TPTP Problem File: SLH0834^1.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_00207_009999__14156232_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1825 ( 662 unt; 553 typ;   0 def)
%            Number of atoms       : 3581 (1865 equ;   0 cnn)
%            Maximal formula atoms :   19 (   2 avg)
%            Number of connectives : 13343 ( 395   ~;  46   |; 354   &;11115   @)
%                                         (   0 <=>;1433  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   23 (   7 avg)
%            Number of types       :  118 ( 117 usr)
%            Number of type conns  : 1690 (1690   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  439 ( 436 usr;  41 con; 0-4 aty)
%            Number of variables   : 4439 ( 263   ^;3908   !; 268   ?;4439   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:30:39.048
%------------------------------------------------------------------------------
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    member2326506860204807626et_nat: produc7112813594591200289et_nat > set_Pr7086239977703175297et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J_J,type,
    member8390928645697427088on_nat: produc7137417655796807271on_nat > set_Pr455547733325246535on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_Mt__Multiset__Omultiset_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    member3421213697272066832od_a_o: produc4365038726796860135od_a_o > set_Pr3669086689860627655od_a_o > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_It__Multiset__Omultiset_Itf__a_J_Mt__Multiset__Omultiset_Itf__a_J_J,type,
    member5199237121806060112iset_a: produc6518373309651786023iset_a > set_Pr79727621955416071iset_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    member8910825520198479363od_a_o: produc1433831906155620204od_a_o > set_Pr94232683564258082od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J,type,
    member7352503068387395495on_nat: produc7088870822282640638on_nat > set_Pr7552745763645690846on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_Mt__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_J_J,type,
    member1022098016950487056on_nat: produc4634397314152313575on_nat > set_Pr8202701763255893447on_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_M_Eo_J_Mt__Product____Type__Oprod_Itf__a_M_Eo_J_J,type,
    member3341120605725091216od_a_o: produc4569562562525714535od_a_o > set_Pr1246879249572150087od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_M_Eo_J_J_J,type,
    member7595723914438784454od_a_o: produc6493764124173775663od_a_o > set_Pr9140199768791955813od_a_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Sum____Type__Osum_It__Product____Type__Oprod_Itf__a_Mt__Multiset____Ordering____NP____Hard__OAnnotation_J_Mt__Nat__Onat_J_Mt__Nat__Onat_J,type,
    member8820252441731870247at_nat: produc8556620195627115390at_nat > set_Pr3345771464143190622at_nat > $o ).

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

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_M_Eo_J,type,
    member2997933407038931562od_a_o: product_prod_a_o > set_Product_prod_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_Mtf__a_J,type,
    member1426531477525435216od_a_a: product_prod_a_a > set_Product_prod_a_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_a: 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_thesis____,type,
    thesis: $o ).

thf(sy_v_x____,type,
    x: a ).

thf(sy_v_xs____,type,
    xs: list_a ).

% Relevant facts (1265)
thf(fact_0_len,axiom,
    ( ( size_s5686234912572693873on_nat @ ns_left )
    = ( size_s5686234912572693873on_nat @ ns_right ) ) ).

% len
thf(fact_1__092_060open_062Inl_A_Ix_M_AUnsigned_J_A_092_060in_062_Aset_Ans__right_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ x @ multis5512618279175645674signed ) ) @ ( set_Su4848624856859743608on_nat @ ns_right ) ).

% \<open>Inl (x, Unsigned) \<in> set ns_right\<close>
thf(fact_2_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_3__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_4__092_060open_062_092_060beta_062_Ax_092_060close_062,axiom,
    beta @ x ).

% \<open>\<beta> x\<close>
thf(fact_5_sum_Oinject_I1_J,axiom,
    ! [X1: produc27318731269267614tation,Y1: produc27318731269267614tation] :
      ( ( ( sum_In3878554002080803550on_nat @ X1 )
        = ( sum_In3878554002080803550on_nat @ Y1 ) )
      = ( X1 = Y1 ) ) ).

% sum.inject(1)
thf(fact_6_old_Osum_Oinject_I1_J,axiom,
    ! [A: produc27318731269267614tation,A2: produc27318731269267614tation] :
      ( ( ( sum_In3878554002080803550on_nat @ A )
        = ( sum_In3878554002080803550on_nat @ A2 ) )
      = ( A = A2 ) ) ).

% old.sum.inject(1)
thf(fact_7_prod_Oinject,axiom,
    ! [X1: list_S5253404628753938141on_nat,X2: produc118098540238352959on_nat,Y1: list_S5253404628753938141on_nat,Y2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ X1 @ X2 )
        = ( produc1545487600272464053on_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

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

% prod.inject
thf(fact_9_prod_Oinject,axiom,
    ! [X1: nat > nat,X2: nat,Y1: nat > nat,Y2: nat] :
      ( ( ( produc72220940542539688at_nat @ X1 @ X2 )
        = ( produc72220940542539688at_nat @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_10_prod_Oinject,axiom,
    ! [X1: a,X2: multis8755981257395730801tation,Y1: a,Y2: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ X1 @ X2 )
        = ( produc7247797765490398872tation @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_11_prod_Oinject,axiom,
    ! [X1: a,X2: $o,Y1: a,Y2: $o] :
      ( ( ( product_Pair_a_o @ X1 @ X2 )
        = ( product_Pair_a_o @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_12_old_Oprod_Oinject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_13_old_Oprod_Oinject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_14_old_Oprod_Oinject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_15_old_Oprod_Oinject,axiom,
    ! [A: a,B: multis8755981257395730801tation,A2: a,B2: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ A @ B )
        = ( produc7247797765490398872tation @ A2 @ B2 ) )
      = ( ( A = A2 )
        & ( B = B2 ) ) ) ).

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

% old.prod.inject
thf(fact_17__092_060open_062Inl_A_Ix_M_AUnsigned_J_A_092_060in_062_Aset_Ans__right_A_092_060union_062_Aset_As__right_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ x @ multis5512618279175645674signed ) ) @ ( sup_su343302011326411951on_nat @ ( set_Su4848624856859743608on_nat @ ns_right ) @ ( set_Su4848624856859743608on_nat @ s_right ) ) ).

% \<open>Inl (x, Unsigned) \<in> set ns_right \<union> set s_right\<close>
thf(fact_18_nmem,axiom,
    ! [B: $o] :
      ~ ( member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ x @ B ) ) @ ( set_Su4848624856859743608on_nat @ ns_left ) ) ).

% nmem
thf(fact_19_sleft,axiom,
    member8476201367316484836on_nat @ s2 @ ( set_Su4848624856859743608on_nat @ s_left ) ).

% sleft
thf(fact_20_Inl__inject,axiom,
    ! [X: produc27318731269267614tation,Y: produc27318731269267614tation] :
      ( ( ( sum_In3878554002080803550on_nat @ X )
        = ( sum_In3878554002080803550on_nat @ Y ) )
     => ( X = Y ) ) ).

% Inl_inject
thf(fact_21_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_22_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_nat_nat] :
      ~ ! [A3: nat,B3: nat] :
          ( Y
         != ( product_Pair_nat_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_23_old_Oprod_Oexhaust,axiom,
    ! [Y: produc8199716216217303280at_nat] :
      ~ ! [A3: nat > nat,B3: nat] :
          ( Y
         != ( produc72220940542539688at_nat @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_24_old_Oprod_Oexhaust,axiom,
    ! [Y: produc27318731269267614tation] :
      ~ ! [A3: a,B3: multis8755981257395730801tation] :
          ( Y
         != ( produc7247797765490398872tation @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_25_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_o] :
      ~ ! [A3: a,B3: $o] :
          ( Y
         != ( product_Pair_a_o @ A3 @ B3 ) ) ).

% old.prod.exhaust
thf(fact_26_surj__pair,axiom,
    ! [P: produc8325617224762090749on_nat] :
    ? [X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat] :
      ( P
      = ( produc1545487600272464053on_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_27_surj__pair,axiom,
    ! [P: product_prod_nat_nat] :
    ? [X3: nat,Y3: nat] :
      ( P
      = ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_28_surj__pair,axiom,
    ! [P: produc8199716216217303280at_nat] :
    ? [X3: nat > nat,Y3: nat] :
      ( P
      = ( produc72220940542539688at_nat @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_29_surj__pair,axiom,
    ! [P: produc27318731269267614tation] :
    ? [X3: a,Y3: multis8755981257395730801tation] :
      ( P
      = ( produc7247797765490398872tation @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_30_surj__pair,axiom,
    ! [P: product_prod_a_o] :
    ? [X3: a,Y3: $o] :
      ( P
      = ( product_Pair_a_o @ X3 @ Y3 ) ) ).

% surj_pair
thf(fact_31_False,axiom,
    ~ b ).

% False
thf(fact_32_sl,axiom,
    ( s2
    = ( multis8230369766971345672_lit_a @ l ) ) ).

% sl
thf(fact_33__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
    = ( ^ [X4: a] : ( member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X4 @ $true ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ) ) ) ).

% \<open>\<beta> \<equiv> \<lambda>x. ms_elem_of_lit (x, True) \<in> set s_left\<close>
thf(fact_34_lxb,axiom,
    ( l
    = ( product_Pair_a_o @ x @ b ) ) ).

% lxb
thf(fact_35_mem,axiom,
    ! [B: $o] : ( member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ x @ B ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ) ).

% mem
thf(fact_36__092_060open_062ms__elem__of__lit_A_Ix_M_ATrue_J_A_092_060in_062_Aset_As__left_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ x @ $true ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ).

% \<open>ms_elem_of_lit (x, True) \<in> set s_left\<close>
thf(fact_37__092_060open_062ms__elem__of__lit_A_Ix_M_AFalse_J_A_092_060in_062_Aset_As__left_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ x @ $false ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ).

% \<open>ms_elem_of_lit (x, False) \<in> set s_left\<close>
thf(fact_38__092_060beta_062__def,axiom,
    ! [X: a] :
      ( ( beta @ X )
      = ( member8476201367316484836on_nat @ ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X @ $true ) ) @ ( set_Su4848624856859743608on_nat @ s_left ) ) ) ).

% \<beta>_def
thf(fact_39__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,
    ~ ! [X3: a,B3: $o] :
        ( l
       != ( product_Pair_a_o @ X3 @ B3 ) ) ).

% \<open>\<And>thesis. (\<And>x b. l = (x, b) \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_40_ms__elem__of__lit_Ocases,axiom,
    ! [X: product_prod_a_o] :
      ( ! [X3: a] :
          ( X
         != ( product_Pair_a_o @ X3 @ $true ) )
     => ~ ! [X3: a] :
            ( X
           != ( product_Pair_a_o @ X3 @ $false ) ) ) ).

% ms_elem_of_lit.cases
thf(fact_41_prod__induct4,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ ( produc7152764266822961111on_nat @ C @ D ) ) ) )
     => ( P2 @ X ) ) ).

% prod_induct4
thf(fact_42_prod__induct3,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,X: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C: produc1714117091004681831on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ C ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_43_prod__cases4,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C: list_P9127870862658512897on_nat,D: list_P9127870862658512897on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ ( produc7152764266822961111on_nat @ C @ D ) ) ) ) ).

% prod_cases4
thf(fact_44_prod__cases3,axiom,
    ! [Y: produc8325617224762090749on_nat] :
      ~ ! [A3: list_S5253404628753938141on_nat,B3: list_S5253404628753938141on_nat,C: produc1714117091004681831on_nat] :
          ( Y
         != ( produc1545487600272464053on_nat @ A3 @ ( produc4176415733022414961on_nat @ B3 @ C ) ) ) ).

% prod_cases3
thf(fact_45_Pair__inject,axiom,
    ! [A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat,A2: list_S5253404628753938141on_nat,B2: produc118098540238352959on_nat] :
      ( ( ( produc1545487600272464053on_nat @ A @ B )
        = ( produc1545487600272464053on_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_46_Pair__inject,axiom,
    ! [A: nat,B: nat,A2: nat,B2: nat] :
      ( ( ( product_Pair_nat_nat @ A @ B )
        = ( product_Pair_nat_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_47_Pair__inject,axiom,
    ! [A: nat > nat,B: nat,A2: nat > nat,B2: nat] :
      ( ( ( produc72220940542539688at_nat @ A @ B )
        = ( produc72220940542539688at_nat @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_48_Pair__inject,axiom,
    ! [A: a,B: multis8755981257395730801tation,A2: a,B2: multis8755981257395730801tation] :
      ( ( ( produc7247797765490398872tation @ A @ B )
        = ( produc7247797765490398872tation @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_49_Pair__inject,axiom,
    ! [A: a,B: $o,A2: a,B2: $o] :
      ( ( ( product_Pair_a_o @ A @ B )
        = ( product_Pair_a_o @ A2 @ B2 ) )
     => ~ ( ( A = A2 )
         => ( B = ~ B2 ) ) ) ).

% Pair_inject
thf(fact_50_prod__cases,axiom,
    ! [P2: produc8325617224762090749on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] : ( P2 @ ( produc1545487600272464053on_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_51_prod__cases,axiom,
    ! [P2: product_prod_nat_nat > $o,P: product_prod_nat_nat] :
      ( ! [A3: nat,B3: nat] : ( P2 @ ( product_Pair_nat_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_52_prod__cases,axiom,
    ! [P2: produc8199716216217303280at_nat > $o,P: produc8199716216217303280at_nat] :
      ( ! [A3: nat > nat,B3: nat] : ( P2 @ ( produc72220940542539688at_nat @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_53_prod__cases,axiom,
    ! [P2: produc27318731269267614tation > $o,P: produc27318731269267614tation] :
      ( ! [A3: a,B3: multis8755981257395730801tation] : ( P2 @ ( produc7247797765490398872tation @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_54_prod__cases,axiom,
    ! [P2: product_prod_a_o > $o,P: product_prod_a_o] :
      ( ! [A3: a,B3: $o] : ( P2 @ ( product_Pair_a_o @ A3 @ B3 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_55_s,axiom,
    ! [T: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ T @ ( set_Su4848624856859743608on_nat @ s_right ) )
     => ? [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ s_left ) )
          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ T ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ) ).

% s
thf(fact_56_all__set__conv__all__nth,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,P2: product_prod_a_o > $o] :
      ( ( ! [X4: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X4 @ ( set_Product_prod_a_o2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1727999245659094451od_a_o @ Xs ) )
           => ( P2 @ ( nth_Product_prod_a_o @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_57_all__set__conv__all__nth,axiom,
    ! [Xs: list_a,P2: a > $o] :
      ( ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
           => ( P2 @ ( nth_a @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_58_all__set__conv__all__nth,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ! [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_59_all__set__conv__all__nth,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( ! [X4: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X4 @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_60_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P2: nat > $o] :
      ( ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( P2 @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_61_all__nth__imp__all__set,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,P2: product_prod_a_o > $o,X: product_prod_a_o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s1727999245659094451od_a_o @ Xs ) )
         => ( P2 @ ( nth_Product_prod_a_o @ Xs @ I3 ) ) )
     => ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_62_all__nth__imp__all__set,axiom,
    ! [Xs: list_a,P2: a > $o,X: a] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs ) )
         => ( P2 @ ( nth_a @ Xs @ I3 ) ) )
     => ( ( member_a @ X @ ( set_a2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_63_all__nth__imp__all__set,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o,X: sum_su4907927656612468301on_nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
         => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) ) )
     => ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_64_all__nth__imp__all__set,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P2: list_P3196869577824546119od_a_o > $o,X: list_P3196869577824546119od_a_o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
         => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) ) )
     => ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_65_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P2: nat > $o,X: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
         => ( P2 @ ( nth_nat @ Xs @ I3 ) ) )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_66_in__set__conv__nth,axiom,
    ! [X: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s1727999245659094451od_a_o @ Xs ) )
            & ( ( nth_Product_prod_a_o @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_67_in__set__conv__nth,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs ) )
            & ( ( nth_a @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_68_in__set__conv__nth,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( nth_Su8471564416988646406on_nat @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_69_in__set__conv__nth,axiom,
    ! [X: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( nth_li3248255784598043854od_a_o @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_70_in__set__conv__nth,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_71_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 ) )
     => ( ! [X3: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_Product_prod_a_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_72_list__ball__nth,axiom,
    ! [N: nat,Xs: list_a,P2: a > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_a @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_73_list__ball__nth,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_74_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 ) )
     => ( ! [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_75_list__ball__nth,axiom,
    ! [N: nat,Xs: list_nat,P2: nat > $o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_nat @ Xs @ N ) ) ) ) ).

% list_ball_nth
thf(fact_76_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_77_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_78_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_79_mem__Collect__eq,axiom,
    ! [A: nat,P2: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_80_mem__Collect__eq,axiom,
    ! [A: a,P2: a > $o] :
      ( ( member_a @ A @ ( collect_a @ P2 ) )
      = ( P2 @ A ) ) ).

% mem_Collect_eq
thf(fact_81_Collect__mem__eq,axiom,
    ! [A4: set_Product_prod_a_o] :
      ( ( collec5729118848859662764od_a_o
        @ ^ [X4: product_prod_a_o] : ( member2997933407038931562od_a_o @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_82_Collect__mem__eq,axiom,
    ! [A4: set_Su266273576555847555on_nat] :
      ( ( collec1912426543561893410on_nat
        @ ^ [X4: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_83_Collect__mem__eq,axiom,
    ! [A4: set_li5011860890973139239od_a_o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X4: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_84_Collect__mem__eq,axiom,
    ! [A4: set_nat] :
      ( ( collect_nat
        @ ^ [X4: nat] : ( member_nat @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_85_Collect__mem__eq,axiom,
    ! [A4: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A4 ) )
      = A4 ) ).

% Collect_mem_eq
thf(fact_86_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_87_nth__mem,axiom,
    ! [N: nat,Xs: list_a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( member_a @ ( nth_a @ Xs @ N ) @ ( set_a2 @ Xs ) ) ) ).

% nth_mem
thf(fact_88_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_89_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_90_nth__mem,axiom,
    ! [N: nat,Xs: list_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ ( nth_nat @ Xs @ N ) @ ( set_nat2 @ Xs ) ) ) ).

% nth_mem
thf(fact_91_nth__equalityI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( ( nth_Su8471564416988646406on_nat @ Xs @ I3 )
              = ( nth_Su8471564416988646406on_nat @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_92_nth__equalityI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( ( nth_li3248255784598043854od_a_o @ Xs @ I3 )
              = ( nth_li3248255784598043854od_a_o @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_93_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_94_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > sum_su4907927656612468301on_nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: sum_su4907927656612468301on_nat] : ( P2 @ I2 @ X5 ) ) )
      = ( ? [Xs2: list_S5253404628753938141on_nat] :
            ( ( ( size_s5686234912572693873on_nat @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P2 @ I2 @ ( nth_Su8471564416988646406on_nat @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_95_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > list_P3196869577824546119od_a_o > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: list_P3196869577824546119od_a_o] : ( P2 @ I2 @ X5 ) ) )
      = ( ? [Xs2: list_l8336264072933094989od_a_o] :
            ( ( ( size_s3379185846251495609od_a_o @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P2 @ I2 @ ( nth_li3248255784598043854od_a_o @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_96_Skolem__list__nth,axiom,
    ! [K: nat,P2: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: nat] : ( P2 @ I2 @ X5 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P2 @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_97_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_S5253404628753938141on_nat,Z: list_S5253404628753938141on_nat] : ( Y4 = Z ) )
    = ( ^ [Xs2: list_S5253404628753938141on_nat,Ys2: list_S5253404628753938141on_nat] :
          ( ( ( size_s5686234912572693873on_nat @ Xs2 )
            = ( size_s5686234912572693873on_nat @ Ys2 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs2 ) )
             => ( ( nth_Su8471564416988646406on_nat @ Xs2 @ I2 )
                = ( nth_Su8471564416988646406on_nat @ Ys2 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_98_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_l8336264072933094989od_a_o,Z: list_l8336264072933094989od_a_o] : ( Y4 = Z ) )
    = ( ^ [Xs2: list_l8336264072933094989od_a_o,Ys2: list_l8336264072933094989od_a_o] :
          ( ( ( size_s3379185846251495609od_a_o @ Xs2 )
            = ( size_s3379185846251495609od_a_o @ Ys2 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
             => ( ( nth_li3248255784598043854od_a_o @ Xs2 @ I2 )
                = ( nth_li3248255784598043854od_a_o @ Ys2 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_99_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_nat,Z: list_nat] : ( Y4 = Z ) )
    = ( ^ [Xs2: list_nat,Ys2: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys2 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I2 )
                = ( nth_nat @ Ys2 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_100_UnCI,axiom,
    ! [C2: product_prod_a_o,B4: set_Product_prod_a_o,A4: set_Product_prod_a_o] :
      ( ( ~ ( member2997933407038931562od_a_o @ C2 @ B4 )
       => ( member2997933407038931562od_a_o @ C2 @ A4 ) )
     => ( member2997933407038931562od_a_o @ C2 @ ( sup_su4274294223881134325od_a_o @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_101_UnCI,axiom,
    ! [C2: list_P3196869577824546119od_a_o,B4: set_li5011860890973139239od_a_o,A4: set_li5011860890973139239od_a_o] :
      ( ( ~ ( member5261019209154752368od_a_o @ C2 @ B4 )
       => ( member5261019209154752368od_a_o @ C2 @ A4 ) )
     => ( member5261019209154752368od_a_o @ C2 @ ( sup_su6115883799403739899od_a_o @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_102_UnCI,axiom,
    ! [C2: nat,B4: set_nat,A4: set_nat] :
      ( ( ~ ( member_nat @ C2 @ B4 )
       => ( member_nat @ C2 @ A4 ) )
     => ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_103_UnCI,axiom,
    ! [C2: a,B4: set_a,A4: set_a] :
      ( ( ~ ( member_a @ C2 @ B4 )
       => ( member_a @ C2 @ A4 ) )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_104_UnCI,axiom,
    ! [C2: sum_su4907927656612468301on_nat,B4: set_Su266273576555847555on_nat,A4: set_Su266273576555847555on_nat] :
      ( ( ~ ( member8476201367316484836on_nat @ C2 @ B4 )
       => ( member8476201367316484836on_nat @ C2 @ A4 ) )
     => ( member8476201367316484836on_nat @ C2 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) ) ) ).

% UnCI
thf(fact_105_Un__iff,axiom,
    ! [C2: product_prod_a_o,A4: set_Product_prod_a_o,B4: set_Product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ C2 @ ( sup_su4274294223881134325od_a_o @ A4 @ B4 ) )
      = ( ( member2997933407038931562od_a_o @ C2 @ A4 )
        | ( member2997933407038931562od_a_o @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_106_Un__iff,axiom,
    ! [C2: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,B4: set_li5011860890973139239od_a_o] :
      ( ( member5261019209154752368od_a_o @ C2 @ ( sup_su6115883799403739899od_a_o @ A4 @ B4 ) )
      = ( ( member5261019209154752368od_a_o @ C2 @ A4 )
        | ( member5261019209154752368od_a_o @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_107_Un__iff,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( ( member_nat @ C2 @ A4 )
        | ( member_nat @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_108_Un__iff,axiom,
    ! [C2: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C2 @ ( sup_sup_set_a @ A4 @ B4 ) )
      = ( ( member_a @ C2 @ A4 )
        | ( member_a @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_109_Un__iff,axiom,
    ! [C2: sum_su4907927656612468301on_nat,A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat] :
      ( ( member8476201367316484836on_nat @ C2 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) )
      = ( ( member8476201367316484836on_nat @ C2 @ A4 )
        | ( member8476201367316484836on_nat @ C2 @ B4 ) ) ) ).

% Un_iff
thf(fact_110_lc,axiom,
    member2997933407038931562od_a_o @ l @ ( set_Product_prod_a_o2 @ c ) ).

% lc
thf(fact_111_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_112_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_113_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_114_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_S5253404628753938141on_nat] :
      ( ( size_s5686234912572693873on_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_115_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_116_Ex__list__of__length,axiom,
    ! [N: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N ) ).

% Ex_list_of_length
thf(fact_117_Un__left__commute,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat,C3: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ A4 @ ( sup_su343302011326411951on_nat @ B4 @ C3 ) )
      = ( sup_su343302011326411951on_nat @ B4 @ ( sup_su343302011326411951on_nat @ A4 @ C3 ) ) ) ).

% Un_left_commute
thf(fact_118_Un__left__absorb,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ A4 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) )
      = ( sup_su343302011326411951on_nat @ A4 @ B4 ) ) ).

% Un_left_absorb
thf(fact_119_Un__commute,axiom,
    ( sup_su343302011326411951on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] : ( sup_su343302011326411951on_nat @ B5 @ A5 ) ) ) ).

% Un_commute
thf(fact_120_Un__absorb,axiom,
    ! [A4: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ A4 @ A4 )
      = A4 ) ).

% Un_absorb
thf(fact_121_Un__assoc,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat,C3: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) @ C3 )
      = ( sup_su343302011326411951on_nat @ A4 @ ( sup_su343302011326411951on_nat @ B4 @ C3 ) ) ) ).

% Un_assoc
thf(fact_122_ball__Un,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ! [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ A4 )
           => ( P2 @ X4 ) )
        & ! [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ B4 )
           => ( P2 @ X4 ) ) ) ) ).

% ball_Un
thf(fact_123_bex__Un,axiom,
    ! [A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat,P2: sum_su4907927656612468301on_nat > $o] :
      ( ( ? [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ A4 )
            & ( P2 @ X4 ) )
        | ? [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ B4 )
            & ( P2 @ X4 ) ) ) ) ).

% bex_Un
thf(fact_124_UnI2,axiom,
    ! [C2: product_prod_a_o,B4: set_Product_prod_a_o,A4: set_Product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ C2 @ B4 )
     => ( member2997933407038931562od_a_o @ C2 @ ( sup_su4274294223881134325od_a_o @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_125_UnI2,axiom,
    ! [C2: list_P3196869577824546119od_a_o,B4: set_li5011860890973139239od_a_o,A4: set_li5011860890973139239od_a_o] :
      ( ( member5261019209154752368od_a_o @ C2 @ B4 )
     => ( member5261019209154752368od_a_o @ C2 @ ( sup_su6115883799403739899od_a_o @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_126_UnI2,axiom,
    ! [C2: nat,B4: set_nat,A4: set_nat] :
      ( ( member_nat @ C2 @ B4 )
     => ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_127_UnI2,axiom,
    ! [C2: a,B4: set_a,A4: set_a] :
      ( ( member_a @ C2 @ B4 )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_128_UnI2,axiom,
    ! [C2: sum_su4907927656612468301on_nat,B4: set_Su266273576555847555on_nat,A4: set_Su266273576555847555on_nat] :
      ( ( member8476201367316484836on_nat @ C2 @ B4 )
     => ( member8476201367316484836on_nat @ C2 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) ) ) ).

% UnI2
thf(fact_129_UnI1,axiom,
    ! [C2: product_prod_a_o,A4: set_Product_prod_a_o,B4: set_Product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ C2 @ A4 )
     => ( member2997933407038931562od_a_o @ C2 @ ( sup_su4274294223881134325od_a_o @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_130_UnI1,axiom,
    ! [C2: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,B4: set_li5011860890973139239od_a_o] :
      ( ( member5261019209154752368od_a_o @ C2 @ A4 )
     => ( member5261019209154752368od_a_o @ C2 @ ( sup_su6115883799403739899od_a_o @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_131_UnI1,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ A4 )
     => ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_132_UnI1,axiom,
    ! [C2: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C2 @ A4 )
     => ( member_a @ C2 @ ( sup_sup_set_a @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_133_UnI1,axiom,
    ! [C2: sum_su4907927656612468301on_nat,A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat] :
      ( ( member8476201367316484836on_nat @ C2 @ A4 )
     => ( member8476201367316484836on_nat @ C2 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) ) ) ).

% UnI1
thf(fact_134_UnE,axiom,
    ! [C2: product_prod_a_o,A4: set_Product_prod_a_o,B4: set_Product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ C2 @ ( sup_su4274294223881134325od_a_o @ A4 @ B4 ) )
     => ( ~ ( member2997933407038931562od_a_o @ C2 @ A4 )
       => ( member2997933407038931562od_a_o @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_135_UnE,axiom,
    ! [C2: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,B4: set_li5011860890973139239od_a_o] :
      ( ( member5261019209154752368od_a_o @ C2 @ ( sup_su6115883799403739899od_a_o @ A4 @ B4 ) )
     => ( ~ ( member5261019209154752368od_a_o @ C2 @ A4 )
       => ( member5261019209154752368od_a_o @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_136_UnE,axiom,
    ! [C2: nat,A4: set_nat,B4: set_nat] :
      ( ( member_nat @ C2 @ ( sup_sup_set_nat @ A4 @ B4 ) )
     => ( ~ ( member_nat @ C2 @ A4 )
       => ( member_nat @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_137_UnE,axiom,
    ! [C2: a,A4: set_a,B4: set_a] :
      ( ( member_a @ C2 @ ( sup_sup_set_a @ A4 @ B4 ) )
     => ( ~ ( member_a @ C2 @ A4 )
       => ( member_a @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_138_UnE,axiom,
    ! [C2: sum_su4907927656612468301on_nat,A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat] :
      ( ( member8476201367316484836on_nat @ C2 @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) )
     => ( ~ ( member8476201367316484836on_nat @ C2 @ A4 )
       => ( member8476201367316484836on_nat @ C2 @ B4 ) ) ) ).

% UnE
thf(fact_139_length__induct,axiom,
    ! [P2: list_S5253404628753938141on_nat > $o,Xs: list_S5253404628753938141on_nat] :
      ( ! [Xs3: list_S5253404628753938141on_nat] :
          ( ! [Ys3: list_S5253404628753938141on_nat] :
              ( ( ord_less_nat @ ( size_s5686234912572693873on_nat @ Ys3 ) @ ( size_s5686234912572693873on_nat @ Xs3 ) )
             => ( P2 @ Ys3 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_140_length__induct,axiom,
    ! [P2: list_l8336264072933094989od_a_o > $o,Xs: list_l8336264072933094989od_a_o] :
      ( ! [Xs3: list_l8336264072933094989od_a_o] :
          ( ! [Ys3: list_l8336264072933094989od_a_o] :
              ( ( ord_less_nat @ ( size_s3379185846251495609od_a_o @ Ys3 ) @ ( size_s3379185846251495609od_a_o @ Xs3 ) )
             => ( P2 @ Ys3 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_141_length__induct,axiom,
    ! [P2: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys3: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys3 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P2 @ Ys3 ) )
         => ( P2 @ Xs3 ) )
     => ( P2 @ Xs ) ) ).

% length_induct
thf(fact_142__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_143_x,axiom,
    member_a @ x @ ( set_a2 @ xs ) ).

% x
thf(fact_144_set__union,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( set_Product_prod_a_o2 @ ( union_5103297149488869051od_a_o @ Xs @ Ys ) )
      = ( sup_su4274294223881134325od_a_o @ ( set_Product_prod_a_o2 @ Xs ) @ ( set_Product_prod_a_o2 @ Ys ) ) ) ).

% set_union
thf(fact_145_set__union,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( set_li2629784761229999580od_a_o @ ( union_386226734345830209od_a_o @ Xs @ Ys ) )
      = ( sup_su6115883799403739899od_a_o @ ( set_li2629784761229999580od_a_o @ Xs ) @ ( set_li2629784761229999580od_a_o @ Ys ) ) ) ).

% set_union
thf(fact_146_set__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( union_nat @ Xs @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).

% set_union
thf(fact_147_set__union,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( union_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_union
thf(fact_148_set__union,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( set_Su4848624856859743608on_nat @ ( union_82372077702528979on_nat @ Xs @ Ys ) )
      = ( sup_su343302011326411951on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ).

% set_union
thf(fact_149_sup_Oright__idem,axiom,
    ! [A: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ ( sup_su343302011326411951on_nat @ A @ B ) @ B )
      = ( sup_su343302011326411951on_nat @ A @ B ) ) ).

% sup.right_idem
thf(fact_150_sup__left__idem,axiom,
    ! [X: set_Su266273576555847555on_nat,Y: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ X @ ( sup_su343302011326411951on_nat @ X @ Y ) )
      = ( sup_su343302011326411951on_nat @ X @ Y ) ) ).

% sup_left_idem
thf(fact_151_sup_Oleft__idem,axiom,
    ! [A: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ A @ ( sup_su343302011326411951on_nat @ A @ B ) )
      = ( sup_su343302011326411951on_nat @ A @ B ) ) ).

% sup.left_idem
thf(fact_152_sup__idem,axiom,
    ! [X: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ X @ X )
      = X ) ).

% sup_idem
thf(fact_153_sup_Oidem,axiom,
    ! [A: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ A @ A )
      = A ) ).

% sup.idem
thf(fact_154_si,axiom,
    member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ s2 @ ( sum_In255631617079920996tation @ i ) ) @ ( set_Pr937530902923597456on_nat @ s ) ).

% si
thf(fact_155_i,axiom,
    member_nat @ i @ ( set_nat2 @ cs ) ).

% i
thf(fact_156_old_Osum_Oinject_I2_J,axiom,
    ! [B: nat,B2: nat] :
      ( ( ( sum_In255631617079920996tation @ B )
        = ( sum_In255631617079920996tation @ B2 ) )
      = ( B = B2 ) ) ).

% old.sum.inject(2)
thf(fact_157_sum_Oinject_I2_J,axiom,
    ! [X2: nat,Y2: nat] :
      ( ( ( sum_In255631617079920996tation @ X2 )
        = ( sum_In255631617079920996tation @ Y2 ) )
      = ( X2 = Y2 ) ) ).

% sum.inject(2)
thf(fact_158__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_159_ccnf,axiom,
    member5261019209154752368od_a_o @ c @ ( set_li2629784761229999580od_a_o @ cnf ) ).

% ccnf
thf(fact_160__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_161_ii,axiom,
    ord_less_nat @ i @ ( size_s3379185846251495609od_a_o @ cnf ) ).

% ii
thf(fact_162__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,
    ~ ! [S: sum_su4907927656612468301on_nat] :
        ( ( member8476201367316484836on_nat @ S @ ( set_Su4848624856859743608on_nat @ s_left ) )
       => ~ ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ S @ ( 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_163__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,
    ? [X3: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ s_left ) )
      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ ( sum_In255631617079920996tation @ i ) ) @ ( set_Pr937530902923597456on_nat @ s ) ) ) ).

% \<open>\<exists>s\<in>set s_left. (s, Inr i) \<in> set S\<close>
thf(fact_164_c__def,axiom,
    ( c
    = ( nth_li3248255784598043854od_a_o @ cnf @ i ) ) ).

% c_def
thf(fact_165_Inr__inject,axiom,
    ! [X: nat,Y: nat] :
      ( ( ( sum_In255631617079920996tation @ X )
        = ( sum_In255631617079920996tation @ Y ) )
     => ( X = Y ) ) ).

% Inr_inject
thf(fact_166_split__sum__all,axiom,
    ( ( ^ [P3: sum_su4907927656612468301on_nat > $o] :
        ! [X6: sum_su4907927656612468301on_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: sum_su4907927656612468301on_nat > $o] :
          ( ! [X4: produc27318731269267614tation] : ( P4 @ ( sum_In3878554002080803550on_nat @ X4 ) )
          & ! [X4: nat] : ( P4 @ ( sum_In255631617079920996tation @ X4 ) ) ) ) ) ).

% split_sum_all
thf(fact_167_split__sum__ex,axiom,
    ( ( ^ [P3: sum_su4907927656612468301on_nat > $o] :
        ? [X6: sum_su4907927656612468301on_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: sum_su4907927656612468301on_nat > $o] :
          ( ? [X4: produc27318731269267614tation] : ( P4 @ ( sum_In3878554002080803550on_nat @ X4 ) )
          | ? [X4: nat] : ( P4 @ ( sum_In255631617079920996tation @ X4 ) ) ) ) ) ).

% split_sum_ex
thf(fact_168_Inr__not__Inl,axiom,
    ! [B: nat,A: produc27318731269267614tation] :
      ( ( sum_In255631617079920996tation @ B )
     != ( sum_In3878554002080803550on_nat @ A ) ) ).

% Inr_not_Inl
thf(fact_169_sumE,axiom,
    ! [S2: sum_su4907927656612468301on_nat] :
      ( ! [X3: produc27318731269267614tation] :
          ( S2
         != ( sum_In3878554002080803550on_nat @ X3 ) )
     => ~ ! [Y3: nat] :
            ( S2
           != ( sum_In255631617079920996tation @ Y3 ) ) ) ).

% sumE
thf(fact_170_old_Osum_Oexhaust,axiom,
    ! [Y: sum_su4907927656612468301on_nat] :
      ( ! [A3: produc27318731269267614tation] :
          ( Y
         != ( sum_In3878554002080803550on_nat @ A3 ) )
     => ~ ! [B3: nat] :
            ( Y
           != ( sum_In255631617079920996tation @ B3 ) ) ) ).

% old.sum.exhaust
thf(fact_171_old_Osum_Odistinct_I1_J,axiom,
    ! [A: produc27318731269267614tation,B2: nat] :
      ( ( sum_In3878554002080803550on_nat @ A )
     != ( sum_In255631617079920996tation @ B2 ) ) ).

% old.sum.distinct(1)
thf(fact_172_old_Osum_Odistinct_I2_J,axiom,
    ! [B2: nat,A: produc27318731269267614tation] :
      ( ( sum_In255631617079920996tation @ B2 )
     != ( sum_In3878554002080803550on_nat @ A ) ) ).

% old.sum.distinct(2)
thf(fact_173_sum_Odistinct_I1_J,axiom,
    ! [X1: produc27318731269267614tation,X2: nat] :
      ( ( sum_In3878554002080803550on_nat @ X1 )
     != ( sum_In255631617079920996tation @ X2 ) ) ).

% sum.distinct(1)
thf(fact_174_inf__sup__aci_I8_J,axiom,
    ! [X: set_Su266273576555847555on_nat,Y: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ X @ ( sup_su343302011326411951on_nat @ X @ Y ) )
      = ( sup_su343302011326411951on_nat @ X @ Y ) ) ).

% inf_sup_aci(8)
thf(fact_175_inf__sup__aci_I7_J,axiom,
    ! [X: set_Su266273576555847555on_nat,Y: set_Su266273576555847555on_nat,Z2: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ X @ ( sup_su343302011326411951on_nat @ Y @ Z2 ) )
      = ( sup_su343302011326411951on_nat @ Y @ ( sup_su343302011326411951on_nat @ X @ Z2 ) ) ) ).

% inf_sup_aci(7)
thf(fact_176_inf__sup__aci_I6_J,axiom,
    ! [X: set_Su266273576555847555on_nat,Y: set_Su266273576555847555on_nat,Z2: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ ( sup_su343302011326411951on_nat @ X @ Y ) @ Z2 )
      = ( sup_su343302011326411951on_nat @ X @ ( sup_su343302011326411951on_nat @ Y @ Z2 ) ) ) ).

% inf_sup_aci(6)
thf(fact_177_inf__sup__aci_I5_J,axiom,
    ( sup_su343302011326411951on_nat
    = ( ^ [X4: set_Su266273576555847555on_nat,Y5: set_Su266273576555847555on_nat] : ( sup_su343302011326411951on_nat @ Y5 @ X4 ) ) ) ).

% inf_sup_aci(5)
thf(fact_178_sup_Oassoc,axiom,
    ! [A: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat,C2: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ ( sup_su343302011326411951on_nat @ A @ B ) @ C2 )
      = ( sup_su343302011326411951on_nat @ A @ ( sup_su343302011326411951on_nat @ B @ C2 ) ) ) ).

% sup.assoc
thf(fact_179_sup__assoc,axiom,
    ! [X: set_Su266273576555847555on_nat,Y: set_Su266273576555847555on_nat,Z2: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ ( sup_su343302011326411951on_nat @ X @ Y ) @ Z2 )
      = ( sup_su343302011326411951on_nat @ X @ ( sup_su343302011326411951on_nat @ Y @ Z2 ) ) ) ).

% sup_assoc
thf(fact_180_sup_Ocommute,axiom,
    ( sup_su343302011326411951on_nat
    = ( ^ [A6: set_Su266273576555847555on_nat,B6: set_Su266273576555847555on_nat] : ( sup_su343302011326411951on_nat @ B6 @ A6 ) ) ) ).

% sup.commute
thf(fact_181_sup__commute,axiom,
    ( sup_su343302011326411951on_nat
    = ( ^ [X4: set_Su266273576555847555on_nat,Y5: set_Su266273576555847555on_nat] : ( sup_su343302011326411951on_nat @ Y5 @ X4 ) ) ) ).

% sup_commute
thf(fact_182_sup_Oleft__commute,axiom,
    ! [B: set_Su266273576555847555on_nat,A: set_Su266273576555847555on_nat,C2: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ B @ ( sup_su343302011326411951on_nat @ A @ C2 ) )
      = ( sup_su343302011326411951on_nat @ A @ ( sup_su343302011326411951on_nat @ B @ C2 ) ) ) ).

% sup.left_commute
thf(fact_183_sup__left__commute,axiom,
    ! [X: set_Su266273576555847555on_nat,Y: set_Su266273576555847555on_nat,Z2: set_Su266273576555847555on_nat] :
      ( ( sup_su343302011326411951on_nat @ X @ ( sup_su343302011326411951on_nat @ Y @ Z2 ) )
      = ( sup_su343302011326411951on_nat @ Y @ ( sup_su343302011326411951on_nat @ X @ Z2 ) ) ) ).

% sup_left_commute
thf(fact_184_less__supI1,axiom,
    ! [X: set_Su266273576555847555on_nat,A: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat] :
      ( ( ord_le5614770093425365463on_nat @ X @ A )
     => ( ord_le5614770093425365463on_nat @ X @ ( sup_su343302011326411951on_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_185_less__supI1,axiom,
    ! [X: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ X @ A )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI1
thf(fact_186_less__supI2,axiom,
    ! [X: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat,A: set_Su266273576555847555on_nat] :
      ( ( ord_le5614770093425365463on_nat @ X @ B )
     => ( ord_le5614770093425365463on_nat @ X @ ( sup_su343302011326411951on_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_187_less__supI2,axiom,
    ! [X: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ X @ B )
     => ( ord_less_nat @ X @ ( sup_sup_nat @ A @ B ) ) ) ).

% less_supI2
thf(fact_188_sup_Oabsorb3,axiom,
    ! [B: set_Su266273576555847555on_nat,A: set_Su266273576555847555on_nat] :
      ( ( ord_le5614770093425365463on_nat @ B @ A )
     => ( ( sup_su343302011326411951on_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_189_sup_Oabsorb3,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( sup_sup_nat @ A @ B )
        = A ) ) ).

% sup.absorb3
thf(fact_190_sup_Oabsorb4,axiom,
    ! [A: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat] :
      ( ( ord_le5614770093425365463on_nat @ A @ B )
     => ( ( sup_su343302011326411951on_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_191_sup_Oabsorb4,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( sup_sup_nat @ A @ B )
        = B ) ) ).

% sup.absorb4
thf(fact_192_sup_Ostrict__boundedE,axiom,
    ! [B: set_Su266273576555847555on_nat,C2: set_Su266273576555847555on_nat,A: set_Su266273576555847555on_nat] :
      ( ( ord_le5614770093425365463on_nat @ ( sup_su343302011326411951on_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_le5614770093425365463on_nat @ B @ A )
         => ~ ( ord_le5614770093425365463on_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_193_sup_Ostrict__boundedE,axiom,
    ! [B: nat,C2: nat,A: nat] :
      ( ( ord_less_nat @ ( sup_sup_nat @ B @ C2 ) @ A )
     => ~ ( ( ord_less_nat @ B @ A )
         => ~ ( ord_less_nat @ C2 @ A ) ) ) ).

% sup.strict_boundedE
thf(fact_194_sup_Ostrict__order__iff,axiom,
    ( ord_le5614770093425365463on_nat
    = ( ^ [B6: set_Su266273576555847555on_nat,A6: set_Su266273576555847555on_nat] :
          ( ( A6
            = ( sup_su343302011326411951on_nat @ A6 @ B6 ) )
          & ( A6 != B6 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_195_sup_Ostrict__order__iff,axiom,
    ( ord_less_nat
    = ( ^ [B6: nat,A6: nat] :
          ( ( A6
            = ( sup_sup_nat @ A6 @ B6 ) )
          & ( A6 != B6 ) ) ) ) ).

% sup.strict_order_iff
thf(fact_196_sup_Ostrict__coboundedI1,axiom,
    ! [C2: set_Su266273576555847555on_nat,A: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat] :
      ( ( ord_le5614770093425365463on_nat @ C2 @ A )
     => ( ord_le5614770093425365463on_nat @ C2 @ ( sup_su343302011326411951on_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_197_sup_Ostrict__coboundedI1,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ C2 @ A )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI1
thf(fact_198_sup_Ostrict__coboundedI2,axiom,
    ! [C2: set_Su266273576555847555on_nat,B: set_Su266273576555847555on_nat,A: set_Su266273576555847555on_nat] :
      ( ( ord_le5614770093425365463on_nat @ C2 @ B )
     => ( ord_le5614770093425365463on_nat @ C2 @ ( sup_su343302011326411951on_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_199_sup_Ostrict__coboundedI2,axiom,
    ! [C2: nat,B: nat,A: nat] :
      ( ( ord_less_nat @ C2 @ B )
     => ( ord_less_nat @ C2 @ ( sup_sup_nat @ A @ B ) ) ) ).

% sup.strict_coboundedI2
thf(fact_200_Inr__Inl__False,axiom,
    ! [X: nat,Y: produc27318731269267614tation] :
      ( ( sum_In255631617079920996tation @ X )
     != ( sum_In3878554002080803550on_nat @ Y ) ) ).

% Inr_Inl_False
thf(fact_201_Inl__Inr__False,axiom,
    ! [X: produc27318731269267614tation,Y: nat] :
      ( ( sum_In3878554002080803550on_nat @ X )
     != ( sum_In255631617079920996tation @ Y ) ) ).

% Inl_Inr_False
thf(fact_202_ms__elem__of__lit_Osimps_I1_J,axiom,
    ! [X: a] :
      ( ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X @ $true ) )
      = ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis8267843437418295310sitive ) ) ) ).

% ms_elem_of_lit.simps(1)
thf(fact_203_ms__elem__of__lit_Osimps_I2_J,axiom,
    ! [X: a] :
      ( ( multis8230369766971345672_lit_a @ ( product_Pair_a_o @ X @ $false ) )
      = ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X @ multis5977116398950053002gative ) ) ) ).

% ms_elem_of_lit.simps(2)
thf(fact_204_obj__sumE,axiom,
    ! [S2: sum_su4907927656612468301on_nat] :
      ( ! [X3: produc27318731269267614tation] :
          ( S2
         != ( sum_In3878554002080803550on_nat @ X3 ) )
     => ~ ! [X3: nat] :
            ( S2
           != ( sum_In255631617079920996tation @ X3 ) ) ) ).

% obj_sumE
thf(fact_205_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_206_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_207_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_208_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_209_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_210_list__ex__length,axiom,
    ( list_e3769318529992910857on_nat
    = ( ^ [P4: sum_su4907927656612468301on_nat > $o,Xs2: list_S5253404628753938141on_nat] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s5686234912572693873on_nat @ Xs2 ) )
          & ( P4 @ ( nth_Su8471564416988646406on_nat @ Xs2 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_211_list__ex__length,axiom,
    ( list_e5403782194016407179od_a_o
    = ( ^ [P4: list_P3196869577824546119od_a_o > $o,Xs2: list_l8336264072933094989od_a_o] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
          & ( P4 @ ( nth_li3248255784598043854od_a_o @ Xs2 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_212_list__ex__length,axiom,
    ( list_ex_nat
    = ( ^ [P4: nat > $o,Xs2: list_nat] :
        ? [N2: nat] :
          ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs2 ) )
          & ( P4 @ ( nth_nat @ Xs2 @ N2 ) ) ) ) ) ).

% list_ex_length
thf(fact_213_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
           => ( member7446769889882171317tation @ ( produc7247797765490398872tation @ ( nth_a @ Xs @ N2 ) @ ( nth_Mu116543695464807850tation @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_214_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs ) )
           => ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ ( nth_a @ Xs @ N2 ) @ ( nth_o @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_215_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_216_listrel__iff__nth,axiom,
    ! [Xs: list_nat_nat,Ys: list_nat,R: set_Pr9093778441882193744at_nat] :
      ( ( member6987746275253522745st_nat @ ( produc7978589510830832328st_nat @ Xs @ Ys ) @ ( listrel_nat_nat_nat @ R ) )
      = ( ( ( size_s8208510060688613859at_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s8208510060688613859at_nat @ Xs ) )
           => ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ ( nth_nat_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_217_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member7885718254550376541_o_nat @ ( produc3563005473579536832_o_nat @ ( nth_li3248255784598043854od_a_o @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_218_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member8910825520198479363od_a_o @ ( produc3946064260745446430od_a_o @ ( nth_nat @ Xs @ N2 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_219_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( member8820252441731870247at_nat @ ( produc7500644556467579062at_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N2 ) @ ( nth_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_220_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
           => ( member7352503068387395495on_nat @ ( produc8397911583705546934on_nat @ ( nth_nat @ Xs @ N2 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_221_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
           => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N2 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_222_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 ) )
        & ! [N2: nat] :
            ( ( ord_less_nat @ N2 @ ( size_s5686234912572693873on_nat @ Xs ) )
           => ( member7595723914438784454od_a_o @ ( produc4183299375811982305od_a_o @ ( nth_Su8471564416988646406on_nat @ Xs @ N2 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ N2 ) ) @ R ) ) ) ) ).

% listrel_iff_nth
thf(fact_223__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_224_length__list__update,axiom,
    ! [Xs: list_S5253404628753938141on_nat,I: nat,X: sum_su4907927656612468301on_nat] :
      ( ( size_s5686234912572693873on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_list_update
thf(fact_225_length__list__update,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,I: nat,X: list_P3196869577824546119od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_list_update
thf(fact_226_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_227_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
      ( ( I != J )
     => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X ) @ J )
        = ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_228_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
      ( ( I != J )
     => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X ) @ J )
        = ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_229_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_230_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_231_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_232_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_233_xs__def,axiom,
    ( xs
    = ( multis4741186478440321083_cnf_a @ cnf ) ) ).

% xs_def
thf(fact_234__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,
    ~ ! [I3: nat] :
        ( ( member_nat @ I3 @ ( set_nat2 @ cs ) )
       => ( ( c
            = ( nth_li3248255784598043854od_a_o @ cnf @ I3 ) )
         => ~ ( ord_less_nat @ I3 @ ( 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_235_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_236_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_237_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ I )
        = X ) ) ).

% nth_list_update_eq
thf(fact_238_Annotation_Odistinct_I5_J,axiom,
    multis8267843437418295310sitive != multis5977116398950053002gative ).

% Annotation.distinct(5)
thf(fact_239_Annotation_Oexhaust,axiom,
    ! [Y: multis8755981257395730801tation] :
      ( ( Y != multis5512618279175645674signed )
     => ( ( Y != multis8267843437418295310sitive )
       => ( Y = multis5977116398950053002gative ) ) ) ).

% Annotation.exhaust
thf(fact_240_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_241_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_242_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_243_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_244_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_245_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_246_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_247_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_248_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_249_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 )
     => ( ! [X3: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e12574053215267589od_a_o @ F @ Xs )
          = ( list_e12574053215267589od_a_o @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_250_list__ex__cong,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > $o,G: sum_su4907927656612468301on_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e3769318529992910857on_nat @ F @ Xs )
          = ( list_e3769318529992910857on_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_251_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 )
     => ( ! [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_e5403782194016407179od_a_o @ F @ Xs )
          = ( list_e5403782194016407179od_a_o @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_252_list__ex__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > $o,G: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_ex_nat @ F @ Xs )
          = ( list_ex_nat @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_253_list__ex__cong,axiom,
    ! [Xs: list_a,Ys: list_a,F: a > $o,G: a > $o] :
      ( ( Xs = Ys )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( list_ex_a @ F @ Xs )
          = ( list_ex_a @ G @ Ys ) ) ) ) ).

% list_ex_cong
thf(fact_254_Annotation_Odistinct_I3_J,axiom,
    multis5512618279175645674signed != multis5977116398950053002gative ).

% Annotation.distinct(3)
thf(fact_255_Annotation_Odistinct_I1_J,axiom,
    multis5512618279175645674signed != multis8267843437418295310sitive ).

% Annotation.distinct(1)
thf(fact_256_set__update__memI,axiom,
    ! [N: nat,Xs: list_P3196869577824546119od_a_o,X: product_prod_a_o] :
      ( ( ord_less_nat @ N @ ( size_s1727999245659094451od_a_o @ Xs ) )
     => ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ ( list_u356469027079241519od_a_o @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_257_set__update__memI,axiom,
    ! [N: nat,Xs: list_a,X: a] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( member_a @ X @ ( set_a2 @ ( list_update_a @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_258_set__update__memI,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ ( list_u8577817230714425439on_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_259_set__update__memI,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_260_set__update__memI,axiom,
    ! [N: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X @ ( set_nat2 @ ( list_update_nat @ Xs @ N @ X ) ) ) ) ).

% set_update_memI
thf(fact_261_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ( list_u8577817230714425439on_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_Su8471564416988646406on_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_262_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( ( list_u329291161401623989od_a_o @ Xs @ I @ X )
          = Xs )
        = ( ( nth_li3248255784598043854od_a_o @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_263_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X ) ) ) ).

% list_update_same_conv
thf(fact_264_nth__list__update,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,J: nat,X: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_Su8471564416988646406on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X ) @ J )
            = ( nth_Su8471564416988646406on_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_265_nth__list__update,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,J: nat,X: 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 @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_li3248255784598043854od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X ) @ J )
            = ( nth_li3248255784598043854od_a_o @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_266_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = X ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_267_ms__elem__of__lit_Oelims,axiom,
    ! [X: product_prod_a_o,Y: sum_su4907927656612468301on_nat] :
      ( ( ( multis8230369766971345672_lit_a @ X )
        = Y )
     => ( ! [X3: a] :
            ( ( X
              = ( product_Pair_a_o @ X3 @ $true ) )
           => ( Y
             != ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis8267843437418295310sitive ) ) ) )
       => ~ ! [X3: a] :
              ( ( X
                = ( product_Pair_a_o @ X3 @ $false ) )
             => ( Y
               != ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5977116398950053002gative ) ) ) ) ) ) ).

% ms_elem_of_lit.elims
thf(fact_268_not__arg__cong__Inr,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ( sum_In255631617079920996tation @ X )
       != ( sum_In255631617079920996tation @ Y ) ) ) ).

% not_arg_cong_Inr
thf(fact_269_NSS,axiom,
    ( nss
    = ( append2865821678206098582on_nat @ ns @ s ) ) ).

% NSS
thf(fact_270_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 ) )
      = ( ? [Y5: sum_su4907927656612468301on_nat,N2: nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ N2 ) @ Y5 ) @ R )
            & ( ord_less_nat @ N2 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( Ys
              = ( list_u8577817230714425439on_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_271_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 ) )
      = ( ? [Y5: list_P3196869577824546119od_a_o,N2: nat] :
            ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ N2 ) @ Y5 ) @ R )
            & ( ord_less_nat @ N2 @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( Ys
              = ( list_u329291161401623989od_a_o @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_272_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) )
      = ( ? [Y5: nat,N2: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N2 ) @ Y5 ) @ R )
            & ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N2 @ Y5 ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_273_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_274_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_275_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_276_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_277__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,
    ? [X3: nat] :
      ( ( member_nat @ X3 @ ( 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 @ X3 ) )
          @ ( set_Product_prod_a_o2 @ ( nth_li3248255784598043854od_a_o @ cnf @ X3 ) ) ) ) ) ).

% \<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_278_assms,axiom,
    ( ( multis4221440410396274297_cnf_a @ cnf )
    = ( produc1545487600272464053on_nat @ left @ ( produc4176415733022414961on_nat @ right @ ( produc7152764266822961111on_nat @ s @ nss ) ) ) ) ).

% assms
thf(fact_279_ms__elem__of__lit_Opelims,axiom,
    ! [X: product_prod_a_o,Y: sum_su4907927656612468301on_nat] :
      ( ( ( multis8230369766971345672_lit_a @ X )
        = Y )
     => ( ( accp_P1185719373787791178od_a_o @ multis101446917994433337_rel_a @ X )
       => ( ! [X3: a] :
              ( ( X
                = ( product_Pair_a_o @ X3 @ $true ) )
             => ( ( Y
                  = ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis8267843437418295310sitive ) ) )
               => ~ ( accp_P1185719373787791178od_a_o @ multis101446917994433337_rel_a @ ( product_Pair_a_o @ X3 @ $true ) ) ) )
         => ~ ! [X3: a] :
                ( ( X
                  = ( product_Pair_a_o @ X3 @ $false ) )
               => ( ( Y
                    = ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X3 @ multis5977116398950053002gative ) ) )
                 => ~ ( accp_P1185719373787791178od_a_o @ multis101446917994433337_rel_a @ ( product_Pair_a_o @ X3 @ $false ) ) ) ) ) ) ) ).

% ms_elem_of_lit.pelims
thf(fact_280_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,S3: set_Pr2535564044409765851on_nat] :
      ( ( list_l4998409290742832897on_nat @ S_list @ Ns_list )
     => ( ( ( size_s5686234912572693873on_nat @ As )
          = ( size_s5686234912572693873on_nat @ Bs ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Bs ) )
             => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I3 ) ) @ NS ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As @ Bs ) @ ( Ns_list @ S3 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_281_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,S3: set_Pr7420778282784443591od_a_o] :
      ( ( list_l1016356893908098451od_a_o @ S_list @ Ns_list )
     => ( ( ( size_s3379185846251495609od_a_o @ As )
          = ( size_s3379185846251495609od_a_o @ Bs ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Bs ) )
             => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I3 ) ) @ NS ) )
         => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As @ Bs ) @ ( Ns_list @ S3 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_282_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,S3: set_Pr1261947904930325089at_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ( ( size_size_list_nat @ As )
          = ( size_size_list_nat @ Bs ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ NS ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S3 @ NS ) ) ) ) ) ).

% list_order_extension.all_ns_imp_ns
thf(fact_283_partition__set,axiom,
    ! [P2: product_prod_a_o > $o,Xs: list_P3196869577824546119od_a_o,Yes: list_P3196869577824546119od_a_o,No: list_P3196869577824546119od_a_o] :
      ( ( ( partit7076252557220421312od_a_o @ P2 @ Xs )
        = ( produc8077463541153823575od_a_o @ Yes @ No ) )
     => ( ( sup_su4274294223881134325od_a_o @ ( set_Product_prod_a_o2 @ Yes ) @ ( set_Product_prod_a_o2 @ No ) )
        = ( set_Product_prod_a_o2 @ Xs ) ) ) ).

% partition_set
thf(fact_284_partition__set,axiom,
    ! [P2: list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Yes: list_l8336264072933094989od_a_o,No: list_l8336264072933094989od_a_o] :
      ( ( ( partit8903024691838806214od_a_o @ P2 @ Xs )
        = ( produc7858527432735436247od_a_o @ Yes @ No ) )
     => ( ( sup_su6115883799403739899od_a_o @ ( set_li2629784761229999580od_a_o @ Yes ) @ ( set_li2629784761229999580od_a_o @ No ) )
        = ( set_li2629784761229999580od_a_o @ Xs ) ) ) ).

% partition_set
thf(fact_285_partition__set,axiom,
    ! [P2: nat > $o,Xs: list_nat,Yes: list_nat,No: list_nat] :
      ( ( ( partition_nat @ P2 @ Xs )
        = ( produc2694037385005941721st_nat @ Yes @ No ) )
     => ( ( sup_sup_set_nat @ ( set_nat2 @ Yes ) @ ( set_nat2 @ No ) )
        = ( set_nat2 @ Xs ) ) ) ).

% partition_set
thf(fact_286_partition__set,axiom,
    ! [P2: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
      ( ( ( partition_a @ P2 @ Xs )
        = ( produc6837034575241423639list_a @ Yes @ No ) )
     => ( ( sup_sup_set_a @ ( set_a2 @ Yes ) @ ( set_a2 @ No ) )
        = ( set_a2 @ Xs ) ) ) ).

% partition_set
thf(fact_287_partition__set,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Yes: list_S5253404628753938141on_nat,No: list_S5253404628753938141on_nat] :
      ( ( ( partit4951714822998333966on_nat @ P2 @ Xs )
        = ( produc2912653624306349395on_nat @ Yes @ No ) )
     => ( ( sup_su343302011326411951on_nat @ ( set_Su4848624856859743608on_nat @ Yes ) @ ( set_Su4848624856859743608on_nat @ No ) )
        = ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% partition_set
thf(fact_288_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_289_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_290_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_291_dist,axiom,
    distin2773651816175405460on_nat @ left ).

% dist
thf(fact_292_image__eqI,axiom,
    ! [B: nat,F: nat > nat,X: nat,A4: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A4 )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_293_image__eqI,axiom,
    ! [B: a,F: nat > a,X: nat,A4: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A4 )
       => ( member_a @ B @ ( image_nat_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_294_image__eqI,axiom,
    ! [B: nat,F: a > nat,X: a,A4: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A4 )
       => ( member_nat @ B @ ( image_a_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_295_image__eqI,axiom,
    ! [B: a,F: a > a,X: a,A4: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A4 )
       => ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_296_image__eqI,axiom,
    ! [B: nat,F: product_prod_a_o > nat,X: product_prod_a_o,A4: set_Product_prod_a_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member2997933407038931562od_a_o @ X @ A4 )
       => ( member_nat @ B @ ( image_8012505923468758860_o_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_297_image__eqI,axiom,
    ! [B: a,F: product_prod_a_o > a,X: product_prod_a_o,A4: set_Product_prod_a_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member2997933407038931562od_a_o @ X @ A4 )
       => ( member_a @ B @ ( image_2171321463282962050_a_o_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_298_image__eqI,axiom,
    ! [B: product_prod_a_o,F: nat > product_prod_a_o,X: nat,A4: set_nat] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_nat @ X @ A4 )
       => ( member2997933407038931562od_a_o @ B @ ( image_5312737573458615594od_a_o @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_299_image__eqI,axiom,
    ! [B: product_prod_a_o,F: a > product_prod_a_o,X: a,A4: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A4 )
       => ( member2997933407038931562od_a_o @ B @ ( image_4279145912174180552od_a_o @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_300_image__eqI,axiom,
    ! [B: nat,F: list_P3196869577824546119od_a_o > nat,X: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5261019209154752368od_a_o @ X @ A4 )
       => ( member_nat @ B @ ( image_5391756926611421138_o_nat @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_301_image__eqI,axiom,
    ! [B: a,F: list_P3196869577824546119od_a_o > a,X: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o] :
      ( ( B
        = ( F @ X ) )
     => ( ( member5261019209154752368od_a_o @ X @ A4 )
       => ( member_a @ B @ ( image_3930418492379151228_a_o_a @ F @ A4 ) ) ) ) ).

% image_eqI
thf(fact_302_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_303_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_304_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_305_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_306_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_307_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_308_append_Oassoc,axiom,
    ! [A: list_P9127870862658512897on_nat,B: list_P9127870862658512897on_nat,C2: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ ( append2865821678206098582on_nat @ A @ B ) @ C2 )
      = ( append2865821678206098582on_nat @ A @ ( append2865821678206098582on_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_309_append_Oassoc,axiom,
    ! [A: list_S5253404628753938141on_nat,B: list_S5253404628753938141on_nat,C2: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ ( append3523812819975775986on_nat @ A @ B ) @ C2 )
      = ( append3523812819975775986on_nat @ A @ ( append3523812819975775986on_nat @ B @ C2 ) ) ) ).

% append.assoc
thf(fact_310_image__ident,axiom,
    ! [Y6: set_nat] :
      ( ( image_nat_nat
        @ ^ [X4: nat] : X4
        @ Y6 )
      = Y6 ) ).

% image_ident
thf(fact_311_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_312_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_313_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_314_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_315_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_316_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_317_distinct__union,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( distin2773651816175405460on_nat @ ( union_82372077702528979on_nat @ Xs @ Ys ) )
      = ( distin2773651816175405460on_nat @ Ys ) ) ).

% distinct_union
thf(fact_318_distinct__union,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( distinct_nat @ ( union_nat @ Xs @ Ys ) )
      = ( distinct_nat @ Ys ) ) ).

% distinct_union
thf(fact_319_set__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( set_Pr937530902923597456on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( sup_su6884158799674119087on_nat @ ( set_Pr937530902923597456on_nat @ Xs ) @ ( set_Pr937530902923597456on_nat @ Ys ) ) ) ).

% set_append
thf(fact_320_set__append,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( set_Product_prod_a_o2 @ ( append2462165029785658588od_a_o @ Xs @ Ys ) )
      = ( sup_su4274294223881134325od_a_o @ ( set_Product_prod_a_o2 @ Xs ) @ ( set_Product_prod_a_o2 @ Ys ) ) ) ).

% set_append
thf(fact_321_set__append,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( set_li2629784761229999580od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) )
      = ( sup_su6115883799403739899od_a_o @ ( set_li2629784761229999580od_a_o @ Xs ) @ ( set_li2629784761229999580od_a_o @ Ys ) ) ) ).

% set_append
thf(fact_322_set__append,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( set_nat2 @ ( append_nat @ Xs @ Ys ) )
      = ( sup_sup_set_nat @ ( set_nat2 @ Xs ) @ ( set_nat2 @ Ys ) ) ) ).

% set_append
thf(fact_323_set__append,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( set_a2 @ ( append_a @ Xs @ Ys ) )
      = ( sup_sup_set_a @ ( set_a2 @ Xs ) @ ( set_a2 @ Ys ) ) ) ).

% set_append
thf(fact_324_set__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( set_Su4848624856859743608on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( sup_su343302011326411951on_nat @ ( set_Su4848624856859743608on_nat @ Xs ) @ ( set_Su4848624856859743608on_nat @ Ys ) ) ) ).

% set_append
thf(fact_325_distinct__product__lists,axiom,
    ! [Xss: list_l8999537460041914605on_nat] :
      ( ! [X3: list_S5253404628753938141on_nat] :
          ( ( member86162308477975540on_nat @ X3 @ ( set_li4371160873219908744on_nat @ Xss ) )
         => ( distin2773651816175405460on_nat @ X3 ) )
     => ( distin7213608244053580452on_nat @ ( produc1748266815408604318on_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_326_distinct__product__lists,axiom,
    ! [Xss: list_list_nat] :
      ( ! [X3: list_nat] :
          ( ( member_list_nat @ X3 @ ( set_list_nat2 @ Xss ) )
         => ( distinct_nat @ X3 ) )
     => ( distinct_list_nat @ ( product_lists_nat @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_327_distinct__product__lists,axiom,
    ! [Xss: list_l8336264072933094989od_a_o] :
      ( ! [X3: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xss ) )
         => ( distin2380507709364613050od_a_o @ X3 ) )
     => ( distin690952206374581440od_a_o @ ( produc3814538932794474544od_a_o @ Xss ) ) ) ).

% distinct_product_lists
thf(fact_328_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_329_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_330_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_331_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_332_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_333_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_334_Compr__image__eq,axiom,
    ! [F: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_nat_nat @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_nat_nat @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_335_Compr__image__eq,axiom,
    ! [F: a > nat,A4: set_a,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_a_nat @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_a_nat @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_336_Compr__image__eq,axiom,
    ! [F: nat > a,A4: set_nat,P2: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_nat_a @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_nat_a @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_337_Compr__image__eq,axiom,
    ! [F: a > a,A4: set_a,P2: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_a_a @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_a_a @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_338_Compr__image__eq,axiom,
    ! [F: nat > product_prod_a_o,A4: set_nat,P2: product_prod_a_o > $o] :
      ( ( collec5729118848859662764od_a_o
        @ ^ [X4: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X4 @ ( image_5312737573458615594od_a_o @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_5312737573458615594od_a_o @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_339_Compr__image__eq,axiom,
    ! [F: a > product_prod_a_o,A4: set_a,P2: product_prod_a_o > $o] :
      ( ( collec5729118848859662764od_a_o
        @ ^ [X4: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X4 @ ( image_4279145912174180552od_a_o @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_4279145912174180552od_a_o @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_340_Compr__image__eq,axiom,
    ! [F: product_prod_a_o > nat,A4: set_Product_prod_a_o,P2: nat > $o] :
      ( ( collect_nat
        @ ^ [X4: nat] :
            ( ( member_nat @ X4 @ ( image_8012505923468758860_o_nat @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_8012505923468758860_o_nat @ F
        @ ( collec5729118848859662764od_a_o
          @ ^ [X4: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_341_Compr__image__eq,axiom,
    ! [F: product_prod_a_o > a,A4: set_Product_prod_a_o,P2: a > $o] :
      ( ( collect_a
        @ ^ [X4: a] :
            ( ( member_a @ X4 @ ( image_2171321463282962050_a_o_a @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_2171321463282962050_a_o_a @ F
        @ ( collec5729118848859662764od_a_o
          @ ^ [X4: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_342_Compr__image__eq,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,A4: set_nat,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X4: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X4 @ ( image_5774815713777330736od_a_o @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_5774815713777330736od_a_o @ F
        @ ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_343_Compr__image__eq,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,A4: set_a,P2: list_P3196869577824546119od_a_o > $o] :
      ( ( collec5117750660557753778od_a_o
        @ ^ [X4: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X4 @ ( image_7865913289429950158od_a_o @ F @ A4 ) )
            & ( P2 @ X4 ) ) )
      = ( image_7865913289429950158od_a_o @ F
        @ ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A4 )
              & ( P2 @ ( F @ X4 ) ) ) ) ) ) ).

% Compr_image_eq
thf(fact_344_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: nat,F: nat > nat] :
      ( ( member_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_345_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: a,F: nat > a] :
      ( ( member_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_nat_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_346_rev__image__eqI,axiom,
    ! [X: a,A4: set_a,B: nat,F: a > nat] :
      ( ( member_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_a_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_347_rev__image__eqI,axiom,
    ! [X: a,A4: set_a,B: a,F: a > a] :
      ( ( member_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_a_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_348_rev__image__eqI,axiom,
    ! [X: product_prod_a_o,A4: set_Product_prod_a_o,B: nat,F: product_prod_a_o > nat] :
      ( ( member2997933407038931562od_a_o @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_8012505923468758860_o_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_349_rev__image__eqI,axiom,
    ! [X: product_prod_a_o,A4: set_Product_prod_a_o,B: a,F: product_prod_a_o > a] :
      ( ( member2997933407038931562od_a_o @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_2171321463282962050_a_o_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_350_rev__image__eqI,axiom,
    ! [X: nat,A4: set_nat,B: product_prod_a_o,F: nat > product_prod_a_o] :
      ( ( member_nat @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member2997933407038931562od_a_o @ B @ ( image_5312737573458615594od_a_o @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_351_rev__image__eqI,axiom,
    ! [X: a,A4: set_a,B: product_prod_a_o,F: a > product_prod_a_o] :
      ( ( member_a @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member2997933407038931562od_a_o @ B @ ( image_4279145912174180552od_a_o @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_352_rev__image__eqI,axiom,
    ! [X: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,B: nat,F: list_P3196869577824546119od_a_o > nat] :
      ( ( member5261019209154752368od_a_o @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_nat @ B @ ( image_5391756926611421138_o_nat @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_353_rev__image__eqI,axiom,
    ! [X: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,B: a,F: list_P3196869577824546119od_a_o > a] :
      ( ( member5261019209154752368od_a_o @ X @ A4 )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_3930418492379151228_a_o_a @ F @ A4 ) ) ) ) ).

% rev_image_eqI
thf(fact_354_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
        @ ^ [X4: product_prod_a_o] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_355_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
        @ ^ [X4: product_prod_a_o] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_356_image__image,axiom,
    ! [F: nat > nat,G: nat > nat,A4: set_nat] :
      ( ( image_nat_nat @ F @ ( image_nat_nat @ G @ A4 ) )
      = ( image_nat_nat
        @ ^ [X4: nat] : ( F @ ( G @ X4 ) )
        @ A4 ) ) ).

% image_image
thf(fact_357_ball__imageD,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o,P2: produc1313295348606329723on_nat > $o] :
      ( ! [X3: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X3 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X7: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X7 @ A4 )
         => ( P2 @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_358_ball__imageD,axiom,
    ! [F: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( image_nat_nat @ F @ A4 ) )
         => ( P2 @ X3 ) )
     => ! [X7: nat] :
          ( ( member_nat @ X7 @ A4 )
         => ( P2 @ ( F @ X7 ) ) ) ) ).

% ball_imageD
thf(fact_359_image__cong,axiom,
    ! [M: set_Product_prod_a_o,N3: set_Product_prod_a_o,F: product_prod_a_o > produc1313295348606329723on_nat,G: product_prod_a_o > produc1313295348606329723on_nat] :
      ( ( M = N3 )
     => ( ! [X3: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X3 @ N3 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_4414724447807847679on_nat @ F @ M )
          = ( image_4414724447807847679on_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_360_image__cong,axiom,
    ! [M: set_nat,N3: set_nat,F: nat > nat,G: nat > nat] :
      ( ( M = N3 )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ N3 )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_nat_nat @ F @ M )
          = ( image_nat_nat @ G @ N3 ) ) ) ) ).

% image_cong
thf(fact_361_bex__imageD,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o,P2: produc1313295348606329723on_nat > $o] :
      ( ? [X7: produc1313295348606329723on_nat] :
          ( ( member5906418179908956708on_nat @ X7 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
          & ( P2 @ X7 ) )
     => ? [X3: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X3 @ A4 )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_362_bex__imageD,axiom,
    ! [F: nat > nat,A4: set_nat,P2: nat > $o] :
      ( ? [X7: nat] :
          ( ( member_nat @ X7 @ ( image_nat_nat @ F @ A4 ) )
          & ( P2 @ X7 ) )
     => ? [X3: nat] :
          ( ( member_nat @ X3 @ A4 )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_363_image__iff,axiom,
    ! [Z2: produc1313295348606329723on_nat,F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o] :
      ( ( member5906418179908956708on_nat @ Z2 @ ( image_4414724447807847679on_nat @ F @ A4 ) )
      = ( ? [X4: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X4 @ A4 )
            & ( Z2
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_364_image__iff,axiom,
    ! [Z2: nat,F: nat > nat,A4: set_nat] :
      ( ( member_nat @ Z2 @ ( image_nat_nat @ F @ A4 ) )
      = ( ? [X4: nat] :
            ( ( member_nat @ X4 @ A4 )
            & ( Z2
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_365_imageI,axiom,
    ! [X: nat,A4: set_nat,F: nat > nat] :
      ( ( member_nat @ X @ A4 )
     => ( member_nat @ ( F @ X ) @ ( image_nat_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_366_imageI,axiom,
    ! [X: nat,A4: set_nat,F: nat > a] :
      ( ( member_nat @ X @ A4 )
     => ( member_a @ ( F @ X ) @ ( image_nat_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_367_imageI,axiom,
    ! [X: a,A4: set_a,F: a > nat] :
      ( ( member_a @ X @ A4 )
     => ( member_nat @ ( F @ X ) @ ( image_a_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_368_imageI,axiom,
    ! [X: a,A4: set_a,F: a > a] :
      ( ( member_a @ X @ A4 )
     => ( member_a @ ( F @ X ) @ ( image_a_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_369_imageI,axiom,
    ! [X: product_prod_a_o,A4: set_Product_prod_a_o,F: product_prod_a_o > nat] :
      ( ( member2997933407038931562od_a_o @ X @ A4 )
     => ( member_nat @ ( F @ X ) @ ( image_8012505923468758860_o_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_370_imageI,axiom,
    ! [X: product_prod_a_o,A4: set_Product_prod_a_o,F: product_prod_a_o > a] :
      ( ( member2997933407038931562od_a_o @ X @ A4 )
     => ( member_a @ ( F @ X ) @ ( image_2171321463282962050_a_o_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_371_imageI,axiom,
    ! [X: nat,A4: set_nat,F: nat > product_prod_a_o] :
      ( ( member_nat @ X @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ X ) @ ( image_5312737573458615594od_a_o @ F @ A4 ) ) ) ).

% imageI
thf(fact_372_imageI,axiom,
    ! [X: a,A4: set_a,F: a > product_prod_a_o] :
      ( ( member_a @ X @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ X ) @ ( image_4279145912174180552od_a_o @ F @ A4 ) ) ) ).

% imageI
thf(fact_373_imageI,axiom,
    ! [X: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,F: list_P3196869577824546119od_a_o > nat] :
      ( ( member5261019209154752368od_a_o @ X @ A4 )
     => ( member_nat @ ( F @ X ) @ ( image_5391756926611421138_o_nat @ F @ A4 ) ) ) ).

% imageI
thf(fact_374_imageI,axiom,
    ! [X: list_P3196869577824546119od_a_o,A4: set_li5011860890973139239od_a_o,F: list_P3196869577824546119od_a_o > a] :
      ( ( member5261019209154752368od_a_o @ X @ A4 )
     => ( member_a @ ( F @ X ) @ ( image_3930418492379151228_a_o_a @ F @ A4 ) ) ) ).

% imageI
thf(fact_375_imageE,axiom,
    ! [B: nat,F: nat > nat,A4: set_nat] :
      ( ( member_nat @ B @ ( image_nat_nat @ F @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_376_imageE,axiom,
    ! [B: nat,F: a > nat,A4: set_a] :
      ( ( member_nat @ B @ ( image_a_nat @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_377_imageE,axiom,
    ! [B: a,F: nat > a,A4: set_nat] :
      ( ( member_a @ B @ ( image_nat_a @ F @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_378_imageE,axiom,
    ! [B: a,F: a > a,A4: set_a] :
      ( ( member_a @ B @ ( image_a_a @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_379_imageE,axiom,
    ! [B: product_prod_a_o,F: nat > product_prod_a_o,A4: set_nat] :
      ( ( member2997933407038931562od_a_o @ B @ ( image_5312737573458615594od_a_o @ F @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_380_imageE,axiom,
    ! [B: product_prod_a_o,F: a > product_prod_a_o,A4: set_a] :
      ( ( member2997933407038931562od_a_o @ B @ ( image_4279145912174180552od_a_o @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_381_imageE,axiom,
    ! [B: nat,F: product_prod_a_o > nat,A4: set_Product_prod_a_o] :
      ( ( member_nat @ B @ ( image_8012505923468758860_o_nat @ F @ A4 ) )
     => ~ ! [X3: product_prod_a_o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member2997933407038931562od_a_o @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_382_imageE,axiom,
    ! [B: a,F: product_prod_a_o > a,A4: set_Product_prod_a_o] :
      ( ( member_a @ B @ ( image_2171321463282962050_a_o_a @ F @ A4 ) )
     => ~ ! [X3: product_prod_a_o] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member2997933407038931562od_a_o @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_383_imageE,axiom,
    ! [B: list_P3196869577824546119od_a_o,F: nat > list_P3196869577824546119od_a_o,A4: set_nat] :
      ( ( member5261019209154752368od_a_o @ B @ ( image_5774815713777330736od_a_o @ F @ A4 ) )
     => ~ ! [X3: nat] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_nat @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_384_imageE,axiom,
    ! [B: list_P3196869577824546119od_a_o,F: a > list_P3196869577824546119od_a_o,A4: set_a] :
      ( ( member5261019209154752368od_a_o @ B @ ( image_7865913289429950158od_a_o @ F @ A4 ) )
     => ~ ! [X3: a] :
            ( ( B
              = ( F @ X3 ) )
           => ~ ( member_a @ X3 @ A4 ) ) ) ).

% imageE
thf(fact_385_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_386_Collect__disj__eq,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,Q: sum_su4907927656612468301on_nat > $o] :
      ( ( collec1912426543561893410on_nat
        @ ^ [X4: sum_su4907927656612468301on_nat] :
            ( ( P2 @ X4 )
            | ( Q @ X4 ) ) )
      = ( sup_su343302011326411951on_nat @ ( collec1912426543561893410on_nat @ P2 ) @ ( collec1912426543561893410on_nat @ Q ) ) ) ).

% Collect_disj_eq
thf(fact_387_sup__set__def,axiom,
    ( sup_su4274294223881134325od_a_o
    = ( ^ [A5: set_Product_prod_a_o,B5: set_Product_prod_a_o] :
          ( collec5729118848859662764od_a_o
          @ ( sup_su1807098469199219240_a_o_o
            @ ^ [X4: product_prod_a_o] : ( member2997933407038931562od_a_o @ X4 @ A5 )
            @ ^ [X4: product_prod_a_o] : ( member2997933407038931562od_a_o @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_388_sup__set__def,axiom,
    ( sup_su6115883799403739899od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o] :
          ( collec5117750660557753778od_a_o
          @ ( sup_su1170899305658069410_a_o_o
            @ ^ [X4: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X4 @ A5 )
            @ ^ [X4: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_389_sup__set__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( sup_sup_nat_o
            @ ^ [X4: nat] : ( member_nat @ X4 @ A5 )
            @ ^ [X4: nat] : ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_390_sup__set__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ( sup_sup_a_o
            @ ^ [X4: a] : ( member_a @ X4 @ A5 )
            @ ^ [X4: a] : ( member_a @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_391_sup__set__def,axiom,
    ( sup_su343302011326411951on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] :
          ( collec1912426543561893410on_nat
          @ ( sup_su2152540609833952918_nat_o
            @ ^ [X4: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X4 @ A5 )
            @ ^ [X4: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X4 @ B5 ) ) ) ) ) ).

% sup_set_def
thf(fact_392_Un__def,axiom,
    ( sup_su4274294223881134325od_a_o
    = ( ^ [A5: set_Product_prod_a_o,B5: set_Product_prod_a_o] :
          ( collec5729118848859662764od_a_o
          @ ^ [X4: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ X4 @ A5 )
              | ( member2997933407038931562od_a_o @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_393_Un__def,axiom,
    ( sup_su6115883799403739899od_a_o
    = ( ^ [A5: set_li5011860890973139239od_a_o,B5: set_li5011860890973139239od_a_o] :
          ( collec5117750660557753778od_a_o
          @ ^ [X4: list_P3196869577824546119od_a_o] :
              ( ( member5261019209154752368od_a_o @ X4 @ A5 )
              | ( member5261019209154752368od_a_o @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_394_Un__def,axiom,
    ( sup_sup_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X4: nat] :
              ( ( member_nat @ X4 @ A5 )
              | ( member_nat @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_395_Un__def,axiom,
    ( sup_sup_set_a
    = ( ^ [A5: set_a,B5: set_a] :
          ( collect_a
          @ ^ [X4: a] :
              ( ( member_a @ X4 @ A5 )
              | ( member_a @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_396_Un__def,axiom,
    ( sup_su343302011326411951on_nat
    = ( ^ [A5: set_Su266273576555847555on_nat,B5: set_Su266273576555847555on_nat] :
          ( collec1912426543561893410on_nat
          @ ^ [X4: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X4 @ A5 )
              | ( member8476201367316484836on_nat @ X4 @ B5 ) ) ) ) ) ).

% Un_def
thf(fact_397_image__Un,axiom,
    ! [F: product_prod_a_o > produc1313295348606329723on_nat,A4: set_Product_prod_a_o,B4: set_Product_prod_a_o] :
      ( ( image_4414724447807847679on_nat @ F @ ( sup_su4274294223881134325od_a_o @ A4 @ B4 ) )
      = ( sup_su6884158799674119087on_nat @ ( image_4414724447807847679on_nat @ F @ A4 ) @ ( image_4414724447807847679on_nat @ F @ B4 ) ) ) ).

% image_Un
thf(fact_398_image__Un,axiom,
    ! [F: nat > nat,A4: set_nat,B4: set_nat] :
      ( ( image_nat_nat @ F @ ( sup_sup_set_nat @ A4 @ B4 ) )
      = ( sup_sup_set_nat @ ( image_nat_nat @ F @ A4 ) @ ( image_nat_nat @ F @ B4 ) ) ) ).

% image_Un
thf(fact_399_image__Un,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,A4: set_Su266273576555847555on_nat,B4: set_Su266273576555847555on_nat] :
      ( ( image_5534209245114917537on_nat @ F @ ( sup_su343302011326411951on_nat @ A4 @ B4 ) )
      = ( sup_su343302011326411951on_nat @ ( image_5534209245114917537on_nat @ F @ A4 ) @ ( image_5534209245114917537on_nat @ F @ B4 ) ) ) ).

% image_Un
thf(fact_400_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_401_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_402_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_403_distinct__conv__nth,axiom,
    ( distin2773651816175405460on_nat
    = ( ^ [Xs2: list_S5253404628753938141on_nat] :
        ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s5686234912572693873on_nat @ Xs2 ) )
             => ( ( I2 != J2 )
               => ( ( nth_Su8471564416988646406on_nat @ Xs2 @ I2 )
                 != ( nth_Su8471564416988646406on_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_404_distinct__conv__nth,axiom,
    ( distin690952206374581440od_a_o
    = ( ^ [Xs2: list_l8336264072933094989od_a_o] :
        ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_s3379185846251495609od_a_o @ Xs2 ) )
             => ( ( I2 != J2 )
               => ( ( nth_li3248255784598043854od_a_o @ Xs2 @ I2 )
                 != ( nth_li3248255784598043854od_a_o @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_405_distinct__conv__nth,axiom,
    ( distinct_nat
    = ( ^ [Xs2: list_nat] :
        ! [I2: nat] :
          ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
         => ! [J2: nat] :
              ( ( ord_less_nat @ J2 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( I2 != J2 )
               => ( ( nth_nat @ Xs2 @ I2 )
                 != ( nth_nat @ Xs2 @ J2 ) ) ) ) ) ) ) ).

% distinct_conv_nth
thf(fact_406_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_407_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_408_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_409_list__update__append1,axiom,
    ! [I: nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat] :
      ( ( ord_less_nat @ I @ ( size_s7490768528479217773on_nat @ Xs ) )
     => ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) @ I @ X )
        = ( append2865821678206098582on_nat @ ( list_u1840724836389618409on_nat @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_410_list__update__append1,axiom,
    ! [I: nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( list_u8577817230714425439on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) @ I @ X )
        = ( append3523812819975775986on_nat @ ( list_u8577817230714425439on_nat @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_411_list__update__append1,axiom,
    ! [I: nat,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( list_u329291161401623989od_a_o @ ( append2159120467191820514od_a_o @ Xs @ Ys ) @ I @ X )
        = ( append2159120467191820514od_a_o @ ( list_u329291161401623989od_a_o @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_412_list__update__append1,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_nat,X: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( list_update_nat @ ( append_nat @ Xs @ Ys ) @ I @ X )
        = ( append_nat @ ( list_update_nat @ Xs @ I @ X ) @ Ys ) ) ) ).

% list_update_append1
thf(fact_413_partition__P,axiom,
    ! [P2: product_prod_a_o > $o,Xs: list_P3196869577824546119od_a_o,Yes: list_P3196869577824546119od_a_o,No: list_P3196869577824546119od_a_o] :
      ( ( ( partit7076252557220421312od_a_o @ P2 @ Xs )
        = ( produc8077463541153823575od_a_o @ Yes @ No ) )
     => ( ! [X7: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X7 @ ( set_Product_prod_a_o2 @ Yes ) )
           => ( P2 @ X7 ) )
        & ! [X7: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X7 @ ( set_Product_prod_a_o2 @ No ) )
           => ~ ( P2 @ X7 ) ) ) ) ).

% partition_P
thf(fact_414_partition__P,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,Xs: list_S5253404628753938141on_nat,Yes: list_S5253404628753938141on_nat,No: list_S5253404628753938141on_nat] :
      ( ( ( partit4951714822998333966on_nat @ P2 @ Xs )
        = ( produc2912653624306349395on_nat @ Yes @ No ) )
     => ( ! [X7: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ Yes ) )
           => ( P2 @ X7 ) )
        & ! [X7: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ No ) )
           => ~ ( P2 @ X7 ) ) ) ) ).

% partition_P
thf(fact_415_partition__P,axiom,
    ! [P2: list_P3196869577824546119od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Yes: list_l8336264072933094989od_a_o,No: list_l8336264072933094989od_a_o] :
      ( ( ( partit8903024691838806214od_a_o @ P2 @ Xs )
        = ( produc7858527432735436247od_a_o @ Yes @ No ) )
     => ( ! [X7: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X7 @ ( set_li2629784761229999580od_a_o @ Yes ) )
           => ( P2 @ X7 ) )
        & ! [X7: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X7 @ ( set_li2629784761229999580od_a_o @ No ) )
           => ~ ( P2 @ X7 ) ) ) ) ).

% partition_P
thf(fact_416_partition__P,axiom,
    ! [P2: nat > $o,Xs: list_nat,Yes: list_nat,No: list_nat] :
      ( ( ( partition_nat @ P2 @ Xs )
        = ( produc2694037385005941721st_nat @ Yes @ No ) )
     => ( ! [X7: nat] :
            ( ( member_nat @ X7 @ ( set_nat2 @ Yes ) )
           => ( P2 @ X7 ) )
        & ! [X7: nat] :
            ( ( member_nat @ X7 @ ( set_nat2 @ No ) )
           => ~ ( P2 @ X7 ) ) ) ) ).

% partition_P
thf(fact_417_partition__P,axiom,
    ! [P2: a > $o,Xs: list_a,Yes: list_a,No: list_a] :
      ( ( ( partition_a @ P2 @ Xs )
        = ( produc6837034575241423639list_a @ Yes @ No ) )
     => ( ! [X7: a] :
            ( ( member_a @ X7 @ ( set_a2 @ Yes ) )
           => ( P2 @ X7 ) )
        & ! [X7: a] :
            ( ( member_a @ X7 @ ( set_a2 @ No ) )
           => ~ ( P2 @ X7 ) ) ) ) ).

% partition_P
thf(fact_418_distinct__Ex1,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,X: product_prod_a_o] :
      ( ( distin2380507709364613050od_a_o @ Xs )
     => ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s1727999245659094451od_a_o @ Xs ) )
            & ( ( nth_Product_prod_a_o @ Xs @ X3 )
              = X )
            & ! [Y7: nat] :
                ( ( ( ord_less_nat @ Y7 @ ( size_s1727999245659094451od_a_o @ Xs ) )
                  & ( ( nth_Product_prod_a_o @ Xs @ Y7 )
                    = X ) )
               => ( Y7 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_419_distinct__Ex1,axiom,
    ! [Xs: list_a,X: a] :
      ( ( distinct_a @ Xs )
     => ( ( member_a @ X @ ( set_a2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_a @ Xs ) )
            & ( ( nth_a @ Xs @ X3 )
              = X )
            & ! [Y7: nat] :
                ( ( ( ord_less_nat @ Y7 @ ( size_size_list_a @ Xs ) )
                  & ( ( nth_a @ Xs @ Y7 )
                    = X ) )
               => ( Y7 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_420_distinct__Ex1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat] :
      ( ( distin2773651816175405460on_nat @ Xs )
     => ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s5686234912572693873on_nat @ Xs ) )
            & ( ( nth_Su8471564416988646406on_nat @ Xs @ X3 )
              = X )
            & ! [Y7: nat] :
                ( ( ( ord_less_nat @ Y7 @ ( size_s5686234912572693873on_nat @ Xs ) )
                  & ( ( nth_Su8471564416988646406on_nat @ Xs @ Y7 )
                    = X ) )
               => ( Y7 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_421_distinct__Ex1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o] :
      ( ( distin690952206374581440od_a_o @ Xs )
     => ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_s3379185846251495609od_a_o @ Xs ) )
            & ( ( nth_li3248255784598043854od_a_o @ Xs @ X3 )
              = X )
            & ! [Y7: nat] :
                ( ( ( ord_less_nat @ Y7 @ ( size_s3379185846251495609od_a_o @ Xs ) )
                  & ( ( nth_li3248255784598043854od_a_o @ Xs @ Y7 )
                    = X ) )
               => ( Y7 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_422_distinct__Ex1,axiom,
    ! [Xs: list_nat,X: nat] :
      ( ( distinct_nat @ Xs )
     => ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
       => ? [X3: nat] :
            ( ( ord_less_nat @ X3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ X3 )
              = X )
            & ! [Y7: nat] :
                ( ( ( ord_less_nat @ Y7 @ ( size_size_list_nat @ Xs ) )
                  & ( ( nth_nat @ Xs @ Y7 )
                    = X ) )
               => ( Y7 = X3 ) ) ) ) ) ).

% distinct_Ex1
thf(fact_423__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_424__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_425__092_060open_062distinct_Aleft_A_061_Adistinct_A_Ins__left_A_064_As__left_J_092_060close_062,axiom,
    ( ( distin2773651816175405460on_nat @ left )
    = ( distin2773651816175405460on_nat @ ( append3523812819975775986on_nat @ ns_left @ s_left ) ) ) ).

% \<open>distinct left = distinct (ns_left @ s_left)\<close>
thf(fact_426_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_427__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_428__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 ) ) )
               => ~ ! [T2: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ T2 @ ( set_Su4848624856859743608on_nat @ S_right ) )
                     => ? [X3: sum_su4907927656612468301on_nat] :
                          ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ S_left ) )
                          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ T2 ) @ ( 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_429__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 ) ) )
          & ! [X7: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
             => ? [Xa: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Xs22 ) )
                  & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa @ X7 ) @ ( 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_430_id__right,axiom,
    ( ( mset_S7907735292211298936on_nat @ right )
    = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ ns_right ) @ ( mset_S7907735292211298936on_nat @ s_right ) ) ) ).

% id_right
thf(fact_431_id__left,axiom,
    ( ( mset_S7907735292211298936on_nat @ left )
    = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ ns_left ) @ ( mset_S7907735292211298936on_nat @ s_left ) ) ) ).

% id_left
thf(fact_432__092_060open_062mset_Aleft_A_061_Amset_A_Ins__left_A_064_As__left_J_092_060close_062,axiom,
    ( ( mset_S7907735292211298936on_nat @ left )
    = ( mset_S7907735292211298936on_nat @ ( append3523812819975775986on_nat @ ns_left @ s_left ) ) ) ).

% \<open>mset left = mset (ns_left @ s_left)\<close>
thf(fact_433_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_434_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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1727999245659094451od_a_o @ Ys12 ) )
               => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ ( nth_Product_prod_a_o @ Xs1 @ I3 ) @ ( nth_Product_prod_a_o @ Ys12 @ I3 ) ) @ NS ) )
           => ( ! [Y3: product_prod_a_o] :
                  ( ( member2997933407038931562od_a_o @ Y3 @ ( set_Product_prod_a_o2 @ Ys23 ) )
                 => ? [X7: product_prod_a_o] :
                      ( ( member2997933407038931562od_a_o @ X7 @ ( set_Product_prod_a_o2 @ Xs23 ) )
                      & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X7 @ Y3 ) @ S3 ) ) )
             => ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ Xs @ Ys ) @ ( multis2022661252535332470od_a_o @ NS @ S3 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_435_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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys12 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs1 @ I3 ) @ ( nth_a @ Ys12 @ I3 ) ) @ NS ) )
           => ( ! [Y3: a] :
                  ( ( member_a @ Y3 @ ( set_a2 @ Ys23 ) )
                 => ? [X7: a] :
                      ( ( member_a @ X7 @ ( set_a2 @ Xs23 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X7 @ Y3 ) @ S3 ) ) )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multis4023584895248362547_ext_a @ NS @ S3 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_436_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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Ys12 ) )
               => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs1 @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Ys12 @ I3 ) ) @ NS ) )
           => ( ! [Y3: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Ys23 ) )
                 => ? [X7: list_P3196869577824546119od_a_o] :
                      ( ( member5261019209154752368od_a_o @ X7 @ ( set_li2629784761229999580od_a_o @ Xs23 ) )
                      & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X7 @ Y3 ) @ S3 ) ) )
             => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis5290879534621336956od_a_o @ NS @ S3 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_437_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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys12 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I3 ) @ ( nth_nat @ Ys12 @ I3 ) ) @ NS ) )
           => ( ! [Y3: nat] :
                  ( ( member_nat @ Y3 @ ( set_nat2 @ Ys23 ) )
                 => ? [X7: nat] :
                      ( ( member_nat @ X7 @ ( set_nat2 @ Xs23 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X7 @ Y3 ) @ S3 ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S3 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_438_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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Ys12 ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs1 @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Ys12 @ I3 ) ) @ NS ) )
           => ( ! [Y3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Ys23 ) )
                 => ? [X7: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ Xs23 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X7 @ Y3 ) @ S3 ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis4183089776494585304on_nat @ NS @ S3 ) ) ) ) ) ) ) ).

% ns_mul_ext_intro
thf(fact_439_ns__mul__ext__elim,axiom,
    ! [Xs: multis8267451372601328135od_a_o,Ys: multis8267451372601328135od_a_o,NS: set_Pr1246879249572150087od_a_o,S3: set_Pr1246879249572150087od_a_o] :
      ( ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ Xs @ Ys ) @ ( multis2022661252535332470od_a_o @ NS @ S3 ) )
     => ? [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 ) )
              & ! [X7: product_prod_a_o] :
                  ( ( member2997933407038931562od_a_o @ X7 @ ( set_Product_prod_a_o2 @ Ys22 ) )
                 => ? [Xa: product_prod_a_o] :
                      ( ( member2997933407038931562od_a_o @ Xa @ ( set_Product_prod_a_o2 @ Xs22 ) )
                      & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_440_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_a,Ys: multiset_a,NS: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multis4023584895248362547_ext_a @ NS @ S3 ) )
     => ? [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 ) )
              & ! [X7: a] :
                  ( ( member_a @ X7 @ ( set_a2 @ Ys22 ) )
                 => ? [Xa: a] :
                      ( ( member_a @ Xa @ ( set_a2 @ Xs22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_441_ns__mul__ext__elim,axiom,
    ! [Xs: multis1714066462791002893od_a_o,Ys: multis1714066462791002893od_a_o,NS: set_Pr7420778282784443591od_a_o,S3: set_Pr7420778282784443591od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis5290879534621336956od_a_o @ NS @ S3 ) )
     => ? [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 ) )
              & ! [X7: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ X7 @ ( set_li2629784761229999580od_a_o @ Ys22 ) )
                 => ? [Xa: list_P3196869577824546119od_a_o] :
                      ( ( member5261019209154752368od_a_o @ Xa @ ( set_li2629784761229999580od_a_o @ Xs22 ) )
                      & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_442_ns__mul__ext__elim,axiom,
    ! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis8831328596575508315xt_nat @ NS @ S3 ) )
     => ? [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 ) )
              & ! [X7: nat] :
                  ( ( member_nat @ X7 @ ( set_nat2 @ Ys22 ) )
                 => ? [Xa: nat] :
                      ( ( member_nat @ Xa @ ( set_nat2 @ Xs22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_443_ns__mul__ext__elim,axiom,
    ! [Xs: multis1856558993800614429on_nat,Ys: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis4183089776494585304on_nat @ NS @ S3 ) )
     => ? [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 ) )
              & ! [X7: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
                 => ? [Xa: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Xs22 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% ns_mul_ext_elim
thf(fact_444__092_060open_062Inl_A_Ix_M_AUnsigned_J_A_092_060in_062_D_Amset_Ans__right_A_L_Amset_As__right_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ x @ multis5512618279175645674signed ) ) @ ( set_ms3922534532180125136on_nat @ ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ ns_right ) @ ( mset_S7907735292211298936on_nat @ s_right ) ) ) ).

% \<open>Inl (x, Unsigned) \<in># mset ns_right + mset s_right\<close>
thf(fact_445_mset__append,axiom,
    ! [Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( mset_P6964650582947932560on_nat @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs ) @ ( mset_P6964650582947932560on_nat @ Ys ) ) ) ).

% mset_append
thf(fact_446_mset__append,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( mset_S7907735292211298936on_nat @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) @ ( mset_S7907735292211298936on_nat @ Ys ) ) ) ).

% mset_append
thf(fact_447_all__ns__ns__mul__ext,axiom,
    ! [As: list_l8336264072933094989od_a_o,Bs: list_l8336264072933094989od_a_o,Ns: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ As )
        = ( size_s3379185846251495609od_a_o @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Bs ) )
           => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I3 ) ) @ Ns ) )
       => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ ( mset_l8139550355095933148od_a_o @ As ) @ ( mset_l8139550355095933148od_a_o @ Bs ) ) @ ( multis5290879534621336956od_a_o @ Ns @ S2 ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_448_all__ns__ns__mul__ext,axiom,
    ! [As: list_nat,Bs: list_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ Ns ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As ) @ ( mset_nat @ Bs ) ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_449_all__ns__ns__mul__ext,axiom,
    ! [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ As )
        = ( size_s5686234912572693873on_nat @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Bs ) )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I3 ) ) @ Ns ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( mset_S7907735292211298936on_nat @ As ) @ ( mset_S7907735292211298936on_nat @ Bs ) ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) ) ) ) ).

% all_ns_ns_mul_ext
thf(fact_450__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_451__092_060open_062Inl_A_Ix_M_AUnsigned_J_A_092_060in_062_D_Amset_Aright_092_060close_062,axiom,
    member8476201367316484836on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ x @ multis5512618279175645674signed ) ) @ ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ right ) ) ).

% \<open>Inl (x, Unsigned) \<in># mset right\<close>
thf(fact_452__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_453_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_454_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_455_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_456_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_457_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_458_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_459_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_460_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_461_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_462_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_463_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_464_self__append__conv2,axiom,
    ! [Y: list_nat,Xs: list_nat] :
      ( ( Y
        = ( append_nat @ Xs @ Y ) )
      = ( Xs = nil_nat ) ) ).

% self_append_conv2
thf(fact_465_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_466_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_467_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_468_append__self__conv2,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Ys )
      = ( Xs = nil_nat ) ) ).

% append_self_conv2
thf(fact_469_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_470_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_471_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_472_self__append__conv,axiom,
    ! [Y: list_nat,Ys: list_nat] :
      ( ( Y
        = ( append_nat @ Y @ Ys ) )
      = ( Ys = nil_nat ) ) ).

% self_append_conv
thf(fact_473_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_474_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_475_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_476_append__self__conv,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( append_nat @ Xs @ Ys )
        = Xs )
      = ( Ys = nil_nat ) ) ).

% append_self_conv
thf(fact_477_append__Nil2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ Xs @ nil_li1372968520647675725od_a_o )
      = Xs ) ).

% append_Nil2
thf(fact_478_append__Nil2,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ Xs @ nil_Su3078906344068544711on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_479_append__Nil2,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ Xs @ nil_Pr8614093721167239297on_nat )
      = Xs ) ).

% append_Nil2
thf(fact_480_append__Nil2,axiom,
    ! [Xs: list_nat] :
      ( ( append_nat @ Xs @ nil_nat )
      = Xs ) ).

% append_Nil2
thf(fact_481_append_Oright__neutral,axiom,
    ! [A: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ A @ nil_li1372968520647675725od_a_o )
      = A ) ).

% append.right_neutral
thf(fact_482_append_Oright__neutral,axiom,
    ! [A: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ A @ nil_Su3078906344068544711on_nat )
      = A ) ).

% append.right_neutral
thf(fact_483_append_Oright__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ A @ nil_Pr8614093721167239297on_nat )
      = A ) ).

% append.right_neutral
thf(fact_484_append_Oright__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ A @ nil_nat )
      = A ) ).

% append.right_neutral
thf(fact_485_list__update__nonempty,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,K: nat,X: list_P3196869577824546119od_a_o] :
      ( ( ( list_u329291161401623989od_a_o @ Xs @ K @ X )
        = nil_li1372968520647675725od_a_o )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% list_update_nonempty
thf(fact_486_list__update__nonempty,axiom,
    ! [Xs: list_S5253404628753938141on_nat,K: nat,X: sum_su4907927656612468301on_nat] :
      ( ( ( list_u8577817230714425439on_nat @ Xs @ K @ X )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% list_update_nonempty
thf(fact_487_list__update__nonempty,axiom,
    ! [Xs: list_P9127870862658512897on_nat,K: nat,X: produc1313295348606329723on_nat] :
      ( ( ( list_u1840724836389618409on_nat @ Xs @ K @ X )
        = nil_Pr8614093721167239297on_nat )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% list_update_nonempty
thf(fact_488_list__update__nonempty,axiom,
    ! [Xs: list_nat,K: nat,X: nat] :
      ( ( ( list_update_nat @ Xs @ K @ X )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% list_update_nonempty
thf(fact_489_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_490_list__ex__simps_I2_J,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o] :
      ~ ( list_e3769318529992910857on_nat @ P2 @ nil_Su3078906344068544711on_nat ) ).

% list_ex_simps(2)
thf(fact_491_list__ex__simps_I2_J,axiom,
    ! [P2: produc1313295348606329723on_nat > $o] :
      ~ ( list_e8248363684495001279on_nat @ P2 @ nil_Pr8614093721167239297on_nat ) ).

% list_ex_simps(2)
thf(fact_492_list__ex__simps_I2_J,axiom,
    ! [P2: nat > $o] :
      ~ ( list_ex_nat @ P2 @ nil_nat ) ).

% list_ex_simps(2)
thf(fact_493_set__mset__union,axiom,
    ! [M: multis1856558993800614429on_nat,N3: multis1856558993800614429on_nat] :
      ( ( set_ms3922534532180125136on_nat @ ( plus_p2586647234311342374on_nat @ M @ N3 ) )
      = ( sup_su343302011326411951on_nat @ ( set_ms3922534532180125136on_nat @ M ) @ ( set_ms3922534532180125136on_nat @ N3 ) ) ) ).

% set_mset_union
thf(fact_494_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_495_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_496_set__mset__mset,axiom,
    ! [Xs: list_nat] :
      ( ( set_mset_nat @ ( mset_nat @ Xs ) )
      = ( set_nat2 @ Xs ) ) ).

% set_mset_mset
thf(fact_497_set__mset__mset,axiom,
    ! [Xs: list_a] :
      ( ( set_mset_a @ ( mset_a @ Xs ) )
      = ( set_a2 @ Xs ) ) ).

% set_mset_mset
thf(fact_498_set__mset__mset,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) )
      = ( set_Su4848624856859743608on_nat @ Xs ) ) ).

% set_mset_mset
thf(fact_499_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_500_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_501_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_502_eq__Nil__appendI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( Xs = Ys )
     => ( Xs
        = ( append_nat @ nil_nat @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_503_append_Oleft__neutral,axiom,
    ! [A: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ nil_li1372968520647675725od_a_o @ A )
      = A ) ).

% append.left_neutral
thf(fact_504_append_Oleft__neutral,axiom,
    ! [A: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_505_append_Oleft__neutral,axiom,
    ! [A: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_506_append_Oleft__neutral,axiom,
    ! [A: list_nat] :
      ( ( append_nat @ nil_nat @ A )
      = A ) ).

% append.left_neutral
thf(fact_507_append__Nil,axiom,
    ! [Ys: list_l8336264072933094989od_a_o] :
      ( ( append2159120467191820514od_a_o @ nil_li1372968520647675725od_a_o @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_508_append__Nil,axiom,
    ! [Ys: list_S5253404628753938141on_nat] :
      ( ( append3523812819975775986on_nat @ nil_Su3078906344068544711on_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_509_append__Nil,axiom,
    ! [Ys: list_P9127870862658512897on_nat] :
      ( ( append2865821678206098582on_nat @ nil_Pr8614093721167239297on_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_510_append__Nil,axiom,
    ! [Ys: list_nat] :
      ( ( append_nat @ nil_nat @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_511_distinct_Osimps_I1_J,axiom,
    distin690952206374581440od_a_o @ nil_li1372968520647675725od_a_o ).

% distinct.simps(1)
thf(fact_512_distinct_Osimps_I1_J,axiom,
    distin2773651816175405460on_nat @ nil_Su3078906344068544711on_nat ).

% distinct.simps(1)
thf(fact_513_distinct_Osimps_I1_J,axiom,
    distin8410642530753306484on_nat @ nil_Pr8614093721167239297on_nat ).

% distinct.simps(1)
thf(fact_514_distinct_Osimps_I1_J,axiom,
    distinct_nat @ nil_nat ).

% distinct.simps(1)
thf(fact_515_in__multiset__in__set,axiom,
    ! [X: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ X @ ( set_ms2137302536914688126od_a_o @ ( mset_P8977776989772351958od_a_o @ Xs ) ) )
      = ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_516_in__multiset__in__set,axiom,
    ! [X: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X @ ( set_ms4142991692181744516od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs ) ) )
      = ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_517_in__multiset__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_mset_nat @ ( mset_nat @ Xs ) ) )
      = ( member_nat @ X @ ( set_nat2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_518_in__multiset__in__set,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_mset_a @ ( mset_a @ Xs ) ) )
      = ( member_a @ X @ ( set_a2 @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_519_in__multiset__in__set,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X @ ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) ) )
      = ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ).

% in_multiset_in_set
thf(fact_520_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: list_P3196869577824546119od_a_o] :
      ( ( list_u329291161401623989od_a_o @ nil_li1372968520647675725od_a_o @ I @ V )
      = nil_li1372968520647675725od_a_o ) ).

% list_update.simps(1)
thf(fact_521_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: sum_su4907927656612468301on_nat] :
      ( ( list_u8577817230714425439on_nat @ nil_Su3078906344068544711on_nat @ I @ V )
      = nil_Su3078906344068544711on_nat ) ).

% list_update.simps(1)
thf(fact_522_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: produc1313295348606329723on_nat] :
      ( ( list_u1840724836389618409on_nat @ nil_Pr8614093721167239297on_nat @ I @ V )
      = nil_Pr8614093721167239297on_nat ) ).

% list_update.simps(1)
thf(fact_523_list__update_Osimps_I1_J,axiom,
    ! [I: nat,V: nat] :
      ( ( list_update_nat @ nil_nat @ I @ V )
      = nil_nat ) ).

% list_update.simps(1)
thf(fact_524_list__update__code_I1_J,axiom,
    ! [I: nat,Y: list_P3196869577824546119od_a_o] :
      ( ( list_u329291161401623989od_a_o @ nil_li1372968520647675725od_a_o @ I @ Y )
      = nil_li1372968520647675725od_a_o ) ).

% list_update_code(1)
thf(fact_525_list__update__code_I1_J,axiom,
    ! [I: nat,Y: sum_su4907927656612468301on_nat] :
      ( ( list_u8577817230714425439on_nat @ nil_Su3078906344068544711on_nat @ I @ Y )
      = nil_Su3078906344068544711on_nat ) ).

% list_update_code(1)
thf(fact_526_list__update__code_I1_J,axiom,
    ! [I: nat,Y: produc1313295348606329723on_nat] :
      ( ( list_u1840724836389618409on_nat @ nil_Pr8614093721167239297on_nat @ I @ Y )
      = nil_Pr8614093721167239297on_nat ) ).

% list_update_code(1)
thf(fact_527_list__update__code_I1_J,axiom,
    ! [I: nat,Y: nat] :
      ( ( list_update_nat @ nil_nat @ I @ Y )
      = nil_nat ) ).

% list_update_code(1)
thf(fact_528_ns__mul__ext__point,axiom,
    ! [As: multis8267451372601328135od_a_o,Bs: multis8267451372601328135od_a_o,NS: set_Pr1246879249572150087od_a_o,S3: set_Pr1246879249572150087od_a_o,B: product_prod_a_o] :
      ( ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ As @ Bs ) @ ( multis2022661252535332470od_a_o @ NS @ S3 ) )
     => ( ( member2997933407038931562od_a_o @ B @ ( set_ms2137302536914688126od_a_o @ Bs ) )
       => ? [X3: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X3 @ ( set_ms2137302536914688126od_a_o @ As ) )
            & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X3 @ B ) @ ( sup_su173060327693730075od_a_o @ NS @ S3 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_529_ns__mul__ext__point,axiom,
    ! [As: multis1714066462791002893od_a_o,Bs: multis1714066462791002893od_a_o,NS: set_Pr7420778282784443591od_a_o,S3: set_Pr7420778282784443591od_a_o,B: list_P3196869577824546119od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ As @ Bs ) @ ( multis5290879534621336956od_a_o @ NS @ S3 ) )
     => ( ( member5261019209154752368od_a_o @ B @ ( set_ms4142991692181744516od_a_o @ Bs ) )
       => ? [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_ms4142991692181744516od_a_o @ As ) )
            & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ B ) @ ( sup_su8024054902183157403od_a_o @ NS @ S3 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_530_ns__mul__ext__point,axiom,
    ! [As: multiset_a,Bs: multiset_a,NS: set_Product_prod_a_a,S3: set_Product_prod_a_a,B: a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ As @ Bs ) @ ( multis4023584895248362547_ext_a @ NS @ S3 ) )
     => ( ( member_a @ B @ ( set_mset_a @ Bs ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( set_mset_a @ As ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ B ) @ ( sup_su3048258781599657691od_a_a @ NS @ S3 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_531_ns__mul__ext__point,axiom,
    ! [As: multiset_nat,Bs: multiset_nat,NS: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,B: nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ As @ Bs ) @ ( multis8831328596575508315xt_nat @ NS @ S3 ) )
     => ( ( member_nat @ B @ ( set_mset_nat @ Bs ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_mset_nat @ As ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ ( sup_su6327502436637775413at_nat @ NS @ S3 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_532_ns__mul__ext__point,axiom,
    ! [As: multis1856558993800614429on_nat,Bs: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat,B: sum_su4907927656612468301on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ As @ Bs ) @ ( multis4183089776494585304on_nat @ NS @ S3 ) )
     => ( ( member8476201367316484836on_nat @ B @ ( set_ms3922534532180125136on_nat @ Bs ) )
       => ? [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ As ) )
            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ B ) @ ( sup_su6884158799674119087on_nat @ NS @ S3 ) ) ) ) ) ).

% ns_mul_ext_point
thf(fact_533_s__mul__ext__point,axiom,
    ! [As: multis8267451372601328135od_a_o,Bs: multis8267451372601328135od_a_o,NS: set_Pr1246879249572150087od_a_o,S3: set_Pr1246879249572150087od_a_o,B: product_prod_a_o] :
      ( ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ As @ Bs ) @ ( multis5035663574087566680od_a_o @ NS @ S3 ) )
     => ( ( member2997933407038931562od_a_o @ B @ ( set_ms2137302536914688126od_a_o @ Bs ) )
       => ? [X3: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ X3 @ ( set_ms2137302536914688126od_a_o @ As ) )
            & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X3 @ B ) @ ( sup_su173060327693730075od_a_o @ NS @ S3 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_534_s__mul__ext__point,axiom,
    ! [As: multis1714066462791002893od_a_o,Bs: multis1714066462791002893od_a_o,NS: set_Pr7420778282784443591od_a_o,S3: set_Pr7420778282784443591od_a_o,B: list_P3196869577824546119od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ As @ Bs ) @ ( multis7840163239927194462od_a_o @ NS @ S3 ) )
     => ( ( member5261019209154752368od_a_o @ B @ ( set_ms4142991692181744516od_a_o @ Bs ) )
       => ? [X3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X3 @ ( set_ms4142991692181744516od_a_o @ As ) )
            & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X3 @ B ) @ ( sup_su8024054902183157403od_a_o @ NS @ S3 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_535_s__mul__ext__point,axiom,
    ! [As: multiset_a,Bs: multiset_a,NS: set_Product_prod_a_a,S3: set_Product_prod_a_a,B: a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ As @ Bs ) @ ( multiset_s_mul_ext_a @ NS @ S3 ) )
     => ( ( member_a @ B @ ( set_mset_a @ Bs ) )
       => ? [X3: a] :
            ( ( member_a @ X3 @ ( set_mset_a @ As ) )
            & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ B ) @ ( sup_su3048258781599657691od_a_a @ NS @ S3 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_536_s__mul__ext__point,axiom,
    ! [As: multiset_nat,Bs: multiset_nat,NS: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat,B: nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ As @ Bs ) @ ( multis4215214144601311993xt_nat @ NS @ S3 ) )
     => ( ( member_nat @ B @ ( set_mset_nat @ Bs ) )
       => ? [X3: nat] :
            ( ( member_nat @ X3 @ ( set_mset_nat @ As ) )
            & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ B ) @ ( sup_su6327502436637775413at_nat @ NS @ S3 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_537_s__mul__ext__point,axiom,
    ! [As: multis1856558993800614429on_nat,Bs: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat,B: sum_su4907927656612468301on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ As @ Bs ) @ ( multis1620213760974180726on_nat @ NS @ S3 ) )
     => ( ( member8476201367316484836on_nat @ B @ ( set_ms3922534532180125136on_nat @ Bs ) )
       => ? [X3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X3 @ ( set_ms3922534532180125136on_nat @ As ) )
            & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ B ) @ ( sup_su6884158799674119087on_nat @ NS @ S3 ) ) ) ) ) ).

% s_mul_ext_point
thf(fact_538_not__listrel1__Nil,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ nil_li1372968520647675725od_a_o ) @ ( listre4186574385322583590od_a_o @ R ) ) ).

% not_listrel1_Nil
thf(fact_539_not__listrel1__Nil,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ nil_Su3078906344068544711on_nat ) @ ( listre1270753918075970734on_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_540_not__listrel1__Nil,axiom,
    ! [Xs: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ~ ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ nil_Pr8614093721167239297on_nat ) @ ( listre7129750038259667930on_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_541_not__listrel1__Nil,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel1_nat @ R ) ) ).

% not_listrel1_Nil
thf(fact_542_not__Nil__listrel1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ~ ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Xs ) @ ( listre4186574385322583590od_a_o @ R ) ) ).

% not_Nil_listrel1
thf(fact_543_not__Nil__listrel1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ~ ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ Xs ) @ ( listre1270753918075970734on_nat @ R ) ) ).

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

% not_Nil_listrel1
thf(fact_545_not__Nil__listrel1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ~ ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel1_nat @ R ) ) ).

% not_Nil_listrel1
thf(fact_546_listrel_ONil,axiom,
    ! [R: set_Pr1261947904930325089at_nat] : ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) @ ( listrel_nat_nat @ R ) ) ).

% listrel.Nil
thf(fact_547_listrel_ONil,axiom,
    ! [R: set_Pr8850484174313502844_o_nat] : ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ nil_li1372968520647675725od_a_o @ nil_nat ) @ ( listre90293961589800166_o_nat @ R ) ) ).

% listrel.Nil
thf(fact_548_listrel_ONil,axiom,
    ! [R: set_Pr94232683564258082od_a_o] : ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ nil_nat @ nil_li1372968520647675725od_a_o ) @ ( listre473352748755709764od_a_o @ R ) ) ).

% listrel.Nil
thf(fact_549_listrel_ONil,axiom,
    ! [R: set_Pr3345771464143190622at_nat] : ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ nil_Su3078906344068544711on_nat @ nil_nat ) @ ( listre6443138976769382928at_nat @ R ) ) ).

% listrel.Nil
thf(fact_550_listrel_ONil,axiom,
    ! [R: set_Pr7552745763645690846on_nat] : ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ nil_nat @ nil_Su3078906344068544711on_nat ) @ ( listre7340406004007350800on_nat @ R ) ) ).

% listrel.Nil
thf(fact_551_listrel_ONil,axiom,
    ! [R: set_Pr7420778282784443591od_a_o] : ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o ) @ ( listre9037316752283237297od_a_o @ R ) ) ).

% listrel.Nil
thf(fact_552_listrel_ONil,axiom,
    ! [R: set_Pr6144483798281041401on_nat] : ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ nil_li1372968520647675725od_a_o @ nil_Su3078906344068544711on_nat ) @ ( listre7573934803181360995on_nat @ R ) ) ).

% listrel.Nil
thf(fact_553_listrel_ONil,axiom,
    ! [R: set_Pr9140199768791955813od_a_o] : ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ nil_Su3078906344068544711on_nat @ nil_li1372968520647675725od_a_o ) @ ( listre1105410954321393415od_a_o @ R ) ) ).

% listrel.Nil
thf(fact_554_listrel_ONil,axiom,
    ! [R: set_Pr2535564044409765851on_nat] : ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ nil_Su3078906344068544711on_nat ) @ ( listre5398654966139722893on_nat @ R ) ) ).

% listrel.Nil
thf(fact_555_listrel_ONil,axiom,
    ! [R: set_Pr3840340505911729584at_nat] : ( member3237407652033183783st_nat @ ( produc6052571375282406026st_nat @ nil_Pr8614093721167239297on_nat @ nil_nat ) @ ( listre6517312349145516570at_nat @ R ) ) ).

% listrel.Nil
thf(fact_556_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ nil_nat @ Xs ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_557_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ nil_li1372968520647675725od_a_o @ Xs ) @ ( listre90293961589800166_o_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_558_listrel__Nil1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr94232683564258082od_a_o] :
      ( ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ nil_nat @ Xs ) @ ( listre473352748755709764od_a_o @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil1
thf(fact_559_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr3345771464143190622at_nat] :
      ( ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ nil_Su3078906344068544711on_nat @ Xs ) @ ( listre6443138976769382928at_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_560_listrel__Nil1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr7552745763645690846on_nat] :
      ( ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ nil_nat @ Xs ) @ ( listre7340406004007350800on_nat @ R ) )
     => ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% listrel_Nil1
thf(fact_561_listrel__Nil1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ Xs ) @ ( listre9037316752283237297od_a_o @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil1
thf(fact_562_listrel__Nil1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr6144483798281041401on_nat] :
      ( ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ nil_li1372968520647675725od_a_o @ Xs ) @ ( listre7573934803181360995on_nat @ R ) )
     => ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% listrel_Nil1
thf(fact_563_listrel__Nil1,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr9140199768791955813od_a_o] :
      ( ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ nil_Su3078906344068544711on_nat @ Xs ) @ ( listre1105410954321393415od_a_o @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil1
thf(fact_564_listrel__Nil1,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ Xs ) @ ( listre5398654966139722893on_nat @ R ) )
     => ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% listrel_Nil1
thf(fact_565_listrel__Nil1,axiom,
    ! [Xs: list_nat,R: set_Pr3840340505911729584at_nat] :
      ( ( member3237407652033183783st_nat @ ( produc6052571375282406026st_nat @ nil_Pr8614093721167239297on_nat @ Xs ) @ ( listre6517312349145516570at_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil1
thf(fact_566_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ nil_nat ) @ ( listrel_nat_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_567_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr94232683564258082od_a_o] :
      ( ( member548352139293716857od_a_o @ ( produc8901617039433855636od_a_o @ Xs @ nil_li1372968520647675725od_a_o ) @ ( listre473352748755709764od_a_o @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_568_listrel__Nil2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr8850484174313502844_o_nat] :
      ( ( member1162260694010923379st_nat @ ( produc4688861313923565270st_nat @ Xs @ nil_nat ) @ ( listre90293961589800166_o_nat @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil2
thf(fact_569_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr7552745763645690846on_nat] :
      ( ( member452587771129616071on_nat @ ( produc9138095300642541526on_nat @ Xs @ nil_Su3078906344068544711on_nat ) @ ( listre7340406004007350800on_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_570_listrel__Nil2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr3345771464143190622at_nat] :
      ( ( member380504172838623943st_nat @ ( produc845777339779355990st_nat @ Xs @ nil_nat ) @ ( listre6443138976769382928at_nat @ R ) )
     => ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% listrel_Nil2
thf(fact_571_listrel__Nil2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr7420778282784443591od_a_o] :
      ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ Xs @ nil_li1372968520647675725od_a_o ) @ ( listre9037316752283237297od_a_o @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil2
thf(fact_572_listrel__Nil2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr9140199768791955813od_a_o] :
      ( ( member2603097825888147004od_a_o @ ( produc1975650536614756183od_a_o @ Xs @ nil_li1372968520647675725od_a_o ) @ ( listre1105410954321393415od_a_o @ R ) )
     => ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% listrel_Nil2
thf(fact_573_listrel__Nil2,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,R: set_Pr6144483798281041401on_nat] :
      ( ( member4418620086355165936on_nat @ ( produc7184742879426471635on_nat @ Xs @ nil_Su3078906344068544711on_nat ) @ ( listre7573934803181360995on_nat @ R ) )
     => ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% listrel_Nil2
thf(fact_574_listrel__Nil2,axiom,
    ! [Xs: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ nil_Su3078906344068544711on_nat ) @ ( listre5398654966139722893on_nat @ R ) )
     => ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% listrel_Nil2
thf(fact_575_listrel__Nil2,axiom,
    ! [Xs: list_nat,R: set_Pr2972819075928174550on_nat] :
      ( ( member9199622813682680237on_nat @ ( produc8802776732007976904on_nat @ Xs @ nil_Pr8614093721167239297on_nat ) @ ( listre7269971596283755512on_nat @ R ) )
     => ( Xs = nil_nat ) ) ).

% listrel_Nil2
thf(fact_576_partition_Osimps_I1_J,axiom,
    ! [P2: list_P3196869577824546119od_a_o > $o] :
      ( ( partit8903024691838806214od_a_o @ P2 @ nil_li1372968520647675725od_a_o )
      = ( produc7858527432735436247od_a_o @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o ) ) ).

% partition.simps(1)
thf(fact_577_partition_Osimps_I1_J,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o] :
      ( ( partit4951714822998333966on_nat @ P2 @ nil_Su3078906344068544711on_nat )
      = ( produc2912653624306349395on_nat @ nil_Su3078906344068544711on_nat @ nil_Su3078906344068544711on_nat ) ) ).

% partition.simps(1)
thf(fact_578_partition_Osimps_I1_J,axiom,
    ! [P2: produc1313295348606329723on_nat > $o] :
      ( ( partit4945910930363278970on_nat @ P2 @ nil_Pr8614093721167239297on_nat )
      = ( produc7152764266822961111on_nat @ nil_Pr8614093721167239297on_nat @ nil_Pr8614093721167239297on_nat ) ) ).

% partition.simps(1)
thf(fact_579_partition_Osimps_I1_J,axiom,
    ! [P2: nat > $o] :
      ( ( partition_nat @ P2 @ nil_nat )
      = ( produc2694037385005941721st_nat @ nil_nat @ nil_nat ) ) ).

% partition.simps(1)
thf(fact_580_nth__mem__mset,axiom,
    ! [I: nat,Ls: list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s1727999245659094451od_a_o @ Ls ) )
     => ( member2997933407038931562od_a_o @ ( nth_Product_prod_a_o @ Ls @ I ) @ ( set_ms2137302536914688126od_a_o @ ( mset_P8977776989772351958od_a_o @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_581_nth__mem__mset,axiom,
    ! [I: nat,Ls: list_a] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Ls ) )
     => ( member_a @ ( nth_a @ Ls @ I ) @ ( set_mset_a @ ( mset_a @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_582_nth__mem__mset,axiom,
    ! [I: nat,Ls: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ I @ ( size_s5686234912572693873on_nat @ Ls ) )
     => ( member8476201367316484836on_nat @ ( nth_Su8471564416988646406on_nat @ Ls @ I ) @ ( set_ms3922534532180125136on_nat @ ( mset_S7907735292211298936on_nat @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_583_nth__mem__mset,axiom,
    ! [I: nat,Ls: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ I @ ( size_s3379185846251495609od_a_o @ Ls ) )
     => ( member5261019209154752368od_a_o @ ( nth_li3248255784598043854od_a_o @ Ls @ I ) @ ( set_ms4142991692181744516od_a_o @ ( mset_l8139550355095933148od_a_o @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_584_nth__mem__mset,axiom,
    ! [I: nat,Ls: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ls ) )
     => ( member_nat @ ( nth_nat @ Ls @ I ) @ ( set_mset_nat @ ( mset_nat @ Ls ) ) ) ) ).

% nth_mem_mset
thf(fact_585_ns__mul__ext__NS__union__S,axiom,
    ! [NS: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( multis4183089776494585304on_nat @ ( sup_su6884158799674119087on_nat @ NS @ S3 ) @ S3 )
      = ( multis4183089776494585304on_nat @ NS @ S3 ) ) ).

% ns_mul_ext_NS_union_S
thf(fact_586_ns__mul__extI__old,axiom,
    ! [A4: multis8267451372601328135od_a_o,Xs: list_P3196869577824546119od_a_o,A22: multis8267451372601328135od_a_o,B4: multis8267451372601328135od_a_o,Ys: list_P3196869577824546119od_a_o,B22: multis8267451372601328135od_a_o,Ns: set_Pr1246879249572150087od_a_o,S2: set_Pr1246879249572150087od_a_o] :
      ( ( A4
        = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Ys ) @ B22 ) )
       => ( ( ( size_s1727999245659094451od_a_o @ Xs )
            = ( size_s1727999245659094451od_a_o @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1727999245659094451od_a_o @ Ys ) )
               => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ ( nth_Product_prod_a_o @ Xs @ I3 ) @ ( nth_Product_prod_a_o @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B3: product_prod_a_o] :
                  ( ( member2997933407038931562od_a_o @ B3 @ ( set_ms2137302536914688126od_a_o @ B22 ) )
                 => ? [A7: product_prod_a_o] :
                      ( ( member2997933407038931562od_a_o @ A7 @ ( set_ms2137302536914688126od_a_o @ A22 ) )
                      & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ A7 @ B3 ) @ S2 ) ) )
             => ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ A4 @ B4 ) @ ( multis2022661252535332470od_a_o @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_587_ns__mul__extI__old,axiom,
    ! [A4: multiset_a,Xs: list_a,A22: multiset_a,B4: multiset_a,Ys: list_a,B22: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( A4
        = ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_plus_multiset_a @ ( mset_a @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_a @ Xs )
            = ( size_size_list_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B3: a] :
                  ( ( member_a @ B3 @ ( set_mset_a @ B22 ) )
                 => ? [A7: a] :
                      ( ( member_a @ A7 @ ( set_mset_a @ A22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A7 @ B3 ) @ S2 ) ) )
             => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_588_ns__mul__extI__old,axiom,
    ! [A4: multis1714066462791002893od_a_o,Xs: list_l8336264072933094989od_a_o,A22: multis1714066462791002893od_a_o,B4: multis1714066462791002893od_a_o,Ys: list_l8336264072933094989od_a_o,B22: multis1714066462791002893od_a_o,Ns: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o] :
      ( ( A4
        = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Ys ) @ B22 ) )
       => ( ( ( size_s3379185846251495609od_a_o @ Xs )
            = ( size_s3379185846251495609od_a_o @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Ys ) )
               => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B3: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ B3 @ ( set_ms4142991692181744516od_a_o @ B22 ) )
                 => ? [A7: list_P3196869577824546119od_a_o] :
                      ( ( member5261019209154752368od_a_o @ A7 @ ( set_ms4142991692181744516od_a_o @ A22 ) )
                      & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A7 @ B3 ) @ S2 ) ) )
             => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ A4 @ B4 ) @ ( multis5290879534621336956od_a_o @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_589_ns__mul__extI__old,axiom,
    ! [A4: multiset_nat,Xs: list_nat,A22: multiset_nat,B4: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( A4
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B3: nat] :
                  ( ( member_nat @ B3 @ ( set_mset_nat @ B22 ) )
                 => ? [A7: nat] :
                      ( ( member_nat @ A7 @ ( set_mset_nat @ A22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B3 ) @ S2 ) ) )
             => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A4 @ B4 ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_590_ns__mul__extI__old,axiom,
    ! [A4: multis1856558993800614429on_nat,Xs: list_S5253404628753938141on_nat,A22: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ys: list_S5253404628753938141on_nat,B22: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( A4
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s5686234912572693873on_nat @ Xs )
            = ( size_s5686234912572693873on_nat @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Ys ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I3 ) ) @ Ns ) )
           => ( ! [B3: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ B3 @ ( set_ms3922534532180125136on_nat @ B22 ) )
                 => ? [A7: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A22 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B3 ) @ S2 ) ) )
             => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) ) ) ) ) ) ) ).

% ns_mul_extI_old
thf(fact_591_mset__eq__setD,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,Ys: list_P3196869577824546119od_a_o] :
      ( ( ( mset_P8977776989772351958od_a_o @ Xs )
        = ( mset_P8977776989772351958od_a_o @ Ys ) )
     => ( ( set_Product_prod_a_o2 @ Xs )
        = ( set_Product_prod_a_o2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_592_mset__eq__setD,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( mset_l8139550355095933148od_a_o @ Xs )
        = ( mset_l8139550355095933148od_a_o @ Ys ) )
     => ( ( set_li2629784761229999580od_a_o @ Xs )
        = ( set_li2629784761229999580od_a_o @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_593_mset__eq__setD,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( set_nat2 @ Xs )
        = ( set_nat2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_594_mset__eq__setD,axiom,
    ! [Xs: list_a,Ys: list_a] :
      ( ( ( mset_a @ Xs )
        = ( mset_a @ Ys ) )
     => ( ( set_a2 @ Xs )
        = ( set_a2 @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_595_mset__eq__setD,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs )
        = ( mset_S7907735292211298936on_nat @ Ys ) )
     => ( ( set_Su4848624856859743608on_nat @ Xs )
        = ( set_Su4848624856859743608on_nat @ Ys ) ) ) ).

% mset_eq_setD
thf(fact_596_mset__eq__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs )
        = ( mset_S7907735292211298936on_nat @ Ys ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% mset_eq_length
thf(fact_597_mset__eq__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( mset_l8139550355095933148od_a_o @ Xs )
        = ( mset_l8139550355095933148od_a_o @ Ys ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% mset_eq_length
thf(fact_598_mset__eq__length,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% mset_eq_length
thf(fact_599_s__mul__ext__elim,axiom,
    ! [Xs: multis4835999835052358849on_nat,Ys: multis4835999835052358849on_nat,NS: set_Pr8202701763255893447on_nat,S3: set_Pr8202701763255893447on_nat] :
      ( ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ Xs @ Ys ) @ ( multis9203262125639806226on_nat @ NS @ S3 ) )
     => ? [Xs12: list_P9127870862658512897on_nat,Xs22: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs12 ) @ ( mset_P6964650582947932560on_nat @ Xs22 ) ) )
          & ? [Ys1: list_P9127870862658512897on_nat,Ys22: list_P9127870862658512897on_nat] :
              ( ( Ys
                = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys1 ) @ ( mset_P6964650582947932560on_nat @ Ys22 ) ) )
              & ( ( size_s7490768528479217773on_nat @ Xs12 )
                = ( size_s7490768528479217773on_nat @ Ys1 ) )
              & ( Xs22 != nil_Pr8614093721167239297on_nat )
              & ! [I4: nat] :
                  ( ( ord_less_nat @ I4 @ ( size_s7490768528479217773on_nat @ Ys1 ) )
                 => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs12 @ I4 ) @ ( nth_Pr3336893728888831618on_nat @ Ys1 @ I4 ) ) @ NS ) )
              & ! [X7: produc1313295348606329723on_nat] :
                  ( ( member5906418179908956708on_nat @ X7 @ ( set_Pr937530902923597456on_nat @ Ys22 ) )
                 => ? [Xa: produc1313295348606329723on_nat] :
                      ( ( member5906418179908956708on_nat @ Xa @ ( set_Pr937530902923597456on_nat @ Xs22 ) )
                      & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_600_s__mul__ext__elim,axiom,
    ! [Xs: multis8267451372601328135od_a_o,Ys: multis8267451372601328135od_a_o,NS: set_Pr1246879249572150087od_a_o,S3: set_Pr1246879249572150087od_a_o] :
      ( ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ Xs @ Ys ) @ ( multis5035663574087566680od_a_o @ NS @ S3 ) )
     => ? [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 ) )
              & ( Xs22 != nil_Product_prod_a_o )
              & ! [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 ) )
              & ! [X7: product_prod_a_o] :
                  ( ( member2997933407038931562od_a_o @ X7 @ ( set_Product_prod_a_o2 @ Ys22 ) )
                 => ? [Xa: product_prod_a_o] :
                      ( ( member2997933407038931562od_a_o @ Xa @ ( set_Product_prod_a_o2 @ Xs22 ) )
                      & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_601_s__mul__ext__elim,axiom,
    ! [Xs: multiset_a,Ys: multiset_a,NS: set_Product_prod_a_a,S3: set_Product_prod_a_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multiset_s_mul_ext_a @ NS @ S3 ) )
     => ? [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 ) )
              & ( Xs22 != nil_a )
              & ! [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 ) )
              & ! [X7: a] :
                  ( ( member_a @ X7 @ ( set_a2 @ Ys22 ) )
                 => ? [Xa: a] :
                      ( ( member_a @ Xa @ ( set_a2 @ Xs22 ) )
                      & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_602_s__mul__ext__elim,axiom,
    ! [Xs: multis1714066462791002893od_a_o,Ys: multis1714066462791002893od_a_o,NS: set_Pr7420778282784443591od_a_o,S3: set_Pr7420778282784443591od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis7840163239927194462od_a_o @ NS @ S3 ) )
     => ? [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 ) )
              & ( Xs22 != nil_li1372968520647675725od_a_o )
              & ! [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 ) )
              & ! [X7: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ X7 @ ( set_li2629784761229999580od_a_o @ Ys22 ) )
                 => ? [Xa: list_P3196869577824546119od_a_o] :
                      ( ( member5261019209154752368od_a_o @ Xa @ ( set_li2629784761229999580od_a_o @ Xs22 ) )
                      & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_603_s__mul__ext__elim,axiom,
    ! [Xs: multiset_nat,Ys: multiset_nat,NS: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S3 ) )
     => ? [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 ) )
              & ( Xs22 != nil_nat )
              & ! [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 ) )
              & ! [X7: nat] :
                  ( ( member_nat @ X7 @ ( set_nat2 @ Ys22 ) )
                 => ? [Xa: nat] :
                      ( ( member_nat @ Xa @ ( set_nat2 @ Xs22 ) )
                      & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_604_s__mul__ext__elim,axiom,
    ! [Xs: multis1856558993800614429on_nat,Ys: multis1856558993800614429on_nat,NS: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis1620213760974180726on_nat @ NS @ S3 ) )
     => ? [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 ) )
              & ( Xs22 != nil_Su3078906344068544711on_nat )
              & ! [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 ) )
              & ! [X7: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ Ys22 ) )
                 => ? [Xa: sum_su4907927656612468301on_nat] :
                      ( ( member8476201367316484836on_nat @ Xa @ ( set_Su4848624856859743608on_nat @ Xs22 ) )
                      & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ Xa @ X7 ) @ S3 ) ) ) ) ) ) ).

% s_mul_ext_elim
thf(fact_605_s__mul__ext__intro,axiom,
    ! [Xs: multis4835999835052358849on_nat,Xs1: list_P9127870862658512897on_nat,Xs23: list_P9127870862658512897on_nat,Ys: multis4835999835052358849on_nat,Ys12: list_P9127870862658512897on_nat,Ys23: list_P9127870862658512897on_nat,NS: set_Pr8202701763255893447on_nat,S3: set_Pr8202701763255893447on_nat] :
      ( ( Xs
        = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Xs1 ) @ ( mset_P6964650582947932560on_nat @ Xs23 ) ) )
     => ( ( Ys
          = ( plus_p7916160393854186104on_nat @ ( mset_P6964650582947932560on_nat @ Ys12 ) @ ( mset_P6964650582947932560on_nat @ Ys23 ) ) )
       => ( ( ( size_s7490768528479217773on_nat @ Xs1 )
            = ( size_s7490768528479217773on_nat @ Ys12 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s7490768528479217773on_nat @ Ys12 ) )
               => ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ ( nth_Pr3336893728888831618on_nat @ Xs1 @ I3 ) @ ( nth_Pr3336893728888831618on_nat @ Ys12 @ I3 ) ) @ NS ) )
           => ( ( Xs23 != nil_Pr8614093721167239297on_nat )
             => ( ! [Y3: produc1313295348606329723on_nat] :
                    ( ( member5906418179908956708on_nat @ Y3 @ ( set_Pr937530902923597456on_nat @ Ys23 ) )
                   => ? [X7: produc1313295348606329723on_nat] :
                        ( ( member5906418179908956708on_nat @ X7 @ ( set_Pr937530902923597456on_nat @ Xs23 ) )
                        & ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X7 @ Y3 ) @ S3 ) ) )
               => ( member8390928645697427088on_nat @ ( produc4043817641637764055on_nat @ Xs @ Ys ) @ ( multis9203262125639806226on_nat @ NS @ S3 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_606_s__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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1727999245659094451od_a_o @ Ys12 ) )
               => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ ( nth_Product_prod_a_o @ Xs1 @ I3 ) @ ( nth_Product_prod_a_o @ Ys12 @ I3 ) ) @ NS ) )
           => ( ( Xs23 != nil_Product_prod_a_o )
             => ( ! [Y3: product_prod_a_o] :
                    ( ( member2997933407038931562od_a_o @ Y3 @ ( set_Product_prod_a_o2 @ Ys23 ) )
                   => ? [X7: product_prod_a_o] :
                        ( ( member2997933407038931562od_a_o @ X7 @ ( set_Product_prod_a_o2 @ Xs23 ) )
                        & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ X7 @ Y3 ) @ S3 ) ) )
               => ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ Xs @ Ys ) @ ( multis5035663574087566680od_a_o @ NS @ S3 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_607_s__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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys12 ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs1 @ I3 ) @ ( nth_a @ Ys12 @ I3 ) ) @ NS ) )
           => ( ( Xs23 != nil_a )
             => ( ! [Y3: a] :
                    ( ( member_a @ Y3 @ ( set_a2 @ Ys23 ) )
                   => ? [X7: a] :
                        ( ( member_a @ X7 @ ( set_a2 @ Xs23 ) )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X7 @ Y3 ) @ S3 ) ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ Xs @ Ys ) @ ( multiset_s_mul_ext_a @ NS @ S3 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_608_s__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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Ys12 ) )
               => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs1 @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Ys12 @ I3 ) ) @ NS ) )
           => ( ( Xs23 != nil_li1372968520647675725od_a_o )
             => ( ! [Y3: list_P3196869577824546119od_a_o] :
                    ( ( member5261019209154752368od_a_o @ Y3 @ ( set_li2629784761229999580od_a_o @ Ys23 ) )
                   => ? [X7: list_P3196869577824546119od_a_o] :
                        ( ( member5261019209154752368od_a_o @ X7 @ ( set_li2629784761229999580od_a_o @ Xs23 ) )
                        & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ X7 @ Y3 ) @ S3 ) ) )
               => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ Xs @ Ys ) @ ( multis7840163239927194462od_a_o @ NS @ S3 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_609_s__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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys12 ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs1 @ I3 ) @ ( nth_nat @ Ys12 @ I3 ) ) @ NS ) )
           => ( ( Xs23 != nil_nat )
             => ( ! [Y3: nat] :
                    ( ( member_nat @ Y3 @ ( set_nat2 @ Ys23 ) )
                   => ? [X7: nat] :
                        ( ( member_nat @ X7 @ ( set_nat2 @ Xs23 ) )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X7 @ Y3 ) @ S3 ) ) )
               => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ Xs @ Ys ) @ ( multis4215214144601311993xt_nat @ NS @ S3 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_610_s__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,S3: 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 ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Ys12 ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs1 @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Ys12 @ I3 ) ) @ NS ) )
           => ( ( Xs23 != nil_Su3078906344068544711on_nat )
             => ( ! [Y3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ Y3 @ ( set_Su4848624856859743608on_nat @ Ys23 ) )
                   => ? [X7: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ X7 @ ( set_Su4848624856859743608on_nat @ Xs23 ) )
                        & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X7 @ Y3 ) @ S3 ) ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ Xs @ Ys ) @ ( multis1620213760974180726on_nat @ NS @ S3 ) ) ) ) ) ) ) ) ).

% s_mul_ext_intro
thf(fact_611_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( mset_nat @ Xs )
        = ( mset_nat @ Ys ) )
     => ( ( distinct_nat @ Xs )
        = ( distinct_nat @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_612_mset__eq__imp__distinct__iff,axiom,
    ! [Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( mset_S7907735292211298936on_nat @ Xs )
        = ( mset_S7907735292211298936on_nat @ Ys ) )
     => ( ( distin2773651816175405460on_nat @ Xs )
        = ( distin2773651816175405460on_nat @ Ys ) ) ) ).

% mset_eq_imp_distinct_iff
thf(fact_613_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_P3196869577824546119od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( distin2380507709364613050od_a_o @ X )
     => ( ( distin2380507709364613050od_a_o @ Y )
       => ( ( ( set_Product_prod_a_o2 @ X )
            = ( set_Product_prod_a_o2 @ Y ) )
          = ( ( mset_P8977776989772351958od_a_o @ X )
            = ( mset_P8977776989772351958od_a_o @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_614_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_l8336264072933094989od_a_o,Y: list_l8336264072933094989od_a_o] :
      ( ( distin690952206374581440od_a_o @ X )
     => ( ( distin690952206374581440od_a_o @ Y )
       => ( ( ( set_li2629784761229999580od_a_o @ X )
            = ( set_li2629784761229999580od_a_o @ Y ) )
          = ( ( mset_l8139550355095933148od_a_o @ X )
            = ( mset_l8139550355095933148od_a_o @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_615_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( distinct_nat @ X )
     => ( ( distinct_nat @ Y )
       => ( ( ( set_nat2 @ X )
            = ( set_nat2 @ Y ) )
          = ( ( mset_nat @ X )
            = ( mset_nat @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_616_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( distinct_a @ X )
     => ( ( distinct_a @ Y )
       => ( ( ( set_a2 @ X )
            = ( set_a2 @ Y ) )
          = ( ( mset_a @ X )
            = ( mset_a @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_617_set__eq__iff__mset__eq__distinct,axiom,
    ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
      ( ( distin2773651816175405460on_nat @ X )
     => ( ( distin2773651816175405460on_nat @ Y )
       => ( ( ( set_Su4848624856859743608on_nat @ X )
            = ( set_Su4848624856859743608on_nat @ Y ) )
          = ( ( mset_S7907735292211298936on_nat @ X )
            = ( mset_S7907735292211298936on_nat @ Y ) ) ) ) ) ).

% set_eq_iff_mset_eq_distinct
thf(fact_618_ns__ns__mul__ext__union__compat,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat,D2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ C3 @ D2 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B4 @ D2 ) ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) ) ) ) ).

% ns_ns_mul_ext_union_compat
thf(fact_619_s__mul__ext__extend__left,axiom,
    ! [B4: multis1856558993800614429on_nat,C3: multis1856558993800614429on_nat,W: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat,A4: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B4 @ C3 ) @ ( multis1620213760974180726on_nat @ W @ S3 ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ B4 ) @ C3 ) @ ( multis1620213760974180726on_nat @ W @ S3 ) ) ) ).

% s_mul_ext_extend_left
thf(fact_620_s__ns__mul__ext,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) ) ) ).

% s_ns_mul_ext
thf(fact_621_s__ns__mul__ext__union__compat,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat,D2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ C3 @ D2 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B4 @ D2 ) ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) ) ) ) ).

% s_ns_mul_ext_union_compat
thf(fact_622_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_623_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_624_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_625_add__less__cancel__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_626_add__less__cancel__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_627_s__mul__extI__old,axiom,
    ! [A4: multis8267451372601328135od_a_o,Xs: list_P3196869577824546119od_a_o,A22: multis8267451372601328135od_a_o,B4: multis8267451372601328135od_a_o,Ys: list_P3196869577824546119od_a_o,B22: multis8267451372601328135od_a_o,Ns: set_Pr1246879249572150087od_a_o,S2: set_Pr1246879249572150087od_a_o] :
      ( ( A4
        = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p3034917348713716670od_a_o @ ( mset_P8977776989772351958od_a_o @ Ys ) @ B22 ) )
       => ( ( ( size_s1727999245659094451od_a_o @ Xs )
            = ( size_s1727999245659094451od_a_o @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s1727999245659094451od_a_o @ Ys ) )
               => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ ( nth_Product_prod_a_o @ Xs @ I3 ) @ ( nth_Product_prod_a_o @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z6798545679276860990od_a_o )
             => ( ! [B3: product_prod_a_o] :
                    ( ( member2997933407038931562od_a_o @ B3 @ ( set_ms2137302536914688126od_a_o @ B22 ) )
                   => ? [A7: product_prod_a_o] :
                        ( ( member2997933407038931562od_a_o @ A7 @ ( set_ms2137302536914688126od_a_o @ A22 ) )
                        & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ A7 @ B3 ) @ S2 ) ) )
               => ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ A4 @ B4 ) @ ( multis5035663574087566680od_a_o @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_628_s__mul__extI__old,axiom,
    ! [A4: multiset_a,Xs: list_a,A22: multiset_a,B4: multiset_a,Ys: list_a,B22: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a] :
      ( ( A4
        = ( plus_plus_multiset_a @ ( mset_a @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_plus_multiset_a @ ( mset_a @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_a @ Xs )
            = ( size_size_list_a @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Ys ) )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ ( nth_a @ Xs @ I3 ) @ ( nth_a @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_zero_multiset_a )
             => ( ! [B3: a] :
                    ( ( member_a @ B3 @ ( set_mset_a @ B22 ) )
                   => ? [A7: a] :
                        ( ( member_a @ A7 @ ( set_mset_a @ A22 ) )
                        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A7 @ B3 ) @ S2 ) ) )
               => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_629_s__mul__extI__old,axiom,
    ! [A4: multis1714066462791002893od_a_o,Xs: list_l8336264072933094989od_a_o,A22: multis1714066462791002893od_a_o,B4: multis1714066462791002893od_a_o,Ys: list_l8336264072933094989od_a_o,B22: multis1714066462791002893od_a_o,Ns: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o] :
      ( ( A4
        = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p8105455965692640580od_a_o @ ( mset_l8139550355095933148od_a_o @ Ys ) @ B22 ) )
       => ( ( ( size_s3379185846251495609od_a_o @ Xs )
            = ( size_s3379185846251495609od_a_o @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Ys ) )
               => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ Xs @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z8905637151584487364od_a_o )
             => ( ! [B3: list_P3196869577824546119od_a_o] :
                    ( ( member5261019209154752368od_a_o @ B3 @ ( set_ms4142991692181744516od_a_o @ B22 ) )
                   => ? [A7: list_P3196869577824546119od_a_o] :
                        ( ( member5261019209154752368od_a_o @ A7 @ ( set_ms4142991692181744516od_a_o @ A22 ) )
                        & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A7 @ B3 ) @ S2 ) ) )
               => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ A4 @ B4 ) @ ( multis7840163239927194462od_a_o @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_630_s__mul__extI__old,axiom,
    ! [A4: multiset_nat,Xs: list_nat,A22: multiset_nat,B4: multiset_nat,Ys: list_nat,B22: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat] :
      ( ( A4
        = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p6334493942879108393et_nat @ ( mset_nat @ Ys ) @ B22 ) )
       => ( ( ( size_size_list_nat @ Xs )
            = ( size_size_list_nat @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Ys ) )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ I3 ) @ ( nth_nat @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z7348594199698428585et_nat )
             => ( ! [B3: nat] :
                    ( ( member_nat @ B3 @ ( set_mset_nat @ B22 ) )
                   => ? [A7: nat] :
                        ( ( member_nat @ A7 @ ( set_mset_nat @ A22 ) )
                        & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B3 ) @ S2 ) ) )
               => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A4 @ B4 ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_631_s__mul__extI__old,axiom,
    ! [A4: multis1856558993800614429on_nat,Xs: list_S5253404628753938141on_nat,A22: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ys: list_S5253404628753938141on_nat,B22: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( A4
        = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) @ A22 ) )
     => ( ( B4
          = ( plus_p2586647234311342374on_nat @ ( mset_S7907735292211298936on_nat @ Ys ) @ B22 ) )
       => ( ( ( size_s5686234912572693873on_nat @ Xs )
            = ( size_s5686234912572693873on_nat @ Ys ) )
         => ( ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Ys ) )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ Xs @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Ys @ I3 ) ) @ Ns ) )
           => ( ( A22 != zero_z8702567803361615526on_nat )
             => ( ! [B3: sum_su4907927656612468301on_nat] :
                    ( ( member8476201367316484836on_nat @ B3 @ ( set_ms3922534532180125136on_nat @ B22 ) )
                   => ? [A7: sum_su4907927656612468301on_nat] :
                        ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A22 ) )
                        & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B3 ) @ S2 ) ) )
               => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) ) ) ) ) ) ) ) ).

% s_mul_extI_old
thf(fact_632_sup__Un__eq,axiom,
    ! [R2: set_Product_prod_a_o,S3: set_Product_prod_a_o] :
      ( ( sup_su1807098469199219240_a_o_o
        @ ^ [X4: product_prod_a_o] : ( member2997933407038931562od_a_o @ X4 @ R2 )
        @ ^ [X4: product_prod_a_o] : ( member2997933407038931562od_a_o @ X4 @ S3 ) )
      = ( ^ [X4: product_prod_a_o] : ( member2997933407038931562od_a_o @ X4 @ ( sup_su4274294223881134325od_a_o @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_633_sup__Un__eq,axiom,
    ! [R2: set_li5011860890973139239od_a_o,S3: set_li5011860890973139239od_a_o] :
      ( ( sup_su1170899305658069410_a_o_o
        @ ^ [X4: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X4 @ R2 )
        @ ^ [X4: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X4 @ S3 ) )
      = ( ^ [X4: list_P3196869577824546119od_a_o] : ( member5261019209154752368od_a_o @ X4 @ ( sup_su6115883799403739899od_a_o @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_634_sup__Un__eq,axiom,
    ! [R2: set_nat,S3: set_nat] :
      ( ( sup_sup_nat_o
        @ ^ [X4: nat] : ( member_nat @ X4 @ R2 )
        @ ^ [X4: nat] : ( member_nat @ X4 @ S3 ) )
      = ( ^ [X4: nat] : ( member_nat @ X4 @ ( sup_sup_set_nat @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_635_sup__Un__eq,axiom,
    ! [R2: set_a,S3: set_a] :
      ( ( sup_sup_a_o
        @ ^ [X4: a] : ( member_a @ X4 @ R2 )
        @ ^ [X4: a] : ( member_a @ X4 @ S3 ) )
      = ( ^ [X4: a] : ( member_a @ X4 @ ( sup_sup_set_a @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_636_sup__Un__eq,axiom,
    ! [R2: set_Su266273576555847555on_nat,S3: set_Su266273576555847555on_nat] :
      ( ( sup_su2152540609833952918_nat_o
        @ ^ [X4: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X4 @ R2 )
        @ ^ [X4: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X4 @ S3 ) )
      = ( ^ [X4: sum_su4907927656612468301on_nat] : ( member8476201367316484836on_nat @ X4 @ ( sup_su343302011326411951on_nat @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq
thf(fact_637_right,axiom,
    ( right
    = ( append3523812819975775986on_nat
      @ ( map_a_2711072540680412167on_nat
        @ ^ [X4: a] : ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5512618279175645674signed ) )
        @ xs )
      @ ( map_na5706685718985795555on_nat @ sum_In255631617079920996tation @ cs ) ) ) ).

% right
thf(fact_638_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_l8336264072933094989od_a_o,Bs: list_l8336264072933094989od_a_o,NS: set_Pr7420778282784443591od_a_o,S3: set_Pr7420778282784443591od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ As )
        = ( size_s3379185846251495609od_a_o @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3379185846251495609od_a_o @ Bs ) )
           => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As @ I3 ) @ ( nth_li3248255784598043854od_a_o @ Bs @ I3 ) ) @ 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 @ S3 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_639_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_nat,Bs: list_nat,NS: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( size_size_list_nat @ As )
        = ( size_size_list_nat @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Bs ) )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As @ I3 ) @ ( nth_nat @ Bs @ I3 ) ) @ 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 @ S3 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_640_mul__ext__list_Oall__ns__imp__ns,axiom,
    ! [As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat,NS: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ As )
        = ( size_s5686234912572693873on_nat @ Bs ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5686234912572693873on_nat @ Bs ) )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As @ I3 ) @ ( nth_Su8471564416988646406on_nat @ Bs @ I3 ) ) @ 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 @ S3 ) ) ) ) ) ) ) ).

% mul_ext_list.all_ns_imp_ns
thf(fact_641_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A4: multis8267451372601328135od_a_o,B4: multis8267451372601328135od_a_o,Ns: set_Pr1246879249572150087od_a_o,S2: set_Pr1246879249572150087od_a_o,C3: multis8267451372601328135od_a_o,D2: multis8267451372601328135od_a_o] :
      ( ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ A4 @ B4 ) @ ( multis2022661252535332470od_a_o @ Ns @ S2 ) )
     => ( ( C3 != zero_z6798545679276860990od_a_o )
       => ( ! [D: product_prod_a_o] :
              ( ( member2997933407038931562od_a_o @ D @ ( set_ms2137302536914688126od_a_o @ D2 ) )
             => ? [C4: product_prod_a_o] :
                  ( ( member2997933407038931562od_a_o @ C4 @ ( set_ms2137302536914688126od_a_o @ C3 ) )
                  & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ C4 @ D ) @ S2 ) ) )
         => ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ ( plus_p3034917348713716670od_a_o @ A4 @ C3 ) @ ( plus_p3034917348713716670od_a_o @ B4 @ D2 ) ) @ ( multis5035663574087566680od_a_o @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_642_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A4: multis1714066462791002893od_a_o,B4: multis1714066462791002893od_a_o,Ns: set_Pr7420778282784443591od_a_o,S2: set_Pr7420778282784443591od_a_o,C3: multis1714066462791002893od_a_o,D2: multis1714066462791002893od_a_o] :
      ( ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ A4 @ B4 ) @ ( multis5290879534621336956od_a_o @ Ns @ S2 ) )
     => ( ( C3 != zero_z8905637151584487364od_a_o )
       => ( ! [D: list_P3196869577824546119od_a_o] :
              ( ( member5261019209154752368od_a_o @ D @ ( set_ms4142991692181744516od_a_o @ D2 ) )
             => ? [C4: list_P3196869577824546119od_a_o] :
                  ( ( member5261019209154752368od_a_o @ C4 @ ( set_ms4142991692181744516od_a_o @ C3 ) )
                  & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ C4 @ D ) @ S2 ) ) )
         => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ ( plus_p8105455965692640580od_a_o @ A4 @ C3 ) @ ( plus_p8105455965692640580od_a_o @ B4 @ D2 ) ) @ ( multis7840163239927194462od_a_o @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_643_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A4: multiset_a,B4: multiset_a,Ns: set_Product_prod_a_a,S2: set_Product_prod_a_a,C3: multiset_a,D2: multiset_a] :
      ( ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ ( multis4023584895248362547_ext_a @ Ns @ S2 ) )
     => ( ( C3 != zero_zero_multiset_a )
       => ( ! [D: a] :
              ( ( member_a @ D @ ( set_mset_a @ D2 ) )
             => ? [C4: a] :
                  ( ( member_a @ C4 @ ( set_mset_a @ C3 ) )
                  & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ C4 @ D ) @ S2 ) ) )
         => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ ( plus_plus_multiset_a @ A4 @ C3 ) @ ( plus_plus_multiset_a @ B4 @ D2 ) ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_644_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A4: multiset_nat,B4: multiset_nat,Ns: set_Pr1261947904930325089at_nat,S2: set_Pr1261947904930325089at_nat,C3: multiset_nat,D2: multiset_nat] :
      ( ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A4 @ B4 ) @ ( multis8831328596575508315xt_nat @ Ns @ S2 ) )
     => ( ( C3 != zero_z7348594199698428585et_nat )
       => ( ! [D: nat] :
              ( ( member_nat @ D @ ( set_mset_nat @ D2 ) )
             => ? [C4: nat] :
                  ( ( member_nat @ C4 @ ( set_mset_nat @ C3 ) )
                  & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ C4 @ D ) @ S2 ) ) )
         => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( plus_p6334493942879108393et_nat @ A4 @ C3 ) @ ( plus_p6334493942879108393et_nat @ B4 @ D2 ) ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_645_ns__s__mul__ext__union__multiset__l,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat,D2: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) )
     => ( ( C3 != zero_z8702567803361615526on_nat )
       => ( ! [D: sum_su4907927656612468301on_nat] :
              ( ( member8476201367316484836on_nat @ D @ ( set_ms3922534532180125136on_nat @ D2 ) )
             => ? [C4: sum_su4907927656612468301on_nat] :
                  ( ( member8476201367316484836on_nat @ C4 @ ( set_ms3922534532180125136on_nat @ C3 ) )
                  & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ C4 @ D ) @ S2 ) ) )
         => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B4 @ D2 ) ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) ) ) ) ) ).

% ns_s_mul_ext_union_multiset_l
thf(fact_646_map__ident,axiom,
    ( ( map_nat_nat
      @ ^ [X4: nat] : X4 )
    = ( ^ [Xs2: list_nat] : Xs2 ) ) ).

% map_ident
thf(fact_647_not__gr__zero,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_648_list_Omap__disc__iff,axiom,
    ! [F: nat > nat,A: list_nat] :
      ( ( ( map_nat_nat @ F @ A )
        = nil_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_649_list_Omap__disc__iff,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,A: list_nat] :
      ( ( ( map_na764632570197019441od_a_o @ F @ A )
        = nil_li1372968520647675725od_a_o )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_650_list_Omap__disc__iff,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,A: list_l8336264072933094989od_a_o] :
      ( ( ( map_li381573783031109843_o_nat @ F @ A )
        = nil_nat )
      = ( A = nil_li1372968520647675725od_a_o ) ) ).

% list.map_disc_iff
thf(fact_651_list_Omap__disc__iff,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,A: list_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ A )
        = nil_Su3078906344068544711on_nat )
      = ( A = nil_nat ) ) ).

% list.map_disc_iff
thf(fact_652_list_Omap__disc__iff,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,A: list_S5253404628753938141on_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ A )
        = nil_nat )
      = ( A = nil_Su3078906344068544711on_nat ) ) ).

% list.map_disc_iff
thf(fact_653_list_Omap__disc__iff,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,A: list_a] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ A )
        = nil_Su3078906344068544711on_nat )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_654_list_Omap__disc__iff,axiom,
    ! [F: a > list_S5253404628753938141on_nat,A: list_a] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ A )
        = nil_li5563851662744376407on_nat )
      = ( A = nil_a ) ) ).

% list.map_disc_iff
thf(fact_655_list_Omap__disc__iff,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,A: list_l8336264072933094989od_a_o] :
      ( ( ( map_li880244354692327940od_a_o @ F @ A )
        = nil_li1372968520647675725od_a_o )
      = ( A = nil_li1372968520647675725od_a_o ) ) ).

% list.map_disc_iff
thf(fact_656_list_Omap__disc__iff,axiom,
    ! [F: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o,A: list_S5253404628753938141on_nat] :
      ( ( ( map_Su446534476614543092od_a_o @ F @ A )
        = nil_li1372968520647675725od_a_o )
      = ( A = nil_Su3078906344068544711on_nat ) ) ).

% list.map_disc_iff
thf(fact_657_list_Omap__disc__iff,axiom,
    ! [F: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat,A: list_l8336264072933094989od_a_o] :
      ( ( ( map_li6915058325474510672on_nat @ F @ A )
        = nil_Su3078906344068544711on_nat )
      = ( A = nil_li1372968520647675725od_a_o ) ) ).

% list.map_disc_iff
thf(fact_658_Nil__is__map__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( nil_nat
        = ( map_nat_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_659_Nil__is__map__conv,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,Xs: list_nat] :
      ( ( nil_li1372968520647675725od_a_o
        = ( map_na764632570197019441od_a_o @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_660_Nil__is__map__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( nil_nat
        = ( map_li381573783031109843_o_nat @ F @ Xs ) )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% Nil_is_map_conv
thf(fact_661_Nil__is__map__conv,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat] :
      ( ( nil_Su3078906344068544711on_nat
        = ( map_na5706685718985795555on_nat @ F @ Xs ) )
      = ( Xs = nil_nat ) ) ).

% Nil_is_map_conv
thf(fact_662_Nil__is__map__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat] :
      ( ( nil_nat
        = ( map_Su4809418691747827683at_nat @ F @ Xs ) )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% Nil_is_map_conv
thf(fact_663_Nil__is__map__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( nil_Su3078906344068544711on_nat
        = ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_664_Nil__is__map__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( nil_li5563851662744376407on_nat
        = ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( Xs = nil_a ) ) ).

% Nil_is_map_conv
thf(fact_665_Nil__is__map__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( nil_li1372968520647675725od_a_o
        = ( map_li880244354692327940od_a_o @ F @ Xs ) )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% Nil_is_map_conv
thf(fact_666_Nil__is__map__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o,Xs: list_S5253404628753938141on_nat] :
      ( ( nil_li1372968520647675725od_a_o
        = ( map_Su446534476614543092od_a_o @ F @ Xs ) )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% Nil_is_map_conv
thf(fact_667_Nil__is__map__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( nil_Su3078906344068544711on_nat
        = ( map_li6915058325474510672on_nat @ F @ Xs ) )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% Nil_is_map_conv
thf(fact_668_map__is__Nil__conv,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_669_map__is__Nil__conv,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,Xs: list_nat] :
      ( ( ( map_na764632570197019441od_a_o @ F @ Xs )
        = nil_li1372968520647675725od_a_o )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_670_map__is__Nil__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( map_li381573783031109843_o_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% map_is_Nil_conv
thf(fact_671_map__is__Nil__conv,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_nat ) ) ).

% map_is_Nil_conv
thf(fact_672_map__is__Nil__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = nil_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% map_is_Nil_conv
thf(fact_673_map__is__Nil__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_674_map__is__Nil__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = nil_li5563851662744376407on_nat )
      = ( Xs = nil_a ) ) ).

% map_is_Nil_conv
thf(fact_675_map__is__Nil__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( map_li880244354692327940od_a_o @ F @ Xs )
        = nil_li1372968520647675725od_a_o )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% map_is_Nil_conv
thf(fact_676_map__is__Nil__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o,Xs: list_S5253404628753938141on_nat] :
      ( ( ( map_Su446534476614543092od_a_o @ F @ Xs )
        = nil_li1372968520647675725od_a_o )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% map_is_Nil_conv
thf(fact_677_map__is__Nil__conv,axiom,
    ! [F: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( ( map_li6915058325474510672on_nat @ F @ Xs )
        = nil_Su3078906344068544711on_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% map_is_Nil_conv
thf(fact_678_map__eq__conv,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat,G: nat > list_P9127870862658512897on_nat] :
      ( ( ( map_na8715451351012060651on_nat @ F @ Xs )
        = ( map_na8715451351012060651on_nat @ G @ Xs ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) ) ) ) ).

% map_eq_conv
thf(fact_679_map__eq__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) ) ) ) ).

% map_eq_conv
thf(fact_680_map__eq__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,G: a > sum_su4907927656612468301on_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Xs ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) ) ) ) ).

% map_eq_conv
thf(fact_681_map__eq__conv,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a,G: a > list_P9127870862658512897on_nat] :
      ( ( ( map_a_7431629901395152583on_nat @ F @ Xs )
        = ( map_a_7431629901395152583on_nat @ G @ Xs ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) ) ) ) ).

% map_eq_conv
thf(fact_682_map__eq__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,G: a > list_S5253404628753938141on_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( map_a_3887198048842487703on_nat @ G @ Xs ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs ) )
           => ( ( F @ X4 )
              = ( G @ X4 ) ) ) ) ) ).

% map_eq_conv
thf(fact_683_length__map,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( size_size_list_nat @ ( map_nat_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_684_length__map,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,Xs: list_nat] :
      ( ( size_s3379185846251495609od_a_o @ ( map_na764632570197019441od_a_o @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_685_length__map,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( size_size_list_nat @ ( map_li381573783031109843_o_nat @ F @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_map
thf(fact_686_length__map,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( size_s5686234912572693873on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_687_length__map,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat] :
      ( ( size_s5686234912572693873on_nat @ ( map_na5706685718985795555on_nat @ F @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_map
thf(fact_688_length__map,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat] :
      ( ( size_size_list_nat @ ( map_Su4809418691747827683at_nat @ F @ Xs ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_map
thf(fact_689_length__map,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( size_s650155532264187265on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( size_size_list_a @ Xs ) ) ).

% length_map
thf(fact_690_length__map,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( size_s3379185846251495609od_a_o @ ( map_li880244354692327940od_a_o @ F @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_map
thf(fact_691_length__map,axiom,
    ! [F: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( size_s5686234912572693873on_nat @ ( map_li6915058325474510672on_nat @ F @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% length_map
thf(fact_692_length__map,axiom,
    ! [F: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o,Xs: list_S5253404628753938141on_nat] :
      ( ( size_s3379185846251495609od_a_o @ ( map_Su446534476614543092od_a_o @ F @ Xs ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% length_map
thf(fact_693_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_694_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_695_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_696_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_697_map__append,axiom,
    ! [F: produc1313295348606329723on_nat > produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( map_Pr6472934937144306180on_nat @ F @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( map_Pr6472934937144306180on_nat @ F @ Xs ) @ ( map_Pr6472934937144306180on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_698_map__append,axiom,
    ! [F: produc1313295348606329723on_nat > sum_su4907927656612468301on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( map_Pr8685107614402573060on_nat @ F @ ( append2865821678206098582on_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( map_Pr8685107614402573060on_nat @ F @ Xs ) @ ( map_Pr8685107614402573060on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_699_map__append,axiom,
    ! [F: sum_su4907927656612468301on_nat > produc1313295348606329723on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( map_Su8438591925558430632on_nat @ F @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( map_Su8438591925558430632on_nat @ F @ Xs ) @ ( map_Su8438591925558430632on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_700_map__append,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( map_Su241112079427911264on_nat @ F @ ( append3523812819975775986on_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( map_Su241112079427911264on_nat @ F @ Xs ) @ ( map_Su241112079427911264on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_701_map__append,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_na8715451351012060651on_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append3839213209832495004on_nat @ ( map_na8715451351012060651on_nat @ F @ Xs ) @ ( map_na8715451351012060651on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_702_map__append,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat] :
      ( ( map_nat_nat @ F @ ( append_nat @ Xs @ Ys ) )
      = ( append_nat @ ( map_nat_nat @ F @ Xs ) @ ( map_nat_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_703_map__append,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Ys: list_a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( append_a @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) @ ( map_a_2711072540680412167on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_704_map__append,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a,Ys: list_a] :
      ( ( map_a_7431629901395152583on_nat @ F @ ( append_a @ Xs @ Ys ) )
      = ( append3839213209832495004on_nat @ ( map_a_7431629901395152583on_nat @ F @ Xs ) @ ( map_a_7431629901395152583on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_705_map__append,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Ys: list_a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( append_a @ Xs @ Ys ) )
      = ( append6174841615030841602on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) @ ( map_a_3887198048842487703on_nat @ F @ Ys ) ) ) ).

% map_append
thf(fact_706_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_707_case__prodI,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] :
      ( ( F @ A @ B )
     => ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_708_case__prodI,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_709_case__prodI,axiom,
    ! [F: ( nat > nat ) > nat > $o,A: nat > nat,B: nat] :
      ( ( F @ A @ B )
     => ( produc1191635805960096411_nat_o @ F @ ( produc72220940542539688at_nat @ A @ B ) ) ) ).

% case_prodI
thf(fact_710_case__prodI,axiom,
    ! [F: a > multis8755981257395730801tation > $o,A: a,B: multis8755981257395730801tation] :
      ( ( F @ A @ B )
     => ( produc8513698947429006227tion_o @ F @ ( produc7247797765490398872tation @ A @ B ) ) ) ).

% case_prodI
thf(fact_711_case__prodI,axiom,
    ! [F: a > $o > $o,A: a,B: $o] :
      ( ( F @ A @ B )
     => ( produc8428829086471790546_a_o_o @ F @ ( product_Pair_a_o @ A @ B ) ) ) ).

% case_prodI
thf(fact_712_case__prodI2,axiom,
    ! [P: produc8325617224762090749on_nat,C2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o] :
      ( ! [A3: list_S5253404628753938141on_nat,B3: produc118098540238352959on_nat] :
          ( ( P
            = ( produc1545487600272464053on_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc7728289706344822350_nat_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_713_case__prodI2,axiom,
    ! [P: product_prod_nat_nat,C2: nat > nat > $o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc6081775807080527818_nat_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_714_case__prodI2,axiom,
    ! [P: produc8199716216217303280at_nat,C2: ( nat > nat ) > nat > $o] :
      ( ! [A3: nat > nat,B3: nat] :
          ( ( P
            = ( produc72220940542539688at_nat @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc1191635805960096411_nat_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_715_case__prodI2,axiom,
    ! [P: produc27318731269267614tation,C2: a > multis8755981257395730801tation > $o] :
      ( ! [A3: a,B3: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc8513698947429006227tion_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_716_case__prodI2,axiom,
    ! [P: product_prod_a_o,C2: a > $o > $o] :
      ( ! [A3: a,B3: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B3 ) )
         => ( C2 @ A3 @ B3 ) )
     => ( produc8428829086471790546_a_o_o @ C2 @ P ) ) ).

% case_prodI2
thf(fact_717_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_718_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_719_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_720_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_721_image__add__0,axiom,
    ! [S3: set_mu5400888710828039123on_nat] :
      ( ( image_9067555771090401601on_nat @ ( plus_p2586647234311342374on_nat @ zero_z8702567803361615526on_nat ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_722_image__add__0,axiom,
    ! [S3: set_nat] :
      ( ( image_nat_nat @ ( plus_plus_nat @ zero_zero_nat ) @ S3 )
      = S3 ) ).

% image_add_0
thf(fact_723_list_Oset__map,axiom,
    ! [F: nat > nat,V: list_nat] :
      ( ( set_nat2 @ ( map_nat_nat @ F @ V ) )
      = ( image_nat_nat @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_724_list_Oset__map,axiom,
    ! [F: a > nat,V: list_a] :
      ( ( set_nat2 @ ( map_a_nat @ F @ V ) )
      = ( image_a_nat @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_725_list_Oset__map,axiom,
    ! [F: nat > a,V: list_nat] :
      ( ( set_a2 @ ( map_nat_a @ F @ V ) )
      = ( image_nat_a @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_726_list_Oset__map,axiom,
    ! [F: a > a,V: list_a] :
      ( ( set_a2 @ ( map_a_a @ F @ V ) )
      = ( image_a_a @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_727_list_Oset__map,axiom,
    ! [F: nat > product_prod_a_o,V: list_nat] :
      ( ( set_Product_prod_a_o2 @ ( map_na1257269399097651883od_a_o @ F @ V ) )
      = ( image_5312737573458615594od_a_o @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_728_list_Oset__map,axiom,
    ! [F: a > product_prod_a_o,V: list_a] :
      ( ( set_Product_prod_a_o2 @ ( map_a_1137562357930033543od_a_o @ F @ V ) )
      = ( image_4279145912174180552od_a_o @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_729_list_Oset__map,axiom,
    ! [F: product_prod_a_o > nat,V: list_P3196869577824546119od_a_o] :
      ( ( set_nat2 @ ( map_Pr3957037749107795149_o_nat @ F @ V ) )
      = ( image_8012505923468758860_o_nat @ F @ ( set_Product_prod_a_o2 @ V ) ) ) ).

% list.set_map
thf(fact_730_list_Oset__map,axiom,
    ! [F: product_prod_a_o > a,V: list_P3196869577824546119od_a_o] :
      ( ( set_a2 @ ( map_Pr8253109945893590849_a_o_a @ F @ V ) )
      = ( image_2171321463282962050_a_o_a @ F @ ( set_Product_prod_a_o2 @ V ) ) ) ).

% list.set_map
thf(fact_731_list_Oset__map,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o,V: list_nat] :
      ( ( set_li2629784761229999580od_a_o @ ( map_na764632570197019441od_a_o @ F @ V ) )
      = ( image_5774815713777330736od_a_o @ F @ ( set_nat2 @ V ) ) ) ).

% list.set_map
thf(fact_732_list_Oset__map,axiom,
    ! [F: a > list_P3196869577824546119od_a_o,V: list_a] :
      ( ( set_li2629784761229999580od_a_o @ ( map_a_5058921920454310925od_a_o @ F @ V ) )
      = ( image_7865913289429950158od_a_o @ F @ ( set_a2 @ V ) ) ) ).

% list.set_map
thf(fact_733_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_734_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_735_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_736_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_737_pair__imageI,axiom,
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_2486076414777270412at_nat @ ( produc6842872674320459806at_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_738_pair__imageI,axiom,
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > a] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_5995000214508162370_nat_a @ ( produc3276484115406849584_nat_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_739_pair__imageI,axiom,
    ! [A: a,B: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > nat] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B ) @ A4 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_5814668043327422325on_nat @ ( produc4465434948709935253on_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_740_pair__imageI,axiom,
    ! [A: a,B: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > a] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_7660857576331368089tion_a @ ( produc4916163002043446649tion_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_741_pair__imageI,axiom,
    ! [A: a,B: $o,A4: set_Product_prod_a_o,F: a > $o > nat] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B ) @ A4 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_8012505923468758860_o_nat @ ( produc578040550328963606_o_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_742_pair__imageI,axiom,
    ! [A: a,B: $o,A4: set_Product_prod_a_o,F: a > $o > a] :
      ( ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_2171321463282962050_a_o_a @ ( produc6357147902488193592_a_o_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_743_pair__imageI,axiom,
    ! [A: nat,B: nat,A4: set_Pr1261947904930325089at_nat,F: nat > nat > product_prod_a_o] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ A @ B ) @ ( image_3028173989285855237od_a_o @ ( produc2980191616527086067od_a_o @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_744_pair__imageI,axiom,
    ! [A: nat > nat,B: nat,A4: set_Pr9093778441882193744at_nat,F: ( nat > nat ) > nat > nat] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ A4 )
     => ( member_nat @ ( F @ A @ B ) @ ( image_1512062188183379323at_nat @ ( produc5210631660310915981at_nat @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_745_pair__imageI,axiom,
    ! [A: nat > nat,B: nat,A4: set_Pr9093778441882193744at_nat,F: ( nat > nat ) > nat > a] :
      ( ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ A @ B ) @ A4 )
     => ( member_a @ ( F @ A @ B ) @ ( image_1723514463707901459_nat_a @ ( produc5914945273577373825_nat_a @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_746_pair__imageI,axiom,
    ! [A: a,B: multis8755981257395730801tation,A4: set_Pr3525538344199662036tation,F: a > multis8755981257395730801tation > product_prod_a_o] :
      ( ( member7446769889882171317tation @ ( produc7247797765490398872tation @ A @ B ) @ A4 )
     => ( member2997933407038931562od_a_o @ ( F @ A @ B ) @ ( image_5815038000605959644od_a_o @ ( produc2267023758550893756od_a_o @ F ) @ A4 ) ) ) ).

% pair_imageI
thf(fact_747_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( map_nat_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_748_nth__map,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > nat] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_nat @ ( map_li381573783031109843_o_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_749_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( map_na764632570197019441od_a_o @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_750_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_Su8471564416988646406on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_751_nth__map,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > nat] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_nat @ ( map_Su4809418691747827683at_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_752_nth__map,axiom,
    ! [N: nat,Xs: list_nat,F: nat > sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Su8471564416988646406on_nat @ ( map_na5706685718985795555on_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_753_nth__map,axiom,
    ! [N: nat,Xs: list_a,F: a > list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ N @ ( size_size_list_a @ Xs ) )
     => ( ( nth_li4091121329102240790on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_a @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_754_nth__map,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( map_li880244354692327940od_a_o @ F @ Xs ) @ N )
        = ( F @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_755_nth__map,axiom,
    ! [N: nat,Xs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o] :
      ( ( ord_less_nat @ N @ ( size_s5686234912572693873on_nat @ Xs ) )
     => ( ( nth_li3248255784598043854od_a_o @ ( map_Su446534476614543092od_a_o @ F @ Xs ) @ N )
        = ( F @ ( nth_Su8471564416988646406on_nat @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_756_nth__map,axiom,
    ! [N: nat,Xs: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat] :
      ( ( ord_less_nat @ N @ ( size_s3379185846251495609od_a_o @ Xs ) )
     => ( ( nth_Su8471564416988646406on_nat @ ( map_li6915058325474510672on_nat @ F @ Xs ) @ N )
        = ( F @ ( nth_li3248255784598043854od_a_o @ Xs @ N ) ) ) ) ).

% nth_map
thf(fact_757_list_Omap__ident,axiom,
    ! [T: list_nat] :
      ( ( map_nat_nat
        @ ^ [X4: nat] : X4
        @ T )
      = T ) ).

% list.map_ident
thf(fact_758_prod_Ocase__distrib,axiom,
    ! [H: product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H @ ( produc2626176000494625587at_nat @ F @ Prod ) )
      = ( produc2626176000494625587at_nat
        @ ^ [X12: nat,X22: nat] : ( H @ ( F @ X12 @ X22 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_759_gr__zeroI,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr_zeroI
thf(fact_760_not__less__zero,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less_zero
thf(fact_761_gr__implies__not__zero,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_762_zero__less__iff__neq__zero,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( N != zero_zero_nat ) ) ).

% zero_less_iff_neq_zero
thf(fact_763_list_Osimps_I8_J,axiom,
    ! [F: nat > nat] :
      ( ( map_nat_nat @ F @ nil_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_764_list_Osimps_I8_J,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat] :
      ( ( map_li381573783031109843_o_nat @ F @ nil_li1372968520647675725od_a_o )
      = nil_nat ) ).

% list.simps(8)
thf(fact_765_list_Osimps_I8_J,axiom,
    ! [F: nat > list_P3196869577824546119od_a_o] :
      ( ( map_na764632570197019441od_a_o @ F @ nil_nat )
      = nil_li1372968520647675725od_a_o ) ).

% list.simps(8)
thf(fact_766_list_Osimps_I8_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat] :
      ( ( map_Su4809418691747827683at_nat @ F @ nil_Su3078906344068544711on_nat )
      = nil_nat ) ).

% list.simps(8)
thf(fact_767_list_Osimps_I8_J,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat] :
      ( ( map_na5706685718985795555on_nat @ F @ nil_nat )
      = nil_Su3078906344068544711on_nat ) ).

% list.simps(8)
thf(fact_768_list_Osimps_I8_J,axiom,
    ! [F: a > sum_su4907927656612468301on_nat] :
      ( ( map_a_2711072540680412167on_nat @ F @ nil_a )
      = nil_Su3078906344068544711on_nat ) ).

% list.simps(8)
thf(fact_769_list_Osimps_I8_J,axiom,
    ! [F: a > list_S5253404628753938141on_nat] :
      ( ( map_a_3887198048842487703on_nat @ F @ nil_a )
      = nil_li5563851662744376407on_nat ) ).

% list.simps(8)
thf(fact_770_list_Osimps_I8_J,axiom,
    ! [F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o] :
      ( ( map_li880244354692327940od_a_o @ F @ nil_li1372968520647675725od_a_o )
      = nil_li1372968520647675725od_a_o ) ).

% list.simps(8)
thf(fact_771_list_Osimps_I8_J,axiom,
    ! [F: list_P3196869577824546119od_a_o > sum_su4907927656612468301on_nat] :
      ( ( map_li6915058325474510672on_nat @ F @ nil_li1372968520647675725od_a_o )
      = nil_Su3078906344068544711on_nat ) ).

% list.simps(8)
thf(fact_772_list_Osimps_I8_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > list_P3196869577824546119od_a_o] :
      ( ( map_Su446534476614543092od_a_o @ F @ nil_Su3078906344068544711on_nat )
      = nil_li1372968520647675725od_a_o ) ).

% list.simps(8)
thf(fact_773_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > list_P9127870862658512897on_nat,G: nat > list_P9127870862658512897on_nat] :
      ( ( X = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_na8715451351012060651on_nat @ F @ X )
          = ( map_na8715451351012060651on_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_774_list_Omap__cong,axiom,
    ! [X: list_nat,Ya: list_nat,F: nat > nat,G: nat > nat] :
      ( ( X = Ya )
     => ( ! [Z3: nat] :
            ( ( member_nat @ Z3 @ ( set_nat2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_775_list_Omap__cong,axiom,
    ! [X: list_a,Ya: list_a,F: a > sum_su4907927656612468301on_nat,G: a > sum_su4907927656612468301on_nat] :
      ( ( X = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_a_2711072540680412167on_nat @ F @ X )
          = ( map_a_2711072540680412167on_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_776_list_Omap__cong,axiom,
    ! [X: list_a,Ya: list_a,F: a > list_P9127870862658512897on_nat,G: a > list_P9127870862658512897on_nat] :
      ( ( X = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_a_7431629901395152583on_nat @ F @ X )
          = ( map_a_7431629901395152583on_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_777_list_Omap__cong,axiom,
    ! [X: list_a,Ya: list_a,F: a > list_S5253404628753938141on_nat,G: a > list_S5253404628753938141on_nat] :
      ( ( X = Ya )
     => ( ! [Z3: a] :
            ( ( member_a @ Z3 @ ( set_a2 @ Ya ) )
           => ( ( F @ Z3 )
              = ( G @ Z3 ) ) )
       => ( ( map_a_3887198048842487703on_nat @ F @ X )
          = ( map_a_3887198048842487703on_nat @ G @ Ya ) ) ) ) ).

% list.map_cong
thf(fact_778_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > list_P9127870862658512897on_nat,G: nat > list_P9127870862658512897on_nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_na8715451351012060651on_nat @ F @ X )
        = ( map_na8715451351012060651on_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_779_list_Omap__cong0,axiom,
    ! [X: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_nat_nat @ F @ X )
        = ( map_nat_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_780_list_Omap__cong0,axiom,
    ! [X: list_a,F: a > sum_su4907927656612468301on_nat,G: a > sum_su4907927656612468301on_nat] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_a_2711072540680412167on_nat @ F @ X )
        = ( map_a_2711072540680412167on_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_781_list_Omap__cong0,axiom,
    ! [X: list_a,F: a > list_P9127870862658512897on_nat,G: a > list_P9127870862658512897on_nat] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_a_7431629901395152583on_nat @ F @ X )
        = ( map_a_7431629901395152583on_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_782_list_Omap__cong0,axiom,
    ! [X: list_a,F: a > list_S5253404628753938141on_nat,G: a > list_S5253404628753938141on_nat] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X ) )
         => ( ( F @ Z3 )
            = ( G @ Z3 ) ) )
     => ( ( map_a_3887198048842487703on_nat @ F @ X )
        = ( map_a_3887198048842487703on_nat @ G @ X ) ) ) ).

% list.map_cong0
thf(fact_783_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa2: list_nat,F: nat > list_P9127870862658512897on_nat,Fa: nat > list_P9127870862658512897on_nat] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_na8715451351012060651on_nat @ F @ X )
          = ( map_na8715451351012060651on_nat @ Fa @ Xa2 ) )
       => ( X = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_784_list_Oinj__map__strong,axiom,
    ! [X: list_nat,Xa2: list_nat,F: nat > nat,Fa: nat > nat] :
      ( ! [Z3: nat,Za: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ X ) )
         => ( ( member_nat @ Za @ ( set_nat2 @ Xa2 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_nat_nat @ F @ X )
          = ( map_nat_nat @ Fa @ Xa2 ) )
       => ( X = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_785_list_Oinj__map__strong,axiom,
    ! [X: list_a,Xa2: list_a,F: a > sum_su4907927656612468301on_nat,Fa: a > sum_su4907927656612468301on_nat] :
      ( ! [Z3: a,Za: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_a_2711072540680412167on_nat @ F @ X )
          = ( map_a_2711072540680412167on_nat @ Fa @ Xa2 ) )
       => ( X = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_786_list_Oinj__map__strong,axiom,
    ! [X: list_a,Xa2: list_a,F: a > list_P9127870862658512897on_nat,Fa: a > list_P9127870862658512897on_nat] :
      ( ! [Z3: a,Za: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_a_7431629901395152583on_nat @ F @ X )
          = ( map_a_7431629901395152583on_nat @ Fa @ Xa2 ) )
       => ( X = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_787_list_Oinj__map__strong,axiom,
    ! [X: list_a,Xa2: list_a,F: a > list_S5253404628753938141on_nat,Fa: a > list_S5253404628753938141on_nat] :
      ( ! [Z3: a,Za: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ X ) )
         => ( ( member_a @ Za @ ( set_a2 @ Xa2 ) )
           => ( ( ( F @ Z3 )
                = ( Fa @ Za ) )
             => ( Z3 = Za ) ) ) )
     => ( ( ( map_a_3887198048842487703on_nat @ F @ X )
          = ( map_a_3887198048842487703on_nat @ Fa @ Xa2 ) )
       => ( X = Xa2 ) ) ) ).

% list.inj_map_strong
thf(fact_788_list_Omap__ident__strong,axiom,
    ! [T: list_P3196869577824546119od_a_o,F: product_prod_a_o > product_prod_a_o] :
      ( ! [Z3: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ Z3 @ ( set_Product_prod_a_o2 @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Pr5742790954183869060od_a_o @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_789_list_Omap__ident__strong,axiom,
    ! [T: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat] :
      ( ! [Z3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ Z3 @ ( set_Su4848624856859743608on_nat @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_Su241112079427911264on_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_790_list_Omap__ident__strong,axiom,
    ! [T: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o] :
      ( ! [Z3: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ Z3 @ ( set_li2629784761229999580od_a_o @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_li880244354692327940od_a_o @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_791_list_Omap__ident__strong,axiom,
    ! [T: list_nat,F: nat > nat] :
      ( ! [Z3: nat] :
          ( ( member_nat @ Z3 @ ( set_nat2 @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_nat_nat @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_792_list_Omap__ident__strong,axiom,
    ! [T: list_a,F: a > a] :
      ( ! [Z3: a] :
          ( ( member_a @ Z3 @ ( set_a2 @ T ) )
         => ( ( F @ Z3 )
            = Z3 ) )
     => ( ( map_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_793_map__ext,axiom,
    ! [Xs: list_nat,F: nat > list_P9127870862658512897on_nat,G: nat > list_P9127870862658512897on_nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_na8715451351012060651on_nat @ F @ Xs )
        = ( map_na8715451351012060651on_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_794_map__ext,axiom,
    ! [Xs: list_nat,F: nat > nat,G: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_795_map__ext,axiom,
    ! [Xs: list_a,F: a > sum_su4907927656612468301on_nat,G: a > sum_su4907927656612468301on_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_796_map__ext,axiom,
    ! [Xs: list_a,F: a > list_P9127870862658512897on_nat,G: a > list_P9127870862658512897on_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_a_7431629901395152583on_nat @ F @ Xs )
        = ( map_a_7431629901395152583on_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_797_map__ext,axiom,
    ! [Xs: list_a,F: a > list_S5253404628753938141on_nat,G: a > list_S5253404628753938141on_nat] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = ( G @ X3 ) ) )
     => ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( map_a_3887198048842487703on_nat @ G @ Xs ) ) ) ).

% map_ext
thf(fact_798_map__idI,axiom,
    ! [Xs: list_P3196869577824546119od_a_o,F: product_prod_a_o > product_prod_a_o] :
      ( ! [X3: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ X3 @ ( set_Product_prod_a_o2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr5742790954183869060od_a_o @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_799_map__idI,axiom,
    ! [Xs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat] :
      ( ! [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Su241112079427911264on_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_800_map__idI,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,F: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o] :
      ( ! [X3: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ X3 @ ( set_li2629784761229999580od_a_o @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_li880244354692327940od_a_o @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_801_map__idI,axiom,
    ! [Xs: list_nat,F: nat > nat] :
      ( ! [X3: nat] :
          ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_nat_nat @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_802_map__idI,axiom,
    ! [Xs: list_a,F: a > a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_a_a @ F @ Xs )
        = Xs ) ) ).

% map_idI
thf(fact_803_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > list_P9127870862658512897on_nat,G: nat > list_P9127870862658512897on_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_na8715451351012060651on_nat @ F @ Xs )
          = ( map_na8715451351012060651on_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_804_map__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,F: nat > nat,G: nat > nat] :
      ( ( Xs = Ys )
     => ( ! [X3: nat] :
            ( ( member_nat @ X3 @ ( set_nat2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_nat_nat @ F @ Xs )
          = ( map_nat_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_805_map__cong,axiom,
    ! [Xs: list_a,Ys: list_a,F: a > sum_su4907927656612468301on_nat,G: a > sum_su4907927656612468301on_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_a_2711072540680412167on_nat @ F @ Xs )
          = ( map_a_2711072540680412167on_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_806_map__cong,axiom,
    ! [Xs: list_a,Ys: list_a,F: a > list_P9127870862658512897on_nat,G: a > list_P9127870862658512897on_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_a_7431629901395152583on_nat @ F @ Xs )
          = ( map_a_7431629901395152583on_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_807_map__cong,axiom,
    ! [Xs: list_a,Ys: list_a,F: a > list_S5253404628753938141on_nat,G: a > list_S5253404628753938141on_nat] :
      ( ( Xs = Ys )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Ys ) )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( map_a_3887198048842487703on_nat @ F @ Xs )
          = ( map_a_3887198048842487703on_nat @ G @ Ys ) ) ) ) ).

% map_cong
thf(fact_808_ex__map__conv,axiom,
    ! [Ys: list_l1726306699111145479on_nat,F: nat > list_P9127870862658512897on_nat] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_na8715451351012060651on_nat @ F @ Xs2 ) ) )
      = ( ! [X4: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Ys ) )
           => ? [Y5: nat] :
                ( X4
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_809_ex__map__conv,axiom,
    ! [Ys: list_l1726306699111145479on_nat,F: a > list_P9127870862658512897on_nat] :
      ( ( ? [Xs2: list_a] :
            ( Ys
            = ( map_a_7431629901395152583on_nat @ F @ Xs2 ) ) )
      = ( ! [X4: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Ys ) )
           => ? [Y5: a] :
                ( X4
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_810_ex__map__conv,axiom,
    ! [Ys: list_l8999537460041914605on_nat,F: a > list_S5253404628753938141on_nat] :
      ( ( ? [Xs2: list_a] :
            ( Ys
            = ( map_a_3887198048842487703on_nat @ F @ Xs2 ) ) )
      = ( ! [X4: list_S5253404628753938141on_nat] :
            ( ( member86162308477975540on_nat @ X4 @ ( set_li4371160873219908744on_nat @ Ys ) )
           => ? [Y5: a] :
                ( X4
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_811_ex__map__conv,axiom,
    ! [Ys: list_S5253404628753938141on_nat,F: a > sum_su4907927656612468301on_nat] :
      ( ( ? [Xs2: list_a] :
            ( Ys
            = ( map_a_2711072540680412167on_nat @ F @ Xs2 ) ) )
      = ( ! [X4: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ X4 @ ( set_Su4848624856859743608on_nat @ Ys ) )
           => ? [Y5: a] :
                ( X4
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_812_ex__map__conv,axiom,
    ! [Ys: list_nat,F: nat > nat] :
      ( ( ? [Xs2: list_nat] :
            ( Ys
            = ( map_nat_nat @ F @ Xs2 ) ) )
      = ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ? [Y5: nat] :
                ( X4
                = ( F @ Y5 ) ) ) ) ) ).

% ex_map_conv
thf(fact_813_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_814_map__eq__imp__length__eq,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o,G: nat > nat,Ys: list_nat] :
      ( ( ( map_li381573783031109843_o_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s3379185846251495609od_a_o @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_815_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: list_P3196869577824546119od_a_o > nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_li381573783031109843_o_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s3379185846251495609od_a_o @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_816_map__eq__imp__length__eq,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,G: a > sum_su4907927656612468301on_nat,Ys: list_a] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_817_map__eq__imp__length__eq,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,G: nat > sum_su4907927656612468301on_nat,Ys: list_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_na5706685718985795555on_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_818_map__eq__imp__length__eq,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
     => ( ( size_s5686234912572693873on_nat @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_819_map__eq__imp__length__eq,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,G: a > sum_su4907927656612468301on_nat,Ys: list_a] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_820_map__eq__imp__length__eq,axiom,
    ! [F: nat > nat,Xs: list_nat,G: sum_su4907927656612468301on_nat > nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Su4809418691747827683at_nat @ G @ Ys ) )
     => ( ( size_size_list_nat @ Xs )
        = ( size_s5686234912572693873on_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_821_map__eq__imp__length__eq,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,G: a > list_S5253404628753938141on_nat,Ys: list_a] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( map_a_3887198048842487703on_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_a @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_822_map__eq__imp__length__eq,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,G: nat > list_S5253404628753938141on_nat,Ys: list_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( map_na4864219606106146163on_nat @ G @ Ys ) )
     => ( ( size_size_list_a @ Xs )
        = ( size_size_list_nat @ Ys ) ) ) ).

% map_eq_imp_length_eq
thf(fact_823_append__eq__map__conv,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,F: produc1313295348606329723on_nat > produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys @ Zs )
        = ( map_Pr6472934937144306180on_nat @ F @ Xs ) )
      = ( ? [Us2: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr6472934937144306180on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Pr6472934937144306180on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_824_append__eq__map__conv,axiom,
    ! [Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat,F: sum_su4907927656612468301on_nat > produc1313295348606329723on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( append2865821678206098582on_nat @ Ys @ Zs )
        = ( map_Su8438591925558430632on_nat @ F @ Xs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Su8438591925558430632on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Su8438591925558430632on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_825_append__eq__map__conv,axiom,
    ! [Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,F: produc1313295348606329723on_nat > sum_su4907927656612468301on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys @ Zs )
        = ( map_Pr8685107614402573060on_nat @ F @ Xs ) )
      = ( ? [Us2: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr8685107614402573060on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Pr8685107614402573060on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_826_append__eq__map__conv,axiom,
    ! [Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( ( append3523812819975775986on_nat @ Ys @ Zs )
        = ( map_Su241112079427911264on_nat @ F @ Xs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Su241112079427911264on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Su241112079427911264on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_827_append__eq__map__conv,axiom,
    ! [Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat,F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( ( append3839213209832495004on_nat @ Ys @ Zs )
        = ( map_na8715451351012060651on_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na8715451351012060651on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_na8715451351012060651on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_828_append__eq__map__conv,axiom,
    ! [Ys: list_nat,Zs: list_nat,F: nat > nat,Xs: list_nat] :
      ( ( ( append_nat @ Ys @ Zs )
        = ( map_nat_nat @ F @ Xs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_829_append__eq__map__conv,axiom,
    ! [Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat,F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( ( append3523812819975775986on_nat @ Ys @ Zs )
        = ( map_a_2711072540680412167on_nat @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_2711072540680412167on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_2711072540680412167on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_830_append__eq__map__conv,axiom,
    ! [Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat,F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( ( append3839213209832495004on_nat @ Ys @ Zs )
        = ( map_a_7431629901395152583on_nat @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_7431629901395152583on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_7431629901395152583on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_831_append__eq__map__conv,axiom,
    ! [Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat,F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( ( append6174841615030841602on_nat @ Ys @ Zs )
        = ( map_a_3887198048842487703on_nat @ F @ Xs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_3887198048842487703on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_3887198048842487703on_nat @ F @ Vs2 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_832_map__eq__append__conv,axiom,
    ! [F: produc1313295348606329723on_nat > produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( map_Pr6472934937144306180on_nat @ F @ Xs )
        = ( append2865821678206098582on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr6472934937144306180on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Pr6472934937144306180on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_833_map__eq__append__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > produc1313295348606329723on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_P9127870862658512897on_nat,Zs: list_P9127870862658512897on_nat] :
      ( ( ( map_Su8438591925558430632on_nat @ F @ Xs )
        = ( append2865821678206098582on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Su8438591925558430632on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Su8438591925558430632on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_834_map__eq__append__conv,axiom,
    ! [F: produc1313295348606329723on_nat > sum_su4907927656612468301on_nat,Xs: list_P9127870862658512897on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( map_Pr8685107614402573060on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_P9127870862658512897on_nat,Vs2: list_P9127870862658512897on_nat] :
            ( ( Xs
              = ( append2865821678206098582on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Pr8685107614402573060on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Pr8685107614402573060on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_835_map__eq__append__conv,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( map_Su241112079427911264on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_S5253404628753938141on_nat,Vs2: list_S5253404628753938141on_nat] :
            ( ( Xs
              = ( append3523812819975775986on_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_Su241112079427911264on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_Su241112079427911264on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_836_map__eq__append__conv,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ( ( map_na8715451351012060651on_nat @ F @ Xs )
        = ( append3839213209832495004on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_na8715451351012060651on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_na8715451351012060651on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_837_map__eq__append__conv,axiom,
    ! [F: nat > nat,Xs: list_nat,Ys: list_nat,Zs: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( append_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_nat,Vs2: list_nat] :
            ( ( Xs
              = ( append_nat @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_nat_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_nat_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_838_map__eq__append__conv,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,Ys: list_S5253404628753938141on_nat,Zs: list_S5253404628753938141on_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( append3523812819975775986on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_2711072540680412167on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_2711072540680412167on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_839_map__eq__append__conv,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a,Ys: list_l1726306699111145479on_nat,Zs: list_l1726306699111145479on_nat] :
      ( ( ( map_a_7431629901395152583on_nat @ F @ Xs )
        = ( append3839213209832495004on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_7431629901395152583on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_7431629901395152583on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_840_map__eq__append__conv,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,Ys: list_l8999537460041914605on_nat,Zs: list_l8999537460041914605on_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( append6174841615030841602on_nat @ Ys @ Zs ) )
      = ( ? [Us2: list_a,Vs2: list_a] :
            ( ( Xs
              = ( append_a @ Us2 @ Vs2 ) )
            & ( Ys
              = ( map_a_3887198048842487703on_nat @ F @ Us2 ) )
            & ( Zs
              = ( map_a_3887198048842487703on_nat @ F @ Vs2 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_841_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat,X1: nat,X2: nat] :
      ( ( produc2626176000494625587at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X2 ) )
      = ( F @ X1 @ X2 ) ) ).

% old.prod.case
thf(fact_842_map__update,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat,K: nat,Y: nat] :
      ( ( map_na8715451351012060651on_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
      = ( list_u184235112954345583on_nat @ ( map_na8715451351012060651on_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_843_map__update,axiom,
    ! [F: nat > nat,Xs: list_nat,K: nat,Y: nat] :
      ( ( map_nat_nat @ F @ ( list_update_nat @ Xs @ K @ Y ) )
      = ( list_update_nat @ ( map_nat_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_844_map__update,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,K: nat,Y: a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( list_update_a @ Xs @ K @ Y ) )
      = ( list_u8577817230714425439on_nat @ ( map_a_2711072540680412167on_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_845_map__update,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a,K: nat,Y: a] :
      ( ( map_a_7431629901395152583on_nat @ F @ ( list_update_a @ Xs @ K @ Y ) )
      = ( list_u184235112954345583on_nat @ ( map_a_7431629901395152583on_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_846_map__update,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,K: nat,Y: a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( list_update_a @ Xs @ K @ Y ) )
      = ( list_u3873310082918383599on_nat @ ( map_a_3887198048842487703on_nat @ F @ Xs ) @ K @ ( F @ Y ) ) ) ).

% map_update
thf(fact_847_case__prodD,axiom,
    ! [F: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,A: list_S5253404628753938141on_nat,B: produc118098540238352959on_nat] :
      ( ( produc7728289706344822350_nat_o @ F @ ( produc1545487600272464053on_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_848_case__prodD,axiom,
    ! [F: nat > nat > $o,A: nat,B: nat] :
      ( ( produc6081775807080527818_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_849_case__prodD,axiom,
    ! [F: ( nat > nat ) > nat > $o,A: nat > nat,B: nat] :
      ( ( produc1191635805960096411_nat_o @ F @ ( produc72220940542539688at_nat @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_850_case__prodD,axiom,
    ! [F: a > multis8755981257395730801tation > $o,A: a,B: multis8755981257395730801tation] :
      ( ( produc8513698947429006227tion_o @ F @ ( produc7247797765490398872tation @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_851_case__prodD,axiom,
    ! [F: a > $o > $o,A: a,B: $o] :
      ( ( produc8428829086471790546_a_o_o @ F @ ( product_Pair_a_o @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_852_case__prodE,axiom,
    ! [C2: list_S5253404628753938141on_nat > produc118098540238352959on_nat > $o,P: produc8325617224762090749on_nat] :
      ( ( produc7728289706344822350_nat_o @ C2 @ P )
     => ~ ! [X3: list_S5253404628753938141on_nat,Y3: produc118098540238352959on_nat] :
            ( ( P
              = ( produc1545487600272464053on_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_853_case__prodE,axiom,
    ! [C2: nat > nat > $o,P: product_prod_nat_nat] :
      ( ( produc6081775807080527818_nat_o @ C2 @ P )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_854_case__prodE,axiom,
    ! [C2: ( nat > nat ) > nat > $o,P: produc8199716216217303280at_nat] :
      ( ( produc1191635805960096411_nat_o @ C2 @ P )
     => ~ ! [X3: nat > nat,Y3: nat] :
            ( ( P
              = ( produc72220940542539688at_nat @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_855_case__prodE,axiom,
    ! [C2: a > multis8755981257395730801tation > $o,P: produc27318731269267614tation] :
      ( ( produc8513698947429006227tion_o @ C2 @ P )
     => ~ ! [X3: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_856_case__prodE,axiom,
    ! [C2: a > $o > $o,P: product_prod_a_o] :
      ( ( produc8428829086471790546_a_o_o @ C2 @ P )
     => ~ ! [X3: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X3 @ Y3 ) )
           => ~ ( C2 @ X3 @ Y3 ) ) ) ).

% case_prodE
thf(fact_857_pos__add__strict,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C2 )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C2 ) ) ) ) ).

% pos_add_strict
thf(fact_858_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C ) )
           => ( C = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_859_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_860_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_861_case__prodE2,axiom,
    ! [Q: product_prod_nat_nat > $o,P2: nat > nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc2626176000494625587at_nat @ P2 @ Z2 ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( Q @ ( P2 @ X3 @ Y3 ) ) ) ) ).

% case_prodE2
thf(fact_862_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat
        @ ^ [X4: nat,Y5: nat] : ( F @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
      = F ) ).

% case_prod_eta
thf(fact_863_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( F @ X3 @ Y3 )
          = ( G @ ( product_Pair_nat_nat @ X3 @ Y3 ) ) )
     => ( ( produc2626176000494625587at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_864_image__set,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( image_nat_nat @ F @ ( set_nat2 @ Xs ) )
      = ( set_nat2 @ ( map_nat_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_865_image__set,axiom,
    ! [F: nat > a,Xs: list_nat] :
      ( ( image_nat_a @ F @ ( set_nat2 @ Xs ) )
      = ( set_a2 @ ( map_nat_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_866_image__set,axiom,
    ! [F: a > nat,Xs: list_a] :
      ( ( image_a_nat @ F @ ( set_a2 @ Xs ) )
      = ( set_nat2 @ ( map_a_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_867_image__set,axiom,
    ! [F: a > a,Xs: list_a] :
      ( ( image_a_a @ F @ ( set_a2 @ Xs ) )
      = ( set_a2 @ ( map_a_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_868_image__set,axiom,
    ! [F: product_prod_a_o > nat,Xs: list_P3196869577824546119od_a_o] :
      ( ( image_8012505923468758860_o_nat @ F @ ( set_Product_prod_a_o2 @ Xs ) )
      = ( set_nat2 @ ( map_Pr3957037749107795149_o_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_869_image__set,axiom,
    ! [F: product_prod_a_o > a,Xs: list_P3196869577824546119od_a_o] :
      ( ( image_2171321463282962050_a_o_a @ F @ ( set_Product_prod_a_o2 @ Xs ) )
      = ( set_a2 @ ( map_Pr8253109945893590849_a_o_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_870_image__set,axiom,
    ! [F: nat > product_prod_a_o,Xs: list_nat] :
      ( ( image_5312737573458615594od_a_o @ F @ ( set_nat2 @ Xs ) )
      = ( set_Product_prod_a_o2 @ ( map_na1257269399097651883od_a_o @ F @ Xs ) ) ) ).

% image_set
thf(fact_871_image__set,axiom,
    ! [F: a > product_prod_a_o,Xs: list_a] :
      ( ( image_4279145912174180552od_a_o @ F @ ( set_a2 @ Xs ) )
      = ( set_Product_prod_a_o2 @ ( map_a_1137562357930033543od_a_o @ F @ Xs ) ) ) ).

% image_set
thf(fact_872_image__set,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o] :
      ( ( image_5391756926611421138_o_nat @ F @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( set_nat2 @ ( map_li381573783031109843_o_nat @ F @ Xs ) ) ) ).

% image_set
thf(fact_873_image__set,axiom,
    ! [F: list_P3196869577824546119od_a_o > a,Xs: list_l8336264072933094989od_a_o] :
      ( ( image_3930418492379151228_a_o_a @ F @ ( set_li2629784761229999580od_a_o @ Xs ) )
      = ( set_a2 @ ( map_li1123427123403511995_a_o_a @ F @ Xs ) ) ) ).

% image_set
thf(fact_874_mul__ext__list_Ons__map,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ S3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ NS )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ 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 @ S3 ) ) ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ 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 @ S3 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_875_mul__ext__list_Ons__map,axiom,
    ! [S3: set_Pr2535564044409765851on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS: set_Pr2535564044409765851on_nat,As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] :
      ( ! [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] :
          ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A3 @ B3 ) @ S3 )
         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ S3 ) )
     => ( ! [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A3 @ B3 ) @ NS )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ 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 @ S3 ) ) ) ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F @ As ) @ ( map_Su241112079427911264on_nat @ F @ 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 @ S3 ) ) ) ) ) ) ) ) ).

% mul_ext_list.ns_map
thf(fact_876_mul__ext__list_Os__map,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S3 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ S3 ) )
     => ( ! [A3: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ NS )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ 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 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S3 ) ) ) ) )
         => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) )
            @ ( collec1570431334306492044st_nat
              @ ( produc6382697403170970154_nat_o
                @ ^ [As2: list_nat,Bs2: list_nat] : ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ ( mset_nat @ As2 ) @ ( mset_nat @ Bs2 ) ) @ ( multis4215214144601311993xt_nat @ NS @ S3 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_877_mul__ext__list_Os__map,axiom,
    ! [S3: set_Pr2535564044409765851on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS: set_Pr2535564044409765851on_nat,As: list_S5253404628753938141on_nat,Bs: list_S5253404628753938141on_nat] :
      ( ! [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] :
          ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A3 @ B3 ) @ S3 )
         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ S3 ) )
     => ( ! [A3: sum_su4907927656612468301on_nat,B3: sum_su4907927656612468301on_nat] :
            ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A3 @ B3 ) @ NS )
           => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ 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 ) ) @ ( multis1620213760974180726on_nat @ NS @ S3 ) ) ) ) )
         => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F @ As ) @ ( map_Su241112079427911264on_nat @ F @ 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 ) ) @ ( multis1620213760974180726on_nat @ NS @ S3 ) ) ) ) ) ) ) ) ).

% mul_ext_list.s_map
thf(fact_878_ns__mul__ext__bottom__uniqueness,axiom,
    ! [A4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ A4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) )
     => ( A4 = zero_z8702567803361615526on_nat ) ) ).

% ns_mul_ext_bottom_uniqueness
thf(fact_879_ns__mul__ext__bottom2,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) )
     => ( ( B4 != zero_z8702567803361615526on_nat )
       => ( A4 != zero_z8702567803361615526on_nat ) ) ) ).

% ns_mul_ext_bottom2
thf(fact_880_ns__mul__ext__bottom,axiom,
    ! [A4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] : ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ zero_z8702567803361615526on_nat ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) ) ).

% ns_mul_ext_bottom
thf(fact_881_s__mul__ext__bottom__strict,axiom,
    ! [A4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ~ ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ zero_z8702567803361615526on_nat @ A4 ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) ) ).

% s_mul_ext_bottom_strict
thf(fact_882_s__mul__ext__bottom,axiom,
    ! [A4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( A4 != zero_z8702567803361615526on_nat )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ zero_z8702567803361615526on_nat ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) ) ) ).

% s_mul_ext_bottom
thf(fact_883_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I2 ) )
              = ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_884_map__equality__iff,axiom,
    ! [F: list_P3196869577824546119od_a_o > nat,Xs: list_l8336264072933094989od_a_o,G: nat > nat,Ys: list_nat] :
      ( ( ( map_li381573783031109843_o_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_s3379185846251495609od_a_o @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_li3248255784598043854od_a_o @ Xs @ I2 ) )
              = ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_885_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: list_P3196869577824546119od_a_o > nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_li381573783031109843_o_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s3379185846251495609od_a_o @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I2 ) )
              = ( G @ ( nth_li3248255784598043854od_a_o @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_886_map__equality__iff,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,G: a > sum_su4907927656612468301on_nat,Ys: list_a] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I2 ) )
              = ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_887_map__equality__iff,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a,G: nat > sum_su4907927656612468301on_nat,Ys: list_nat] :
      ( ( ( map_a_2711072540680412167on_nat @ F @ Xs )
        = ( map_na5706685718985795555on_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I2 ) )
              = ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_888_map__equality__iff,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,G: nat > nat,Ys: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( map_nat_nat @ G @ Ys ) )
      = ( ( ( size_s5686234912572693873on_nat @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_Su8471564416988646406on_nat @ Xs @ I2 ) )
              = ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_889_map__equality__iff,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,G: a > sum_su4907927656612468301on_nat,Ys: list_a] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( map_a_2711072540680412167on_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I2 ) )
              = ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_890_map__equality__iff,axiom,
    ! [F: nat > nat,Xs: list_nat,G: sum_su4907927656612468301on_nat > nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( map_Su4809418691747827683at_nat @ G @ Ys ) )
      = ( ( ( size_size_list_nat @ Xs )
          = ( size_s5686234912572693873on_nat @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Ys ) )
           => ( ( F @ ( nth_nat @ Xs @ I2 ) )
              = ( G @ ( nth_Su8471564416988646406on_nat @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_891_map__equality__iff,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,G: a > list_S5253404628753938141on_nat,Ys: list_a] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( map_a_3887198048842487703on_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_a @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I2 ) )
              = ( G @ ( nth_a @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_892_map__equality__iff,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a,G: nat > list_S5253404628753938141on_nat,Ys: list_nat] :
      ( ( ( map_a_3887198048842487703on_nat @ F @ Xs )
        = ( map_na4864219606106146163on_nat @ G @ Ys ) )
      = ( ( ( size_size_list_a @ Xs )
          = ( size_size_list_nat @ Ys ) )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ys ) )
           => ( ( F @ ( nth_a @ Xs @ I2 ) )
              = ( G @ ( nth_nat @ Ys @ I2 ) ) ) ) ) ) ).

% map_equality_iff
thf(fact_893_list__order__extension_Ons__map,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,S3: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ! [A3: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ S3 ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ NS )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ NS ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( Ns_list @ S3 @ NS ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) ) @ ( Ns_list @ S3 @ NS ) ) ) ) ) ) ).

% list_order_extension.ns_map
thf(fact_894_list__order__extension_Os__map,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,S3: set_Pr1261947904930325089at_nat,F: nat > nat,NS: set_Pr1261947904930325089at_nat,As: list_nat,Bs: list_nat] :
      ( ( list_l792762466888043652on_nat @ S_list @ Ns_list )
     => ( ! [A3: nat,B3: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ S3 )
           => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ S3 ) )
       => ( ! [A3: nat,B3: nat] :
              ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A3 @ B3 ) @ NS )
             => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F @ A3 ) @ ( F @ B3 ) ) @ NS ) )
         => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As @ Bs ) @ ( S_list @ S3 @ NS ) )
           => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F @ As ) @ ( map_nat_nat @ F @ Bs ) ) @ ( S_list @ S3 @ NS ) ) ) ) ) ) ).

% list_order_extension.s_map
thf(fact_895_pred__equals__eq2,axiom,
    ! [R2: set_Pr6244131468446592861on_nat,S3: set_Pr6244131468446592861on_nat] :
      ( ( ( ^ [X4: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_896_pred__equals__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_897_pred__equals__eq2,axiom,
    ! [R2: set_Pr9093778441882193744at_nat,S3: set_Pr9093778441882193744at_nat] :
      ( ( ( ^ [X4: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_898_pred__equals__eq2,axiom,
    ! [R2: set_Pr3525538344199662036tation,S3: set_Pr3525538344199662036tation] :
      ( ( ( ^ [X4: a,Y5: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: a,Y5: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_899_pred__equals__eq2,axiom,
    ! [R2: set_Product_prod_a_o,S3: set_Product_prod_a_o] :
      ( ( ( ^ [X4: a,Y5: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X4 @ Y5 ) @ R2 ) )
        = ( ^ [X4: a,Y5: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X4 @ Y5 ) @ S3 ) ) )
      = ( R2 = S3 ) ) ).

% pred_equals_eq2
thf(fact_900_sup__Un__eq2,axiom,
    ! [R2: set_Pr6244131468446592861on_nat,S3: set_Pr6244131468446592861on_nat] :
      ( ( sup_su5577399213107430846_nat_o
        @ ^ [X4: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y5 ) @ R2 )
        @ ^ [X4: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: list_S5253404628753938141on_nat,Y5: produc118098540238352959on_nat] : ( member3464510148199751334on_nat @ ( produc1545487600272464053on_nat @ X4 @ Y5 ) @ ( sup_su3554973707900075825on_nat @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_901_sup__Un__eq2,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( sup_sup_nat_nat_o
        @ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
        @ ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: nat,Y5: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ ( sup_su6327502436637775413at_nat @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_902_sup__Un__eq2,axiom,
    ! [R2: set_Pr9093778441882193744at_nat,S3: set_Pr9093778441882193744at_nat] :
      ( ( sup_su4798059109528701835_nat_o
        @ ^ [X4: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y5 ) @ R2 )
        @ ^ [X4: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: nat > nat,Y5: nat] : ( member7226740684066999833at_nat @ ( produc72220940542539688at_nat @ X4 @ Y5 ) @ ( sup_su5262723077604177188at_nat @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_903_sup__Un__eq2,axiom,
    ! [R2: set_Pr3525538344199662036tation,S3: set_Pr3525538344199662036tation] :
      ( ( sup_su8722909587282799537tion_o
        @ ^ [X4: a,Y5: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X4 @ Y5 ) @ R2 )
        @ ^ [X4: a,Y5: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: a,Y5: multis8755981257395730801tation] : ( member7446769889882171317tation @ ( produc7247797765490398872tation @ X4 @ Y5 ) @ ( sup_su7069682352510443776tation @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_904_sup__Un__eq2,axiom,
    ! [R2: set_Product_prod_a_o,S3: set_Product_prod_a_o] :
      ( ( sup_sup_a_o_o
        @ ^ [X4: a,Y5: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X4 @ Y5 ) @ R2 )
        @ ^ [X4: a,Y5: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X4 @ Y5 ) @ S3 ) )
      = ( ^ [X4: a,Y5: $o] : ( member2997933407038931562od_a_o @ ( product_Pair_a_o @ X4 @ Y5 ) @ ( sup_su4274294223881134325od_a_o @ R2 @ S3 ) ) ) ) ).

% sup_Un_eq2
thf(fact_905_all__s__s__mul__ext,axiom,
    ! [A4: multis8267451372601328135od_a_o,B4: multis8267451372601328135od_a_o,S2: set_Pr1246879249572150087od_a_o,Ns: set_Pr1246879249572150087od_a_o] :
      ( ( A4 != zero_z6798545679276860990od_a_o )
     => ( ! [B3: product_prod_a_o] :
            ( ( member2997933407038931562od_a_o @ B3 @ ( set_ms2137302536914688126od_a_o @ B4 ) )
           => ? [A7: product_prod_a_o] :
                ( ( member2997933407038931562od_a_o @ A7 @ ( set_ms2137302536914688126od_a_o @ A4 ) )
                & ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ A7 @ B3 ) @ S2 ) ) )
       => ( member3421213697272066832od_a_o @ ( produc5737648263999282007od_a_o @ A4 @ B4 ) @ ( multis5035663574087566680od_a_o @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_906_all__s__s__mul__ext,axiom,
    ! [A4: multis1714066462791002893od_a_o,B4: multis1714066462791002893od_a_o,S2: set_Pr7420778282784443591od_a_o,Ns: set_Pr7420778282784443591od_a_o] :
      ( ( A4 != zero_z8905637151584487364od_a_o )
     => ( ! [B3: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ B3 @ ( set_ms4142991692181744516od_a_o @ B4 ) )
           => ? [A7: list_P3196869577824546119od_a_o] :
                ( ( member5261019209154752368od_a_o @ A7 @ ( set_ms4142991692181744516od_a_o @ A4 ) )
                & ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A7 @ B3 ) @ S2 ) ) )
       => ( member627220060761961104od_a_o @ ( produc7484134505636919767od_a_o @ A4 @ B4 ) @ ( multis7840163239927194462od_a_o @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_907_all__s__s__mul__ext,axiom,
    ! [A4: multiset_a,B4: multiset_a,S2: set_Product_prod_a_a,Ns: set_Product_prod_a_a] :
      ( ( A4 != zero_zero_multiset_a )
     => ( ! [B3: a] :
            ( ( member_a @ B3 @ ( set_mset_a @ B4 ) )
           => ? [A7: a] :
                ( ( member_a @ A7 @ ( set_mset_a @ A4 ) )
                & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A7 @ B3 ) @ S2 ) ) )
       => ( member5199237121806060112iset_a @ ( produc654756711066625303iset_a @ A4 @ B4 ) @ ( multiset_s_mul_ext_a @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_908_all__s__s__mul__ext,axiom,
    ! [A4: multiset_nat,B4: multiset_nat,S2: set_Pr1261947904930325089at_nat,Ns: set_Pr1261947904930325089at_nat] :
      ( ( A4 != zero_z7348594199698428585et_nat )
     => ( ! [B3: nat] :
            ( ( member_nat @ B3 @ ( set_mset_nat @ B4 ) )
           => ? [A7: nat] :
                ( ( member_nat @ A7 @ ( set_mset_nat @ A4 ) )
                & ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B3 ) @ S2 ) ) )
       => ( member2326506860204807626et_nat @ ( produc2735455520514455641et_nat @ A4 @ B4 ) @ ( multis4215214144601311993xt_nat @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_909_all__s__s__mul__ext,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,S2: set_Pr2535564044409765851on_nat,Ns: set_Pr2535564044409765851on_nat] :
      ( ( A4 != zero_z8702567803361615526on_nat )
     => ( ! [B3: sum_su4907927656612468301on_nat] :
            ( ( member8476201367316484836on_nat @ B3 @ ( set_ms3922534532180125136on_nat @ B4 ) )
           => ? [A7: sum_su4907927656612468301on_nat] :
                ( ( member8476201367316484836on_nat @ A7 @ ( set_ms3922534532180125136on_nat @ A4 ) )
                & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B3 ) @ S2 ) ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) ) ) ) ).

% all_s_s_mul_ext
thf(fact_910_s__mul__ext__ne__extend__left,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,C3: multis1856558993800614429on_nat,W: set_Pr2535564044409765851on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( A4 != zero_z8702567803361615526on_nat )
     => ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ B4 @ C3 ) @ ( multis4183089776494585304on_nat @ W @ S3 ) )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ B4 ) @ C3 ) @ ( multis1620213760974180726on_nat @ W @ S3 ) ) ) ) ).

% s_mul_ext_ne_extend_left
thf(fact_911_mul__ext__list_Olist__order__extension__axioms,axiom,
    ( list_l4998409290742832897on_nat
    @ ^ [S4: set_Pr2535564044409765851on_nat,Ns2: set_Pr2535564044409765851on_nat] :
        ( 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 ) ) @ ( multis1620213760974180726on_nat @ Ns2 @ S4 ) ) ) )
    @ ^ [S4: set_Pr2535564044409765851on_nat,Ns2: set_Pr2535564044409765851on_nat] :
        ( 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 @ Ns2 @ S4 ) ) ) ) ) ).

% mul_ext_list.list_order_extension_axioms
thf(fact_912_add__mono__thms__linordered__field_I5_J,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_mono_thms_linordered_field(5)
thf(fact_913_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: nat,J: nat,K: nat,L3: nat] :
      ( ( ( I = J )
        & ( ord_less_nat @ K @ L3 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_914_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: nat,J: nat,K: nat,L3: nat] :
      ( ( ( ord_less_nat @ I @ J )
        & ( K = L3 ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ ( plus_plus_nat @ J @ L3 ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_915_add__strict__mono,axiom,
    ! [A: nat,B: nat,C2: nat,D3: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C2 @ D3 )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ D3 ) ) ) ) ).

% add_strict_mono
thf(fact_916_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) ) ) ).

% add_strict_left_mono
thf(fact_917_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) ) ) ).

% add_strict_right_mono
thf(fact_918_add__less__imp__less__left,axiom,
    ! [C2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C2 @ A ) @ ( plus_plus_nat @ C2 @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_919_add__less__imp__less__right,axiom,
    ! [A: nat,C2: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C2 ) @ ( plus_plus_nat @ B @ C2 ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_920_nat__add__left__cancel__less,axiom,
    ! [K: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M2 ) @ ( plus_plus_nat @ K @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% nat_add_left_cancel_less
thf(fact_921_split__cong,axiom,
    ! [Q2: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat,P: product_prod_nat_nat] :
      ( ! [X3: nat,Y3: nat] :
          ( ( ( product_Pair_nat_nat @ X3 @ Y3 )
            = Q2 )
         => ( ( F @ X3 @ Y3 )
            = ( G @ X3 @ Y3 ) ) )
     => ( ( P = Q2 )
       => ( ( produc2626176000494625587at_nat @ F @ P )
          = ( produc2626176000494625587at_nat @ G @ Q2 ) ) ) ) ).

% split_cong
thf(fact_922_s__mul__ext__self__extend__left,axiom,
    ! [A4: multis1856558993800614429on_nat,W: set_Pr2535564044409765851on_nat,B4: multis1856558993800614429on_nat,S3: set_Pr2535564044409765851on_nat] :
      ( ( A4 != zero_z8702567803361615526on_nat )
     => ( ( locall2418947207864277812on_nat @ W @ B4 )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ B4 ) @ B4 ) @ ( multis1620213760974180726on_nat @ W @ S3 ) ) ) ) ).

% s_mul_ext_self_extend_left
thf(fact_923_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_924_list__order__extension__def,axiom,
    ( list_l4998409290742832897on_nat
    = ( ^ [S_list2: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list2: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat] :
          ( ! [S5: set_Pr2535564044409765851on_nat,NS2: set_Pr2535564044409765851on_nat] :
              ( ( order_760306688390729023on_nat @ S5 @ NS2 )
             => ( order_7380660629604060367on_nat @ ( S_list2 @ S5 @ NS2 ) @ ( Ns_list2 @ S5 @ NS2 ) ) )
          & ! [S5: set_Pr2535564044409765851on_nat,F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS2: set_Pr2535564044409765851on_nat,As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat] :
              ( ! [A6: sum_su4907927656612468301on_nat,B6: sum_su4907927656612468301on_nat] :
                  ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A6 @ B6 ) @ S5 )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ S5 ) )
             => ( ! [A6: sum_su4907927656612468301on_nat,B6: sum_su4907927656612468301on_nat] :
                    ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A6 @ B6 ) @ NS2 )
                   => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ NS2 ) )
               => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs2 ) @ ( S_list2 @ S5 @ NS2 ) )
                 => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F2 @ As2 ) @ ( map_Su241112079427911264on_nat @ F2 @ Bs2 ) ) @ ( S_list2 @ S5 @ NS2 ) ) ) ) )
          & ! [S5: set_Pr2535564044409765851on_nat,F2: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS2: set_Pr2535564044409765851on_nat,As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat] :
              ( ! [A6: sum_su4907927656612468301on_nat,B6: sum_su4907927656612468301on_nat] :
                  ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A6 @ B6 ) @ S5 )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ S5 ) )
             => ( ! [A6: sum_su4907927656612468301on_nat,B6: sum_su4907927656612468301on_nat] :
                    ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A6 @ B6 ) @ NS2 )
                   => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ NS2 ) )
               => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS2 ) )
                 => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F2 @ As2 ) @ ( map_Su241112079427911264on_nat @ F2 @ Bs2 ) ) @ ( Ns_list2 @ S5 @ NS2 ) ) ) ) )
          & ! [As2: list_S5253404628753938141on_nat,Bs2: list_S5253404628753938141on_nat,NS2: set_Pr2535564044409765851on_nat,S5: set_Pr2535564044409765851on_nat] :
              ( ( ( size_s5686234912572693873on_nat @ As2 )
                = ( size_s5686234912572693873on_nat @ Bs2 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s5686234912572693873on_nat @ Bs2 ) )
                   => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As2 @ I2 ) @ ( nth_Su8471564416988646406on_nat @ Bs2 @ I2 ) ) @ NS2 ) )
               => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS2 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_925_list__order__extension__def,axiom,
    ( list_l1016356893908098451od_a_o
    = ( ^ [S_list2: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o,Ns_list2: set_Pr7420778282784443591od_a_o > set_Pr7420778282784443591od_a_o > set_Pr6888711208267297863od_a_o] :
          ( ! [S5: set_Pr7420778282784443591od_a_o,NS2: set_Pr7420778282784443591od_a_o] :
              ( ( order_627342493205700821od_a_o @ S5 @ NS2 )
             => ( order_6632892376036510427od_a_o @ ( S_list2 @ S5 @ NS2 ) @ ( Ns_list2 @ S5 @ NS2 ) ) )
          & ! [S5: set_Pr7420778282784443591od_a_o,F2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,NS2: set_Pr7420778282784443591od_a_o,As2: list_l8336264072933094989od_a_o,Bs2: list_l8336264072933094989od_a_o] :
              ( ! [A6: list_P3196869577824546119od_a_o,B6: list_P3196869577824546119od_a_o] :
                  ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A6 @ B6 ) @ S5 )
                 => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ S5 ) )
             => ( ! [A6: list_P3196869577824546119od_a_o,B6: list_P3196869577824546119od_a_o] :
                    ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A6 @ B6 ) @ NS2 )
                   => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ NS2 ) )
               => ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As2 @ Bs2 ) @ ( S_list2 @ S5 @ NS2 ) )
                 => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( map_li880244354692327940od_a_o @ F2 @ As2 ) @ ( map_li880244354692327940od_a_o @ F2 @ Bs2 ) ) @ ( S_list2 @ S5 @ NS2 ) ) ) ) )
          & ! [S5: set_Pr7420778282784443591od_a_o,F2: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,NS2: set_Pr7420778282784443591od_a_o,As2: list_l8336264072933094989od_a_o,Bs2: list_l8336264072933094989od_a_o] :
              ( ! [A6: list_P3196869577824546119od_a_o,B6: list_P3196869577824546119od_a_o] :
                  ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A6 @ B6 ) @ S5 )
                 => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ S5 ) )
             => ( ! [A6: list_P3196869577824546119od_a_o,B6: list_P3196869577824546119od_a_o] :
                    ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A6 @ B6 ) @ NS2 )
                   => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ NS2 ) )
               => ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS2 ) )
                 => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( map_li880244354692327940od_a_o @ F2 @ As2 ) @ ( map_li880244354692327940od_a_o @ F2 @ Bs2 ) ) @ ( Ns_list2 @ S5 @ NS2 ) ) ) ) )
          & ! [As2: list_l8336264072933094989od_a_o,Bs2: list_l8336264072933094989od_a_o,NS2: set_Pr7420778282784443591od_a_o,S5: set_Pr7420778282784443591od_a_o] :
              ( ( ( size_s3379185846251495609od_a_o @ As2 )
                = ( size_s3379185846251495609od_a_o @ Bs2 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_s3379185846251495609od_a_o @ Bs2 ) )
                   => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As2 @ I2 ) @ ( nth_li3248255784598043854od_a_o @ Bs2 @ I2 ) ) @ NS2 ) )
               => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS2 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_926_list__order__extension__def,axiom,
    ( list_l792762466888043652on_nat
    = ( ^ [S_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list2: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
          ( ! [S5: set_Pr1261947904930325089at_nat,NS2: set_Pr1261947904930325089at_nat] :
              ( ( order_7802623214671921858ir_nat @ S5 @ NS2 )
             => ( order_2144575240049343698st_nat @ ( S_list2 @ S5 @ NS2 ) @ ( Ns_list2 @ S5 @ NS2 ) ) )
          & ! [S5: set_Pr1261947904930325089at_nat,F2: nat > nat,NS2: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
              ( ! [A6: nat,B6: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ S5 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ S5 ) )
             => ( ! [A6: nat,B6: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ NS2 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ NS2 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( S_list2 @ S5 @ NS2 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F2 @ As2 ) @ ( map_nat_nat @ F2 @ Bs2 ) ) @ ( S_list2 @ S5 @ NS2 ) ) ) ) )
          & ! [S5: set_Pr1261947904930325089at_nat,F2: nat > nat,NS2: set_Pr1261947904930325089at_nat,As2: list_nat,Bs2: list_nat] :
              ( ! [A6: nat,B6: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ S5 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ S5 ) )
             => ( ! [A6: nat,B6: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ B6 ) @ NS2 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F2 @ A6 ) @ ( F2 @ B6 ) ) @ NS2 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS2 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F2 @ As2 ) @ ( map_nat_nat @ F2 @ Bs2 ) ) @ ( Ns_list2 @ S5 @ NS2 ) ) ) ) )
          & ! [As2: list_nat,Bs2: list_nat,NS2: set_Pr1261947904930325089at_nat,S5: set_Pr1261947904930325089at_nat] :
              ( ( ( size_size_list_nat @ As2 )
                = ( size_size_list_nat @ Bs2 ) )
             => ( ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Bs2 ) )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As2 @ I2 ) @ ( nth_nat @ Bs2 @ I2 ) ) @ NS2 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As2 @ Bs2 ) @ ( Ns_list2 @ S5 @ NS2 ) ) ) ) ) ) ) ).

% list_order_extension_def
thf(fact_927_less__nat__zero__code,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_928_neq0__conv,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% neq0_conv
thf(fact_929_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_930_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z2: nat,C2: nat > nat > set_nat] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_nat @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z2 @ ( produc6189476227299908564et_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_931_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z2: a,C2: nat > nat > set_a] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc8052394783829583760_set_a @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_932_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z2: nat,C2: a > multis8755981257395730801tation > set_nat] :
      ( ! [A3: a,B3: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B3 ) )
         => ( member_nat @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z2 @ ( produc6722172017951998027et_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_933_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z2: a,C2: a > multis8755981257395730801tation > set_a] :
      ( ! [A3: a,B3: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc7412255387654812377_set_a @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_934_mem__case__prodI2,axiom,
    ! [P: product_prod_a_o,Z2: nat,C2: a > $o > set_nat] :
      ( ! [A3: a,B3: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B3 ) )
         => ( member_nat @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z2 @ ( produc8120172443933582796et_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_935_mem__case__prodI2,axiom,
    ! [P: product_prod_a_o,Z2: a,C2: a > $o > set_a] :
      ( ! [A3: a,B3: $o] :
          ( ( P
            = ( product_Pair_a_o @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc6584190720350841752_set_a @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_936_mem__case__prodI2,axiom,
    ! [P: product_prod_nat_nat,Z2: product_prod_a_o,C2: nat > nat > set_Product_prod_a_o] :
      ( ! [A3: nat,B3: nat] :
          ( ( P
            = ( product_Pair_nat_nat @ A3 @ B3 ) )
         => ( member2997933407038931562od_a_o @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member2997933407038931562od_a_o @ Z2 @ ( produc2779750408732127059od_a_o @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_937_mem__case__prodI2,axiom,
    ! [P: produc8199716216217303280at_nat,Z2: nat,C2: ( nat > nat ) > nat > set_nat] :
      ( ! [A3: nat > nat,B3: nat] :
          ( ( P
            = ( produc72220940542539688at_nat @ A3 @ B3 ) )
         => ( member_nat @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_nat @ Z2 @ ( produc7086537269843436355et_nat @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_938_mem__case__prodI2,axiom,
    ! [P: produc8199716216217303280at_nat,Z2: a,C2: ( nat > nat ) > nat > set_a] :
      ( ! [A3: nat > nat,B3: nat] :
          ( ( P
            = ( produc72220940542539688at_nat @ A3 @ B3 ) )
         => ( member_a @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member_a @ Z2 @ ( produc1640476547790970337_set_a @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_939_mem__case__prodI2,axiom,
    ! [P: produc27318731269267614tation,Z2: product_prod_a_o,C2: a > multis8755981257395730801tation > set_Product_prod_a_o] :
      ( ! [A3: a,B3: multis8755981257395730801tation] :
          ( ( P
            = ( produc7247797765490398872tation @ A3 @ B3 ) )
         => ( member2997933407038931562od_a_o @ Z2 @ ( C2 @ A3 @ B3 ) ) )
     => ( member2997933407038931562od_a_o @ Z2 @ ( produc8399654089035374108od_a_o @ C2 @ P ) ) ) ).

% mem_case_prodI2
thf(fact_940_mem__case__prodI,axiom,
    ! [Z2: nat,C2: nat > nat > set_nat,A: nat,B: nat] :
      ( ( member_nat @ Z2 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc6189476227299908564et_nat @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_941_mem__case__prodI,axiom,
    ! [Z2: a,C2: nat > nat > set_a,A: nat,B: nat] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc8052394783829583760_set_a @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_942_mem__case__prodI,axiom,
    ! [Z2: nat,C2: a > multis8755981257395730801tation > set_nat,A: a,B: multis8755981257395730801tation] :
      ( ( member_nat @ Z2 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc6722172017951998027et_nat @ C2 @ ( produc7247797765490398872tation @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_943_mem__case__prodI,axiom,
    ! [Z2: a,C2: a > multis8755981257395730801tation > set_a,A: a,B: multis8755981257395730801tation] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc7412255387654812377_set_a @ C2 @ ( produc7247797765490398872tation @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_944_mem__case__prodI,axiom,
    ! [Z2: nat,C2: a > $o > set_nat,A: a,B: $o] :
      ( ( member_nat @ Z2 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc8120172443933582796et_nat @ C2 @ ( product_Pair_a_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_945_mem__case__prodI,axiom,
    ! [Z2: a,C2: a > $o > set_a,A: a,B: $o] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc6584190720350841752_set_a @ C2 @ ( product_Pair_a_o @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_946_mem__case__prodI,axiom,
    ! [Z2: product_prod_a_o,C2: nat > nat > set_Product_prod_a_o,A: nat,B: nat] :
      ( ( member2997933407038931562od_a_o @ Z2 @ ( C2 @ A @ B ) )
     => ( member2997933407038931562od_a_o @ Z2 @ ( produc2779750408732127059od_a_o @ C2 @ ( product_Pair_nat_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_947_mem__case__prodI,axiom,
    ! [Z2: nat,C2: ( nat > nat ) > nat > set_nat,A: nat > nat,B: nat] :
      ( ( member_nat @ Z2 @ ( C2 @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc7086537269843436355et_nat @ C2 @ ( produc72220940542539688at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_948_mem__case__prodI,axiom,
    ! [Z2: a,C2: ( nat > nat ) > nat > set_a,A: nat > nat,B: nat] :
      ( ( member_a @ Z2 @ ( C2 @ A @ B ) )
     => ( member_a @ Z2 @ ( produc1640476547790970337_set_a @ C2 @ ( produc72220940542539688at_nat @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_949_mem__case__prodI,axiom,
    ! [Z2: product_prod_a_o,C2: a > multis8755981257395730801tation > set_Product_prod_a_o,A: a,B: multis8755981257395730801tation] :
      ( ( member2997933407038931562od_a_o @ Z2 @ ( C2 @ A @ B ) )
     => ( member2997933407038931562od_a_o @ Z2 @ ( produc8399654089035374108od_a_o @ C2 @ ( produc7247797765490398872tation @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_950_length__0__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( ( size_s7490768528479217773on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Pr8614093721167239297on_nat ) ) ).

% length_0_conv
thf(fact_951_length__0__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_Su3078906344068544711on_nat ) ) ).

% length_0_conv
thf(fact_952_length__0__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_li1372968520647675725od_a_o ) ) ).

% length_0_conv
thf(fact_953_length__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = zero_zero_nat )
      = ( Xs = nil_nat ) ) ).

% length_0_conv
thf(fact_954_add__gr__0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M2 @ N ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M2 )
        | ( ord_less_nat @ zero_zero_nat @ N ) ) ) ).

% add_gr_0
thf(fact_955_length__greater__0__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s7490768528479217773on_nat @ Xs ) )
      = ( Xs != nil_Pr8614093721167239297on_nat ) ) ).

% length_greater_0_conv
thf(fact_956_length__greater__0__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s5686234912572693873on_nat @ Xs ) )
      = ( Xs != nil_Su3078906344068544711on_nat ) ) ).

% length_greater_0_conv
thf(fact_957_length__greater__0__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ Xs ) )
      = ( Xs != nil_li1372968520647675725od_a_o ) ) ).

% length_greater_0_conv
thf(fact_958_length__greater__0__conv,axiom,
    ! [Xs: list_nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) )
      = ( Xs != nil_nat ) ) ).

% length_greater_0_conv
thf(fact_959_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_960_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_961_gr0I,axiom,
    ! [N: nat] :
      ( ( N != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N ) ) ).

% gr0I
thf(fact_962_not__gr0,axiom,
    ! [N: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N ) )
      = ( N = zero_zero_nat ) ) ).

% not_gr0
thf(fact_963_not__less0,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% not_less0
thf(fact_964_less__zeroE,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ zero_zero_nat ) ).

% less_zeroE
thf(fact_965_gr__implies__not0,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( N != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_966_infinite__descent0,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ( P2 @ zero_zero_nat )
     => ( ! [N4: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N4 )
           => ( ~ ( P2 @ N4 )
             => ? [M3: nat] :
                  ( ( ord_less_nat @ M3 @ N4 )
                  & ~ ( P2 @ M3 ) ) ) )
       => ( P2 @ N ) ) ) ).

% infinite_descent0
thf(fact_967_mem__case__prodE,axiom,
    ! [Z2: nat,C2: nat > nat > set_nat,P: product_prod_nat_nat] :
      ( ( member_nat @ Z2 @ ( produc6189476227299908564et_nat @ C2 @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_968_mem__case__prodE,axiom,
    ! [Z2: a,C2: nat > nat > set_a,P: product_prod_nat_nat] :
      ( ( member_a @ Z2 @ ( produc8052394783829583760_set_a @ C2 @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_969_mem__case__prodE,axiom,
    ! [Z2: nat,C2: a > multis8755981257395730801tation > set_nat,P: produc27318731269267614tation] :
      ( ( member_nat @ Z2 @ ( produc6722172017951998027et_nat @ C2 @ P ) )
     => ~ ! [X3: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_970_mem__case__prodE,axiom,
    ! [Z2: a,C2: a > multis8755981257395730801tation > set_a,P: produc27318731269267614tation] :
      ( ( member_a @ Z2 @ ( produc7412255387654812377_set_a @ C2 @ P ) )
     => ~ ! [X3: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_971_mem__case__prodE,axiom,
    ! [Z2: nat,C2: a > $o > set_nat,P: product_prod_a_o] :
      ( ( member_nat @ Z2 @ ( produc8120172443933582796et_nat @ C2 @ P ) )
     => ~ ! [X3: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_972_mem__case__prodE,axiom,
    ! [Z2: a,C2: a > $o > set_a,P: product_prod_a_o] :
      ( ( member_a @ Z2 @ ( produc6584190720350841752_set_a @ C2 @ P ) )
     => ~ ! [X3: a,Y3: $o] :
            ( ( P
              = ( product_Pair_a_o @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_973_mem__case__prodE,axiom,
    ! [Z2: product_prod_a_o,C2: nat > nat > set_Product_prod_a_o,P: product_prod_nat_nat] :
      ( ( member2997933407038931562od_a_o @ Z2 @ ( produc2779750408732127059od_a_o @ C2 @ P ) )
     => ~ ! [X3: nat,Y3: nat] :
            ( ( P
              = ( product_Pair_nat_nat @ X3 @ Y3 ) )
           => ~ ( member2997933407038931562od_a_o @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_974_mem__case__prodE,axiom,
    ! [Z2: nat,C2: ( nat > nat ) > nat > set_nat,P: produc8199716216217303280at_nat] :
      ( ( member_nat @ Z2 @ ( produc7086537269843436355et_nat @ C2 @ P ) )
     => ~ ! [X3: nat > nat,Y3: nat] :
            ( ( P
              = ( produc72220940542539688at_nat @ X3 @ Y3 ) )
           => ~ ( member_nat @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_975_mem__case__prodE,axiom,
    ! [Z2: a,C2: ( nat > nat ) > nat > set_a,P: produc8199716216217303280at_nat] :
      ( ( member_a @ Z2 @ ( produc1640476547790970337_set_a @ C2 @ P ) )
     => ~ ! [X3: nat > nat,Y3: nat] :
            ( ( P
              = ( produc72220940542539688at_nat @ X3 @ Y3 ) )
           => ~ ( member_a @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_976_mem__case__prodE,axiom,
    ! [Z2: product_prod_a_o,C2: a > multis8755981257395730801tation > set_Product_prod_a_o,P: produc27318731269267614tation] :
      ( ( member2997933407038931562od_a_o @ Z2 @ ( produc8399654089035374108od_a_o @ C2 @ P ) )
     => ~ ! [X3: a,Y3: multis8755981257395730801tation] :
            ( ( P
              = ( produc7247797765490398872tation @ X3 @ Y3 ) )
           => ~ ( member2997933407038931562od_a_o @ Z2 @ ( C2 @ X3 @ Y3 ) ) ) ) ).

% mem_case_prodE
thf(fact_977_case__prod__Pair__iden,axiom,
    ! [P: produc8325617224762090749on_nat] :
      ( ( produc4035448539481142195on_nat @ produc1545487600272464053on_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_978_case__prod__Pair__iden,axiom,
    ! [P: produc8199716216217303280at_nat] :
      ( ( produc3780431343476367475at_nat @ produc72220940542539688at_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_979_case__prod__Pair__iden,axiom,
    ! [P: produc27318731269267614tation] :
      ( ( produc1859402794938042595tation @ produc7247797765490398872tation @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_980_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_981_case__prod__Pair__iden,axiom,
    ! [P: product_prod_nat_nat] :
      ( ( produc2626176000494625587at_nat @ product_Pair_nat_nat @ P )
      = P ) ).

% case_prod_Pair_iden
thf(fact_982_SN__order__pair_Omul__ext__SN__order__pair,axiom,
    ! [S3: set_Pr2535564044409765851on_nat,NS: set_Pr2535564044409765851on_nat] :
      ( ( order_760306688390729023on_nat @ S3 @ NS )
     => ( order_8476787420702163983on_nat @ ( multis1620213760974180726on_nat @ NS @ S3 ) @ ( multis4183089776494585304on_nat @ NS @ S3 ) ) ) ).

% SN_order_pair.mul_ext_SN_order_pair
thf(fact_983_list_Osize_I3_J,axiom,
    ( ( size_s7490768528479217773on_nat @ nil_Pr8614093721167239297on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_984_list_Osize_I3_J,axiom,
    ( ( size_s5686234912572693873on_nat @ nil_Su3078906344068544711on_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_985_list_Osize_I3_J,axiom,
    ( ( size_s3379185846251495609od_a_o @ nil_li1372968520647675725od_a_o )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_986_list_Osize_I3_J,axiom,
    ( ( size_size_list_nat @ nil_nat )
    = zero_zero_nat ) ).

% list.size(3)
thf(fact_987_nat__neq__iff,axiom,
    ! [M2: nat,N: nat] :
      ( ( M2 != N )
      = ( ( ord_less_nat @ M2 @ N )
        | ( ord_less_nat @ N @ M2 ) ) ) ).

% nat_neq_iff
thf(fact_988_less__not__refl,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_not_refl
thf(fact_989_less__not__refl2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( M2 != N ) ) ).

% less_not_refl2
thf(fact_990_less__not__refl3,axiom,
    ! [S2: nat,T: nat] :
      ( ( ord_less_nat @ S2 @ T )
     => ( S2 != T ) ) ).

% less_not_refl3
thf(fact_991_less__irrefl__nat,axiom,
    ! [N: nat] :
      ~ ( ord_less_nat @ N @ N ) ).

% less_irrefl_nat
thf(fact_992_nat__less__induct,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ! [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
             => ( P2 @ M3 ) )
         => ( P2 @ N4 ) )
     => ( P2 @ N ) ) ).

% nat_less_induct
thf(fact_993_infinite__descent,axiom,
    ! [P2: nat > $o,N: nat] :
      ( ! [N4: nat] :
          ( ~ ( P2 @ N4 )
         => ? [M3: nat] :
              ( ( ord_less_nat @ M3 @ N4 )
              & ~ ( P2 @ M3 ) ) )
     => ( P2 @ N ) ) ).

% infinite_descent
thf(fact_994_linorder__neqE__nat,axiom,
    ! [X: nat,Y: nat] :
      ( ( X != Y )
     => ( ~ ( ord_less_nat @ X @ Y )
       => ( ord_less_nat @ Y @ X ) ) ) ).

% linorder_neqE_nat
thf(fact_995_less__add__eq__less,axiom,
    ! [K: nat,L3: nat,M2: nat,N: nat] :
      ( ( ord_less_nat @ K @ L3 )
     => ( ( ( plus_plus_nat @ M2 @ L3 )
          = ( plus_plus_nat @ K @ N ) )
       => ( ord_less_nat @ M2 @ N ) ) ) ).

% less_add_eq_less
thf(fact_996_trans__less__add2,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M2 @ J ) ) ) ).

% trans_less_add2
thf(fact_997_trans__less__add1,axiom,
    ! [I: nat,J: nat,M2: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M2 ) ) ) ).

% trans_less_add1
thf(fact_998_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_999_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_1000_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_1001_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_1002_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_1003_size__neq__size__imp__neq,axiom,
    ! [X: list_S5253404628753938141on_nat,Y: list_S5253404628753938141on_nat] :
      ( ( ( size_s5686234912572693873on_nat @ X )
       != ( size_s5686234912572693873on_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1004_size__neq__size__imp__neq,axiom,
    ! [X: list_l8336264072933094989od_a_o,Y: list_l8336264072933094989od_a_o] :
      ( ( ( size_s3379185846251495609od_a_o @ X )
       != ( size_s3379185846251495609od_a_o @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1005_size__neq__size__imp__neq,axiom,
    ! [X: multis8755981257395730801tation,Y: multis8755981257395730801tation] :
      ( ( ( size_s7536720479165623301tation @ X )
       != ( size_s7536720479165623301tation @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1006_size__neq__size__imp__neq,axiom,
    ! [X: char,Y: char] :
      ( ( ( size_size_char @ X )
       != ( size_size_char @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1007_size__neq__size__imp__neq,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X )
       != ( size_size_list_nat @ Y ) )
     => ( X != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_1008_length__pos__if__in__set,axiom,
    ! [X: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ X @ ( set_Product_prod_a_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s1727999245659094451od_a_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1009_length__pos__if__in__set,axiom,
    ! [X: a,Xs: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_a @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1010_length__pos__if__in__set,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ X @ ( set_Su4848624856859743608on_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5686234912572693873on_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1011_length__pos__if__in__set,axiom,
    ! [X: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ X @ ( set_li2629784761229999580od_a_o @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1012_length__pos__if__in__set,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1013_ns__mul__ext__refl__local,axiom,
    ! [Ns: set_Pr2535564044409765851on_nat,A4: multis1856558993800614429on_nat,S2: set_Pr2535564044409765851on_nat] :
      ( ( locall2418947207864277812on_nat @ Ns @ A4 )
     => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ A4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) ) ) ).

% ns_mul_ext_refl_local
thf(fact_1014_mul__ext__list_Oextension,axiom,
    ! [S3: set_Pr2535564044409765851on_nat,NS: set_Pr2535564044409765851on_nat] :
      ( ( order_760306688390729023on_nat @ S3 @ NS )
     => ( order_7380660629604060367on_nat
        @ ( 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 ) ) @ ( multis1620213760974180726on_nat @ NS @ S3 ) ) ) )
        @ ( 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 @ S3 ) ) ) ) ) ) ).

% mul_ext_list.extension
thf(fact_1015_ns__mul__ext__union__compat,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) )
     => ( ( locall2418947207864277812on_nat @ Ns @ C3 )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B4 @ C3 ) ) @ ( multis4183089776494585304on_nat @ Ns @ S2 ) ) ) ) ).

% ns_mul_ext_union_compat
thf(fact_1016_s__mul__ext__union__compat,axiom,
    ! [A4: multis1856558993800614429on_nat,B4: multis1856558993800614429on_nat,Ns: set_Pr2535564044409765851on_nat,S2: set_Pr2535564044409765851on_nat,C3: multis1856558993800614429on_nat] :
      ( ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ A4 @ B4 ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) )
     => ( ( locall2418947207864277812on_nat @ Ns @ C3 )
       => ( member4278285197320822468on_nat @ ( produc5200047356169809875on_nat @ ( plus_p2586647234311342374on_nat @ A4 @ C3 ) @ ( plus_p2586647234311342374on_nat @ B4 @ C3 ) ) @ ( multis1620213760974180726on_nat @ Ns @ S2 ) ) ) ) ).

% s_mul_ext_union_compat
thf(fact_1017_list__order__extension_Ointro,axiom,
    ! [S_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat,Ns_list: set_Pr2535564044409765851on_nat > set_Pr2535564044409765851on_nat > set_Pr7828957961435280891on_nat] :
      ( ! [S6: set_Pr2535564044409765851on_nat,NS3: set_Pr2535564044409765851on_nat] :
          ( ( order_760306688390729023on_nat @ S6 @ NS3 )
         => ( order_7380660629604060367on_nat @ ( S_list @ S6 @ NS3 ) @ ( Ns_list @ S6 @ NS3 ) ) )
     => ( ! [S6: set_Pr2535564044409765851on_nat,F3: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS3: set_Pr2535564044409765851on_nat,As3: list_S5253404628753938141on_nat,Bs3: list_S5253404628753938141on_nat] :
            ( ! [A7: sum_su4907927656612468301on_nat,B7: sum_su4907927656612468301on_nat] :
                ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B7 ) @ S6 )
               => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ S6 ) )
           => ( ! [A7: sum_su4907927656612468301on_nat,B7: sum_su4907927656612468301on_nat] :
                  ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B7 ) @ NS3 )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ NS3 ) )
             => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As3 @ Bs3 ) @ ( S_list @ S6 @ NS3 ) )
               => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F3 @ As3 ) @ ( map_Su241112079427911264on_nat @ F3 @ Bs3 ) ) @ ( S_list @ S6 @ NS3 ) ) ) ) )
       => ( ! [S6: set_Pr2535564044409765851on_nat,F3: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,NS3: set_Pr2535564044409765851on_nat,As3: list_S5253404628753938141on_nat,Bs3: list_S5253404628753938141on_nat] :
              ( ! [A7: sum_su4907927656612468301on_nat,B7: sum_su4907927656612468301on_nat] :
                  ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B7 ) @ S6 )
                 => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ S6 ) )
             => ( ! [A7: sum_su4907927656612468301on_nat,B7: sum_su4907927656612468301on_nat] :
                    ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A7 @ B7 ) @ NS3 )
                   => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ NS3 ) )
               => ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As3 @ Bs3 ) @ ( Ns_list @ S6 @ NS3 ) )
                 => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( map_Su241112079427911264on_nat @ F3 @ As3 ) @ ( map_Su241112079427911264on_nat @ F3 @ Bs3 ) ) @ ( Ns_list @ S6 @ NS3 ) ) ) ) )
         => ( ! [As3: list_S5253404628753938141on_nat,Bs3: list_S5253404628753938141on_nat,NS3: set_Pr2535564044409765851on_nat,S6: set_Pr2535564044409765851on_nat] :
                ( ( ( size_s5686234912572693873on_nat @ As3 )
                  = ( size_s5686234912572693873on_nat @ Bs3 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s5686234912572693873on_nat @ Bs3 ) )
                     => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ ( nth_Su8471564416988646406on_nat @ As3 @ I4 ) @ ( nth_Su8471564416988646406on_nat @ Bs3 @ I4 ) ) @ NS3 ) )
                 => ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ As3 @ Bs3 ) @ ( Ns_list @ S6 @ NS3 ) ) ) )
           => ( list_l4998409290742832897on_nat @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_1018_list__order__extension_Ointro,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] :
      ( ! [S6: set_Pr7420778282784443591od_a_o,NS3: set_Pr7420778282784443591od_a_o] :
          ( ( order_627342493205700821od_a_o @ S6 @ NS3 )
         => ( order_6632892376036510427od_a_o @ ( S_list @ S6 @ NS3 ) @ ( Ns_list @ S6 @ NS3 ) ) )
     => ( ! [S6: set_Pr7420778282784443591od_a_o,F3: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,NS3: set_Pr7420778282784443591od_a_o,As3: list_l8336264072933094989od_a_o,Bs3: list_l8336264072933094989od_a_o] :
            ( ! [A7: list_P3196869577824546119od_a_o,B7: list_P3196869577824546119od_a_o] :
                ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A7 @ B7 ) @ S6 )
               => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ S6 ) )
           => ( ! [A7: list_P3196869577824546119od_a_o,B7: list_P3196869577824546119od_a_o] :
                  ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A7 @ B7 ) @ NS3 )
                 => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ NS3 ) )
             => ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As3 @ Bs3 ) @ ( S_list @ S6 @ NS3 ) )
               => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( map_li880244354692327940od_a_o @ F3 @ As3 ) @ ( map_li880244354692327940od_a_o @ F3 @ Bs3 ) ) @ ( S_list @ S6 @ NS3 ) ) ) ) )
       => ( ! [S6: set_Pr7420778282784443591od_a_o,F3: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o,NS3: set_Pr7420778282784443591od_a_o,As3: list_l8336264072933094989od_a_o,Bs3: list_l8336264072933094989od_a_o] :
              ( ! [A7: list_P3196869577824546119od_a_o,B7: list_P3196869577824546119od_a_o] :
                  ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A7 @ B7 ) @ S6 )
                 => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ S6 ) )
             => ( ! [A7: list_P3196869577824546119od_a_o,B7: list_P3196869577824546119od_a_o] :
                    ( ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A7 @ B7 ) @ NS3 )
                   => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ NS3 ) )
               => ( ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As3 @ Bs3 ) @ ( Ns_list @ S6 @ NS3 ) )
                 => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ ( map_li880244354692327940od_a_o @ F3 @ As3 ) @ ( map_li880244354692327940od_a_o @ F3 @ Bs3 ) ) @ ( Ns_list @ S6 @ NS3 ) ) ) ) )
         => ( ! [As3: list_l8336264072933094989od_a_o,Bs3: list_l8336264072933094989od_a_o,NS3: set_Pr7420778282784443591od_a_o,S6: set_Pr7420778282784443591od_a_o] :
                ( ( ( size_s3379185846251495609od_a_o @ As3 )
                  = ( size_s3379185846251495609od_a_o @ Bs3 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_s3379185846251495609od_a_o @ Bs3 ) )
                     => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ ( nth_li3248255784598043854od_a_o @ As3 @ I4 ) @ ( nth_li3248255784598043854od_a_o @ Bs3 @ I4 ) ) @ NS3 ) )
                 => ( member6499739831149278864od_a_o @ ( produc7858527432735436247od_a_o @ As3 @ Bs3 ) @ ( Ns_list @ S6 @ NS3 ) ) ) )
           => ( list_l1016356893908098451od_a_o @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_1019_list__order__extension_Ointro,axiom,
    ! [S_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat,Ns_list: set_Pr1261947904930325089at_nat > set_Pr1261947904930325089at_nat > set_Pr3451248702717554689st_nat] :
      ( ! [S6: set_Pr1261947904930325089at_nat,NS3: set_Pr1261947904930325089at_nat] :
          ( ( order_7802623214671921858ir_nat @ S6 @ NS3 )
         => ( order_2144575240049343698st_nat @ ( S_list @ S6 @ NS3 ) @ ( Ns_list @ S6 @ NS3 ) ) )
     => ( ! [S6: set_Pr1261947904930325089at_nat,F3: nat > nat,NS3: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
            ( ! [A7: nat,B7: nat] :
                ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B7 ) @ S6 )
               => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ S6 ) )
           => ( ! [A7: nat,B7: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B7 ) @ NS3 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ NS3 ) )
             => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( S_list @ S6 @ NS3 ) )
               => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F3 @ As3 ) @ ( map_nat_nat @ F3 @ Bs3 ) ) @ ( S_list @ S6 @ NS3 ) ) ) ) )
       => ( ! [S6: set_Pr1261947904930325089at_nat,F3: nat > nat,NS3: set_Pr1261947904930325089at_nat,As3: list_nat,Bs3: list_nat] :
              ( ! [A7: nat,B7: nat] :
                  ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B7 ) @ S6 )
                 => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ S6 ) )
             => ( ! [A7: nat,B7: nat] :
                    ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A7 @ B7 ) @ NS3 )
                   => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( F3 @ A7 ) @ ( F3 @ B7 ) ) @ NS3 ) )
               => ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list @ S6 @ NS3 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( map_nat_nat @ F3 @ As3 ) @ ( map_nat_nat @ F3 @ Bs3 ) ) @ ( Ns_list @ S6 @ NS3 ) ) ) ) )
         => ( ! [As3: list_nat,Bs3: list_nat,NS3: set_Pr1261947904930325089at_nat,S6: set_Pr1261947904930325089at_nat] :
                ( ( ( size_size_list_nat @ As3 )
                  = ( size_size_list_nat @ Bs3 ) )
               => ( ! [I4: nat] :
                      ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Bs3 ) )
                     => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ As3 @ I4 ) @ ( nth_nat @ Bs3 @ I4 ) ) @ NS3 ) )
                 => ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ As3 @ Bs3 ) @ ( Ns_list @ S6 @ NS3 ) ) ) )
           => ( list_l792762466888043652on_nat @ S_list @ Ns_list ) ) ) ) ) ).

% list_order_extension.intro
thf(fact_1020__092_060open_062_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_A_Iconcat_A_Imap_A_I_092_060lambda_062i_O_Amap_A_I_092_060lambda_062l_O_A_Ims__elem__of__lit_Al_M_AInr_Ai_J_J_A_Icnf_A_B_Ai_J_J_Acs_J_J_092_060close_062,axiom,
    ! [T: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ T @ ( set_Su4848624856859743608on_nat @ s_right ) )
     => ? [X3: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ X3 @ ( set_Su4848624856859743608on_nat @ s_left ) )
          & ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X3 @ T )
            @ ( set_Pr937530902923597456on_nat
              @ ( concat937493773106420796on_nat
                @ ( map_na8715451351012060651on_nat
                  @ ^ [I2: nat] :
                      ( map_Pr6945619049609468478on_nat
                      @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I2 ) )
                      @ ( nth_li3248255784598043854od_a_o @ cnf @ I2 ) )
                  @ cs ) ) ) ) ) ) ).

% \<open>\<And>t. t \<in> set s_right \<Longrightarrow> \<exists>s\<in>set s_left. (s, t) \<in> set (concat (map (\<lambda>i. map (\<lambda>l. (ms_elem_of_lit l, Inr i)) (cnf ! i)) cs))\<close>
thf(fact_1021_S,axiom,
    ( s
    = ( concat937493773106420796on_nat
      @ ( map_na8715451351012060651on_nat
        @ ^ [I2: nat] :
            ( map_Pr6945619049609468478on_nat
            @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I2 ) )
            @ ( nth_li3248255784598043854od_a_o @ cnf @ I2 ) )
        @ cs ) ) ) ).

% S
thf(fact_1022_cs__def,axiom,
    ( cs
    = ( upt @ zero_zero_nat @ ( size_s3379185846251495609od_a_o @ cnf ) ) ) ).

% cs_def
thf(fact_1023_res,axiom,
    ( ( produc1545487600272464053on_nat
      @ ( concat6161509665823917772on_nat
        @ ( map_a_3887198048842487703on_nat
          @ ^ [X4: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
          @ xs ) )
      @ ( produc4176415733022414961on_nat
        @ ( append3523812819975775986on_nat
          @ ( map_a_2711072540680412167on_nat
            @ ^ [X4: a] : ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5512618279175645674signed ) )
            @ xs )
          @ ( map_na5706685718985795555on_nat @ sum_In255631617079920996tation @ cs ) )
        @ ( produc7152764266822961111on_nat
          @ ( concat937493773106420796on_nat
            @ ( map_na8715451351012060651on_nat
              @ ^ [I2: nat] :
                  ( map_Pr6945619049609468478on_nat
                  @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I2 ) )
                  @ ( nth_li3248255784598043854od_a_o @ cnf @ I2 ) )
              @ cs ) )
          @ ( append2865821678206098582on_nat
            @ ( concat937493773106420796on_nat
              @ ( map_a_7431629901395152583on_nat
                @ ^ [X4: a] : ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis8267843437418295310sitive ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5512618279175645674signed ) ) ) @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5977116398950053002gative ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5512618279175645674signed ) ) ) @ nil_Pr8614093721167239297on_nat ) )
                @ xs ) )
            @ ( concat937493773106420796on_nat
              @ ( map_na8715451351012060651on_nat
                @ ^ [I2: nat] :
                    ( map_Pr6945619049609468478on_nat
                    @ ^ [L2: product_prod_a_o] : ( produc6704844767664046899on_nat @ ( multis8230369766971345672_lit_a @ L2 ) @ ( sum_In255631617079920996tation @ I2 ) )
                    @ ( nth_li3248255784598043854od_a_o @ cnf @ I2 ) )
                @ cs ) ) ) ) ) )
    = ( produc1545487600272464053on_nat @ left @ ( produc4176415733022414961on_nat @ right @ ( produc7152764266822961111on_nat @ s @ nss ) ) ) ) ).

% res
thf(fact_1024_locally__refl__def,axiom,
    ( locall9097760533475055130od_a_o
    = ( ^ [R3: set_Pr1246879249572150087od_a_o,A5: multis8267451372601328135od_a_o] :
        ! [A6: product_prod_a_o] :
          ( ( member2997933407038931562od_a_o @ A6 @ ( set_ms2137302536914688126od_a_o @ A5 ) )
         => ( member3341120605725091216od_a_o @ ( produc3235384523528429271od_a_o @ A6 @ A6 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_1025_locally__refl__def,axiom,
    ( locall4369668106379569952od_a_o
    = ( ^ [R3: set_Pr7420778282784443591od_a_o,A5: multis1714066462791002893od_a_o] :
        ! [A6: list_P3196869577824546119od_a_o] :
          ( ( member5261019209154752368od_a_o @ A6 @ ( set_ms4142991692181744516od_a_o @ A5 ) )
         => ( member5440785652130980624od_a_o @ ( produc8077463541153823575od_a_o @ A6 @ A6 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_1026_locally__refl__def,axiom,
    ( locally_refl_a
    = ( ^ [R3: set_Product_prod_a_a,A5: multiset_a] :
        ! [A6: a] :
          ( ( member_a @ A6 @ ( set_mset_a @ A5 ) )
         => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A6 @ A6 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_1027_locally__refl__def,axiom,
    ( locall2418947207864277812on_nat
    = ( ^ [R3: set_Pr2535564044409765851on_nat,A5: multis1856558993800614429on_nat] :
        ! [A6: sum_su4907927656612468301on_nat] :
          ( ( member8476201367316484836on_nat @ A6 @ ( set_ms3922534532180125136on_nat @ A5 ) )
         => ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ A6 @ A6 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_1028_locally__refl__def,axiom,
    ( locally_refl_nat
    = ( ^ [R3: set_Pr1261947904930325089at_nat,A5: multiset_nat] :
        ! [A6: nat] :
          ( ( member_nat @ A6 @ ( set_mset_nat @ A5 ) )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A6 @ A6 ) @ R3 ) ) ) ) ).

% locally_refl_def
thf(fact_1029_NS__def,axiom,
    ( ns
    = ( concat937493773106420796on_nat
      @ ( map_a_7431629901395152583on_nat
        @ ^ [X4: a] : ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis8267843437418295310sitive ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5512618279175645674signed ) ) ) @ ( cons_P1582596238967110705on_nat @ ( produc6704844767664046899on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5977116398950053002gative ) ) @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5512618279175645674signed ) ) ) @ nil_Pr8614093721167239297on_nat ) )
        @ xs ) ) ) ).

% NS_def
thf(fact_1030_list_Oinject,axiom,
    ! [X21: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat,Y21: sum_su4907927656612468301on_nat,Y22: list_S5253404628753938141on_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X21 @ X222 )
        = ( cons_S6797538523646374167on_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_1031_list_Oinject,axiom,
    ! [X21: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat,Y21: produc1313295348606329723on_nat,Y22: list_P9127870862658512897on_nat] :
      ( ( ( cons_P1582596238967110705on_nat @ X21 @ X222 )
        = ( cons_P1582596238967110705on_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_1032_list_Oinject,axiom,
    ! [X21: nat,X222: list_nat,Y21: nat,Y22: list_nat] :
      ( ( ( cons_nat @ X21 @ X222 )
        = ( cons_nat @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_1033_size__union,axiom,
    ! [M: multis1856558993800614429on_nat,N3: multis1856558993800614429on_nat] :
      ( ( size_s708034413772738737on_nat @ ( plus_p2586647234311342374on_nat @ M @ N3 ) )
      = ( plus_plus_nat @ ( size_s708034413772738737on_nat @ M ) @ ( size_s708034413772738737on_nat @ N3 ) ) ) ).

% size_union
thf(fact_1034_size__mset,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( size_s708034413772738737on_nat @ ( mset_S7907735292211298936on_nat @ Xs ) )
      = ( size_s5686234912572693873on_nat @ Xs ) ) ).

% size_mset
thf(fact_1035_size__mset,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( size_s2843428218691353977od_a_o @ ( mset_l8139550355095933148od_a_o @ Xs ) )
      = ( size_s3379185846251495609od_a_o @ Xs ) ) ).

% size_mset
thf(fact_1036_size__mset,axiom,
    ! [Xs: list_nat] :
      ( ( size_s5917832649809541300et_nat @ ( mset_nat @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% size_mset
thf(fact_1037_concat__append,axiom,
    ! [Xs: list_l1726306699111145479on_nat,Ys: list_l1726306699111145479on_nat] :
      ( ( concat937493773106420796on_nat @ ( append3839213209832495004on_nat @ Xs @ Ys ) )
      = ( append2865821678206098582on_nat @ ( concat937493773106420796on_nat @ Xs ) @ ( concat937493773106420796on_nat @ Ys ) ) ) ).

% concat_append
thf(fact_1038_concat__append,axiom,
    ! [Xs: list_l8999537460041914605on_nat,Ys: list_l8999537460041914605on_nat] :
      ( ( concat6161509665823917772on_nat @ ( append6174841615030841602on_nat @ Xs @ Ys ) )
      = ( append3523812819975775986on_nat @ ( concat6161509665823917772on_nat @ Xs ) @ ( concat6161509665823917772on_nat @ Ys ) ) ) ).

% concat_append
thf(fact_1039_list__ex__simps_I1_J,axiom,
    ! [P2: sum_su4907927656612468301on_nat > $o,X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( list_e3769318529992910857on_nat @ P2 @ ( cons_S6797538523646374167on_nat @ X @ Xs ) )
      = ( ( P2 @ X )
        | ( list_e3769318529992910857on_nat @ P2 @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_1040_list__ex__simps_I1_J,axiom,
    ! [P2: produc1313295348606329723on_nat > $o,X: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( list_e8248363684495001279on_nat @ P2 @ ( cons_P1582596238967110705on_nat @ X @ Xs ) )
      = ( ( P2 @ X )
        | ( list_e8248363684495001279on_nat @ P2 @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_1041_list__ex__simps_I1_J,axiom,
    ! [P2: nat > $o,X: nat,Xs: list_nat] :
      ( ( list_ex_nat @ P2 @ ( cons_nat @ X @ Xs ) )
      = ( ( P2 @ X )
        | ( list_ex_nat @ P2 @ Xs ) ) ) ).

% list_ex_simps(1)
thf(fact_1042_append1__eq__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X @ nil_li1372968520647675725od_a_o ) )
        = ( append2159120467191820514od_a_o @ Ys @ ( cons_l9179691953475787517od_a_o @ Y @ nil_li1372968520647675725od_a_o ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1043_append1__eq__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ( ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X @ nil_Su3078906344068544711on_nat ) )
        = ( append3523812819975775986on_nat @ Ys @ ( cons_S6797538523646374167on_nat @ Y @ nil_Su3078906344068544711on_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1044_append1__eq__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat] :
      ( ( ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X @ nil_Pr8614093721167239297on_nat ) )
        = ( append2865821678206098582on_nat @ Ys @ ( cons_P1582596238967110705on_nat @ Y @ nil_Pr8614093721167239297on_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1045_append1__eq__conv,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( ( append_nat @ Xs @ ( cons_nat @ X @ nil_nat ) )
        = ( append_nat @ Ys @ ( cons_nat @ Y @ nil_nat ) ) )
      = ( ( Xs = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_1046_nth__Cons__0,axiom,
    ! [X: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( nth_li3248255784598043854od_a_o @ ( cons_l9179691953475787517od_a_o @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_1047_nth__Cons__0,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( cons_S6797538523646374167on_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_1048_nth__Cons__0,axiom,
    ! [X: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( cons_P1582596238967110705on_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_1049_nth__Cons__0,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( nth_nat @ ( cons_nat @ X @ Xs ) @ zero_zero_nat )
      = X ) ).

% nth_Cons_0
thf(fact_1050_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_1051_Nil__eq__concat__conv,axiom,
    ! [Xss: list_l7431997615882138323od_a_o] :
      ( ( nil_li1372968520647675725od_a_o
        = ( concat3529464442532842888od_a_o @ Xss ) )
      = ( ! [X4: list_l8336264072933094989od_a_o] :
            ( ( member8885487059802017782od_a_o @ X4 @ ( set_li3106133282984605794od_a_o @ Xss ) )
           => ( X4 = nil_li1372968520647675725od_a_o ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_1052_Nil__eq__concat__conv,axiom,
    ! [Xss: list_l8999537460041914605on_nat] :
      ( ( nil_Su3078906344068544711on_nat
        = ( concat6161509665823917772on_nat @ Xss ) )
      = ( ! [X4: list_S5253404628753938141on_nat] :
            ( ( member86162308477975540on_nat @ X4 @ ( set_li4371160873219908744on_nat @ Xss ) )
           => ( X4 = nil_Su3078906344068544711on_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_1053_Nil__eq__concat__conv,axiom,
    ! [Xss: list_l1726306699111145479on_nat] :
      ( ( nil_Pr8614093721167239297on_nat
        = ( concat937493773106420796on_nat @ Xss ) )
      = ( ! [X4: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Xss ) )
           => ( X4 = nil_Pr8614093721167239297on_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_1054_Nil__eq__concat__conv,axiom,
    ! [Xss: list_list_nat] :
      ( ( nil_nat
        = ( concat_nat @ Xss ) )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss ) )
           => ( X4 = nil_nat ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_1055_Nil__eq__concat__conv,axiom,
    ! [Xss: list_l8336264072933094989od_a_o] :
      ( ( nil_Product_prod_a_o
        = ( concat7947350394041715842od_a_o @ Xss ) )
      = ( ! [X4: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X4 @ ( set_li2629784761229999580od_a_o @ Xss ) )
           => ( X4 = nil_Product_prod_a_o ) ) ) ) ).

% Nil_eq_concat_conv
thf(fact_1056_concat__eq__Nil__conv,axiom,
    ! [Xss: list_l7431997615882138323od_a_o] :
      ( ( ( concat3529464442532842888od_a_o @ Xss )
        = nil_li1372968520647675725od_a_o )
      = ( ! [X4: list_l8336264072933094989od_a_o] :
            ( ( member8885487059802017782od_a_o @ X4 @ ( set_li3106133282984605794od_a_o @ Xss ) )
           => ( X4 = nil_li1372968520647675725od_a_o ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_1057_concat__eq__Nil__conv,axiom,
    ! [Xss: list_l8999537460041914605on_nat] :
      ( ( ( concat6161509665823917772on_nat @ Xss )
        = nil_Su3078906344068544711on_nat )
      = ( ! [X4: list_S5253404628753938141on_nat] :
            ( ( member86162308477975540on_nat @ X4 @ ( set_li4371160873219908744on_nat @ Xss ) )
           => ( X4 = nil_Su3078906344068544711on_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_1058_concat__eq__Nil__conv,axiom,
    ! [Xss: list_l1726306699111145479on_nat] :
      ( ( ( concat937493773106420796on_nat @ Xss )
        = nil_Pr8614093721167239297on_nat )
      = ( ! [X4: list_P9127870862658512897on_nat] :
            ( ( member783415501022416426on_nat @ X4 @ ( set_li8154783781463198870on_nat @ Xss ) )
           => ( X4 = nil_Pr8614093721167239297on_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_1059_concat__eq__Nil__conv,axiom,
    ! [Xss: list_list_nat] :
      ( ( ( concat_nat @ Xss )
        = nil_nat )
      = ( ! [X4: list_nat] :
            ( ( member_list_nat @ X4 @ ( set_list_nat2 @ Xss ) )
           => ( X4 = nil_nat ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_1060_concat__eq__Nil__conv,axiom,
    ! [Xss: list_l8336264072933094989od_a_o] :
      ( ( ( concat7947350394041715842od_a_o @ Xss )
        = nil_Product_prod_a_o )
      = ( ! [X4: list_P3196869577824546119od_a_o] :
            ( ( member5261019209154752368od_a_o @ X4 @ ( set_li2629784761229999580od_a_o @ Xss ) )
           => ( X4 = nil_Product_prod_a_o ) ) ) ) ).

% concat_eq_Nil_conv
thf(fact_1061_nth__append__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( nth_Pr3336893728888831618on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X @ Ys ) ) @ ( size_s7490768528479217773on_nat @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_1062_nth__append__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( nth_Su8471564416988646406on_nat @ ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X @ Ys ) ) @ ( size_s5686234912572693873on_nat @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_1063_nth__append__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( nth_li3248255784598043854od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X @ Ys ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_1064_nth__append__length,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat] :
      ( ( nth_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) )
      = X ) ).

% nth_append_length
thf(fact_1065_list__update__length,axiom,
    ! [Xs: list_P9127870862658512897on_nat,X: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat] :
      ( ( list_u1840724836389618409on_nat @ ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ X @ Ys ) ) @ ( size_s7490768528479217773on_nat @ Xs ) @ Y )
      = ( append2865821678206098582on_nat @ Xs @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1066_list__update__length,axiom,
    ! [Xs: list_S5253404628753938141on_nat,X: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat] :
      ( ( list_u8577817230714425439on_nat @ ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ X @ Ys ) ) @ ( size_s5686234912572693873on_nat @ Xs ) @ Y )
      = ( append3523812819975775986on_nat @ Xs @ ( cons_S6797538523646374167on_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1067_list__update__length,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,X: list_P3196869577824546119od_a_o,Ys: list_l8336264072933094989od_a_o,Y: list_P3196869577824546119od_a_o] :
      ( ( list_u329291161401623989od_a_o @ ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ X @ Ys ) ) @ ( size_s3379185846251495609od_a_o @ Xs ) @ Y )
      = ( append2159120467191820514od_a_o @ Xs @ ( cons_l9179691953475787517od_a_o @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1068_list__update__length,axiom,
    ! [Xs: list_nat,X: nat,Ys: list_nat,Y: nat] :
      ( ( list_update_nat @ ( append_nat @ Xs @ ( cons_nat @ X @ Ys ) ) @ ( size_size_list_nat @ Xs ) @ Y )
      = ( append_nat @ Xs @ ( cons_nat @ Y @ Ys ) ) ) ).

% list_update_length
thf(fact_1069_Cons__listrel1__Cons,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat,R: set_Pr2535564044409765851on_nat] :
      ( ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ ( cons_S6797538523646374167on_nat @ X @ Xs ) @ ( cons_S6797538523646374167on_nat @ Y @ Ys ) ) @ ( listre1270753918075970734on_nat @ R ) )
      = ( ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member4221009242412147780on_nat @ ( produc2912653624306349395on_nat @ Xs @ Ys ) @ ( listre1270753918075970734on_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_1070_Cons__listrel1__Cons,axiom,
    ! [X: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat,R: set_Pr8202701763255893447on_nat] :
      ( ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ ( cons_P1582596238967110705on_nat @ X @ Xs ) @ ( cons_P1582596238967110705on_nat @ Y @ Ys ) ) @ ( listre7129750038259667930on_nat @ R ) )
      = ( ( ( member1022098016950487056on_nat @ ( produc3570076344267071063on_nat @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member4629988367768146576on_nat @ ( produc7152764266822961111on_nat @ Xs @ Ys ) @ ( listre7129750038259667930on_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_1071_Cons__listrel1__Cons,axiom,
    ! [X: nat,Xs: list_nat,Y: nat,Ys: list_nat,R: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ ( cons_nat @ X @ Xs ) @ ( cons_nat @ Y @ Ys ) ) @ ( listrel1_nat @ R ) )
      = ( ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
          & ( Xs = Ys ) )
        | ( ( X = Y )
          & ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_1072_left,axiom,
    ( left
    = ( concat6161509665823917772on_nat
      @ ( map_a_3887198048842487703on_nat
        @ ^ [X4: a] : ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis8267843437418295310sitive ) ) @ ( cons_S6797538523646374167on_nat @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ X4 @ multis5977116398950053002gative ) ) @ nil_Su3078906344068544711on_nat ) )
        @ xs ) ) ) ).

% left
thf(fact_1073_concat__map__singleton,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,Xs: list_nat] :
      ( ( concat4341727598008994882on_nat
        @ ( map_na8454508660465608945on_nat
          @ ^ [X4: nat] : ( cons_l3028799188195208631on_nat @ ( F @ X4 ) @ nil_li2682182165152866823on_nat )
          @ Xs ) )
      = ( map_na8715451351012060651on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1074_concat__map__singleton,axiom,
    ! [F: a > list_P9127870862658512897on_nat,Xs: list_a] :
      ( ( concat4341727598008994882on_nat
        @ ( map_a_8347779969077652429on_nat
          @ ^ [X4: a] : ( cons_l3028799188195208631on_nat @ ( F @ X4 ) @ nil_li2682182165152866823on_nat )
          @ Xs ) )
      = ( map_a_7431629901395152583on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1075_concat__map__singleton,axiom,
    ! [F: a > list_S5253404628753938141on_nat,Xs: list_a] :
      ( ( concat4774721536428481500on_nat
        @ ( map_a_3820070144034313383on_nat
          @ ^ [X4: a] : ( cons_l5857885737047467175on_nat @ ( F @ X4 ) @ nil_li5563851662744376407on_nat )
          @ Xs ) )
      = ( map_a_3887198048842487703on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1076_concat__map__singleton,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,Xs: list_a] :
      ( ( concat6161509665823917772on_nat
        @ ( map_a_3887198048842487703on_nat
          @ ^ [X4: a] : ( cons_S6797538523646374167on_nat @ ( F @ X4 ) @ nil_Su3078906344068544711on_nat )
          @ Xs ) )
      = ( map_a_2711072540680412167on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1077_concat__map__singleton,axiom,
    ! [F: nat > produc1313295348606329723on_nat,Xs: list_nat] :
      ( ( concat937493773106420796on_nat
        @ ( map_na8715451351012060651on_nat
          @ ^ [X4: nat] : ( cons_P1582596238967110705on_nat @ ( F @ X4 ) @ nil_Pr8614093721167239297on_nat )
          @ Xs ) )
      = ( map_na2641141678045226597on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1078_concat__map__singleton,axiom,
    ! [F: a > produc1313295348606329723on_nat,Xs: list_a] :
      ( ( concat937493773106420796on_nat
        @ ( map_a_7431629901395152583on_nat
          @ ^ [X4: a] : ( cons_P1582596238967110705on_nat @ ( F @ X4 ) @ nil_Pr8614093721167239297on_nat )
          @ Xs ) )
      = ( map_a_2212652920793933121on_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1079_concat__map__singleton,axiom,
    ! [F: nat > nat,Xs: list_nat] :
      ( ( concat_nat
        @ ( map_nat_list_nat
          @ ^ [X4: nat] : ( cons_nat @ ( F @ X4 ) @ nil_nat )
          @ Xs ) )
      = ( map_nat_nat @ F @ Xs ) ) ).

% concat_map_singleton
thf(fact_1080_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_1081_not__Cons__self2,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( cons_S6797538523646374167on_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_1082_not__Cons__self2,axiom,
    ! [X: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( cons_P1582596238967110705on_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_1083_not__Cons__self2,axiom,
    ! [X: nat,Xs: list_nat] :
      ( ( cons_nat @ X @ Xs )
     != Xs ) ).

% not_Cons_self2
thf(fact_1084_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_1085_transpose_Ocases,axiom,
    ! [X: list_l7431997615882138323od_a_o] :
      ( ( X != nil_li6902169125192129875od_a_o )
     => ( ! [Xss2: list_l7431997615882138323od_a_o] :
            ( X
           != ( cons_l8465095039735648515od_a_o @ nil_li1372968520647675725od_a_o @ Xss2 ) )
       => ~ ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Xss2: list_l7431997615882138323od_a_o] :
              ( X
             != ( cons_l8465095039735648515od_a_o @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_1086_transpose_Ocases,axiom,
    ! [X: list_l8336264072933094989od_a_o] :
      ( ( X != nil_li1372968520647675725od_a_o )
     => ( ! [Xss2: list_l8336264072933094989od_a_o] :
            ( X
           != ( cons_l9179691953475787517od_a_o @ nil_Product_prod_a_o @ Xss2 ) )
       => ~ ! [X3: product_prod_a_o,Xs3: list_P3196869577824546119od_a_o,Xss2: list_l8336264072933094989od_a_o] :
              ( X
             != ( cons_l9179691953475787517od_a_o @ ( cons_P8471407633269037175od_a_o @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_1087_transpose_Ocases,axiom,
    ! [X: list_l8999537460041914605on_nat] :
      ( ( X != nil_li5563851662744376407on_nat )
     => ( ! [Xss2: list_l8999537460041914605on_nat] :
            ( X
           != ( cons_l5857885737047467175on_nat @ nil_Su3078906344068544711on_nat @ Xss2 ) )
       => ~ ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Xss2: list_l8999537460041914605on_nat] :
              ( X
             != ( cons_l5857885737047467175on_nat @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_1088_transpose_Ocases,axiom,
    ! [X: list_l1726306699111145479on_nat] :
      ( ( X != nil_li2682182165152866823on_nat )
     => ( ! [Xss2: list_l1726306699111145479on_nat] :
            ( X
           != ( cons_l3028799188195208631on_nat @ nil_Pr8614093721167239297on_nat @ Xss2 ) )
       => ~ ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Xss2: list_l1726306699111145479on_nat] :
              ( X
             != ( cons_l3028799188195208631on_nat @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_1089_transpose_Ocases,axiom,
    ! [X: list_list_nat] :
      ( ( X != nil_list_nat )
     => ( ! [Xss2: list_list_nat] :
            ( X
           != ( cons_list_nat @ nil_nat @ Xss2 ) )
       => ~ ! [X3: nat,Xs3: list_nat,Xss2: list_list_nat] :
              ( X
             != ( cons_list_nat @ ( cons_nat @ X3 @ Xs3 ) @ Xss2 ) ) ) ) ).

% transpose.cases
thf(fact_1090_list_Odistinct_I1_J,axiom,
    ! [X21: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o] :
      ( nil_li1372968520647675725od_a_o
     != ( cons_l9179691953475787517od_a_o @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_1091_list_Odistinct_I1_J,axiom,
    ! [X21: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] :
      ( nil_Su3078906344068544711on_nat
     != ( cons_S6797538523646374167on_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_1092_list_Odistinct_I1_J,axiom,
    ! [X21: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat] :
      ( nil_Pr8614093721167239297on_nat
     != ( cons_P1582596238967110705on_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_1093_list_Odistinct_I1_J,axiom,
    ! [X21: nat,X222: list_nat] :
      ( nil_nat
     != ( cons_nat @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_1094_list_OdiscI,axiom,
    ! [List: list_l8336264072933094989od_a_o,X21: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o] :
      ( ( List
        = ( cons_l9179691953475787517od_a_o @ X21 @ X222 ) )
     => ( List != nil_li1372968520647675725od_a_o ) ) ).

% list.discI
thf(fact_1095_list_OdiscI,axiom,
    ! [List: list_S5253404628753938141on_nat,X21: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] :
      ( ( List
        = ( cons_S6797538523646374167on_nat @ X21 @ X222 ) )
     => ( List != nil_Su3078906344068544711on_nat ) ) ).

% list.discI
thf(fact_1096_list_OdiscI,axiom,
    ! [List: list_P9127870862658512897on_nat,X21: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat] :
      ( ( List
        = ( cons_P1582596238967110705on_nat @ X21 @ X222 ) )
     => ( List != nil_Pr8614093721167239297on_nat ) ) ).

% list.discI
thf(fact_1097_list_OdiscI,axiom,
    ! [List: list_nat,X21: nat,X222: list_nat] :
      ( ( List
        = ( cons_nat @ X21 @ X222 ) )
     => ( List != nil_nat ) ) ).

% list.discI
thf(fact_1098_list_Oexhaust,axiom,
    ! [Y: list_l8336264072933094989od_a_o] :
      ( ( Y != nil_li1372968520647675725od_a_o )
     => ~ ! [X212: list_P3196869577824546119od_a_o,X223: list_l8336264072933094989od_a_o] :
            ( Y
           != ( cons_l9179691953475787517od_a_o @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_1099_list_Oexhaust,axiom,
    ! [Y: list_S5253404628753938141on_nat] :
      ( ( Y != nil_Su3078906344068544711on_nat )
     => ~ ! [X212: sum_su4907927656612468301on_nat,X223: list_S5253404628753938141on_nat] :
            ( Y
           != ( cons_S6797538523646374167on_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_1100_list_Oexhaust,axiom,
    ! [Y: list_P9127870862658512897on_nat] :
      ( ( Y != nil_Pr8614093721167239297on_nat )
     => ~ ! [X212: produc1313295348606329723on_nat,X223: list_P9127870862658512897on_nat] :
            ( Y
           != ( cons_P1582596238967110705on_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_1101_list_Oexhaust,axiom,
    ! [Y: list_nat] :
      ( ( Y != nil_nat )
     => ~ ! [X212: nat,X223: list_nat] :
            ( Y
           != ( cons_nat @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_1102_min__list_Ocases,axiom,
    ! [X: list_nat] :
      ( ! [X3: nat,Xs3: list_nat] :
          ( X
         != ( cons_nat @ X3 @ Xs3 ) )
     => ( X = nil_nat ) ) ).

% min_list.cases
thf(fact_1103_remdups__adj_Ocases,axiom,
    ! [X: list_l8336264072933094989od_a_o] :
      ( ( X != nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o] :
            ( X
           != ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) )
       => ~ ! [X3: list_P3196869577824546119od_a_o,Y3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o] :
              ( X
             != ( cons_l9179691953475787517od_a_o @ X3 @ ( cons_l9179691953475787517od_a_o @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1104_remdups__adj_Ocases,axiom,
    ! [X: list_S5253404628753938141on_nat] :
      ( ( X != nil_Su3078906344068544711on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat] :
            ( X
           != ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) )
       => ~ ! [X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( X
             != ( cons_S6797538523646374167on_nat @ X3 @ ( cons_S6797538523646374167on_nat @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1105_remdups__adj_Ocases,axiom,
    ! [X: list_P9127870862658512897on_nat] :
      ( ( X != nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat] :
            ( X
           != ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) )
       => ~ ! [X3: produc1313295348606329723on_nat,Y3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
              ( X
             != ( cons_P1582596238967110705on_nat @ X3 @ ( cons_P1582596238967110705on_nat @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1106_remdups__adj_Ocases,axiom,
    ! [X: list_nat] :
      ( ( X != nil_nat )
     => ( ! [X3: nat] :
            ( X
           != ( cons_nat @ X3 @ nil_nat ) )
       => ~ ! [X3: nat,Y3: nat,Xs3: list_nat] :
              ( X
             != ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ).

% remdups_adj.cases
thf(fact_1107_neq__Nil__conv,axiom,
    ! [Xs: list_l8336264072933094989od_a_o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
      = ( ? [Y5: list_P3196869577824546119od_a_o,Ys2: list_l8336264072933094989od_a_o] :
            ( Xs
            = ( cons_l9179691953475787517od_a_o @ Y5 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_1108_neq__Nil__conv,axiom,
    ! [Xs: list_S5253404628753938141on_nat] :
      ( ( Xs != nil_Su3078906344068544711on_nat )
      = ( ? [Y5: sum_su4907927656612468301on_nat,Ys2: list_S5253404628753938141on_nat] :
            ( Xs
            = ( cons_S6797538523646374167on_nat @ Y5 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_1109_neq__Nil__conv,axiom,
    ! [Xs: list_P9127870862658512897on_nat] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
      = ( ? [Y5: produc1313295348606329723on_nat,Ys2: list_P9127870862658512897on_nat] :
            ( Xs
            = ( cons_P1582596238967110705on_nat @ Y5 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_1110_neq__Nil__conv,axiom,
    ! [Xs: list_nat] :
      ( ( Xs != nil_nat )
      = ( ? [Y5: nat,Ys2: list_nat] :
            ( Xs
            = ( cons_nat @ Y5 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_1111_list__induct2_H,axiom,
    ! [P2: list_nat > list_nat > $o,Xs: list_nat,Ys: list_nat] :
      ( ( P2 @ nil_nat @ nil_nat )
     => ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P2 @ nil_nat @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: nat,Xs3: list_nat,Y3: nat,Ys4: list_nat] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1112_list__induct2_H,axiom,
    ! [P2: list_l8336264072933094989od_a_o > list_nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_nat] :
      ( ( P2 @ nil_li1372968520647675725od_a_o @ nil_nat )
     => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o] : ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P2 @ nil_li1372968520647675725od_a_o @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Y3: nat,Ys4: list_nat] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1113_list__induct2_H,axiom,
    ! [P2: list_nat > list_l8336264072933094989od_a_o > $o,Xs: list_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( P2 @ nil_nat @ nil_li1372968520647675725od_a_o )
     => ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_li1372968520647675725od_a_o )
       => ( ! [Y3: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] : ( P2 @ nil_nat @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys4 ) )
         => ( ! [X3: nat,Xs3: list_nat,Y3: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1114_list__induct2_H,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_nat] :
      ( ( P2 @ nil_Su3078906344068544711on_nat @ nil_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] : ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P2 @ nil_Su3078906344068544711on_nat @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y3: nat,Ys4: list_nat] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1115_list__induct2_H,axiom,
    ! [P2: list_nat > list_S5253404628753938141on_nat > $o,Xs: list_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( P2 @ nil_nat @ nil_Su3078906344068544711on_nat )
     => ( ! [X3: nat,Xs3: list_nat] : ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ nil_Su3078906344068544711on_nat )
       => ( ! [Y3: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] : ( P2 @ nil_nat @ ( cons_S6797538523646374167on_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: nat,Xs3: list_nat,Y3: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_nat @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1116_list__induct2_H,axiom,
    ! [P2: list_l8336264072933094989od_a_o > list_l8336264072933094989od_a_o > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_l8336264072933094989od_a_o] :
      ( ( P2 @ nil_li1372968520647675725od_a_o @ nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o] : ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ nil_li1372968520647675725od_a_o )
       => ( ! [Y3: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] : ( P2 @ nil_li1372968520647675725od_a_o @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys4 ) )
         => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Y3: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1117_list__induct2_H,axiom,
    ! [P2: list_l8336264072933094989od_a_o > list_S5253404628753938141on_nat > $o,Xs: list_l8336264072933094989od_a_o,Ys: list_S5253404628753938141on_nat] :
      ( ( P2 @ nil_li1372968520647675725od_a_o @ nil_Su3078906344068544711on_nat )
     => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o] : ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ nil_Su3078906344068544711on_nat )
       => ( ! [Y3: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] : ( P2 @ nil_li1372968520647675725od_a_o @ ( cons_S6797538523646374167on_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o,Y3: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1118_list__induct2_H,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_l8336264072933094989od_a_o > $o,Xs: list_S5253404628753938141on_nat,Ys: list_l8336264072933094989od_a_o] :
      ( ( P2 @ nil_Su3078906344068544711on_nat @ nil_li1372968520647675725od_a_o )
     => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] : ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ nil_li1372968520647675725od_a_o )
       => ( ! [Y3: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] : ( P2 @ nil_Su3078906344068544711on_nat @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys4 ) )
         => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y3: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_l9179691953475787517od_a_o @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1119_list__induct2_H,axiom,
    ! [P2: list_S5253404628753938141on_nat > list_S5253404628753938141on_nat > $o,Xs: list_S5253404628753938141on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( P2 @ nil_Su3078906344068544711on_nat @ nil_Su3078906344068544711on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] : ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ nil_Su3078906344068544711on_nat )
       => ( ! [Y3: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] : ( P2 @ nil_Su3078906344068544711on_nat @ ( cons_S6797538523646374167on_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat,Y3: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) @ ( cons_S6797538523646374167on_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1120_list__induct2_H,axiom,
    ! [P2: list_P9127870862658512897on_nat > list_nat > $o,Xs: list_P9127870862658512897on_nat,Ys: list_nat] :
      ( ( P2 @ nil_Pr8614093721167239297on_nat @ nil_nat )
     => ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] : ( P2 @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ nil_nat )
       => ( ! [Y3: nat,Ys4: list_nat] : ( P2 @ nil_Pr8614093721167239297on_nat @ ( cons_nat @ Y3 @ Ys4 ) )
         => ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat,Y3: nat,Ys4: list_nat] :
                ( ( P2 @ Xs3 @ Ys4 )
               => ( P2 @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) @ ( cons_nat @ Y3 @ Ys4 ) ) )
           => ( P2 @ Xs @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_1121_list__nonempty__induct,axiom,
    ! [Xs: list_l8336264072933094989od_a_o,P2: list_l8336264072933094989od_a_o > $o] :
      ( ( Xs != nil_li1372968520647675725od_a_o )
     => ( ! [X3: list_P3196869577824546119od_a_o] : ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) )
       => ( ! [X3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o] :
              ( ( Xs3 != nil_li1372968520647675725od_a_o )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( cons_l9179691953475787517od_a_o @ X3 @ Xs3 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1122_list__nonempty__induct,axiom,
    ! [Xs: list_S5253404628753938141on_nat,P2: list_S5253404628753938141on_nat > $o] :
      ( ( Xs != nil_Su3078906344068544711on_nat )
     => ( ! [X3: sum_su4907927656612468301on_nat] : ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) )
       => ( ! [X3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( ( Xs3 != nil_Su3078906344068544711on_nat )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( cons_S6797538523646374167on_nat @ X3 @ Xs3 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1123_list__nonempty__induct,axiom,
    ! [Xs: list_P9127870862658512897on_nat,P2: list_P9127870862658512897on_nat > $o] :
      ( ( Xs != nil_Pr8614093721167239297on_nat )
     => ( ! [X3: produc1313295348606329723on_nat] : ( P2 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) )
       => ( ! [X3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
              ( ( Xs3 != nil_Pr8614093721167239297on_nat )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( cons_P1582596238967110705on_nat @ X3 @ Xs3 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1124_list__nonempty__induct,axiom,
    ! [Xs: list_nat,P2: list_nat > $o] :
      ( ( Xs != nil_nat )
     => ( ! [X3: nat] : ( P2 @ ( cons_nat @ X3 @ nil_nat ) )
       => ( ! [X3: nat,Xs3: list_nat] :
              ( ( Xs3 != nil_nat )
             => ( ( P2 @ Xs3 )
               => ( P2 @ ( cons_nat @ X3 @ Xs3 ) ) ) )
         => ( P2 @ Xs ) ) ) ) ).

% list_nonempty_induct
thf(fact_1125_sorted__wrt_Ocases,axiom,
    ! [X: produc8998563774250336198od_a_o] :
      ( ! [P5: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o] :
          ( X
         != ( produc6784936493919748534od_a_o @ P5 @ nil_li1372968520647675725od_a_o ) )
     => ~ ! [P5: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,X3: list_P3196869577824546119od_a_o,Ys4: list_l8336264072933094989od_a_o] :
            ( X
           != ( produc6784936493919748534od_a_o @ P5 @ ( cons_l9179691953475787517od_a_o @ X3 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1126_sorted__wrt_Ocases,axiom,
    ! [X: produc3191134443298988280on_nat] :
      ( ! [P5: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
          ( X
         != ( produc3262228428711020402on_nat @ P5 @ nil_Su3078906344068544711on_nat ) )
     => ~ ! [P5: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,X3: sum_su4907927656612468301on_nat,Ys4: list_S5253404628753938141on_nat] :
            ( X
           != ( produc3262228428711020402on_nat @ P5 @ ( cons_S6797538523646374167on_nat @ X3 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1127_sorted__wrt_Ocases,axiom,
    ! [X: produc2264144286178290682on_nat] :
      ( ! [P5: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o] :
          ( X
         != ( produc6237247764877134954on_nat @ P5 @ nil_Pr8614093721167239297on_nat ) )
     => ~ ! [P5: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,X3: produc1313295348606329723on_nat,Ys4: list_P9127870862658512897on_nat] :
            ( X
           != ( produc6237247764877134954on_nat @ P5 @ ( cons_P1582596238967110705on_nat @ X3 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1128_sorted__wrt_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ~ ! [P5: nat > nat > $o,X3: nat,Ys4: list_nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ Ys4 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_1129_successively_Ocases,axiom,
    ! [X: produc8998563774250336198od_a_o] :
      ( ! [P5: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o] :
          ( X
         != ( produc6784936493919748534od_a_o @ P5 @ nil_li1372968520647675725od_a_o ) )
     => ( ! [P5: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,X3: list_P3196869577824546119od_a_o] :
            ( X
           != ( produc6784936493919748534od_a_o @ P5 @ ( cons_l9179691953475787517od_a_o @ X3 @ nil_li1372968520647675725od_a_o ) ) )
       => ~ ! [P5: list_P3196869577824546119od_a_o > list_P3196869577824546119od_a_o > $o,X3: list_P3196869577824546119od_a_o,Y3: list_P3196869577824546119od_a_o,Xs3: list_l8336264072933094989od_a_o] :
              ( X
             != ( produc6784936493919748534od_a_o @ P5 @ ( cons_l9179691953475787517od_a_o @ X3 @ ( cons_l9179691953475787517od_a_o @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1130_successively_Ocases,axiom,
    ! [X: produc3191134443298988280on_nat] :
      ( ! [P5: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o] :
          ( X
         != ( produc3262228428711020402on_nat @ P5 @ nil_Su3078906344068544711on_nat ) )
     => ( ! [P5: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,X3: sum_su4907927656612468301on_nat] :
            ( X
           != ( produc3262228428711020402on_nat @ P5 @ ( cons_S6797538523646374167on_nat @ X3 @ nil_Su3078906344068544711on_nat ) ) )
       => ~ ! [P5: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat > $o,X3: sum_su4907927656612468301on_nat,Y3: sum_su4907927656612468301on_nat,Xs3: list_S5253404628753938141on_nat] :
              ( X
             != ( produc3262228428711020402on_nat @ P5 @ ( cons_S6797538523646374167on_nat @ X3 @ ( cons_S6797538523646374167on_nat @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1131_successively_Ocases,axiom,
    ! [X: produc2264144286178290682on_nat] :
      ( ! [P5: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o] :
          ( X
         != ( produc6237247764877134954on_nat @ P5 @ nil_Pr8614093721167239297on_nat ) )
     => ( ! [P5: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,X3: produc1313295348606329723on_nat] :
            ( X
           != ( produc6237247764877134954on_nat @ P5 @ ( cons_P1582596238967110705on_nat @ X3 @ nil_Pr8614093721167239297on_nat ) ) )
       => ~ ! [P5: produc1313295348606329723on_nat > produc1313295348606329723on_nat > $o,X3: produc1313295348606329723on_nat,Y3: produc1313295348606329723on_nat,Xs3: list_P9127870862658512897on_nat] :
              ( X
             != ( produc6237247764877134954on_nat @ P5 @ ( cons_P1582596238967110705on_nat @ X3 @ ( cons_P1582596238967110705on_nat @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1132_successively_Ocases,axiom,
    ! [X: produc254973753779126261st_nat] :
      ( ! [P5: nat > nat > $o] :
          ( X
         != ( produc4727192421694094319st_nat @ P5 @ nil_nat ) )
     => ( ! [P5: nat > nat > $o,X3: nat] :
            ( X
           != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ nil_nat ) ) )
       => ~ ! [P5: nat > nat > $o,X3: nat,Y3: nat,Xs3: list_nat] :
              ( X
             != ( produc4727192421694094319st_nat @ P5 @ ( cons_nat @ X3 @ ( cons_nat @ Y3 @ Xs3 ) ) ) ) ) ) ).

% successively.cases
thf(fact_1133_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_a_o,X222: list_P3196869577824546119od_a_o,X21: product_prod_a_o] :
      ( ( member2997933407038931562od_a_o @ Y @ ( set_Product_prod_a_o2 @ X222 ) )
     => ( member2997933407038931562od_a_o @ Y @ ( set_Product_prod_a_o2 @ ( cons_P8471407633269037175od_a_o @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1134_list_Oset__intros_I2_J,axiom,
    ! [Y: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o,X21: list_P3196869577824546119od_a_o] :
      ( ( member5261019209154752368od_a_o @ Y @ ( set_li2629784761229999580od_a_o @ X222 ) )
     => ( member5261019209154752368od_a_o @ Y @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1135_list_Oset__intros_I2_J,axiom,
    ! [Y: a,X222: list_a,X21: a] :
      ( ( member_a @ Y @ ( set_a2 @ X222 ) )
     => ( member_a @ Y @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1136_list_Oset__intros_I2_J,axiom,
    ! [Y: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat,X21: sum_su4907927656612468301on_nat] :
      ( ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ X222 ) )
     => ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1137_list_Oset__intros_I2_J,axiom,
    ! [Y: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat,X21: produc1313295348606329723on_nat] :
      ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ X222 ) )
     => ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1138_list_Oset__intros_I2_J,axiom,
    ! [Y: nat,X222: list_nat,X21: nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ X222 ) )
     => ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_1139_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_a_o,X222: list_P3196869577824546119od_a_o] : ( member2997933407038931562od_a_o @ X21 @ ( set_Product_prod_a_o2 @ ( cons_P8471407633269037175od_a_o @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1140_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P3196869577824546119od_a_o,X222: list_l8336264072933094989od_a_o] : ( member5261019209154752368od_a_o @ X21 @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1141_list_Oset__intros_I1_J,axiom,
    ! [X21: a,X222: list_a] : ( member_a @ X21 @ ( set_a2 @ ( cons_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1142_list_Oset__intros_I1_J,axiom,
    ! [X21: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] : ( member8476201367316484836on_nat @ X21 @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1143_list_Oset__intros_I1_J,axiom,
    ! [X21: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat] : ( member5906418179908956708on_nat @ X21 @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1144_list_Oset__intros_I1_J,axiom,
    ! [X21: nat,X222: list_nat] : ( member_nat @ X21 @ ( set_nat2 @ ( cons_nat @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_1145_list_Oset__cases,axiom,
    ! [E: product_prod_a_o,A: list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ E @ ( set_Product_prod_a_o2 @ A ) )
     => ( ! [Z22: list_P3196869577824546119od_a_o] :
            ( A
           != ( cons_P8471407633269037175od_a_o @ E @ Z22 ) )
       => ~ ! [Z1: product_prod_a_o,Z22: list_P3196869577824546119od_a_o] :
              ( ( A
                = ( cons_P8471407633269037175od_a_o @ Z1 @ Z22 ) )
             => ~ ( member2997933407038931562od_a_o @ E @ ( set_Product_prod_a_o2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1146_list_Oset__cases,axiom,
    ! [E: list_P3196869577824546119od_a_o,A: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ E @ ( set_li2629784761229999580od_a_o @ A ) )
     => ( ! [Z22: list_l8336264072933094989od_a_o] :
            ( A
           != ( cons_l9179691953475787517od_a_o @ E @ Z22 ) )
       => ~ ! [Z1: list_P3196869577824546119od_a_o,Z22: list_l8336264072933094989od_a_o] :
              ( ( A
                = ( cons_l9179691953475787517od_a_o @ Z1 @ Z22 ) )
             => ~ ( member5261019209154752368od_a_o @ E @ ( set_li2629784761229999580od_a_o @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1147_list_Oset__cases,axiom,
    ! [E: a,A: list_a] :
      ( ( member_a @ E @ ( set_a2 @ A ) )
     => ( ! [Z22: list_a] :
            ( A
           != ( cons_a @ E @ Z22 ) )
       => ~ ! [Z1: a,Z22: list_a] :
              ( ( A
                = ( cons_a @ Z1 @ Z22 ) )
             => ~ ( member_a @ E @ ( set_a2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1148_list_Oset__cases,axiom,
    ! [E: sum_su4907927656612468301on_nat,A: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ E @ ( set_Su4848624856859743608on_nat @ A ) )
     => ( ! [Z22: list_S5253404628753938141on_nat] :
            ( A
           != ( cons_S6797538523646374167on_nat @ E @ Z22 ) )
       => ~ ! [Z1: sum_su4907927656612468301on_nat,Z22: list_S5253404628753938141on_nat] :
              ( ( A
                = ( cons_S6797538523646374167on_nat @ Z1 @ Z22 ) )
             => ~ ( member8476201367316484836on_nat @ E @ ( set_Su4848624856859743608on_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1149_list_Oset__cases,axiom,
    ! [E: produc1313295348606329723on_nat,A: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ E @ ( set_Pr937530902923597456on_nat @ A ) )
     => ( ! [Z22: list_P9127870862658512897on_nat] :
            ( A
           != ( cons_P1582596238967110705on_nat @ E @ Z22 ) )
       => ~ ! [Z1: produc1313295348606329723on_nat,Z22: list_P9127870862658512897on_nat] :
              ( ( A
                = ( cons_P1582596238967110705on_nat @ Z1 @ Z22 ) )
             => ~ ( member5906418179908956708on_nat @ E @ ( set_Pr937530902923597456on_nat @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1150_list_Oset__cases,axiom,
    ! [E: nat,A: list_nat] :
      ( ( member_nat @ E @ ( set_nat2 @ A ) )
     => ( ! [Z22: list_nat] :
            ( A
           != ( cons_nat @ E @ Z22 ) )
       => ~ ! [Z1: nat,Z22: list_nat] :
              ( ( A
                = ( cons_nat @ Z1 @ Z22 ) )
             => ~ ( member_nat @ E @ ( set_nat2 @ Z22 ) ) ) ) ) ).

% list.set_cases
thf(fact_1151_set__ConsD,axiom,
    ! [Y: product_prod_a_o,X: product_prod_a_o,Xs: list_P3196869577824546119od_a_o] :
      ( ( member2997933407038931562od_a_o @ Y @ ( set_Product_prod_a_o2 @ ( cons_P8471407633269037175od_a_o @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member2997933407038931562od_a_o @ Y @ ( set_Product_prod_a_o2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1152_set__ConsD,axiom,
    ! [Y: list_P3196869577824546119od_a_o,X: list_P3196869577824546119od_a_o,Xs: list_l8336264072933094989od_a_o] :
      ( ( member5261019209154752368od_a_o @ Y @ ( set_li2629784761229999580od_a_o @ ( cons_l9179691953475787517od_a_o @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member5261019209154752368od_a_o @ Y @ ( set_li2629784761229999580od_a_o @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1153_set__ConsD,axiom,
    ! [Y: a,X: a,Xs: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_a @ Y @ ( set_a2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1154_set__ConsD,axiom,
    ! [Y: sum_su4907927656612468301on_nat,X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat] :
      ( ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ ( cons_S6797538523646374167on_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member8476201367316484836on_nat @ Y @ ( set_Su4848624856859743608on_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1155_set__ConsD,axiom,
    ! [Y: produc1313295348606329723on_nat,X: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat] :
      ( ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ ( cons_P1582596238967110705on_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member5906418179908956708on_nat @ Y @ ( set_Pr937530902923597456on_nat @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1156_set__ConsD,axiom,
    ! [Y: nat,X: nat,Xs: list_nat] :
      ( ( member_nat @ Y @ ( set_nat2 @ ( cons_nat @ X @ Xs ) ) )
     => ( ( Y = X )
        | ( member_nat @ Y @ ( set_nat2 @ Xs ) ) ) ) ).

% set_ConsD
thf(fact_1157_list_Osimps_I9_J,axiom,
    ! [F: nat > nat,X21: nat,X222: list_nat] :
      ( ( map_nat_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_nat_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1158_list_Osimps_I9_J,axiom,
    ! [F: a > sum_su4907927656612468301on_nat,X21: a,X222: list_a] :
      ( ( map_a_2711072540680412167on_nat @ F @ ( cons_a @ X21 @ X222 ) )
      = ( cons_S6797538523646374167on_nat @ ( F @ X21 ) @ ( map_a_2711072540680412167on_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1159_list_Osimps_I9_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,X21: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] :
      ( ( map_Su4809418691747827683at_nat @ F @ ( cons_S6797538523646374167on_nat @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_Su4809418691747827683at_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1160_list_Osimps_I9_J,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,X21: nat,X222: list_nat] :
      ( ( map_na5706685718985795555on_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_S6797538523646374167on_nat @ ( F @ X21 ) @ ( map_na5706685718985795555on_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1161_list_Osimps_I9_J,axiom,
    ! [F: a > list_S5253404628753938141on_nat,X21: a,X222: list_a] :
      ( ( map_a_3887198048842487703on_nat @ F @ ( cons_a @ X21 @ X222 ) )
      = ( cons_l5857885737047467175on_nat @ ( F @ X21 ) @ ( map_a_3887198048842487703on_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1162_list_Osimps_I9_J,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,X21: sum_su4907927656612468301on_nat,X222: list_S5253404628753938141on_nat] :
      ( ( map_Su241112079427911264on_nat @ F @ ( cons_S6797538523646374167on_nat @ X21 @ X222 ) )
      = ( cons_S6797538523646374167on_nat @ ( F @ X21 ) @ ( map_Su241112079427911264on_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1163_list_Osimps_I9_J,axiom,
    ! [F: produc1313295348606329723on_nat > nat,X21: produc1313295348606329723on_nat,X222: list_P9127870862658512897on_nat] :
      ( ( map_Pr1888482430906987655at_nat @ F @ ( cons_P1582596238967110705on_nat @ X21 @ X222 ) )
      = ( cons_nat @ ( F @ X21 ) @ ( map_Pr1888482430906987655at_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1164_list_Osimps_I9_J,axiom,
    ! [F: nat > produc1313295348606329723on_nat,X21: nat,X222: list_nat] :
      ( ( map_na2641141678045226597on_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_P1582596238967110705on_nat @ ( F @ X21 ) @ ( map_na2641141678045226597on_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1165_list_Osimps_I9_J,axiom,
    ! [F: a > list_P9127870862658512897on_nat,X21: a,X222: list_a] :
      ( ( map_a_7431629901395152583on_nat @ F @ ( cons_a @ X21 @ X222 ) )
      = ( cons_l3028799188195208631on_nat @ ( F @ X21 ) @ ( map_a_7431629901395152583on_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1166_list_Osimps_I9_J,axiom,
    ! [F: nat > list_P9127870862658512897on_nat,X21: nat,X222: list_nat] :
      ( ( map_na8715451351012060651on_nat @ F @ ( cons_nat @ X21 @ X222 ) )
      = ( cons_l3028799188195208631on_nat @ ( F @ X21 ) @ ( map_na8715451351012060651on_nat @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1167_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: nat > nat,Ys: list_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_nat_nat @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_nat_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1168_Cons__eq__map__D,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: a > sum_su4907927656612468301on_nat,Ys: list_a] :
      ( ( ( cons_S6797538523646374167on_nat @ X @ Xs )
        = ( map_a_2711072540680412167on_nat @ F @ Ys ) )
     => ? [Z3: a,Zs2: list_a] :
          ( ( Ys
            = ( cons_a @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_a_2711072540680412167on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1169_Cons__eq__map__D,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: nat > sum_su4907927656612468301on_nat,Ys: list_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X @ Xs )
        = ( map_na5706685718985795555on_nat @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_na5706685718985795555on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1170_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: sum_su4907927656612468301on_nat > nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_Su4809418691747827683at_nat @ F @ Ys ) )
     => ? [Z3: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Ys
            = ( cons_S6797538523646374167on_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Su4809418691747827683at_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1171_Cons__eq__map__D,axiom,
    ! [X: list_S5253404628753938141on_nat,Xs: list_l8999537460041914605on_nat,F: a > list_S5253404628753938141on_nat,Ys: list_a] :
      ( ( ( cons_l5857885737047467175on_nat @ X @ Xs )
        = ( map_a_3887198048842487703on_nat @ F @ Ys ) )
     => ? [Z3: a,Zs2: list_a] :
          ( ( Ys
            = ( cons_a @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_a_3887198048842487703on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1172_Cons__eq__map__D,axiom,
    ! [X: sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( cons_S6797538523646374167on_nat @ X @ Xs )
        = ( map_Su241112079427911264on_nat @ F @ Ys ) )
     => ? [Z3: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Ys
            = ( cons_S6797538523646374167on_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Su241112079427911264on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1173_Cons__eq__map__D,axiom,
    ! [X: produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,F: nat > produc1313295348606329723on_nat,Ys: list_nat] :
      ( ( ( cons_P1582596238967110705on_nat @ X @ Xs )
        = ( map_na2641141678045226597on_nat @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_na2641141678045226597on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1174_Cons__eq__map__D,axiom,
    ! [X: nat,Xs: list_nat,F: produc1313295348606329723on_nat > nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( cons_nat @ X @ Xs )
        = ( map_Pr1888482430906987655at_nat @ F @ Ys ) )
     => ? [Z3: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Ys
            = ( cons_P1582596238967110705on_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_Pr1888482430906987655at_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1175_Cons__eq__map__D,axiom,
    ! [X: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,F: a > list_P9127870862658512897on_nat,Ys: list_a] :
      ( ( ( cons_l3028799188195208631on_nat @ X @ Xs )
        = ( map_a_7431629901395152583on_nat @ F @ Ys ) )
     => ? [Z3: a,Zs2: list_a] :
          ( ( Ys
            = ( cons_a @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_a_7431629901395152583on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1176_Cons__eq__map__D,axiom,
    ! [X: list_P9127870862658512897on_nat,Xs: list_l1726306699111145479on_nat,F: nat > list_P9127870862658512897on_nat,Ys: list_nat] :
      ( ( ( cons_l3028799188195208631on_nat @ X @ Xs )
        = ( map_na8715451351012060651on_nat @ F @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Ys
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( X
            = ( F @ Z3 ) )
          & ( Xs
            = ( map_na8715451351012060651on_nat @ F @ Zs2 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1177_map__eq__Cons__D,axiom,
    ! [F: sum_su4907927656612468301on_nat > sum_su4907927656612468301on_nat,Xs: list_S5253404628753938141on_nat,Y: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( map_Su241112079427911264on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys ) )
     => ? [Z3: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( cons_S6797538523646374167on_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Su241112079427911264on_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1178_map__eq__Cons__D,axiom,
    ! [F: produc1313295348606329723on_nat > sum_su4907927656612468301on_nat,Xs: list_P9127870862658512897on_nat,Y: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( map_Pr8685107614402573060on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys ) )
     => ? [Z3: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( cons_P1582596238967110705on_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Pr8685107614402573060on_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1179_map__eq__Cons__D,axiom,
    ! [F: nat > sum_su4907927656612468301on_nat,Xs: list_nat,Y: sum_su4907927656612468301on_nat,Ys: list_S5253404628753938141on_nat] :
      ( ( ( map_na5706685718985795555on_nat @ F @ Xs )
        = ( cons_S6797538523646374167on_nat @ Y @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_na5706685718985795555on_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1180_map__eq__Cons__D,axiom,
    ! [F: sum_su4907927656612468301on_nat > produc1313295348606329723on_nat,Xs: list_S5253404628753938141on_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( map_Su8438591925558430632on_nat @ F @ Xs )
        = ( cons_P1582596238967110705on_nat @ Y @ Ys ) )
     => ? [Z3: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( cons_S6797538523646374167on_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Su8438591925558430632on_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1181_map__eq__Cons__D,axiom,
    ! [F: produc1313295348606329723on_nat > produc1313295348606329723on_nat,Xs: list_P9127870862658512897on_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( map_Pr6472934937144306180on_nat @ F @ Xs )
        = ( cons_P1582596238967110705on_nat @ Y @ Ys ) )
     => ? [Z3: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( cons_P1582596238967110705on_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Pr6472934937144306180on_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1182_map__eq__Cons__D,axiom,
    ! [F: nat > produc1313295348606329723on_nat,Xs: list_nat,Y: produc1313295348606329723on_nat,Ys: list_P9127870862658512897on_nat] :
      ( ( ( map_na2641141678045226597on_nat @ F @ Xs )
        = ( cons_P1582596238967110705on_nat @ Y @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_na2641141678045226597on_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1183_map__eq__Cons__D,axiom,
    ! [F: sum_su4907927656612468301on_nat > nat,Xs: list_S5253404628753938141on_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_Su4809418691747827683at_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z3: sum_su4907927656612468301on_nat,Zs2: list_S5253404628753938141on_nat] :
          ( ( Xs
            = ( cons_S6797538523646374167on_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Su4809418691747827683at_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1184_map__eq__Cons__D,axiom,
    ! [F: produc1313295348606329723on_nat > nat,Xs: list_P9127870862658512897on_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_Pr1888482430906987655at_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z3: produc1313295348606329723on_nat,Zs2: list_P9127870862658512897on_nat] :
          ( ( Xs
            = ( cons_P1582596238967110705on_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_Pr1888482430906987655at_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1185_map__eq__Cons__D,axiom,
    ! [F: nat > nat,Xs: list_nat,Y: nat,Ys: list_nat] :
      ( ( ( map_nat_nat @ F @ Xs )
        = ( cons_nat @ Y @ Ys ) )
     => ? [Z3: nat,Zs2: list_nat] :
          ( ( Xs
            = ( cons_nat @ Z3 @ Zs2 ) )
          & ( ( F @ Z3 )
            = Y )
          & ( ( map_nat_nat @ F @ Zs2 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1186_map__add__upt,axiom,
    ! [N: nat,M2: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N )
        @ ( upt @ zero_zero_nat @ M2 ) )
      = ( upt @ N @ ( plus_plus_nat @ M2 @ N ) ) ) ).

% map_add_upt
thf(fact_1187_Annotation_Osize_I4_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size(4)
thf(fact_1188_Annotation_Osize_I5_J,axiom,
    ( ( size_s7536720479165623301tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size(5)
thf(fact_1189_Annotation_Osize_I6_J,axiom,
    ( ( size_s7536720479165623301tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size(6)
thf(fact_1190_size__char__eq__0,axiom,
    ( size_size_char
    = ( ^ [C5: char] : zero_zero_nat ) ) ).

% size_char_eq_0
thf(fact_1191_Annotation_Osize__gen_I2_J,axiom,
    ( ( multis6502627171228953925tation @ multis8267843437418295310sitive )
    = zero_zero_nat ) ).

% Annotation.size_gen(2)
thf(fact_1192_Annotation_Osize__gen_I1_J,axiom,
    ( ( multis6502627171228953925tation @ multis5512618279175645674signed )
    = zero_zero_nat ) ).

% Annotation.size_gen(1)
thf(fact_1193_Annotation_Osize__gen_I3_J,axiom,
    ( ( multis6502627171228953925tation @ multis5977116398950053002gative )
    = zero_zero_nat ) ).

% Annotation.size_gen(3)
thf(fact_1194_lessI,axiom,
    ! [N: nat] : ( ord_less_nat @ N @ ( suc @ N ) ) ).

% lessI
thf(fact_1195_Suc__mono,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% Suc_mono
thf(fact_1196_Suc__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_eq
thf(fact_1197_tl__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( tl_nat @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ N ) ) ).

% tl_upt
thf(fact_1198_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_1199_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_1200_zero__less__Suc,axiom,
    ! [N: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N ) ) ).

% zero_less_Suc
thf(fact_1201_less__Suc0,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ ( suc @ zero_zero_nat ) )
      = ( N = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_1202_zero__less__diff,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N @ M2 ) )
      = ( ord_less_nat @ M2 @ N ) ) ).

% zero_less_diff
thf(fact_1203_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_1204_diff__Suc__less,axiom,
    ! [N: nat,I: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ord_less_nat @ ( minus_minus_nat @ N @ ( suc @ I ) ) @ N ) ) ).

% diff_Suc_less
thf(fact_1205_diff__less__mono2,axiom,
    ! [M2: nat,N: nat,L3: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ord_less_nat @ M2 @ L3 )
       => ( ord_less_nat @ ( minus_minus_nat @ L3 @ N ) @ ( minus_minus_nat @ L3 @ M2 ) ) ) ) ).

% diff_less_mono2
thf(fact_1206_less__imp__diff__less,axiom,
    ! [J: nat,K: nat,N: nat] :
      ( ( ord_less_nat @ J @ K )
     => ( ord_less_nat @ ( minus_minus_nat @ J @ N ) @ K ) ) ).

% less_imp_diff_less
thf(fact_1207_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_1208_Suc__lessD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ N )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_lessD
thf(fact_1209_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_1210_Suc__lessI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ( ( suc @ M2 )
         != N )
       => ( ord_less_nat @ ( suc @ M2 ) @ N ) ) ) ).

% Suc_lessI
thf(fact_1211_less__SucE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
     => ( ~ ( ord_less_nat @ M2 @ N )
       => ( M2 = N ) ) ) ).

% less_SucE
thf(fact_1212_less__SucI,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ( ord_less_nat @ M2 @ ( suc @ N ) ) ) ).

% less_SucI
thf(fact_1213_Ex__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P2 @ I2 ) ) )
      = ( ( P2 @ N )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P2 @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_1214_less__Suc__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( ord_less_nat @ M2 @ N )
        | ( M2 = N ) ) ) ).

% less_Suc_eq
thf(fact_1215_not__less__eq,axiom,
    ! [M2: nat,N: nat] :
      ( ( ~ ( ord_less_nat @ M2 @ N ) )
      = ( ord_less_nat @ N @ ( suc @ M2 ) ) ) ).

% not_less_eq
thf(fact_1216_All__less__Suc,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P2 @ I2 ) ) )
      = ( ( P2 @ N )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P2 @ I2 ) ) ) ) ).

% All_less_Suc
thf(fact_1217_Suc__less__eq2,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ ( suc @ N ) @ M2 )
      = ( ? [M4: nat] :
            ( ( M2
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_1218_less__antisym,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
       => ( M2 = N ) ) ) ).

% less_antisym
thf(fact_1219_Suc__less__SucD,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ ( suc @ M2 ) @ ( suc @ N ) )
     => ( ord_less_nat @ M2 @ N ) ) ).

% Suc_less_SucD
thf(fact_1220_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_1221_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J3: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J3 )
             => ( ( ord_less_nat @ J3 @ K2 )
               => ( ( P2 @ I3 @ J3 )
                 => ( ( P2 @ J3 @ K2 )
                   => ( P2 @ I3 @ K2 ) ) ) ) )
         => ( P2 @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_1222_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P2: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P2 @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P2 @ ( suc @ I3 ) )
               => ( P2 @ I3 ) ) )
         => ( P2 @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_1223_not__less__less__Suc__eq,axiom,
    ! [N: nat,M2: nat] :
      ( ~ ( ord_less_nat @ N @ M2 )
     => ( ( ord_less_nat @ N @ ( suc @ M2 ) )
        = ( N = M2 ) ) ) ).

% not_less_less_Suc_eq
thf(fact_1224_diff__less__Suc,axiom,
    ! [M2: nat,N: nat] : ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ ( suc @ M2 ) ) ).

% diff_less_Suc
thf(fact_1225_Suc__diff__Suc,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ N @ M2 )
     => ( ( suc @ ( minus_minus_nat @ M2 @ ( suc @ N ) ) )
        = ( minus_minus_nat @ M2 @ N ) ) ) ).

% Suc_diff_Suc
thf(fact_1226_upt__conv__Cons__Cons,axiom,
    ! [M2: nat,N: nat,Ns: list_nat,Q2: nat] :
      ( ( ( cons_nat @ M2 @ ( cons_nat @ N @ Ns ) )
        = ( upt @ M2 @ Q2 ) )
      = ( ( cons_nat @ N @ Ns )
        = ( upt @ ( suc @ M2 ) @ Q2 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_1227_map__decr__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat
        @ ^ [N2: nat] : ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
      = ( upt @ M2 @ N ) ) ).

% map_decr_upt
thf(fact_1228_diff__less,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( ord_less_nat @ zero_zero_nat @ M2 )
       => ( ord_less_nat @ ( minus_minus_nat @ M2 @ N ) @ M2 ) ) ) ).

% diff_less
thf(fact_1229_add__diff__inverse__nat,axiom,
    ! [M2: nat,N: nat] :
      ( ~ ( ord_less_nat @ M2 @ N )
     => ( ( plus_plus_nat @ N @ ( minus_minus_nat @ M2 @ N ) )
        = M2 ) ) ).

% add_diff_inverse_nat
thf(fact_1230_less__diff__conv,axiom,
    ! [I: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_1231_less__Suc__eq__0__disj,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ ( suc @ N ) )
      = ( ( M2 = zero_zero_nat )
        | ? [J2: nat] :
            ( ( M2
              = ( suc @ J2 ) )
            & ( ord_less_nat @ J2 @ N ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_1232_gr0__implies__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ? [M5: nat] :
          ( N
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_1233_All__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
           => ( P2 @ I2 ) ) )
      = ( ( P2 @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
           => ( P2 @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1234_gr0__conv__Suc,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
      = ( ? [M6: nat] :
            ( N
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1235_Ex__less__Suc2,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N ) )
            & ( P2 @ I2 ) ) )
      = ( ( P2 @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N )
            & ( P2 @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1236_less__imp__Suc__add,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ? [K2: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_1237_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N2: nat] :
        ? [K3: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_1238_less__add__Suc2,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M2 @ I ) ) ) ).

% less_add_Suc2
thf(fact_1239_less__add__Suc1,axiom,
    ! [I: nat,M2: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M2 ) ) ) ).

% less_add_Suc1
thf(fact_1240_less__natE,axiom,
    ! [M2: nat,N: nat] :
      ( ( ord_less_nat @ M2 @ N )
     => ~ ! [Q3: nat] :
            ( N
           != ( suc @ ( plus_plus_nat @ M2 @ Q3 ) ) ) ) ).

% less_natE
thf(fact_1241_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_1242_map__Suc__upt,axiom,
    ! [M2: nat,N: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M2 @ N ) )
      = ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).

% map_Suc_upt
thf(fact_1243_nat__diff__split__asm,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P2 @ zero_zero_nat ) )
            | ? [D4: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D4 ) )
                & ~ ( P2 @ D4 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_1244_nat__diff__split,axiom,
    ! [P2: nat > $o,A: nat,B: nat] :
      ( ( P2 @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P2 @ zero_zero_nat ) )
        & ! [D4: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D4 ) )
           => ( P2 @ D4 ) ) ) ) ).

% nat_diff_split
thf(fact_1245_upt__rec,axiom,
    ( upt
    = ( ^ [I2: nat,J2: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J2 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J2 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_1246_upt__rec__numeral,axiom,
    ! [M2: num,N: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M2 ) ) @ ( numeral_numeral_nat @ N ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
       => ( ( upt @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_1247_Euclidean__Division_Odivmod__nat__if,axiom,
    ( euclidean_divmod_nat
    = ( ^ [M6: nat,N2: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N2 = zero_zero_nat )
            | ( ord_less_nat @ M6 @ N2 ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q4: nat] : ( product_Pair_nat_nat @ ( suc @ Q4 ) )
            @ ( euclidean_divmod_nat @ ( minus_minus_nat @ M6 @ N2 ) @ N2 ) ) ) ) ) ).

% Euclidean_Division.divmod_nat_if
thf(fact_1248_nat__gcd_Ocases,axiom,
    ! [X: product_prod_nat_nat] :
      ~ ! [X3: nat,Y3: nat] :
          ( X
         != ( product_Pair_nat_nat @ X3 @ Y3 ) ) ).

% nat_gcd.cases
thf(fact_1249_inf__concat__simple_Ocases,axiom,
    ! [X: produc8199716216217303280at_nat] :
      ( ! [F3: nat > nat] :
          ( X
         != ( produc72220940542539688at_nat @ F3 @ zero_zero_nat ) )
     => ~ ! [F3: nat > nat,N4: nat] :
            ( X
           != ( produc72220940542539688at_nat @ F3 @ ( suc @ N4 ) ) ) ) ).

% inf_concat_simple.cases
thf(fact_1250_inf__concat__simple_Osimps_I2_J,axiom,
    ! [F: nat > nat,N: nat] :
      ( ( inf_concat_simple @ F @ ( suc @ N ) )
      = ( produc2626176000494625587at_nat
        @ ^ [I2: nat,J2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J2 ) @ ( F @ I2 ) ) @ ( product_Pair_nat_nat @ I2 @ ( suc @ J2 ) ) @ ( product_Pair_nat_nat @ ( suc @ I2 ) @ zero_zero_nat ) )
        @ ( inf_concat_simple @ F @ N ) ) ) ).

% inf_concat_simple.simps(2)
thf(fact_1251_inf__concat__simple_Oelims,axiom,
    ! [X: nat > nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( inf_concat_simple @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y
           != ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) )
       => ~ ! [N4: nat] :
              ( ( Xa2
                = ( suc @ N4 ) )
             => ( Y
               != ( produc2626176000494625587at_nat
                  @ ^ [I2: nat,J2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J2 ) @ ( X @ I2 ) ) @ ( product_Pair_nat_nat @ I2 @ ( suc @ J2 ) ) @ ( product_Pair_nat_nat @ ( suc @ I2 ) @ zero_zero_nat ) )
                  @ ( inf_concat_simple @ X @ N4 ) ) ) ) ) ) ).

% inf_concat_simple.elims
thf(fact_1252_inf__concat__simple__surj__zero,axiom,
    ! [F: nat > nat,I: nat] :
    ? [K2: nat] :
      ( ( inf_concat_simple @ F @ K2 )
      = ( product_Pair_nat_nat @ I @ zero_zero_nat ) ) ).

% inf_concat_simple_surj_zero
thf(fact_1253_inf__concat__simple_Osimps_I1_J,axiom,
    ! [F: nat > nat] :
      ( ( inf_concat_simple @ F @ zero_zero_nat )
      = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ).

% inf_concat_simple.simps(1)
thf(fact_1254_inf__concat__simple__surj,axiom,
    ! [J: nat,F: nat > nat,I: nat] :
      ( ( ord_less_nat @ J @ ( F @ I ) )
     => ? [K2: nat] :
          ( ( inf_concat_simple @ F @ K2 )
          = ( product_Pair_nat_nat @ I @ J ) ) ) ).

% inf_concat_simple_surj
thf(fact_1255_inf__concat__simple__add,axiom,
    ! [F: nat > nat,K: nat,I: nat,J: nat,L3: nat] :
      ( ( ( inf_concat_simple @ F @ K )
        = ( product_Pair_nat_nat @ I @ J ) )
     => ( ( ord_less_nat @ ( plus_plus_nat @ J @ L3 ) @ ( F @ I ) )
       => ( ( inf_concat_simple @ F @ ( plus_plus_nat @ K @ L3 ) )
          = ( product_Pair_nat_nat @ I @ ( plus_plus_nat @ J @ L3 ) ) ) ) ) ).

% inf_concat_simple_add
thf(fact_1256_zero__notin__Suc__image,axiom,
    ! [A4: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A4 ) ) ).

% zero_notin_Suc_image
thf(fact_1257_less__one,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ N @ one_one_nat )
      = ( N = zero_zero_nat ) ) ).

% less_one
thf(fact_1258_Suc__diff__1,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( suc @ ( minus_minus_nat @ N @ one_one_nat ) )
        = N ) ) ).

% Suc_diff_1
thf(fact_1259_nat__induct__non__zero,axiom,
    ! [N: nat,P2: nat > $o] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( P2 @ one_one_nat )
       => ( ! [N4: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ N4 )
             => ( ( P2 @ N4 )
               => ( P2 @ ( suc @ N4 ) ) ) )
         => ( P2 @ N ) ) ) ) ).

% nat_induct_non_zero
thf(fact_1260_Suc__diff__eq__diff__pred,axiom,
    ! [N: nat,M2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( ( minus_minus_nat @ ( suc @ M2 ) @ N )
        = ( minus_minus_nat @ M2 @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_1261_Suc__pred_H,axiom,
    ! [N: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N )
     => ( N
        = ( suc @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ).

% Suc_pred'
thf(fact_1262_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_1263_drop__upt,axiom,
    ! [M2: nat,I: nat,J: nat] :
      ( ( drop_nat @ M2 @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M2 ) @ J ) ) ).

% drop_upt
thf(fact_1264_inf__concat__simple_Opelims,axiom,
    ! [X: nat > nat,Xa2: nat,Y: product_prod_nat_nat] :
      ( ( ( inf_concat_simple @ X @ Xa2 )
        = Y )
     => ( ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X @ Xa2 ) )
       => ( ( ( Xa2 = zero_zero_nat )
           => ( ( Y
                = ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) )
             => ~ ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X @ zero_zero_nat ) ) ) )
         => ~ ! [N4: nat] :
                ( ( Xa2
                  = ( suc @ N4 ) )
               => ( ( Y
                    = ( produc2626176000494625587at_nat
                      @ ^ [I2: nat,J2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_nat @ ( suc @ J2 ) @ ( X @ I2 ) ) @ ( product_Pair_nat_nat @ I2 @ ( suc @ J2 ) ) @ ( product_Pair_nat_nat @ ( suc @ I2 ) @ zero_zero_nat ) )
                      @ ( inf_concat_simple @ X @ N4 ) ) )
                 => ~ ( accp_P6412015083367660281at_nat @ inf_co2821209343098213861le_rel @ ( produc72220940542539688at_nat @ X @ ( suc @ N4 ) ) ) ) ) ) ) ) ).

% inf_concat_simple.pelims

% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X @ Y )
      = X ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [S7: sum_su4907927656612468301on_nat] :
      ( ( member5906418179908956708on_nat @ ( produc6704844767664046899on_nat @ S7 @ ( sum_In3878554002080803550on_nat @ ( produc7247797765490398872tation @ x @ multis5512618279175645674signed ) ) ) @ ( set_Pr937530902923597456on_nat @ ns ) )
     => ( ( member8476201367316484836on_nat @ S7 @ ( set_Su4848624856859743608on_nat @ ns_left ) )
       => thesis ) ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------