%                                         (   0 <=>;1490  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   22 (   7 avg)
%            Number of types       :  173 ( 172 usr)
%            Number of type conns  : 2342 (2342   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  654 ( 651 usr;  51 con; 0-3 aty)
%            Number of variables   : 4330 ( 443   ^;3701   !; 186   ?;4330   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:35.994
%------------------------------------------------------------------------------
% Could-be-implicit typings (172)
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    member5323513488200936985od_a_o: produc6642649676046662896od_a_o > set_Pr8005900414960172880od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J,type,
    member7446769889882171317tation: produc27318731269267614tation > set_Pr3525538344199662036tation > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Nat__Onat_J,type,
    member5724188588386418708_a_nat: product_prod_a_nat > set_Pr4934435412358123699_a_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    member965141594903935763od_a_o: produc415918608035012970od_a_o > set_Pr168325451112473930od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    member891341725418739729on_nat: produc7439768882380032634on_nat > set_Pr4636495525820713520on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_a > $o ).

thf(sy_c_member_001t__Set__Oset_It__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    member2068264008706428752od_a_o: set_li5011860890973139239od_a_o > set_se7281003012898648327od_a_o > $o ).

thf(sy_c_member_001t__Set__Oset_It__Nat__Onat_J,type,
    member_set_nat: set_nat > set_set_nat > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    member8594096980020929482od_a_o: set_Product_prod_a_o > set_se5048426422324813441od_a_o > $o ).

thf(sy_c_member_001t__Set__Oset_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    member7157586822906380314on_nat: set_Su266273576555847555on_nat > set_se4637380921049203129on_nat > $o ).

thf(sy_c_member_001t__Set__Oset_Itf__a_J,type,
    member_set_a: set_a > set_set_a > $o ).

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

thf(sy_c_member_001tf__a,type,
    member_a2: a > set_a > $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__092_060beta_062____,type,
    beta: a > $o ).

thf(sy_v_b____,type,
    b: $o ).

thf(sy_v_c____,type,
    c: list_P3196869577824546119od_a_o ).

thf(sy_v_cnf,type,
    cnf: list_l8336264072933094989od_a_o ).

thf(sy_v_cs____,type,
    cs: list_nat ).

thf(sy_v_i____,type,
    i: nat ).

thf(sy_v_l____,type,
    l: product_prod_a_o ).

thf(sy_v_left,type,
    left: list_S5253404628753938141on_nat ).

thf(sy_v_ns__left____,type,
    ns_left: list_S5253404628753938141on_nat ).

thf(sy_v_ns__right____,type,
    ns_right: list_S5253404628753938141on_nat ).

thf(sy_v_right,type,
    right: list_S5253404628753938141on_nat ).

thf(sy_v_s____,type,
    s2: sum_su4907927656612468301on_nat ).

thf(sy_v_s__left____,type,
    s_left: list_S5253404628753938141on_nat ).

thf(sy_v_s__right____,type,
    s_right: list_S5253404628753938141on_nat ).

thf(sy_v_x____,type,
    x: a ).

thf(sy_v_xs____,type,
    xs: list_a ).

% Relevant facts (1264)
thf(fact_0_mem,axiom,
    member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ x @ b ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ).

% mem
thf(fact_1__092_060open_062_092_060beta_062_Ax_A_092_060noteq_062_Ab_092_060close_062,axiom,
    ( ( beta @ x )
   != b ) ).

% \<open>\<beta> x \<noteq> b\<close>
thf(fact_2_sleft,axiom,
    member8476201367316484836on_nat @ s2 @ ( set_Su4848624856859743608on_nat @ s_left ) ).

% sleft
thf(fact_3_lxb,axiom,
    ( l
    = ( product_Pair_a_o @ x @ b ) ) ).

% lxb
thf(fact_4__092_060open_062_092_060beta_062_A_092_060equiv_062_A_092_060lambda_062x_O_Ams__elem__of__lit_A_Ix_M_ATrue_J_A_092_060in_062_Aset_As__left_092_060close_062,axiom,
    ( beta
    = ( ^ [X: a] : ( member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X @ $true ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ) ) ) ).

% \<open>\<beta> \<equiv> \<lambda>x. ms_elem_of_lit (x, True) \<in> set s_left\<close>
thf(fact_5__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062x_Ab_O_Al_A_061_A_Ix_M_Ab_J_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [X2: a,B: $o] :
        ( l
       != ( product_Pair_a_o @ X2 @ B ) ) ).

% \<open>\<And>thesis. (\<And>x b. l = (x, b) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_6__092_060open_062_092_060beta_062_Ax_092_060close_062,axiom,
    beta @ x ).

% \<open>\<beta> x\<close>
thf(fact_7__092_060beta_062__def,axiom,
    ! [X3: a] :
      ( ( beta @ X3 )
      = ( member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X3 @ $true ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ) ) ).

% \<beta>_def
thf(fact_8_ms__elem__of__lit_Ocases,axiom,
    ! [X3: product_prod_a_o] :
      ( ! [X2: a] :
          ( X3
         != ( product_Pair_a_o @ X2 @ $true ) )
     => ~ ! [X2: a] :
            ( X3
           != ( product_Pair_a_o @ X2 @ $false ) ) ) ).

% ms_elem_of_lit.cases
thf(fact_9_sl,axiom,
    ( s2
    = ( multis8230369766971345672_lit_a @ l ) ) ).

% sl
thf(fact_10_prod_Oinject,axiom,
    ! [X1: list_P9127870862658512897on_nat,X22: list_P9127870862658512897on_nat,Y1: list_P9127870862658512897on_nat,Y2: list_P9127870862658512897on_nat] :
      ( ( ( produc7152764266822961111on_nat @ X1 @ X22 )
        = ( produc7152764266822961111on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc1714117091004681831on_nat,Y1: list_S5253404628753938141on_nat,Y2: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ X1 @ X22 )
        = ( produc4176415733022414961on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_12_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X22: produc118098540238352959on_nat,Y1: list_S5253404628753938141on_nat,Y2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ X1 @ X22 )
        = ( produc1545487600272464053on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_13_prod_Oinject,axiom,
    ! [X1: a,X22: multis8755981257395730801tation,Y1: a,Y2: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ X1 @ X22 )
        = ( produc7247797765490398872tation @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_14_prod_Oinject,axiom,
    ! [X1: a,X22: $o,Y1: a,Y2: $o] :
      ( ( ( product_Pair_a_o @ X1 @ X22 )
        = ( product_Pair_a_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat,A2: list_P9127870862658512897on_nat,B3: list_P9127870862658512897on_nat] :
      ( ( ( produc7152764266822961111on_nat @ A @ B2 )
        = ( produc7152764266822961111on_nat @ A2 @ B3 ) )
      = ( ( A = A2 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_16_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat,A2: list_S5253404628753938141on_nat,B3: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B2 )
        = ( produc4176415733022414961on_nat @ A2 @ B3 ) )
      = ( ( A = A2 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_17_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B2 )
        = ( produc1545487600272464053on_nat @ A2 @ B3 ) )
      = ( ( A = A2 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_18_old_Oprod_Oinject,axiom,
    ! [A: a,B2: multis8755981257395730801tation,A2: a,B3: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ A @ B2 )
        = ( produc7247797765490398872tation @ A2 @ B3 ) )
      = ( ( A = A2 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_19_old_Oprod_Oinject,axiom,
    ! [A: a,B2: $o,A2: a,B3: $o] :
      ( ( ( product_Pair_a_o @ A @ B2 )
        = ( product_Pair_a_o @ A2 @ B3 ) )
      = ( ( A = A2 )
        & ( B2 = B3 ) ) ) ).

% old.prod.inject
thf(fact_20_old_Oprod_Oexhaust,axiom,
    ! [Y: produc1714117091004681831on_nat] :
      ~ ! [A3: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc7152764266822961111on_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_21_old_Oprod_Oexhaust,axiom,
    ! [Y: produc118098540238352959on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat] :
          ( Y
         != ( produc4176415733022414961on_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_22_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_23_old_Oprod_Oexhaust,axiom,
    ! [Y: produc27318731269267614tation] :
      ~ ! [A3: a,B: multis8755981257395730801tation] :
          ( Y
         != ( produc7247797765490398872tation @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_24_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_o] :
      ~ ! [A3: a,B: $o] :
          ( Y
         != ( product_Pair_a_o @ A3 @ B ) ) ).

% old.prod.exhaust
thf(fact_25_surj__pair,axiom,
    ! [P: produc1714117091004681831on_nat] :
    ? [X2: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
      ( P
      = ( produc7152764266822961111on_nat @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_26_surj__pair,axiom,
    ! [P: produc118098540238352959on_nat] :
    ? [X2: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat] :
      ( P
      = ( produc4176415733022414961on_nat @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_27_surj__pair,axiom,
    ! [P: produc8325617224762090749on_nat] :
    ? [X2: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat] :
      ( P
      = ( produc1545487600272464053on_nat @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_28_surj__pair,axiom,
    ! [P: produc27318731269267614tation] :
    ? [X2: a,Y3: multis8755981257395730801tation] :
      ( P
      = ( produc7247797765490398872tation @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_29_surj__pair,axiom,
    ! [P: product_prod_a_o] :
    ? [X2: a,Y3: $o] :
      ( P
      = ( product_Pair_a_o @ X2 @ Y3 ) ) ).

% surj_pair
thf(fact_30_prod__cases,axiom,
    ! [P2: produc1714117091004681831on_nat > $o,P: produc1714117091004681831on_nat] :
      ( ! [A3: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat] : ( P2 @ ( produc7152764266822961111on_nat @ A3 @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_31_prod__cases,axiom,
    ! [P2: produc118098540238352959on_nat > $o,P: produc118098540238352959on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat] : ( P2 @ ( produc4176415733022414961on_nat @ A3 @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_32_prod__cases,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A3 @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_33_prod__cases,axiom,
    ! [P2: produc27318731269267614tation > $o,P: produc27318731269267614tation] :
      ( ! [A3: a,B: multis8755981257395730801tation] : ( P2 @ ( produc7247797765490398872tation @ A3 @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_34_prod__cases,axiom,
    ! [P2: product_prod_a_o > $o,P: product_prod_a_o] :
      ( ! [A3: a,B: $o] : ( P2 @ ( product_Pair_a_o @ A3 @ B ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_35_Pair__inject,axiom,
    ! [A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat,A2: list_P9127870862658512897on_nat,B3: list_P9127870862658512897on_nat] :
      ( ( ( produc7152764266822961111on_nat @ A @ B2 )
        = ( produc7152764266822961111on_nat @ A2 @ B3 ) )
     => ~ ( ( A = A2 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_36_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat,A2: list_S5253404628753938141on_nat,B3: produc1714117091004681831on_nat] :
      ( ( ( produc4176415733022414961on_nat @ A @ B2 )
        = ( produc4176415733022414961on_nat @ A2 @ B3 ) )
     => ~ ( ( A = A2 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_37_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B2 )
        = ( produc1545487600272464053on_nat @ A2 @ B3 ) )
     => ~ ( ( A = A2 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_38_Pair__inject,axiom,
    ! [A: a,B2: multis8755981257395730801tation,A2: a,B3: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ A @ B2 )
        = ( produc7247797765490398872tation @ A2 @ B3 ) )
     => ~ ( ( A = A2 )
         => ( B2 != B3 ) ) ) ).

% Pair_inject
thf(fact_39_Pair__inject,axiom,
    ! [A: a,B2: $o,A2: a,B3: $o] :
      ( ( ( product_Pair_a_o @ A @ B2 )
        = ( product_Pair_a_o @ A2 @ B3 ) )
     => ~ ( ( A = A2 )
         => ( B2 = ~ B3 ) ) ) ).

% Pair_inject
thf(fact_40_prod__cases3,axiom,
    ! [Y: produc118098540238352959on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc4176415733022414961on_nat @ A3 @ ( produc7152764266822961111on_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_41_prod__cases3,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,C: produc1714117091004681831on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B @ C ) ) ) ).

% prod_cases3
thf(fact_42_prod__cases4,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,C: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B @ ( produc7152764266822961111on_nat @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_43_False,axiom,
    ~ b ).

% False
thf(fact_44_prod__induct4,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X3: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,C: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B @ ( produc7152764266822961111on_nat @ C @ D ) ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct4
thf(fact_45_prod__induct3,axiom,
    ! [P2: produc118098540238352959on_nat > $o,X3: produc118098540238352959on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B: list_P9127870862658512897on_nat,C: list_P9127870862658512897on_nat] : ( P2 @ ( produc4176415733022414961on_nat @ A3 @ ( produc7152764266822961111on_nat @ B @ C ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_46_prod__induct3,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X3: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,C: produc1714117091004681831on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B @ C ) ) )
     => ( P2 @ X3 ) ) ).

% prod_induct3
thf(fact_47_lc,axiom,
    member2997933407038931562od_a_o @ l @ ( set_Product_prod_a_o2 @ c ) ).

% lc
thf(fact_48__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062l_O_A_092_060lbrakk_062l_A_092_060in_062_Aset_Ac_059_As_A_061_Ams__elem__of__lit_Al_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [L: product_prod_a_o] :
        ( ( member2997933407038931562od_a_o @ L @ ( set_Product_prod_a_o2 @ c ) )
       => ( s2
         != ( multis8230369766971345672_lit_a @ L ) ) ) ).

% \<open>\<And>thesis. (\<And>l. \<lbrakk>l \<in> set c; s = ms_elem_of_lit l\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_49_ssubst__Pair__rhs,axiom,
    ! [R: list_P9127870862658512897on_nat,S: list_P9127870862658512897on_nat,R2: set_Pr8586234676504894535on_nat,S2: list_P9127870862658512897on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_50_ssubst__Pair__rhs,axiom,
    ! [R: list_S5253404628753938141on_nat,S: produc1714117091004681831on_nat,R2: set_Pr737724919871395061on_nat,S2: produc1714117091004681831on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_51_ssubst__Pair__rhs,axiom,
    ! [R: list_S5253404628753938141on_nat,S: produc118098540238352959on_nat,R2: set_Pr6244131468446592861on_nat,S2: produc118098540238352959on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_52_ssubst__Pair__rhs,axiom,
    ! [R: a,S: multis8755981257395730801tation,R2: set_Pr3525538344199662036tation,S2: multis8755981257395730801tation] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_53_ssubst__Pair__rhs,axiom,
    ! [R: a,S: $o,R2: set_Product_prod_a_o,S2: $o] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ R @ S ) @ R2 )
     => ( ( S2 = S )
       => ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ R @ S2 ) @ R2 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_54_curryI,axiom,
    ! [F: produc1714117091004681831on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( F @ ( produc7152764266822961111on_nat @ A @ B2 ) )
     => ( produc1961822403333099213_nat_o @ F @ A @ B2 ) ) ).

% curryI
thf(fact_55_curryI,axiom,
    ! [F: produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( F @ ( produc4176415733022414961on_nat @ A @ B2 ) )
     => ( produc4151365118158361241_nat_o @ F @ A @ B2 ) ) ).

% curryI
thf(fact_56_curryI,axiom,
    ! [F: produc8325617224762090749on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( F @ ( produc1545487600272464053on_nat @ A @ B2 ) )
     => ( produc4058710642246024687_nat_o @ F @ A @ B2 ) ) ).

% curryI
thf(fact_57_curryI,axiom,
    ! [F: produc27318731269267614tation > $o,A: a,B2: multis8755981257395730801tation] :
      ( ( F @ ( produc7247797765490398872tation @ A @ B2 ) )
     => ( produc3701758970522427058tion_o @ F @ A @ B2 ) ) ).

% curryI
thf(fact_58_curryI,axiom,
    ! [F: product_prod_a_o > $o,A: a,B2: $o] :
      ( ( F @ ( product_Pair_a_o @ A @ B2 ) )
     => ( product_curry_a_o_o @ F @ A @ B2 ) ) ).

% curryI
thf(fact_59_swap__simp,axiom,
    ! [X3: produc1714117091004681831on_nat,Y: list_S5253404628753938141on_nat] :
      ( ( produc5255540369398868045on_nat @ ( produc7669470456564080109on_nat @ X3 @ Y ) )
      = ( produc4176415733022414961on_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_60_swap__simp,axiom,
    ! [X3: produc118098540238352959on_nat,Y: list_S5253404628753938141on_nat] :
      ( ( produc2713330623810446677on_nat @ ( produc4211132126058264501on_nat @ X3 @ Y ) )
      = ( produc1545487600272464053on_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_61_swap__simp,axiom,
    ! [X3: multis8755981257395730801tation,Y: a] :
      ( ( produc5759830441326725076tion_a @ ( produc8803423008725155188tion_a @ X3 @ Y ) )
      = ( produc7247797765490398872tation @ Y @ X3 ) ) ).

% swap_simp
thf(fact_62_swap__simp,axiom,
    ! [X3: $o,Y: a] :
      ( ( product_swap_o_a @ ( product_Pair_o_a @ X3 @ Y ) )
      = ( product_Pair_a_o @ Y @ X3 ) ) ).

% swap_simp
thf(fact_63_swap__simp,axiom,
    ! [X3: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( produc8057558996099302775on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) )
      = ( produc7152764266822961111on_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_64_swap__simp,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
      ( ( produc1762485645857202897on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y ) )
      = ( produc7669470456564080109on_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_65_swap__simp,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
      ( ( produc47686098024646229on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y ) )
      = ( produc4211132126058264501on_nat @ Y @ X3 ) ) ).

% swap_simp
thf(fact_66_swap__simp,axiom,
    ! [X3: a,Y: multis8755981257395730801tation] :
      ( ( produc4204205198091968760tation @ ( produc7247797765490398872tation @ X3 @ Y ) )
      = ( produc8803423008725155188tion_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_67_swap__simp,axiom,
    ! [X3: a,Y: $o] :
      ( ( product_swap_a_o @ ( product_Pair_a_o @ X3 @ Y ) )
      = ( product_Pair_o_a @ Y @ X3 ) ) ).

% swap_simp
thf(fact_68_x,axiom,
    member_a2 @ x @ ( set_a2 @ xs ) ).

% x
thf(fact_69_curryD,axiom,
    ! [F: produc1714117091004681831on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( produc1961822403333099213_nat_o @ F @ A @ B2 )
     => ( F @ ( produc7152764266822961111on_nat @ A @ B2 ) ) ) ).

% curryD
thf(fact_70_curryD,axiom,
    ! [F: produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( produc4151365118158361241_nat_o @ F @ A @ B2 )
     => ( F @ ( produc4176415733022414961on_nat @ A @ B2 ) ) ) ).

% curryD
thf(fact_71_curryD,axiom,
    ! [F: produc8325617224762090749on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( produc4058710642246024687_nat_o @ F @ A @ B2 )
     => ( F @ ( produc1545487600272464053on_nat @ A @ B2 ) ) ) ).

% curryD
thf(fact_72_curryD,axiom,
    ! [F: produc27318731269267614tation > $o,A: a,B2: multis8755981257395730801tation] :
      ( ( produc3701758970522427058tion_o @ F @ A @ B2 )
     => ( F @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ).

% curryD
thf(fact_73_curryD,axiom,
    ! [F: product_prod_a_o > $o,A: a,B2: $o] :
      ( ( product_curry_a_o_o @ F @ A @ B2 )
     => ( F @ ( product_Pair_a_o @ A @ B2 ) ) ) ).

% curryD
thf(fact_74_curryE,axiom,
    ! [F: produc1714117091004681831on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( produc1961822403333099213_nat_o @ F @ A @ B2 )
     => ( F @ ( produc7152764266822961111on_nat @ A @ B2 ) ) ) ).

% curryE
thf(fact_75_curryE,axiom,
    ! [F: produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( produc4151365118158361241_nat_o @ F @ A @ B2 )
     => ( F @ ( produc4176415733022414961on_nat @ A @ B2 ) ) ) ).

% curryE
thf(fact_76_curryE,axiom,
    ! [F: produc8325617224762090749on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( produc4058710642246024687_nat_o @ F @ A @ B2 )
     => ( F @ ( produc1545487600272464053on_nat @ A @ B2 ) ) ) ).

% curryE
thf(fact_77_curryE,axiom,
    ! [F: produc27318731269267614tation > $o,A: a,B2: multis8755981257395730801tation] :
      ( ( produc3701758970522427058tion_o @ F @ A @ B2 )
     => ( F @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ).

% curryE
thf(fact_78_curryE,axiom,
    ! [F: product_prod_a_o > $o,A: a,B2: $o] :
      ( ( product_curry_a_o_o @ F @ A @ B2 )
     => ( F @ ( product_Pair_a_o @ A @ B2 ) ) ) ).

% curryE
thf(fact_79_xs__def,axiom,
    ( xs
    = ( multis4741186478440321083_cnf_a @ cnf ) ) ).

% xs_def
thf(fact_80_ccnf,axiom,
    member5261019209154752368od_a_o @ c @ ( set_li2629784761229999580od_a_o @ cnf ) ).

% ccnf
thf(fact_81_mem__Collect__eq,axiom,
    ! [A: product_prod_a_o,P2: product_prod_a_o > $o] :
      ( ( member2997933407038931562od_a_o @ A @ ( collec5729118848859662764od_a_o @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_82_mem__Collect__eq,axiom,
    ! [A: sum_su4907927656612468301on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( member8476201367316484836on_nat @ A @ ( collec1912426543561893410on_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_83_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_84_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat2 @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_85_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a2 @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_86_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_a_o] :
      ( ( collec5729118848859662764od_a_o
        @ ^ [X: product_prod_a_o] : ( member2997933407038931562od_a_o @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_87_Collect__mem__eq,axiom,
    ! [A4: set_Su266273576555847555on_nat] :
      ( ( collec1912426543561893410on_nat
        @ ^ [X: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_88_Collect__mem__eq,axiom,
    ! [A4: set_li5011860890973139239od_a_o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_89_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat2 @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_90_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X: a] : ( member_a2 @ X @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_91_pair__in__swap__image,axiom,
    ! [Y: produc1714117091004681831on_nat,X3: list_S5253404628753938141on_nat,A4: set_Pr737724919871395061on_nat] :
      ( ( member4065180873401903370on_nat @ ( produc7669470456564080109on_nat @ Y @ X3 ) @ ( image_3839552252155826745on_nat @ produc1762485645857202897on_nat @ A4 ) )
      = ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_92_pair__in__swap__image,axiom,
    ! [Y: produc118098540238352959on_nat,X3: list_S5253404628753938141on_nat,A4: set_Pr6244131468446592861on_nat] :
      ( ( member5746340541560749734on_nat @ ( produc4211132126058264501on_nat @ Y @ X3 ) @ ( image_212404780875737157on_nat @ produc47686098024646229on_nat @ A4 ) )
      = ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_93_pair__in__swap__image,axiom,
    ! [Y: multis8755981257395730801tation,X3: a,A4: set_Pr3525538344199662036tation] :
      ( ( member5794953671549508825tion_a @ ( produc8803423008725155188tion_a @ Y @ X3 ) @ ( image_8196844605755565287tion_a @ produc4204205198091968760tation @ A4 ) )
      = ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_94_pair__in__swap__image,axiom,
    ! [Y: $o,X3: a,A4: set_Product_prod_a_o] :
      ( ( member1718822050171266614od_o_a @ ( product_Pair_o_a @ Y @ X3 ) @ ( image_7544959513298650385od_o_a @ product_swap_a_o @ A4 ) )
      = ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_95_pair__in__swap__image,axiom,
    ! [Y: list_P9127870862658512897on_nat,X3: list_P9127870862658512897on_nat,A4: set_Pr8586234676504894535on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Y @ X3 ) @ ( image_3067255937158254533on_nat @ produc8057558996099302775on_nat @ A4 ) )
      = ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_96_pair__in__swap__image,axiom,
    ! [Y: list_S5253404628753938141on_nat,X3: produc1714117091004681831on_nat,A4: set_Pr637138698490276009on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ Y @ X3 ) @ ( image_5485573264706132025on_nat @ produc5255540369398868045on_nat @ A4 ) )
      = ( member4065180873401903370on_nat @ ( produc7669470456564080109on_nat @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_97_pair__in__swap__image,axiom,
    ! [Y: list_S5253404628753938141on_nat,X3: produc118098540238352959on_nat,A4: set_Pr8090687219299372893on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ Y @ X3 ) @ ( image_2326488914019723333on_nat @ produc2713330623810446677on_nat @ A4 ) )
      = ( member5746340541560749734on_nat @ ( produc4211132126058264501on_nat @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_98_pair__in__swap__image,axiom,
    ! [Y: a,X3: multis8755981257395730801tation,A4: set_Pr5255243115176348920tion_a] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ Y @ X3 ) @ ( image_2776320019631618791tation @ produc5759830441326725076tion_a @ A4 ) )
      = ( member5794953671549508825tion_a @ ( produc8803423008725155188tion_a @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_99_pair__in__swap__image,axiom,
    ! [Y: a,X3: $o,A4: set_Product_prod_o_a] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ Y @ X3 ) @ ( image_28068756556222073od_a_o @ product_swap_o_a @ A4 ) )
      = ( member1718822050171266614od_o_a @ ( product_Pair_o_a @ X3 @ Y ) @ A4 ) ) ).

% pair_in_swap_image
thf(fact_100_apsnd__conv,axiom,
    ! [F: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat,X3: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( produc1615944377057951811on_nat @ F @ ( produc7152764266822961111on_nat @ X3 @ Y ) )
      = ( produc7152764266822961111on_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_101_apsnd__conv,axiom,
    ! [F: produc1714117091004681831on_nat > produc1714117091004681831on_nat,X3: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
      ( ( produc1780334326586307739on_nat @ F @ ( produc4176415733022414961on_nat @ X3 @ Y ) )
      = ( produc4176415733022414961on_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_102_apsnd__conv,axiom,
    ! [F: produc1714117091004681831on_nat > produc118098540238352959on_nat,X3: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat] :
      ( ( produc8328242075292354183on_nat @ F @ ( produc4176415733022414961on_nat @ X3 @ Y ) )
      = ( produc1545487600272464053on_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_103_apsnd__conv,axiom,
    ! [F: produc118098540238352959on_nat > produc1714117091004681831on_nat,X3: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
      ( ( produc7597149594502495431on_nat @ F @ ( produc1545487600272464053on_nat @ X3 @ Y ) )
      = ( produc4176415733022414961on_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_104_apsnd__conv,axiom,
    ! [F: produc118098540238352959on_nat > produc118098540238352959on_nat,X3: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat] :
      ( ( produc2968767120564268379on_nat @ F @ ( produc1545487600272464053on_nat @ X3 @ Y ) )
      = ( produc1545487600272464053on_nat @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_105_apsnd__conv,axiom,
    ! [F: multis8755981257395730801tation > multis8755981257395730801tation,X3: a,Y: multis8755981257395730801tation] :
      ( ( produc5851446926483149468tion_a @ F @ ( produc7247797765490398872tation @ X3 @ Y ) )
      = ( produc7247797765490398872tation @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_106_apsnd__conv,axiom,
    ! [F: multis8755981257395730801tation > $o,X3: a,Y: multis8755981257395730801tation] :
      ( ( produc1483440316738056661on_o_a @ F @ ( produc7247797765490398872tation @ X3 @ Y ) )
      = ( product_Pair_a_o @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_107_apsnd__conv,axiom,
    ! [F: $o > multis8755981257395730801tation,X3: a,Y: $o] :
      ( ( produc6275676691024199601tion_a @ F @ ( product_Pair_a_o @ X3 @ Y ) )
      = ( produc7247797765490398872tation @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_108_apsnd__conv,axiom,
    ! [F: $o > $o,X3: a,Y: $o] :
      ( ( product_apsnd_o_o_a @ F @ ( product_Pair_a_o @ X3 @ Y ) )
      = ( product_Pair_a_o @ X3 @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_109_zip__same,axiom,
    ! [A: product_prod_a_o,B2: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ A @ B2 ) @ ( set_Pr3821420946794936316od_a_o @ ( zip_Pr351013771977258487od_a_o @ Xs @ Xs ) ) )
      = ( ( member2997933407038931562od_a_o @ A @ ( set_Product_prod_a_o2 @ Xs ) )
        & ( A = B2 ) ) ) ).

% zip_same
thf(fact_110_zip__same,axiom,
    ! [A: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ B2 ) @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Xs ) ) )
      = ( ( member8476201367316484836on_nat @ A @ ( set_Su4848624856859743608on_nat @ Xs ) )
        & ( A = B2 ) ) ) ).

% zip_same
thf(fact_111_zip__same,axiom,
    ! [A: list_P3196869577824546119od_a_o,B2: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A @ B2 ) @ ( set_Pr7142968438029117820od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs @ Xs ) ) )
      = ( ( member5261019209154752368od_a_o @ A @ ( set_li2629784761229999580od_a_o @ Xs ) )
        & ( A = B2 ) ) ) ).

% zip_same
thf(fact_112_zip__same,axiom,
    ! [A: nat,B2: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat2 @ A @ ( set_nat2 @ Xs ) )
        & ( A = B2 ) ) ) ).

% zip_same
thf(fact_113_zip__same,axiom,
    ! [A: a,B2: a,Xs: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Xs ) ) )
      = ( ( member_a2 @ A @ ( set_a2 @ Xs ) )
        & ( A = B2 ) ) ) ).

% zip_same
thf(fact_114_zip__same,axiom,
    ! [A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B2 ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Xs ) ) )
      = ( ( member783415501022416426on_nat @ A @ ( set_li8154783781463198870on_nat @ Xs ) )
        & ( A = B2 ) ) ) ).

% zip_same
thf(fact_115_in__set__zipE,axiom,
    ! [X3: nat,Y: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_116_in__set__zipE,axiom,
    ! [X3: nat,Y: a,Xs: list_nat,Ys: list_a] :
      ( ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X3 @ Y ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member_a2 @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_117_in__set__zipE,axiom,
    ! [X3: a,Y: nat,Xs: list_a,Ys: list_nat] :
      ( ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_118_in__set__zipE,axiom,
    ! [X3: a,Y: a,Xs: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs @ Ys ) ) )
     => ~ ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_a2 @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_119_in__set__zipE,axiom,
    ! [X3: a,Y: multis8755981257395730801tation,Xs: list_a,Ys: list_M7834926499894101121tation] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X3 @ Y ) @ ( set_Pr5992250453084334153tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) ) )
     => ~ ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member7672564098010296200tation @ Y @ ( set_Mu7203219757648765980tation @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_120_in__set__zipE,axiom,
    ! [X3: a,Y: $o,Xs: list_a,Ys: list_o] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X3 @ Y ) @ ( set_Product_prod_a_o2 @ ( zip_a_o @ Xs @ Ys ) ) )
     => ~ ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member_o @ Y @ ( set_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_121_in__set__zipE,axiom,
    ! [X3: product_prod_a_o,Y: nat,Xs: list_P3196869577824546119od_a_o,Ys: list_nat] :
      ( ( member6380975538497366487_o_nat @ ( produc8040076513301755962_o_nat @ X3 @ Y ) @ ( set_Pr8647722510924136043_o_nat @ ( zip_Pr7459748363961616666_o_nat @ Xs @ Ys ) ) )
     => ~ ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
         => ~ ( member_nat2 @ Y @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_122_in__set__zipE,axiom,
    ! [X3: product_prod_a_o,Y: a,Xs: list_P3196869577824546119od_a_o,Ys: list_a] :
      ( ( member5411179314388669517_a_o_a @ ( produc8966842053957846292_a_o_a @ X3 @ Y ) @ ( set_Pr8855197574426033337_a_o_a @ ( zip_Pr6655751374316219444_a_o_a @ Xs @ Ys ) ) )
     => ~ ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
         => ~ ( member_a2 @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_123_in__set__zipE,axiom,
    ! [X3: nat,Y: product_prod_a_o,Xs: list_nat,Ys: list_P3196869577824546119od_a_o] :
      ( ( member5556949019511437309od_a_o @ ( produc5340308163291612696od_a_o @ X3 @ Y ) @ ( set_Pr7823695991938206865od_a_o @ ( zip_na4759980013951473400od_a_o @ Xs @ Ys ) ) )
     => ~ ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
         => ~ ( member2997933407038931562od_a_o @ Y @ ( set_Product_prod_a_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_124_in__set__zipE,axiom,
    ! [X3: a,Y: product_prod_a_o,Xs: list_a,Ys: list_P3196869577824546119od_a_o] :
      ( ( member965141594903935763od_a_o @ ( produc1851294465994288986od_a_o @ X3 @ Y ) @ ( set_Pr4409159854941299583od_a_o @ ( zip_a_8763575823207437946od_a_o @ Xs @ Ys ) ) )
     => ~ ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
         => ~ ( member2997933407038931562od_a_o @ Y @ ( set_Product_prod_a_o2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_125_set__zip__leftD,axiom,
    ! [X3: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) )
     => ( member783415501022416426on_nat @ X3 @ ( set_li8154783781463198870on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_126_set__zip__leftD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y ) @ ( set_Pr8183371313083159530on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) )
     => ( member86162308477975540on_nat @ X3 @ ( set_li4371160873219908744on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_127_set__zip__leftD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y ) @ ( set_Pr4584522749497345810on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) )
     => ( member86162308477975540on_nat @ X3 @ ( set_li4371160873219908744on_nat @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_128_set__zip__leftD,axiom,
    ! [X3: a,Y: multis8755981257395730801tation,Xs: list_a,Ys: list_M7834926499894101121tation] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X3 @ Y ) @ ( set_Pr5992250453084334153tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) ) )
     => ( member_a2 @ X3 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_129_set__zip__leftD,axiom,
    ! [X3: a,Y: $o,Xs: list_a,Ys: list_o] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X3 @ Y ) @ ( set_Product_prod_a_o2 @ ( zip_a_o @ Xs @ Ys ) ) )
     => ( member_a2 @ X3 @ ( set_a2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_130_set__zip__rightD,axiom,
    ! [X3: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ ( set_Pr6705596042943559932on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) ) )
     => ( member783415501022416426on_nat @ Y @ ( set_li8154783781463198870on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_131_set__zip__rightD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y: produc1714117091004681831on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P8734936519038225773on_nat] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X3 @ Y ) @ ( set_Pr8183371313083159530on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) ) )
     => ( member4629988367768146576on_nat @ Y @ ( set_Pr6705596042943559932on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_132_set__zip__rightD,axiom,
    ! [X3: list_S5253404628753938141on_nat,Y: produc118098540238352959on_nat,Xs: list_l8999537460041914605on_nat,Ys: list_P5508476027986415439on_nat] :
      ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X3 @ Y ) @ ( set_Pr4584522749497345810on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) ) )
     => ( member4585374397974051670on_nat @ Y @ ( set_Pr8183371313083159530on_nat @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_133_set__zip__rightD,axiom,
    ! [X3: a,Y: multis8755981257395730801tation,Xs: list_a,Ys: list_M7834926499894101121tation] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X3 @ Y ) @ ( set_Pr5992250453084334153tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) ) )
     => ( member7672564098010296200tation @ Y @ ( set_Mu7203219757648765980tation @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_134_set__zip__rightD,axiom,
    ! [X3: a,Y: $o,Xs: list_a,Ys: list_o] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X3 @ Y ) @ ( set_Product_prod_a_o2 @ ( zip_a_o @ Xs @ Ys ) ) )
     => ( member_o @ Y @ ( set_o2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_135_mem__case__prodE,axiom,
    ! [Z: nat,C2: a > multis8755981257395730801tation > set_nat,P: produc27318731269267614tation] :
      ( ( member_nat2 @ Z @ ( produc6722172017951998027et_nat @ C2 @ P ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X2 @ Y3 ) )
           => ~ ( member_nat2 @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_136_mem__case__prodE,axiom,
    ! [Z: a,C2: a > multis8755981257395730801tation > set_a,P: produc27318731269267614tation] :
      ( ( member_a2 @ Z @ ( produc7412255387654812377_set_a @ C2 @ P ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X2 @ Y3 ) )
           => ~ ( member_a2 @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_137_mem__case__prodE,axiom,
    ! [Z: nat,C2: a > $o > set_nat,P: product_prod_a_o] :
      ( ( member_nat2 @ Z @ ( produc8120172443933582796et_nat @ C2 @ P ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X2 @ Y3 ) )
           => ~ ( member_nat2 @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_138_mem__case__prodE,axiom,
    ! [Z: a,C2: a > $o > set_a,P: product_prod_a_o] :
      ( ( member_a2 @ Z @ ( produc6584190720350841752_set_a @ C2 @ P ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X2 @ Y3 ) )
           => ~ ( member_a2 @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_139_mem__case__prodE,axiom,
    ! [Z: product_prod_a_o,C2: a > multis8755981257395730801tation > set_Product_prod_a_o,P: produc27318731269267614tation] :
      ( ( member2997933407038931562od_a_o @ Z @ ( produc8399654089035374108od_a_o @ C2 @ P ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X2 @ Y3 ) )
           => ~ ( member2997933407038931562od_a_o @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_140_mem__case__prodE,axiom,
    ! [Z: product_prod_a_o,C2: a > $o > set_Product_prod_a_o,P: product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ Z @ ( produc1948407148953235291od_a_o @ C2 @ P ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X2 @ Y3 ) )
           => ~ ( member2997933407038931562od_a_o @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_141_mem__case__prodE,axiom,
    ! [Z: list_P3196869577824546119od_a_o,C2: a > multis8755981257395730801tation > set_li5011860890973139239od_a_o,P: produc27318731269267614tation] :
      ( ( member5261019209154752368od_a_o @ Z @ ( produc2219104145104074914od_a_o @ C2 @ P ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X2 @ Y3 ) )
           => ~ ( member5261019209154752368od_a_o @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_142_mem__case__prodE,axiom,
    ! [Z: list_P3196869577824546119od_a_o,C2: a > $o > set_li5011860890973139239od_a_o,P: product_prod_a_o] :
      ( ( member5261019209154752368od_a_o @ Z @ ( produc929350387168271969od_a_o @ C2 @ P ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X2 @ Y3 ) )
           => ~ ( member5261019209154752368od_a_o @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_143_mem__case__prodE,axiom,
    ! [Z: sum_su4907927656612468301on_nat,C2: a > multis8755981257395730801tation > set_Su266273576555847555on_nat,P: produc27318731269267614tation] :
      ( ( member8476201367316484836on_nat @ Z @ ( produc6506295005197259848on_nat @ C2 @ P ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X2 @ Y3 ) )
           => ~ ( member8476201367316484836on_nat @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_144_mem__case__prodE,axiom,
    ! [Z: sum_su4907927656612468301on_nat,C2: a > $o > set_Su266273576555847555on_nat,P: product_prod_a_o] :
      ( ( member8476201367316484836on_nat @ Z @ ( produc2666714314943159753on_nat @ C2 @ P ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X2 @ Y3 ) )
           => ~ ( member8476201367316484836on_nat @ Z @ ( C2 @ X2 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_145_image__eqI,axiom,
    ! [B2: nat,F: nat > nat,X3: nat,A4: set_nat] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_nat2 @ X3 @ A4 )
       => ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_146_image__eqI,axiom,
    ! [B2: a,F: nat > a,X3: nat,A4: set_nat] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_nat2 @ X3 @ A4 )
       => ( member_a2 @ B2 @ ( image_nat_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_147_image__eqI,axiom,
    ! [B2: nat,F: a > nat,X3: a,A4: set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_a2 @ X3 @ A4 )
       => ( member_nat2 @ B2 @ ( image_a_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_148_image__eqI,axiom,
    ! [B2: a,F: a > a,X3: a,A4: set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_a2 @ X3 @ A4 )
       => ( member_a2 @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_149_image__eqI,axiom,
    ! [B2: nat,F: product_prod_a_o > nat,X3: product_prod_a_o,A4: set_Product_prod_a_o] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member2997933407038931562od_a_o @ X3 @ A4 )
       => ( member_nat2 @ B2 @ ( image_8012505923468758860_o_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_150_image__eqI,axiom,
    ! [B2: a,F: product_prod_a_o > a,X3: product_prod_a_o,A4: set_Product_prod_a_o] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member2997933407038931562od_a_o @ X3 @ A4 )
       => ( member_a2 @ B2 @ ( image_2171321463282962050_a_o_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_151_image__eqI,axiom,
    ! [B2: product_prod_a_o,F: nat > product_prod_a_o,X3: nat,A4: set_nat] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_nat2 @ X3 @ A4 )
       => ( member2997933407038931562od_a_o @ B2 @ ( image_5312737573458615594od_a_o @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_152_image__eqI,axiom,
    ! [B2: product_prod_a_o,F: a > product_prod_a_o,X3: a,A4: set_a] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member_a2 @ X3 @ A4 )
       => ( member2997933407038931562od_a_o @ B2 @ ( image_4279145912174180552od_a_o @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_153_image__eqI,axiom,
    ! [B2: nat,F: list_P3196869577824546119od_a_o > nat,X3: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member5261019209154752368od_a_o @ X3 @ A4 )
       => ( member_nat2 @ B2 @ ( image_5391756926611421138_o_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_154_image__eqI,axiom,
    ! [B2: a,F: list_P3196869577824546119od_a_o > a,X3: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o] :
      ( ( B2
        = ( F @ X3 ) )
     => ( ( member5261019209154752368od_a_o @ X3 @ A4 )
       => ( member_a2 @ B2 @ ( image_3930418492379151228_a_o_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_155_c__def,axiom,
    ( c
    = ( nth_li3248255784598043854od_a_o @ cnf @ i ) ) ).

% c_def
thf(fact_156_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_M7834926499894101121tation,Y: multis8755981257395730801tation] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3663592329813318933tation @ Ys ) )
     => ( ( member7672564098010296200tation @ Y @ ( set_Mu7203219757648765980tation @ Ys ) )
       => ~ ! [X2: a] :
              ~ ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X2 @ Y ) @ ( set_Pr5992250453084334153tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_157_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_a,Ys: list_o,Y: $o] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y @ ( set_o2 @ Ys ) )
       => ~ ! [X2: a] :
              ~ ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X2 @ Y ) @ ( set_Product_prod_a_o2 @ ( zip_a_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_158_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_a,Y: a] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a2 @ Y @ ( set_a2 @ Ys ) )
       => ~ ! [X2: nat] :
              ~ ( member8962352052110095674_nat_a @ ( product_Pair_nat_a @ X2 @ Y ) @ ( set_Pr4163146838226711502_nat_a @ ( zip_nat_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_159_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Y: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat2 @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X2: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X2 @ Y ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_160_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_P3196869577824546119od_a_o,Y: product_prod_a_o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s1727999245659094451od_a_o @ Ys ) )
     => ( ( member2997933407038931562od_a_o @ Y @ ( set_Product_prod_a_o2 @ Ys ) )
       => ~ ! [X2: nat] :
              ~ ( member5556949019511437309od_a_o @ ( produc5340308163291612696od_a_o @ X2 @ Y ) @ ( set_Pr7823695991938206865od_a_o @ ( zip_na4759980013951473400od_a_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_161_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_a,Y: a] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a2 @ Y @ ( set_a2 @ Ys ) )
       => ~ ! [X2: list_P3196869577824546119od_a_o] :
              ~ ( member9175574212068111175_a_o_a @ ( produc2646396952918671630_a_o_a @ X2 @ Y ) @ ( set_Pr303663273329636787_a_o_a @ ( zip_li7249964909781669934_a_o_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_162_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,Y: nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat2 @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X2: list_P3196869577824546119od_a_o] :
              ~ ( member7885718254550376541_o_nat @ ( produc3563005473579536832_o_nat @ X2 @ Y ) @ ( set_Pr7830758801870232305_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_163_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( member5261019209154752368od_a_o @ Y @ ( set_li2629784761229999580od_a_o @ Ys ) )
       => ~ ! [X2: nat] :
              ~ ( member8910825520198479363od_a_o @ ( produc3946064260745446430od_a_o @ X2 @ Y ) @ ( set_Pr8855866067518335127od_a_o @ ( zip_na1422845649429636350od_a_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_164_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_a,Y: a] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_a @ Ys ) )
     => ( ( member_a2 @ Y @ ( set_a2 @ Ys ) )
       => ~ ! [X2: sum_su4907927656612468301on_nat] :
              ~ ( member4790692447696258045_nat_a @ ( produc4044987999758810008_nat_a @ X2 @ Y ) @ ( set_Pr8216944003675722385_nat_a @ ( zip_Su1389992033985510520_nat_a @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_165_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,Y: nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat2 @ Y @ ( set_nat2 @ Ys ) )
       => ~ ! [X2: sum_su4907927656612468301on_nat] :
              ~ ( member8820252441731870247at_nat @ ( produc7500644556467579062at_nat @ X2 @ Y ) @ ( set_Pr1192295990000706195at_nat @ ( zip_Su8633780382783139798at_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_166_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_M7834926499894101121tation,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3663592329813318933tation @ Ys ) )
     => ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: multis8755981257395730801tation] :
              ~ ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X3 @ Y3 ) @ ( set_Pr5992250453084334153tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_167_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_o,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: $o] :
              ~ ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X3 @ Y3 ) @ ( set_Product_prod_a_o2 @ ( zip_a_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_168_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_nat,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member5724188588386418708_a_nat @ ( product_Pair_a_nat @ X3 @ Y3 ) @ ( set_Pr924983374503034536_a_nat @ ( zip_a_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_169_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_nat,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_170_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_nat,X3: product_prod_a_o] :
      ( ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member6380975538497366487_o_nat @ ( produc8040076513301755962_o_nat @ X3 @ Y3 ) @ ( set_Pr8647722510924136043_o_nat @ ( zip_Pr7459748363961616666_o_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_171_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_l8336264072933094989od_a_o,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: list_P3196869577824546119od_a_o] :
              ~ ( member5323513488200936985od_a_o @ ( produc6581891749969470560od_a_o @ X3 @ Y3 ) @ ( set_Pr5674974586317238405od_a_o @ ( zip_a_1962087669977693056od_a_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_172_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,X3: list_P3196869577824546119od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member7885718254550376541_o_nat @ ( produc3563005473579536832_o_nat @ X3 @ Y3 ) @ ( set_Pr7830758801870232305_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_173_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
       => ~ ! [Y3: list_P3196869577824546119od_a_o] :
              ~ ( member8910825520198479363od_a_o @ ( produc3946064260745446430od_a_o @ X3 @ Y3 ) @ ( set_Pr8855866067518335127od_a_o @ ( zip_na1422845649429636350od_a_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_174_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_a,Ys: list_S5253404628753938141on_nat,X3: a] :
      ( ( ( size_size_list_a @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
       => ~ ! [Y3: sum_su4907927656612468301on_nat] :
              ~ ( member891341725418739729on_nat @ ( produc1716885640609124852on_nat @ X3 @ Y3 ) @ ( set_Pr4317593281398204069on_nat @ ( zip_a_8285261711690601172on_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_175_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ~ ! [Y3: nat] :
              ~ ( member8820252441731870247at_nat @ ( produc7500644556467579062at_nat @ X3 @ Y3 ) @ ( set_Pr1192295990000706195at_nat @ ( zip_Su8633780382783139798at_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_176_insort__insert__key__triv,axiom,
    ! [F: product_prod_a_o > nat,X3: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member_nat2 @ ( F @ X3 ) @ ( image_8012505923468758860_o_nat @ F @ ( set_Product_prod_a_o2 @ Xs ) ) )
     => ( ( linord243265191649646095_o_nat @ F @ X3 @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_177_insort__insert__key__triv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,X3: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member_nat2 @ ( F @ X3 ) @ ( image_7049302322420781092at_nat @ F @ ( set_Su4848624856859743608on_nat @ Xs ) ) )
     => ( ( linord3889321027028539681at_nat @ F @ X3 @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_178_insort__insert__key__triv,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,X3: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member_nat2 @ ( F @ X3 ) @ ( image_5391756926611421138_o_nat @ F @ ( set_li2629784761229999580od_a_o @ Xs ) ) )
     => ( ( linord7176415375258435861_o_nat @ F @ X3 @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_179_insort__insert__key__triv,axiom,
    ! [F: nat > nat,X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ ( F @ X3 ) @ ( image_nat_nat @ F @ ( set_nat2 @ Xs ) ) )
     => ( ( linord1921536354676448932at_nat @ F @ X3 @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_180_insort__insert__key__triv,axiom,
    ! [F: a > nat,X3: a,Xs: list_a] :
      ( ( member_nat2 @ ( F @ X3 ) @ ( image_a_nat @ F @ ( set_a2 @ Xs ) ) )
     => ( ( linord1046132949341221836_a_nat @ F @ X3 @ Xs )
        = Xs ) ) ).

% insort_insert_key_triv
thf(fact_181_s,axiom,
    ! [T: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ T @ ( set_Su4848624856859743608on_nat @ s_right ) )
     => ? [X2: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ s_left ) )
          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ T ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ).

% s
thf(fact_182_in__set__member,axiom,
    ! [X3: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
      = ( member3960567427143945692od_a_o @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_183_in__set__member,axiom,
    ! [X3: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( member4696886135494049266on_nat @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_184_in__set__member,axiom,
    ! [X3: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( member6642339946199276002od_a_o @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_185_in__set__member,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
      = ( member_nat @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_186_in__set__member,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
      = ( member_a @ Xs @ X3 ) ) ).

% in_set_member
thf(fact_187_Inf_OINF__cong,axiom,
    ! [A4: set_Product_prod_a_o,B4: set_Product_prod_a_o,C3: product_prod_a_o > produc1313295348606329723on_nat,D2: product_prod_a_o > produc1313295348606329723on_nat,Inf: set_Pr2535564044409765851on_nat > produc1313295348606329723on_nat] :
      ( ( A4 = B4 )
     => ( ! [X2: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X2 @ B4 )
           => ( ( C3 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Inf @ ( image_4414724447807847679on_nat @ C3 @ A4 ) )
          = ( Inf @ ( image_4414724447807847679on_nat @ D2 @ B4 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_188_Sup_OSUP__cong,axiom,
    ! [A4: set_Product_prod_a_o,B4: set_Product_prod_a_o,C3: product_prod_a_o > produc1313295348606329723on_nat,D2: product_prod_a_o > produc1313295348606329723on_nat,Sup: set_Pr2535564044409765851on_nat > produc1313295348606329723on_nat] :
      ( ( A4 = B4 )
     => ( ! [X2: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X2 @ B4 )
           => ( ( C3 @ X2 )
              = ( D2 @ X2 ) ) )
       => ( ( Sup @ ( image_4414724447807847679on_nat @ C3 @ A4 ) )
          = ( Sup @ ( image_4414724447807847679on_nat @ D2 @ B4 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_189_i,axiom,
    member_nat2 @ i @ ( set_nat2 @ cs ) ).

% i
thf(fact_190_UNIV__I,axiom,
    ! [X3: product_prod_a_o] : ( member2997933407038931562od_a_o @ X3 @ top_to6843115102692631153od_a_o ) ).

% UNIV_I
thf(fact_191_UNIV__I,axiom,
    ! [X3: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X3 @ top_to7488204556702531123on_nat ) ).

% UNIV_I
thf(fact_192_UNIV__I,axiom,
    ! [X3: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X3 @ top_to324007776275943031od_a_o ) ).

% UNIV_I
thf(fact_193_UNIV__I,axiom,
    ! [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ).

% UNIV_I
thf(fact_194_UNIV__I,axiom,
    ! [X3: a] : ( member_a2 @ X3 @ top_top_set_a ) ).

% UNIV_I
thf(fact_195_len,axiom,
    ( ( size_s5686234912572693873on_nat @ ns_left )
    = ( size_s5686234912572693873on_nat @ ns_right ) ) ).

% len
thf(fact_196_ii,axiom,
    ord_less_nat @ i @ ( size_s3379185846251495609od_a_o @ cnf ) ).

% ii
thf(fact_197_si,axiom,
    member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ s2 @ ( sum_In255631617079920996tation @ i ) ) @ ( set_Pr937530902923597456on_nat @ s ) ).

% si
thf(fact_198_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_199_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_200_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N ) ).

% Ex_list_of_length
thf(fact_201_neq__if__length__neq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
       != ( size_s5686234912572693873on_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_202_neq__if__length__neq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
       != ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_203_neq__if__length__neq,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
       != ( size_size_list_nat @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_204_UNIV__witness,axiom,
    ? [X2: product_prod_a_o] : ( member2997933407038931562od_a_o @ X2 @ top_to6843115102692631153od_a_o ) ).

% UNIV_witness
thf(fact_205_UNIV__witness,axiom,
    ? [X2: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X2 @ top_to7488204556702531123on_nat ) ).

% UNIV_witness
thf(fact_206_UNIV__witness,axiom,
    ? [X2: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X2 @ top_to324007776275943031od_a_o ) ).

% UNIV_witness
thf(fact_207_UNIV__witness,axiom,
    ? [X2: nat] : ( member_nat2 @ X2 @ top_top_set_nat ) ).

% UNIV_witness
thf(fact_208_UNIV__witness,axiom,
    ? [X2: a] : ( member_a2 @ X2 @ top_top_set_a ) ).

% UNIV_witness
thf(fact_209_UNIV__eq__I,axiom,
    ! [A4: set_Product_prod_a_o] :
      ( ! [X2: product_prod_a_o] : ( member2997933407038931562od_a_o @ X2 @ A4 )
     => ( top_to6843115102692631153od_a_o = A4 ) ) ).

% UNIV_eq_I
thf(fact_210_UNIV__eq__I,axiom,
    ! [A4: set_Su266273576555847555on_nat] :
      ( ! [X2: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X2 @ A4 )
     => ( top_to7488204556702531123on_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_211_UNIV__eq__I,axiom,
    ! [A4: set_li5011860890973139239od_a_o] :
      ( ! [X2: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X2 @ A4 )
     => ( top_to324007776275943031od_a_o = A4 ) ) ).

% UNIV_eq_I
thf(fact_212_UNIV__eq__I,axiom,
    ! [A4: set_nat] :
      ( ! [X2: nat] : ( member_nat2 @ X2 @ A4 )
     => ( top_top_set_nat = A4 ) ) ).

% UNIV_eq_I
thf(fact_213_UNIV__eq__I,axiom,
    ! [A4: set_a] :
      ( ! [X2: a] : ( member_a2 @ X2 @ A4 )
     => ( top_top_set_a = A4 ) ) ).

% UNIV_eq_I
thf(fact_214_range__eqI,axiom,
    ! [B2: produc1313295348606329723on_nat,F: product_prod_a_o > produc1313295348606329723on_nat,X3: product_prod_a_o] :
      ( ( B2
        = ( F @ X3 ) )
     => ( member5906418179908956708on_nat @ B2 @ ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o ) ) ) ).

% range_eqI
thf(fact_215_rangeI,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,X3: product_prod_a_o] : ( member5906418179908956708on_nat @ ( F @ X3 ) @ ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o ) ) ).

% rangeI
thf(fact_216_rev__image__eqI,axiom,
    ! [X3: nat,A4: set_nat,B2: nat,F: nat > nat] :
      ( ( member_nat2 @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_217_rev__image__eqI,axiom,
    ! [X3: nat,A4: set_nat,B2: a,F: nat > a] :
      ( ( member_nat2 @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_a2 @ B2 @ ( image_nat_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_218_rev__image__eqI,axiom,
    ! [X3: a,A4: set_a,B2: nat,F: a > nat] :
      ( ( member_a2 @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_nat2 @ B2 @ ( image_a_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_219_rev__image__eqI,axiom,
    ! [X3: a,A4: set_a,B2: a,F: a > a] :
      ( ( member_a2 @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_a2 @ B2 @ ( image_a_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_220_rev__image__eqI,axiom,
    ! [X3: product_prod_a_o,A4: set_Product_prod_a_o,B2: nat,F: product_prod_a_o > nat] :
      ( ( member2997933407038931562od_a_o @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_nat2 @ B2 @ ( image_8012505923468758860_o_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_221_rev__image__eqI,axiom,
    ! [X3: product_prod_a_o,A4: set_Product_prod_a_o,B2: a,F: product_prod_a_o > a] :
      ( ( member2997933407038931562od_a_o @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_a2 @ B2 @ ( image_2171321463282962050_a_o_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_222_rev__image__eqI,axiom,
    ! [X3: nat,A4: set_nat,B2: product_prod_a_o,F: nat > product_prod_a_o] :
      ( ( member_nat2 @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member2997933407038931562od_a_o @ B2 @ ( image_5312737573458615594od_a_o @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_223_rev__image__eqI,axiom,
    ! [X3: a,A4: set_a,B2: product_prod_a_o,F: a > product_prod_a_o] :
      ( ( member_a2 @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member2997933407038931562od_a_o @ B2 @ ( image_4279145912174180552od_a_o @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_224_rev__image__eqI,axiom,
    ! [X3: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,B2: nat,F: list_P3196869577824546119od_a_o > nat] :
      ( ( member5261019209154752368od_a_o @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_nat2 @ B2 @ ( image_5391756926611421138_o_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_225_rev__image__eqI,axiom,
    ! [X3: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,B2: a,F: list_P3196869577824546119od_a_o > a] :
      ( ( member5261019209154752368od_a_o @ X3 @ A4 )
     => ( ( B2
          = ( F @ X3 ) )
       => ( member_a2 @ B2 @ ( image_3930418492379151228_a_o_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_226_ball__imageD,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o,P2: produc1313295348606329723on_nat > $o] :
      ( ! [X2: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X2 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
         => ( P2 @ X2 ) )
     => ! [X4: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X4 @ A4 )
         => ( P2 @ ( F @ X4 ) ) ) ) ).

% ball_imageD
thf(fact_227_image__cong,axiom,
    ! [M: set_Product_prod_a_o,N2: set_Product_prod_a_o,F: product_prod_a_o > produc1313295348606329723on_nat,G: product_prod_a_o > produc1313295348606329723on_nat] :
      ( ( M = N2 )
     => ( ! [X2: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X2 @ N2 )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( image_4414724447807847679on_nat @ F @ M )
          = ( image_4414724447807847679on_nat @ G @ N2 ) ) ) ) ).

% image_cong
thf(fact_228_bex__imageD,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X4: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X4 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
          & ( P2 @ X4 ) )
     => ? [X2: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X2 @ A4 )
          & ( P2 @ ( F @ X2 ) ) ) ) ).

% bex_imageD
thf(fact_229_image__iff,axiom,
    ! [Z: produc1313295348606329723on_nat,F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o] :
      ( ( member5906418179908956708on_nat @ Z @ ( image_4414724447807847679on_nat @ F @ A4 ) )
      = ( ? [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ A4 )
            & ( Z
              = ( F @ X ) ) ) ) ) ).

% image_iff
thf(fact_230_imageI,axiom,
    ! [X3: nat,A4: set_nat,F: nat > nat] :
      ( ( member_nat2 @ X3 @ A4 )
     => ( member_nat2 @ ( F @ X3 ) @ ( image_nat_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_231_imageI,axiom,
    ! [X3: nat,A4: set_nat,F: nat > a] :
      ( ( member_nat2 @ X3 @ A4 )
     => ( member_a2 @ ( F @ X3 ) @ ( image_nat_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_232_imageI,axiom,
    ! [X3: a,A4: set_a,F: a > nat] :
      ( ( member_a2 @ X3 @ A4 )
     => ( member_nat2 @ ( F @ X3 ) @ ( image_a_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_233_imageI,axiom,
    ! [X3: a,A4: set_a,F: a > a] :
      ( ( member_a2 @ X3 @ A4 )
     => ( member_a2 @ ( F @ X3 ) @ ( image_a_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_234_imageI,axiom,
    ! [X3: product_prod_a_o,A4: set_Product_prod_a_o,F: product_prod_a_o > nat] :
      ( ( member2997933407038931562od_a_o @ X3 @ A4 )
     => ( member_nat2 @ ( F @ X3 ) @ ( image_8012505923468758860_o_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_235_imageI,axiom,
    ! [X3: product_prod_a_o,A4: set_Product_prod_a_o,F: product_prod_a_o > a] :
      ( ( member2997933407038931562od_a_o @ X3 @ A4 )
     => ( member_a2 @ ( F @ X3 ) @ ( image_2171321463282962050_a_o_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_236_imageI,axiom,
    ! [X3: nat,A4: set_nat,F: nat > product_prod_a_o] :
      ( ( member_nat2 @ X3 @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ X3 ) @ ( image_5312737573458615594od_a_o @ F @ A4 ) ) ) ).

% imageI
thf(fact_237_imageI,axiom,
    ! [X3: a,A4: set_a,F: a > product_prod_a_o] :
      ( ( member_a2 @ X3 @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ X3 ) @ ( image_4279145912174180552od_a_o @ F @ A4 ) ) ) ).

% imageI
thf(fact_238_imageI,axiom,
    ! [X3: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,F: list_P3196869577824546119od_a_o > nat] :
      ( ( member5261019209154752368od_a_o @ X3 @ A4 )
     => ( member_nat2 @ ( F @ X3 ) @ ( image_5391756926611421138_o_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_239_imageI,axiom,
    ! [X3: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,F: list_P3196869577824546119od_a_o > a] :
      ( ( member5261019209154752368od_a_o @ X3 @ A4 )
     => ( member_a2 @ ( F @ X3 ) @ ( image_3930418492379151228_a_o_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_240__092_060open_062_092_060exists_062s_092_060in_062set_As__left_O_A_Is_M_AInr_Ai_J_A_092_060in_062_Aset_AS_092_060close_062,axiom,
    ? [X2: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ s_left ) )
      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ ( sum_In255631617079920996tation @ i ) ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ).

% \<open>\<exists>s\<in>set s_left. (s, Inr i) \<in> set S\<close>
thf(fact_241__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062s_O_A_092_060lbrakk_062s_A_092_060in_062_Aset_As__left_059_A_Is_M_AInr_Ai_J_A_092_060in_062_Aset_AS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [S3: sum_su4907927656612468301on_nat] :
        ( ( member8476201367316484836on_nat @ S3 @ ( set_Su4848624856859743608on_nat @ s_left ) )
       => ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ S3 @ ( sum_In255631617079920996tation @ i ) ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ).

% \<open>\<And>thesis. (\<And>s. \<lbrakk>s \<in> set s_left; (s, Inr i) \<in> set S\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_242__092_060open_062Inr_Ai_A_092_060in_062_Aset_As__right_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( sum_In255631617079920996tation @ i ) @ ( set_Su4848624856859743608on_nat @ s_right ) ).

% \<open>Inr i \<in> set s_right\<close>
thf(fact_243_iso__tuple__UNIV__I,axiom,
    ! [X3: product_prod_a_o] : ( member2997933407038931562od_a_o @ X3 @ top_to6843115102692631153od_a_o ) ).

% iso_tuple_UNIV_I
thf(fact_244_iso__tuple__UNIV__I,axiom,
    ! [X3: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X3 @ top_to7488204556702531123on_nat ) ).

% iso_tuple_UNIV_I
thf(fact_245_iso__tuple__UNIV__I,axiom,
    ! [X3: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X3 @ top_to324007776275943031od_a_o ) ).

% iso_tuple_UNIV_I
thf(fact_246_iso__tuple__UNIV__I,axiom,
    ! [X3: nat] : ( member_nat2 @ X3 @ top_top_set_nat ) ).

% iso_tuple_UNIV_I
thf(fact_247_iso__tuple__UNIV__I,axiom,
    ! [X3: a] : ( member_a2 @ X3 @ top_top_set_a ) ).

% iso_tuple_UNIV_I
thf(fact_248_surj__def,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat] :
      ( ( ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o )
        = top_to9060292713886516523on_nat )
      = ( ! [Y4: produc1313295348606329723on_nat] :
          ? [X: product_prod_a_o] :
            ( Y4
            = ( F @ X ) ) ) ) ).

% surj_def
thf(fact_249_surjI,axiom,
    ! [G: product_prod_a_o > produc1313295348606329723on_nat,F: produc1313295348606329723on_nat > product_prod_a_o] :
      ( ! [X2: produc1313295348606329723on_nat] :
          ( ( G @ ( F @ X2 ) )
          = X2 )
     => ( ( image_4414724447807847679on_nat @ G @ top_to6843115102692631153od_a_o )
        = top_to9060292713886516523on_nat ) ) ).

% surjI
thf(fact_250_surjE,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat] :
      ( ( ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o )
        = top_to9060292713886516523on_nat )
     => ~ ! [X2: product_prod_a_o] :
            ( Y
           != ( F @ X2 ) ) ) ).

% surjE
thf(fact_251_surjD,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,Y: produc1313295348606329723on_nat] :
      ( ( ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o )
        = top_to9060292713886516523on_nat )
     => ? [X2: product_prod_a_o] :
          ( Y
          = ( F @ X2 ) ) ) ).

% surjD
thf(fact_252_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_M7834926499894101121tation] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3663592329813318933tation @ Ys ) )
       => ( ( nth_Pr7991355429936432087tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) @ I )
          = ( produc7247797765490398872tation @ ( nth_a @ Xs @ I ) @ ( nth_Mu116543695464807850tation @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_253_nth__zip,axiom,
    ! [I: nat,Xs: list_a,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Product_prod_a_o @ ( zip_a_o @ Xs @ Ys ) @ I )
          = ( product_Pair_a_o @ ( nth_a @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_254_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7617993195940197384at_nat @ ( zip_nat_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_255_nth__zip,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr7540282607350343039_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs @ Ys ) @ I )
          = ( produc3563005473579536832_o_nat @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_256_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( ( nth_Pr8565389872998445861od_a_o @ ( zip_na1422845649429636350od_a_o @ Xs @ Ys ) @ I )
          = ( produc3946064260745446430od_a_o @ ( nth_nat @ Xs @ I ) @ ( nth_li3248255784598043854od_a_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_257_nth__zip,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr6603386610713772677at_nat @ ( zip_Su8633780382783139798at_nat @ Xs @ Ys ) @ I )
          = ( produc7500644556467579062at_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_258_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys ) )
       => ( ( nth_Pr5135637237369297925on_nat @ ( zip_na307675373166331862on_nat @ Xs @ Ys ) @ I )
          = ( produc8397911583705546934on_nat @ ( nth_nat @ Xs @ I ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_259_nth__zip,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( ( nth_Pr8178810406531800174od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs @ Ys ) @ I )
          = ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) @ ( nth_li3248255784598043854od_a_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_260_nth__zip,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( ( nth_Pr2746116061037069032od_a_o @ ( zip_Su2363964881504851137od_a_o @ Xs @ Ys ) @ I )
          = ( produc4183299375811982305od_a_o @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) @ ( nth_li3248255784598043854od_a_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_261_nth__zip,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ys ) )
       => ( ( nth_Pr6207575030832400764on_nat @ ( zip_li8832488730364818717on_nat @ Xs @ Ys ) @ I )
          = ( produc1428451187817174077on_nat @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_262__092_060open_062Inr_Ai_A_092_060in_062_Aset_Ans__right_A_092_060or_062_AInr_Ai_A_092_060in_062_Aset_As__right_092_060close_062,axiom,
    ( ( member8476201367316484836on_nat @ ( sum_In255631617079920996tation @ i ) @ ( set_Su4848624856859743608on_nat @ ns_right ) )
    | ( member8476201367316484836on_nat @ ( sum_In255631617079920996tation @ i ) @ ( set_Su4848624856859743608on_nat @ s_right ) ) ) ).

% \<open>Inr i \<in> set ns_right \<or> Inr i \<in> set s_right\<close>
thf(fact_263__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062i_O_A_092_060lbrakk_062i_A_092_060in_062_Aset_Acs_059_Ac_A_061_Acnf_A_B_Ai_059_Ai_A_060_Alength_Acnf_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [I2: nat] :
        ( ( member_nat2 @ I2 @ ( set_nat2 @ cs ) )
       => ( ( c
            = ( nth_li3248255784598043854od_a_o @ cnf @ I2 ) )
         => ~ ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ) ).

% \<open>\<And>thesis. (\<And>i. \<lbrakk>i \<in> set cs; c = cnf ! i; i < length cnf\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_264_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_265_less__imp__neq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% less_imp_neq
thf(fact_266_order_Oasym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order.asym
thf(fact_267_ord__eq__less__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A = B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_eq_less_trans
thf(fact_268_ord__less__eq__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% ord_less_eq_trans
thf(fact_269_less__induct,axiom,
    ! [P2: nat > $o,A: nat] :
      ( ! [X2: nat] :
          ( ! [Y5: nat] :
              ( ( ord_less_nat @ Y5 @ X2 )
             => ( P2 @ Y5 ) )
         => ( P2 @ X2 ) )
     => ( P2 @ A ) ) ).

% less_induct
thf(fact_270_antisym__conv3,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv3
thf(fact_271_linorder__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( X3 != Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_cases
thf(fact_272_dual__order_Oasym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ~ ( ord_less_nat @ A @ B2 ) ) ).

% dual_order.asym
thf(fact_273_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_274_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X5: nat] : ( P3 @ X5 ) )
    = ( ^ [P4: nat > $o] :
        ? [N3: nat] :
          ( ( P4 @ N3 )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ~ ( P4 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_275_linorder__less__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A3: nat,B: nat] :
          ( ( ord_less_nat @ A3 @ B )
         => ( P2 @ A3 @ B ) )
     => ( ! [A3: nat] : ( P2 @ A3 @ A3 )
       => ( ! [A3: nat,B: nat] :
              ( ( P2 @ B @ A3 )
             => ( P2 @ A3 @ B ) )
         => ( P2 @ A @ B2 ) ) ) ) ).

% linorder_less_wlog
thf(fact_276_order_Ostrict__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans
thf(fact_277_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ( ord_less_nat @ Y @ X3 )
        | ( X3 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_278_dual__order_Ostrict__trans,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_279_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( A != B2 ) ) ).

% order.strict_implies_not_eq
thf(fact_280_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( A != B2 ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_281_linorder__neqE,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE
thf(fact_282_order__less__asym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_asym
thf(fact_283_linorder__neq__iff,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
      = ( ( ord_less_nat @ X3 @ Y )
        | ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_284_order__less__asym_H,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ~ ( ord_less_nat @ B2 @ A ) ) ).

% order_less_asym'
thf(fact_285_order__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_trans
thf(fact_286_ord__eq__less__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_nat @ X2 @ Y3 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_287_ord__less__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_nat @ X2 @ Y3 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_less_eq_subst
thf(fact_288_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_289_order__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_nat @ X2 @ Y3 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_subst1
thf(fact_290_order__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_nat @ X2 @ Y3 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_subst2
thf(fact_291_order__less__not__sym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_not_sym
thf(fact_292_order__less__imp__triv,axiom,
    ! [X3: nat,Y: nat,P2: $o] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ X3 )
       => P2 ) ) ).

% order_less_imp_triv
thf(fact_293_linorder__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
      | ( X3 = Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_less_linear
thf(fact_294_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( X3 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_295_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( Y != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_296_order__less__imp__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ~ ( ord_less_nat @ Y @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_297_length__induct,axiom,
    ! [P2: list_S5253404628753938141on_nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ! [Xs2: list_S5253404628753938141on_nat] :
          ( ! [Ys2: list_S5253404628753938141on_nat] :
              ( ( ord_less_nat @ ( size_s5686234912572693873on_nat @ Ys2 ) @ ( size_s5686234912572693873on_nat @ Xs2 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_298_length__induct,axiom,
    ! [P2: list_l8336264072933094989od_a_o > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ! [Xs2: list_l8336264072933094989od_a_o] :
          ( ! [Ys2: list_l8336264072933094989od_a_o] :
              ( ( ord_less_nat @ ( size_s3379185846251495609od_a_o @ Ys2 ) @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_299_length__induct,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P2 @ Ys2 ) )
         => ( P2 @ Xs2 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_300_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_S5253404628753938141on_nat,Z2: list_S5253404628753938141on_nat] : ( Y6 = Z2 ) )
    = ( ^ [Xs3: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( ( nth_Su8471564416988646406on_nat @ Xs3 @ I3 )
                = ( nth_Su8471564416988646406on_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_301_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_l8336264072933094989od_a_o,Z2: list_l8336264072933094989od_a_o] : ( Y6 = Z2 ) )
    = ( ^ [Xs3: list_l8336264072933094989od_a_o,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( ( nth_li3248255784598043854od_a_o @ Xs3 @ I3 )
                = ( nth_li3248255784598043854od_a_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_302_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z2: list_nat] : ( Y6 = Z2 ) )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_303_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > sum_su4907927656612468301on_nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: sum_su4907927656612468301on_nat] : ( P2 @ I3 @ X6 ) ) )
      = ( ? [Xs3: list_S5253404628753938141on_nat] :
            ( ( ( size_s5686234912572693873on_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_304_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > list_P3196869577824546119od_a_o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: list_P3196869577824546119od_a_o] : ( P2 @ I3 @ X6 ) ) )
      = ( ? [Xs3: list_l8336264072933094989od_a_o] :
            ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_305_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X6: nat] : ( P2 @ I3 @ X6 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P2 @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_306_nth__equalityI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( ( nth_Su8471564416988646406on_nat @ Xs @ I2 )
              = ( nth_Su8471564416988646406on_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_307_nth__equalityI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( ( nth_li3248255784598043854od_a_o @ Xs @ I2 )
              = ( nth_li3248255784598043854od_a_o @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_308_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I2 )
              = ( nth_nat @ Ys @ I2 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_309_nth__mem,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( member2997933407038931562od_a_o @ ( nth_Product_prod_a_o @ Xs @ N ) @ ( set_Product_prod_a_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_310_nth__mem,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( member_a2 @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_311_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_312_nth__mem,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( member5261019209154752368od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ).

% nth_mem
thf(fact_313_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_314_list__ball__nth,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o,P2: product_prod_a_o > $o] :
      ( ( ord_less_nat @ N @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( ! [X2: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X2 @ ( set_Product_prod_a_o2 @ Xs ) )
           => ( P2 @ X2 ) )
       => ( P2 @ ( nth_Product_prod_a_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_315_list__ball__nth,axiom,
    ! [N: nat,Xs: list_a,P2: a > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ! [X2: a] :
            ( ( member_a2 @ X2 @ ( set_a2 @ Xs ) )
           => ( P2 @ X2 ) )
       => ( P2 @ ( nth_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_316_list__ball__nth,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ! [X2: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P2 @ X2 ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_317_list__ball__nth,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ! [X2: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( P2 @ X2 ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_318_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P2: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_nat2 @ Xs ) )
           => ( P2 @ X2 ) )
       => ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_319_in__set__conv__nth,axiom,
    ! [X3: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1727999245659094451od_a_o @ Xs ) )
            & ( ( nth_Product_prod_a_o @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_320_in__set__conv__nth,axiom,
    ! [X3: a,Xs: list_a] :
      ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
            & ( ( nth_a @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_321_in__set__conv__nth,axiom,
    ! [X3: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( nth_Su8471564416988646406on_nat @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_322_in__set__conv__nth,axiom,
    ! [X3: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( nth_li3248255784598043854od_a_o @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_323_in__set__conv__nth,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_324_all__nth__imp__all__set,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,P2: product_prod_a_o > $o,X3: product_prod_a_o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s1727999245659094451od_a_o @ Xs ) )
         => ( P2 @ ( nth_Product_prod_a_o @ Xs @ I2 ) ) )
     => ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
       => ( P2 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_325_all__nth__imp__all__set,axiom,
    ! [Xs: list_a,P2: a > $o,X3: a] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
         => ( P2 @ ( nth_a @ Xs @ I2 ) ) )
     => ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
       => ( P2 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_326_all__nth__imp__all__set,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o,X3: sum_su4907927656612468301on_nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
         => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ I2 ) ) )
     => ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( P2 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_327_all__nth__imp__all__set,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > $o,X3: list_P3196869577824546119od_a_o] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
         => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ I2 ) ) )
     => ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
       => ( P2 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_328_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: nat > $o,X3: nat] :
      ( ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
         => ( P2 @ ( nth_nat @ Xs @ I2 ) ) )
     => ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
       => ( P2 @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_329_all__set__conv__all__nth,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,P2: product_prod_a_o > $o] :
      ( ( ! [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) )
           => ( P2 @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s1727999245659094451od_a_o @ Xs ) )
           => ( P2 @ ( nth_Product_prod_a_o @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_330_all__set__conv__all__nth,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ( ! [X: a] :
            ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
           => ( P2 @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
           => ( P2 @ ( nth_a @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_331_all__set__conv__all__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ! [X: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P2 @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_332_all__set__conv__all__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( ! [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( P2 @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_333_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
           => ( P2 @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( P2 @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_334_ns,axiom,
    ! [I: nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ ns_right ) )
     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ ns_left @ I ) @ ( nth_Su8471564416988646406on_nat @ ns_right @ I ) ) @ ( set_Pr937530902923597456on_nat @ ns ) ) ) ).

% ns
thf(fact_335__092_060open_062_092_060exists_062x_092_060in_062set_Acs_O_A_Is_M_AInr_Ai_J_A_092_060in_062_A_I_092_060lambda_062l_O_A_Ims__elem__of__lit_Al_M_AInr_Ax_J_J_A_096_Aset_A_Icnf_A_B_Ax_J_092_060close_062,axiom,
    ? [X2: nat] :
      ( ( member_nat2 @ X2 @ ( set_nat2 @ cs ) )
      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ s2 @ ( sum_In255631617079920996tation @ i ) )
        @ ( image_4414724447807847679on_nat
          @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ X2 ) )
          @ ( set_Product_prod_a_o2 @ ( nth_li3248255784598043854od_a_o @ cnf @ X2 ) ) ) ) ) ).

% \<open>\<exists>x\<in>set cs. (s, Inr i) \<in> (\<lambda>l. (ms_elem_of_lit l, Inr x)) ` set (cnf ! x)\<close>
thf(fact_336_sum_Oinject_I2_J,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( sum_In255631617079920996tation @ X22 )
        = ( sum_In255631617079920996tation @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% sum.inject(2)
thf(fact_337_old_Osum_Oinject_I2_J,axiom,
    ! [B2: nat,B3: nat] :
      ( ( ( sum_In255631617079920996tation @ B2 )
        = ( sum_In255631617079920996tation @ B3 ) )
      = ( B2 = B3 ) ) ).

% old.sum.inject(2)
thf(fact_338_in__set__product__lists__length,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Xss: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ Xs @ ( set_li2629784761229999580od_a_o @ ( produc3814538932794474544od_a_o @ Xss ) ) )
     => ( ( size_s1727999245659094451od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_339_in__set__product__lists__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Xss: list_l8999537460041914605on_nat] :
      ( ( member86162308477975540on_nat @ Xs @ ( set_li4371160873219908744on_nat @ ( produc1748266815408604318on_nat @ Xss ) ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s650155532264187265on_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_340_in__set__product__lists__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Xss: list_l7431997615882138323od_a_o] :
      ( ( member8885487059802017782od_a_o @ Xs @ ( set_li3106133282984605794od_a_o @ ( produc6266365578930819638od_a_o @ Xss ) ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s1509257606055081791od_a_o @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_341_in__set__product__lists__length,axiom,
    ! [Xs: list_nat,Xss: list_list_nat] :
      ( ( member_list_nat @ Xs @ ( set_list_nat2 @ ( product_lists_nat @ Xss ) ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3023201423986296836st_nat @ Xss ) ) ) ).

% in_set_product_lists_length
thf(fact_342_set__swap,axiom,
    ! [I: nat,Xs: list_P3196869577824546119od_a_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s1727999245659094451od_a_o @ Xs ) )
       => ( ( set_Product_prod_a_o2 @ ( list_u356469027079241519od_a_o @ ( list_u356469027079241519od_a_o @ Xs @ I @ ( nth_Product_prod_a_o @ Xs @ J ) ) @ J @ ( nth_Product_prod_a_o @ Xs @ I ) ) )
          = ( set_Product_prod_a_o2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_343_set__swap,axiom,
    ! [I: nat,Xs: list_a,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_a @ Xs ) )
       => ( ( set_a2 @ ( list_update_a @ ( list_update_a @ Xs @ I @ ( nth_a @ Xs @ J ) ) @ J @ ( nth_a @ Xs @ I ) ) )
          = ( set_a2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_344_set__swap,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) @ J @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) ) )
          = ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% set_swap
thf(fact_345_set__swap,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( set_li2629784761229999580od_a_o @ ( list_u329291161401623989od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) @ J @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) ) )
          = ( set_li2629784761229999580od_a_o @ Xs ) ) ) ) ).

% set_swap
thf(fact_346_set__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( set_nat2 @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( set_nat2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_347_list__ex__length,axiom,
    ( list_e3769318529992910857on_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > $o,Xs3: list_S5253404628753938141on_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
          & ( P4 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_348_list__ex__length,axiom,
    ( list_e5403782194016407179od_a_o
    = ( ^ [P4: list_P3196869577824546119od_a_o > $o,Xs3: list_l8336264072933094989od_a_o] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
          & ( P4 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_349_list__ex__length,axiom,
    ( list_ex_nat
    = ( ^ [P4: nat > $o,Xs3: list_nat] :
        ? [N3: nat] :
          ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs3 ) )
          & ( P4 @ ( nth_nat @ Xs3 @ N3 ) ) ) ) ) ).

% list_ex_length
thf(fact_350_length__n__lists__elem,axiom,
    ! [Ys: list_P3196869577824546119od_a_o,N: nat,Xs: list_P3196869577824546119od_a_o] :
      ( ( member5261019209154752368od_a_o @ Ys @ ( set_li2629784761229999580od_a_o @ ( n_list4838562191945764047od_a_o @ N @ Xs ) ) )
     => ( ( size_s1727999245659094451od_a_o @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_351_length__n__lists__elem,axiom,
    ! [Ys: list_S5253404628753938141on_nat,N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member86162308477975540on_nat @ Ys @ ( set_li4371160873219908744on_nat @ ( n_list6604620681149647551on_nat @ N @ Xs ) ) )
     => ( ( size_s5686234912572693873on_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_352_length__n__lists__elem,axiom,
    ! [Ys: list_l8336264072933094989od_a_o,N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( member8885487059802017782od_a_o @ Ys @ ( set_li3106133282984605794od_a_o @ ( n_list931218472929919829od_a_o @ N @ Xs ) ) )
     => ( ( size_s3379185846251495609od_a_o @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_353_length__n__lists__elem,axiom,
    ! [Ys: list_nat,N: nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) ) )
     => ( ( size_size_list_nat @ Ys )
        = N ) ) ).

% length_n_lists_elem
thf(fact_354_listrel__iff__nth,axiom,
    ! [Xs: list_a,Ys: list_M7834926499894101121tation,R: set_Pr3525538344199662036tation] :
      ( ( member1315478125161133003tation @ ( produc8106702317871607470tation @ Xs @ Ys ) @ ( listre7259538626121459390tation @ R ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_s3663592329813318933tation @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
           => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ ( nth_a @ Xs @ N3 ) @ ( nth_Mu116543695464807850tation @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_355_listrel__iff__nth,axiom,
    ! [Xs: list_a,Ys: list_o,R: set_Product_prod_a_o] :
      ( ( member5884096760728255722list_o @ ( produc997889681483320753list_o @ Xs @ Ys ) @ ( listrel_a_o @ R ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs ) )
           => ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ ( nth_a @ Xs @ N3 ) @ ( nth_o @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_356_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_357_listrel__iff__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,R: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ Xs @ Ys ) @ ( listre90293961589800166_o_nat @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member7885718254550376541_o_nat @ ( produc3563005473579536832_o_nat @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_358_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr94232683564258082od_a_o] :
      ( ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ Xs @ Ys ) @ ( listre473352748755709764od_a_o @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member8910825520198479363od_a_o @ ( produc3946064260745446430od_a_o @ ( nth_nat @ Xs @ N3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_359_listrel__iff__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,R: set_Pr3345771464143190622at_nat] :
      ( ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ Xs @ Ys ) @ ( listre6443138976769382928at_nat @ R ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( member8820252441731870247at_nat @ ( produc7500644556467579062at_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N3 ) @ ( nth_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_360_listrel__iff__nth,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr7552745763645690846on_nat] :
      ( ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ Xs @ Ys ) @ ( listre7340406004007350800on_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
           => ( member7352503068387395495on_nat @ ( produc8397911583705546934on_nat @ ( nth_nat @ Xs @ N3 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_361_listrel__iff__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre9037316752283237297od_a_o @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_362_listrel__iff__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr9140199768791955813od_a_o] :
      ( ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ Xs @ Ys ) @ ( listre1105410954321393415od_a_o @ R ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( member7595723914438784454od_a_o @ ( produc4183299375811982305od_a_o @ ( nth_Su8471564416988646406on_nat @ Xs @ N3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_363_listrel__iff__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,R: set_Pr6144483798281041401on_nat] :
      ( ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ Xs @ Ys ) @ ( listre7573934803181360995on_nat @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member1833810847379340378on_nat @ ( produc1428451187817174077on_nat @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ N3 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_364_map__prod__surj,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,G: $o > sum_su4907927656612468301on_nat] :
      ( ( ( image_6104031138955651590on_nat @ F @ top_top_set_a )
        = top_to7488204556702531123on_nat )
     => ( ( ( image_1218510692399648672on_nat @ G @ top_top_set_o )
          = top_to7488204556702531123on_nat )
       => ( ( image_4414724447807847679on_nat @ ( produc4673986130646558416on_nat @ F @ G ) @ top_to6843115102692631153od_a_o )
          = top_to9060292713886516523on_nat ) ) ) ).

% map_prod_surj
thf(fact_365_map__prod__surj,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,G: product_prod_a_o > produc1313295348606329723on_nat] :
      ( ( ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o )
        = top_to9060292713886516523on_nat )
     => ( ( ( image_4414724447807847679on_nat @ G @ top_to6843115102692631153od_a_o )
          = top_to9060292713886516523on_nat )
       => ( ( image_4395559148152102469on_nat @ ( produc7138782500827752938on_nat @ F @ G ) @ top_to8745644026400245911od_a_o )
          = top_to8814693007203093271on_nat ) ) ) ).

% map_prod_surj
thf(fact_366_case__prodI2,axiom,
    ! [P: produc1714117091004681831on_nat,C2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o] :
      ( ! [A3: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat] :
          ( ( P
            = ( produc7152764266822961111on_nat @ A3 @ B ) )
         => ( C2 @ A3 @ B ) )
     => ( produc4129062015429275820_nat_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_367_case__prodI2,axiom,
    ! [P: produc118098540238352959on_nat,C2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o] :
      ( ! [A3: list_S5253404628753938141on_nat,B: produc1714117091004681831on_nat] :
          ( ( P
            = ( produc4176415733022414961on_nat @ A3 @ B ) )
         => ( C2 @ A3 @ B ) )
     => ( produc1360909077854731642_nat_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_368_case__prodI2,axiom,
    ! [P: produc8325617224762090749on_nat,C2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ! [A3: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] :
          ( ( P
            = ( produc1545487600272464053on_nat @ A3 @ B ) )
         => ( C2 @ A3 @ B ) )
     => ( produc7728289706344822350_nat_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_369_case__prodI2,axiom,
    ! [P: produc27318731269267614tation,C2: a > multis8755981257395730801tation > $o] :
      ( ! [A3: a,B: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B ) )
         => ( C2 @ A3 @ B ) )
     => ( produc8513698947429006227tion_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_370_case__prodI2,axiom,
    ! [P: product_prod_a_o,C2: a > $o > $o] :
      ( ! [A3: a,B: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B ) )
         => ( C2 @ A3 @ B ) )
     => ( produc8428829086471790546_a_o_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_371_case__prodI,axiom,
    ! [F: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( F @ A @ B2 )
     => ( produc4129062015429275820_nat_o @ F @ ( produc7152764266822961111on_nat @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_372_case__prodI,axiom,
    ! [F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( F @ A @ B2 )
     => ( produc1360909077854731642_nat_o @ F @ ( produc4176415733022414961on_nat @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_373_case__prodI,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( F @ A @ B2 )
     => ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_374_case__prodI,axiom,
    ! [F: a > multis8755981257395730801tation > $o,A: a,B2: multis8755981257395730801tation] :
      ( ( F @ A @ B2 )
     => ( produc8513698947429006227tion_o @ F @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_375_case__prodI,axiom,
    ! [F: a > $o > $o,A: a,B2: $o] :
      ( ( F @ A @ B2 )
     => ( produc8428829086471790546_a_o_o @ F @ ( product_Pair_a_o @ A @ B2 ) ) ) ).

% case_prodI
thf(fact_376_map__prod__simp,axiom,
    ! [F: a > a,G: multis8755981257395730801tation > multis8755981257395730801tation,A: a,B2: multis8755981257395730801tation] :
      ( ( produc7564068419863864794tation @ F @ G @ ( produc7247797765490398872tation @ A @ B2 ) )
      = ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_377_map__prod__simp,axiom,
    ! [F: a > a,G: multis8755981257395730801tation > $o,A: a,B2: multis8755981257395730801tation] :
      ( ( produc3829317145882942831tion_o @ F @ G @ ( produc7247797765490398872tation @ A @ B2 ) )
      = ( product_Pair_a_o @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_378_map__prod__simp,axiom,
    ! [F: a > a,G: $o > multis8755981257395730801tation,A: a,B2: $o] :
      ( ( produc1440123698213052563tation @ F @ G @ ( product_Pair_a_o @ A @ B2 ) )
      = ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_379_map__prod__simp,axiom,
    ! [F: a > a,G: $o > $o,A: a,B2: $o] :
      ( ( produc7047141983197783286_a_o_o @ F @ G @ ( product_Pair_a_o @ A @ B2 ) )
      = ( product_Pair_a_o @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_380_map__prod__simp,axiom,
    ! [F: list_P9127870862658512897on_nat > a,G: list_P9127870862658512897on_nat > multis8755981257395730801tation,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( produc4159954741969166591tation @ F @ G @ ( produc7152764266822961111on_nat @ A @ B2 ) )
      = ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_381_map__prod__simp,axiom,
    ! [F: list_P9127870862658512897on_nat > a,G: list_P9127870862658512897on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( produc2995880458181291914_nat_o @ F @ G @ ( produc7152764266822961111on_nat @ A @ B2 ) )
      = ( product_Pair_a_o @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_382_map__prod__simp,axiom,
    ! [F: a > list_P9127870862658512897on_nat,G: multis8755981257395730801tation > list_P9127870862658512897on_nat,A: a,B2: multis8755981257395730801tation] :
      ( ( produc2773900488858266773on_nat @ F @ G @ ( produc7247797765490398872tation @ A @ B2 ) )
      = ( produc7152764266822961111on_nat @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_383_map__prod__simp,axiom,
    ! [F: a > list_P9127870862658512897on_nat,G: $o > list_P9127870862658512897on_nat,A: a,B2: $o] :
      ( ( produc4118371380427577954on_nat @ F @ G @ ( product_Pair_a_o @ A @ B2 ) )
      = ( produc7152764266822961111on_nat @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_384_map__prod__simp,axiom,
    ! [F: list_S5253404628753938141on_nat > a,G: produc1714117091004681831on_nat > multis8755981257395730801tation,A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( produc743305375745534255tation @ F @ G @ ( produc4176415733022414961on_nat @ A @ B2 ) )
      = ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_385_map__prod__simp,axiom,
    ! [F: list_S5253404628753938141on_nat > a,G: produc1714117091004681831on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( produc3007302543126082394_nat_o @ F @ G @ ( produc4176415733022414961on_nat @ A @ B2 ) )
      = ( product_Pair_a_o @ ( F @ A ) @ ( G @ B2 ) ) ) ).

% map_prod_simp
thf(fact_386_length__list__update,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I: nat,X3: sum_su4907927656612468301on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X3 ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_list_update
thf(fact_387_length__list__update,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I: nat,X3: list_P3196869577824546119od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X3 ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_list_update
thf(fact_388_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X3 ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_389_list__update__id,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I: nat] :
      ( ( list_u329291161401623989od_a_o @ Xs @ I @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_390_list__update__id,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I: nat] :
      ( ( list_u8577817230714425439on_nat @ Xs @ I @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_391_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_392_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o] :
      ( ( I != J )
     => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X3 ) @ J )
        = ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_393_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( I != J )
     => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X3 ) @ J )
        = ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_394_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X3: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_395_mem__case__prodI,axiom,
    ! [Z: nat,C2: a > multis8755981257395730801tation > set_nat,A: a,B2: multis8755981257395730801tation] :
      ( ( member_nat2 @ Z @ ( C2 @ A @ B2 ) )
     => ( member_nat2 @ Z @ ( produc6722172017951998027et_nat @ C2 @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_396_mem__case__prodI,axiom,
    ! [Z: a,C2: a > multis8755981257395730801tation > set_a,A: a,B2: multis8755981257395730801tation] :
      ( ( member_a2 @ Z @ ( C2 @ A @ B2 ) )
     => ( member_a2 @ Z @ ( produc7412255387654812377_set_a @ C2 @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_397_mem__case__prodI,axiom,
    ! [Z: nat,C2: a > $o > set_nat,A: a,B2: $o] :
      ( ( member_nat2 @ Z @ ( C2 @ A @ B2 ) )
     => ( member_nat2 @ Z @ ( produc8120172443933582796et_nat @ C2 @ ( product_Pair_a_o @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_398_mem__case__prodI,axiom,
    ! [Z: a,C2: a > $o > set_a,A: a,B2: $o] :
      ( ( member_a2 @ Z @ ( C2 @ A @ B2 ) )
     => ( member_a2 @ Z @ ( produc6584190720350841752_set_a @ C2 @ ( product_Pair_a_o @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_399_mem__case__prodI,axiom,
    ! [Z: product_prod_a_o,C2: a > multis8755981257395730801tation > set_Product_prod_a_o,A: a,B2: multis8755981257395730801tation] :
      ( ( member2997933407038931562od_a_o @ Z @ ( C2 @ A @ B2 ) )
     => ( member2997933407038931562od_a_o @ Z @ ( produc8399654089035374108od_a_o @ C2 @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_400_mem__case__prodI,axiom,
    ! [Z: product_prod_a_o,C2: a > $o > set_Product_prod_a_o,A: a,B2: $o] :
      ( ( member2997933407038931562od_a_o @ Z @ ( C2 @ A @ B2 ) )
     => ( member2997933407038931562od_a_o @ Z @ ( produc1948407148953235291od_a_o @ C2 @ ( product_Pair_a_o @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_401_mem__case__prodI,axiom,
    ! [Z: list_P3196869577824546119od_a_o,C2: a > multis8755981257395730801tation > set_li5011860890973139239od_a_o,A: a,B2: multis8755981257395730801tation] :
      ( ( member5261019209154752368od_a_o @ Z @ ( C2 @ A @ B2 ) )
     => ( member5261019209154752368od_a_o @ Z @ ( produc2219104145104074914od_a_o @ C2 @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_402_mem__case__prodI,axiom,
    ! [Z: list_P3196869577824546119od_a_o,C2: a > $o > set_li5011860890973139239od_a_o,A: a,B2: $o] :
      ( ( member5261019209154752368od_a_o @ Z @ ( C2 @ A @ B2 ) )
     => ( member5261019209154752368od_a_o @ Z @ ( produc929350387168271969od_a_o @ C2 @ ( product_Pair_a_o @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_403_mem__case__prodI,axiom,
    ! [Z: sum_su4907927656612468301on_nat,C2: a > multis8755981257395730801tation > set_Su266273576555847555on_nat,A: a,B2: multis8755981257395730801tation] :
      ( ( member8476201367316484836on_nat @ Z @ ( C2 @ A @ B2 ) )
     => ( member8476201367316484836on_nat @ Z @ ( produc6506295005197259848on_nat @ C2 @ ( produc7247797765490398872tation @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_404_mem__case__prodI,axiom,
    ! [Z: sum_su4907927656612468301on_nat,C2: a > $o > set_Su266273576555847555on_nat,A: a,B2: $o] :
      ( ( member8476201367316484836on_nat @ Z @ ( C2 @ A @ B2 ) )
     => ( member8476201367316484836on_nat @ Z @ ( produc2666714314943159753on_nat @ C2 @ ( product_Pair_a_o @ A @ B2 ) ) ) ) ).

% mem_case_prodI
thf(fact_405_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z: nat,C2: a > multis8755981257395730801tation > set_nat] :
      ( ! [A3: a,B: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B ) )
         => ( member_nat2 @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member_nat2 @ Z @ ( produc6722172017951998027et_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_406_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z: a,C2: a > multis8755981257395730801tation > set_a] :
      ( ! [A3: a,B: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B ) )
         => ( member_a2 @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member_a2 @ Z @ ( produc7412255387654812377_set_a @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_407_mem__case__prodI2,axiom,
    ! [P: product_prod_a_o,Z: nat,C2: a > $o > set_nat] :
      ( ! [A3: a,B: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B ) )
         => ( member_nat2 @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member_nat2 @ Z @ ( produc8120172443933582796et_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_408_mem__case__prodI2,axiom,
    ! [P: product_prod_a_o,Z: a,C2: a > $o > set_a] :
      ( ! [A3: a,B: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B ) )
         => ( member_a2 @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member_a2 @ Z @ ( produc6584190720350841752_set_a @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_409_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z: product_prod_a_o,C2: a > multis8755981257395730801tation > set_Product_prod_a_o] :
      ( ! [A3: a,B: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B ) )
         => ( member2997933407038931562od_a_o @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member2997933407038931562od_a_o @ Z @ ( produc8399654089035374108od_a_o @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_410_mem__case__prodI2,axiom,
    ! [P: product_prod_a_o,Z: product_prod_a_o,C2: a > $o > set_Product_prod_a_o] :
      ( ! [A3: a,B: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B ) )
         => ( member2997933407038931562od_a_o @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member2997933407038931562od_a_o @ Z @ ( produc1948407148953235291od_a_o @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_411_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z: list_P3196869577824546119od_a_o,C2: a > multis8755981257395730801tation > set_li5011860890973139239od_a_o] :
      ( ! [A3: a,B: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B ) )
         => ( member5261019209154752368od_a_o @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member5261019209154752368od_a_o @ Z @ ( produc2219104145104074914od_a_o @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_412_mem__case__prodI2,axiom,
    ! [P: product_prod_a_o,Z: list_P3196869577824546119od_a_o,C2: a > $o > set_li5011860890973139239od_a_o] :
      ( ! [A3: a,B: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B ) )
         => ( member5261019209154752368od_a_o @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member5261019209154752368od_a_o @ Z @ ( produc929350387168271969od_a_o @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_413_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z: sum_su4907927656612468301on_nat,C2: a > multis8755981257395730801tation > set_Su266273576555847555on_nat] :
      ( ! [A3: a,B: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B ) )
         => ( member8476201367316484836on_nat @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member8476201367316484836on_nat @ Z @ ( produc6506295005197259848on_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_414_mem__case__prodI2,axiom,
    ! [P: product_prod_a_o,Z: sum_su4907927656612468301on_nat,C2: a > $o > set_Su266273576555847555on_nat] :
      ( ! [A3: a,B: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B ) )
         => ( member8476201367316484836on_nat @ Z @ ( C2 @ A3 @ B ) ) )
     => ( member8476201367316484836on_nat @ Z @ ( produc2666714314943159753on_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_415_map__prod__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,R2: set_Pr3525538344199662036tation,F: a > a,G: multis8755981257395730801tation > multis8755981257395730801tation] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ R2 )
     => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_625288787233451971tation @ ( produc7564068419863864794tation @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_416_map__prod__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,R2: set_Pr3525538344199662036tation,F: a > a,G: multis8755981257395730801tation > $o] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ R2 )
     => ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_5815038000605959644od_a_o @ ( produc3829317145882942831tion_o @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_417_map__prod__imageI,axiom,
    ! [A: a,B2: $o,R2: set_Product_prod_a_o,F: a > a,G: $o > multis8755981257395730801tation] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ R2 )
     => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_4101429262907221530tation @ ( produc1440123698213052563tation @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_418_map__prod__imageI,axiom,
    ! [A: a,B2: $o,R2: set_Product_prod_a_o,F: a > a,G: $o > $o] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ R2 )
     => ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_8824070870166315333od_a_o @ ( produc7047141983197783286_a_o_o @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_419_map__prod__imageI,axiom,
    ! [A: a,B2: $o,R2: set_Product_prod_a_o,F: a > sum_su4907927656612468301on_nat,G: $o > sum_su4907927656612468301on_nat] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ R2 )
     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_4414724447807847679on_nat @ ( produc4673986130646558416on_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_420_map__prod__imageI,axiom,
    ! [A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat,R2: set_Pr8586234676504894535on_nat,F: list_P9127870862658512897on_nat > a,G: list_P9127870862658512897on_nat > multis8755981257395730801tation] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B2 ) @ R2 )
     => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_3019096259419038912tation @ ( produc4159954741969166591tation @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_421_map__prod__imageI,axiom,
    ! [A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat,R2: set_Pr8586234676504894535on_nat,F: list_P9127870862658512897on_nat > a,G: list_P9127870862658512897on_nat > $o] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B2 ) @ R2 )
     => ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_3830984041189458975od_a_o @ ( produc2995880458181291914_nat_o @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_422_map__prod__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,R2: set_Pr3525538344199662036tation,F: a > list_P9127870862658512897on_nat,G: multis8755981257395730801tation > list_P9127870862658512897on_nat] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ R2 )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_4245465781113571586on_nat @ ( produc2773900488858266773on_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_423_map__prod__imageI,axiom,
    ! [A: a,B2: $o,R2: set_Product_prod_a_o,F: a > list_P9127870862658512897on_nat,G: $o > list_P9127870862658512897on_nat] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ R2 )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_3684678123239247339on_nat @ ( produc4118371380427577954on_nat @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_424_map__prod__imageI,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat,R2: set_Pr737724919871395061on_nat,F: list_S5253404628753938141on_nat > a,G: produc1714117091004681831on_nat > multis8755981257395730801tation] :
      ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ A @ B2 ) @ R2 )
     => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ ( F @ A ) @ ( G @ B2 ) ) @ ( image_6940753310563843812tation @ ( produc743305375745534255tation @ F @ G ) @ R2 ) ) ) ).

% map_prod_imageI
thf(fact_425_pair__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > nat] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ A4 )
     => ( member_nat2 @ ( F @ A @ B2 ) @ ( image_5814668043327422325on_nat @ ( produc4465434948709935253on_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_426_pair__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > a] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ A4 )
     => ( member_a2 @ ( F @ A @ B2 ) @ ( image_7660857576331368089tion_a @ ( produc4916163002043446649tion_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_427_pair__imageI,axiom,
    ! [A: a,B2: $o,A4: set_Product_prod_a_o,F: a > $o > nat] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ A4 )
     => ( member_nat2 @ ( F @ A @ B2 ) @ ( image_8012505923468758860_o_nat @ ( produc578040550328963606_o_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_428_pair__imageI,axiom,
    ! [A: a,B2: $o,A4: set_Product_prod_a_o,F: a > $o > a] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ A4 )
     => ( member_a2 @ ( F @ A @ B2 ) @ ( image_2171321463282962050_a_o_a @ ( produc6357147902488193592_a_o_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_429_pair__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > product_prod_a_o] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ A @ B2 ) @ ( image_5815038000605959644od_a_o @ ( produc2267023758550893756od_a_o @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_430_pair__imageI,axiom,
    ! [A: a,B2: $o,A4: set_Product_prod_a_o,F: a > $o > product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ A @ B2 ) @ ( image_8824070870166315333od_a_o @ ( produc453674083619213819od_a_o @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_431_pair__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > list_P3196869577824546119od_a_o] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ A4 )
     => ( member5261019209154752368od_a_o @ ( F @ A @ B2 ) @ ( image_8789645372280709602od_a_o @ ( produc3635115104032628930od_a_o @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_432_pair__imageI,axiom,
    ! [A: a,B2: $o,A4: set_Product_prod_a_o,F: a > $o > list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ A4 )
     => ( member5261019209154752368od_a_o @ ( F @ A @ B2 ) @ ( image_322316094668268747od_a_o @ ( produc2369997295210398337od_a_o @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_433_pair__imageI,axiom,
    ! [A: a,B2: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > sum_su4907927656612468301on_nat] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B2 ) @ A4 )
     => ( member8476201367316484836on_nat @ ( F @ A @ B2 ) @ ( image_524866980406113778on_nat @ ( produc2540041555698278162on_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_434_pair__imageI,axiom,
    ! [A: a,B2: $o,A4: set_Product_prod_a_o,F: a > $o > sum_su4907927656612468301on_nat] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B2 ) @ A4 )
     => ( member8476201367316484836on_nat @ ( F @ A @ B2 ) @ ( image_7473851654885147081on_nat @ ( produc8300224648618142867on_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_435_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_436_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_437_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_438_Compr__image__eq,axiom,
    ! [F: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat2 @ X @ ( image_nat_nat @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_439_Compr__image__eq,axiom,
    ! [F: a > nat,A4: set_a,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat2 @ X @ ( image_a_nat @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_a_nat @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_440_Compr__image__eq,axiom,
    ! [F: nat > a,A4: set_nat,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a2 @ X @ ( image_nat_a @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_nat_a @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_441_Compr__image__eq,axiom,
    ! [F: a > a,A4: set_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a2 @ X @ ( image_a_a @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_442_Compr__image__eq,axiom,
    ! [F: nat > product_prod_a_o,A4: set_nat,P2: product_prod_a_o > $o] :
      ( ( collec5729118848859662764od_a_o
        @ ^ [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ ( image_5312737573458615594od_a_o @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_5312737573458615594od_a_o @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_443_Compr__image__eq,axiom,
    ! [F: a > product_prod_a_o,A4: set_a,P2: product_prod_a_o > $o] :
      ( ( collec5729118848859662764od_a_o
        @ ^ [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ ( image_4279145912174180552od_a_o @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_4279145912174180552od_a_o @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_444_Compr__image__eq,axiom,
    ! [F: product_prod_a_o > nat,A4: set_Product_prod_a_o,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat2 @ X @ ( image_8012505923468758860_o_nat @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_8012505923468758860_o_nat @ F
        @ ( collec5729118848859662764od_a_o
          @ ^ [X: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_445_Compr__image__eq,axiom,
    ! [F: product_prod_a_o > a,A4: set_Product_prod_a_o,P2: a > $o] :
      ( ( collect_a
        @ ^ [X: a] :
            ( ( member_a2 @ X @ ( image_2171321463282962050_a_o_a @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_2171321463282962050_a_o_a @ F
        @ ( collec5729118848859662764od_a_o
          @ ^ [X: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_446_Compr__image__eq,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,A4: set_nat,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ ( image_5774815713777330736od_a_o @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_5774815713777330736od_a_o @ F
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_447_Compr__image__eq,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,A4: set_a,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ ( image_7865913289429950158od_a_o @ F @ A4 ) )
            & ( P2 @ X ) ) )
      = ( image_7865913289429950158od_a_o @ F
        @ ( collect_a
          @ ^ [X: a] :
              ( ( member_a2 @ X @ A4 )
              & ( P2 @ ( F @ X ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_448_image__image,axiom,
    ! [F: produc1313295348606329723on_nat > produc1313295348606329723on_nat,G: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o] :
      ( ( image_9087143737194713285on_nat @ F @ ( image_4414724447807847679on_nat @ G @ A4 ) )
      = ( image_4414724447807847679on_nat
        @ ^ [X: product_prod_a_o] : ( F @ ( G @ X ) )
        @ A4 ) ) ).

% image_image
thf(fact_449_image__image,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,G: product_prod_a_o > product_prod_a_o,A4: set_Product_prod_a_o] :
      ( ( image_4414724447807847679on_nat @ F @ ( image_8824070870166315333od_a_o @ G @ A4 ) )
      = ( image_4414724447807847679on_nat
        @ ^ [X: product_prod_a_o] : ( F @ ( G @ X ) )
        @ A4 ) ) ).

% image_image
thf(fact_450_imageE,axiom,
    ! [B2: nat,F: nat > nat,A4: set_nat] :
      ( ( member_nat2 @ B2 @ ( image_nat_nat @ F @ A4 ) )
     => ~ ! [X2: nat] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_nat2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_451_imageE,axiom,
    ! [B2: nat,F: a > nat,A4: set_a] :
      ( ( member_nat2 @ B2 @ ( image_a_nat @ F @ A4 ) )
     => ~ ! [X2: a] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_a2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_452_imageE,axiom,
    ! [B2: a,F: nat > a,A4: set_nat] :
      ( ( member_a2 @ B2 @ ( image_nat_a @ F @ A4 ) )
     => ~ ! [X2: nat] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_nat2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_453_imageE,axiom,
    ! [B2: a,F: a > a,A4: set_a] :
      ( ( member_a2 @ B2 @ ( image_a_a @ F @ A4 ) )
     => ~ ! [X2: a] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_a2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_454_imageE,axiom,
    ! [B2: product_prod_a_o,F: nat > product_prod_a_o,A4: set_nat] :
      ( ( member2997933407038931562od_a_o @ B2 @ ( image_5312737573458615594od_a_o @ F @ A4 ) )
     => ~ ! [X2: nat] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_nat2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_455_imageE,axiom,
    ! [B2: product_prod_a_o,F: a > product_prod_a_o,A4: set_a] :
      ( ( member2997933407038931562od_a_o @ B2 @ ( image_4279145912174180552od_a_o @ F @ A4 ) )
     => ~ ! [X2: a] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_a2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_456_imageE,axiom,
    ! [B2: nat,F: product_prod_a_o > nat,A4: set_Product_prod_a_o] :
      ( ( member_nat2 @ B2 @ ( image_8012505923468758860_o_nat @ F @ A4 ) )
     => ~ ! [X2: product_prod_a_o] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member2997933407038931562od_a_o @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_457_imageE,axiom,
    ! [B2: a,F: product_prod_a_o > a,A4: set_Product_prod_a_o] :
      ( ( member_a2 @ B2 @ ( image_2171321463282962050_a_o_a @ F @ A4 ) )
     => ~ ! [X2: product_prod_a_o] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member2997933407038931562od_a_o @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_458_imageE,axiom,
    ! [B2: list_P3196869577824546119od_a_o,F: nat > list_P3196869577824546119od_a_o,A4: set_nat] :
      ( ( member5261019209154752368od_a_o @ B2 @ ( image_5774815713777330736od_a_o @ F @ A4 ) )
     => ~ ! [X2: nat] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_nat2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_459_imageE,axiom,
    ! [B2: list_P3196869577824546119od_a_o,F: a > list_P3196869577824546119od_a_o,A4: set_a] :
      ( ( member5261019209154752368od_a_o @ B2 @ ( image_7865913289429950158od_a_o @ F @ A4 ) )
     => ~ ! [X2: a] :
            ( ( B2
              = ( F @ X2 ) )
           => ~ ( member_a2 @ X2 @ A4 ) ) ) ).

% imageE
thf(fact_460_case__prodE,axiom,
    ! [C2: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,P: produc1714117091004681831on_nat] :
      ( ( produc4129062015429275820_nat_o @ C2 @ P )
     => ~ ! [X2: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
            ( ( P
              = ( produc7152764266822961111on_nat @ X2 @ Y3 ) )
           => ~ ( C2 @ X2 @ Y3 ) ) ) ).

% case_prodE
thf(fact_461_case__prodE,axiom,
    ! [C2: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,P: produc118098540238352959on_nat] :
      ( ( produc1360909077854731642_nat_o @ C2 @ P )
     => ~ ! [X2: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat] :
            ( ( P
              = ( produc4176415733022414961on_nat @ X2 @ Y3 ) )
           => ~ ( C2 @ X2 @ Y3 ) ) ) ).

% case_prodE
thf(fact_462_case__prodE,axiom,
    ! [C2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ( produc7728289706344822350_nat_o @ C2 @ P )
     => ~ ! [X2: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat] :
            ( ( P
              = ( produc1545487600272464053on_nat @ X2 @ Y3 ) )
           => ~ ( C2 @ X2 @ Y3 ) ) ) ).

% case_prodE
thf(fact_463_case__prodE,axiom,
    ! [C2: a > multis8755981257395730801tation > $o,P: produc27318731269267614tation] :
      ( ( produc8513698947429006227tion_o @ C2 @ P )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X2 @ Y3 ) )
           => ~ ( C2 @ X2 @ Y3 ) ) ) ).

% case_prodE
thf(fact_464_case__prodE,axiom,
    ! [C2: a > $o > $o,P: product_prod_a_o] :
      ( ( produc8428829086471790546_a_o_o @ C2 @ P )
     => ~ ! [X2: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X2 @ Y3 ) )
           => ~ ( C2 @ X2 @ Y3 ) ) ) ).

% case_prodE
thf(fact_465_case__prodD,axiom,
    ! [F: list_P9127870862658512897on_nat > list_P9127870862658512897on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( produc4129062015429275820_nat_o @ F @ ( produc7152764266822961111on_nat @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_466_case__prodD,axiom,
    ! [F: list_S5253404628753938141on_nat > produc1714117091004681831on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc1714117091004681831on_nat] :
      ( ( produc1360909077854731642_nat_o @ F @ ( produc4176415733022414961on_nat @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_467_case__prodD,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_468_case__prodD,axiom,
    ! [F: a > multis8755981257395730801tation > $o,A: a,B2: multis8755981257395730801tation] :
      ( ( produc8513698947429006227tion_o @ F @ ( produc7247797765490398872tation @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_469_case__prodD,axiom,
    ! [F: a > $o > $o,A: a,B2: $o] :
      ( ( produc8428829086471790546_a_o_o @ F @ ( product_Pair_a_o @ A @ B2 ) )
     => ( F @ A @ B2 ) ) ).

% case_prodD
thf(fact_470_rangeE,axiom,
    ! [B2: produc1313295348606329723on_nat,F: product_prod_a_o > produc1313295348606329723on_nat] :
      ( ( member5906418179908956708on_nat @ B2 @ ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o ) )
     => ~ ! [X2: product_prod_a_o] :
            ( B2
           != ( F @ X2 ) ) ) ).

% rangeE
thf(fact_471_range__composition,axiom,
    ! [F: produc1313295348606329723on_nat > produc1313295348606329723on_nat,G: product_prod_a_o > produc1313295348606329723on_nat] :
      ( ( image_4414724447807847679on_nat
        @ ^ [X: product_prod_a_o] : ( F @ ( G @ X ) )
        @ top_to6843115102692631153od_a_o )
      = ( image_9087143737194713285on_nat @ F @ ( image_4414724447807847679on_nat @ G @ top_to6843115102692631153od_a_o ) ) ) ).

% range_composition
thf(fact_472_range__composition,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,G: product_prod_a_o > product_prod_a_o] :
      ( ( image_4414724447807847679on_nat
        @ ^ [X: product_prod_a_o] : ( F @ ( G @ X ) )
        @ top_to6843115102692631153od_a_o )
      = ( image_4414724447807847679on_nat @ F @ ( image_8824070870166315333od_a_o @ G @ top_to6843115102692631153od_a_o ) ) ) ).

% range_composition
thf(fact_473_zip__update,axiom,
    ! [Xs: list_l1726306699111145479on_nat,I: nat,X3: list_P9127870862658512897on_nat,Ys: list_l1726306699111145479on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( zip_li3935435632236560119on_nat @ ( list_u184235112954345583on_nat @ Xs @ I @ X3 ) @ ( list_u184235112954345583on_nat @ Ys @ I @ Y ) )
      = ( list_u3278723242984596181on_nat @ ( zip_li3935435632236560119on_nat @ Xs @ Ys ) @ I @ ( produc7152764266822961111on_nat @ X3 @ Y ) ) ) ).

% zip_update
thf(fact_474_zip__update,axiom,
    ! [Xs: list_l8999537460041914605on_nat,I: nat,X3: list_S5253404628753938141on_nat,Ys: list_P8734936519038225773on_nat,Y: produc1714117091004681831on_nat] :
      ( ( zip_li8749512630040955217on_nat @ ( list_u3873310082918383599on_nat @ Xs @ I @ X3 ) @ ( list_u3278723242984596181on_nat @ Ys @ I @ Y ) )
      = ( list_u3336313086739624785on_nat @ ( zip_li8749512630040955217on_nat @ Xs @ Ys ) @ I @ ( produc4176415733022414961on_nat @ X3 @ Y ) ) ) ).

% zip_update
thf(fact_475_zip__update,axiom,
    ! [Xs: list_l8999537460041914605on_nat,I: nat,X3: list_S5253404628753938141on_nat,Ys: list_P5508476027986415439on_nat,Y: produc118098540238352959on_nat] :
      ( ( zip_li7388484927123455957on_nat @ ( list_u3873310082918383599on_nat @ Xs @ I @ X3 ) @ ( list_u3336313086739624785on_nat @ Ys @ I @ Y ) )
      = ( list_u550736303359258219on_nat @ ( zip_li7388484927123455957on_nat @ Xs @ Ys ) @ I @ ( produc1545487600272464053on_nat @ X3 @ Y ) ) ) ).

% zip_update
thf(fact_476_zip__update,axiom,
    ! [Xs: list_a,I: nat,X3: a,Ys: list_M7834926499894101121tation,Y: multis8755981257395730801tation] :
      ( ( zip_a_5995912112572712824tation @ ( list_update_a @ Xs @ I @ X3 ) @ ( list_u8956197407191130243tation @ Ys @ I @ Y ) )
      = ( list_u9175788670554990000tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) @ I @ ( produc7247797765490398872tation @ X3 @ Y ) ) ) ).

% zip_update
thf(fact_477_zip__update,axiom,
    ! [Xs: list_a,I: nat,X3: a,Ys: list_o,Y: $o] :
      ( ( zip_a_o @ ( list_update_a @ Xs @ I @ X3 ) @ ( list_update_o @ Ys @ I @ Y ) )
      = ( list_u356469027079241519od_a_o @ ( zip_a_o @ Xs @ Ys ) @ I @ ( product_Pair_a_o @ X3 @ Y ) ) ) ).

% zip_update
thf(fact_478_insort__insert__triv,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( linord1921536354676448932at_nat
          @ ^ [X: nat] : X
          @ X3
          @ Xs )
        = Xs ) ) ).

% insort_insert_triv
thf(fact_479_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_480_listrel__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre5398654966139722893on_nat @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_481_listrel__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr9140199768791955813od_a_o] :
      ( ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ Xs @ Ys ) @ ( listre1105410954321393415od_a_o @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_482_listrel__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,R: set_Pr3345771464143190622at_nat] :
      ( ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ Xs @ Ys ) @ ( listre6443138976769382928at_nat @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_483_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,R: set_Pr6144483798281041401on_nat] :
      ( ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ Xs @ Ys ) @ ( listre7573934803181360995on_nat @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_484_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre9037316752283237297od_a_o @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_485_listrel__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,R: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ Xs @ Ys ) @ ( listre90293961589800166_o_nat @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_486_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr7552745763645690846on_nat] :
      ( ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ Xs @ Ys ) @ ( listre7340406004007350800on_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_487_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr94232683564258082od_a_o] :
      ( ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ Xs @ Ys ) @ ( listre473352748755709764od_a_o @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_488_listrel__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel_eq_len
thf(fact_489_list__ex__cong,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o,F: product_prod_a_o > $o,G: product_prod_a_o > $o] :
      ( ( Xs = Ys )
     => ( ! [X2: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X2 @ ( set_Product_prod_a_o2 @ Ys ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( list_e12574053215267589od_a_o @ F @ Xs )
          = ( list_e12574053215267589od_a_o @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_490_list__ex__cong,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > $o,G: sum_su4907927656612468301on_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X2: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ Ys ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( list_e3769318529992910857on_nat @ F @ Xs )
          = ( list_e3769318529992910857on_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_491_list__ex__cong,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > $o,G: list_P3196869577824546119od_a_o > $o] :
      ( ( Xs = Ys )
     => ( ! [X2: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Ys ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( list_e5403782194016407179od_a_o @ F @ Xs )
          = ( list_e5403782194016407179od_a_o @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_492_list__ex__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X2: nat] :
            ( ( member_nat2 @ X2 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( list_ex_nat @ F @ Xs )
          = ( list_ex_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_493_list__ex__cong,axiom,
    ! [Xs: list_a,Ys: list_a,F: a > $o,G: a > $o] :
      ( ( Xs = Ys )
     => ( ! [X2: a] :
            ( ( member_a2 @ X2 @ ( set_a2 @ Ys ) )
           => ( ( F @ X2 )
              = ( G @ X2 ) ) )
       => ( ( list_ex_a @ F @ Xs )
          = ( list_ex_a @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_494_prod__fun__imageE,axiom,
    ! [C2: produc27318731269267614tation,F: a > a,G: multis8755981257395730801tation > multis8755981257395730801tation,R2: set_Pr3525538344199662036tation] :
      ( ( member7446769889882171317tation @ C2 @ ( image_625288787233451971tation @ ( produc7564068419863864794tation @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( C2
              = ( produc7247797765490398872tation @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_495_prod__fun__imageE,axiom,
    ! [C2: produc27318731269267614tation,F: a > a,G: $o > multis8755981257395730801tation,R2: set_Product_prod_a_o] :
      ( ( member7446769889882171317tation @ C2 @ ( image_4101429262907221530tation @ ( produc1440123698213052563tation @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( C2
              = ( produc7247797765490398872tation @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_496_prod__fun__imageE,axiom,
    ! [C2: product_prod_a_o,F: a > a,G: multis8755981257395730801tation > $o,R2: set_Pr3525538344199662036tation] :
      ( ( member2997933407038931562od_a_o @ C2 @ ( image_5815038000605959644od_a_o @ ( produc3829317145882942831tion_o @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( C2
              = ( product_Pair_a_o @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_497_prod__fun__imageE,axiom,
    ! [C2: product_prod_a_o,F: a > a,G: $o > $o,R2: set_Product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ C2 @ ( image_8824070870166315333od_a_o @ ( produc7047141983197783286_a_o_o @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( C2
              = ( product_Pair_a_o @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_498_prod__fun__imageE,axiom,
    ! [C2: produc1313295348606329723on_nat,F: a > sum_su4907927656612468301on_nat,G: $o > sum_su4907927656612468301on_nat,R2: set_Product_prod_a_o] :
      ( ( member5906418179908956708on_nat @ C2 @ ( image_4414724447807847679on_nat @ ( produc4673986130646558416on_nat @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( C2
              = ( produc6704844767664046899on_nat @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_499_prod__fun__imageE,axiom,
    ! [C2: produc1714117091004681831on_nat,F: a > list_P9127870862658512897on_nat,G: multis8755981257395730801tation > list_P9127870862658512897on_nat,R2: set_Pr3525538344199662036tation] :
      ( ( member4629988367768146576on_nat @ C2 @ ( image_4245465781113571586on_nat @ ( produc2773900488858266773on_nat @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( C2
              = ( produc7152764266822961111on_nat @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_500_prod__fun__imageE,axiom,
    ! [C2: produc1714117091004681831on_nat,F: a > list_P9127870862658512897on_nat,G: $o > list_P9127870862658512897on_nat,R2: set_Product_prod_a_o] :
      ( ( member4629988367768146576on_nat @ C2 @ ( image_3684678123239247339on_nat @ ( produc4118371380427577954on_nat @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: $o] :
            ( ( C2
              = ( produc7152764266822961111on_nat @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_501_prod__fun__imageE,axiom,
    ! [C2: produc27318731269267614tation,F: list_P9127870862658512897on_nat > a,G: list_P9127870862658512897on_nat > multis8755981257395730801tation,R2: set_Pr8586234676504894535on_nat] :
      ( ( member7446769889882171317tation @ C2 @ ( image_3019096259419038912tation @ ( produc4159954741969166591tation @ F @ G ) @ R2 ) )
     => ~ ! [X2: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
            ( ( C2
              = ( produc7247797765490398872tation @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_502_prod__fun__imageE,axiom,
    ! [C2: product_prod_a_o,F: list_P9127870862658512897on_nat > a,G: list_P9127870862658512897on_nat > $o,R2: set_Pr8586234676504894535on_nat] :
      ( ( member2997933407038931562od_a_o @ C2 @ ( image_3830984041189458975od_a_o @ ( produc2995880458181291914_nat_o @ F @ G ) @ R2 ) )
     => ~ ! [X2: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
            ( ( C2
              = ( product_Pair_a_o @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_503_prod__fun__imageE,axiom,
    ! [C2: produc118098540238352959on_nat,F: a > list_S5253404628753938141on_nat,G: multis8755981257395730801tation > produc1714117091004681831on_nat,R2: set_Pr3525538344199662036tation] :
      ( ( member4585374397974051670on_nat @ C2 @ ( image_649891390038869860on_nat @ ( produc775701553971539087on_nat @ F @ G ) @ R2 ) )
     => ~ ! [X2: a,Y3: multis8755981257395730801tation] :
            ( ( C2
              = ( produc4176415733022414961on_nat @ ( F @ X2 ) @ ( G @ Y3 ) ) )
           => ~ ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X2 @ Y3 ) @ R2 ) ) ) ).

% prod_fun_imageE
thf(fact_504_set__update__memI,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o,X3: product_prod_a_o] :
      ( ( ord_less_nat @ N @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ ( list_u356469027079241519od_a_o @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_505_set__update__memI,axiom,
    ! [N: nat,Xs: list_a,X3: a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( member_a2 @ X3 @ ( set_a2 @ ( list_update_a @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_506_set__update__memI,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_507_set__update__memI,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_508_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X3: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat2 @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_509_nth__list__update,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,J: nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X3 ) @ J )
            = ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_510_nth__list__update,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,J: nat,X3: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X3 ) @ J )
            = ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_511_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_512_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ( list_u8577817230714425439on_nat @ Xs @ I @ X3 )
          = Xs )
        = ( ( nth_Su8471564416988646406on_nat @ Xs @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_513_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ( list_u329291161401623989od_a_o @ Xs @ I @ X3 )
          = Xs )
        = ( ( nth_li3248255784598043854od_a_o @ Xs @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_514_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X3 )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_515_Inr__inject,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ( sum_In255631617079920996tation @ X3 )
        = ( sum_In255631617079920996tation @ Y ) )
     => ( X3 = Y ) ) ).

% Inr_inject
thf(fact_516_top__empty__eq2,axiom,
    ( top_to3006227208522020094_nat_o
    = ( ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ top_to3600439413794707863on_nat ) ) ) ).

% top_empty_eq2
thf(fact_517_top__empty__eq2,axiom,
    ( top_to6788304175586553730_nat_o
    = ( ^ [X: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y4 ) @ top_to5493003302315727781on_nat ) ) ) ).

% top_empty_eq2
thf(fact_518_top__empty__eq2,axiom,
    ( top_to7109364567023080250_nat_o
    = ( ^ [X: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y4 ) @ top_to3590780582172288173on_nat ) ) ) ).

% top_empty_eq2
thf(fact_519_top__empty__eq2,axiom,
    ( top_to6157075553826403125tion_o
    = ( ^ [X: a,Y4: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X @ Y4 ) @ top_to4352242710280458372tation ) ) ) ).

% top_empty_eq2
thf(fact_520_top__empty__eq2,axiom,
    ( top_top_a_o_o
    = ( ^ [X: a,Y4: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X @ Y4 ) @ top_to6843115102692631153od_a_o ) ) ) ).

% top_empty_eq2
thf(fact_521_top__empty__eq,axiom,
    ( top_to5018766571977796524_a_o_o
    = ( ^ [X: product_prod_a_o] : ( member2997933407038931562od_a_o @ X @ top_to6843115102692631153od_a_o ) ) ) ).

% top_empty_eq
thf(fact_522_top__empty__eq,axiom,
    ( top_to412742067730463250_nat_o
    = ( ^ [X: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X @ top_to7488204556702531123on_nat ) ) ) ).

% top_empty_eq
thf(fact_523_top__empty__eq,axiom,
    ( top_to5610533107075744038_a_o_o
    = ( ^ [X: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X @ top_to324007776275943031od_a_o ) ) ) ).

% top_empty_eq
thf(fact_524_top__empty__eq,axiom,
    ( top_top_nat_o
    = ( ^ [X: nat] : ( member_nat2 @ X @ top_top_set_nat ) ) ) ).

% top_empty_eq
thf(fact_525_top__empty__eq,axiom,
    ( top_top_a_o
    = ( ^ [X: a] : ( member_a2 @ X @ top_top_set_a ) ) ) ).

% top_empty_eq
thf(fact_526_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,N3: nat] :
            ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs @ N3 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s129639936549468787on_nat @ Xs ) )
            & ( Ys
              = ( list_u184235112954345583on_nat @ Xs @ N3 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_527_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,N3: nat] :
            ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs @ N3 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s7490768528479217773on_nat @ Xs ) )
            & ( Ys
              = ( list_u1840724836389618409on_nat @ Xs @ N3 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_528_listrel1__iff__update,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
      = ( ? [Y4: sum_su4907927656612468301on_nat,N3: nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N3 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( Ys
              = ( list_u8577817230714425439on_nat @ Xs @ N3 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_529_listrel1__iff__update,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre4186574385322583590od_a_o @ R ) )
      = ( ? [Y4: list_P3196869577824546119od_a_o,N3: nat] :
            ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N3 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( Ys
              = ( list_u329291161401623989od_a_o @ Xs @ N3 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_530_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
      = ( ? [Y4: nat,N3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N3 ) @ Y4 ) @ R )
            & ( ord_less_nat @ N3 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N3 @ Y4 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_531_listrel__iff__zip,axiom,
    ! [Xs: list_a,Ys: list_M7834926499894101121tation,R: set_Pr3525538344199662036tation] :
      ( ( member1315478125161133003tation @ ( produc8106702317871607470tation @ Xs @ Ys ) @ ( listre7259538626121459390tation @ R ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_s3663592329813318933tation @ Ys ) )
        & ! [X: produc27318731269267614tation] :
            ( ( member7446769889882171317tation @ X @ ( set_Pr5992250453084334153tation @ ( zip_a_5995912112572712824tation @ Xs @ Ys ) ) )
           => ( produc8513698947429006227tion_o
              @ ^ [Y4: a,Z3: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_532_listrel__iff__zip,axiom,
    ! [Xs: list_a,Ys: list_o,R: set_Product_prod_a_o] :
      ( ( member5884096760728255722list_o @ ( produc997889681483320753list_o @ Xs @ Ys ) @ ( listrel_a_o @ R ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_o @ Ys ) )
        & ! [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ ( zip_a_o @ Xs @ Ys ) ) )
           => ( produc8428829086471790546_a_o_o
              @ ^ [Y4: a,Z3: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_533_listrel__iff__zip,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel_nat_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
           => ( produc6081775807080527818_nat_o
              @ ^ [Y4: nat,Z3: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_534_listrel__iff__zip,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,R: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ Xs @ Ys ) @ ( listre90293961589800166_o_nat @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [X: produc408724640507517382_o_nat] :
            ( ( member7885718254550376541_o_nat @ X @ ( set_Pr7830758801870232305_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs @ Ys ) ) )
           => ( produc4957984460634321003_nat_o
              @ ^ [Y4: list_P3196869577824546119od_a_o,Z3: nat] : ( member7885718254550376541_o_nat @ ( produc3563005473579536832_o_nat @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_535_listrel__iff__zip,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr94232683564258082od_a_o] :
      ( ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ Xs @ Ys ) @ ( listre473352748755709764od_a_o @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [X: produc1433831906155620204od_a_o] :
            ( ( member8910825520198479363od_a_o @ X @ ( set_Pr8855866067518335127od_a_o @ ( zip_na1422845649429636350od_a_o @ Xs @ Ys ) ) )
           => ( produc8470836043472241293_a_o_o
              @ ^ [Y4: nat,Z3: list_P3196869577824546119od_a_o] : ( member8910825520198479363od_a_o @ ( produc3946064260745446430od_a_o @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_536_listrel__iff__zip,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,R: set_Pr3345771464143190622at_nat] :
      ( ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ Xs @ Ys ) @ ( listre6443138976769382928at_nat @ R ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [X: produc8556620195627115390at_nat] :
            ( ( member8820252441731870247at_nat @ X @ ( set_Pr1192295990000706195at_nat @ ( zip_Su8633780382783139798at_nat @ Xs @ Ys ) ) )
           => ( produc2579979265757742093_nat_o
              @ ^ [Y4: sum_su4907927656612468301on_nat,Z3: nat] : ( member8820252441731870247at_nat @ ( produc7500644556467579062at_nat @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_537_listrel__iff__zip,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr7552745763645690846on_nat] :
      ( ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ Xs @ Ys ) @ ( listre7340406004007350800on_nat @ R ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [X: produc7088870822282640638on_nat] :
            ( ( member7352503068387395495on_nat @ X @ ( set_Pr8947918653511007251on_nat @ ( zip_na307675373166331862on_nat @ Xs @ Ys ) ) )
           => ( produc2700354003219672077_nat_o
              @ ^ [Y4: nat,Z3: sum_su4907927656612468301on_nat] : ( member7352503068387395495on_nat @ ( produc8397911583705546934on_nat @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_538_listrel__iff__zip,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre9037316752283237297od_a_o @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [X: produc8520174026553075431od_a_o] :
            ( ( member5440785652130980624od_a_o @ X @ ( set_Pr7142968438029117820od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs @ Ys ) ) )
           => ( produc731694360491113772_a_o_o
              @ ^ [Y4: list_P3196869577824546119od_a_o,Z3: list_P3196869577824546119od_a_o] : ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_539_listrel__iff__zip,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,R: set_Pr9140199768791955813od_a_o] :
      ( ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ Xs @ Ys ) @ ( listre1105410954321393415od_a_o @ R ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [X: produc6493764124173775663od_a_o] :
            ( ( member7595723914438784454od_a_o @ X @ ( set_Pr6587213113994854490od_a_o @ ( zip_Su2363964881504851137od_a_o @ Xs @ Ys ) ) )
           => ( produc7187800550436554762_a_o_o
              @ ^ [Y4: sum_su4907927656612468301on_nat,Z3: list_P3196869577824546119od_a_o] : ( member7595723914438784454od_a_o @ ( produc4183299375811982305od_a_o @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_540_listrel__iff__zip,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,R: set_Pr6144483798281041401on_nat] :
      ( ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ Xs @ Ys ) @ ( listre7573934803181360995on_nat @ R ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [X: produc731851057114331587on_nat] :
            ( ( member1833810847379340378on_nat @ X @ ( set_Pr825300046935410414on_nat @ ( zip_li8832488730364818717on_nat @ Xs @ Ys ) ) )
           => ( produc963041392342583470_nat_o
              @ ^ [Y4: list_P3196869577824546119od_a_o,Z3: sum_su4907927656612468301on_nat] : ( member1833810847379340378on_nat @ ( produc1428451187817174077on_nat @ Y4 @ Z3 ) @ R )
              @ X ) ) ) ) ).

% listrel_iff_zip
thf(fact_541_listrelp__listrel__eq,axiom,
    ! [R: set_Pr8202701763255893447on_nat] :
      ( ( listre1281383508592798463on_nat
        @ ^ [X: produc1313295348606329723on_nat,Y4: produc1313295348606329723on_nat] : ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X @ Y4 ) @ R ) )
      = ( ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ ( listre6301251474747857073on_nat @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_542_listrelp__listrel__eq,axiom,
    ! [R: set_Pr8586234676504894535on_nat] :
      ( ( listre889148463277894783on_nat
        @ ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ R ) )
      = ( ^ [X: list_l1726306699111145479on_nat,Y4: list_l1726306699111145479on_nat] : ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ X @ Y4 ) @ ( listre5026495727120537649on_nat @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_543_listrelp__listrel__eq,axiom,
    ! [R: set_Pr737724919871395061on_nat] :
      ( ( listre6179979345059730889on_nat
        @ ^ [X: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y4 ) @ R ) )
      = ( ^ [X: list_l8999537460041914605on_nat,Y4: list_P8734936519038225773on_nat] : ( member8961008115305732172on_nat @ ( produc3669147386793386087on_nat @ X @ Y4 ) @ ( listre1486203369474292887on_nat @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_544_listrelp__listrel__eq,axiom,
    ! [R: set_Pr6244131468446592861on_nat] :
      ( ( listre5391820689435064669on_nat
        @ ^ [X: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y4 ) @ R ) )
      = ( ^ [X: list_l8999537460041914605on_nat,Y4: list_P5508476027986415439on_nat] : ( member9005765888383368902on_nat @ ( produc3605646969090963669on_nat @ X @ Y4 ) @ ( listre1567628990915239951on_nat @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_545_listrelp__listrel__eq,axiom,
    ! [R: set_Pr3525538344199662036tation] :
      ( ( listre5863515767970726896tation
        @ ^ [X: a,Y4: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X @ Y4 ) @ R ) )
      = ( ^ [X: list_a,Y4: list_M7834926499894101121tation] : ( member1315478125161133003tation @ ( produc8106702317871607470tation @ X @ Y4 ) @ ( listre7259538626121459390tation @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_546_listrelp__listrel__eq,axiom,
    ! [R: set_Product_prod_a_o] :
      ( ( listrelp_a_o
        @ ^ [X: a,Y4: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X @ Y4 ) @ R ) )
      = ( ^ [X: list_a,Y4: list_o] : ( member5884096760728255722list_o @ ( produc997889681483320753list_o @ X @ Y4 ) @ ( listrel_a_o @ R ) ) ) ) ).

% listrelp_listrel_eq
thf(fact_547_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,As: list_P9127870862658512897on_nat,Bs: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S4: set_Pr8202701763255893447on_nat] :
      ( ( list_l5400313399564392903on_nat @ S_list @ Ns_list )
     => ( ( ( size_s7490768528479217773on_nat @ As )
          = ( size_s7490768528479217773on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Bs ) )
             => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ As @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As @ Bs ) @ ( Ns_list @ S4 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_548_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,As: list_l1726306699111145479on_nat,Bs: list_l1726306699111145479on_nat,NS: set_Pr8586234676504894535on_nat,S4: set_Pr8586234676504894535on_nat] :
      ( ( list_l2763215806797416525on_nat @ S_list @ Ns_list )
     => ( ( ( size_s129639936549468787on_nat @ As )
          = ( size_s129639936549468787on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Bs ) )
             => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ As @ I2 ) @ ( nth_li1302981970230039432on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ As @ Bs ) @ ( Ns_list @ S4 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_549_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S4: set_Pr2535564044409765851on_nat] :
      ( ( list_l4998409290742832897on_nat @ S_list @ Ns_list )
     => ( ( ( size_s5686234912572693873on_nat @ As )
          = ( size_s5686234912572693873on_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Bs ) )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As @ Bs ) @ ( Ns_list @ S4 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_550_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o,Ns_list: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o,As: list_l8336264072933094989od_a_o,Bs: list_l8336264072933094989od_a_o,NS: set_Pr7420778282784443591od_a_o,S4: set_Pr7420778282784443591od_a_o] :
      ( ( list_l1016356893908098451od_a_o @ S_list @ Ns_list )
     => ( ( ( size_s3379185846251495609od_a_o @ As )
          = ( size_s3379185846251495609od_a_o @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Bs ) )
             => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I2 ) ) @ NS ) )
         => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As @ Bs ) @ ( Ns_list @ S4 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_551_list__order__extension_Oall__ns__imp__ns,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ( ( size_size_list_nat @ As )
          = ( size_size_list_nat @ Bs ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ NS ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S4 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_552_set__n__lists,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( set_list_a2 @ ( n_lists_a @ N @ Xs ) )
      = ( collect_list_a
        @ ^ [Ys3: list_a] :
            ( ( ( size_size_list_a @ Ys3 )
              = N )
            & ( ord_less_eq_set_a @ ( set_a2 @ Ys3 ) @ ( set_a2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_553_set__n__lists,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o] :
      ( ( set_li2629784761229999580od_a_o @ ( n_list4838562191945764047od_a_o @ N @ Xs ) )
      = ( collec5117750660557753778od_a_o
        @ ^ [Ys3: list_P3196869577824546119od_a_o] :
            ( ( ( size_s1727999245659094451od_a_o @ Ys3 )
              = N )
            & ( ord_le8613718869425792193od_a_o @ ( set_Product_prod_a_o2 @ Ys3 ) @ ( set_Product_prod_a_o2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_554_set__n__lists,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat] :
      ( ( set_li4371160873219908744on_nat @ ( n_list6604620681149647551on_nat @ N @ Xs ) )
      = ( collec7158715823234635826on_nat
        @ ^ [Ys3: list_S5253404628753938141on_nat] :
            ( ( ( size_s5686234912572693873on_nat @ Ys3 )
              = N )
            & ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Ys3 ) @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_555_set__n__lists,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( set_li3106133282984605794od_a_o @ ( n_list931218472929919829od_a_o @ N @ Xs ) )
      = ( collec3723402900366632248od_a_o
        @ ^ [Ys3: list_l8336264072933094989od_a_o] :
            ( ( ( size_s3379185846251495609od_a_o @ Ys3 )
              = N )
            & ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ Ys3 ) @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_556_set__n__lists,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( set_list_nat2 @ ( n_lists_nat @ N @ Xs ) )
      = ( collect_list_nat
        @ ^ [Ys3: list_nat] :
            ( ( ( size_size_list_nat @ Ys3 )
              = N )
            & ( ord_less_eq_set_nat @ ( set_nat2 @ Ys3 ) @ ( set_nat2 @ Xs ) ) ) ) ) ).

% set_n_lists
thf(fact_557_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_558_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_559_subsetI,axiom,
    ! [A4: set_Product_prod_a_o,B4: set_Product_prod_a_o] :
      ( ! [X2: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X2 @ A4 )
         => ( member2997933407038931562od_a_o @ X2 @ B4 ) )
     => ( ord_le8613718869425792193od_a_o @ A4 @ B4 ) ) ).

% subsetI
thf(fact_560_subsetI,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat] :
      ( ! [X2: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X2 @ A4 )
         => ( member8476201367316484836on_nat @ X2 @ B4 ) )
     => ( ord_le6595998496543023075on_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_561_subsetI,axiom,
    ! [A4: set_li5011860890973139239od_a_o,B4: set_li5011860890973139239od_a_o] :
      ( ! [X2: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X2 @ A4 )
         => ( member5261019209154752368od_a_o @ X2 @ B4 ) )
     => ( ord_le1271121679792220359od_a_o @ A4 @ B4 ) ) ).

% subsetI
thf(fact_562_subsetI,axiom,
    ! [A4: set_nat,B4: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A4 )
         => ( member_nat2 @ X2 @ B4 ) )
     => ( ord_less_eq_set_nat @ A4 @ B4 ) ) ).

% subsetI
thf(fact_563_subsetI,axiom,
    ! [A4: set_a,B4: set_a] :
      ( ! [X2: a] :
          ( ( member_a2 @ X2 @ A4 )
         => ( member_a2 @ X2 @ B4 ) )
     => ( ord_less_eq_set_a @ A4 @ B4 ) ) ).

% subsetI
thf(fact_564_prop__restrict,axiom,
    ! [X3: product_prod_a_o,Z4: set_Product_prod_a_o,X7: set_Product_prod_a_o,P2: product_prod_a_o > $o] :
      ( ( member2997933407038931562od_a_o @ X3 @ Z4 )
     => ( ( ord_le8613718869425792193od_a_o @ Z4
          @ ( collec5729118848859662764od_a_o
            @ ^ [X: product_prod_a_o] :
                ( ( member2997933407038931562od_a_o @ X @ X7 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_565_prop__restrict,axiom,
    ! [X3: sum_su4907927656612468301on_nat,Z4: set_Su266273576555847555on_nat,X7: set_Su266273576555847555on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( member8476201367316484836on_nat @ X3 @ Z4 )
     => ( ( ord_le6595998496543023075on_nat @ Z4
          @ ( collec1912426543561893410on_nat
            @ ^ [X: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ X @ X7 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_566_prop__restrict,axiom,
    ! [X3: list_P3196869577824546119od_a_o,Z4: set_li5011860890973139239od_a_o,X7: set_li5011860890973139239od_a_o,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( member5261019209154752368od_a_o @ X3 @ Z4 )
     => ( ( ord_le1271121679792220359od_a_o @ Z4
          @ ( collec5117750660557753778od_a_o
            @ ^ [X: list_P3196869577824546119od_a_o] :
                ( ( member5261019209154752368od_a_o @ X @ X7 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_567_prop__restrict,axiom,
    ! [X3: nat,Z4: set_nat,X7: set_nat,P2: nat > $o] :
      ( ( member_nat2 @ X3 @ Z4 )
     => ( ( ord_less_eq_set_nat @ Z4
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat2 @ X @ X7 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_568_prop__restrict,axiom,
    ! [X3: a,Z4: set_a,X7: set_a,P2: a > $o] :
      ( ( member_a2 @ X3 @ Z4 )
     => ( ( ord_less_eq_set_a @ Z4
          @ ( collect_a
            @ ^ [X: a] :
                ( ( member_a2 @ X @ X7 )
                & ( P2 @ X ) ) ) )
       => ( P2 @ X3 ) ) ) ).

% prop_restrict
thf(fact_569_Collect__restrict,axiom,
    ! [X7: set_Product_prod_a_o,P2: product_prod_a_o > $o] :
      ( ord_le8613718869425792193od_a_o
      @ ( collec5729118848859662764od_a_o
        @ ^ [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ X7 )
            & ( P2 @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_570_Collect__restrict,axiom,
    ! [X7: set_Su266273576555847555on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ord_le6595998496543023075on_nat
      @ ( collec1912426543561893410on_nat
        @ ^ [X: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X @ X7 )
            & ( P2 @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_571_Collect__restrict,axiom,
    ! [X7: set_li5011860890973139239od_a_o,P2: list_P3196869577824546119od_a_o > $o] :
      ( ord_le1271121679792220359od_a_o
      @ ( collec5117750660557753778od_a_o
        @ ^ [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ X7 )
            & ( P2 @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_572_Collect__restrict,axiom,
    ! [X7: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat2 @ X @ X7 )
            & ( P2 @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_573_Collect__restrict,axiom,
    ! [X7: set_a,P2: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X: a] :
            ( ( member_a2 @ X @ X7 )
            & ( P2 @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_574_Collect__subset,axiom,
    ! [A4: set_Product_prod_a_o,P2: product_prod_a_o > $o] :
      ( ord_le8613718869425792193od_a_o
      @ ( collec5729118848859662764od_a_o
        @ ^ [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ A4 )
            & ( P2 @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_575_Collect__subset,axiom,
    ! [A4: set_Su266273576555847555on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ord_le6595998496543023075on_nat
      @ ( collec1912426543561893410on_nat
        @ ^ [X: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X @ A4 )
            & ( P2 @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_576_Collect__subset,axiom,
    ! [A4: set_li5011860890973139239od_a_o,P2: list_P3196869577824546119od_a_o > $o] :
      ( ord_le1271121679792220359od_a_o
      @ ( collec5117750660557753778od_a_o
        @ ^ [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ A4 )
            & ( P2 @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_577_Collect__subset,axiom,
    ! [A4: set_nat,P2: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat2 @ X @ A4 )
            & ( P2 @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_578_Collect__subset,axiom,
    ! [A4: set_a,P2: a > $o] :
      ( ord_less_eq_set_a
      @ ( collect_a
        @ ^ [X: a] :
            ( ( member_a2 @ X @ A4 )
            & ( P2 @ X ) ) )
      @ A4 ) ).

% Collect_subset
thf(fact_579_subrelI,axiom,
    ! [R: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ! [X2: list_P9127870862658512897on_nat,Y3: list_P9127870862658512897on_nat] :
          ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X2 @ Y3 ) @ R )
         => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X2 @ Y3 ) @ S ) )
     => ( ord_le2174946625535087591on_nat @ R @ S ) ) ).

% subrelI
thf(fact_580_subrelI,axiom,
    ! [R: set_Pr737724919871395061on_nat,S: set_Pr737724919871395061on_nat] :
      ( ! [X2: list_S5253404628753938141on_nat,Y3: produc1714117091004681831on_nat] :
          ( ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y3 ) @ R )
         => ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X2 @ Y3 ) @ S ) )
     => ( ord_le248320493815342421on_nat @ R @ S ) ) ).

% subrelI
thf(fact_581_subrelI,axiom,
    ! [R: set_Pr6244131468446592861on_nat,S: set_Pr6244131468446592861on_nat] :
      ( ! [X2: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat] :
          ( ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y3 ) @ R )
         => ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X2 @ Y3 ) @ S ) )
     => ( ord_le3399424890108663549on_nat @ R @ S ) ) ).

% subrelI
thf(fact_582_subrelI,axiom,
    ! [R: set_Pr3525538344199662036tation,S: set_Pr3525538344199662036tation] :
      ( ! [X2: a,Y3: multis8755981257395730801tation] :
          ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X2 @ Y3 ) @ R )
         => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X2 @ Y3 ) @ S ) )
     => ( ord_le1700925397483502132tation @ R @ S ) ) ).

% subrelI
thf(fact_583_subrelI,axiom,
    ! [R: set_Product_prod_a_o,S: set_Product_prod_a_o] :
      ( ! [X2: a,Y3: $o] :
          ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X2 @ Y3 ) @ R )
         => ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X2 @ Y3 ) @ S ) )
     => ( ord_le8613718869425792193od_a_o @ R @ S ) ) ).

% subrelI
thf(fact_584_subset__iff,axiom,
    ( ord_le8613718869425792193od_a_o
    = ( ^ [A5: set_Product_prod_a_o,B5: set_Product_prod_a_o] :
        ! [T2: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ T2 @ A5 )
         => ( member2997933407038931562od_a_o @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_585_subset__iff,axiom,
    ( ord_le6595998496543023075on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] :
        ! [T2: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ T2 @ A5 )
         => ( member8476201367316484836on_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_586_subset__iff,axiom,
    ( ord_le1271121679792220359od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o] :
        ! [T2: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ T2 @ A5 )
         => ( member5261019209154752368od_a_o @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_587_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [T2: nat] :
          ( ( member_nat2 @ T2 @ A5 )
         => ( member_nat2 @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_588_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
        ! [T2: a] :
          ( ( member_a2 @ T2 @ A5 )
         => ( member_a2 @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_589_subset__eq,axiom,
    ( ord_le8613718869425792193od_a_o
    = ( ^ [A5: set_Product_prod_a_o,B5: set_Product_prod_a_o] :
        ! [X: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X @ A5 )
         => ( member2997933407038931562od_a_o @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_590_subset__eq,axiom,
    ( ord_le6595998496543023075on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] :
        ! [X: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X @ A5 )
         => ( member8476201367316484836on_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_591_subset__eq,axiom,
    ( ord_le1271121679792220359od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o] :
        ! [X: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X @ A5 )
         => ( member5261019209154752368od_a_o @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_592_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [X: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( member_nat2 @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_593_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
        ! [X: a] :
          ( ( member_a2 @ X @ A5 )
         => ( member_a2 @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_594_Ball__def,axiom,
    ( ball_P9042141085894431851od_a_o
    = ( ^ [A5: set_Product_prod_a_o,P4: product_prod_a_o > $o] :
        ! [X: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X @ A5 )
         => ( P4 @ X ) ) ) ) ).

% Ball_def
thf(fact_595_Ball__def,axiom,
    ( ball_S7161496081633768995on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,P4: sum_su4907927656612468301on_nat > $o] :
        ! [X: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X @ A5 )
         => ( P4 @ X ) ) ) ) ).

% Ball_def
thf(fact_596_Ball__def,axiom,
    ( ball_l6150250797124963569od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,P4: list_P3196869577824546119od_a_o > $o] :
        ! [X: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X @ A5 )
         => ( P4 @ X ) ) ) ) ).

% Ball_def
thf(fact_597_Ball__def,axiom,
    ( ball_nat
    = ( ^ [A5: set_nat,P4: nat > $o] :
        ! [X: nat] :
          ( ( member_nat2 @ X @ A5 )
         => ( P4 @ X ) ) ) ) ).

% Ball_def
thf(fact_598_Ball__def,axiom,
    ( ball_a
    = ( ^ [A5: set_a,P4: a > $o] :
        ! [X: a] :
          ( ( member_a2 @ X @ A5 )
         => ( P4 @ X ) ) ) ) ).

% Ball_def
thf(fact_599_subsetD,axiom,
    ! [A4: set_Product_prod_a_o,B4: set_Product_prod_a_o,C2: product_prod_a_o] :
      ( ( ord_le8613718869425792193od_a_o @ A4 @ B4 )
     => ( ( member2997933407038931562od_a_o @ C2 @ A4 )
       => ( member2997933407038931562od_a_o @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_600_subsetD,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat,C2: sum_su4907927656612468301on_nat] :
      ( ( ord_le6595998496543023075on_nat @ A4 @ B4 )
     => ( ( member8476201367316484836on_nat @ C2 @ A4 )
       => ( member8476201367316484836on_nat @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_601_subsetD,axiom,
    ! [A4: set_li5011860890973139239od_a_o,B4: set_li5011860890973139239od_a_o,C2: list_P3196869577824546119od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ A4 @ B4 )
     => ( ( member5261019209154752368od_a_o @ C2 @ A4 )
       => ( member5261019209154752368od_a_o @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_602_subsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat2 @ C2 @ A4 )
       => ( member_nat2 @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_603_subsetD,axiom,
    ! [A4: set_a,B4: set_a,C2: a] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ( member_a2 @ C2 @ A4 )
       => ( member_a2 @ C2 @ B4 ) ) ) ).

% subsetD
thf(fact_604_in__mono,axiom,
    ! [A4: set_Product_prod_a_o,B4: set_Product_prod_a_o,X3: product_prod_a_o] :
      ( ( ord_le8613718869425792193od_a_o @ A4 @ B4 )
     => ( ( member2997933407038931562od_a_o @ X3 @ A4 )
       => ( member2997933407038931562od_a_o @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_605_in__mono,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ord_le6595998496543023075on_nat @ A4 @ B4 )
     => ( ( member8476201367316484836on_nat @ X3 @ A4 )
       => ( member8476201367316484836on_nat @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_606_in__mono,axiom,
    ! [A4: set_li5011860890973139239od_a_o,B4: set_li5011860890973139239od_a_o,X3: list_P3196869577824546119od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ A4 @ B4 )
     => ( ( member5261019209154752368od_a_o @ X3 @ A4 )
       => ( member5261019209154752368od_a_o @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_607_in__mono,axiom,
    ! [A4: set_nat,B4: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A4 @ B4 )
     => ( ( member_nat2 @ X3 @ A4 )
       => ( member_nat2 @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_608_in__mono,axiom,
    ! [A4: set_a,B4: set_a,X3: a] :
      ( ( ord_less_eq_set_a @ A4 @ B4 )
     => ( ( member_a2 @ X3 @ A4 )
       => ( member_a2 @ X3 @ B4 ) ) ) ).

% in_mono
thf(fact_609_order__antisym__conv,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% order_antisym_conv
thf(fact_610_linorder__le__cases,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_le_cases
thf(fact_611_ord__le__eq__subst,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ( F @ B2 )
          = C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% ord_le_eq_subst
thf(fact_612_ord__eq__le__subst,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( A
        = ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_613_linorder__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_linear
thf(fact_614_order__eq__refl,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 = Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_eq_refl
thf(fact_615_order__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_subst2
thf(fact_616_order__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_subst1
thf(fact_617_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_618_antisym,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ A )
       => ( A = B2 ) ) ) ).

% antisym
thf(fact_619_dual__order_Otrans,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_eq_nat @ C2 @ A ) ) ) ).

% dual_order.trans
thf(fact_620_dual__order_Oantisym,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( A = B2 ) ) ) ).

% dual_order.antisym
thf(fact_621_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_622_linorder__wlog,axiom,
    ! [P2: nat > nat > $o,A: nat,B2: nat] :
      ( ! [A3: nat,B: nat] :
          ( ( ord_less_eq_nat @ A3 @ B )
         => ( P2 @ A3 @ B ) )
     => ( ! [A3: nat,B: nat] :
            ( ( P2 @ B @ A3 )
           => ( P2 @ A3 @ B ) )
       => ( P2 @ A @ B2 ) ) ) ).

% linorder_wlog
thf(fact_623_order__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X3 @ Z ) ) ) ).

% order_trans
thf(fact_624_order_Otrans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% order.trans
thf(fact_625_order__antisym,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X3 )
       => ( X3 = Y ) ) ) ).

% order_antisym
thf(fact_626_ord__le__eq__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( B2 = C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_le_eq_trans
thf(fact_627_ord__eq__le__trans,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( A = B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_eq_nat @ A @ C2 ) ) ) ).

% ord_eq_le_trans
thf(fact_628_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 ) )
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_629_le__cases3,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_630_nle__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B2 ) )
      = ( ( ord_less_eq_nat @ B2 @ A )
        & ( B2 != A ) ) ) ).

% nle_le
thf(fact_631_leD,axiom,
    ! [Y: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y ) ) ).

% leD
thf(fact_632_leI,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ Y @ X3 ) ) ).

% leI
thf(fact_633_nless__le,axiom,
    ! [A: nat,B2: nat] :
      ( ( ~ ( ord_less_nat @ A @ B2 ) )
      = ( ~ ( ord_less_eq_nat @ A @ B2 )
        | ( A = B2 ) ) ) ).

% nless_le
thf(fact_634_antisym__conv1,axiom,
    ! [X3: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ X3 @ Y )
        = ( X3 = Y ) ) ) ).

% antisym_conv1
thf(fact_635_antisym__conv2,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ~ ( ord_less_nat @ X3 @ Y ) )
        = ( X3 = Y ) ) ) ).

% antisym_conv2
thf(fact_636_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ~ ( ord_less_eq_nat @ Y4 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_637_not__le__imp__less,axiom,
    ! [Y: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X3 )
     => ( ord_less_nat @ X3 @ Y ) ) ).

% not_le_imp_less
thf(fact_638_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_nat @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% order.order_iff_strict
thf(fact_639_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% order.strict_iff_order
thf(fact_640_order_Ostrict__trans1,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans1
thf(fact_641_order_Ostrict__trans2,axiom,
    ! [A: nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ord_less_nat @ A @ C2 ) ) ) ).

% order.strict_trans2
thf(fact_642_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A6: nat,B6: nat] :
          ( ( ord_less_eq_nat @ A6 @ B6 )
          & ~ ( ord_less_eq_nat @ B6 @ A6 ) ) ) ) ).

% order.strict_iff_not
thf(fact_643_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( ord_less_nat @ B6 @ A6 )
          | ( A6 = B6 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_644_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ( A6 != B6 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_645_dual__order_Ostrict__trans1,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_eq_nat @ B2 @ A )
     => ( ( ord_less_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_646_dual__order_Ostrict__trans2,axiom,
    ! [B2: nat,A: nat,C2: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ( ord_less_eq_nat @ C2 @ B2 )
       => ( ord_less_nat @ C2 @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_647_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( ord_less_eq_nat @ B6 @ A6 )
          & ~ ( ord_less_eq_nat @ A6 @ B6 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_648_order_Ostrict__implies__order,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ord_less_eq_nat @ A @ B2 ) ) ).

% order.strict_implies_order
thf(fact_649_dual__order_Ostrict__implies__order,axiom,
    ! [B2: nat,A: nat] :
      ( ( ord_less_nat @ B2 @ A )
     => ( ord_less_eq_nat @ B2 @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_650_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_nat @ X @ Y4 )
          | ( X = Y4 ) ) ) ) ).

% order_le_less
thf(fact_651_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y4: nat] :
          ( ( ord_less_eq_nat @ X @ Y4 )
          & ( X != Y4 ) ) ) ) ).

% order_less_le
thf(fact_652_linorder__not__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y ) )
      = ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_not_le
thf(fact_653_linorder__not__less,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X3 ) ) ).

% linorder_not_less
thf(fact_654_order__less__imp__le,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ord_less_eq_nat @ X3 @ Y ) ) ).

% order_less_imp_le
thf(fact_655_order__le__neq__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( A != B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_le_neq_trans
thf(fact_656_order__neq__le__trans,axiom,
    ! [A: nat,B2: nat] :
      ( ( A != B2 )
     => ( ( ord_less_eq_nat @ A @ B2 )
       => ( ord_less_nat @ A @ B2 ) ) ) ).

% order_neq_le_trans
thf(fact_657_order__le__less__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_658_order__less__le__trans,axiom,
    ! [X3: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X3 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X3 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_659_order__le__less__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_nat @ X2 @ Y3 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_660_order__le__less__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_eq_nat @ A @ B2 )
     => ( ( ord_less_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_le_less_subst2
thf(fact_661_order__less__le__subst1,axiom,
    ! [A: nat,F: nat > nat,B2: nat,C2: nat] :
      ( ( ord_less_nat @ A @ ( F @ B2 ) )
     => ( ( ord_less_eq_nat @ B2 @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X2 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C2 ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_662_order__less__le__subst2,axiom,
    ! [A: nat,B2: nat,F: nat > nat,C2: nat] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( ord_less_eq_nat @ ( F @ B2 ) @ C2 )
       => ( ! [X2: nat,Y3: nat] :
              ( ( ord_less_nat @ X2 @ Y3 )
             => ( ord_less_nat @ ( F @ X2 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C2 ) ) ) ) ).

% order_less_le_subst2
thf(fact_663_linorder__le__less__linear,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
      | ( ord_less_nat @ Y @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_664_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y )
     => ( ( ord_less_nat @ X3 @ Y )
        | ( X3 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_665_image__mono,axiom,
    ! [A4: set_Product_prod_a_o,B4: set_Product_prod_a_o,F: product_prod_a_o > produc1313295348606329723on_nat] :
      ( ( ord_le8613718869425792193od_a_o @ A4 @ B4 )
     => ( ord_le6961793561435735931on_nat @ ( image_4414724447807847679on_nat @ F @ A4 ) @ ( image_4414724447807847679on_nat @ F @ B4 ) ) ) ).

% image_mono
thf(fact_666_image__subsetI,axiom,
    ! [A4: set_nat,F: nat > nat,B4: set_nat] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A4 )
         => ( member_nat2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_nat_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_667_image__subsetI,axiom,
    ! [A4: set_nat,F: nat > a,B4: set_a] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A4 )
         => ( member_a2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_nat_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_668_image__subsetI,axiom,
    ! [A4: set_a,F: a > nat,B4: set_nat] :
      ( ! [X2: a] :
          ( ( member_a2 @ X2 @ A4 )
         => ( member_nat2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_a_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_669_image__subsetI,axiom,
    ! [A4: set_a,F: a > a,B4: set_a] :
      ( ! [X2: a] :
          ( ( member_a2 @ X2 @ A4 )
         => ( member_a2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_a_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_670_image__subsetI,axiom,
    ! [A4: set_Product_prod_a_o,F: product_prod_a_o > nat,B4: set_nat] :
      ( ! [X2: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X2 @ A4 )
         => ( member_nat2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_8012505923468758860_o_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_671_image__subsetI,axiom,
    ! [A4: set_Product_prod_a_o,F: product_prod_a_o > a,B4: set_a] :
      ( ! [X2: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X2 @ A4 )
         => ( member_a2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_2171321463282962050_a_o_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_672_image__subsetI,axiom,
    ! [A4: set_nat,F: nat > product_prod_a_o,B4: set_Product_prod_a_o] :
      ( ! [X2: nat] :
          ( ( member_nat2 @ X2 @ A4 )
         => ( member2997933407038931562od_a_o @ ( F @ X2 ) @ B4 ) )
     => ( ord_le8613718869425792193od_a_o @ ( image_5312737573458615594od_a_o @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_673_image__subsetI,axiom,
    ! [A4: set_a,F: a > product_prod_a_o,B4: set_Product_prod_a_o] :
      ( ! [X2: a] :
          ( ( member_a2 @ X2 @ A4 )
         => ( member2997933407038931562od_a_o @ ( F @ X2 ) @ B4 ) )
     => ( ord_le8613718869425792193od_a_o @ ( image_4279145912174180552od_a_o @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_674_image__subsetI,axiom,
    ! [A4: set_li5011860890973139239od_a_o,F: list_P3196869577824546119od_a_o > nat,B4: set_nat] :
      ( ! [X2: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X2 @ A4 )
         => ( member_nat2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_nat @ ( image_5391756926611421138_o_nat @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_675_image__subsetI,axiom,
    ! [A4: set_li5011860890973139239od_a_o,F: list_P3196869577824546119od_a_o > a,B4: set_a] :
      ( ! [X2: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X2 @ A4 )
         => ( member_a2 @ ( F @ X2 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_3930418492379151228_a_o_a @ F @ A4 ) @ B4 ) ) ).

% image_subsetI
thf(fact_676_subset__imageE,axiom,
    ! [B4: set_Pr2535564044409765851on_nat,F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o] :
      ( ( ord_le6961793561435735931on_nat @ B4 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
     => ~ ! [C4: set_Product_prod_a_o] :
            ( ( ord_le8613718869425792193od_a_o @ C4 @ A4 )
           => ( B4
             != ( image_4414724447807847679on_nat @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_677_image__subset__iff,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o,B4: set_Pr2535564044409765851on_nat] :
      ( ( ord_le6961793561435735931on_nat @ ( image_4414724447807847679on_nat @ F @ A4 ) @ B4 )
      = ( ! [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ A4 )
           => ( member5906418179908956708on_nat @ ( F @ X ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_678_subset__image__iff,axiom,
    ! [B4: set_Pr2535564044409765851on_nat,F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o] :
      ( ( ord_le6961793561435735931on_nat @ B4 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
      = ( ? [AA: set_Product_prod_a_o] :
            ( ( ord_le8613718869425792193od_a_o @ AA @ A4 )
            & ( B4
              = ( image_4414724447807847679on_nat @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_679_subset__code_I1_J,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,B4: set_Product_prod_a_o] :
      ( ( ord_le8613718869425792193od_a_o @ ( set_Product_prod_a_o2 @ Xs ) @ B4 )
      = ( ! [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) )
           => ( member2997933407038931562od_a_o @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_680_subset__code_I1_J,axiom,
    ! [Xs: list_S5253404628753938141on_nat,B4: set_Su266273576555847555on_nat] :
      ( ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ B4 )
      = ( ! [X: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( member8476201367316484836on_nat @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_681_subset__code_I1_J,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,B4: set_li5011860890973139239od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ Xs ) @ B4 )
      = ( ! [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( member5261019209154752368od_a_o @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_682_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B4: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B4 )
      = ( ! [X: nat] :
            ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat2 @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_683_subset__code_I1_J,axiom,
    ! [Xs: list_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ B4 )
      = ( ! [X: a] :
            ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
           => ( member_a2 @ X @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_684_image__Collect__subsetI,axiom,
    ! [P2: product_prod_a_o > $o,F: product_prod_a_o > produc1313295348606329723on_nat,B4: set_Pr2535564044409765851on_nat] :
      ( ! [X2: product_prod_a_o] :
          ( ( P2 @ X2 )
         => ( member5906418179908956708on_nat @ ( F @ X2 ) @ B4 ) )
     => ( ord_le6961793561435735931on_nat @ ( image_4414724447807847679on_nat @ F @ ( collec5729118848859662764od_a_o @ P2 ) ) @ B4 ) ) ).

% image_Collect_subsetI
thf(fact_685_range__subsetD,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,B4: set_Pr2535564044409765851on_nat,I: product_prod_a_o] :
      ( ( ord_le6961793561435735931on_nat @ ( image_4414724447807847679on_nat @ F @ top_to6843115102692631153od_a_o ) @ B4 )
     => ( member5906418179908956708on_nat @ ( F @ I ) @ B4 ) ) ).

% range_subsetD
thf(fact_686_set__update__subsetI,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,A4: set_Product_prod_a_o,X3: product_prod_a_o,I: nat] :
      ( ( ord_le8613718869425792193od_a_o @ ( set_Product_prod_a_o2 @ Xs ) @ A4 )
     => ( ( member2997933407038931562od_a_o @ X3 @ A4 )
       => ( ord_le8613718869425792193od_a_o @ ( set_Product_prod_a_o2 @ ( list_u356469027079241519od_a_o @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_687_set__update__subsetI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,A4: set_Su266273576555847555on_nat,X3: sum_su4907927656612468301on_nat,I: nat] :
      ( ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ A4 )
     => ( ( member8476201367316484836on_nat @ X3 @ A4 )
       => ( ord_le6595998496543023075on_nat @ ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_688_set__update__subsetI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,A4: set_li5011860890973139239od_a_o,X3: list_P3196869577824546119od_a_o,I: nat] :
      ( ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ Xs ) @ A4 )
     => ( ( member5261019209154752368od_a_o @ X3 @ A4 )
       => ( ord_le1271121679792220359od_a_o @ ( set_li2629784761229999580od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_689_set__update__subsetI,axiom,
    ! [Xs: list_nat,A4: set_nat,X3: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A4 )
     => ( ( member_nat2 @ X3 @ A4 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_690_set__update__subsetI,axiom,
    ! [Xs: list_a,A4: set_a,X3: a,I: nat] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs ) @ A4 )
     => ( ( member_a2 @ X3 @ A4 )
       => ( ord_less_eq_set_a @ ( set_a2 @ ( list_update_a @ Xs @ I @ X3 ) ) @ A4 ) ) ) ).

% set_update_subsetI
thf(fact_691_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_692_listrel1__eq__len,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_693_listrel1__eq__len,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ Ys ) @ ( listre4186574385322583590od_a_o @ R ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_694_listrel1__eq__len,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_695_list__eq__iff__zip__eq,axiom,
    ( ( ^ [Y6: list_S5253404628753938141on_nat,Z2: list_S5253404628753938141on_nat] : ( Y6 = Z2 ) )
    = ( ^ [Xs3: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [X: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs3 @ Ys3 ) ) )
             => ( produc208325271403635792_nat_o
                @ ^ [Y6: sum_su4907927656612468301on_nat,Z2: sum_su4907927656612468301on_nat] : ( Y6 = Z2 )
                @ X ) ) ) ) ) ).

% list_eq_iff_zip_eq
thf(fact_696_list__eq__iff__zip__eq,axiom,
    ( ( ^ [Y6: list_l8336264072933094989od_a_o,Z2: list_l8336264072933094989od_a_o] : ( Y6 = Z2 ) )
    = ( ^ [Xs3: list_l8336264072933094989od_a_o,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [X: produc8520174026553075431od_a_o] :
              ( ( member5440785652130980624od_a_o @ X @ ( set_Pr7142968438029117820od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs3 @ Ys3 ) ) )
             => ( produc731694360491113772_a_o_o
                @ ^ [Y6: list_P3196869577824546119od_a_o,Z2: list_P3196869577824546119od_a_o] : ( Y6 = Z2 )
                @ X ) ) ) ) ) ).

% list_eq_iff_zip_eq
thf(fact_697_list__eq__iff__zip__eq,axiom,
    ( ( ^ [Y6: list_nat,Z2: list_nat] : ( Y6 = Z2 ) )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs3 @ Ys3 ) ) )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y6: nat,Z2: nat] : ( Y6 = Z2 )
                @ X ) ) ) ) ) ).

% list_eq_iff_zip_eq
thf(fact_698_listrel__def,axiom,
    ( listre5026495727120537649on_nat
    = ( ^ [R3: set_Pr8586234676504894535on_nat] :
          ( collec6800664810089915218on_nat
          @ ( produc2872587184234751276_nat_o
            @ ( listre889148463277894783on_nat
              @ ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_699_listrel__def,axiom,
    ( listre1486203369474292887on_nat
    = ( ^ [R3: set_Pr737724919871395061on_nat] :
          ( collec8397000633925766794on_nat
          @ ( produc4405497983287177220_nat_o
            @ ( listre6179979345059730889on_nat
              @ ^ [X: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_700_listrel__def,axiom,
    ( listre1567628990915239951on_nat
    = ( ^ [R3: set_Pr6244131468446592861on_nat] :
          ( collec5137301729278706184on_nat
          @ ( produc5331718163378463278_nat_o
            @ ( listre5391820689435064669on_nat
              @ ^ [X: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_701_listrel__def,axiom,
    ( listre7259538626121459390tation
    = ( ^ [R3: set_Pr3525538344199662036tation] :
          ( collec8193118915346073097tation
          @ ( produc5118108073676854781tion_o
            @ ( listre5863515767970726896tation
              @ ^ [X: a,Y4: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_702_listrel__def,axiom,
    ( listrel_a_o
    = ( ^ [R3: set_Product_prod_a_o] :
          ( collec722012601292870188list_o
          @ ( produc6238658148480694866st_o_o
            @ ( listrelp_a_o
              @ ^ [X: a,Y4: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X @ Y4 ) @ R3 ) ) ) ) ) ) ).

% listrel_def
thf(fact_703_case__prod__Pair__iden,axiom,
    ! [P: produc1714117091004681831on_nat] :
      ( ( produc6274880072277425531on_nat @ produc7152764266822961111on_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_704_case__prod__Pair__iden,axiom,
    ! [P: produc118098540238352959on_nat] :
      ( ( produc7619365177700836829on_nat @ produc4176415733022414961on_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_705_case__prod__Pair__iden,axiom,
    ! [P: produc8325617224762090749on_nat] :
      ( ( produc4035448539481142195on_nat @ produc1545487600272464053on_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_706_case__prod__Pair__iden,axiom,
    ! [P: produc27318731269267614tation] :
      ( ( produc1859402794938042595tation @ produc7247797765490398872tation @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_707_case__prod__Pair__iden,axiom,
    ! [P: product_prod_a_o] :
      ( ( produc453674083619213819od_a_o @ product_Pair_a_o @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_708_pred__equals__eq2,axiom,
    ! [R2: set_Pr8586234676504894535on_nat,S4: set_Pr8586234676504894535on_nat] :
      ( ( ( ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ R2 ) )
        = ( ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_709_pred__equals__eq2,axiom,
    ! [R2: set_Pr737724919871395061on_nat,S4: set_Pr737724919871395061on_nat] :
      ( ( ( ^ [X: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y4 ) @ R2 ) )
        = ( ^ [X: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y4 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_710_pred__equals__eq2,axiom,
    ! [R2: set_Pr6244131468446592861on_nat,S4: set_Pr6244131468446592861on_nat] :
      ( ( ( ^ [X: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y4 ) @ R2 ) )
        = ( ^ [X: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y4 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_711_pred__equals__eq2,axiom,
    ! [R2: set_Pr3525538344199662036tation,S4: set_Pr3525538344199662036tation] :
      ( ( ( ^ [X: a,Y4: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X @ Y4 ) @ R2 ) )
        = ( ^ [X: a,Y4: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X @ Y4 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_712_pred__equals__eq2,axiom,
    ! [R2: set_Product_prod_a_o,S4: set_Product_prod_a_o] :
      ( ( ( ^ [X: a,Y4: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X @ Y4 ) @ R2 ) )
        = ( ^ [X: a,Y4: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X @ Y4 ) @ S4 ) ) )
      = ( R2 = S4 ) ) ).

% pred_equals_eq2
thf(fact_713_listrel1p__def,axiom,
    ( listre5892843376277566186on_nat
    = ( ^ [R3: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,Xs3: list_P9127870862658512897on_nat,Ys3: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs3 @ Ys3 ) @ ( listre7129750038259667930on_nat @ ( collec8365270904964013138on_nat @ ( produc3964550740795057708_nat_o @ R3 ) ) ) ) ) ) ).

% listrel1p_def
thf(fact_714_all__subset__image,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o,P2: set_Pr2535564044409765851on_nat > $o] :
      ( ( ! [B5: set_Pr2535564044409765851on_nat] :
            ( ( ord_le6961793561435735931on_nat @ B5 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
           => ( P2 @ B5 ) ) )
      = ( ! [B5: set_Product_prod_a_o] :
            ( ( ord_le8613718869425792193od_a_o @ B5 @ A4 )
           => ( P2 @ ( image_4414724447807847679on_nat @ F @ B5 ) ) ) ) ) ).

% all_subset_image
thf(fact_715_complete__interval,axiom,
    ! [A: nat,B2: nat,P2: nat > $o] :
      ( ( ord_less_nat @ A @ B2 )
     => ( ( P2 @ A )
       => ( ~ ( P2 @ B2 )
         => ? [C: nat] :
              ( ( ord_less_eq_nat @ A @ C )
              & ( ord_less_eq_nat @ C @ B2 )
              & ! [X4: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X4 )
                    & ( ord_less_nat @ X4 @ C ) )
                 => ( P2 @ X4 ) )
              & ! [D3: nat] :
                  ( ! [X2: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X2 )
                        & ( ord_less_nat @ X2 @ D3 ) )
                     => ( P2 @ X2 ) )
                 => ( ord_less_eq_nat @ D3 @ C ) ) ) ) ) ) ).

% complete_interval
thf(fact_716_verit__comp__simplify1_I3_J,axiom,
    ! [B3: nat,A2: nat] :
      ( ( ~ ( ord_less_eq_nat @ B3 @ A2 ) )
      = ( ord_less_nat @ A2 @ B3 ) ) ).

% verit_comp_simplify1(3)
thf(fact_717_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ~ ( ord_less_eq_nat @ X4 @ T ) ) ).

% pinf(6)
thf(fact_718_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( ord_less_eq_nat @ T @ X4 ) ) ).

% pinf(8)
thf(fact_719_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( ord_less_eq_nat @ X4 @ T ) ) ).

% minf(6)
thf(fact_720_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ~ ( ord_less_eq_nat @ T @ X4 ) ) ).

% minf(8)
thf(fact_721_list__update__beyond,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I: nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ord_less_eq_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ I )
     => ( ( list_u8577817230714425439on_nat @ Xs @ I @ X3 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_722_list__update__beyond,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I: nat,X3: list_P3196869577824546119od_a_o] :
      ( ( ord_less_eq_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ I )
     => ( ( list_u329291161401623989od_a_o @ Xs @ I @ X3 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_723_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X3 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_724_psubsetD,axiom,
    ! [A4: set_Product_prod_a_o,B4: set_Product_prod_a_o,C2: product_prod_a_o] :
      ( ( ord_le1356998911005946317od_a_o @ A4 @ B4 )
     => ( ( member2997933407038931562od_a_o @ C2 @ A4 )
       => ( member2997933407038931562od_a_o @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_725_psubsetD,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat,C2: sum_su4907927656612468301on_nat] :
      ( ( ord_le5614770093425365463on_nat @ A4 @ B4 )
     => ( ( member8476201367316484836on_nat @ C2 @ A4 )
       => ( member8476201367316484836on_nat @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_726_psubsetD,axiom,
    ! [A4: set_li5011860890973139239od_a_o,B4: set_li5011860890973139239od_a_o,C2: list_P3196869577824546119od_a_o] :
      ( ( ord_le6036158148124213203od_a_o @ A4 @ B4 )
     => ( ( member5261019209154752368od_a_o @ C2 @ A4 )
       => ( member5261019209154752368od_a_o @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_727_psubsetD,axiom,
    ! [A4: set_nat,B4: set_nat,C2: nat] :
      ( ( ord_less_set_nat @ A4 @ B4 )
     => ( ( member_nat2 @ C2 @ A4 )
       => ( member_nat2 @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_728_psubsetD,axiom,
    ! [A4: set_a,B4: set_a,C2: a] :
      ( ( ord_less_set_a @ A4 @ B4 )
     => ( ( member_a2 @ C2 @ A4 )
       => ( member_a2 @ C2 @ B4 ) ) ) ).

% psubsetD
thf(fact_729_less__set__def,axiom,
    ( ord_le1356998911005946317od_a_o
    = ( ^ [A5: set_Product_prod_a_o,B5: set_Product_prod_a_o] :
          ( ord_le5737457564498941776_a_o_o
          @ ^ [X: product_prod_a_o] : ( member2997933407038931562od_a_o @ X @ A5 )
          @ ^ [X: product_prod_a_o] : ( member2997933407038931562od_a_o @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_730_less__set__def,axiom,
    ( ord_le5614770093425365463on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] :
          ( ord_le4231893053037816686_nat_o
          @ ^ [X: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X @ A5 )
          @ ^ [X: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_731_less__set__def,axiom,
    ( ord_le6036158148124213203od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o] :
          ( ord_le4638316749336627914_a_o_o
          @ ^ [X: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X @ A5 )
          @ ^ [X: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_732_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat2 @ X @ A5 )
          @ ^ [X: nat] : ( member_nat2 @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_733_less__set__def,axiom,
    ( ord_less_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( ord_less_a_o
          @ ^ [X: a] : ( member_a2 @ X @ A5 )
          @ ^ [X: a] : ( member_a2 @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_734_nat__descend__induct,axiom,
    ! [N: nat,P2: nat > $o,M3: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N @ K2 )
         => ( P2 @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P2 @ I4 ) )
             => ( P2 @ K2 ) ) )
       => ( P2 @ M3 ) ) ) ).

% nat_descend_induct
thf(fact_735_pred__subset__eq2,axiom,
    ! [R2: set_Pr8586234676504894535on_nat,S4: set_Pr8586234676504894535on_nat] :
      ( ( ord_le2713980917179944782_nat_o
        @ ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ R2 )
        @ ^ [X: list_P9127870862658512897on_nat,Y4: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ Y4 ) @ S4 ) )
      = ( ord_le2174946625535087591on_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_736_pred__subset__eq2,axiom,
    ! [R2: set_Pr737724919871395061on_nat,S4: set_Pr737724919871395061on_nat] :
      ( ( ord_le4954966455738030386_nat_o
        @ ^ [X: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y4 ) @ R2 )
        @ ^ [X: list_S5253404628753938141on_nat,Y4: produc1714117091004681831on_nat] : ( member4585374397974051670on_nat @ ( produc4176415733022414961on_nat @ X @ Y4 ) @ S4 ) )
      = ( ord_le248320493815342421on_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_737_pred__subset__eq2,axiom,
    ! [R2: set_Pr6244131468446592861on_nat,S4: set_Pr6244131468446592861on_nat] :
      ( ( ord_le291120284566496650_nat_o
        @ ^ [X: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y4 ) @ R2 )
        @ ^ [X: list_S5253404628753938141on_nat,Y4: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X @ Y4 ) @ S4 ) )
      = ( ord_le3399424890108663549on_nat @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_738_pred__subset__eq2,axiom,
    ! [R2: set_Pr3525538344199662036tation,S4: set_Pr3525538344199662036tation] :
      ( ( ord_le3255747248521444581tion_o
        @ ^ [X: a,Y4: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X @ Y4 ) @ R2 )
        @ ^ [X: a,Y4: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X @ Y4 ) @ S4 ) )
      = ( ord_le1700925397483502132tation @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_739_pred__subset__eq2,axiom,
    ! [R2: set_Product_prod_a_o,S4: set_Product_prod_a_o] :
      ( ( ord_less_eq_a_o_o
        @ ^ [X: a,Y4: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X @ Y4 ) @ R2 )
        @ ^ [X: a,Y4: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X @ Y4 ) @ S4 ) )
      = ( ord_le8613718869425792193od_a_o @ R2 @ S4 ) ) ).

% pred_subset_eq2
thf(fact_740_less__eq__set__def,axiom,
    ( ord_le8613718869425792193od_a_o
    = ( ^ [A5: set_Product_prod_a_o,B5: set_Product_prod_a_o] :
          ( ord_le5118620054153227612_a_o_o
          @ ^ [X: product_prod_a_o] : ( member2997933407038931562od_a_o @ X @ A5 )
          @ ^ [X: product_prod_a_o] : ( member2997933407038931562od_a_o @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_741_less__eq__set__def,axiom,
    ( ord_le6595998496543023075on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] :
          ( ord_le5750405997874517090_nat_o
          @ ^ [X: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X @ A5 )
          @ ^ [X: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_742_less__eq__set__def,axiom,
    ( ord_le1271121679792220359od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o] :
          ( ord_le6360763901695748822_a_o_o
          @ ^ [X: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X @ A5 )
          @ ^ [X: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_743_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat2 @ X @ A5 )
          @ ^ [X: nat] : ( member_nat2 @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_744_less__eq__set__def,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( ord_less_eq_a_o
          @ ^ [X: a] : ( member_a2 @ X @ A5 )
          @ ^ [X: a] : ( member_a2 @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_745_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_746_pinf_I1_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z6: nat] :
        ! [X2: nat] :
          ( ( ord_less_nat @ Z6 @ X2 )
         => ( ( P2 @ X2 )
            = ( P5 @ X2 ) ) )
     => ( ? [Z6: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z6 @ X2 )
           => ( ( Q @ X2 )
              = ( Q2 @ X2 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( ( P2 @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_747_pinf_I2_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z6: nat] :
        ! [X2: nat] :
          ( ( ord_less_nat @ Z6 @ X2 )
         => ( ( P2 @ X2 )
            = ( P5 @ X2 ) ) )
     => ( ? [Z6: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z6 @ X2 )
           => ( ( Q @ X2 )
              = ( Q2 @ X2 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z5 @ X4 )
           => ( ( ( P2 @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_748_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(3)
thf(fact_749_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( X4 != T ) ) ).

% pinf(4)
thf(fact_750_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ~ ( ord_less_nat @ X4 @ T ) ) ).

% pinf(5)
thf(fact_751_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ Z5 @ X4 )
     => ( ord_less_nat @ T @ X4 ) ) ).

% pinf(7)
thf(fact_752_minf_I1_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z6: nat] :
        ! [X2: nat] :
          ( ( ord_less_nat @ X2 @ Z6 )
         => ( ( P2 @ X2 )
            = ( P5 @ X2 ) ) )
     => ( ? [Z6: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z6 )
           => ( ( Q @ X2 )
              = ( Q2 @ X2 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( ( P2 @ X4 )
                & ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                & ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(1)
thf(fact_753_minf_I2_J,axiom,
    ! [P2: nat > $o,P5: nat > $o,Q: nat > $o,Q2: nat > $o] :
      ( ? [Z6: nat] :
        ! [X2: nat] :
          ( ( ord_less_nat @ X2 @ Z6 )
         => ( ( P2 @ X2 )
            = ( P5 @ X2 ) ) )
     => ( ? [Z6: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z6 )
           => ( ( Q @ X2 )
              = ( Q2 @ X2 ) ) )
       => ? [Z5: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z5 )
           => ( ( ( P2 @ X4 )
                | ( Q @ X4 ) )
              = ( ( P5 @ X4 )
                | ( Q2 @ X4 ) ) ) ) ) ) ).

% minf(2)
thf(fact_754_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(3)
thf(fact_755_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( X4 != T ) ) ).

% minf(4)
thf(fact_756_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ( ord_less_nat @ X4 @ T ) ) ).

% minf(5)
thf(fact_757_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z5: nat] :
    ! [X4: nat] :
      ( ( ord_less_nat @ X4 @ Z5 )
     => ~ ( ord_less_nat @ T @ X4 ) ) ).

% minf(7)
thf(fact_758_subset__subseqs,axiom,
    ! [X7: set_Product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( ord_le8613718869425792193od_a_o @ X7 @ ( set_Product_prod_a_o2 @ Xs ) )
     => ( member8594096980020929482od_a_o @ X7 @ ( image_8445419443616214687od_a_o @ set_Product_prod_a_o2 @ ( set_li2629784761229999580od_a_o @ ( subseq3821054824087877782od_a_o @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_759_subset__subseqs,axiom,
    ! [X7: set_Su266273576555847555on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ord_le6595998496543023075on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( member7157586822906380314on_nat @ X7 @ ( image_4802297152779768167on_nat @ set_Su4848624856859743608on_nat @ ( set_li4371160873219908744on_nat @ ( subseq3316081458638881720on_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_760_subset__subseqs,axiom,
    ! [X7: set_li5011860890973139239od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_le1271121679792220359od_a_o @ X7 @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( member2068264008706428752od_a_o @ X7 @ ( image_866151878574500639od_a_o @ set_li2629784761229999580od_a_o @ ( set_li3106133282984605794od_a_o @ ( subseq4901587760977928092od_a_o @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_761_subset__subseqs,axiom,
    ! [X7: set_nat,Xs: list_nat] :
      ( ( ord_less_eq_set_nat @ X7 @ ( set_nat2 @ Xs ) )
     => ( member_set_nat @ X7 @ ( image_1775855109352712557et_nat @ set_nat2 @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_762_subset__subseqs,axiom,
    ! [X7: set_a,Xs: list_a] :
      ( ( ord_less_eq_set_a @ X7 @ ( set_a2 @ Xs ) )
     => ( member_set_a @ X7 @ ( image_list_a_set_a @ set_a2 @ ( set_list_a2 @ ( subseqs_a @ Xs ) ) ) ) ) ).

% subset_subseqs
thf(fact_763_list__all2__iff,axiom,
    ( list_all2_a_o
    = ( ^ [P4: a > $o > $o,Xs3: list_a,Ys3: list_o] :
          ( ( ( size_size_list_a @ Xs3 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [X: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ ( zip_a_o @ Xs3 @ Ys3 ) ) )
             => ( produc8428829086471790546_a_o_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_764_list__all2__iff,axiom,
    ( list_a1658445451649724195on_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs3: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [X: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ X @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ Xs3 @ Ys3 ) ) )
             => ( produc208325271403635792_nat_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_765_list__all2__iff,axiom,
    ( list_a1062525910057997297od_a_o
    = ( ^ [P4: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs3: list_S5253404628753938141on_nat,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [X: produc6493764124173775663od_a_o] :
              ( ( member7595723914438784454od_a_o @ X @ ( set_Pr6587213113994854490od_a_o @ ( zip_Su2363964881504851137od_a_o @ Xs3 @ Ys3 ) ) )
             => ( produc7187800550436554762_a_o_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_766_list__all2__iff,axiom,
    ( list_a5295772765654029478at_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > nat > $o,Xs3: list_S5253404628753938141on_nat,Ys3: list_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [X: produc8556620195627115390at_nat] :
              ( ( member8820252441731870247at_nat @ X @ ( set_Pr1192295990000706195at_nat @ ( zip_Su8633780382783139798at_nat @ Xs3 @ Ys3 ) ) )
             => ( produc2579979265757742093_nat_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_767_list__all2__iff,axiom,
    ( list_a7531049758917964877on_nat
    = ( ^ [P4: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs3: list_l8336264072933094989od_a_o,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [X: produc731851057114331587on_nat] :
              ( ( member1833810847379340378on_nat @ X @ ( set_Pr825300046935410414on_nat @ ( zip_li8832488730364818717on_nat @ Xs3 @ Ys3 ) ) )
             => ( produc963041392342583470_nat_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_768_list__all2__iff,axiom,
    ( list_a6864181014682439495od_a_o
    = ( ^ [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs3: list_l8336264072933094989od_a_o,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [X: produc8520174026553075431od_a_o] :
              ( ( member5440785652130980624od_a_o @ X @ ( set_Pr7142968438029117820od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs3 @ Ys3 ) ) )
             => ( produc731694360491113772_a_o_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_769_list__all2__iff,axiom,
    ( list_a3027799476954313168_o_nat
    = ( ^ [P4: list_P3196869577824546119od_a_o > nat > $o,Xs3: list_l8336264072933094989od_a_o,Ys3: list_nat] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [X: produc408724640507517382_o_nat] :
              ( ( member7885718254550376541_o_nat @ X @ ( set_Pr7830758801870232305_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs3 @ Ys3 ) ) )
             => ( produc4957984460634321003_nat_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_770_list__all2__iff,axiom,
    ( list_a6193039792891997350on_nat
    = ( ^ [P4: nat > sum_su4907927656612468301on_nat > $o,Xs3: list_nat,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [X: produc7088870822282640638on_nat] :
              ( ( member7352503068387395495on_nat @ X @ ( set_Pr8947918653511007251on_nat @ ( zip_na307675373166331862on_nat @ Xs3 @ Ys3 ) ) )
             => ( produc2700354003219672077_nat_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_771_list__all2__iff,axiom,
    ( list_a3410858264120222766od_a_o
    = ( ^ [P4: nat > list_P3196869577824546119od_a_o > $o,Xs3: list_nat,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [X: produc1433831906155620204od_a_o] :
              ( ( member8910825520198479363od_a_o @ X @ ( set_Pr8855866067518335127od_a_o @ ( zip_na1422845649429636350od_a_o @ Xs3 @ Ys3 ) ) )
             => ( produc8470836043472241293_a_o_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_772_list__all2__iff,axiom,
    ( list_all2_nat_nat
    = ( ^ [P4: nat > nat > $o,Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs3 @ Ys3 ) ) )
             => ( produc6081775807080527818_nat_o @ P4 @ X ) ) ) ) ) ).

% list_all2_iff
thf(fact_773_distinct__swap,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( distin2773651816175405460on_nat @ ( list_u8577817230714425439on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) @ J @ ( nth_Su8471564416988646406on_nat @ Xs @ I ) ) )
          = ( distin2773651816175405460on_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_774_distinct__swap,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( distin690952206374581440od_a_o @ ( list_u329291161401623989od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) @ J @ ( nth_li3248255784598043854od_a_o @ Xs @ I ) ) )
          = ( distin690952206374581440od_a_o @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_775_distinct__swap,axiom,
    ! [I: nat,Xs: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
       => ( ( distinct_nat @ ( list_update_nat @ ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ J ) ) @ J @ ( nth_nat @ Xs @ I ) ) )
          = ( distinct_nat @ Xs ) ) ) ) ).

% distinct_swap
thf(fact_776__092_060open_062ns__mul__ext_A_Iset_ANSS_J_A_Iset_AS_J_A_061_Ans__mul__ext_A_Iset_ANS_J_A_Iset_AS_J_092_060close_062,axiom,
    ( ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ nss ) @ ( set_Pr937530902923597456on_nat @ s ) )
    = ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ ns ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ).

% \<open>ns_mul_ext (set NSS) (set S) = ns_mul_ext (set NS) (set S)\<close>
thf(fact_777_subseqs__distinctD,axiom,
    ! [Ys: list_nat,Xs: list_nat] :
      ( ( member_list_nat @ Ys @ ( set_list_nat2 @ ( subseqs_nat @ Xs ) ) )
     => ( ( distinct_nat @ Xs )
       => ( distinct_nat @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_778_subseqs__distinctD,axiom,
    ! [Ys: list_P3196869577824546119od_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member5261019209154752368od_a_o @ Ys @ ( set_li2629784761229999580od_a_o @ ( subseq3821054824087877782od_a_o @ Xs ) ) )
     => ( ( distin2380507709364613050od_a_o @ Xs )
       => ( distin2380507709364613050od_a_o @ Ys ) ) ) ).

% subseqs_distinctD
thf(fact_779_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
    ! [Xs: list_nat] :
      ( ( distinct_nat @ Xs )
     => ( distinct_nat @ Xs ) ) ).

% sorted_list_of_set.distinct_if_distinct_map
thf(fact_780_ordering__top_Oextremum__uniqueI,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
       => ( A = Top ) ) ) ).

% ordering_top.extremum_uniqueI
thf(fact_781_ordering__top_Onot__eq__extremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( A != Top )
        = ( Less @ A @ Top ) ) ) ).

% ordering_top.not_eq_extremum
thf(fact_782_ordering__top_Oextremum__unique,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( ( Less_eq @ Top @ A )
        = ( A = Top ) ) ) ).

% ordering_top.extremum_unique
thf(fact_783_ordering__top_Oextremum__strict,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ~ ( Less @ Top @ A ) ) ).

% ordering_top.extremum_strict
thf(fact_784_ordering__top_Oextremum,axiom,
    ! [Less_eq: nat > nat > $o,Less: nat > nat > $o,Top: nat,A: nat] :
      ( ( ordering_top_nat @ Less_eq @ Less @ Top )
     => ( Less_eq @ A @ Top ) ) ).

% ordering_top.extremum
thf(fact_785_refl__onD2,axiom,
    ! [A4: set_Product_prod_a_o,R: set_Pr1246879249572150087od_a_o,X3: product_prod_a_o,Y: product_prod_a_o] :
      ( ( refl_o3745565406092796480od_a_o @ A4 @ R )
     => ( ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X3 @ Y ) @ R )
       => ( member2997933407038931562od_a_o @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_786_refl__onD2,axiom,
    ! [A4: set_Su266273576555847555on_nat,R: set_Pr2535564044409765851on_nat,X3: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ( refl_o6374727810006644366on_nat @ A4 @ R )
     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y ) @ R )
       => ( member8476201367316484836on_nat @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_787_refl__onD2,axiom,
    ! [A4: set_li5011860890973139239od_a_o,R: set_Pr7420778282784443591od_a_o,X3: list_P3196869577824546119od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( refl_o1202476486973598790od_a_o @ A4 @ R )
     => ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ Y ) @ R )
       => ( member5261019209154752368od_a_o @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_788_refl__onD2,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
       => ( member_nat2 @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_789_refl__onD2,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
       => ( member_a2 @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_790_refl__onD2,axiom,
    ! [A4: set_li1748721617258726113on_nat,R: set_Pr8586234676504894535on_nat,X3: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( refl_o794705341054444288on_nat @ A4 @ R )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ R )
       => ( member783415501022416426on_nat @ Y @ A4 ) ) ) ).

% refl_onD2
thf(fact_791_refl__onD1,axiom,
    ! [A4: set_Product_prod_a_o,R: set_Pr1246879249572150087od_a_o,X3: product_prod_a_o,Y: product_prod_a_o] :
      ( ( refl_o3745565406092796480od_a_o @ A4 @ R )
     => ( ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X3 @ Y ) @ R )
       => ( member2997933407038931562od_a_o @ X3 @ A4 ) ) ) ).

% refl_onD1
thf(fact_792_refl__onD1,axiom,
    ! [A4: set_Su266273576555847555on_nat,R: set_Pr2535564044409765851on_nat,X3: sum_su4907927656612468301on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ( refl_o6374727810006644366on_nat @ A4 @ R )
     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ Y ) @ R )
       => ( member8476201367316484836on_nat @ X3 @ A4 ) ) ) ).

% refl_onD1
thf(fact_793_refl__onD1,axiom,
    ! [A4: set_li5011860890973139239od_a_o,R: set_Pr7420778282784443591od_a_o,X3: list_P3196869577824546119od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( refl_o1202476486973598790od_a_o @ A4 @ R )
     => ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ Y ) @ R )
       => ( member5261019209154752368od_a_o @ X3 @ A4 ) ) ) ).

% refl_onD1
thf(fact_794_refl__onD1,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,X3: nat,Y: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y ) @ R )
       => ( member_nat2 @ X3 @ A4 ) ) ) ).

% refl_onD1
thf(fact_795_refl__onD1,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,X3: a,Y: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y ) @ R )
       => ( member_a2 @ X3 @ A4 ) ) ) ).

% refl_onD1
thf(fact_796_refl__onD1,axiom,
    ! [A4: set_li1748721617258726113on_nat,R: set_Pr8586234676504894535on_nat,X3: list_P9127870862658512897on_nat,Y: list_P9127870862658512897on_nat] :
      ( ( refl_o794705341054444288on_nat @ A4 @ R )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X3 @ Y ) @ R )
       => ( member783415501022416426on_nat @ X3 @ A4 ) ) ) ).

% refl_onD1
thf(fact_797_refl__onD,axiom,
    ! [A4: set_Product_prod_a_o,R: set_Pr1246879249572150087od_a_o,A: product_prod_a_o] :
      ( ( refl_o3745565406092796480od_a_o @ A4 @ R )
     => ( ( member2997933407038931562od_a_o @ A @ A4 )
       => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_798_refl__onD,axiom,
    ! [A4: set_Su266273576555847555on_nat,R: set_Pr2535564044409765851on_nat,A: sum_su4907927656612468301on_nat] :
      ( ( refl_o6374727810006644366on_nat @ A4 @ R )
     => ( ( member8476201367316484836on_nat @ A @ A4 )
       => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_799_refl__onD,axiom,
    ! [A4: set_li5011860890973139239od_a_o,R: set_Pr7420778282784443591od_a_o,A: list_P3196869577824546119od_a_o] :
      ( ( refl_o1202476486973598790od_a_o @ A4 @ R )
     => ( ( member5261019209154752368od_a_o @ A @ A4 )
       => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_800_refl__onD,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member_nat2 @ A @ A4 )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_801_refl__onD,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,A: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member_a2 @ A @ A4 )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_802_refl__onD,axiom,
    ! [A4: set_li1748721617258726113on_nat,R: set_Pr8586234676504894535on_nat,A: list_P9127870862658512897on_nat] :
      ( ( refl_o794705341054444288on_nat @ A4 @ R )
     => ( ( member783415501022416426on_nat @ A @ A4 )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ A ) @ R ) ) ) ).

% refl_onD
thf(fact_803_list__all2__same,axiom,
    ! [P2: product_prod_a_o > product_prod_a_o > $o,Xs: list_P3196869577824546119od_a_o] :
      ( ( list_a2434814095049548487od_a_o @ P2 @ Xs @ Xs )
      = ( ! [X: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) )
           => ( P2 @ X @ X ) ) ) ) ).

% list_all2_same
thf(fact_804_list__all2__same,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ Xs @ Xs )
      = ( ! [X: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P2 @ X @ X ) ) ) ) ).

% list_all2_same
thf(fact_805_list__all2__same,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ Xs @ Xs )
      = ( ! [X: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( P2 @ X @ X ) ) ) ) ).

% list_all2_same
thf(fact_806_list__all2__same,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat] :
      ( ( list_all2_nat_nat @ P2 @ Xs @ Xs )
      = ( ! [X: nat] :
            ( ( member_nat2 @ X @ ( set_nat2 @ Xs ) )
           => ( P2 @ X @ X ) ) ) ) ).

% list_all2_same
thf(fact_807_list__all2__same,axiom,
    ! [P2: a > a > $o,Xs: list_a] :
      ( ( list_all2_a_a @ P2 @ Xs @ Xs )
      = ( ! [X: a] :
            ( ( member_a2 @ X @ ( set_a2 @ Xs ) )
           => ( P2 @ X @ X ) ) ) ) ).

% list_all2_same
thf(fact_808_list_Orel__refl__strong,axiom,
    ! [X3: list_P3196869577824546119od_a_o,Ra: product_prod_a_o > product_prod_a_o > $o] :
      ( ! [Z5: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ Z5 @ ( set_Product_prod_a_o2 @ X3 ) )
         => ( Ra @ Z5 @ Z5 ) )
     => ( list_a2434814095049548487od_a_o @ Ra @ X3 @ X3 ) ) ).

% list.rel_refl_strong
thf(fact_809_list_Orel__refl__strong,axiom,
    ! [X3: list_S5253404628753938141on_nat,Ra: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
      ( ! [Z5: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ Z5 @ ( set_Su4848624856859743608on_nat @ X3 ) )
         => ( Ra @ Z5 @ Z5 ) )
     => ( list_a1658445451649724195on_nat @ Ra @ X3 @ X3 ) ) ).

% list.rel_refl_strong
thf(fact_810_list_Orel__refl__strong,axiom,
    ! [X3: list_l8336264072933094989od_a_o,Ra: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o] :
      ( ! [Z5: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ Z5 @ ( set_li2629784761229999580od_a_o @ X3 ) )
         => ( Ra @ Z5 @ Z5 ) )
     => ( list_a6864181014682439495od_a_o @ Ra @ X3 @ X3 ) ) ).

% list.rel_refl_strong
thf(fact_811_list_Orel__refl__strong,axiom,
    ! [X3: list_nat,Ra: nat > nat > $o] :
      ( ! [Z5: nat] :
          ( ( member_nat2 @ Z5 @ ( set_nat2 @ X3 ) )
         => ( Ra @ Z5 @ Z5 ) )
     => ( list_all2_nat_nat @ Ra @ X3 @ X3 ) ) ).

% list.rel_refl_strong
thf(fact_812_list_Orel__refl__strong,axiom,
    ! [X3: list_a,Ra: a > a > $o] :
      ( ! [Z5: a] :
          ( ( member_a2 @ Z5 @ ( set_a2 @ X3 ) )
         => ( Ra @ Z5 @ Z5 ) )
     => ( list_all2_a_a @ Ra @ X3 @ X3 ) ) ).

% list.rel_refl_strong
thf(fact_813_list_Orel__mono__strong,axiom,
    ! [R2: nat > nat > $o,X3: list_nat,Y: list_nat,Ra: nat > nat > $o] :
      ( ( list_all2_nat_nat @ R2 @ X3 @ Y )
     => ( ! [Z5: nat,Yb: nat] :
            ( ( member_nat2 @ Z5 @ ( set_nat2 @ X3 ) )
           => ( ( member_nat2 @ Yb @ ( set_nat2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_all2_nat_nat @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_814_list_Orel__mono__strong,axiom,
    ! [R2: nat > a > $o,X3: list_nat,Y: list_a,Ra: nat > a > $o] :
      ( ( list_all2_nat_a @ R2 @ X3 @ Y )
     => ( ! [Z5: nat,Yb: a] :
            ( ( member_nat2 @ Z5 @ ( set_nat2 @ X3 ) )
           => ( ( member_a2 @ Yb @ ( set_a2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_all2_nat_a @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_815_list_Orel__mono__strong,axiom,
    ! [R2: a > nat > $o,X3: list_a,Y: list_nat,Ra: a > nat > $o] :
      ( ( list_all2_a_nat @ R2 @ X3 @ Y )
     => ( ! [Z5: a,Yb: nat] :
            ( ( member_a2 @ Z5 @ ( set_a2 @ X3 ) )
           => ( ( member_nat2 @ Yb @ ( set_nat2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_all2_a_nat @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_816_list_Orel__mono__strong,axiom,
    ! [R2: a > a > $o,X3: list_a,Y: list_a,Ra: a > a > $o] :
      ( ( list_all2_a_a @ R2 @ X3 @ Y )
     => ( ! [Z5: a,Yb: a] :
            ( ( member_a2 @ Z5 @ ( set_a2 @ X3 ) )
           => ( ( member_a2 @ Yb @ ( set_a2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_all2_a_a @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_817_list_Orel__mono__strong,axiom,
    ! [R2: product_prod_a_o > nat > $o,X3: list_P3196869577824546119od_a_o,Y: list_nat,Ra: product_prod_a_o > nat > $o] :
      ( ( list_a6036629288510939210_o_nat @ R2 @ X3 @ Y )
     => ( ! [Z5: product_prod_a_o,Yb: nat] :
            ( ( member2997933407038931562od_a_o @ Z5 @ ( set_Product_prod_a_o2 @ X3 ) )
           => ( ( member_nat2 @ Yb @ ( set_nat2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_a6036629288510939210_o_nat @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_818_list_Orel__mono__strong,axiom,
    ! [R2: product_prod_a_o > a > $o,X3: list_P3196869577824546119od_a_o,Y: list_a,Ra: product_prod_a_o > a > $o] :
      ( ( list_a8388632826625720580_a_o_a @ R2 @ X3 @ Y )
     => ( ! [Z5: product_prod_a_o,Yb: a] :
            ( ( member2997933407038931562od_a_o @ Z5 @ ( set_Product_prod_a_o2 @ X3 ) )
           => ( ( member_a2 @ Yb @ ( set_a2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_a8388632826625720580_a_o_a @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_819_list_Orel__mono__strong,axiom,
    ! [R2: nat > product_prod_a_o > $o,X3: list_nat,Y: list_P3196869577824546119od_a_o,Ra: nat > product_prod_a_o > $o] :
      ( ( list_a3336860938500795944od_a_o @ R2 @ X3 @ Y )
     => ( ! [Z5: nat,Yb: product_prod_a_o] :
            ( ( member_nat2 @ Z5 @ ( set_nat2 @ X3 ) )
           => ( ( member2997933407038931562od_a_o @ Yb @ ( set_Product_prod_a_o2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_a3336860938500795944od_a_o @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_820_list_Orel__mono__strong,axiom,
    ! [R2: a > product_prod_a_o > $o,X3: list_a,Y: list_P3196869577824546119od_a_o,Ra: a > product_prod_a_o > $o] :
      ( ( list_a1273085238662163274od_a_o @ R2 @ X3 @ Y )
     => ( ! [Z5: a,Yb: product_prod_a_o] :
            ( ( member_a2 @ Z5 @ ( set_a2 @ X3 ) )
           => ( ( member2997933407038931562od_a_o @ Yb @ ( set_Product_prod_a_o2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_a1273085238662163274od_a_o @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_821_list_Orel__mono__strong,axiom,
    ! [R2: list_P3196869577824546119od_a_o > nat > $o,X3: list_l8336264072933094989od_a_o,Y: list_nat,Ra: list_P3196869577824546119od_a_o > nat > $o] :
      ( ( list_a3027799476954313168_o_nat @ R2 @ X3 @ Y )
     => ( ! [Z5: list_P3196869577824546119od_a_o,Yb: nat] :
            ( ( member5261019209154752368od_a_o @ Z5 @ ( set_li2629784761229999580od_a_o @ X3 ) )
           => ( ( member_nat2 @ Yb @ ( set_nat2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_a3027799476954313168_o_nat @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_822_list_Orel__mono__strong,axiom,
    ! [R2: list_P3196869577824546119od_a_o > a > $o,X3: list_l8336264072933094989od_a_o,Y: list_a,Ra: list_P3196869577824546119od_a_o > a > $o] :
      ( ( list_a5221360748940363006_a_o_a @ R2 @ X3 @ Y )
     => ( ! [Z5: list_P3196869577824546119od_a_o,Yb: a] :
            ( ( member5261019209154752368od_a_o @ Z5 @ ( set_li2629784761229999580od_a_o @ X3 ) )
           => ( ( member_a2 @ Yb @ ( set_a2 @ Y ) )
             => ( ( R2 @ Z5 @ Yb )
               => ( Ra @ Z5 @ Yb ) ) ) )
       => ( list_a5221360748940363006_a_o_a @ Ra @ X3 @ Y ) ) ) ).

% list.rel_mono_strong
thf(fact_823_list_Orel__cong,axiom,
    ! [X3: list_nat,Ya: list_nat,Y: list_nat,Xa: list_nat,R2: nat > nat > $o,Ra: nat > nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: nat,Yb: nat] :
              ( ( member_nat2 @ Z5 @ ( set_nat2 @ Ya ) )
             => ( ( member_nat2 @ Yb @ ( set_nat2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_all2_nat_nat @ R2 @ X3 @ Y )
            = ( list_all2_nat_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_824_list_Orel__cong,axiom,
    ! [X3: list_nat,Ya: list_nat,Y: list_a,Xa: list_a,R2: nat > a > $o,Ra: nat > a > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: nat,Yb: a] :
              ( ( member_nat2 @ Z5 @ ( set_nat2 @ Ya ) )
             => ( ( member_a2 @ Yb @ ( set_a2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_all2_nat_a @ R2 @ X3 @ Y )
            = ( list_all2_nat_a @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_825_list_Orel__cong,axiom,
    ! [X3: list_a,Ya: list_a,Y: list_nat,Xa: list_nat,R2: a > nat > $o,Ra: a > nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: a,Yb: nat] :
              ( ( member_a2 @ Z5 @ ( set_a2 @ Ya ) )
             => ( ( member_nat2 @ Yb @ ( set_nat2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_all2_a_nat @ R2 @ X3 @ Y )
            = ( list_all2_a_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_826_list_Orel__cong,axiom,
    ! [X3: list_a,Ya: list_a,Y: list_a,Xa: list_a,R2: a > a > $o,Ra: a > a > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: a,Yb: a] :
              ( ( member_a2 @ Z5 @ ( set_a2 @ Ya ) )
             => ( ( member_a2 @ Yb @ ( set_a2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_all2_a_a @ R2 @ X3 @ Y )
            = ( list_all2_a_a @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_827_list_Orel__cong,axiom,
    ! [X3: list_P3196869577824546119od_a_o,Ya: list_P3196869577824546119od_a_o,Y: list_nat,Xa: list_nat,R2: product_prod_a_o > nat > $o,Ra: product_prod_a_o > nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: product_prod_a_o,Yb: nat] :
              ( ( member2997933407038931562od_a_o @ Z5 @ ( set_Product_prod_a_o2 @ Ya ) )
             => ( ( member_nat2 @ Yb @ ( set_nat2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_a6036629288510939210_o_nat @ R2 @ X3 @ Y )
            = ( list_a6036629288510939210_o_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_828_list_Orel__cong,axiom,
    ! [X3: list_P3196869577824546119od_a_o,Ya: list_P3196869577824546119od_a_o,Y: list_a,Xa: list_a,R2: product_prod_a_o > a > $o,Ra: product_prod_a_o > a > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: product_prod_a_o,Yb: a] :
              ( ( member2997933407038931562od_a_o @ Z5 @ ( set_Product_prod_a_o2 @ Ya ) )
             => ( ( member_a2 @ Yb @ ( set_a2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_a8388632826625720580_a_o_a @ R2 @ X3 @ Y )
            = ( list_a8388632826625720580_a_o_a @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_829_list_Orel__cong,axiom,
    ! [X3: list_nat,Ya: list_nat,Y: list_P3196869577824546119od_a_o,Xa: list_P3196869577824546119od_a_o,R2: nat > product_prod_a_o > $o,Ra: nat > product_prod_a_o > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: nat,Yb: product_prod_a_o] :
              ( ( member_nat2 @ Z5 @ ( set_nat2 @ Ya ) )
             => ( ( member2997933407038931562od_a_o @ Yb @ ( set_Product_prod_a_o2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_a3336860938500795944od_a_o @ R2 @ X3 @ Y )
            = ( list_a3336860938500795944od_a_o @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_830_list_Orel__cong,axiom,
    ! [X3: list_a,Ya: list_a,Y: list_P3196869577824546119od_a_o,Xa: list_P3196869577824546119od_a_o,R2: a > product_prod_a_o > $o,Ra: a > product_prod_a_o > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: a,Yb: product_prod_a_o] :
              ( ( member_a2 @ Z5 @ ( set_a2 @ Ya ) )
             => ( ( member2997933407038931562od_a_o @ Yb @ ( set_Product_prod_a_o2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_a1273085238662163274od_a_o @ R2 @ X3 @ Y )
            = ( list_a1273085238662163274od_a_o @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_831_list_Orel__cong,axiom,
    ! [X3: list_l8336264072933094989od_a_o,Ya: list_l8336264072933094989od_a_o,Y: list_nat,Xa: list_nat,R2: list_P3196869577824546119od_a_o > nat > $o,Ra: list_P3196869577824546119od_a_o > nat > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: list_P3196869577824546119od_a_o,Yb: nat] :
              ( ( member5261019209154752368od_a_o @ Z5 @ ( set_li2629784761229999580od_a_o @ Ya ) )
             => ( ( member_nat2 @ Yb @ ( set_nat2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_a3027799476954313168_o_nat @ R2 @ X3 @ Y )
            = ( list_a3027799476954313168_o_nat @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_832_list_Orel__cong,axiom,
    ! [X3: list_l8336264072933094989od_a_o,Ya: list_l8336264072933094989od_a_o,Y: list_a,Xa: list_a,R2: list_P3196869577824546119od_a_o > a > $o,Ra: list_P3196869577824546119od_a_o > a > $o] :
      ( ( X3 = Ya )
     => ( ( Y = Xa )
       => ( ! [Z5: list_P3196869577824546119od_a_o,Yb: a] :
              ( ( member5261019209154752368od_a_o @ Z5 @ ( set_li2629784761229999580od_a_o @ Ya ) )
             => ( ( member_a2 @ Yb @ ( set_a2 @ Xa ) )
               => ( ( R2 @ Z5 @ Yb )
                  = ( Ra @ Z5 @ Yb ) ) ) )
         => ( ( list_a5221360748940363006_a_o_a @ R2 @ X3 @ Y )
            = ( list_a5221360748940363006_a_o_a @ Ra @ Ya @ Xa ) ) ) ) ) ).

% list.rel_cong
thf(fact_833_list__all2__lengthD,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ Xs @ Ys )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_834_list__all2__lengthD,axiom,
    ! [P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( list_a1062525910057997297od_a_o @ P2 @ Xs @ Ys )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_835_list__all2__lengthD,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ Xs @ Ys )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_836_list__all2__lengthD,axiom,
    ! [P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat] :
      ( ( list_a7531049758917964877on_nat @ P2 @ Xs @ Ys )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_837_list__all2__lengthD,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ Xs @ Ys )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_838_list__all2__lengthD,axiom,
    ! [P2: list_P3196869577824546119od_a_o > nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_nat] :
      ( ( list_a3027799476954313168_o_nat @ P2 @ Xs @ Ys )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_839_list__all2__lengthD,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,Xs: list_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ Xs @ Ys )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_840_list__all2__lengthD,axiom,
    ! [P2: nat > list_P3196869577824546119od_a_o > $o,Xs: list_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( list_a3410858264120222766od_a_o @ P2 @ Xs @ Ys )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_841_list__all2__lengthD,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( list_all2_nat_nat @ P2 @ Xs @ Ys )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% list_all2_lengthD
thf(fact_842_distinct__product__lists,axiom,
    ! [Xss: list_list_nat] :
      ( ! [X2: list_nat] :
          ( ( member_list_nat @ X2 @ ( set_list_nat2 @ Xss ) )
         => ( distinct_nat @ X2 ) )
     => ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_843_distinct__product__lists,axiom,
    ! [Xss: list_l8336264072933094989od_a_o] :
      ( ! [X2: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X2 @ ( set_li2629784761229999580od_a_o @ Xss ) )
         => ( distin2380507709364613050od_a_o @ X2 ) )
     => ( distin690952206374581440od_a_o @ ( produc3814538932794474544od_a_o @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_844_reflD,axiom,
    ! [R: set_Pr8586234676504894535on_nat,A: list_P9127870862658512897on_nat] :
      ( ( refl_o794705341054444288on_nat @ top_to667412791749894193on_nat @ R )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ A ) @ R ) ) ).

% reflD
thf(fact_845_reflI,axiom,
    ! [R: set_Pr8586234676504894535on_nat] :
      ( ! [X2: list_P9127870862658512897on_nat] : ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X2 @ X2 ) @ R )
     => ( refl_o794705341054444288on_nat @ top_to667412791749894193on_nat @ R ) ) ).

% reflI
thf(fact_846_subseqs__refl,axiom,
    ! [Xs: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ Xs @ ( set_li2629784761229999580od_a_o @ ( subseq3821054824087877782od_a_o @ Xs ) ) ) ).

% subseqs_refl
thf(fact_847_list__all2__conv__all__nth,axiom,
    ( list_a1658445451649724195on_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs3: list_S5253404628753938141on_nat,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( P4 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_848_list__all2__conv__all__nth,axiom,
    ( list_a1062525910057997297od_a_o
    = ( ^ [P4: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs3: list_S5253404628753938141on_nat,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( P4 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_849_list__all2__conv__all__nth,axiom,
    ( list_a5295772765654029478at_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > nat > $o,Xs3: list_S5253404628753938141on_nat,Ys3: list_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( P4 @ ( nth_Su8471564416988646406on_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_850_list__all2__conv__all__nth,axiom,
    ( list_a7531049758917964877on_nat
    = ( ^ [P4: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs3: list_l8336264072933094989od_a_o,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( P4 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_851_list__all2__conv__all__nth,axiom,
    ( list_a6864181014682439495od_a_o
    = ( ^ [P4: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs3: list_l8336264072933094989od_a_o,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( P4 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_852_list__all2__conv__all__nth,axiom,
    ( list_a3027799476954313168_o_nat
    = ( ^ [P4: list_P3196869577824546119od_a_o > nat > $o,Xs3: list_l8336264072933094989od_a_o,Ys3: list_nat] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( P4 @ ( nth_li3248255784598043854od_a_o @ Xs3 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_853_list__all2__conv__all__nth,axiom,
    ( list_a6193039792891997350on_nat
    = ( ^ [P4: nat > sum_su4907927656612468301on_nat > $o,Xs3: list_nat,Ys3: list_S5253404628753938141on_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_s5686234912572693873on_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( P4 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_854_list__all2__conv__all__nth,axiom,
    ( list_a3410858264120222766od_a_o
    = ( ^ [P4: nat > list_P3196869577824546119od_a_o > $o,Xs3: list_nat,Ys3: list_l8336264072933094989od_a_o] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_s3379185846251495609od_a_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( P4 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_855_list__all2__conv__all__nth,axiom,
    ( list_all2_nat_nat
    = ( ^ [P4: nat > nat > $o,Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( P4 @ ( nth_nat @ Xs3 @ I3 ) @ ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_all2_conv_all_nth
thf(fact_856_list__all2__all__nthI,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
      ( ( ( size_s5686234912572693873on_nat @ A )
        = ( size_s5686234912572693873on_nat @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s5686234912572693873on_nat @ A ) )
           => ( P2 @ ( nth_Su8471564416988646406on_nat @ A @ N4 ) @ ( nth_Su8471564416988646406on_nat @ B2 @ N4 ) ) )
       => ( list_a1658445451649724195on_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_857_list__all2__all__nthI,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: list_l8336264072933094989od_a_o,P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o] :
      ( ( ( size_s5686234912572693873on_nat @ A )
        = ( size_s3379185846251495609od_a_o @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s5686234912572693873on_nat @ A ) )
           => ( P2 @ ( nth_Su8471564416988646406on_nat @ A @ N4 ) @ ( nth_li3248255784598043854od_a_o @ B2 @ N4 ) ) )
       => ( list_a1062525910057997297od_a_o @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_858_list__all2__all__nthI,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: list_nat,P2: sum_su4907927656612468301on_nat > nat > $o] :
      ( ( ( size_s5686234912572693873on_nat @ A )
        = ( size_size_list_nat @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s5686234912572693873on_nat @ A ) )
           => ( P2 @ ( nth_Su8471564416988646406on_nat @ A @ N4 ) @ ( nth_nat @ B2 @ N4 ) ) )
       => ( list_a5295772765654029478at_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_859_list__all2__all__nthI,axiom,
    ! [A: list_l8336264072933094989od_a_o,B2: list_S5253404628753938141on_nat,P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ A )
        = ( size_s5686234912572693873on_nat @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s3379185846251495609od_a_o @ A ) )
           => ( P2 @ ( nth_li3248255784598043854od_a_o @ A @ N4 ) @ ( nth_Su8471564416988646406on_nat @ B2 @ N4 ) ) )
       => ( list_a7531049758917964877on_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_860_list__all2__all__nthI,axiom,
    ! [A: list_l8336264072933094989od_a_o,B2: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ A )
        = ( size_s3379185846251495609od_a_o @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s3379185846251495609od_a_o @ A ) )
           => ( P2 @ ( nth_li3248255784598043854od_a_o @ A @ N4 ) @ ( nth_li3248255784598043854od_a_o @ B2 @ N4 ) ) )
       => ( list_a6864181014682439495od_a_o @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_861_list__all2__all__nthI,axiom,
    ! [A: list_l8336264072933094989od_a_o,B2: list_nat,P2: list_P3196869577824546119od_a_o > nat > $o] :
      ( ( ( size_s3379185846251495609od_a_o @ A )
        = ( size_size_list_nat @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_s3379185846251495609od_a_o @ A ) )
           => ( P2 @ ( nth_li3248255784598043854od_a_o @ A @ N4 ) @ ( nth_nat @ B2 @ N4 ) ) )
       => ( list_a3027799476954313168_o_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_862_list__all2__all__nthI,axiom,
    ! [A: list_nat,B2: list_S5253404628753938141on_nat,P2: nat > sum_su4907927656612468301on_nat > $o] :
      ( ( ( size_size_list_nat @ A )
        = ( size_s5686234912572693873on_nat @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_nat @ A ) )
           => ( P2 @ ( nth_nat @ A @ N4 ) @ ( nth_Su8471564416988646406on_nat @ B2 @ N4 ) ) )
       => ( list_a6193039792891997350on_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_863_list__all2__all__nthI,axiom,
    ! [A: list_nat,B2: list_l8336264072933094989od_a_o,P2: nat > list_P3196869577824546119od_a_o > $o] :
      ( ( ( size_size_list_nat @ A )
        = ( size_s3379185846251495609od_a_o @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_nat @ A ) )
           => ( P2 @ ( nth_nat @ A @ N4 ) @ ( nth_li3248255784598043854od_a_o @ B2 @ N4 ) ) )
       => ( list_a3410858264120222766od_a_o @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_864_list__all2__all__nthI,axiom,
    ! [A: list_nat,B2: list_nat,P2: nat > nat > $o] :
      ( ( ( size_size_list_nat @ A )
        = ( size_size_list_nat @ B2 ) )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ N4 @ ( size_size_list_nat @ A ) )
           => ( P2 @ ( nth_nat @ A @ N4 ) @ ( nth_nat @ B2 @ N4 ) ) )
       => ( list_all2_nat_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2_all_nthI
thf(fact_865_list__all2__nthD2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,P: nat] :
      ( ( list_a7531049758917964877on_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s5686234912572693873on_nat @ Ys ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ P ) @ ( nth_Su8471564416988646406on_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_866_list__all2__nthD2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,P: nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s5686234912572693873on_nat @ Ys ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ P ) @ ( nth_Su8471564416988646406on_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_867_list__all2__nthD2,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,Xs: list_nat,Ys: list_S5253404628753938141on_nat,P: nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s5686234912572693873on_nat @ Ys ) )
       => ( P2 @ ( nth_nat @ Xs @ P ) @ ( nth_Su8471564416988646406on_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_868_list__all2__nthD2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,P: nat] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ P ) @ ( nth_li3248255784598043854od_a_o @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_869_list__all2__nthD2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,P: nat] :
      ( ( list_a1062525910057997297od_a_o @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ P ) @ ( nth_li3248255784598043854od_a_o @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_870_list__all2__nthD2,axiom,
    ! [P2: nat > list_P3196869577824546119od_a_o > $o,Xs: list_nat,Ys: list_l8336264072933094989od_a_o,P: nat] :
      ( ( list_a3410858264120222766od_a_o @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s3379185846251495609od_a_o @ Ys ) )
       => ( P2 @ ( nth_nat @ Xs @ P ) @ ( nth_li3248255784598043854od_a_o @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_871_list__all2__nthD2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_nat,P: nat] :
      ( ( list_a3027799476954313168_o_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_size_list_nat @ Ys ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ P ) @ ( nth_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_872_list__all2__nthD2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_nat,P: nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_size_list_nat @ Ys ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ P ) @ ( nth_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_873_list__all2__nthD2,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat,Ys: list_nat,P: nat] :
      ( ( list_all2_nat_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_size_list_nat @ Ys ) )
       => ( P2 @ ( nth_nat @ Xs @ P ) @ ( nth_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD2
thf(fact_874_list__all2__nthD,axiom,
    ! [P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,P: nat] :
      ( ( list_a1062525910057997297od_a_o @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ P ) @ ( nth_li3248255784598043854od_a_o @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_875_list__all2__nthD,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,P: nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ P ) @ ( nth_Su8471564416988646406on_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_876_list__all2__nthD,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_nat,P: nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ P ) @ ( nth_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_877_list__all2__nthD,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,P: nat] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ P ) @ ( nth_li3248255784598043854od_a_o @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_878_list__all2__nthD,axiom,
    ! [P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,P: nat] :
      ( ( list_a7531049758917964877on_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ P ) @ ( nth_Su8471564416988646406on_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_879_list__all2__nthD,axiom,
    ! [P2: list_P3196869577824546119od_a_o > nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_nat,P: nat] :
      ( ( list_a3027799476954313168_o_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ P ) @ ( nth_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_880_list__all2__nthD,axiom,
    ! [P2: nat > list_P3196869577824546119od_a_o > $o,Xs: list_nat,Ys: list_l8336264072933094989od_a_o,P: nat] :
      ( ( list_a3410858264120222766od_a_o @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_size_list_nat @ Xs ) )
       => ( P2 @ ( nth_nat @ Xs @ P ) @ ( nth_li3248255784598043854od_a_o @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_881_list__all2__nthD,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,Xs: list_nat,Ys: list_S5253404628753938141on_nat,P: nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_size_list_nat @ Xs ) )
       => ( P2 @ ( nth_nat @ Xs @ P ) @ ( nth_Su8471564416988646406on_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_882_list__all2__nthD,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat,Ys: list_nat,P: nat] :
      ( ( list_all2_nat_nat @ P2 @ Xs @ Ys )
     => ( ( ord_less_nat @ P @ ( size_size_list_nat @ Xs ) )
       => ( P2 @ ( nth_nat @ Xs @ P ) @ ( nth_nat @ Ys @ P ) ) ) ) ).

% list_all2_nthD
thf(fact_883_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I: nat,J: nat] :
      ( ( distin2773651816175405460on_nat @ Xs )
     => ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s5686234912572693873on_nat @ Xs ) )
         => ( ( ( nth_Su8471564416988646406on_nat @ Xs @ I )
              = ( nth_Su8471564416988646406on_nat @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_884_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I: nat,J: nat] :
      ( ( distin690952206374581440od_a_o @ Xs )
     => ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Xs ) )
         => ( ( ( nth_li3248255784598043854od_a_o @ Xs @ I )
              = ( nth_li3248255784598043854od_a_o @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_885_nth__eq__iff__index__eq,axiom,
    ! [Xs: list_nat,I: nat,J: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
       => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Xs ) )
         => ( ( ( nth_nat @ Xs @ I )
              = ( nth_nat @ Xs @ J ) )
            = ( I = J ) ) ) ) ) ).

% nth_eq_iff_index_eq
thf(fact_886_distinct__conv__nth,axiom,
    ( distin2773651816175405460on_nat
    = ( ^ [Xs3: list_S5253404628753938141on_nat] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( ( I3 != J2 )
               => ( ( nth_Su8471564416988646406on_nat @ Xs3 @ I3 )
                 != ( nth_Su8471564416988646406on_nat @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_887_distinct__conv__nth,axiom,
    ( distin690952206374581440od_a_o
    = ( ^ [Xs3: list_l8336264072933094989od_a_o] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( ( I3 != J2 )
               => ( ( nth_li3248255784598043854od_a_o @ Xs3 @ I3 )
                 != ( nth_li3248255784598043854od_a_o @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_888_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs3: list_nat] :
        ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( I3 != J2 )
               => ( ( nth_nat @ Xs3 @ I3 )
                 != ( nth_nat @ Xs3 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_889_refl__on__def_H,axiom,
    ( refl_o3745565406092796480od_a_o
    = ( ^ [A5: set_Product_prod_a_o,R3: set_Pr1246879249572150087od_a_o] :
          ( ! [X: produc4569562562525714535od_a_o] :
              ( ( member3341120605725091216od_a_o @ X @ R3 )
             => ( produc6297817303439074732_a_o_o
                @ ^ [Y4: product_prod_a_o,Z3: product_prod_a_o] :
                    ( ( member2997933407038931562od_a_o @ Y4 @ A5 )
                    & ( member2997933407038931562od_a_o @ Z3 @ A5 ) )
                @ X ) )
          & ! [X: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ X @ A5 )
             => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X @ X ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_890_refl__on__def_H,axiom,
    ( refl_o6374727810006644366on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,R3: set_Pr2535564044409765851on_nat] :
          ( ! [X: produc1313295348606329723on_nat] :
              ( ( member5906418179908956708on_nat @ X @ R3 )
             => ( produc208325271403635792_nat_o
                @ ^ [Y4: sum_su4907927656612468301on_nat,Z3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y4 @ A5 )
                    & ( member8476201367316484836on_nat @ Z3 @ A5 ) )
                @ X ) )
          & ! [X: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X @ A5 )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ X ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_891_refl__on__def_H,axiom,
    ( refl_o1202476486973598790od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,R3: set_Pr7420778282784443591od_a_o] :
          ( ! [X: produc8520174026553075431od_a_o] :
              ( ( member5440785652130980624od_a_o @ X @ R3 )
             => ( produc731694360491113772_a_o_o
                @ ^ [Y4: list_P3196869577824546119od_a_o,Z3: list_P3196869577824546119od_a_o] :
                    ( ( member5261019209154752368od_a_o @ Y4 @ A5 )
                    & ( member5261019209154752368od_a_o @ Z3 @ A5 ) )
                @ X ) )
          & ! [X: list_P3196869577824546119od_a_o] :
              ( ( member5261019209154752368od_a_o @ X @ A5 )
             => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X @ X ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_892_refl__on__def_H,axiom,
    ( refl_on_nat
    = ( ^ [A5: set_nat,R3: set_Pr1261947904930325089at_nat] :
          ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ R3 )
             => ( produc6081775807080527818_nat_o
                @ ^ [Y4: nat,Z3: nat] :
                    ( ( member_nat2 @ Y4 @ A5 )
                    & ( member_nat2 @ Z3 @ A5 ) )
                @ X ) )
          & ! [X: nat] :
              ( ( member_nat2 @ X @ A5 )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ X ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_893_refl__on__def_H,axiom,
    ( refl_on_a
    = ( ^ [A5: set_a,R3: set_Product_prod_a_a] :
          ( ! [X: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ X @ R3 )
             => ( produc6436628058953941356_a_a_o
                @ ^ [Y4: a,Z3: a] :
                    ( ( member_a2 @ Y4 @ A5 )
                    & ( member_a2 @ Z3 @ A5 ) )
                @ X ) )
          & ! [X: a] :
              ( ( member_a2 @ X @ A5 )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ X ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_894_refl__on__def_H,axiom,
    ( refl_o794705341054444288on_nat
    = ( ^ [A5: set_li1748721617258726113on_nat,R3: set_Pr8586234676504894535on_nat] :
          ( ! [X: produc1714117091004681831on_nat] :
              ( ( member4629988367768146576on_nat @ X @ R3 )
             => ( produc4129062015429275820_nat_o
                @ ^ [Y4: list_P9127870862658512897on_nat,Z3: list_P9127870862658512897on_nat] :
                    ( ( member783415501022416426on_nat @ Y4 @ A5 )
                    & ( member783415501022416426on_nat @ Z3 @ A5 ) )
                @ X ) )
          & ! [X: list_P9127870862658512897on_nat] :
              ( ( member783415501022416426on_nat @ X @ A5 )
             => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X @ X ) @ R3 ) ) ) ) ) ).

% refl_on_def'
thf(fact_895_product__lists__set,axiom,
    ! [Xss: list_l8999537460041914605on_nat] :
      ( ( set_li4371160873219908744on_nat @ ( produc1748266815408604318on_nat @ Xss ) )
      = ( collec7158715823234635826on_nat
        @ ^ [Xs3: list_S5253404628753938141on_nat] :
            ( list_a6440322313061018803on_nat
            @ ^ [X: sum_su4907927656612468301on_nat,Ys3: list_S5253404628753938141on_nat] : ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Ys3 ) )
            @ Xs3
            @ Xss ) ) ) ).

% product_lists_set
thf(fact_896_product__lists__set,axiom,
    ! [Xss: list_l7431997615882138323od_a_o] :
      ( ( set_li3106133282984605794od_a_o @ ( produc6266365578930819638od_a_o @ Xss ) )
      = ( collec3723402900366632248od_a_o
        @ ^ [Xs3: list_l8336264072933094989od_a_o] :
            ( list_a2210580654873286733od_a_o
            @ ^ [X: list_P3196869577824546119od_a_o,Ys3: list_l8336264072933094989od_a_o] : ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Ys3 ) )
            @ Xs3
            @ Xss ) ) ) ).

% product_lists_set
thf(fact_897_product__lists__set,axiom,
    ! [Xss: list_list_nat] :
      ( ( set_list_nat2 @ ( product_lists_nat @ Xss ) )
      = ( collect_list_nat
        @ ^ [Xs3: list_nat] :
            ( list_a5155878676884504761st_nat
            @ ^ [X: nat,Ys3: list_nat] : ( member_nat2 @ X @ ( set_nat2 @ Ys3 ) )
            @ Xs3
            @ Xss ) ) ) ).

% product_lists_set
thf(fact_898_product__lists__set,axiom,
    ! [Xss: list_list_a] :
      ( ( set_list_a2 @ ( product_lists_a @ Xss ) )
      = ( collect_list_a
        @ ^ [Xs3: list_a] :
            ( list_all2_a_list_a
            @ ^ [X: a,Ys3: list_a] : ( member_a2 @ X @ ( set_a2 @ Ys3 ) )
            @ Xs3
            @ Xss ) ) ) ).

% product_lists_set
thf(fact_899_product__lists__set,axiom,
    ! [Xss: list_l8336264072933094989od_a_o] :
      ( ( set_li2629784761229999580od_a_o @ ( produc3814538932794474544od_a_o @ Xss ) )
      = ( collec5117750660557753778od_a_o
        @ ^ [Xs3: list_P3196869577824546119od_a_o] :
            ( list_a5010313301513502029od_a_o
            @ ^ [X: product_prod_a_o,Ys3: list_P3196869577824546119od_a_o] : ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Ys3 ) )
            @ Xs3
            @ Xss ) ) ) ).

% product_lists_set
thf(fact_900_list__all2I,axiom,
    ! [A: list_a,B2: list_o,P2: a > $o > $o] :
      ( ! [X2: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X2 @ ( set_Product_prod_a_o2 @ ( zip_a_o @ A @ B2 ) ) )
         => ( produc8428829086471790546_a_o_o @ P2 @ X2 ) )
     => ( ( ( size_size_list_a @ A )
          = ( size_size_list_o @ B2 ) )
       => ( list_all2_a_o @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_901_list__all2I,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
      ( ! [X2: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X2 @ ( set_Pr937530902923597456on_nat @ ( zip_Su8130003265398491731on_nat @ A @ B2 ) ) )
         => ( produc208325271403635792_nat_o @ P2 @ X2 ) )
     => ( ( ( size_s5686234912572693873on_nat @ A )
          = ( size_s5686234912572693873on_nat @ B2 ) )
       => ( list_a1658445451649724195on_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_902_list__all2I,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: list_l8336264072933094989od_a_o,P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o] :
      ( ! [X2: produc6493764124173775663od_a_o] :
          ( ( member7595723914438784454od_a_o @ X2 @ ( set_Pr6587213113994854490od_a_o @ ( zip_Su2363964881504851137od_a_o @ A @ B2 ) ) )
         => ( produc7187800550436554762_a_o_o @ P2 @ X2 ) )
     => ( ( ( size_s5686234912572693873on_nat @ A )
          = ( size_s3379185846251495609od_a_o @ B2 ) )
       => ( list_a1062525910057997297od_a_o @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_903_list__all2I,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: list_nat,P2: sum_su4907927656612468301on_nat > nat > $o] :
      ( ! [X2: produc8556620195627115390at_nat] :
          ( ( member8820252441731870247at_nat @ X2 @ ( set_Pr1192295990000706195at_nat @ ( zip_Su8633780382783139798at_nat @ A @ B2 ) ) )
         => ( produc2579979265757742093_nat_o @ P2 @ X2 ) )
     => ( ( ( size_s5686234912572693873on_nat @ A )
          = ( size_size_list_nat @ B2 ) )
       => ( list_a5295772765654029478at_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_904_list__all2I,axiom,
    ! [A: list_l8336264072933094989od_a_o,B2: list_S5253404628753938141on_nat,P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o] :
      ( ! [X2: produc731851057114331587on_nat] :
          ( ( member1833810847379340378on_nat @ X2 @ ( set_Pr825300046935410414on_nat @ ( zip_li8832488730364818717on_nat @ A @ B2 ) ) )
         => ( produc963041392342583470_nat_o @ P2 @ X2 ) )
     => ( ( ( size_s3379185846251495609od_a_o @ A )
          = ( size_s5686234912572693873on_nat @ B2 ) )
       => ( list_a7531049758917964877on_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_905_list__all2I,axiom,
    ! [A: list_l8336264072933094989od_a_o,B2: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o] :
      ( ! [X2: produc8520174026553075431od_a_o] :
          ( ( member5440785652130980624od_a_o @ X2 @ ( set_Pr7142968438029117820od_a_o @ ( zip_li3837685382659732087od_a_o @ A @ B2 ) ) )
         => ( produc731694360491113772_a_o_o @ P2 @ X2 ) )
     => ( ( ( size_s3379185846251495609od_a_o @ A )
          = ( size_s3379185846251495609od_a_o @ B2 ) )
       => ( list_a6864181014682439495od_a_o @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_906_list__all2I,axiom,
    ! [A: list_l8336264072933094989od_a_o,B2: list_nat,P2: list_P3196869577824546119od_a_o > nat > $o] :
      ( ! [X2: produc408724640507517382_o_nat] :
          ( ( member7885718254550376541_o_nat @ X2 @ ( set_Pr7830758801870232305_o_nat @ ( zip_li1039786862263726752_o_nat @ A @ B2 ) ) )
         => ( produc4957984460634321003_nat_o @ P2 @ X2 ) )
     => ( ( ( size_s3379185846251495609od_a_o @ A )
          = ( size_size_list_nat @ B2 ) )
       => ( list_a3027799476954313168_o_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_907_list__all2I,axiom,
    ! [A: list_nat,B2: list_S5253404628753938141on_nat,P2: nat > sum_su4907927656612468301on_nat > $o] :
      ( ! [X2: produc7088870822282640638on_nat] :
          ( ( member7352503068387395495on_nat @ X2 @ ( set_Pr8947918653511007251on_nat @ ( zip_na307675373166331862on_nat @ A @ B2 ) ) )
         => ( produc2700354003219672077_nat_o @ P2 @ X2 ) )
     => ( ( ( size_size_list_nat @ A )
          = ( size_s5686234912572693873on_nat @ B2 ) )
       => ( list_a6193039792891997350on_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_908_list__all2I,axiom,
    ! [A: list_nat,B2: list_l8336264072933094989od_a_o,P2: nat > list_P3196869577824546119od_a_o > $o] :
      ( ! [X2: produc1433831906155620204od_a_o] :
          ( ( member8910825520198479363od_a_o @ X2 @ ( set_Pr8855866067518335127od_a_o @ ( zip_na1422845649429636350od_a_o @ A @ B2 ) ) )
         => ( produc8470836043472241293_a_o_o @ P2 @ X2 ) )
     => ( ( ( size_size_list_nat @ A )
          = ( size_s3379185846251495609od_a_o @ B2 ) )
       => ( list_a3410858264120222766od_a_o @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_909_list__all2I,axiom,
    ! [A: list_nat,B2: list_nat,P2: nat > nat > $o] :
      ( ! [X2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X2 @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ A @ B2 ) ) )
         => ( produc6081775807080527818_nat_o @ P2 @ X2 ) )
     => ( ( ( size_size_list_nat @ A )
          = ( size_size_list_nat @ B2 ) )
       => ( list_all2_nat_nat @ P2 @ A @ B2 ) ) ) ).

% list_all2I
thf(fact_910_distinct__Ex1,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,X3: product_prod_a_o] :
      ( ( distin2380507709364613050od_a_o @ Xs )
     => ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
       => ? [X2: nat] :
            ( ( ord_less_nat @ X2 @ ( size_s1727999245659094451od_a_o @ Xs ) )
            & ( ( nth_Product_prod_a_o @ Xs @ X2 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s1727999245659094451od_a_o @ Xs ) )
                  & ( ( nth_Product_prod_a_o @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_911_distinct__Ex1,axiom,
    ! [Xs: list_a,X3: a] :
      ( ( distinct_a @ Xs )
     => ( ( member_a2 @ X3 @ ( set_a2 @ Xs ) )
       => ? [X2: nat] :
            ( ( ord_less_nat @ X2 @ ( size_size_list_a @ Xs ) )
            & ( ( nth_a @ Xs @ X2 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_a @ Xs ) )
                  & ( ( nth_a @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_912_distinct__Ex1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( distin2773651816175405460on_nat @ Xs )
     => ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ? [X2: nat] :
            ( ( ord_less_nat @ X2 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( nth_Su8471564416988646406on_nat @ Xs @ X2 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s5686234912572693873on_nat @ Xs ) )
                  & ( ( nth_Su8471564416988646406on_nat @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_913_distinct__Ex1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o] :
      ( ( distin690952206374581440od_a_o @ Xs )
     => ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
       => ? [X2: nat] :
            ( ( ord_less_nat @ X2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( nth_li3248255784598043854od_a_o @ Xs @ X2 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_s3379185846251495609od_a_o @ Xs ) )
                  & ( ( nth_li3248255784598043854od_a_o @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_914_distinct__Ex1,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat2 @ X3 @ ( set_nat2 @ Xs ) )
       => ? [X2: nat] :
            ( ( ord_less_nat @ X2 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X2 )
              = X3 )
            & ! [Y5: nat] :
                ( ( ( ord_less_nat @ Y5 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y5 )
                    = X3 ) )
               => ( Y5 = X2 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_915_NSS,axiom,
    ( nss
    = ( append2865821678206098582on_nat @ ns @ s ) ) ).

% NSS
thf(fact_916_ns__mul__ext__refl,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat,A4: multis1856558993800614429on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( refl_o6374727810006644366on_nat @ top_to7488204556702531123on_nat @ Ns )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ A4 ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ).

% ns_mul_ext_refl
thf(fact_917_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_918_refl__on__domain,axiom,
    ! [A4: set_Product_prod_a_o,R: set_Pr1246879249572150087od_a_o,A: product_prod_a_o,B2: product_prod_a_o] :
      ( ( refl_o3745565406092796480od_a_o @ A4 @ R )
     => ( ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ A @ B2 ) @ R )
       => ( ( member2997933407038931562od_a_o @ A @ A4 )
          & ( member2997933407038931562od_a_o @ B2 @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_919_refl__on__domain,axiom,
    ! [A4: set_Su266273576555847555on_nat,R: set_Pr2535564044409765851on_nat,A: sum_su4907927656612468301on_nat,B2: sum_su4907927656612468301on_nat] :
      ( ( refl_o6374727810006644366on_nat @ A4 @ R )
     => ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A @ B2 ) @ R )
       => ( ( member8476201367316484836on_nat @ A @ A4 )
          & ( member8476201367316484836on_nat @ B2 @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_920_refl__on__domain,axiom,
    ! [A4: set_li5011860890973139239od_a_o,R: set_Pr7420778282784443591od_a_o,A: list_P3196869577824546119od_a_o,B2: list_P3196869577824546119od_a_o] :
      ( ( refl_o1202476486973598790od_a_o @ A4 @ R )
     => ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A @ B2 ) @ R )
       => ( ( member5261019209154752368od_a_o @ A @ A4 )
          & ( member5261019209154752368od_a_o @ B2 @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_921_refl__on__domain,axiom,
    ! [A4: set_nat,R: set_Pr1261947904930325089at_nat,A: nat,B2: nat] :
      ( ( refl_on_nat @ A4 @ R )
     => ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B2 ) @ R )
       => ( ( member_nat2 @ A @ A4 )
          & ( member_nat2 @ B2 @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_922_refl__on__domain,axiom,
    ! [A4: set_a,R: set_Product_prod_a_a,A: a,B2: a] :
      ( ( refl_on_a @ A4 @ R )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A @ B2 ) @ R )
       => ( ( member_a2 @ A @ A4 )
          & ( member_a2 @ B2 @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_923_refl__on__domain,axiom,
    ! [A4: set_li1748721617258726113on_nat,R: set_Pr8586234676504894535on_nat,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat] :
      ( ( refl_o794705341054444288on_nat @ A4 @ R )
     => ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ A @ B2 ) @ R )
       => ( ( member783415501022416426on_nat @ A @ A4 )
          & ( member783415501022416426on_nat @ B2 @ A4 ) ) ) ) ).

% refl_on_domain
thf(fact_924_assms,axiom,
    ( ( multis4221440410396274297_cnf_a @ cnf )
    = ( produc1545487600272464053on_nat @ left @ ( produc4176415733022414961on_nat @ right @ ( produc7152764266822961111on_nat @ s @ nss ) ) ) ) ).

% assms
thf(fact_925_append_Oassoc,axiom,
    ! [A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat,C2: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ A @ B2 ) @ C2 )
      = ( append2865821678206098582on_nat @ A @ ( append2865821678206098582on_nat @ B2 @ C2 ) ) ) ).

% append.assoc
thf(fact_926_append_Oassoc,axiom,
    ! [A: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,C2: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( append3523812819975775986on_nat @ A @ B2 ) @ C2 )
      = ( append3523812819975775986on_nat @ A @ ( append3523812819975775986on_nat @ B2 @ C2 ) ) ) ).

% append.assoc
thf(fact_927_append_Oassoc,axiom,
    ! [A: list_nat,B2: list_nat,C2: list_nat] :
      ( ( append_nat @ ( append_nat @ A @ B2 ) @ C2 )
      = ( append_nat @ A @ ( append_nat @ B2 @ C2 ) ) ) ).

% append.assoc
thf(fact_928_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_929_append__assoc,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ Zs )
      = ( append3523812819975775986on_nat @ Xs @ ( append3523812819975775986on_nat @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_930_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_931_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_932_append__same__eq,axiom,
    ! [Ys: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys @ Xs )
        = ( append3523812819975775986on_nat @ Zs @ Xs ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_933_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_934_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_935_same__append__eq,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys )
        = ( append3523812819975775986on_nat @ Xs @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_936_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_937_append__eq__append__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Us: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
      ( ( ( ( size_s7490768528479217773on_nat @ Xs )
          = ( size_s7490768528479217773on_nat @ Ys ) )
        | ( ( size_s7490768528479217773on_nat @ Us )
          = ( size_s7490768528479217773on_nat @ Vs ) ) )
     => ( ( ( append2865821678206098582on_nat @ Xs @ Us )
          = ( append2865821678206098582on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_938_append__eq__append__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        | ( ( size_s5686234912572693873on_nat @ Us )
          = ( size_s5686234912572693873on_nat @ Vs ) ) )
     => ( ( ( append3523812819975775986on_nat @ Xs @ Us )
          = ( append3523812819975775986on_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_939_append__eq__append__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Us: list_l8336264072933094989od_a_o,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        | ( ( size_s3379185846251495609od_a_o @ Us )
          = ( size_s3379185846251495609od_a_o @ Vs ) ) )
     => ( ( ( append2159120467191820514od_a_o @ Xs @ Us )
          = ( append2159120467191820514od_a_o @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_940_append__eq__append__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        | ( ( size_size_list_nat @ Us )
          = ( size_size_list_nat @ Vs ) ) )
     => ( ( ( append_nat @ Xs @ Us )
          = ( append_nat @ Ys @ Vs ) )
        = ( ( Xs = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_941_list__ex__append,axiom,
    ! [P2: produc1313295348606329723on_nat > $o,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( list_e8248363684495001279on_nat @ P2 @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( ( list_e8248363684495001279on_nat @ P2 @ Xs )
        | ( list_e8248363684495001279on_nat @ P2 @ Ys ) ) ) ).

% list_ex_append
thf(fact_942_list__ex__append,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( list_e3769318529992910857on_nat @ P2 @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( ( list_e3769318529992910857on_nat @ P2 @ Xs )
        | ( list_e3769318529992910857on_nat @ P2 @ Ys ) ) ) ).

% list_ex_append
thf(fact_943_list__ex__append,axiom,
    ! [P2: nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( list_ex_nat @ P2 @ ( append_nat @ Xs @ Ys ) )
      = ( ( list_ex_nat @ P2 @ Xs )
        | ( list_ex_nat @ P2 @ Ys ) ) ) ).

% list_ex_append
thf(fact_944_zip__append,axiom,
    ! [Xs: list_nat,Us: list_nat,Ys: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_nat_nat @ ( append_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append985823374593552924at_nat @ ( zip_nat_nat @ Xs @ Us ) @ ( zip_nat_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_945_zip__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Us: list_nat,Ys: list_l8336264072933094989od_a_o,Vs: list_nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_li1039786862263726752_o_nat @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append4227299948903008363_o_nat @ ( zip_li1039786862263726752_o_nat @ Xs @ Us ) @ ( zip_li1039786862263726752_o_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_946_zip__append,axiom,
    ! [Xs: list_nat,Us: list_l8336264072933094989od_a_o,Ys: list_nat,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Us ) )
     => ( ( zip_na1422845649429636350od_a_o @ ( append_nat @ Xs @ Ys ) @ ( append2159120467191820514od_a_o @ Us @ Vs ) )
        = ( append5252407214551111185od_a_o @ ( zip_na1422845649429636350od_a_o @ Xs @ Us ) @ ( zip_na1422845649429636350od_a_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_947_zip__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Us: list_nat,Ys: list_S5253404628753938141on_nat,Vs: list_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Su8633780382783139798at_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append7566965782333837465at_nat @ ( zip_Su8633780382783139798at_nat @ Xs @ Us ) @ ( zip_Su8633780382783139798at_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_948_zip__append,axiom,
    ! [Xs: list_nat,Us: list_S5253404628753938141on_nat,Ys: list_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Us ) )
     => ( ( zip_na307675373166331862on_nat @ ( append_nat @ Xs @ Ys ) @ ( append3523812819975775986on_nat @ Us @ Vs ) )
        = ( append6099216408989362713on_nat @ ( zip_na307675373166331862on_nat @ Xs @ Us ) @ ( zip_na307675373166331862on_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_949_zip__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Us: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Us ) )
     => ( ( zip_li3837685382659732087od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append2159120467191820514od_a_o @ Us @ Vs ) )
        = ( append3616137071411550338od_a_o @ ( zip_li3837685382659732087od_a_o @ Xs @ Us ) @ ( zip_li3837685382659732087od_a_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_950_zip__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Us: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Us ) )
     => ( ( zip_Su2363964881504851137od_a_o @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( append2159120467191820514od_a_o @ Us @ Vs ) )
        = ( append5996814989978490324od_a_o @ ( zip_Su2363964881504851137od_a_o @ Xs @ Us ) @ ( zip_Su2363964881504851137od_a_o @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_951_zip__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Us: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,Vs: list_S5253404628753938141on_nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s5686234912572693873on_nat @ Us ) )
     => ( ( zip_li8832488730364818717on_nat @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( append3523812819975775986on_nat @ Us @ Vs ) )
        = ( append234901922919046248on_nat @ ( zip_li8832488730364818717on_nat @ Xs @ Us ) @ ( zip_li8832488730364818717on_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_952_zip__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Us ) )
     => ( ( zip_Su8130003265398491731on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( append3523812819975775986on_nat @ Us @ Vs ) )
        = ( append2865821678206098582on_nat @ ( zip_Su8130003265398491731on_nat @ Xs @ Us ) @ ( zip_Su8130003265398491731on_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_953_zip__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Us: list_nat,Ys: list_P9127870862658512897on_nat,Vs: list_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_size_list_nat @ Us ) )
     => ( ( zip_Pr6524550728014728404at_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ ( append_nat @ Us @ Vs ) )
        = ( append5573366755791373471at_nat @ ( zip_Pr6524550728014728404at_nat @ Xs @ Us ) @ ( zip_Pr6524550728014728404at_nat @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_954_append__eq__appendI,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Xs1: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Us: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append2865821678206098582on_nat @ Xs1 @ Us ) )
       => ( ( append2865821678206098582on_nat @ Xs @ Ys )
          = ( append2865821678206098582on_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_955_append__eq__appendI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Xs1: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Us: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append3523812819975775986on_nat @ Xs1 @ Us ) )
       => ( ( append3523812819975775986on_nat @ Xs @ Ys )
          = ( append3523812819975775986on_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_956_append__eq__appendI,axiom,
    ! [Xs: list_nat,Xs1: list_nat,Zs: list_nat,Ys: list_nat,Us: list_nat] :
      ( ( ( append_nat @ Xs @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_nat @ Xs1 @ Us ) )
       => ( ( append_nat @ Xs @ Ys )
          = ( append_nat @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_957_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 ) )
      = ( ? [Us2: list_P9127870862658512897on_nat] :
            ( ( ( Xs
                = ( append2865821678206098582on_nat @ Zs @ Us2 ) )
              & ( ( append2865821678206098582on_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append2865821678206098582on_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append2865821678206098582on_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_958_append__eq__append__conv2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,Ts: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys )
        = ( append3523812819975775986on_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_S5253404628753938141on_nat] :
            ( ( ( Xs
                = ( append3523812819975775986on_nat @ Zs @ Us2 ) )
              & ( ( append3523812819975775986on_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append3523812819975775986on_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append3523812819975775986on_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_959_append__eq__append__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat,Zs: list_nat,Ts: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = ( append_nat @ Zs @ Ts ) )
      = ( ? [Us2: list_nat] :
            ( ( ( Xs
                = ( append_nat @ Zs @ Us2 ) )
              & ( ( append_nat @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_nat @ Xs @ Us2 )
                = Zs )
              & ( Ys
                = ( append_nat @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_960_list__all2__appendI,axiom,
    ! [P2: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_P9127870862658512897on_nat,C2: list_P9127870862658512897on_nat,D4: list_P9127870862658512897on_nat] :
      ( ( list_a1726649567543124551on_nat @ P2 @ A @ B2 )
     => ( ( list_a1726649567543124551on_nat @ P2 @ C2 @ D4 )
       => ( list_a1726649567543124551on_nat @ P2 @ ( append2865821678206098582on_nat @ A @ C2 ) @ ( append2865821678206098582on_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_961_list__all2__appendI,axiom,
    ! [P2: produc1313295348606329723on_nat > sum_su4907927656612468301on_nat > $o,A: list_P9127870862658512897on_nat,B2: list_S5253404628753938141on_nat,C2: list_P9127870862658512897on_nat,D4: list_S5253404628753938141on_nat] :
      ( ( list_a2660791513809139329on_nat @ P2 @ A @ B2 )
     => ( ( list_a2660791513809139329on_nat @ P2 @ C2 @ D4 )
       => ( list_a2660791513809139329on_nat @ P2 @ ( append2865821678206098582on_nat @ A @ C2 ) @ ( append3523812819975775986on_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_962_list__all2__appendI,axiom,
    ! [P2: produc1313295348606329723on_nat > nat > $o,A: list_P9127870862658512897on_nat,B2: list_nat,C2: list_P9127870862658512897on_nat,D4: list_nat] :
      ( ( list_a1112363324972753924at_nat @ P2 @ A @ B2 )
     => ( ( list_a1112363324972753924at_nat @ P2 @ C2 @ D4 )
       => ( list_a1112363324972753924at_nat @ P2 @ ( append2865821678206098582on_nat @ A @ C2 ) @ ( append_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_963_list__all2__appendI,axiom,
    ! [P2: sum_su4907927656612468301on_nat > produc1313295348606329723on_nat > $o,A: list_S5253404628753938141on_nat,B2: list_P9127870862658512897on_nat,C2: list_S5253404628753938141on_nat,D4: list_P9127870862658512897on_nat] :
      ( ( list_a2414275824964996901on_nat @ P2 @ A @ B2 )
     => ( ( list_a2414275824964996901on_nat @ P2 @ C2 @ D4 )
       => ( list_a2414275824964996901on_nat @ P2 @ ( append3523812819975775986on_nat @ A @ C2 ) @ ( append2865821678206098582on_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_964_list__all2__appendI,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,A: list_S5253404628753938141on_nat,B2: list_S5253404628753938141on_nat,C2: list_S5253404628753938141on_nat,D4: list_S5253404628753938141on_nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ A @ B2 )
     => ( ( list_a1658445451649724195on_nat @ P2 @ C2 @ D4 )
       => ( list_a1658445451649724195on_nat @ P2 @ ( append3523812819975775986on_nat @ A @ C2 ) @ ( append3523812819975775986on_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_965_list__all2__appendI,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,A: list_S5253404628753938141on_nat,B2: list_nat,C2: list_S5253404628753938141on_nat,D4: list_nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ A @ B2 )
     => ( ( list_a5295772765654029478at_nat @ P2 @ C2 @ D4 )
       => ( list_a5295772765654029478at_nat @ P2 @ ( append3523812819975775986on_nat @ A @ C2 ) @ ( append_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_966_list__all2__appendI,axiom,
    ! [P2: nat > produc1313295348606329723on_nat > $o,A: list_nat,B2: list_P9127870862658512897on_nat,C2: list_nat,D4: list_P9127870862658512897on_nat] :
      ( ( list_a1865022572110992866on_nat @ P2 @ A @ B2 )
     => ( ( list_a1865022572110992866on_nat @ P2 @ C2 @ D4 )
       => ( list_a1865022572110992866on_nat @ P2 @ ( append_nat @ A @ C2 ) @ ( append2865821678206098582on_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_967_list__all2__appendI,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,A: list_nat,B2: list_S5253404628753938141on_nat,C2: list_nat,D4: list_S5253404628753938141on_nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ A @ B2 )
     => ( ( list_a6193039792891997350on_nat @ P2 @ C2 @ D4 )
       => ( list_a6193039792891997350on_nat @ P2 @ ( append_nat @ A @ C2 ) @ ( append3523812819975775986on_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_968_list__all2__appendI,axiom,
    ! [P2: nat > nat > $o,A: list_nat,B2: list_nat,C2: list_nat,D4: list_nat] :
      ( ( list_all2_nat_nat @ P2 @ A @ B2 )
     => ( ( list_all2_nat_nat @ P2 @ C2 @ D4 )
       => ( list_all2_nat_nat @ P2 @ ( append_nat @ A @ C2 ) @ ( append_nat @ B2 @ D4 ) ) ) ) ).

% list_all2_appendI
thf(fact_969_list__all2__append2,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( list_all2_nat_nat @ P2 @ Xs @ ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us2 )
              = ( size_size_list_nat @ Ys ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_size_list_nat @ Zs ) )
            & ( list_all2_nat_nat @ P2 @ Us2 @ Ys )
            & ( list_all2_nat_nat @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_970_list__all2__append2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_nat,Zs: list_nat] :
      ( ( list_a3027799476954313168_o_nat @ P2 @ Xs @ ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_l8336264072933094989od_a_o,Vs2: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Us2 @ Vs2 ) )
            & ( ( size_s3379185846251495609od_a_o @ Us2 )
              = ( size_size_list_nat @ Ys ) )
            & ( ( size_s3379185846251495609od_a_o @ Vs2 )
              = ( size_size_list_nat @ Zs ) )
            & ( list_a3027799476954313168_o_nat @ P2 @ Us2 @ Ys )
            & ( list_a3027799476954313168_o_nat @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_971_list__all2__append2,axiom,
    ! [P2: nat > list_P3196869577824546119od_a_o > $o,Xs: list_nat,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( list_a3410858264120222766od_a_o @ P2 @ Xs @ ( append2159120467191820514od_a_o @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us2 )
              = ( size_s3379185846251495609od_a_o @ Ys ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_s3379185846251495609od_a_o @ Zs ) )
            & ( list_a3410858264120222766od_a_o @ P2 @ Us2 @ Ys )
            & ( list_a3410858264120222766od_a_o @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_972_list__all2__append2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_nat,Zs: list_nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ Xs @ ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s5686234912572693873on_nat @ Us2 )
              = ( size_size_list_nat @ Ys ) )
            & ( ( size_s5686234912572693873on_nat @ Vs2 )
              = ( size_size_list_nat @ Zs ) )
            & ( list_a5295772765654029478at_nat @ P2 @ Us2 @ Ys )
            & ( list_a5295772765654029478at_nat @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_973_list__all2__append2,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,Xs: list_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ Xs @ ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us2 )
              = ( size_s5686234912572693873on_nat @ Ys ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_s5686234912572693873on_nat @ Zs ) )
            & ( list_a6193039792891997350on_nat @ P2 @ Us2 @ Ys )
            & ( list_a6193039792891997350on_nat @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_974_list__all2__append2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ Xs @ ( append2159120467191820514od_a_o @ Ys @ Zs ) )
      = ( ? [Us2: list_l8336264072933094989od_a_o,Vs2: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Us2 @ Vs2 ) )
            & ( ( size_s3379185846251495609od_a_o @ Us2 )
              = ( size_s3379185846251495609od_a_o @ Ys ) )
            & ( ( size_s3379185846251495609od_a_o @ Vs2 )
              = ( size_s3379185846251495609od_a_o @ Zs ) )
            & ( list_a6864181014682439495od_a_o @ P2 @ Us2 @ Ys )
            & ( list_a6864181014682439495od_a_o @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_975_list__all2__append2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( list_a1062525910057997297od_a_o @ P2 @ Xs @ ( append2159120467191820514od_a_o @ Ys @ Zs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s5686234912572693873on_nat @ Us2 )
              = ( size_s3379185846251495609od_a_o @ Ys ) )
            & ( ( size_s5686234912572693873on_nat @ Vs2 )
              = ( size_s3379185846251495609od_a_o @ Zs ) )
            & ( list_a1062525910057997297od_a_o @ P2 @ Us2 @ Ys )
            & ( list_a1062525910057997297od_a_o @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_976_list__all2__append2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( list_a7531049758917964877on_nat @ P2 @ Xs @ ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_l8336264072933094989od_a_o,Vs2: list_l8336264072933094989od_a_o] :
            ( ( Xs
              = ( append2159120467191820514od_a_o @ Us2 @ Vs2 ) )
            & ( ( size_s3379185846251495609od_a_o @ Us2 )
              = ( size_s5686234912572693873on_nat @ Ys ) )
            & ( ( size_s3379185846251495609od_a_o @ Vs2 )
              = ( size_s5686234912572693873on_nat @ Zs ) )
            & ( list_a7531049758917964877on_nat @ P2 @ Us2 @ Ys )
            & ( list_a7531049758917964877on_nat @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_977_list__all2__append2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ Xs @ ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s5686234912572693873on_nat @ Us2 )
              = ( size_s5686234912572693873on_nat @ Ys ) )
            & ( ( size_s5686234912572693873on_nat @ Vs2 )
              = ( size_s5686234912572693873on_nat @ Zs ) )
            & ( list_a1658445451649724195on_nat @ P2 @ Us2 @ Ys )
            & ( list_a1658445451649724195on_nat @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_978_list__all2__append2,axiom,
    ! [P2: produc1313295348606329723on_nat > nat > $o,Xs: list_P9127870862658512897on_nat,Ys: list_nat,Zs: list_nat] :
      ( ( list_a1112363324972753924at_nat @ P2 @ Xs @ ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s7490768528479217773on_nat @ Us2 )
              = ( size_size_list_nat @ Ys ) )
            & ( ( size_s7490768528479217773on_nat @ Vs2 )
              = ( size_size_list_nat @ Zs ) )
            & ( list_a1112363324972753924at_nat @ P2 @ Us2 @ Ys )
            & ( list_a1112363324972753924at_nat @ P2 @ Vs2 @ Zs ) ) ) ) ).

% list_all2_append2
thf(fact_979_list__all2__append1,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( list_all2_nat_nat @ P2 @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Zs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us2 )
              = ( size_size_list_nat @ Xs ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_size_list_nat @ Ys ) )
            & ( list_all2_nat_nat @ P2 @ Xs @ Us2 )
            & ( list_all2_nat_nat @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_980_list__all2__append1,axiom,
    ! [P2: nat > list_P3196869577824546119od_a_o > $o,Xs: list_nat,Ys: list_nat,Zs: list_l8336264072933094989od_a_o] :
      ( ( list_a3410858264120222766od_a_o @ P2 @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_l8336264072933094989od_a_o,Vs2: list_l8336264072933094989od_a_o] :
            ( ( Zs
              = ( append2159120467191820514od_a_o @ Us2 @ Vs2 ) )
            & ( ( size_s3379185846251495609od_a_o @ Us2 )
              = ( size_size_list_nat @ Xs ) )
            & ( ( size_s3379185846251495609od_a_o @ Vs2 )
              = ( size_size_list_nat @ Ys ) )
            & ( list_a3410858264120222766od_a_o @ P2 @ Xs @ Us2 )
            & ( list_a3410858264120222766od_a_o @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_981_list__all2__append1,axiom,
    ! [P2: list_P3196869577824546119od_a_o > nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_nat] :
      ( ( list_a3027799476954313168_o_nat @ P2 @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Zs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us2 )
              = ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_s3379185846251495609od_a_o @ Ys ) )
            & ( list_a3027799476954313168_o_nat @ P2 @ Xs @ Us2 )
            & ( list_a3027799476954313168_o_nat @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_982_list__all2__append1,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Zs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s5686234912572693873on_nat @ Us2 )
              = ( size_size_list_nat @ Xs ) )
            & ( ( size_s5686234912572693873on_nat @ Vs2 )
              = ( size_size_list_nat @ Ys ) )
            & ( list_a6193039792891997350on_nat @ P2 @ Xs @ Us2 )
            & ( list_a6193039792891997350on_nat @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_983_list__all2__append1,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Zs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( ( size_size_list_nat @ Us2 )
              = ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( size_size_list_nat @ Vs2 )
              = ( size_s5686234912572693873on_nat @ Ys ) )
            & ( list_a5295772765654029478at_nat @ P2 @ Xs @ Us2 )
            & ( list_a5295772765654029478at_nat @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_984_list__all2__append1,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_l8336264072933094989od_a_o] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_l8336264072933094989od_a_o,Vs2: list_l8336264072933094989od_a_o] :
            ( ( Zs
              = ( append2159120467191820514od_a_o @ Us2 @ Vs2 ) )
            & ( ( size_s3379185846251495609od_a_o @ Us2 )
              = ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( size_s3379185846251495609od_a_o @ Vs2 )
              = ( size_s3379185846251495609od_a_o @ Ys ) )
            & ( list_a6864181014682439495od_a_o @ P2 @ Xs @ Us2 )
            & ( list_a6864181014682439495od_a_o @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_985_list__all2__append1,axiom,
    ! [P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,Zs: list_S5253404628753938141on_nat] :
      ( ( list_a7531049758917964877on_nat @ P2 @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Zs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s5686234912572693873on_nat @ Us2 )
              = ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( size_s5686234912572693873on_nat @ Vs2 )
              = ( size_s3379185846251495609od_a_o @ Ys ) )
            & ( list_a7531049758917964877on_nat @ P2 @ Xs @ Us2 )
            & ( list_a7531049758917964877on_nat @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_986_list__all2__append1,axiom,
    ! [P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_l8336264072933094989od_a_o] :
      ( ( list_a1062525910057997297od_a_o @ P2 @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_l8336264072933094989od_a_o,Vs2: list_l8336264072933094989od_a_o] :
            ( ( Zs
              = ( append2159120467191820514od_a_o @ Us2 @ Vs2 ) )
            & ( ( size_s3379185846251495609od_a_o @ Us2 )
              = ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( size_s3379185846251495609od_a_o @ Vs2 )
              = ( size_s5686234912572693873on_nat @ Ys ) )
            & ( list_a1062525910057997297od_a_o @ P2 @ Xs @ Us2 )
            & ( list_a1062525910057997297od_a_o @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_987_list__all2__append1,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Zs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s5686234912572693873on_nat @ Us2 )
              = ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( size_s5686234912572693873on_nat @ Vs2 )
              = ( size_s5686234912572693873on_nat @ Ys ) )
            & ( list_a1658445451649724195on_nat @ P2 @ Xs @ Us2 )
            & ( list_a1658445451649724195on_nat @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_988_list__all2__append1,axiom,
    ! [P2: nat > produc1313295348606329723on_nat > $o,Xs: list_nat,Ys: list_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( list_a1865022572110992866on_nat @ P2 @ ( append_nat @ Xs @ Ys ) @ Zs )
      = ( ? [Us2: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
            ( ( Zs
              = ( append2865821678206098582on_nat @ Us2 @ Vs2 ) )
            & ( ( size_s7490768528479217773on_nat @ Us2 )
              = ( size_size_list_nat @ Xs ) )
            & ( ( size_s7490768528479217773on_nat @ Vs2 )
              = ( size_size_list_nat @ Ys ) )
            & ( list_a1865022572110992866on_nat @ P2 @ Xs @ Us2 )
            & ( list_a1865022572110992866on_nat @ P2 @ Ys @ Vs2 ) ) ) ) ).

% list_all2_append1
thf(fact_989_list__all2__append,axiom,
    ! [Xs: list_nat,Ys: list_nat,P2: nat > nat > $o,Us: list_nat,Vs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( list_all2_nat_nat @ P2 @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) )
        = ( ( list_all2_nat_nat @ P2 @ Xs @ Ys )
          & ( list_all2_nat_nat @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_990_list__all2__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat,P2: list_P3196869577824546119od_a_o > nat > $o,Us: list_l8336264072933094989od_a_o,Vs: list_nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( list_a3027799476954313168_o_nat @ P2 @ ( append2159120467191820514od_a_o @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) )
        = ( ( list_a3027799476954313168_o_nat @ P2 @ Xs @ Ys )
          & ( list_a3027799476954313168_o_nat @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_991_list__all2__append,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o,P2: nat > list_P3196869577824546119od_a_o > $o,Us: list_nat,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( list_a3410858264120222766od_a_o @ P2 @ ( append_nat @ Xs @ Us ) @ ( append2159120467191820514od_a_o @ Ys @ Vs ) )
        = ( ( list_a3410858264120222766od_a_o @ P2 @ Xs @ Ys )
          & ( list_a3410858264120222766od_a_o @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_992_list__all2__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat,P2: sum_su4907927656612468301on_nat > nat > $o,Us: list_S5253404628753938141on_nat,Vs: list_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( list_a5295772765654029478at_nat @ P2 @ ( append3523812819975775986on_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) )
        = ( ( list_a5295772765654029478at_nat @ P2 @ Xs @ Ys )
          & ( list_a5295772765654029478at_nat @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_993_list__all2__append,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat,P2: nat > sum_su4907927656612468301on_nat > $o,Us: list_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( list_a6193039792891997350on_nat @ P2 @ ( append_nat @ Xs @ Us ) @ ( append3523812819975775986on_nat @ Ys @ Vs ) )
        = ( ( list_a6193039792891997350on_nat @ P2 @ Xs @ Ys )
          & ( list_a6193039792891997350on_nat @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_994_list__all2__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Us: list_l8336264072933094989od_a_o,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( list_a6864181014682439495od_a_o @ P2 @ ( append2159120467191820514od_a_o @ Xs @ Us ) @ ( append2159120467191820514od_a_o @ Ys @ Vs ) )
        = ( ( list_a6864181014682439495od_a_o @ P2 @ Xs @ Ys )
          & ( list_a6864181014682439495od_a_o @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_995_list__all2__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o,P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Us: list_S5253404628753938141on_nat,Vs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ( list_a1062525910057997297od_a_o @ P2 @ ( append3523812819975775986on_nat @ Xs @ Us ) @ ( append2159120467191820514od_a_o @ Ys @ Vs ) )
        = ( ( list_a1062525910057997297od_a_o @ P2 @ Xs @ Ys )
          & ( list_a1062525910057997297od_a_o @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_996_list__all2__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat,P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Us: list_l8336264072933094989od_a_o,Vs: list_S5253404628753938141on_nat] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( list_a7531049758917964877on_nat @ P2 @ ( append2159120467191820514od_a_o @ Xs @ Us ) @ ( append3523812819975775986on_nat @ Ys @ Vs ) )
        = ( ( list_a7531049758917964877on_nat @ P2 @ Xs @ Ys )
          & ( list_a7531049758917964877on_nat @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_997_list__all2__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ( list_a1658445451649724195on_nat @ P2 @ ( append3523812819975775986on_nat @ Xs @ Us ) @ ( append3523812819975775986on_nat @ Ys @ Vs ) )
        = ( ( list_a1658445451649724195on_nat @ P2 @ Xs @ Ys )
          & ( list_a1658445451649724195on_nat @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_998_list__all2__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_nat,P2: produc1313295348606329723on_nat > nat > $o,Us: list_P9127870862658512897on_nat,Vs: list_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( list_a1112363324972753924at_nat @ P2 @ ( append2865821678206098582on_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) )
        = ( ( list_a1112363324972753924at_nat @ P2 @ Xs @ Ys )
          & ( list_a1112363324972753924at_nat @ P2 @ Us @ Vs ) ) ) ) ).

% list_all2_append
thf(fact_999_append__listrel1I,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat,Us: list_S5253404628753938141on_nat,Vs: list_S5253404628753938141on_nat] :
      ( ( ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Us @ Vs ) @ ( listre1270753918075970734on_nat @ R ) ) ) )
     => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( append3523812819975775986on_nat @ Xs @ Us ) @ ( append3523812819975775986on_nat @ Ys @ Vs ) ) @ ( listre1270753918075970734on_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_1000_append__listrel1I,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat,Us: list_nat,Vs: list_nat] :
      ( ( ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Us @ Vs ) @ ( listrel1_nat @ R ) ) ) )
     => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( append_nat @ Xs @ Us ) @ ( append_nat @ Ys @ Vs ) ) @ ( listrel1_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_1001_append__listrel1I,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat,Us: list_P9127870862658512897on_nat,Vs: list_P9127870862658512897on_nat] :
      ( ( ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) )
          & ( Us = Vs ) )
        | ( ( Xs = Ys )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Us @ Vs ) @ ( listre7129750038259667930on_nat @ R ) ) ) )
     => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( append2865821678206098582on_nat @ Xs @ Us ) @ ( append2865821678206098582on_nat @ Ys @ Vs ) ) @ ( listre7129750038259667930on_nat @ R ) ) ) ).

% append_listrel1I
thf(fact_1002_list__update__append1,axiom,
    ! [I: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X3: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ I @ X3 )
        = ( append2865821678206098582on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I @ X3 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1003_list__update__append1,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( list_u8577817230714425439on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ I @ X3 )
        = ( append3523812819975775986on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X3 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1004_list__update__append1,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,X3: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( list_u329291161401623989od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ I @ X3 )
        = ( append2159120467191820514od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X3 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1005_list__update__append1,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X3 )
        = ( append_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_1006__092_060open_062_Imset_Aleft_M_Amset_Aright_J_A_092_060in_062_Ans__mul__ext_A_Iset_ANS_J_A_Iset_AS_J_092_060close_062,axiom,
    member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ left ) @ ( mset_S7907735292211298936on_nat @ right ) ) @ ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ ns ) @ ( set_Pr937530902923597456on_nat @ s ) ) ).

% \<open>(mset left, mset right) \<in> ns_mul_ext (set NS) (set S)\<close>
thf(fact_1007__092_060open_062_Imset_Aleft_M_Amset_Aright_J_A_092_060in_062_Ans__mul__ext_A_Iset_ANSS_J_A_Iset_AS_J_092_060close_062,axiom,
    member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ left ) @ ( mset_S7907735292211298936on_nat @ right ) ) @ ( multis4183089776494585304on_nat @ ( set_Pr937530902923597456on_nat @ nss ) @ ( set_Pr937530902923597456on_nat @ s ) ) ).

% \<open>(mset left, mset right) \<in> ns_mul_ext (set NSS) (set S)\<close>
thf(fact_1008_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_1009__092_060open_062_Imset_Aleft_M_Amset_Aright_J_A_092_060in_062_Ans__mul__ext_A_Iset_ANSS_J_A_Iset_AS_J_A_092_060or_062_A_Imset_Aleft_M_Amset_Aright_J_A_092_060in_062_As__mul__ext_A_Iset_ANSS_J_A_Iset_AS_J_092_060close_062,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 ) ) ) ) ).

% \<open>(mset left, mset right) \<in> ns_mul_ext (set NSS) (set S) \<or> (mset left, mset right) \<in> s_mul_ext (set NSS) (set S)\<close>
thf(fact_1010__092_060open_062_092_060exists_062xs1_Axs2_Ays1_Ays2_O_Amset_Aleft_A_061_Amset_Axs1_A_L_Amset_Axs2_A_092_060and_062_Amset_Aright_A_061_Amset_Ays1_A_L_Amset_Ays2_A_092_060and_062_Alength_Axs1_A_061_Alength_Ays1_A_092_060and_062_A_I_092_060forall_062i_060length_Ays1_O_A_Ixs1_A_B_Ai_M_Ays1_A_B_Ai_J_A_092_060in_062_Aset_ANS_J_A_092_060and_062_A_I_092_060forall_062y_092_060in_062set_Ays2_O_A_092_060exists_062x_092_060in_062set_Axs2_O_A_Ix_M_Ay_J_A_092_060in_062_Aset_AS_J_092_060close_062,axiom,
    ? [Xs12: list_S5253404628753938141on_nat,Xs22: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ left )
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs12 ) @ ( mset_S7907735292211298936on_nat @ Xs22 ) ) )
      & ? [Ys1: list_S5253404628753938141on_nat,Ys22: list_S5253404628753938141on_nat] :
          ( ( ( mset_S7907735292211298936on_nat @ right )
            = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys1 ) @ ( mset_S7907735292211298936on_nat @ Ys22 ) ) )
          & ( ( size_s5686234912572693873on_nat @ Xs12 )
            = ( size_s5686234912572693873on_nat @ Ys1 ) )
          & ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ ( size_s5686234912572693873on_nat @ Ys1 ) )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs12 @ I4 ) @ ( nth_Su8471564416988646406on_nat @ Ys1 @ I4 ) ) @ ( set_Pr937530902923597456on_nat @ ns ) ) )
          & ! [X4: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
             => ? [Xa2: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Xa2 @ ( set_Su4848624856859743608on_nat @ Xs22 ) )
                  & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa2 @ X4 ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ) ) ).

% \<open>\<exists>xs1 xs2 ys1 ys2. mset left = mset xs1 + mset xs2 \<and> mset right = mset ys1 + mset ys2 \<and> length xs1 = length ys1 \<and> (\<forall>i<length ys1. (xs1 ! i, ys1 ! i) \<in> set NS) \<and> (\<forall>y\<in>set ys2. \<exists>x\<in>set xs2. (x, y) \<in> set S)\<close>
thf(fact_1011_id__right,axiom,
    ( ( mset_S7907735292211298936on_nat @ right )
    = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ ns_right ) @ ( mset_S7907735292211298936on_nat @ s_right ) ) ) ).

% id_right
thf(fact_1012_id__left,axiom,
    ( ( mset_S7907735292211298936on_nat @ left )
    = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ ns_left ) @ ( mset_S7907735292211298936on_nat @ s_left ) ) ) ).

% id_left
thf(fact_1013__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062ns__left_As__left_Ans__right_As__right_O_A_092_060lbrakk_062mset_Aleft_A_061_Amset_Ans__left_A_L_Amset_As__left_059_Amset_Aright_A_061_Amset_Ans__right_A_L_Amset_As__right_059_Alength_Ans__left_A_061_Alength_Ans__right_059_A_092_060And_062i_O_Ai_A_060_Alength_Ans__right_A_092_060Longrightarrow_062_A_Ins__left_A_B_Ai_M_Ans__right_A_B_Ai_J_A_092_060in_062_Aset_ANS_059_A_092_060And_062t_O_At_A_092_060in_062_Aset_As__right_A_092_060Longrightarrow_062_A_092_060exists_062s_092_060in_062set_As__left_O_A_Is_M_At_J_A_092_060in_062_Aset_AS_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Ns_left: list_S5253404628753938141on_nat,S_left: list_S5253404628753938141on_nat] :
        ( ( ( mset_S7907735292211298936on_nat @ left )
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ns_left ) @ ( mset_S7907735292211298936on_nat @ S_left ) ) )
       => ! [Ns_right: list_S5253404628753938141on_nat,S_right: list_S5253404628753938141on_nat] :
            ( ( ( mset_S7907735292211298936on_nat @ right )
              = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ns_right ) @ ( mset_S7907735292211298936on_nat @ S_right ) ) )
           => ( ( ( size_s5686234912572693873on_nat @ Ns_left )
                = ( size_s5686234912572693873on_nat @ Ns_right ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( size_s5686234912572693873on_nat @ Ns_right ) )
                   => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Ns_left @ I4 ) @ ( nth_Su8471564416988646406on_nat @ Ns_right @ I4 ) ) @ ( set_Pr937530902923597456on_nat @ ns ) ) )
               => ~ ! [T3: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ T3 @ ( set_Su4848624856859743608on_nat @ S_right ) )
                     => ? [X2: sum_su4907927656612468301on_nat] :
                          ( ( member8476201367316484836on_nat @ X2 @ ( set_Su4848624856859743608on_nat @ S_left ) )
                          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X2 @ T3 ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ) ) ) ) ).

% \<open>\<And>thesis. (\<And>ns_left s_left ns_right s_right. \<lbrakk>mset left = mset ns_left + mset s_left; mset right = mset ns_right + mset s_right; length ns_left = length ns_right; \<And>i. i < length ns_right \<Longrightarrow> (ns_left ! i, ns_right ! i) \<in> set NS; \<And>t. t \<in> set s_right \<Longrightarrow> \<exists>s\<in>set s_left. (s, t) \<in> set S\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_1014_all__ns__ns__mul__ext,axiom,
    ! [As: list_l1726306699111145479on_nat,Bs: list_l1726306699111145479on_nat,Ns: set_Pr8586234676504894535on_nat,S: set_Pr8586234676504894535on_nat] :
      ( ( ( size_s129639936549468787on_nat @ As )
        = ( size_s129639936549468787on_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Bs ) )
           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ As @ I2 ) @ ( nth_li1302981970230039432on_nat @ Bs @ I2 ) ) @ Ns ) )
       => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( mset_l2262431794812116886on_nat @ As ) @ ( mset_l2262431794812116886on_nat @ Bs ) ) @ ( multis664119935410451510on_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_1015_all__ns__ns__mul__ext,axiom,
    ! [As: list_l8336264072933094989od_a_o,Bs: list_l8336264072933094989od_a_o,Ns: set_Pr7420778282784443591od_a_o,S: set_Pr7420778282784443591od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ As )
        = ( size_s3379185846251495609od_a_o @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Bs ) )
           => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I2 ) ) @ Ns ) )
       => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ ( mset_l8139550355095933148od_a_o @ As ) @ ( mset_l8139550355095933148od_a_o @ Bs ) ) @ ( multis5290879534621336956od_a_o @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_1016_all__ns__ns__mul__ext,axiom,
    ! [As: list_nat,Bs: list_nat,Ns: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ Ns ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As ) @ ( mset_nat @ Bs ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_1017_all__ns__ns__mul__ext,axiom,
    ! [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat,Ns: set_Pr2535564044409765851on_nat,S: set_Pr2535564044409765851on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ As )
        = ( size_s5686234912572693873on_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Bs ) )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I2 ) ) @ Ns ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis4183089776494585304on_nat @ Ns @ S ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_1018_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_P9127870862658512897on_nat,Bs: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S4: set_Pr8202701763255893447on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ As )
        = ( size_s7490768528479217773on_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s7490768528479217773on_nat @ Bs ) )
           => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ As @ I2 ) @ ( nth_Pr3336893728888831618on_nat @ Bs @ I2 ) ) @ NS ) )
       => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ As @ Bs )
          @ ( collec5627915860394808530on_nat
            @ ( produc4129062015429275820_nat_o
              @ ^ [As2: list_P9127870862658512897on_nat,Bs2: list_P9127870862658512897on_nat] : ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ ( mset_P6964650582947932560on_nat @ As2 ) @ ( mset_P6964650582947932560on_nat @ Bs2 ) ) @ ( multis762940313840963120on_nat @ NS @ S4 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_1019_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_l1726306699111145479on_nat,Bs: list_l1726306699111145479on_nat,NS: set_Pr8586234676504894535on_nat,S4: set_Pr8586234676504894535on_nat] :
      ( ( ( size_s129639936549468787on_nat @ As )
        = ( size_s129639936549468787on_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Bs ) )
           => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ As @ I2 ) @ ( nth_li1302981970230039432on_nat @ Bs @ I2 ) ) @ NS ) )
       => ( member7797982458869347088on_nat @ ( produc4739740463051851607on_nat @ As @ Bs )
          @ ( collec6800664810089915218on_nat
            @ ( produc2872587184234751276_nat_o
              @ ^ [As2: list_l1726306699111145479on_nat,Bs2: list_l1726306699111145479on_nat] : ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ ( mset_l2262431794812116886on_nat @ As2 ) @ ( mset_l2262431794812116886on_nat @ Bs2 ) ) @ ( multis664119935410451510on_nat @ NS @ S4 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_1020_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_l8336264072933094989od_a_o,Bs: list_l8336264072933094989od_a_o,NS: set_Pr7420778282784443591od_a_o,S4: set_Pr7420778282784443591od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ As )
        = ( size_s3379185846251495609od_a_o @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Bs ) )
           => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I2 ) ) @ NS ) )
       => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As @ Bs )
          @ ( collec4835922272736728786od_a_o
            @ ( produc763134688753909420_a_o_o
              @ ^ [As2: list_l8336264072933094989od_a_o,Bs2: list_l8336264072933094989od_a_o] : ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ ( mset_l8139550355095933148od_a_o @ As2 ) @ ( mset_l8139550355095933148od_a_o @ Bs2 ) ) @ ( multis5290879534621336956od_a_o @ NS @ S4 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_1021_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I2 ) @ ( nth_nat @ Bs @ I2 ) ) @ NS ) )
       => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs )
          @ ( collec1570431334306492044st_nat
            @ ( produc6382697403170970154_nat_o
              @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis8831328596575508315xt_nat @ NS @ S4 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_1022_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S4: set_Pr2535564044409765851on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ As )
        = ( size_s5686234912572693873on_nat @ Bs ) )
     => ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Bs ) )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I2 ) ) @ NS ) )
       => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As @ Bs )
          @ ( collec1293568243156533638on_nat
            @ ( produc7163990014243126832_nat_o
              @ ^ [As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As2 ) @ ( mset_S7907735292211298936on_nat @ Bs2 ) ) @ ( multis4183089776494585304on_nat @ NS @ S4 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_1023_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_1024_ns__mul__ext__intro,axiom,
    ! [Xs: multis8267451372601328135od_a_o,Xs1: list_P3196869577824546119od_a_o,Xs23: list_P3196869577824546119od_a_o,Ys: multis8267451372601328135od_a_o,Ys12: list_P3196869577824546119od_a_o,Ys23: list_P3196869577824546119od_a_o,NS: set_Pr1246879249572150087od_a_o,S4: set_Pr1246879249572150087od_a_o] :
      ( ( Xs
        = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Xs1 ) @ ( mset_P8977776989772351958od_a_o @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Ys12 ) @ ( mset_P8977776989772351958od_a_o @ Ys23 ) ) )
       => ( ( ( size_s1727999245659094451od_a_o @ Xs1 )
            = ( size_s1727999245659094451od_a_o @ Ys12 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s1727999245659094451od_a_o @ Ys12 ) )
               => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ ( nth_Product_prod_a_o @ Xs1 @ I2 ) @ ( nth_Product_prod_a_o @ Ys12 @ I2 ) ) @ NS ) )
           => ( ! [Y3: product_prod_a_o] :
                  ( ( member2997933407038931562od_a_o @ Y3 @ ( set_Product_prod_a_o2 @ Ys23 ) )
                 => ? [X4: product_prod_a_o] :
                      ( ( member2997933407038931562od_a_o @ X4 @ ( set_Product_prod_a_o2 @ Xs23 ) )
                      & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X4 @ Y3 ) @ S4 ) ) )
             => ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ Xs @ Ys ) @ ( multis2022661252535332470od_a_o @ NS @ S4 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_1025_ns__mul__ext__intro,axiom,
    ! [Xs: multiset_a,Xs1: list_a,Xs23: list_a,Ys: multiset_a,Ys12: list_a,Ys23: list_a,NS: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
      ( ( Xs
        = ( plus_plus_multiset_a @ ( mset_a @ Xs1 ) @ ( mset_a @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_plus_multiset_a @ ( mset_a @ Ys12 ) @ ( mset_a @ Ys23 ) ) )
       => ( ( ( size_size_list_a @ Xs1 )
            = ( size_size_list_a @ Ys12 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys12 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs1 @ I2 ) @ ( nth_a @ Ys12 @ I2 ) ) @ NS ) )
           => ( ! [Y3: a] :
                  ( ( member_a2 @ Y3 @ ( set_a2 @ Ys23 ) )
                 => ? [X4: a] :
                      ( ( member_a2 @ X4 @ ( set_a2 @ Xs23 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X4 @ Y3 ) @ S4 ) ) )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multis4023584895248362547_ext_a @ NS @ S4 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_1026_ns__mul__ext__intro,axiom,
    ! [Xs: multis5168333519641040071on_nat,Xs1: list_l1726306699111145479on_nat,Xs23: list_l1726306699111145479on_nat,Ys: multis5168333519641040071on_nat,Ys12: list_l1726306699111145479on_nat,Ys23: list_l1726306699111145479on_nat,NS: set_Pr8586234676504894535on_nat,S4: set_Pr8586234676504894535on_nat] :
      ( ( Xs
        = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs1 ) @ ( mset_l2262431794812116886on_nat @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys12 ) @ ( mset_l2262431794812116886on_nat @ Ys23 ) ) )
       => ( ( ( size_s129639936549468787on_nat @ Xs1 )
            = ( size_s129639936549468787on_nat @ Ys12 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s129639936549468787on_nat @ Ys12 ) )
               => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs1 @ I2 ) @ ( nth_li1302981970230039432on_nat @ Ys12 @ I2 ) ) @ NS ) )
           => ( ! [Y3: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ Y3 @ ( set_li8154783781463198870on_nat @ Ys23 ) )
                 => ? [X4: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Xs23 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ X4 @ Y3 ) @ S4 ) ) )
             => ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Xs @ Ys ) @ ( multis664119935410451510on_nat @ NS @ S4 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_1027_ns__mul__ext__intro,axiom,
    ! [Xs: multis1714066462791002893od_a_o,Xs1: list_l8336264072933094989od_a_o,Xs23: list_l8336264072933094989od_a_o,Ys: multis1714066462791002893od_a_o,Ys12: list_l8336264072933094989od_a_o,Ys23: list_l8336264072933094989od_a_o,NS: set_Pr7420778282784443591od_a_o,S4: set_Pr7420778282784443591od_a_o] :
      ( ( Xs
        = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs1 ) @ ( mset_l8139550355095933148od_a_o @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Ys12 ) @ ( mset_l8139550355095933148od_a_o @ Ys23 ) ) )
       => ( ( ( size_s3379185846251495609od_a_o @ Xs1 )
            = ( size_s3379185846251495609od_a_o @ Ys12 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Ys12 ) )
               => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs1 @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Ys12 @ I2 ) ) @ NS ) )
           => ( ! [Y3: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Ys23 ) )
                 => ? [X4: list_P3196869577824546119od_a_o] :
                      ( ( member5261019209154752368od_a_o @ X4 @ ( set_li2629784761229999580od_a_o @ Xs23 ) )
                      & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X4 @ Y3 ) @ S4 ) ) )
             => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis5290879534621336956od_a_o @ NS @ S4 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_1028_ns__mul__ext__intro,axiom,
    ! [Xs: multiset_nat,Xs1: list_nat,Xs23: list_nat,Ys: multiset_nat,Ys12: list_nat,Ys23: list_nat,NS: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( Xs
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs1 ) @ ( mset_nat @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys12 ) @ ( mset_nat @ Ys23 ) ) )
       => ( ( ( size_size_list_nat @ Xs1 )
            = ( size_size_list_nat @ Ys12 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys12 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I2 ) @ ( nth_nat @ Ys12 @ I2 ) ) @ NS ) )
           => ( ! [Y3: nat] :
                  ( ( member_nat2 @ Y3 @ ( set_nat2 @ Ys23 ) )
                 => ? [X4: nat] :
                      ( ( member_nat2 @ X4 @ ( set_nat2 @ Xs23 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y3 ) @ S4 ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S4 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_1029_ns__mul__ext__intro,axiom,
    ! [Xs: multis1856558993800614429on_nat,Xs1: list_S5253404628753938141on_nat,Xs23: list_S5253404628753938141on_nat,Ys: multis1856558993800614429on_nat,Ys12: list_S5253404628753938141on_nat,Ys23: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S4: set_Pr2535564044409765851on_nat] :
      ( ( Xs
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs1 ) @ ( mset_S7907735292211298936on_nat @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys12 ) @ ( mset_S7907735292211298936on_nat @ Ys23 ) ) )
       => ( ( ( size_s5686234912572693873on_nat @ Xs1 )
            = ( size_s5686234912572693873on_nat @ Ys12 ) )
         => ( ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Ys12 ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs1 @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Ys12 @ I2 ) ) @ NS ) )
           => ( ! [Y3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Ys23 ) )
                 => ? [X4: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs23 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X4 @ Y3 ) @ S4 ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis4183089776494585304on_nat @ NS @ S4 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_1030_ns__mul__ext__elim,axiom,
    ! [Xs: multis8267451372601328135od_a_o,Ys: multis8267451372601328135od_a_o,NS: set_Pr1246879249572150087od_a_o,S4: set_Pr1246879249572150087od_a_o] :
      ( ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ Xs @ Ys ) @ ( multis2022661252535332470od_a_o @ NS @ S4 ) )
     => ? [Xs12: list_P3196869577824546119od_a_o,Xs22: list_P3196869577824546119od_a_o] :
          ( ( Xs
            = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Xs12 ) @ ( mset_P8977776989772351958od_a_o @ Xs22 ) ) )
          & ? [Ys1: list_P3196869577824546119od_a_o,Ys22: list_P3196869577824546119od_a_o] :
              ( ( Ys
                = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Ys1 ) @ ( mset_P8977776989772351958od_a_o @ Ys22 ) ) )
              & ( ( size_s1727999245659094451od_a_o @ Xs12 )
                = ( size_s1727999245659094451od_a_o @ Ys1 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s1727999245659094451od_a_o @ Ys1 ) )
                 => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ ( nth_Product_prod_a_o @ Xs12 @ I4 ) @ ( nth_Product_prod_a_o @ Ys1 @ I4 ) ) @ NS ) )
              & ! [X4: product_prod_a_o] :
                  ( ( member2997933407038931562od_a_o @ X4 @ ( set_Product_prod_a_o2 @ Ys22 ) )
                 => ? [Xa2: product_prod_a_o] :
                      ( ( member2997933407038931562od_a_o @ Xa2 @ ( set_Product_prod_a_o2 @ Xs22 ) )
                      & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ Xa2 @ X4 ) @ S4 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_1031_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_a,Ys: multiset_a,NS: set_Product_prod_a_a,S4: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multis4023584895248362547_ext_a @ NS @ S4 ) )
     => ? [Xs12: list_a,Xs22: list_a] :
          ( ( Xs
            = ( plus_plus_multiset_a @ ( mset_a @ Xs12 ) @ ( mset_a @ Xs22 ) ) )
          & ? [Ys1: list_a,Ys22: list_a] :
              ( ( Ys
                = ( plus_plus_multiset_a @ ( mset_a @ Ys1 ) @ ( mset_a @ Ys22 ) ) )
              & ( ( size_size_list_a @ Xs12 )
                = ( size_size_list_a @ Ys1 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_a @ Ys1 ) )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs12 @ I4 ) @ ( nth_a @ Ys1 @ I4 ) ) @ NS ) )
              & ! [X4: a] :
                  ( ( member_a2 @ X4 @ ( set_a2 @ Ys22 ) )
                 => ? [Xa2: a] :
                      ( ( member_a2 @ Xa2 @ ( set_a2 @ Xs22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa2 @ X4 ) @ S4 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_1032_ns__mul__ext__elim,axiom,
    ! [Xs: multis5168333519641040071on_nat,Ys: multis5168333519641040071on_nat,NS: set_Pr8586234676504894535on_nat,S4: set_Pr8586234676504894535on_nat] :
      ( ( member6183217143914783504on_nat @ ( produc3872347347025921879on_nat @ Xs @ Ys ) @ ( multis664119935410451510on_nat @ NS @ S4 ) )
     => ? [Xs12: list_l1726306699111145479on_nat,Xs22: list_l1726306699111145479on_nat] :
          ( ( Xs
            = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Xs12 ) @ ( mset_l2262431794812116886on_nat @ Xs22 ) ) )
          & ? [Ys1: list_l1726306699111145479on_nat,Ys22: list_l1726306699111145479on_nat] :
              ( ( Ys
                = ( plus_p7957816880051667710on_nat @ ( mset_l2262431794812116886on_nat @ Ys1 ) @ ( mset_l2262431794812116886on_nat @ Ys22 ) ) )
              & ( ( size_s129639936549468787on_nat @ Xs12 )
                = ( size_s129639936549468787on_nat @ Ys1 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s129639936549468787on_nat @ Ys1 ) )
                 => ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( nth_li1302981970230039432on_nat @ Xs12 @ I4 ) @ ( nth_li1302981970230039432on_nat @ Ys1 @ I4 ) ) @ NS ) )
              & ! [X4: list_P9127870862658512897on_nat] :
                  ( ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Ys22 ) )
                 => ? [Xa2: list_P9127870862658512897on_nat] :
                      ( ( member783415501022416426on_nat @ Xa2 @ ( set_li8154783781463198870on_nat @ Xs22 ) )
                      & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xa2 @ X4 ) @ S4 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_1033_ns__mul__ext__elim,axiom,
    ! [Xs: multis1714066462791002893od_a_o,Ys: multis1714066462791002893od_a_o,NS: set_Pr7420778282784443591od_a_o,S4: set_Pr7420778282784443591od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis5290879534621336956od_a_o @ NS @ S4 ) )
     => ? [Xs12: list_l8336264072933094989od_a_o,Xs22: list_l8336264072933094989od_a_o] :
          ( ( Xs
            = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs12 ) @ ( mset_l8139550355095933148od_a_o @ Xs22 ) ) )
          & ? [Ys1: list_l8336264072933094989od_a_o,Ys22: list_l8336264072933094989od_a_o] :
              ( ( Ys
                = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Ys1 ) @ ( mset_l8139550355095933148od_a_o @ Ys22 ) ) )
              & ( ( size_s3379185846251495609od_a_o @ Xs12 )
                = ( size_s3379185846251495609od_a_o @ Ys1 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s3379185846251495609od_a_o @ Ys1 ) )
                 => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs12 @ I4 ) @ ( nth_li3248255784598043854od_a_o @ Ys1 @ I4 ) ) @ NS ) )
              & ! [X4: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ X4 @ ( set_li2629784761229999580od_a_o @ Ys22 ) )
                 => ? [Xa2: list_P3196869577824546119od_a_o] :
                      ( ( member5261019209154752368od_a_o @ Xa2 @ ( set_li2629784761229999580od_a_o @ Xs22 ) )
                      & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ Xa2 @ X4 ) @ S4 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_1034_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S4: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S4 ) )
     => ? [Xs12: list_nat,Xs22: list_nat] :
          ( ( Xs
            = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs12 ) @ ( mset_nat @ Xs22 ) ) )
          & ? [Ys1: list_nat,Ys22: list_nat] :
              ( ( Ys
                = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys1 ) @ ( mset_nat @ Ys22 ) ) )
              & ( ( size_size_list_nat @ Xs12 )
                = ( size_size_list_nat @ Ys1 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Ys1 ) )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs12 @ I4 ) @ ( nth_nat @ Ys1 @ I4 ) ) @ NS ) )
              & ! [X4: nat] :
                  ( ( member_nat2 @ X4 @ ( set_nat2 @ Ys22 ) )
                 => ? [Xa2: nat] :
                      ( ( member_nat2 @ Xa2 @ ( set_nat2 @ Xs22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa2 @ X4 ) @ S4 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_1035_ns__mul__ext__elim,axiom,
    ! [Xs: multis1856558993800614429on_nat,Ys: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S4: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis4183089776494585304on_nat @ NS @ S4 ) )
     => ? [Xs12: list_S5253404628753938141on_nat,Xs22: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs12 ) @ ( mset_S7907735292211298936on_nat @ Xs22 ) ) )
          & ? [Ys1: list_S5253404628753938141on_nat,Ys22: list_S5253404628753938141on_nat] :
              ( ( Ys
                = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys1 ) @ ( mset_S7907735292211298936on_nat @ Ys22 ) ) )
              & ( ( size_s5686234912572693873on_nat @ Xs12 )
                = ( size_s5686234912572693873on_nat @ Ys1 ) )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s5686234912572693873on_nat @ Ys1 ) )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs12 @ I4 ) @ ( nth_Su8471564416988646406on_nat @ Ys1 @ I4 ) ) @ NS ) )
              & ! [X4: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
                 => ? [Xa2: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Xa2 @ ( set_Su4848624856859743608on_nat @ Xs22 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa2 @ X4 ) @ S4 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_1036__092_060open_062Inr_Ai_A_092_060in_062_D_Amset_Ans__right_A_L_Amset_As__right_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( sum_In255631617079920996tation @ i ) @ ( set_ms3922534532180125136on_nat @ ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ ns_right ) @ ( mset_S7907735292211298936on_nat @ s_right ) ) ) ).

% \<open>Inr i \<in># mset ns_right + mset s_right\<close>
thf(fact_1037_mset__swap,axiom,
    ! [I: nat,Ls: list_S5253404628753938141on_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( 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 @ I ) ) @ I @ ( nth_Su8471564416988646406on_nat @ Ls @ J ) ) )
          = ( mset_S7907735292211298936on_nat @ Ls ) ) ) ) ).

% mset_swap
thf(fact_1038_mset__swap,axiom,
    ! [I: nat,Ls: list_l8336264072933094989od_a_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Ls ) )
     => ( ( ord_less_nat @ J @ ( size_s3379185846251495609od_a_o @ Ls ) )
       => ( ( mset_l8139550355095933148od_a_o @ ( list_u329291161401623989od_a_o @ ( list_u329291161401623989od_a_o @ Ls @ J @ ( nth_li3248255784598043854od_a_o @ Ls @ I ) ) @ I @ ( nth_li3248255784598043854od_a_o @ Ls @ J ) ) )
          = ( mset_l8139550355095933148od_a_o @ Ls ) ) ) ) ).

% mset_swap
thf(fact_1039_mset__swap,axiom,
    ! [I: nat,Ls: list_nat,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ls ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_nat @ Ls ) )
       => ( ( mset_nat @ ( list_update_nat @ ( list_update_nat @ Ls @ J @ ( nth_nat @ Ls @ I ) ) @ I @ ( nth_nat @ Ls @ J ) ) )
          = ( mset_nat @ Ls ) ) ) ) ).

% mset_swap
thf(fact_1040_append_Oright__neutral,axiom,
    ! [A: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ A @ nil_li1372968520647675725od_a_o )
      = A ) ).

% append.right_neutral
thf(fact_1041_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_1042_append_Oright__neutral,axiom,
    ! [A: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ A @ nil_Su3078906344068544711on_nat )
      = A ) ).

% append.right_neutral
thf(fact_1043_append_Oright__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ A @ nil_Pr8614093721167239297on_nat )
      = A ) ).

% append.right_neutral
thf(fact_1044_append__Nil2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ Xs @ nil_li1372968520647675725od_a_o )
      = Xs ) ).

% append_Nil2
thf(fact_1045_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_1046_append__Nil2,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ Xs @ nil_Su3078906344068544711on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_1047_append__Nil2,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ Xs @ nil_Pr8614093721167239297on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_1048_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_1049_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_1050_append__self__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_Su3078906344068544711on_nat ) ) ).

% append_self_conv
thf(fact_1051_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_1052_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_1053_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_1054_self__append__conv,axiom,
    ! [Y: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( Y
        = ( append3523812819975775986on_nat @ Y @ Ys ) )
      = ( Ys = nil_Su3078906344068544711on_nat ) ) ).

% self_append_conv
thf(fact_1055_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_1056_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_1057_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_1058_append__self__conv2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% append_self_conv2
thf(fact_1059_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_1060_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_1061_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_1062_self__append__conv2,axiom,
    ! [Y: list_S5253404628753938141on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( Y
        = ( append3523812819975775986on_nat @ Xs @ Y ) )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% self_append_conv2
thf(fact_1063_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_1064_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_1065_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_1066_Nil__is__append__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( nil_Su3078906344068544711on_nat
        = ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        & ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% Nil_is_append_conv
thf(fact_1067_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_1068_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_1069_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_1070_append__is__Nil__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ Ys )
        = nil_Su3078906344068544711on_nat )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        & ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% append_is_Nil_conv
thf(fact_1071_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_1072_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_1073_length__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% length_append
thf(fact_1074_length__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% length_append
thf(fact_1075_length__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( size_size_list_nat @ ( append_nat @ Xs @ Ys ) )
      = ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_append
thf(fact_1076_list__all2__Nil,axiom,
    ! [P2: nat > nat > $o,Ys: list_nat] :
      ( ( list_all2_nat_nat @ P2 @ nil_nat @ Ys )
      = ( Ys = nil_nat ) ) ).

% list_all2_Nil
thf(fact_1077_list__all2__Nil,axiom,
    ! [P2: list_P3196869577824546119od_a_o > nat > $o,Ys: list_nat] :
      ( ( list_a3027799476954313168_o_nat @ P2 @ nil_li1372968520647675725od_a_o @ Ys )
      = ( Ys = nil_nat ) ) ).

% list_all2_Nil
thf(fact_1078_list__all2__Nil,axiom,
    ! [P2: nat > list_P3196869577824546119od_a_o > $o,Ys: list_l8336264072933094989od_a_o] :
      ( ( list_a3410858264120222766od_a_o @ P2 @ nil_nat @ Ys )
      = ( Ys = nil_li1372968520647675725od_a_o ) ) ).

% list_all2_Nil
thf(fact_1079_list__all2__Nil,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,Ys: list_S5253404628753938141on_nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ nil_nat @ Ys )
      = ( Ys = nil_Su3078906344068544711on_nat ) ) ).

% list_all2_Nil
thf(fact_1080_list__all2__Nil,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,Ys: list_nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ nil_Su3078906344068544711on_nat @ Ys )
      = ( Ys = nil_nat ) ) ).

% list_all2_Nil
thf(fact_1081_list__all2__Nil,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Ys: list_l8336264072933094989od_a_o] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ nil_li1372968520647675725od_a_o @ Ys )
      = ( Ys = nil_li1372968520647675725od_a_o ) ) ).

% list_all2_Nil
thf(fact_1082_list__all2__Nil,axiom,
    ! [P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Ys: list_S5253404628753938141on_nat] :
      ( ( list_a7531049758917964877on_nat @ P2 @ nil_li1372968520647675725od_a_o @ Ys )
      = ( Ys = nil_Su3078906344068544711on_nat ) ) ).

% list_all2_Nil
thf(fact_1083_list__all2__Nil,axiom,
    ! [P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Ys: list_l8336264072933094989od_a_o] :
      ( ( list_a1062525910057997297od_a_o @ P2 @ nil_Su3078906344068544711on_nat @ Ys )
      = ( Ys = nil_li1372968520647675725od_a_o ) ) ).

% list_all2_Nil
thf(fact_1084_list__all2__Nil,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Ys: list_S5253404628753938141on_nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ nil_Su3078906344068544711on_nat @ Ys )
      = ( Ys = nil_Su3078906344068544711on_nat ) ) ).

% list_all2_Nil
thf(fact_1085_list__all2__Nil,axiom,
    ! [P2: nat > produc1313295348606329723on_nat > $o,Ys: list_P9127870862658512897on_nat] :
      ( ( list_a1865022572110992866on_nat @ P2 @ nil_nat @ Ys )
      = ( Ys = nil_Pr8614093721167239297on_nat ) ) ).

% list_all2_Nil
thf(fact_1086_list__all2__Nil2,axiom,
    ! [P2: nat > nat > $o,Xs: list_nat] :
      ( ( list_all2_nat_nat @ P2 @ Xs @ nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_all2_Nil2
thf(fact_1087_list__all2__Nil2,axiom,
    ! [P2: nat > list_P3196869577824546119od_a_o > $o,Xs: list_nat] :
      ( ( list_a3410858264120222766od_a_o @ P2 @ Xs @ nil_li1372968520647675725od_a_o )
      = ( Xs = nil_nat ) ) ).

% list_all2_Nil2
thf(fact_1088_list__all2__Nil2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > nat > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ( list_a3027799476954313168_o_nat @ P2 @ Xs @ nil_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% list_all2_Nil2
thf(fact_1089_list__all2__Nil2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ( list_a5295772765654029478at_nat @ P2 @ Xs @ nil_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% list_all2_Nil2
thf(fact_1090_list__all2__Nil2,axiom,
    ! [P2: nat > sum_su4907927656612468301on_nat > $o,Xs: list_nat] :
      ( ( list_a6193039792891997350on_nat @ P2 @ Xs @ nil_Su3078906344068544711on_nat )
      = ( Xs = nil_nat ) ) ).

% list_all2_Nil2
thf(fact_1091_list__all2__Nil2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ( list_a6864181014682439495od_a_o @ P2 @ Xs @ nil_li1372968520647675725od_a_o )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% list_all2_Nil2
thf(fact_1092_list__all2__Nil2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o > $o,Xs: list_S5253404628753938141on_nat] :
      ( ( list_a1062525910057997297od_a_o @ P2 @ Xs @ nil_li1372968520647675725od_a_o )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% list_all2_Nil2
thf(fact_1093_list__all2__Nil2,axiom,
    ! [P2: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ( list_a7531049758917964877on_nat @ P2 @ Xs @ nil_Su3078906344068544711on_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% list_all2_Nil2
thf(fact_1094_list__all2__Nil2,axiom,
    ! [P2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ( list_a1658445451649724195on_nat @ P2 @ Xs @ nil_Su3078906344068544711on_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% list_all2_Nil2
thf(fact_1095_list__all2__Nil2,axiom,
    ! [P2: produc1313295348606329723on_nat > nat > $o,Xs: list_P9127870862658512897on_nat] :
      ( ( list_a1112363324972753924at_nat @ P2 @ Xs @ nil_nat )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% list_all2_Nil2
thf(fact_1096_zip__Nil,axiom,
    ! [Ys: list_S5253404628753938141on_nat] :
      ( ( zip_Su8130003265398491731on_nat @ nil_Su3078906344068544711on_nat @ Ys )
      = nil_Pr8614093721167239297on_nat ) ).

% zip_Nil
thf(fact_1097_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( nil_Pr5478986624290739719at_nat
        = ( zip_nat_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1098_Nil__eq__zip__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat] :
      ( ( nil_Pr367109023944153152_o_nat
        = ( zip_li1039786862263726752_o_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1099_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( nil_Pr1392216289592255974od_a_o
        = ( zip_na1422845649429636350od_a_o @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1100_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( nil_Pr4053236792990210564on_nat
        = ( zip_na307675373166331862on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1101_Nil__eq__zip__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat] :
      ( ( nil_Pr5520986166334685316at_nat
        = ( zip_Su8633780382783139798at_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        | ( Ys = nil_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1102_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_1103_Nil__eq__zip__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat] :
      ( ( nil_Pr3096418252878553149on_nat
        = ( zip_li8832488730364818717on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        | ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1104_Nil__eq__zip__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( nil_Pr8858331319937997225od_a_o
        = ( zip_Su2363964881504851137od_a_o @ Xs @ Ys ) )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        | ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1105_Nil__eq__zip__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( nil_Pr8614093721167239297on_nat
        = ( zip_Su8130003265398491731on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        | ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1106_Nil__eq__zip__iff,axiom,
    ! [Xs: list_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( nil_Pr7411674412305980442on_nat
        = ( zip_na7277209975152967346on_nat @ Xs @ Ys ) )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr8614093721167239297on_nat ) ) ) ).

% Nil_eq_zip_iff
thf(fact_1107_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( zip_nat_nat @ Xs @ Ys )
        = nil_Pr5478986624290739719at_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1108_zip__eq__Nil__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_nat] :
      ( ( ( zip_li1039786862263726752_o_nat @ Xs @ Ys )
        = nil_Pr367109023944153152_o_nat )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1109_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( zip_na1422845649429636350od_a_o @ Xs @ Ys )
        = nil_Pr1392216289592255974od_a_o )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1110_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( zip_na307675373166331862on_nat @ Xs @ Ys )
        = nil_Pr4053236792990210564on_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1111_zip__eq__Nil__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_nat] :
      ( ( ( zip_Su8633780382783139798at_nat @ Xs @ Ys )
        = nil_Pr5520986166334685316at_nat )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        | ( Ys = nil_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1112_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_1113_zip__eq__Nil__iff,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat] :
      ( ( ( zip_li8832488730364818717on_nat @ Xs @ Ys )
        = nil_Pr3096418252878553149on_nat )
      = ( ( Xs = nil_li1372968520647675725od_a_o )
        | ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1114_zip__eq__Nil__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( zip_Su2363964881504851137od_a_o @ Xs @ Ys )
        = nil_Pr8858331319937997225od_a_o )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        | ( Ys = nil_li1372968520647675725od_a_o ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1115_zip__eq__Nil__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( zip_Su8130003265398491731on_nat @ Xs @ Ys )
        = nil_Pr8614093721167239297on_nat )
      = ( ( Xs = nil_Su3078906344068544711on_nat )
        | ( Ys = nil_Su3078906344068544711on_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1116_zip__eq__Nil__iff,axiom,
    ! [Xs: list_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( zip_na7277209975152967346on_nat @ Xs @ Ys )
        = nil_Pr7411674412305980442on_nat )
      = ( ( Xs = nil_nat )
        | ( Ys = nil_Pr8614093721167239297on_nat ) ) ) ).

% zip_eq_Nil_iff
thf(fact_1117__092_060open_062Inr_Ai_A_092_060in_062_D_Amset_Aright_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( sum_In255631617079920996tation @ i ) @ ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ right ) ) ).

% \<open>Inr i \<in># mset right\<close>
thf(fact_1118_list__update__nonempty,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,K: nat,X3: list_P3196869577824546119od_a_o] :
      ( ( ( list_u329291161401623989od_a_o @ Xs @ K @ X3 )
        = nil_li1372968520647675725od_a_o )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% list_update_nonempty
thf(fact_1119_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X3: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X3 )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_1120_list__update__nonempty,axiom,
    ! [Xs: list_S5253404628753938141on_nat,K: nat,X3: sum_su4907927656612468301on_nat] :
      ( ( ( list_u8577817230714425439on_nat @ Xs @ K @ X3 )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% list_update_nonempty
thf(fact_1121_list__update__nonempty,axiom,
    ! [Xs: list_P9127870862658512897on_nat,K: nat,X3: produc1313295348606329723on_nat] :
      ( ( ( list_u1840724836389618409on_nat @ Xs @ K @ X3 )
        = nil_Pr8614093721167239297on_nat )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% list_update_nonempty
thf(fact_1122_list__ex__simps_I2_J,axiom,
    ! [P2: list_P3196869577824546119od_a_o > $o] :
      ~ ( list_e5403782194016407179od_a_o @ P2 @ nil_li1372968520647675725od_a_o ) ).

% list_ex_simps(2)
thf(fact_1123_list__ex__simps_I2_J,axiom,
    ! [P2: nat > $o] :
      ~ ( list_ex_nat @ P2 @ nil_nat ) ).

% list_ex_simps(2)
thf(fact_1124_list__ex__simps_I2_J,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o] :
      ~ ( list_e3769318529992910857on_nat @ P2 @ nil_Su3078906344068544711on_nat ) ).

% list_ex_simps(2)
thf(fact_1125_list__ex__simps_I2_J,axiom,
    ! [P2: produc1313295348606329723on_nat > $o] :
      ~ ( list_e8248363684495001279on_nat @ P2 @ nil_Pr8614093721167239297on_nat ) ).

% list_ex_simps(2)
thf(fact_1126_set__mset__mset,axiom,
    ! [Xs: list_P3196869577824546119od_a_o] :
      ( ( set_ms2137302536914688126od_a_o @ ( mset_P8977776989772351958od_a_o @ Xs ) )
      = ( set_Product_prod_a_o2 @ Xs ) ) ).

% set_mset_mset
thf(fact_1127_set__mset__mset,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( set_ms4142991692181744516od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs ) )
      = ( set_li2629784761229999580od_a_o @ Xs ) ) ).

% set_mset_mset
thf(fact_1128_set__mset__mset,axiom,
    ! [Xs: list_nat] :
      ( ( set_mset_nat @ ( mset_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_mset_mset
thf(fact_1129_set__mset__mset,axiom,
    ! [Xs: list_a] :
      ( ( set_mset_a @ ( mset_a @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% set_mset_mset
thf(fact_1130_set__mset__mset,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) )
      = ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% set_mset_mset
thf(fact_1131_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_1132_nth__append__length__plus,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,N: nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s5686234912572693873on_nat @ Xs ) @ N ) )
      = ( nth_Su8471564416988646406on_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_1133_nth__append__length__plus,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,N: nat] :
      ( ( nth_li3248255784598043854od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_s3379185846251495609od_a_o @ Xs ) @ N ) )
      = ( nth_li3248255784598043854od_a_o @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_1134_nth__append__length__plus,axiom,
    ! [Xs: list_nat,Ys: list_nat,N: nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ Ys ) @ ( plus_plus_nat @ ( size_size_list_nat @ Xs ) @ N ) )
      = ( nth_nat @ Ys @ N ) ) ).

% nth_append_length_plus
thf(fact_1135_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_1136_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1137_eq__Nil__appendI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_1138_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_1139_append_Oleft__neutral,axiom,
    ! [A: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ nil_li1372968520647675725od_a_o @ A )
      = A ) ).

% append.left_neutral
thf(fact_1140_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_1141_append_Oleft__neutral,axiom,
    ! [A: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_1142_append_Oleft__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_1143_append__Nil,axiom,
    ! [Ys: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ nil_li1372968520647675725od_a_o @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1144_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1145_append__Nil,axiom,
    ! [Ys: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1146_append__Nil,axiom,
    ! [Ys: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_1147_nat__add__left__cancel__less,axiom,
    ! [K: nat,M3: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M3 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_1148_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M3: nat,K: nat] :
      ( ! [M4: nat,N4: nat] :
          ( ( ord_less_nat @ M4 @ N4 )
         => ( ord_less_nat @ ( F @ M4 ) @ ( F @ N4 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M3 ) @ K ) @ ( F @ ( plus_plus_nat @ M3 @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_1149_linorder__neqE__nat,axiom,
    ! [X3: nat,Y: nat] :
      ( ( X3 != Y )
     => ( ~ ( ord_less_nat @ X3 @ Y )
       => ( ord_less_nat @ Y @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_1150_infinite__descent,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ~ ( P2 @ N4 )
         => ? [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
              & ~ ( P2 @ M5 ) ) )
     => ( P2 @ N ) ) ).

% infinite_descent
thf(fact_1151_nat__less__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M5: nat] :
              ( ( ord_less_nat @ M5 @ N4 )
             => ( P2 @ M5 ) )
         => ( P2 @ N4 ) )
     => ( P2 @ N ) ) ).

% nat_less_induct
thf(fact_1152_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_1153_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_1154_less__not__refl2,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_nat @ N @ M3 )
     => ( M3 != N ) ) ).

% less_not_refl2
thf(fact_1155_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_1156_nat__neq__iff,axiom,
    ! [M3: nat,N: nat] :
      ( ( M3 != N )
      = ( ( ord_less_nat @ M3 @ N )
        | ( ord_less_nat @ N @ M3 ) ) ) ).

% nat_neq_iff
thf(fact_1157_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I2: nat,J3: nat] :
          ( ( ord_less_nat @ I2 @ J3 )
         => ( ord_less_nat @ ( F @ I2 ) @ ( F @ J3 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1158_le__neq__implies__less,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ( M3 != N )
       => ( ord_less_nat @ M3 @ N ) ) ) ).

% le_neq_implies_less
thf(fact_1159_less__or__eq__imp__le,axiom,
    ! [M3: nat,N: nat] :
      ( ( ( ord_less_nat @ M3 @ N )
        | ( M3 = N ) )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% less_or_eq_imp_le
thf(fact_1160_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N3: nat] :
          ( ( ord_less_nat @ M2 @ N3 )
          | ( M2 = N3 ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1161_less__imp__le__nat,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ord_less_eq_nat @ M3 @ N ) ) ).

% less_imp_le_nat
thf(fact_1162_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M2 @ N3 )
          & ( M2 != N3 ) ) ) ) ).

% nat_less_le
thf(fact_1163_less__add__eq__less,axiom,
    ! [K: nat,L3: nat,M3: nat,N: nat] :
      ( ( ord_less_nat @ K @ L3 )
     => ( ( ( plus_plus_nat @ M3 @ L3 )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M3 @ N ) ) ) ).

% less_add_eq_less
thf(fact_1164_trans__less__add2,axiom,
    ! [I: nat,J: nat,M3: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M3 @ J ) ) ) ).

% trans_less_add2
thf(fact_1165_trans__less__add1,axiom,
    ! [I: nat,J: nat,M3: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M3 ) ) ) ).

% trans_less_add1
thf(fact_1166_add__less__mono1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_1167_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_1168_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_1169_add__less__mono,axiom,
    ! [I: nat,J: nat,K: nat,L3: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ K @ L3 )
       => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ) ).

% add_less_mono
thf(fact_1170_add__lessD1,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ K )
     => ( ord_less_nat @ I @ K ) ) ).

% add_lessD1
thf(fact_1171_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_1172_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_1173_bot__nat__0_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ A ) ) ).

% bot_nat_0.not_eq_extremum
thf(fact_1174_add__gr__0,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M3 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M3 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_1175_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P2 @ N4 )
             => ? [M5: nat] :
                  ( ( ord_less_nat @ M5 @ N4 )
                  & ~ ( P2 @ M5 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_1176_gr__implies__not0,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1177_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1178_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_1179_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1180_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_1181_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1182_ex__least__nat__le,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1183_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1184_bot__nat__0_Oordering__top__axioms,axiom,
    ( ordering_top_nat
    @ ^ [X: nat,Y4: nat] : ( ord_less_eq_nat @ Y4 @ X )
    @ ^ [X: nat,Y4: nat] : ( ord_less_nat @ Y4 @ X )
    @ zero_zero_nat ) ).

% bot_nat_0.ordering_top_axioms
thf(fact_1185_S,axiom,
    ( s
    = ( concat937493773106420796on_nat
      @ ( map_na8715451351012060651on_nat
        @ ^ [I3: nat] :
            ( map_Pr6945619049609468478on_nat
            @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I3 ) )
            @ ( nth_li3248255784598043854od_a_o @ cnf @ I3 ) )
        @ cs ) ) ) ).

% S
thf(fact_1186_cs__def,axiom,
    ( cs
    = ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ).

% cs_def
thf(fact_1187_right,axiom,
    ( right
    = ( append3523812819975775986on_nat
      @ ( map_a_2711072540680412167on_nat
        @ ^ [X: a] : ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) )
        @ xs )
      @ ( map_na5706685718985795555on_nat @ sum_In255631617079920996tation @ cs ) ) ) ).

% right
thf(fact_1188_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_1189_upt__eq__Nil__conv,axiom,
    ! [I: nat,J: nat] :
      ( ( ( upt @ I @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I ) ) ) ).

% upt_eq_Nil_conv
thf(fact_1190_nth__upt,axiom,
    ! [I: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I @ J ) @ K )
        = ( plus_plus_nat @ I @ K ) ) ) ).

% nth_upt
thf(fact_1191_map__add__upt,axiom,
    ! [N: nat,M3: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N )
        @ ( upt @ zero_zero_nat @ M3 ) )
      = ( upt @ N @ ( plus_plus_nat @ M3 @ N ) ) ) ).

% map_add_upt
thf(fact_1192_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_1193_Annotation_Osize_I4_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size(4)
thf(fact_1194_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_1195_upt__add__eq__append,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_1196_res,axiom,
    ( ( produc1545487600272464053on_nat
      @ ( concat6161509665823917772on_nat
        @ ( map_a_3887198048842487703on_nat
          @ ^ [X: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
          @ xs ) )
      @ ( produc4176415733022414961on_nat
        @ ( append3523812819975775986on_nat
          @ ( map_a_2711072540680412167on_nat
            @ ^ [X: a] : ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) )
            @ xs )
          @ ( map_na5706685718985795555on_nat @ sum_In255631617079920996tation @ cs ) )
        @ ( produc7152764266822961111on_nat
          @ ( concat937493773106420796on_nat
            @ ( map_na8715451351012060651on_nat
              @ ^ [I3: nat] :
                  ( map_Pr6945619049609468478on_nat
                  @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I3 ) )
                  @ ( nth_li3248255784598043854od_a_o @ cnf @ I3 ) )
              @ cs ) )
          @ ( append2865821678206098582on_nat
            @ ( concat937493773106420796on_nat
              @ ( map_a_7431629901395152583on_nat
                @ ^ [X: a] : ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ nil_Pr8614093721167239297on_nat ) )
                @ xs ) )
            @ ( concat937493773106420796on_nat
              @ ( map_na8715451351012060651on_nat
                @ ^ [I3: nat] :
                    ( map_Pr6945619049609468478on_nat
                    @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I3 ) )
                    @ ( nth_li3248255784598043854od_a_o @ cnf @ I3 ) )
                @ cs ) ) ) ) ) )
    = ( produc1545487600272464053on_nat @ left @ ( produc4176415733022414961on_nat @ right @ ( produc7152764266822961111on_nat @ s @ nss ) ) ) ) ).

% res
thf(fact_1197_left,axiom,
    ( left
    = ( concat6161509665823917772on_nat
      @ ( map_a_3887198048842487703on_nat
        @ ^ [X: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
        @ xs ) ) ) ).

% left
thf(fact_1198_NS__def,axiom,
    ( ns
    = ( concat937493773106420796on_nat
      @ ( map_a_7431629901395152583on_nat
        @ ^ [X: a] : ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5512618279175645674signed ) ) ) @ nil_Pr8614093721167239297on_nat ) )
        @ xs ) ) ) ).

% NS_def
thf(fact_1199_Annotation_Osize_I6_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size(6)
thf(fact_1200_Annotation_Osize_I5_J,axiom,
    ( ( size_s7536720479165623301tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size(5)
thf(fact_1201_Annotation_Oexhaust,axiom,
    ! [Y: multis8755981257395730801tation] :
      ( ( Y != multis5512618279175645674signed )
     => ( ( Y != multis8267843437418295310sitive )
       => ( Y = multis5977116398950053002gative ) ) ) ).

% Annotation.exhaust
thf(fact_1202_Annotation_Odistinct_I5_J,axiom,
    multis8267843437418295310sitive != multis5977116398950053002gative ).

% Annotation.distinct(5)
thf(fact_1203_Annotation_Odistinct_I3_J,axiom,
    multis5512618279175645674signed != multis5977116398950053002gative ).

% Annotation.distinct(3)
thf(fact_1204_Annotation_Odistinct_I1_J,axiom,
    multis5512618279175645674signed != multis8267843437418295310sitive ).

% Annotation.distinct(1)
thf(fact_1205_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C5: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_1206_Annotation_Osize__gen_I2_J,axiom,
    ( ( multis6502627171228953925tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size_gen(2)
thf(fact_1207_Annotation_Osize__gen_I1_J,axiom,
    ( ( multis6502627171228953925tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size_gen(1)
thf(fact_1208_Annotation_Osize__gen_I3_J,axiom,
    ( ( multis6502627171228953925tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size_gen(3)
thf(fact_1209_Suc__less__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% Suc_less_eq
thf(fact_1210_Suc__mono,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1211_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1212_tl__upt,axiom,
    ! [M3: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M3 @ N ) )
      = ( upt @ ( suc @ M3 ) @ N ) ) ).

% tl_upt
thf(fact_1213_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_1214_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1215_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1216_upt__conv__Cons,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( upt @ I @ J )
        = ( cons_nat @ I @ ( upt @ ( suc @ I ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_1217_upt__conv__Cons__Cons,axiom,
    ! [M3: nat,N: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M3 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M3 @ Q3 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M3 ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1218_Ex__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1219_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M2: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1220_All__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P2 @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1221_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M4: nat] :
          ( N
          = ( suc @ M4 ) ) ) ).

% gr0_implies_Suc
thf(fact_1222_less__Suc__eq__0__disj,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ ( suc @ N ) )
      = ( ( M3 = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M3
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1223_Suc__leI,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ord_less_eq_nat @ ( suc @ M3 ) @ N ) ) ).

% Suc_leI
thf(fact_1224_Suc__le__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
      = ( ord_less_nat @ M3 @ N ) ) ).

% Suc_le_eq
thf(fact_1225_dec__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ I )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P2 @ N4 )
                 => ( P2 @ ( suc @ N4 ) ) ) ) )
         => ( P2 @ J ) ) ) ) ).

% dec_induct
thf(fact_1226_inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( P2 @ J )
       => ( ! [N4: nat] :
              ( ( ord_less_eq_nat @ I @ N4 )
             => ( ( ord_less_nat @ N4 @ J )
               => ( ( P2 @ ( suc @ N4 ) )
                 => ( P2 @ N4 ) ) ) )
         => ( P2 @ I ) ) ) ) ).

% inc_induct
thf(fact_1227_Suc__le__lessD,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M3 ) @ N )
     => ( ord_less_nat @ M3 @ N ) ) ).

% Suc_le_lessD
thf(fact_1228_le__less__Suc__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ( ord_less_nat @ N @ ( suc @ M3 ) )
        = ( N = M3 ) ) ) ).

% le_less_Suc_eq
thf(fact_1229_less__Suc__eq__le,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ ( suc @ N ) )
      = ( ord_less_eq_nat @ M3 @ N ) ) ).

% less_Suc_eq_le
thf(fact_1230_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N3: nat] : ( ord_less_eq_nat @ ( suc @ N3 ) ) ) ) ).

% less_eq_Suc_le
thf(fact_1231_le__imp__less__Suc,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_eq_nat @ M3 @ N )
     => ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).

% le_imp_less_Suc
thf(fact_1232_less__natE,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ~ ! [Q4: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M3 @ Q4 ) ) ) ) ).

% less_natE
thf(fact_1233_less__add__Suc1,axiom,
    ! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M3 ) ) ) ).

% less_add_Suc1
thf(fact_1234_less__add__Suc2,axiom,
    ! [I: nat,M3: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M3 @ I ) ) ) ).

% less_add_Suc2
thf(fact_1235_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N3: nat] :
        ? [K3: nat] :
          ( N3
          = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1236_less__imp__Suc__add,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M3 @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1237_not__less__less__Suc__eq,axiom,
    ! [N: nat,M3: nat] :
      ( ~ ( ord_less_nat @ N @ M3 )
     => ( ( ord_less_nat @ N @ ( suc @ M3 ) )
        = ( N = M3 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1238_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] :
            ( ( J
              = ( suc @ I2 ) )
           => ( P2 @ I2 ) )
       => ( ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ J )
             => ( ( P2 @ ( suc @ I2 ) )
               => ( P2 @ I2 ) ) )
         => ( P2 @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1239_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I2: nat] : ( P2 @ I2 @ ( suc @ I2 ) )
       => ( ! [I2: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I2 @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P2 @ I2 @ J3 )
                 => ( ( P2 @ J3 @ K2 )
                   => ( P2 @ I2 @ K2 ) ) ) ) )
         => ( P2 @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1240_less__trans__Suc,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_1241_Suc__less__SucD,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M3 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M3 @ N ) ) ).

% Suc_less_SucD
thf(fact_1242_less__antisym,axiom,
    ! [N: nat,M3: nat] :
      ( ~ ( ord_less_nat @ N @ M3 )
     => ( ( ord_less_nat @ N @ ( suc @ M3 ) )
       => ( M3 = N ) ) ) ).

% less_antisym
thf(fact_1243_Suc__less__eq2,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M3 )
      = ( ? [M6: nat] :
            ( ( M3
              = ( suc @ M6 ) )
            & ( ord_less_nat @ N @ M6 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1244_All__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
           => ( P2 @ I3 ) ) )
      = ( ( P2 @ N )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
           => ( P2 @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_1245_not__less__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M3 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M3 ) ) ) ).

% not_less_eq
thf(fact_1246_less__Suc__eq,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M3 @ N )
        | ( M3 = N ) ) ) ).

% less_Suc_eq
thf(fact_1247_Ex__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N ) )
            & ( P2 @ I3 ) ) )
      = ( ( P2 @ N )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N )
            & ( P2 @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1248_less__SucI,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ord_less_nat @ M3 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1249_less__SucE,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M3 @ N )
       => ( M3 = N ) ) ) ).

% less_SucE
thf(fact_1250_Suc__lessI,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ M3 @ N )
     => ( ( ( suc @ M3 )
         != N )
       => ( ord_less_nat @ ( suc @ M3 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1251_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J3: nat] :
            ( ( ord_less_nat @ I @ J3 )
           => ( K
             != ( suc @ J3 ) ) ) ) ).

% Suc_lessE
thf(fact_1252_Suc__lessD,axiom,
    ! [M3: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M3 ) @ N )
     => ( ord_less_nat @ M3 @ N ) ) ).

% Suc_lessD
thf(fact_1253_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J3: nat] :
              ( ( ord_less_nat @ I @ J3 )
             => ( K
               != ( suc @ J3 ) ) ) ) ) ).

% Nat.lessE
thf(fact_1254_map__Suc__upt,axiom,
    ! [M3: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M3 @ N ) )
      = ( upt @ ( suc @ M3 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_1255_ex__least__nat__less,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ N )
     => ( ~ ( P2 @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P2 @ I4 ) )
            & ( P2 @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1256_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J2 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1257_upt__Suc,axiom,
    ! [I: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I @ J )
       => ( ( upt @ I @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_1258_upt__Suc__append,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( upt @ I @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_1259_upt__rec__numeral,axiom,
    ! [M3: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M3 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M3 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M3 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_1260_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_1261_zero__less__diff,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M3 ) )
      = ( ord_less_nat @ M3 @ N ) ) ).

% zero_less_diff
thf(fact_1262_Suc__pred,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) )
        = N ) ) ).

% Suc_pred
thf(fact_1263_Suc__diff__Suc,axiom,
    ! [N: nat,M3: nat] :
      ( ( ord_less_nat @ N @ M3 )
     => ( ( suc @ ( minus_minus_nat @ M3 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M3 @ N ) ) ) ).

% Suc_diff_Suc

% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X3 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X3 @ Y )
      = X3 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ x @ $true ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ).

%------------------------------------------------------------------------------