%------------------------------------------------------------------------------
% 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    : Query_Optimization/0009_Dtree/prob_03594_153963__15183426_1 [Des23]

% Status   : Theorem
% Rating   : ? v8.2.0
% Syntax   : Number of formulae    : 1761 ( 476 unt; 481 typ;   0 def)
%            Number of atoms       : 4001 (1835 equ;   0 cnn)
%            Maximal formula atoms :   16 (   3 avg)
%            Number of connectives : 14835 ( 813   ~;  63   |; 488   &;11689   @)
%                                         (   0 <=>;1782  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   29 (   8 avg)
%            Number of types       :   83 (  82 usr)
%            Number of type conns  : 1176 (1176   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  402 ( 399 usr;  27 con; 0-3 aty)
%            Number of variables   : 4693 (  30   ^;4178   !; 485   ?;4693   :)
% SPC      : TH0_THM_EQU_NAR

% Comments : This file was generated by Isabelle (most likely Sledgehammer)
%            2023-01-18 16:06:38.719
%------------------------------------------------------------------------------
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    the_el8589169213296894365od_a_b: set_Product_prod_a_b > product_prod_a_b ).

thf(sy_c_Set_Othe__elem_001tf__a,type,
    the_elem_a: set_a > a ).

thf(sy_c_Vertex__Walk_Ojoinable_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    vertex7469968122360900522od_a_b: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o ).

thf(sy_c_Vertex__Walk_Ovwalk__arcs_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    vertex3879532237292798496od_a_b: list_P1467974523212632366od_a_b > list_P3463435030753280173od_a_b ).

thf(sy_c_Vertex__Walk_Ovwalk__arcs_001tf__a,type,
    vertex_vwalk_arcs_a: list_a > list_P1396940483166286381od_a_a ).

thf(sy_c_Vertex__Walk_Ovwalk__arcs__rel_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    vertex3308915910108218357od_a_b: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o ).

thf(sy_c_Vertex__Walk_Ovwalk__join_001t__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    vertex9110485820359517623od_a_b: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b ).

thf(sy_c_Vertex__Walk_Ovwalk__join_001tf__a,type,
    vertex_vwalk_join_a: list_a > list_a > list_a ).

thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    accp_l5219092091033086903od_a_b: ( list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o ) > list_P1467974523212632366od_a_b > $o ).

thf(sy_c_Wellfounded_Oaccp_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J,type,
    accp_P2269134446667985584od_a_b: ( produc7397007428618769063od_a_b > produc7397007428618769063od_a_b > $o ) > produc7397007428618769063od_a_b > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_Itf__a_Mtf__b_J_M_062_It__Product____Type__Oprod_Itf__a_Mtf__b_J_M_Eo_J_J,type,
    member6766302768593690647_a_b_o: ( product_prod_a_b > product_prod_a_b > $o ) > set_Pr5159192339024047694_a_b_o > $o ).

thf(sy_c_member_001t__Dtree__Odtree_Itf__a_Mtf__b_J,type,
    member_dtree_a_b: dtree_a_b > set_dtree_a_b > $o ).

thf(sy_c_member_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member6895035109809442519od_a_b: list_P1467974523212632366od_a_b > set_li6306622547289818894od_a_b > $o ).

thf(sy_c_member_001t__List__Olist_Itf__a_J,type,
    member_list_a: list_a > set_list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member4408059462661755003od_a_b: produc2878020351943352036od_a_b > set_Pr9099570220598255002od_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J_Mtf__a_J,type,
    member6586581021983584763_b_a_a: produc9157795540071588452_b_a_a > set_Pr1532811110425372442_b_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member8860753849636704977od_a_b: produc1969440603286172218od_a_b > set_Pr4653771035052550384od_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__a_J_Mtf__a_J,type,
    member5016570112408633425_a_a_a: produc5194124205918388666_a_a_a > set_Pr290715669063433840_a_a_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Dtree__Odtree_Itf__a_Mtf__b_J_Mtf__b_J,type,
    member4380921116106875537_a_b_b: produc4558475209616630778_a_b_b > set_Pr3012420139608375472_a_b_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_Mt__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_J,type,
    member3051990972100052432od_a_b: produc7397007428618769063od_a_b > set_Pr3015088788472578951od_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__a_J_J,type,
    member8191768239178080336list_a: produc9164743771328383783list_a > set_Pr4048851178543822343list_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_Itf__a_J_Mt__List__Olist_Itf__b_J_J,type,
    member8262802279224426321list_b: produc12405774519953960list_b > set_Pr1527666660254786696list_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member5065438833300252112od_a_b: produc2233330766403592615od_a_b > set_Pr5493900063595384711od_a_b > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Product____Type__Oprod_Itf__a_Mtf__b_J_Mtf__a_J,type,
    member2194267517197600294_a_b_a: produc3167243049347903741_a_b_a > set_Pr2303819121155477725_a_b_a > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_Itf__a_Mt__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member3142156093895948538od_a_b: produc4115131626046251985od_a_b > set_Pr3008899384982559153od_a_b > $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__Product____Type__Oprod_Itf__a_Mtf__b_J,type,
    member1426531481828664017od_a_b: product_prod_a_b > set_Product_prod_a_b > $o ).

thf(sy_c_member_001t__Set__Oset_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J,type,
    member1887650552762594865od_a_b: set_Product_prod_a_b > set_se3214616458824132456od_a_b > $o ).

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

thf(sy_c_member_001tf__a,type,
    member_a: a > set_a > $o ).

thf(sy_c_member_001tf__b,type,
    member_b: b > set_b > $o ).

thf(sy_v_as,type,
    as: list_P1467974523212632366od_a_b ).

thf(sy_v_bs,type,
    bs: list_P1467974523212632366od_a_b ).

thf(sy_v_e,type,
    e: b ).

thf(sy_v_r,type,
    r: a ).

thf(sy_v_t,type,
    t: dtree_a_b ).

% Relevant facts (1274)
thf(fact_0_dtree_Oinject,axiom,
    ! [X1: product_prod_a_b,X2: fset_P6924693328471822228od_a_b,Y1: product_prod_a_b,Y2: fset_P6924693328471822228od_a_b] :
      ( ( ( node_P1936840572307073598od_a_b @ X1 @ X2 )
        = ( node_P1936840572307073598od_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% dtree.inject
thf(fact_1_dtree_Oinject,axiom,
    ! [X1: a,X2: fset_P8459658061218866410od_a_b,Y1: a,Y2: fset_P8459658061218866410od_a_b] :
      ( ( ( node_a4529940054729032296od_a_b @ X1 @ X2 )
        = ( node_a4529940054729032296od_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% dtree.inject
thf(fact_2_dtree_Oinject,axiom,
    ! [X1: product_prod_a_b,X2: fset_P1695881499225648404_b_a_a,Y1: product_prod_a_b,Y2: fset_P1695881499225648404_b_a_a] :
      ( ( ( node_P9154982556985054868_a_b_a @ X1 @ X2 )
        = ( node_P9154982556985054868_a_b_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% dtree.inject
thf(fact_3_dtree_Oinject,axiom,
    ! [X1: a,X2: fset_P2559403164575059562_a_a_a,Y1: a,Y2: fset_P2559403164575059562_a_a_a] :
      ( ( ( node_a_a @ X1 @ X2 )
        = ( node_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% dtree.inject
thf(fact_4_dtree_Oinject,axiom,
    ! [X1: a,X2: fset_P5281107635120001194_a_b_b,Y1: a,Y2: fset_P5281107635120001194_a_b_b] :
      ( ( ( node_a_b @ X1 @ X2 )
        = ( node_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% dtree.inject
thf(fact_5__092_060open_062_092_060exists_062xs_O_Astrict__subtree_A_INode_Ar_Axs_J_At_A_092_060and_062_Adverts_A_INode_Ar_Axs_J_A_061_Ainsert_Ar_A_Ifst_A_096_Aset_Abs_J_092_060close_062,axiom,
    ? [Xs: fset_P5281107635120001194_a_b_b] :
      ( ( strict_subtree_a_b @ ( node_a_b @ r @ Xs ) @ t )
      & ( ( dverts_a_b @ ( node_a_b @ r @ Xs ) )
        = ( insert_a2 @ r @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ bs ) ) ) ) ) ).

% \<open>\<exists>xs. strict_subtree (Node r xs) t \<and> dverts (Node r xs) = insert r (fst ` set bs)\<close>
thf(fact_6_dtree_Oset__intros_I1_J,axiom,
    ! [X1: product_prod_a_b,X2: fset_P6924693328471822228od_a_b] : ( member1426531481828664017od_a_b @ X1 @ ( dverts2678499813339623674od_a_b @ ( node_P1936840572307073598od_a_b @ X1 @ X2 ) ) ) ).

% dtree.set_intros(1)
thf(fact_7_dtree_Oset__intros_I1_J,axiom,
    ! [X1: a,X2: fset_P8459658061218866410od_a_b] : ( member_a @ X1 @ ( dverts1623436559482070308od_a_b @ ( node_a4529940054729032296od_a_b @ X1 @ X2 ) ) ) ).

% dtree.set_intros(1)
thf(fact_8_dtree_Oset__intros_I1_J,axiom,
    ! [X1: product_prod_a_b,X2: fset_P1695881499225648404_b_a_a] : ( member1426531481828664017od_a_b @ X1 @ ( dverts6248479061738092880_a_b_a @ ( node_P9154982556985054868_a_b_a @ X1 @ X2 ) ) ) ).

% dtree.set_intros(1)
thf(fact_9_dtree_Oset__intros_I1_J,axiom,
    ! [X1: a,X2: fset_P2559403164575059562_a_a_a] : ( member_a @ X1 @ ( dverts_a_a @ ( node_a_a @ X1 @ X2 ) ) ) ).

% dtree.set_intros(1)
thf(fact_10_dtree_Oset__intros_I1_J,axiom,
    ! [X1: a,X2: fset_P5281107635120001194_a_b_b] : ( member_a @ X1 @ ( dverts_a_b @ ( node_a_b @ X1 @ X2 ) ) ) ).

% dtree.set_intros(1)
thf(fact_11_dtree_Oexhaust,axiom,
    ! [Y: dtree_2253494508914376569od_a_b] :
      ~ ! [X12: product_prod_a_b,X22: fset_P6924693328471822228od_a_b] :
          ( Y
         != ( node_P1936840572307073598od_a_b @ X12 @ X22 ) ) ).

% dtree.exhaust
thf(fact_12_dtree_Oexhaust,axiom,
    ! [Y: dtree_2020077848175485859od_a_b] :
      ~ ! [X12: a,X22: fset_P8459658061218866410od_a_b] :
          ( Y
         != ( node_a4529940054729032296od_a_b @ X12 @ X22 ) ) ).

% dtree.exhaust
thf(fact_13_dtree_Oexhaust,axiom,
    ! [Y: dtree_1072189271477137615_a_b_a] :
      ~ ! [X12: product_prod_a_b,X22: fset_P1695881499225648404_b_a_a] :
          ( Y
         != ( node_P9154982556985054868_a_b_a @ X12 @ X22 ) ) ).

% dtree.exhaust
thf(fact_14_dtree_Oexhaust,axiom,
    ! [Y: dtree_a_a] :
      ~ ! [X12: a,X22: fset_P2559403164575059562_a_a_a] :
          ( Y
         != ( node_a_a @ X12 @ X22 ) ) ).

% dtree.exhaust
thf(fact_15_dtree_Oexhaust,axiom,
    ! [Y: dtree_a_b] :
      ~ ! [X12: a,X22: fset_P5281107635120001194_a_b_b] :
          ( Y
         != ( node_a_b @ X12 @ X22 ) ) ).

% dtree.exhaust
thf(fact_16_dverts__mset_Ocases,axiom,
    ! [X: dtree_2253494508914376569od_a_b] :
      ~ ! [R: product_prod_a_b,Xs: fset_P6924693328471822228od_a_b] :
          ( X
         != ( node_P1936840572307073598od_a_b @ R @ Xs ) ) ).

% dverts_mset.cases
thf(fact_17_dverts__mset_Ocases,axiom,
    ! [X: dtree_2020077848175485859od_a_b] :
      ~ ! [R: a,Xs: fset_P8459658061218866410od_a_b] :
          ( X
         != ( node_a4529940054729032296od_a_b @ R @ Xs ) ) ).

% dverts_mset.cases
thf(fact_18_dverts__mset_Ocases,axiom,
    ! [X: dtree_1072189271477137615_a_b_a] :
      ~ ! [R: product_prod_a_b,Xs: fset_P1695881499225648404_b_a_a] :
          ( X
         != ( node_P9154982556985054868_a_b_a @ R @ Xs ) ) ).

% dverts_mset.cases
thf(fact_19_dverts__mset_Ocases,axiom,
    ! [X: dtree_a_a] :
      ~ ! [R: a,Xs: fset_P2559403164575059562_a_a_a] :
          ( X
         != ( node_a_a @ R @ Xs ) ) ).

% dverts_mset.cases
thf(fact_20_dverts__mset_Ocases,axiom,
    ! [X: dtree_a_b] :
      ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b] :
          ( X
         != ( node_a_b @ R @ Xs ) ) ).

% dverts_mset.cases
thf(fact_21_assms_I2_J,axiom,
    ( ( append5335208823350062147od_a_b @ as @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ r @ e ) @ bs ) )
    = ( dtree_to_list_a_b @ t ) ) ).

% assms(2)
thf(fact_22_list_Oinject,axiom,
    ! [X21: a,X222: list_a,Y21: a,Y22: list_a] :
      ( ( ( cons_a @ X21 @ X222 )
        = ( cons_a @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_23_list_Oinject,axiom,
    ! [X21: list_P1467974523212632366od_a_b,X222: list_l8778361445553822004od_a_b,Y21: list_P1467974523212632366od_a_b,Y22: list_l8778361445553822004od_a_b] :
      ( ( ( cons_l907386227682280420od_a_b @ X21 @ X222 )
        = ( cons_l907386227682280420od_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_24_list_Oinject,axiom,
    ! [X21: produc2233330766403592615od_a_b,X222: list_P3463435030753280173od_a_b,Y21: produc2233330766403592615od_a_b,Y22: list_P3463435030753280173od_a_b] :
      ( ( ( cons_P8222935586017696605od_a_b @ X21 @ X222 )
        = ( cons_P8222935586017696605od_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_25_list_Oinject,axiom,
    ! [X21: b,X222: list_b,Y21: b,Y22: list_b] :
      ( ( ( cons_b @ X21 @ X222 )
        = ( cons_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_26_list_Oinject,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b,Y21: product_prod_a_b,Y22: list_P1467974523212632366od_a_b] :
      ( ( ( cons_P7316939131009794654od_a_b @ X21 @ X222 )
        = ( cons_P7316939131009794654od_a_b @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% list.inject
thf(fact_27_image__eqI,axiom,
    ! [B: a,F: product_prod_a_b > a,X: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( B
        = ( F @ X ) )
     => ( ( member1426531481828664017od_a_b @ X @ A )
       => ( member_a @ B @ ( image_2802296252294471259_a_b_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_28_image__eqI,axiom,
    ! [B: a,F: a > a,X: a,A: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_a @ B @ ( image_a_a2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_29_image__eqI,axiom,
    ! [B: product_prod_a_b,F: a > product_prod_a_b,X: a,A: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member1426531481828664017od_a_b @ B @ ( image_7400625786893224495od_a_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_30_image__eqI,axiom,
    ! [B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b,X: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( B
        = ( F @ X ) )
     => ( ( member1426531481828664017od_a_b @ X @ A )
       => ( member1426531481828664017od_a_b @ B @ ( image_3300603549555413765od_a_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_31_image__eqI,axiom,
    ! [B: b,F: a > b,X: a,A: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_b @ B @ ( image_a_b2 @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_32_image__eqI,axiom,
    ! [B: a,F: b > a,X: b,A: set_b] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_b @ X @ A )
       => ( member_a @ B @ ( image_b_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_33_image__eqI,axiom,
    ! [B: b,F: b > b,X: b,A: set_b] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_b @ X @ A )
       => ( member_b @ B @ ( image_b_b @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_34_image__eqI,axiom,
    ! [B: list_a,F: a > list_a,X: a,A: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_35_image__eqI,axiom,
    ! [B: set_a,F: a > set_a,X: a,A: set_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_a @ X @ A )
       => ( member_set_a @ B @ ( image_a_set_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_36_image__eqI,axiom,
    ! [B: a,F: list_a > a,X: list_a,A: set_list_a] :
      ( ( B
        = ( F @ X ) )
     => ( ( member_list_a @ X @ A )
       => ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).

% image_eqI
thf(fact_37_prod_Oinject,axiom,
    ! [X1: list_a,X2: list_a,Y1: list_a,Y2: list_a] :
      ( ( ( produc6837034575241423639list_a @ X1 @ X2 )
        = ( produc6837034575241423639list_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_38_prod_Oinject,axiom,
    ! [X1: product_prod_a_b > product_prod_a_b > $o,X2: list_P1467974523212632366od_a_b,Y1: product_prod_a_b > product_prod_a_b > $o,Y2: list_P1467974523212632366od_a_b] :
      ( ( ( produc7704224619955794135od_a_b @ X1 @ X2 )
        = ( produc7704224619955794135od_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_39_prod_Oinject,axiom,
    ! [X1: a,X2: product_prod_a_b,Y1: a,Y2: product_prod_a_b] :
      ( ( ( produc431845345726502849od_a_b @ X1 @ X2 )
        = ( produc431845345726502849od_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_40_prod_Oinject,axiom,
    ! [X1: a,X2: list_P1467974523212632366od_a_b,Y1: a,Y2: list_P1467974523212632366od_a_b] :
      ( ( ( produc6643264353547226055od_a_b @ X1 @ X2 )
        = ( produc6643264353547226055od_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_41_prod_Oinject,axiom,
    ! [X1: a,X2: a,Y1: a,Y2: a] :
      ( ( ( product_Pair_a_a @ X1 @ X2 )
        = ( product_Pair_a_a @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_42_prod_Oinject,axiom,
    ! [X1: a,X2: b,Y1: a,Y2: b] :
      ( ( ( product_Pair_a_b @ X1 @ X2 )
        = ( product_Pair_a_b @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X2 = Y2 ) ) ) ).

% prod.inject
thf(fact_43_old_Oprod_Oinject,axiom,
    ! [A2: list_a,B: list_a,A3: list_a,B2: list_a] :
      ( ( ( produc6837034575241423639list_a @ A2 @ B )
        = ( produc6837034575241423639list_a @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_44_old_Oprod_Oinject,axiom,
    ! [A2: product_prod_a_b > product_prod_a_b > $o,B: list_P1467974523212632366od_a_b,A3: product_prod_a_b > product_prod_a_b > $o,B2: list_P1467974523212632366od_a_b] :
      ( ( ( produc7704224619955794135od_a_b @ A2 @ B )
        = ( produc7704224619955794135od_a_b @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_45_old_Oprod_Oinject,axiom,
    ! [A2: a,B: product_prod_a_b,A3: a,B2: product_prod_a_b] :
      ( ( ( produc431845345726502849od_a_b @ A2 @ B )
        = ( produc431845345726502849od_a_b @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_46_old_Oprod_Oinject,axiom,
    ! [A2: a,B: list_P1467974523212632366od_a_b,A3: a,B2: list_P1467974523212632366od_a_b] :
      ( ( ( produc6643264353547226055od_a_b @ A2 @ B )
        = ( produc6643264353547226055od_a_b @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_47_old_Oprod_Oinject,axiom,
    ! [A2: a,B: a,A3: a,B2: a] :
      ( ( ( product_Pair_a_a @ A2 @ B )
        = ( product_Pair_a_a @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_48_old_Oprod_Oinject,axiom,
    ! [A2: a,B: b,A3: a,B2: b] :
      ( ( ( product_Pair_a_b @ A2 @ B )
        = ( product_Pair_a_b @ A3 @ B2 ) )
      = ( ( A2 = A3 )
        & ( B = B2 ) ) ) ).

% old.prod.inject
thf(fact_49_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P321204300973800749list_a] :
      ( ~ ( wf_lis1674799249052484319list_a @ X )
     => ~ ! [V: list_a,E: list_a,Xs: list_P321204300973800749list_a] :
            ( ( X
              = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ V @ E ) @ Xs ) )
           => ( ~ ( member_list_a @ V @ ( image_5971222721807453474list_a @ produc3698117735987127555list_a @ ( set_Pr2906193453920572092list_a @ Xs ) ) )
              & ( wf_lis1674799249052484319list_a @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_50_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P8094385941167427565od_a_b] :
      ( ~ ( wf_lis7776600765086547103od_a_b @ X )
     => ~ ! [V: product_prod_a_b > product_prod_a_b > $o,E: list_P1467974523212632366od_a_b,Xs: list_P8094385941167427565od_a_b] :
            ( ( X
              = ( cons_P3561991119763190045od_a_b @ ( produc7704224619955794135od_a_b @ V @ E ) @ Xs ) )
           => ( ~ ( member6766302768593690647_a_b_o @ V @ ( image_9098747296146071628_a_b_o @ produc8595103656739649731od_a_b @ ( set_Pr7552351584573503100od_a_b @ Xs ) ) )
              & ( wf_lis7776600765086547103od_a_b @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_51_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P8504624544267313367od_a_b] :
      ( ~ ( wf_lis4492982056392339337od_a_b @ X )
     => ~ ! [V: a,E: product_prod_a_b,Xs: list_P8504624544267313367od_a_b] :
            ( ( X
              = ( cons_P135045198529856391od_a_b @ ( produc431845345726502849od_a_b @ V @ E ) @ Xs ) )
           => ( ~ ( member_a @ V @ ( image_4017424819796311410_a_b_a @ produc6516300543326982573od_a_b @ ( set_Pr7079953345493216102od_a_b @ Xs ) ) )
              & ( wf_lis4492982056392339337od_a_b @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_52_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P5700792096213675741od_a_b] :
      ( ~ ( wf_lis8345819997161463695od_a_b @ X )
     => ~ ! [V: a,E: list_P1467974523212632366od_a_b,Xs: list_P5700792096213675741od_a_b] :
            ( ( X
              = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ V @ E ) @ Xs ) )
           => ( ~ ( member_a @ V @ ( image_2222990102783010156_a_b_a @ produc7763512175944540083od_a_b @ ( set_Pr187010687427835756od_a_b @ Xs ) ) )
              & ( wf_lis8345819997161463695od_a_b @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_53_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P3463435030753280173od_a_b] :
      ( ~ ( wf_lis873957137097608799od_a_b @ X )
     => ~ ! [V: product_prod_a_b,E: product_prod_a_b,Xs: list_P3463435030753280173od_a_b] :
            ( ( X
              = ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ V @ E ) @ Xs ) )
           => ( ~ ( member1426531481828664017od_a_b @ V @ ( image_2926283117629707974od_a_b @ produc5013601669340818563od_a_b @ ( set_Pr5228040199190236220od_a_b @ Xs ) ) )
              & ( wf_lis873957137097608799od_a_b @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_54_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P5360711134766023690_a_b_b] :
      ( ~ ( wf_lis4446142942163678948_a_b_b @ X )
     => ~ ! [V: dtree_a_b,E: b,Xs: list_P5360711134766023690_a_b_b] :
            ( ( X
              = ( cons_P2785074014425598660_a_b_b @ ( produc331601717337510060_a_b_b @ V @ E ) @ Xs ) )
           => ( ~ ( member_dtree_a_b @ V @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( set_Pr8239206241032387877_a_b_b @ Xs ) ) )
              & ( wf_lis4446142942163678948_a_b_b @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_55_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P1396940483166286381od_a_a] :
      ( ~ ( wf_list_verts_a_a @ X )
     => ~ ! [V: a,E: a,Xs: list_P1396940483166286381od_a_a] :
            ( ( X
              = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ V @ E ) @ Xs ) )
           => ( ~ ( member_a @ V @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xs ) ) )
              & ( wf_list_verts_a_a @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_56_wf__list__verts_Oelims_I3_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ~ ( wf_list_verts_a_b @ X )
     => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
            ( ( X
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
           => ( ~ ( member_a @ V @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
              & ( wf_list_verts_a_b @ Xs ) ) ) ) ).

% wf_list_verts.elims(3)
thf(fact_57_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: list_a,E2: list_a,Xs2: list_P321204300973800749list_a] :
      ( ( wf_lis1674799249052484319list_a @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member_list_a @ V2 @ ( image_5971222721807453474list_a @ produc3698117735987127555list_a @ ( set_Pr2906193453920572092list_a @ Xs2 ) ) )
        & ( wf_lis1674799249052484319list_a @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_58_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: product_prod_a_b > product_prod_a_b > $o,E2: list_P1467974523212632366od_a_b,Xs2: list_P8094385941167427565od_a_b] :
      ( ( wf_lis7776600765086547103od_a_b @ ( cons_P3561991119763190045od_a_b @ ( produc7704224619955794135od_a_b @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member6766302768593690647_a_b_o @ V2 @ ( image_9098747296146071628_a_b_o @ produc8595103656739649731od_a_b @ ( set_Pr7552351584573503100od_a_b @ Xs2 ) ) )
        & ( wf_lis7776600765086547103od_a_b @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_59_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: a,E2: product_prod_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( wf_lis4492982056392339337od_a_b @ ( cons_P135045198529856391od_a_b @ ( produc431845345726502849od_a_b @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member_a @ V2 @ ( image_4017424819796311410_a_b_a @ produc6516300543326982573od_a_b @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) ) )
        & ( wf_lis4492982056392339337od_a_b @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_60_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: a,E2: list_P1467974523212632366od_a_b,Xs2: list_P5700792096213675741od_a_b] :
      ( ( wf_lis8345819997161463695od_a_b @ ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member_a @ V2 @ ( image_2222990102783010156_a_b_a @ produc7763512175944540083od_a_b @ ( set_Pr187010687427835756od_a_b @ Xs2 ) ) )
        & ( wf_lis8345819997161463695od_a_b @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_61_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: product_prod_a_b,E2: product_prod_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( wf_lis873957137097608799od_a_b @ ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member1426531481828664017od_a_b @ V2 @ ( image_2926283117629707974od_a_b @ produc5013601669340818563od_a_b @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) ) )
        & ( wf_lis873957137097608799od_a_b @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_62_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: dtree_a_b,E2: b,Xs2: list_P5360711134766023690_a_b_b] :
      ( ( wf_lis4446142942163678948_a_b_b @ ( cons_P2785074014425598660_a_b_b @ ( produc331601717337510060_a_b_b @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member_dtree_a_b @ V2 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( set_Pr8239206241032387877_a_b_b @ Xs2 ) ) )
        & ( wf_lis4446142942163678948_a_b_b @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_63_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: a,E2: a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( wf_list_verts_a_a @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member_a @ V2 @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xs2 ) ) )
        & ( wf_list_verts_a_a @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_64_wf__list__verts_Osimps_I2_J,axiom,
    ! [V2: a,E2: b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( wf_list_verts_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member_a @ V2 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) ) )
        & ( wf_list_verts_a_b @ Xs2 ) ) ) ).

% wf_list_verts.simps(2)
thf(fact_65_fstI,axiom,
    ! [X: produc9164743771328383783list_a,Y: list_a,Z: list_a] :
      ( ( X
        = ( produc6837034575241423639list_a @ Y @ Z ) )
     => ( ( produc3698117735987127555list_a @ X )
        = Y ) ) ).

% fstI
thf(fact_66_fstI,axiom,
    ! [X: produc3334210663054469735od_a_b,Y: product_prod_a_b > product_prod_a_b > $o,Z: list_P1467974523212632366od_a_b] :
      ( ( X
        = ( produc7704224619955794135od_a_b @ Y @ Z ) )
     => ( ( produc8595103656739649731od_a_b @ X )
        = Y ) ) ).

% fstI
thf(fact_67_fstI,axiom,
    ! [X: produc4115131626046251985od_a_b,Y: a,Z: product_prod_a_b] :
      ( ( X
        = ( produc431845345726502849od_a_b @ Y @ Z ) )
     => ( ( produc6516300543326982573od_a_b @ X )
        = Y ) ) ).

% fstI
thf(fact_68_fstI,axiom,
    ! [X: produc2639161875728825943od_a_b,Y: a,Z: list_P1467974523212632366od_a_b] :
      ( ( X
        = ( produc6643264353547226055od_a_b @ Y @ Z ) )
     => ( ( produc7763512175944540083od_a_b @ X )
        = Y ) ) ).

% fstI
thf(fact_69_fstI,axiom,
    ! [X: produc2233330766403592615od_a_b,Y: product_prod_a_b,Z: product_prod_a_b] :
      ( ( X
        = ( produc6550459592057863831od_a_b @ Y @ Z ) )
     => ( ( produc5013601669340818563od_a_b @ X )
        = Y ) ) ).

% fstI
thf(fact_70_fstI,axiom,
    ! [X: produc4558475209616630778_a_b_b,Y: dtree_a_b,Z: b] :
      ( ( X
        = ( produc331601717337510060_a_b_b @ Y @ Z ) )
     => ( ( produc697780174058963904_a_b_b @ X )
        = Y ) ) ).

% fstI
thf(fact_71_fstI,axiom,
    ! [X: product_prod_a_a,Y: a,Z: a] :
      ( ( X
        = ( product_Pair_a_a @ Y @ Z ) )
     => ( ( product_fst_a_a @ X )
        = Y ) ) ).

% fstI
thf(fact_72_fstI,axiom,
    ! [X: product_prod_a_b,Y: a,Z: b] :
      ( ( X
        = ( product_Pair_a_b @ Y @ Z ) )
     => ( ( product_fst_a_b @ X )
        = Y ) ) ).

% fstI
thf(fact_73_fst__eqD,axiom,
    ! [X: list_a,Y: list_a,A2: list_a] :
      ( ( ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_74_fst__eqD,axiom,
    ! [X: product_prod_a_b > product_prod_a_b > $o,Y: list_P1467974523212632366od_a_b,A2: product_prod_a_b > product_prod_a_b > $o] :
      ( ( ( produc8595103656739649731od_a_b @ ( produc7704224619955794135od_a_b @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_75_fst__eqD,axiom,
    ! [X: a,Y: product_prod_a_b,A2: a] :
      ( ( ( produc6516300543326982573od_a_b @ ( produc431845345726502849od_a_b @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_76_fst__eqD,axiom,
    ! [X: a,Y: list_P1467974523212632366od_a_b,A2: a] :
      ( ( ( produc7763512175944540083od_a_b @ ( produc6643264353547226055od_a_b @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_77_fst__eqD,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,A2: product_prod_a_b] :
      ( ( ( produc5013601669340818563od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_78_fst__eqD,axiom,
    ! [X: dtree_a_b,Y: b,A2: dtree_a_b] :
      ( ( ( produc697780174058963904_a_b_b @ ( produc331601717337510060_a_b_b @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_79_fst__eqD,axiom,
    ! [X: a,Y: a,A2: a] :
      ( ( ( product_fst_a_a @ ( product_Pair_a_a @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_80_fst__eqD,axiom,
    ! [X: a,Y: b,A2: a] :
      ( ( ( product_fst_a_b @ ( product_Pair_a_b @ X @ Y ) )
        = A2 )
     => ( X = A2 ) ) ).

% fst_eqD
thf(fact_81_fst__conv,axiom,
    ! [X1: list_a,X2: list_a] :
      ( ( produc3698117735987127555list_a @ ( produc6837034575241423639list_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_82_fst__conv,axiom,
    ! [X1: product_prod_a_b > product_prod_a_b > $o,X2: list_P1467974523212632366od_a_b] :
      ( ( produc8595103656739649731od_a_b @ ( produc7704224619955794135od_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_83_fst__conv,axiom,
    ! [X1: a,X2: product_prod_a_b] :
      ( ( produc6516300543326982573od_a_b @ ( produc431845345726502849od_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_84_fst__conv,axiom,
    ! [X1: a,X2: list_P1467974523212632366od_a_b] :
      ( ( produc7763512175944540083od_a_b @ ( produc6643264353547226055od_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_85_fst__conv,axiom,
    ! [X1: product_prod_a_b,X2: product_prod_a_b] :
      ( ( produc5013601669340818563od_a_b @ ( produc6550459592057863831od_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_86_fst__conv,axiom,
    ! [X1: dtree_a_b,X2: b] :
      ( ( produc697780174058963904_a_b_b @ ( produc331601717337510060_a_b_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_87_fst__conv,axiom,
    ! [X1: a,X2: a] :
      ( ( product_fst_a_a @ ( product_Pair_a_a @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_88_fst__conv,axiom,
    ! [X1: a,X2: b] :
      ( ( product_fst_a_b @ ( product_Pair_a_b @ X1 @ X2 ) )
      = X1 ) ).

% fst_conv
thf(fact_89_eq__fst__iff,axiom,
    ! [A2: list_a,P: produc9164743771328383783list_a] :
      ( ( A2
        = ( produc3698117735987127555list_a @ P ) )
      = ( ? [B3: list_a] :
            ( P
            = ( produc6837034575241423639list_a @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_90_eq__fst__iff,axiom,
    ! [A2: product_prod_a_b > product_prod_a_b > $o,P: produc3334210663054469735od_a_b] :
      ( ( A2
        = ( produc8595103656739649731od_a_b @ P ) )
      = ( ? [B3: list_P1467974523212632366od_a_b] :
            ( P
            = ( produc7704224619955794135od_a_b @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_91_eq__fst__iff,axiom,
    ! [A2: a,P: produc4115131626046251985od_a_b] :
      ( ( A2
        = ( produc6516300543326982573od_a_b @ P ) )
      = ( ? [B3: product_prod_a_b] :
            ( P
            = ( produc431845345726502849od_a_b @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_92_eq__fst__iff,axiom,
    ! [A2: a,P: produc2639161875728825943od_a_b] :
      ( ( A2
        = ( produc7763512175944540083od_a_b @ P ) )
      = ( ? [B3: list_P1467974523212632366od_a_b] :
            ( P
            = ( produc6643264353547226055od_a_b @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_93_eq__fst__iff,axiom,
    ! [A2: product_prod_a_b,P: produc2233330766403592615od_a_b] :
      ( ( A2
        = ( produc5013601669340818563od_a_b @ P ) )
      = ( ? [B3: product_prod_a_b] :
            ( P
            = ( produc6550459592057863831od_a_b @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_94_eq__fst__iff,axiom,
    ! [A2: dtree_a_b,P: produc4558475209616630778_a_b_b] :
      ( ( A2
        = ( produc697780174058963904_a_b_b @ P ) )
      = ( ? [B3: b] :
            ( P
            = ( produc331601717337510060_a_b_b @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_95_eq__fst__iff,axiom,
    ! [A2: a,P: product_prod_a_a] :
      ( ( A2
        = ( product_fst_a_a @ P ) )
      = ( ? [B3: a] :
            ( P
            = ( product_Pair_a_a @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_96_eq__fst__iff,axiom,
    ! [A2: a,P: product_prod_a_b] :
      ( ( A2
        = ( product_fst_a_b @ P ) )
      = ( ? [B3: b] :
            ( P
            = ( product_Pair_a_b @ A2 @ B3 ) ) ) ) ).

% eq_fst_iff
thf(fact_97_list_Oset__intros_I2_J,axiom,
    ! [Y: list_a,X222: list_list_a,X21: list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ X222 ) )
     => ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_98_list_Oset__intros_I2_J,axiom,
    ! [Y: set_Product_prod_a_b,X222: list_s6539019640784088206od_a_b,X21: set_Product_prod_a_b] :
      ( ( member1887650552762594865od_a_b @ Y @ ( set_se8479788141692617885od_a_b @ X222 ) )
     => ( member1887650552762594865od_a_b @ Y @ ( set_se8479788141692617885od_a_b @ ( cons_s5806323297083630014od_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_99_list_Oset__intros_I2_J,axiom,
    ! [Y: set_a,X222: list_set_a,X21: set_a] :
      ( ( member_set_a @ Y @ ( set_set_a2 @ X222 ) )
     => ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_100_list_Oset__intros_I2_J,axiom,
    ! [Y: produc4115131626046251985od_a_b,X222: list_P8504624544267313367od_a_b,X21: produc4115131626046251985od_a_b] :
      ( ( member3142156093895948538od_a_b @ Y @ ( set_Pr7079953345493216102od_a_b @ X222 ) )
     => ( member3142156093895948538od_a_b @ Y @ ( set_Pr7079953345493216102od_a_b @ ( cons_P135045198529856391od_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_101_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_a_a,X222: list_P1396940483166286381od_a_a,X21: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ X222 ) )
     => ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_102_list_Oset__intros_I2_J,axiom,
    ! [Y: list_P1467974523212632366od_a_b,X222: list_l8778361445553822004od_a_b,X21: list_P1467974523212632366od_a_b] :
      ( ( member6895035109809442519od_a_b @ Y @ ( set_li5045283583078295363od_a_b @ X222 ) )
     => ( member6895035109809442519od_a_b @ Y @ ( set_li5045283583078295363od_a_b @ ( cons_l907386227682280420od_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_103_list_Oset__intros_I2_J,axiom,
    ! [Y: produc2233330766403592615od_a_b,X222: list_P3463435030753280173od_a_b,X21: produc2233330766403592615od_a_b] :
      ( ( member5065438833300252112od_a_b @ Y @ ( set_Pr5228040199190236220od_a_b @ X222 ) )
     => ( member5065438833300252112od_a_b @ Y @ ( set_Pr5228040199190236220od_a_b @ ( cons_P8222935586017696605od_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_104_list_Oset__intros_I2_J,axiom,
    ! [Y: b,X222: list_b,X21: b] :
      ( ( member_b @ Y @ ( set_b2 @ X222 ) )
     => ( member_b @ Y @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_105_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_106_list_Oset__intros_I2_J,axiom,
    ! [Y: product_prod_a_b,X222: list_P1467974523212632366od_a_b,X21: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ X222 ) )
     => ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) ) ) ) ).

% list.set_intros(2)
thf(fact_107_insertCI,axiom,
    ! [A2: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( ~ ( member1426531477525435216od_a_a @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_108_insertCI,axiom,
    ! [A2: list_P1467974523212632366od_a_b,B4: set_li6306622547289818894od_a_b,B: list_P1467974523212632366od_a_b] :
      ( ( ~ ( member6895035109809442519od_a_b @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member6895035109809442519od_a_b @ A2 @ ( insert6792547294658701054od_a_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_109_insertCI,axiom,
    ! [A2: list_a,B4: set_list_a,B: list_a] :
      ( ( ~ ( member_list_a @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_list_a @ A2 @ ( insert_list_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_110_insertCI,axiom,
    ! [A2: set_Product_prod_a_b,B4: set_se3214616458824132456od_a_b,B: set_Product_prod_a_b] :
      ( ( ~ ( member1887650552762594865od_a_b @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member1887650552762594865od_a_b @ A2 @ ( insert985587154976485848od_a_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_111_insertCI,axiom,
    ! [A2: set_a,B4: set_set_a,B: set_a] :
      ( ( ~ ( member_set_a @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_set_a @ A2 @ ( insert_set_a @ B @ B4 ) ) ) ).

% insertCI
thf(fact_112_insertCI,axiom,
    ! [A2: b,B4: set_b,B: b] :
      ( ( ~ ( member_b @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_b @ A2 @ ( insert_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_113_insertCI,axiom,
    ! [A2: a,B4: set_a,B: a] :
      ( ( ~ ( member_a @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member_a @ A2 @ ( insert_a2 @ B @ B4 ) ) ) ).

% insertCI
thf(fact_114_insertCI,axiom,
    ! [A2: product_prod_a_b,B4: set_Product_prod_a_b,B: product_prod_a_b] :
      ( ( ~ ( member1426531481828664017od_a_b @ A2 @ B4 )
       => ( A2 = B ) )
     => ( member1426531481828664017od_a_b @ A2 @ ( insert4534936386344385144od_a_b @ B @ B4 ) ) ) ).

% insertCI
thf(fact_115_insert__iff,axiom,
    ! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ A ) )
      = ( ( A2 = B )
        | ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_116_insert__iff,axiom,
    ! [A2: list_P1467974523212632366od_a_b,B: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b] :
      ( ( member6895035109809442519od_a_b @ A2 @ ( insert6792547294658701054od_a_b @ B @ A ) )
      = ( ( A2 = B )
        | ( member6895035109809442519od_a_b @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_117_insert__iff,axiom,
    ! [A2: list_a,B: list_a,A: set_list_a] :
      ( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
      = ( ( A2 = B )
        | ( member_list_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_118_insert__iff,axiom,
    ! [A2: set_Product_prod_a_b,B: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( member1887650552762594865od_a_b @ A2 @ ( insert985587154976485848od_a_b @ B @ A ) )
      = ( ( A2 = B )
        | ( member1887650552762594865od_a_b @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_119_insert__iff,axiom,
    ! [A2: set_a,B: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
      = ( ( A2 = B )
        | ( member_set_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_120_insert__iff,axiom,
    ! [A2: b,B: b,A: set_b] :
      ( ( member_b @ A2 @ ( insert_b @ B @ A ) )
      = ( ( A2 = B )
        | ( member_b @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_121_insert__iff,axiom,
    ! [A2: a,B: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a2 @ B @ A ) )
      = ( ( A2 = B )
        | ( member_a @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_122_insert__iff,axiom,
    ! [A2: product_prod_a_b,B: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ ( insert4534936386344385144od_a_b @ B @ A ) )
      = ( ( A2 = B )
        | ( member1426531481828664017od_a_b @ A2 @ A ) ) ) ).

% insert_iff
thf(fact_123_insert__absorb2,axiom,
    ! [X: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ X @ A ) )
      = ( insert4534936382041156343od_a_a @ X @ A ) ) ).

% insert_absorb2
thf(fact_124_insert__absorb2,axiom,
    ! [X: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b] :
      ( ( insert6792547294658701054od_a_b @ X @ ( insert6792547294658701054od_a_b @ X @ A ) )
      = ( insert6792547294658701054od_a_b @ X @ A ) ) ).

% insert_absorb2
thf(fact_125_insert__absorb2,axiom,
    ! [X: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( insert985587154976485848od_a_b @ X @ ( insert985587154976485848od_a_b @ X @ A ) )
      = ( insert985587154976485848od_a_b @ X @ A ) ) ).

% insert_absorb2
thf(fact_126_insert__absorb2,axiom,
    ! [X: set_a,A: set_set_a] :
      ( ( insert_set_a @ X @ ( insert_set_a @ X @ A ) )
      = ( insert_set_a @ X @ A ) ) ).

% insert_absorb2
thf(fact_127_insert__absorb2,axiom,
    ! [X: b,A: set_b] :
      ( ( insert_b @ X @ ( insert_b @ X @ A ) )
      = ( insert_b @ X @ A ) ) ).

% insert_absorb2
thf(fact_128_insert__absorb2,axiom,
    ! [X: a,A: set_a] :
      ( ( insert_a2 @ X @ ( insert_a2 @ X @ A ) )
      = ( insert_a2 @ X @ A ) ) ).

% insert_absorb2
thf(fact_129_append_Oassoc,axiom,
    ! [A2: list_a,B: list_a,C: list_a] :
      ( ( append_a @ ( append_a @ A2 @ B ) @ C )
      = ( append_a @ A2 @ ( append_a @ B @ C ) ) ) ).

% append.assoc
thf(fact_130_append_Oassoc,axiom,
    ! [A2: list_P3463435030753280173od_a_b,B: list_P3463435030753280173od_a_b,C: list_P3463435030753280173od_a_b] :
      ( ( append3447563168971123522od_a_b @ ( append3447563168971123522od_a_b @ A2 @ B ) @ C )
      = ( append3447563168971123522od_a_b @ A2 @ ( append3447563168971123522od_a_b @ B @ C ) ) ) ).

% append.assoc
thf(fact_131_append_Oassoc,axiom,
    ! [A2: list_l8778361445553822004od_a_b,B: list_l8778361445553822004od_a_b,C: list_l8778361445553822004od_a_b] :
      ( ( append7557446516274476873od_a_b @ ( append7557446516274476873od_a_b @ A2 @ B ) @ C )
      = ( append7557446516274476873od_a_b @ A2 @ ( append7557446516274476873od_a_b @ B @ C ) ) ) ).

% append.assoc
thf(fact_132_append_Oassoc,axiom,
    ! [A2: list_b,B: list_b,C: list_b] :
      ( ( append_b @ ( append_b @ A2 @ B ) @ C )
      = ( append_b @ A2 @ ( append_b @ B @ C ) ) ) ).

% append.assoc
thf(fact_133_append_Oassoc,axiom,
    ! [A2: list_P1467974523212632366od_a_b,B: list_P1467974523212632366od_a_b,C: list_P1467974523212632366od_a_b] :
      ( ( append5335208823350062147od_a_b @ ( append5335208823350062147od_a_b @ A2 @ B ) @ C )
      = ( append5335208823350062147od_a_b @ A2 @ ( append5335208823350062147od_a_b @ B @ C ) ) ) ).

% append.assoc
thf(fact_134_append__assoc,axiom,
    ! [Xs2: list_a,Ys: list_a,Zs: list_a] :
      ( ( append_a @ ( append_a @ Xs2 @ Ys ) @ Zs )
      = ( append_a @ Xs2 @ ( append_a @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_135_append__assoc,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,Ys: list_P3463435030753280173od_a_b,Zs: list_P3463435030753280173od_a_b] :
      ( ( append3447563168971123522od_a_b @ ( append3447563168971123522od_a_b @ Xs2 @ Ys ) @ Zs )
      = ( append3447563168971123522od_a_b @ Xs2 @ ( append3447563168971123522od_a_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_136_append__assoc,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b,Zs: list_l8778361445553822004od_a_b] :
      ( ( append7557446516274476873od_a_b @ ( append7557446516274476873od_a_b @ Xs2 @ Ys ) @ Zs )
      = ( append7557446516274476873od_a_b @ Xs2 @ ( append7557446516274476873od_a_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_137_append__assoc,axiom,
    ! [Xs2: list_b,Ys: list_b,Zs: list_b] :
      ( ( append_b @ ( append_b @ Xs2 @ Ys ) @ Zs )
      = ( append_b @ Xs2 @ ( append_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_138_append__assoc,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( append5335208823350062147od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ Zs )
      = ( append5335208823350062147od_a_b @ Xs2 @ ( append5335208823350062147od_a_b @ Ys @ Zs ) ) ) ).

% append_assoc
thf(fact_139_append__same__eq,axiom,
    ! [Ys: list_a,Xs2: list_a,Zs: list_a] :
      ( ( ( append_a @ Ys @ Xs2 )
        = ( append_a @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_140_append__same__eq,axiom,
    ! [Ys: list_P3463435030753280173od_a_b,Xs2: list_P3463435030753280173od_a_b,Zs: list_P3463435030753280173od_a_b] :
      ( ( ( append3447563168971123522od_a_b @ Ys @ Xs2 )
        = ( append3447563168971123522od_a_b @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_141_append__same__eq,axiom,
    ! [Ys: list_l8778361445553822004od_a_b,Xs2: list_l8778361445553822004od_a_b,Zs: list_l8778361445553822004od_a_b] :
      ( ( ( append7557446516274476873od_a_b @ Ys @ Xs2 )
        = ( append7557446516274476873od_a_b @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_142_append__same__eq,axiom,
    ! [Ys: list_b,Xs2: list_b,Zs: list_b] :
      ( ( ( append_b @ Ys @ Xs2 )
        = ( append_b @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_143_append__same__eq,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Ys @ Xs2 )
        = ( append5335208823350062147od_a_b @ Zs @ Xs2 ) )
      = ( Ys = Zs ) ) ).

% append_same_eq
thf(fact_144_same__append__eq,axiom,
    ! [Xs2: list_a,Ys: list_a,Zs: list_a] :
      ( ( ( append_a @ Xs2 @ Ys )
        = ( append_a @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_145_same__append__eq,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,Ys: list_P3463435030753280173od_a_b,Zs: list_P3463435030753280173od_a_b] :
      ( ( ( append3447563168971123522od_a_b @ Xs2 @ Ys )
        = ( append3447563168971123522od_a_b @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_146_same__append__eq,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b,Zs: list_l8778361445553822004od_a_b] :
      ( ( ( append7557446516274476873od_a_b @ Xs2 @ Ys )
        = ( append7557446516274476873od_a_b @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_147_same__append__eq,axiom,
    ! [Xs2: list_b,Ys: list_b,Zs: list_b] :
      ( ( ( append_b @ Xs2 @ Ys )
        = ( append_b @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_148_same__append__eq,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ Ys )
        = ( append5335208823350062147od_a_b @ Xs2 @ Zs ) )
      = ( Ys = Zs ) ) ).

% same_append_eq
thf(fact_149_insert__image,axiom,
    ! [X: a,A: set_a,F: a > a] :
      ( ( member_a @ X @ A )
     => ( ( insert_a2 @ ( F @ X ) @ ( image_a_a2 @ F @ A ) )
        = ( image_a_a2 @ F @ A ) ) ) ).

% insert_image
thf(fact_150_insert__image,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,F: product_prod_a_b > a] :
      ( ( member1426531481828664017od_a_b @ X @ A )
     => ( ( insert_a2 @ ( F @ X ) @ ( image_2802296252294471259_a_b_a @ F @ A ) )
        = ( image_2802296252294471259_a_b_a @ F @ A ) ) ) ).

% insert_image
thf(fact_151_insert__image,axiom,
    ! [X: a,A: set_a,F: a > b] :
      ( ( member_a @ X @ A )
     => ( ( insert_b @ ( F @ X ) @ ( image_a_b2 @ F @ A ) )
        = ( image_a_b2 @ F @ A ) ) ) ).

% insert_image
thf(fact_152_insert__image,axiom,
    ! [X: b,A: set_b,F: b > a] :
      ( ( member_b @ X @ A )
     => ( ( insert_a2 @ ( F @ X ) @ ( image_b_a @ F @ A ) )
        = ( image_b_a @ F @ A ) ) ) ).

% insert_image
thf(fact_153_insert__image,axiom,
    ! [X: b,A: set_b,F: b > b] :
      ( ( member_b @ X @ A )
     => ( ( insert_b @ ( F @ X ) @ ( image_b_b @ F @ A ) )
        = ( image_b_b @ F @ A ) ) ) ).

% insert_image
thf(fact_154_insert__image,axiom,
    ! [X: a,A: set_a,F: a > set_a] :
      ( ( member_a @ X @ A )
     => ( ( insert_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) )
        = ( image_a_set_a @ F @ A ) ) ) ).

% insert_image
thf(fact_155_insert__image,axiom,
    ! [X: list_a,A: set_list_a,F: list_a > a] :
      ( ( member_list_a @ X @ A )
     => ( ( insert_a2 @ ( F @ X ) @ ( image_list_a_a @ F @ A ) )
        = ( image_list_a_a @ F @ A ) ) ) ).

% insert_image
thf(fact_156_insert__image,axiom,
    ! [X: list_a,A: set_list_a,F: list_a > b] :
      ( ( member_list_a @ X @ A )
     => ( ( insert_b @ ( F @ X ) @ ( image_list_a_b @ F @ A ) )
        = ( image_list_a_b @ F @ A ) ) ) ).

% insert_image
thf(fact_157_insert__image,axiom,
    ! [X: set_a,A: set_set_a,F: set_a > a] :
      ( ( member_set_a @ X @ A )
     => ( ( insert_a2 @ ( F @ X ) @ ( image_set_a_a @ F @ A ) )
        = ( image_set_a_a @ F @ A ) ) ) ).

% insert_image
thf(fact_158_insert__image,axiom,
    ! [X: set_a,A: set_set_a,F: set_a > b] :
      ( ( member_set_a @ X @ A )
     => ( ( insert_b @ ( F @ X ) @ ( image_set_a_b @ F @ A ) )
        = ( image_set_a_b @ F @ A ) ) ) ).

% insert_image
thf(fact_159_image__insert,axiom,
    ! [F: a > a,A2: a,B4: set_a] :
      ( ( image_a_a2 @ F @ ( insert_a2 @ A2 @ B4 ) )
      = ( insert_a2 @ ( F @ A2 ) @ ( image_a_a2 @ F @ B4 ) ) ) ).

% image_insert
thf(fact_160_image__insert,axiom,
    ! [F: product_prod_a_b > a,A2: product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( image_2802296252294471259_a_b_a @ F @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) )
      = ( insert_a2 @ ( F @ A2 ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ).

% image_insert
thf(fact_161_image__insert,axiom,
    ! [F: a > b,A2: a,B4: set_a] :
      ( ( image_a_b2 @ F @ ( insert_a2 @ A2 @ B4 ) )
      = ( insert_b @ ( F @ A2 ) @ ( image_a_b2 @ F @ B4 ) ) ) ).

% image_insert
thf(fact_162_image__insert,axiom,
    ! [F: b > a,A2: b,B4: set_b] :
      ( ( image_b_a @ F @ ( insert_b @ A2 @ B4 ) )
      = ( insert_a2 @ ( F @ A2 ) @ ( image_b_a @ F @ B4 ) ) ) ).

% image_insert
thf(fact_163_image__insert,axiom,
    ! [F: b > b,A2: b,B4: set_b] :
      ( ( image_b_b @ F @ ( insert_b @ A2 @ B4 ) )
      = ( insert_b @ ( F @ A2 ) @ ( image_b_b @ F @ B4 ) ) ) ).

% image_insert
thf(fact_164_image__insert,axiom,
    ! [F: a > set_a,A2: a,B4: set_a] :
      ( ( image_a_set_a @ F @ ( insert_a2 @ A2 @ B4 ) )
      = ( insert_set_a @ ( F @ A2 ) @ ( image_a_set_a @ F @ B4 ) ) ) ).

% image_insert
thf(fact_165_image__insert,axiom,
    ! [F: set_a > a,A2: set_a,B4: set_set_a] :
      ( ( image_set_a_a @ F @ ( insert_set_a @ A2 @ B4 ) )
      = ( insert_a2 @ ( F @ A2 ) @ ( image_set_a_a @ F @ B4 ) ) ) ).

% image_insert
thf(fact_166_image__insert,axiom,
    ! [F: set_a > b,A2: set_a,B4: set_set_a] :
      ( ( image_set_a_b @ F @ ( insert_set_a @ A2 @ B4 ) )
      = ( insert_b @ ( F @ A2 ) @ ( image_set_a_b @ F @ B4 ) ) ) ).

% image_insert
thf(fact_167_image__insert,axiom,
    ! [F: b > set_a,A2: b,B4: set_b] :
      ( ( image_b_set_a @ F @ ( insert_b @ A2 @ B4 ) )
      = ( insert_set_a @ ( F @ A2 ) @ ( image_b_set_a @ F @ B4 ) ) ) ).

% image_insert
thf(fact_168_image__insert,axiom,
    ! [F: a > product_prod_a_b,A2: a,B4: set_a] :
      ( ( image_7400625786893224495od_a_b @ F @ ( insert_a2 @ A2 @ B4 ) )
      = ( insert4534936386344385144od_a_b @ ( F @ A2 ) @ ( image_7400625786893224495od_a_b @ F @ B4 ) ) ) ).

% image_insert
thf(fact_169_list_Osimps_I15_J,axiom,
    ! [X21: set_Product_prod_a_b,X222: list_s6539019640784088206od_a_b] :
      ( ( set_se8479788141692617885od_a_b @ ( cons_s5806323297083630014od_a_b @ X21 @ X222 ) )
      = ( insert985587154976485848od_a_b @ X21 @ ( set_se8479788141692617885od_a_b @ X222 ) ) ) ).

% list.simps(15)
thf(fact_170_list_Osimps_I15_J,axiom,
    ! [X21: set_a,X222: list_set_a] :
      ( ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) )
      = ( insert_set_a @ X21 @ ( set_set_a2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_171_list_Osimps_I15_J,axiom,
    ! [X21: produc4115131626046251985od_a_b,X222: list_P8504624544267313367od_a_b] :
      ( ( set_Pr7079953345493216102od_a_b @ ( cons_P135045198529856391od_a_b @ X21 @ X222 ) )
      = ( insert6030560416357929377od_a_b @ X21 @ ( set_Pr7079953345493216102od_a_b @ X222 ) ) ) ).

% list.simps(15)
thf(fact_172_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_a_a,X222: list_P1396940483166286381od_a_a] :
      ( ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) )
      = ( insert4534936382041156343od_a_a @ X21 @ ( set_Product_prod_a_a2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_173_list_Osimps_I15_J,axiom,
    ! [X21: list_P1467974523212632366od_a_b,X222: list_l8778361445553822004od_a_b] :
      ( ( set_li5045283583078295363od_a_b @ ( cons_l907386227682280420od_a_b @ X21 @ X222 ) )
      = ( insert6792547294658701054od_a_b @ X21 @ ( set_li5045283583078295363od_a_b @ X222 ) ) ) ).

% list.simps(15)
thf(fact_174_list_Osimps_I15_J,axiom,
    ! [X21: produc2233330766403592615od_a_b,X222: list_P3463435030753280173od_a_b] :
      ( ( set_Pr5228040199190236220od_a_b @ ( cons_P8222935586017696605od_a_b @ X21 @ X222 ) )
      = ( insert7668604973114392439od_a_b @ X21 @ ( set_Pr5228040199190236220od_a_b @ X222 ) ) ) ).

% list.simps(15)
thf(fact_175_list_Osimps_I15_J,axiom,
    ! [X21: b,X222: list_b] :
      ( ( set_b2 @ ( cons_b @ X21 @ X222 ) )
      = ( insert_b @ X21 @ ( set_b2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_176_list_Osimps_I15_J,axiom,
    ! [X21: a,X222: list_a] :
      ( ( set_a2 @ ( cons_a @ X21 @ X222 ) )
      = ( insert_a2 @ X21 @ ( set_a2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_177_list_Osimps_I15_J,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) )
      = ( insert4534936386344385144od_a_b @ X21 @ ( set_Product_prod_a_b2 @ X222 ) ) ) ).

% list.simps(15)
thf(fact_178_insertE,axiom,
    ! [A2: product_prod_a_a,B: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ A ) )
     => ( ( A2 != B )
       => ( member1426531477525435216od_a_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_179_insertE,axiom,
    ! [A2: list_P1467974523212632366od_a_b,B: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b] :
      ( ( member6895035109809442519od_a_b @ A2 @ ( insert6792547294658701054od_a_b @ B @ A ) )
     => ( ( A2 != B )
       => ( member6895035109809442519od_a_b @ A2 @ A ) ) ) ).

% insertE
thf(fact_180_insertE,axiom,
    ! [A2: list_a,B: list_a,A: set_list_a] :
      ( ( member_list_a @ A2 @ ( insert_list_a @ B @ A ) )
     => ( ( A2 != B )
       => ( member_list_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_181_insertE,axiom,
    ! [A2: set_Product_prod_a_b,B: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( member1887650552762594865od_a_b @ A2 @ ( insert985587154976485848od_a_b @ B @ A ) )
     => ( ( A2 != B )
       => ( member1887650552762594865od_a_b @ A2 @ A ) ) ) ).

% insertE
thf(fact_182_insertE,axiom,
    ! [A2: set_a,B: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ ( insert_set_a @ B @ A ) )
     => ( ( A2 != B )
       => ( member_set_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_183_insertE,axiom,
    ! [A2: b,B: b,A: set_b] :
      ( ( member_b @ A2 @ ( insert_b @ B @ A ) )
     => ( ( A2 != B )
       => ( member_b @ A2 @ A ) ) ) ).

% insertE
thf(fact_184_insertE,axiom,
    ! [A2: a,B: a,A: set_a] :
      ( ( member_a @ A2 @ ( insert_a2 @ B @ A ) )
     => ( ( A2 != B )
       => ( member_a @ A2 @ A ) ) ) ).

% insertE
thf(fact_185_insertE,axiom,
    ! [A2: product_prod_a_b,B: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ ( insert4534936386344385144od_a_b @ B @ A ) )
     => ( ( A2 != B )
       => ( member1426531481828664017od_a_b @ A2 @ A ) ) ) ).

% insertE
thf(fact_186_insertI1,axiom,
    ! [A2: product_prod_a_a,B4: set_Product_prod_a_a] : ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ A2 @ B4 ) ) ).

% insertI1
thf(fact_187_insertI1,axiom,
    ! [A2: list_P1467974523212632366od_a_b,B4: set_li6306622547289818894od_a_b] : ( member6895035109809442519od_a_b @ A2 @ ( insert6792547294658701054od_a_b @ A2 @ B4 ) ) ).

% insertI1
thf(fact_188_insertI1,axiom,
    ! [A2: list_a,B4: set_list_a] : ( member_list_a @ A2 @ ( insert_list_a @ A2 @ B4 ) ) ).

% insertI1
thf(fact_189_insertI1,axiom,
    ! [A2: set_Product_prod_a_b,B4: set_se3214616458824132456od_a_b] : ( member1887650552762594865od_a_b @ A2 @ ( insert985587154976485848od_a_b @ A2 @ B4 ) ) ).

% insertI1
thf(fact_190_insertI1,axiom,
    ! [A2: set_a,B4: set_set_a] : ( member_set_a @ A2 @ ( insert_set_a @ A2 @ B4 ) ) ).

% insertI1
thf(fact_191_insertI1,axiom,
    ! [A2: b,B4: set_b] : ( member_b @ A2 @ ( insert_b @ A2 @ B4 ) ) ).

% insertI1
thf(fact_192_insertI1,axiom,
    ! [A2: a,B4: set_a] : ( member_a @ A2 @ ( insert_a2 @ A2 @ B4 ) ) ).

% insertI1
thf(fact_193_insertI1,axiom,
    ! [A2: product_prod_a_b,B4: set_Product_prod_a_b] : ( member1426531481828664017od_a_b @ A2 @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) ) ).

% insertI1
thf(fact_194_insertI2,axiom,
    ! [A2: product_prod_a_a,B4: set_Product_prod_a_a,B: product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ B4 )
     => ( member1426531477525435216od_a_a @ A2 @ ( insert4534936382041156343od_a_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_195_insertI2,axiom,
    ! [A2: list_P1467974523212632366od_a_b,B4: set_li6306622547289818894od_a_b,B: list_P1467974523212632366od_a_b] :
      ( ( member6895035109809442519od_a_b @ A2 @ B4 )
     => ( member6895035109809442519od_a_b @ A2 @ ( insert6792547294658701054od_a_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_196_insertI2,axiom,
    ! [A2: list_a,B4: set_list_a,B: list_a] :
      ( ( member_list_a @ A2 @ B4 )
     => ( member_list_a @ A2 @ ( insert_list_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_197_insertI2,axiom,
    ! [A2: set_Product_prod_a_b,B4: set_se3214616458824132456od_a_b,B: set_Product_prod_a_b] :
      ( ( member1887650552762594865od_a_b @ A2 @ B4 )
     => ( member1887650552762594865od_a_b @ A2 @ ( insert985587154976485848od_a_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_198_insertI2,axiom,
    ! [A2: set_a,B4: set_set_a,B: set_a] :
      ( ( member_set_a @ A2 @ B4 )
     => ( member_set_a @ A2 @ ( insert_set_a @ B @ B4 ) ) ) ).

% insertI2
thf(fact_199_insertI2,axiom,
    ! [A2: b,B4: set_b,B: b] :
      ( ( member_b @ A2 @ B4 )
     => ( member_b @ A2 @ ( insert_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_200_insertI2,axiom,
    ! [A2: a,B4: set_a,B: a] :
      ( ( member_a @ A2 @ B4 )
     => ( member_a @ A2 @ ( insert_a2 @ B @ B4 ) ) ) ).

% insertI2
thf(fact_201_insertI2,axiom,
    ! [A2: product_prod_a_b,B4: set_Product_prod_a_b,B: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ B4 )
     => ( member1426531481828664017od_a_b @ A2 @ ( insert4534936386344385144od_a_b @ B @ B4 ) ) ) ).

% insertI2
thf(fact_202_Set_Oset__insert,axiom,
    ! [X: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ A )
     => ~ ! [B5: set_Product_prod_a_a] :
            ( ( A
              = ( insert4534936382041156343od_a_a @ X @ B5 ) )
           => ( member1426531477525435216od_a_a @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_203_Set_Oset__insert,axiom,
    ! [X: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b] :
      ( ( member6895035109809442519od_a_b @ X @ A )
     => ~ ! [B5: set_li6306622547289818894od_a_b] :
            ( ( A
              = ( insert6792547294658701054od_a_b @ X @ B5 ) )
           => ( member6895035109809442519od_a_b @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_204_Set_Oset__insert,axiom,
    ! [X: list_a,A: set_list_a] :
      ( ( member_list_a @ X @ A )
     => ~ ! [B5: set_list_a] :
            ( ( A
              = ( insert_list_a @ X @ B5 ) )
           => ( member_list_a @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_205_Set_Oset__insert,axiom,
    ! [X: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( member1887650552762594865od_a_b @ X @ A )
     => ~ ! [B5: set_se3214616458824132456od_a_b] :
            ( ( A
              = ( insert985587154976485848od_a_b @ X @ B5 ) )
           => ( member1887650552762594865od_a_b @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_206_Set_Oset__insert,axiom,
    ! [X: set_a,A: set_set_a] :
      ( ( member_set_a @ X @ A )
     => ~ ! [B5: set_set_a] :
            ( ( A
              = ( insert_set_a @ X @ B5 ) )
           => ( member_set_a @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_207_Set_Oset__insert,axiom,
    ! [X: b,A: set_b] :
      ( ( member_b @ X @ A )
     => ~ ! [B5: set_b] :
            ( ( A
              = ( insert_b @ X @ B5 ) )
           => ( member_b @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_208_Set_Oset__insert,axiom,
    ! [X: a,A: set_a] :
      ( ( member_a @ X @ A )
     => ~ ! [B5: set_a] :
            ( ( A
              = ( insert_a2 @ X @ B5 ) )
           => ( member_a @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_209_Set_Oset__insert,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ A )
     => ~ ! [B5: set_Product_prod_a_b] :
            ( ( A
              = ( insert4534936386344385144od_a_b @ X @ B5 ) )
           => ( member1426531481828664017od_a_b @ X @ B5 ) ) ) ).

% Set.set_insert
thf(fact_210_insert__ident,axiom,
    ! [X: product_prod_a_a,A: set_Product_prod_a_a,B4: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ X @ A )
     => ( ~ ( member1426531477525435216od_a_a @ X @ B4 )
       => ( ( ( insert4534936382041156343od_a_a @ X @ A )
            = ( insert4534936382041156343od_a_a @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_211_insert__ident,axiom,
    ! [X: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b,B4: set_li6306622547289818894od_a_b] :
      ( ~ ( member6895035109809442519od_a_b @ X @ A )
     => ( ~ ( member6895035109809442519od_a_b @ X @ B4 )
       => ( ( ( insert6792547294658701054od_a_b @ X @ A )
            = ( insert6792547294658701054od_a_b @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_212_insert__ident,axiom,
    ! [X: list_a,A: set_list_a,B4: set_list_a] :
      ( ~ ( member_list_a @ X @ A )
     => ( ~ ( member_list_a @ X @ B4 )
       => ( ( ( insert_list_a @ X @ A )
            = ( insert_list_a @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_213_insert__ident,axiom,
    ! [X: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b,B4: set_se3214616458824132456od_a_b] :
      ( ~ ( member1887650552762594865od_a_b @ X @ A )
     => ( ~ ( member1887650552762594865od_a_b @ X @ B4 )
       => ( ( ( insert985587154976485848od_a_b @ X @ A )
            = ( insert985587154976485848od_a_b @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_214_insert__ident,axiom,
    ! [X: set_a,A: set_set_a,B4: set_set_a] :
      ( ~ ( member_set_a @ X @ A )
     => ( ~ ( member_set_a @ X @ B4 )
       => ( ( ( insert_set_a @ X @ A )
            = ( insert_set_a @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_215_insert__ident,axiom,
    ! [X: b,A: set_b,B4: set_b] :
      ( ~ ( member_b @ X @ A )
     => ( ~ ( member_b @ X @ B4 )
       => ( ( ( insert_b @ X @ A )
            = ( insert_b @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_216_insert__ident,axiom,
    ! [X: a,A: set_a,B4: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ~ ( member_a @ X @ B4 )
       => ( ( ( insert_a2 @ X @ A )
            = ( insert_a2 @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_217_insert__ident,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ A )
     => ( ~ ( member1426531481828664017od_a_b @ X @ B4 )
       => ( ( ( insert4534936386344385144od_a_b @ X @ A )
            = ( insert4534936386344385144od_a_b @ X @ B4 ) )
          = ( A = B4 ) ) ) ) ).

% insert_ident
thf(fact_218_insert__absorb,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ A )
     => ( ( insert4534936382041156343od_a_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_219_insert__absorb,axiom,
    ! [A2: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b] :
      ( ( member6895035109809442519od_a_b @ A2 @ A )
     => ( ( insert6792547294658701054od_a_b @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_220_insert__absorb,axiom,
    ! [A2: list_a,A: set_list_a] :
      ( ( member_list_a @ A2 @ A )
     => ( ( insert_list_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_221_insert__absorb,axiom,
    ! [A2: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( member1887650552762594865od_a_b @ A2 @ A )
     => ( ( insert985587154976485848od_a_b @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_222_insert__absorb,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ A )
     => ( ( insert_set_a @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_223_insert__absorb,axiom,
    ! [A2: b,A: set_b] :
      ( ( member_b @ A2 @ A )
     => ( ( insert_b @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_224_insert__absorb,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( insert_a2 @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_225_insert__absorb,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ( insert4534936386344385144od_a_b @ A2 @ A )
        = A ) ) ).

% insert_absorb
thf(fact_226_insert__eq__iff,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a,B: product_prod_a_a,B4: set_Product_prod_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ A2 @ A )
     => ( ~ ( member1426531477525435216od_a_a @ B @ B4 )
       => ( ( ( insert4534936382041156343od_a_a @ A2 @ A )
            = ( insert4534936382041156343od_a_a @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_Product_prod_a_a] :
                  ( ( A
                    = ( insert4534936382041156343od_a_a @ B @ C2 ) )
                  & ~ ( member1426531477525435216od_a_a @ B @ C2 )
                  & ( B4
                    = ( insert4534936382041156343od_a_a @ A2 @ C2 ) )
                  & ~ ( member1426531477525435216od_a_a @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_227_insert__eq__iff,axiom,
    ! [A2: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b,B: list_P1467974523212632366od_a_b,B4: set_li6306622547289818894od_a_b] :
      ( ~ ( member6895035109809442519od_a_b @ A2 @ A )
     => ( ~ ( member6895035109809442519od_a_b @ B @ B4 )
       => ( ( ( insert6792547294658701054od_a_b @ A2 @ A )
            = ( insert6792547294658701054od_a_b @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_li6306622547289818894od_a_b] :
                  ( ( A
                    = ( insert6792547294658701054od_a_b @ B @ C2 ) )
                  & ~ ( member6895035109809442519od_a_b @ B @ C2 )
                  & ( B4
                    = ( insert6792547294658701054od_a_b @ A2 @ C2 ) )
                  & ~ ( member6895035109809442519od_a_b @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_228_insert__eq__iff,axiom,
    ! [A2: list_a,A: set_list_a,B: list_a,B4: set_list_a] :
      ( ~ ( member_list_a @ A2 @ A )
     => ( ~ ( member_list_a @ B @ B4 )
       => ( ( ( insert_list_a @ A2 @ A )
            = ( insert_list_a @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_list_a] :
                  ( ( A
                    = ( insert_list_a @ B @ C2 ) )
                  & ~ ( member_list_a @ B @ C2 )
                  & ( B4
                    = ( insert_list_a @ A2 @ C2 ) )
                  & ~ ( member_list_a @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_229_insert__eq__iff,axiom,
    ! [A2: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b,B: set_Product_prod_a_b,B4: set_se3214616458824132456od_a_b] :
      ( ~ ( member1887650552762594865od_a_b @ A2 @ A )
     => ( ~ ( member1887650552762594865od_a_b @ B @ B4 )
       => ( ( ( insert985587154976485848od_a_b @ A2 @ A )
            = ( insert985587154976485848od_a_b @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_se3214616458824132456od_a_b] :
                  ( ( A
                    = ( insert985587154976485848od_a_b @ B @ C2 ) )
                  & ~ ( member1887650552762594865od_a_b @ B @ C2 )
                  & ( B4
                    = ( insert985587154976485848od_a_b @ A2 @ C2 ) )
                  & ~ ( member1887650552762594865od_a_b @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_230_insert__eq__iff,axiom,
    ! [A2: set_a,A: set_set_a,B: set_a,B4: set_set_a] :
      ( ~ ( member_set_a @ A2 @ A )
     => ( ~ ( member_set_a @ B @ B4 )
       => ( ( ( insert_set_a @ A2 @ A )
            = ( insert_set_a @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_set_a] :
                  ( ( A
                    = ( insert_set_a @ B @ C2 ) )
                  & ~ ( member_set_a @ B @ C2 )
                  & ( B4
                    = ( insert_set_a @ A2 @ C2 ) )
                  & ~ ( member_set_a @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_231_insert__eq__iff,axiom,
    ! [A2: b,A: set_b,B: b,B4: set_b] :
      ( ~ ( member_b @ A2 @ A )
     => ( ~ ( member_b @ B @ B4 )
       => ( ( ( insert_b @ A2 @ A )
            = ( insert_b @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_b] :
                  ( ( A
                    = ( insert_b @ B @ C2 ) )
                  & ~ ( member_b @ B @ C2 )
                  & ( B4
                    = ( insert_b @ A2 @ C2 ) )
                  & ~ ( member_b @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_232_insert__eq__iff,axiom,
    ! [A2: a,A: set_a,B: a,B4: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ~ ( member_a @ B @ B4 )
       => ( ( ( insert_a2 @ A2 @ A )
            = ( insert_a2 @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_a] :
                  ( ( A
                    = ( insert_a2 @ B @ C2 ) )
                  & ~ ( member_a @ B @ C2 )
                  & ( B4
                    = ( insert_a2 @ A2 @ C2 ) )
                  & ~ ( member_a @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_233_insert__eq__iff,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B: product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ~ ( member1426531481828664017od_a_b @ B @ B4 )
       => ( ( ( insert4534936386344385144od_a_b @ A2 @ A )
            = ( insert4534936386344385144od_a_b @ B @ B4 ) )
          = ( ( ( A2 = B )
             => ( A = B4 ) )
            & ( ( A2 != B )
             => ? [C2: set_Product_prod_a_b] :
                  ( ( A
                    = ( insert4534936386344385144od_a_b @ B @ C2 ) )
                  & ~ ( member1426531481828664017od_a_b @ B @ C2 )
                  & ( B4
                    = ( insert4534936386344385144od_a_b @ A2 @ C2 ) )
                  & ~ ( member1426531481828664017od_a_b @ A2 @ C2 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_234_insert__commute,axiom,
    ! [X: product_prod_a_a,Y: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( insert4534936382041156343od_a_a @ X @ ( insert4534936382041156343od_a_a @ Y @ A ) )
      = ( insert4534936382041156343od_a_a @ Y @ ( insert4534936382041156343od_a_a @ X @ A ) ) ) ).

% insert_commute
thf(fact_235_insert__commute,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b] :
      ( ( insert6792547294658701054od_a_b @ X @ ( insert6792547294658701054od_a_b @ Y @ A ) )
      = ( insert6792547294658701054od_a_b @ Y @ ( insert6792547294658701054od_a_b @ X @ A ) ) ) ).

% insert_commute
thf(fact_236_insert__commute,axiom,
    ! [X: set_Product_prod_a_b,Y: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( insert985587154976485848od_a_b @ X @ ( insert985587154976485848od_a_b @ Y @ A ) )
      = ( insert985587154976485848od_a_b @ Y @ ( insert985587154976485848od_a_b @ X @ A ) ) ) ).

% insert_commute
thf(fact_237_insert__commute,axiom,
    ! [X: set_a,Y: set_a,A: set_set_a] :
      ( ( insert_set_a @ X @ ( insert_set_a @ Y @ A ) )
      = ( insert_set_a @ Y @ ( insert_set_a @ X @ A ) ) ) ).

% insert_commute
thf(fact_238_insert__commute,axiom,
    ! [X: b,Y: b,A: set_b] :
      ( ( insert_b @ X @ ( insert_b @ Y @ A ) )
      = ( insert_b @ Y @ ( insert_b @ X @ A ) ) ) ).

% insert_commute
thf(fact_239_insert__commute,axiom,
    ! [X: a,Y: a,A: set_a] :
      ( ( insert_a2 @ X @ ( insert_a2 @ Y @ A ) )
      = ( insert_a2 @ Y @ ( insert_a2 @ X @ A ) ) ) ).

% insert_commute
thf(fact_240_append__eq__appendI,axiom,
    ! [Xs2: list_a,Xs1: list_a,Zs: list_a,Ys: list_a,Us: list_a] :
      ( ( ( append_a @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_a @ Xs1 @ Us ) )
       => ( ( append_a @ Xs2 @ Ys )
          = ( append_a @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_241_append__eq__appendI,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,Xs1: list_P3463435030753280173od_a_b,Zs: list_P3463435030753280173od_a_b,Ys: list_P3463435030753280173od_a_b,Us: list_P3463435030753280173od_a_b] :
      ( ( ( append3447563168971123522od_a_b @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append3447563168971123522od_a_b @ Xs1 @ Us ) )
       => ( ( append3447563168971123522od_a_b @ Xs2 @ Ys )
          = ( append3447563168971123522od_a_b @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_242_append__eq__appendI,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,Xs1: list_l8778361445553822004od_a_b,Zs: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b,Us: list_l8778361445553822004od_a_b] :
      ( ( ( append7557446516274476873od_a_b @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append7557446516274476873od_a_b @ Xs1 @ Us ) )
       => ( ( append7557446516274476873od_a_b @ Xs2 @ Ys )
          = ( append7557446516274476873od_a_b @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_243_append__eq__appendI,axiom,
    ! [Xs2: list_b,Xs1: list_b,Zs: list_b,Ys: list_b,Us: list_b] :
      ( ( ( append_b @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append_b @ Xs1 @ Us ) )
       => ( ( append_b @ Xs2 @ Ys )
          = ( append_b @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_244_append__eq__appendI,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Xs1: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Us: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ Xs1 )
        = Zs )
     => ( ( Ys
          = ( append5335208823350062147od_a_b @ Xs1 @ Us ) )
       => ( ( append5335208823350062147od_a_b @ Xs2 @ Ys )
          = ( append5335208823350062147od_a_b @ Zs @ Us ) ) ) ) ).

% append_eq_appendI
thf(fact_245_mk__disjoint__insert,axiom,
    ! [A2: product_prod_a_a,A: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ A2 @ A )
     => ? [B5: set_Product_prod_a_a] :
          ( ( A
            = ( insert4534936382041156343od_a_a @ A2 @ B5 ) )
          & ~ ( member1426531477525435216od_a_a @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_246_mk__disjoint__insert,axiom,
    ! [A2: list_P1467974523212632366od_a_b,A: set_li6306622547289818894od_a_b] :
      ( ( member6895035109809442519od_a_b @ A2 @ A )
     => ? [B5: set_li6306622547289818894od_a_b] :
          ( ( A
            = ( insert6792547294658701054od_a_b @ A2 @ B5 ) )
          & ~ ( member6895035109809442519od_a_b @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_247_mk__disjoint__insert,axiom,
    ! [A2: list_a,A: set_list_a] :
      ( ( member_list_a @ A2 @ A )
     => ? [B5: set_list_a] :
          ( ( A
            = ( insert_list_a @ A2 @ B5 ) )
          & ~ ( member_list_a @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_248_mk__disjoint__insert,axiom,
    ! [A2: set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( member1887650552762594865od_a_b @ A2 @ A )
     => ? [B5: set_se3214616458824132456od_a_b] :
          ( ( A
            = ( insert985587154976485848od_a_b @ A2 @ B5 ) )
          & ~ ( member1887650552762594865od_a_b @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_249_mk__disjoint__insert,axiom,
    ! [A2: set_a,A: set_set_a] :
      ( ( member_set_a @ A2 @ A )
     => ? [B5: set_set_a] :
          ( ( A
            = ( insert_set_a @ A2 @ B5 ) )
          & ~ ( member_set_a @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_250_mk__disjoint__insert,axiom,
    ! [A2: b,A: set_b] :
      ( ( member_b @ A2 @ A )
     => ? [B5: set_b] :
          ( ( A
            = ( insert_b @ A2 @ B5 ) )
          & ~ ( member_b @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_251_mk__disjoint__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ? [B5: set_a] :
          ( ( A
            = ( insert_a2 @ A2 @ B5 ) )
          & ~ ( member_a @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_252_mk__disjoint__insert,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ A )
     => ? [B5: set_Product_prod_a_b] :
          ( ( A
            = ( insert4534936386344385144od_a_b @ A2 @ B5 ) )
          & ~ ( member1426531481828664017od_a_b @ A2 @ B5 ) ) ) ).

% mk_disjoint_insert
thf(fact_253_append__eq__append__conv2,axiom,
    ! [Xs2: list_a,Ys: list_a,Zs: list_a,Ts: list_a] :
      ( ( ( append_a @ Xs2 @ Ys )
        = ( append_a @ Zs @ Ts ) )
      = ( ? [Us2: list_a] :
            ( ( ( Xs2
                = ( append_a @ Zs @ Us2 ) )
              & ( ( append_a @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_a @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append_a @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_254_append__eq__append__conv2,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,Ys: list_P3463435030753280173od_a_b,Zs: list_P3463435030753280173od_a_b,Ts: list_P3463435030753280173od_a_b] :
      ( ( ( append3447563168971123522od_a_b @ Xs2 @ Ys )
        = ( append3447563168971123522od_a_b @ Zs @ Ts ) )
      = ( ? [Us2: list_P3463435030753280173od_a_b] :
            ( ( ( Xs2
                = ( append3447563168971123522od_a_b @ Zs @ Us2 ) )
              & ( ( append3447563168971123522od_a_b @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append3447563168971123522od_a_b @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append3447563168971123522od_a_b @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_255_append__eq__append__conv2,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b,Zs: list_l8778361445553822004od_a_b,Ts: list_l8778361445553822004od_a_b] :
      ( ( ( append7557446516274476873od_a_b @ Xs2 @ Ys )
        = ( append7557446516274476873od_a_b @ Zs @ Ts ) )
      = ( ? [Us2: list_l8778361445553822004od_a_b] :
            ( ( ( Xs2
                = ( append7557446516274476873od_a_b @ Zs @ Us2 ) )
              & ( ( append7557446516274476873od_a_b @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append7557446516274476873od_a_b @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append7557446516274476873od_a_b @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_256_append__eq__append__conv2,axiom,
    ! [Xs2: list_b,Ys: list_b,Zs: list_b,Ts: list_b] :
      ( ( ( append_b @ Xs2 @ Ys )
        = ( append_b @ Zs @ Ts ) )
      = ( ? [Us2: list_b] :
            ( ( ( Xs2
                = ( append_b @ Zs @ Us2 ) )
              & ( ( append_b @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append_b @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append_b @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_257_append__eq__append__conv2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,Ts: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ Ys )
        = ( append5335208823350062147od_a_b @ Zs @ Ts ) )
      = ( ? [Us2: list_P1467974523212632366od_a_b] :
            ( ( ( Xs2
                = ( append5335208823350062147od_a_b @ Zs @ Us2 ) )
              & ( ( append5335208823350062147od_a_b @ Us2 @ Ys )
                = Ts ) )
            | ( ( ( append5335208823350062147od_a_b @ Xs2 @ Us2 )
                = Zs )
              & ( Ys
                = ( append5335208823350062147od_a_b @ Us2 @ Ts ) ) ) ) ) ) ).

% append_eq_append_conv2
thf(fact_258_Cons__eq__appendI,axiom,
    ! [X: a,Xs1: list_a,Ys: list_a,Xs2: list_a,Zs: list_a] :
      ( ( ( cons_a @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append_a @ Xs1 @ Zs ) )
       => ( ( cons_a @ X @ Xs2 )
          = ( append_a @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_259_Cons__eq__appendI,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs1: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b,Xs2: list_l8778361445553822004od_a_b,Zs: list_l8778361445553822004od_a_b] :
      ( ( ( cons_l907386227682280420od_a_b @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append7557446516274476873od_a_b @ Xs1 @ Zs ) )
       => ( ( cons_l907386227682280420od_a_b @ X @ Xs2 )
          = ( append7557446516274476873od_a_b @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_260_Cons__eq__appendI,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs1: list_P3463435030753280173od_a_b,Ys: list_P3463435030753280173od_a_b,Xs2: list_P3463435030753280173od_a_b,Zs: list_P3463435030753280173od_a_b] :
      ( ( ( cons_P8222935586017696605od_a_b @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append3447563168971123522od_a_b @ Xs1 @ Zs ) )
       => ( ( cons_P8222935586017696605od_a_b @ X @ Xs2 )
          = ( append3447563168971123522od_a_b @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_261_Cons__eq__appendI,axiom,
    ! [X: b,Xs1: list_b,Ys: list_b,Xs2: list_b,Zs: list_b] :
      ( ( ( cons_b @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append_b @ Xs1 @ Zs ) )
       => ( ( cons_b @ X @ Xs2 )
          = ( append_b @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_262_Cons__eq__appendI,axiom,
    ! [X: product_prod_a_b,Xs1: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( cons_P7316939131009794654od_a_b @ X @ Xs1 )
        = Ys )
     => ( ( Xs2
          = ( append5335208823350062147od_a_b @ Xs1 @ Zs ) )
       => ( ( cons_P7316939131009794654od_a_b @ X @ Xs2 )
          = ( append5335208823350062147od_a_b @ Ys @ Zs ) ) ) ) ).

% Cons_eq_appendI
thf(fact_263_append__Cons,axiom,
    ! [X: a,Xs2: list_a,Ys: list_a] :
      ( ( append_a @ ( cons_a @ X @ Xs2 ) @ Ys )
      = ( cons_a @ X @ ( append_a @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_264_append__Cons,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b] :
      ( ( append7557446516274476873od_a_b @ ( cons_l907386227682280420od_a_b @ X @ Xs2 ) @ Ys )
      = ( cons_l907386227682280420od_a_b @ X @ ( append7557446516274476873od_a_b @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_265_append__Cons,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b,Ys: list_P3463435030753280173od_a_b] :
      ( ( append3447563168971123522od_a_b @ ( cons_P8222935586017696605od_a_b @ X @ Xs2 ) @ Ys )
      = ( cons_P8222935586017696605od_a_b @ X @ ( append3447563168971123522od_a_b @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_266_append__Cons,axiom,
    ! [X: b,Xs2: list_b,Ys: list_b] :
      ( ( append_b @ ( cons_b @ X @ Xs2 ) @ Ys )
      = ( cons_b @ X @ ( append_b @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_267_append__Cons,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( append5335208823350062147od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys )
      = ( cons_P7316939131009794654od_a_b @ X @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) ) ) ).

% append_Cons
thf(fact_268_dtail_Ocases,axiom,
    ! [X: produc1898233765819362879od_a_b] :
      ~ ! [R: product_prod_a_b,Xs: fset_P6924693328471822228od_a_b,Def: product_prod_a_b > product_prod_a_b] :
          ( X
         != ( produc4243744949930134775od_a_b @ ( node_P1936840572307073598od_a_b @ R @ Xs ) @ Def ) ) ).

% dtail.cases
thf(fact_269_dtail_Ocases,axiom,
    ! [X: produc3356957763246339775_a_b_a] :
      ~ ! [R: a,Xs: fset_P8459658061218866410od_a_b,Def: product_prod_a_b > a] :
          ( X
         != ( produc2560940082450600567_a_b_a @ ( node_a4529940054729032296od_a_b @ R @ Xs ) @ Def ) ) ).

% dtail.cases
thf(fact_270_dtail_Ocases,axiom,
    ! [X: produc6632176389122652351od_a_b] :
      ~ ! [R: product_prod_a_b,Xs: fset_P1695881499225648404_b_a_a,Def: a > product_prod_a_b] :
          ( X
         != ( produc6078967021198179959od_a_b @ ( node_P9154982556985054868_a_b_a @ R @ Xs ) @ Def ) ) ).

% dtail.cases
thf(fact_271_dtail_Ocases,axiom,
    ! [X: produc40638974386926911_a_a_a] :
      ~ ! [R: a,Xs: fset_P2559403164575059562_a_a_a,Def: a > a] :
          ( X
         != ( produc3102611097480951799_a_a_a @ ( node_a_a @ R @ Xs ) @ Def ) ) ).

% dtail.cases
thf(fact_272_dtail_Ocases,axiom,
    ! [X: produc1083523234014712191_b_b_a] :
      ~ ! [R: a,Xs: fset_P5281107635120001194_a_b_b,Def: b > a] :
          ( X
         != ( produc1993688775741047735_b_b_a @ ( node_a_b @ R @ Xs ) @ Def ) ) ).

% dtail.cases
thf(fact_273_is__subtree_Ocases,axiom,
    ! [X: produc4982129085209094099od_a_b] :
      ~ ! [X3: dtree_2253494508914376569od_a_b,R: product_prod_a_b,Xs: fset_P6924693328471822228od_a_b] :
          ( X
         != ( produc1922829451070556043od_a_b @ X3 @ ( node_P1936840572307073598od_a_b @ R @ Xs ) ) ) ).

% is_subtree.cases
thf(fact_274_is__subtree_Ocases,axiom,
    ! [X: produc3050359278047883047od_a_b] :
      ~ ! [X3: dtree_2020077848175485859od_a_b,R: a,Xs: fset_P8459658061218866410od_a_b] :
          ( X
         != ( produc2795810003198522847od_a_b @ X3 @ ( node_a4529940054729032296od_a_b @ R @ Xs ) ) ) ).

% is_subtree.cases
thf(fact_275_is__subtree_Ocases,axiom,
    ! [X: produc3458261246270610303_a_b_a] :
      ~ ! [X3: dtree_1072189271477137615_a_b_a,R: product_prod_a_b,Xs: fset_P1695881499225648404_b_a_a] :
          ( X
         != ( produc2960903658549983287_a_b_a @ X3 @ ( node_P9154982556985054868_a_b_a @ R @ Xs ) ) ) ).

% is_subtree.cases
thf(fact_276_is__subtree_Ocases,axiom,
    ! [X: produc8861388899196491475ee_a_a] :
      ~ ! [X3: dtree_a_a,R: a,Xs: fset_P2559403164575059562_a_a_a] :
          ( X
         != ( produc2336797772351871627ee_a_a @ X3 @ ( node_a_a @ R @ Xs ) ) ) ).

% is_subtree.cases
thf(fact_277_is__subtree_Ocases,axiom,
    ! [X: produc5177672665255943253ee_a_b] :
      ~ ! [X3: dtree_a_b,R: a,Xs: fset_P5281107635120001194_a_b_b] :
          ( X
         != ( produc7805419539522982029ee_a_b @ X3 @ ( node_a_b @ R @ Xs ) ) ) ).

% is_subtree.cases
thf(fact_278_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ( ? [X4: produc4115131626046251985od_a_b] :
            ( ( member3142156093895948538od_a_b @ X4 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P8504624544267313367od_a_b,X4: produc4115131626046251985od_a_b] :
            ( ? [Zs2: list_P8504624544267313367od_a_b] :
                ( Xs2
                = ( append4796217420789006444od_a_b @ Ys2 @ ( cons_P135045198529856391od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: produc4115131626046251985od_a_b] :
                ( ( member3142156093895948538od_a_b @ Y3 @ ( set_Pr7079953345493216102od_a_b @ Ys2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_279_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ( ? [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P1396940483166286381od_a_a,X4: product_prod_a_a] :
            ( ? [Zs2: list_P1396940483166286381od_a_a] :
                ( Xs2
                = ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Ys2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_280_split__list__first__prop__iff,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ( ? [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_a,X4: a] :
            ( ? [Zs2: list_a] :
                ( Xs2
                = ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: a] :
                ( ( member_a @ Y3 @ ( set_a2 @ Ys2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_281_split__list__first__prop__iff,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ( ? [X4: list_P1467974523212632366od_a_b] :
            ( ( member6895035109809442519od_a_b @ X4 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_l8778361445553822004od_a_b,X4: list_P1467974523212632366od_a_b] :
            ( ? [Zs2: list_l8778361445553822004od_a_b] :
                ( Xs2
                = ( append7557446516274476873od_a_b @ Ys2 @ ( cons_l907386227682280420od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: list_P1467974523212632366od_a_b] :
                ( ( member6895035109809442519od_a_b @ Y3 @ ( set_li5045283583078295363od_a_b @ Ys2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_282_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ( ? [X4: produc2233330766403592615od_a_b] :
            ( ( member5065438833300252112od_a_b @ X4 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P3463435030753280173od_a_b,X4: produc2233330766403592615od_a_b] :
            ( ? [Zs2: list_P3463435030753280173od_a_b] :
                ( Xs2
                = ( append3447563168971123522od_a_b @ Ys2 @ ( cons_P8222935586017696605od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: produc2233330766403592615od_a_b] :
                ( ( member5065438833300252112od_a_b @ Y3 @ ( set_Pr5228040199190236220od_a_b @ Ys2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_283_split__list__first__prop__iff,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ( ? [X4: b] :
            ( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_b,X4: b] :
            ( ? [Zs2: list_b] :
                ( Xs2
                = ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: b] :
                ( ( member_b @ Y3 @ ( set_b2 @ Ys2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_284_split__list__first__prop__iff,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ( ? [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P1467974523212632366od_a_b,X4: product_prod_a_b] :
            ( ? [Zs2: list_P1467974523212632366od_a_b] :
                ( Xs2
                = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: product_prod_a_b] :
                ( ( member1426531481828664017od_a_b @ Y3 @ ( set_Product_prod_a_b2 @ Ys2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_first_prop_iff
thf(fact_285_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ( ? [X4: produc4115131626046251985od_a_b] :
            ( ( member3142156093895948538od_a_b @ X4 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P8504624544267313367od_a_b,X4: produc4115131626046251985od_a_b,Zs2: list_P8504624544267313367od_a_b] :
            ( ( Xs2
              = ( append4796217420789006444od_a_b @ Ys2 @ ( cons_P135045198529856391od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: produc4115131626046251985od_a_b] :
                ( ( member3142156093895948538od_a_b @ Y3 @ ( set_Pr7079953345493216102od_a_b @ Zs2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_286_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ( ? [X4: product_prod_a_a] :
            ( ( member1426531477525435216od_a_a @ X4 @ ( set_Product_prod_a_a2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P1396940483166286381od_a_a,X4: product_prod_a_a,Zs2: list_P1396940483166286381od_a_a] :
            ( ( Xs2
              = ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: product_prod_a_a] :
                ( ( member1426531477525435216od_a_a @ Y3 @ ( set_Product_prod_a_a2 @ Zs2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_287_split__list__last__prop__iff,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ( ? [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_a,X4: a,Zs2: list_a] :
            ( ( Xs2
              = ( append_a @ Ys2 @ ( cons_a @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: a] :
                ( ( member_a @ Y3 @ ( set_a2 @ Zs2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_288_split__list__last__prop__iff,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ( ? [X4: list_P1467974523212632366od_a_b] :
            ( ( member6895035109809442519od_a_b @ X4 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_l8778361445553822004od_a_b,X4: list_P1467974523212632366od_a_b,Zs2: list_l8778361445553822004od_a_b] :
            ( ( Xs2
              = ( append7557446516274476873od_a_b @ Ys2 @ ( cons_l907386227682280420od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: list_P1467974523212632366od_a_b] :
                ( ( member6895035109809442519od_a_b @ Y3 @ ( set_li5045283583078295363od_a_b @ Zs2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_289_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ( ? [X4: produc2233330766403592615od_a_b] :
            ( ( member5065438833300252112od_a_b @ X4 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P3463435030753280173od_a_b,X4: produc2233330766403592615od_a_b,Zs2: list_P3463435030753280173od_a_b] :
            ( ( Xs2
              = ( append3447563168971123522od_a_b @ Ys2 @ ( cons_P8222935586017696605od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: produc2233330766403592615od_a_b] :
                ( ( member5065438833300252112od_a_b @ Y3 @ ( set_Pr5228040199190236220od_a_b @ Zs2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_290_split__list__last__prop__iff,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ( ? [X4: b] :
            ( ( member_b @ X4 @ ( set_b2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_b,X4: b,Zs2: list_b] :
            ( ( Xs2
              = ( append_b @ Ys2 @ ( cons_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: b] :
                ( ( member_b @ Y3 @ ( set_b2 @ Zs2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_291_split__list__last__prop__iff,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ( ? [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
            & ( P2 @ X4 ) ) )
      = ( ? [Ys2: list_P1467974523212632366od_a_b,X4: product_prod_a_b,Zs2: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X4 @ Zs2 ) ) )
            & ( P2 @ X4 )
            & ! [Y3: product_prod_a_b] :
                ( ( member1426531481828664017od_a_b @ Y3 @ ( set_Product_prod_a_b2 @ Zs2 ) )
               => ~ ( P2 @ Y3 ) ) ) ) ) ).

% split_list_last_prop_iff
thf(fact_292_in__set__conv__decomp__first,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
      = ( ? [Ys2: list_list_a,Zs2: list_list_a] :
            ( ( Xs2
              = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_293_in__set__conv__decomp__first,axiom,
    ! [X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_s6539019640784088206od_a_b,Zs2: list_s6539019640784088206od_a_b] :
            ( ( Xs2
              = ( append8009123847654438819od_a_b @ Ys2 @ ( cons_s5806323297083630014od_a_b @ X @ Zs2 ) ) )
            & ~ ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_294_in__set__conv__decomp__first,axiom,
    ! [X: set_a,Xs2: list_set_a] :
      ( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
      = ( ? [Ys2: list_set_a,Zs2: list_set_a] :
            ( ( Xs2
              = ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) )
            & ~ ( member_set_a @ X @ ( set_set_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_295_in__set__conv__decomp__first,axiom,
    ! [X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_P8504624544267313367od_a_b,Zs2: list_P8504624544267313367od_a_b] :
            ( ( Xs2
              = ( append4796217420789006444od_a_b @ Ys2 @ ( cons_P135045198529856391od_a_b @ X @ Zs2 ) ) )
            & ~ ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_296_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
      = ( ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
            ( ( Xs2
              = ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
            & ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_297_in__set__conv__decomp__first,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_l8778361445553822004od_a_b,Zs2: list_l8778361445553822004od_a_b] :
            ( ( Xs2
              = ( append7557446516274476873od_a_b @ Ys2 @ ( cons_l907386227682280420od_a_b @ X @ Zs2 ) ) )
            & ~ ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_298_in__set__conv__decomp__first,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_P3463435030753280173od_a_b,Zs2: list_P3463435030753280173od_a_b] :
            ( ( Xs2
              = ( append3447563168971123522od_a_b @ Ys2 @ ( cons_P8222935586017696605od_a_b @ X @ Zs2 ) ) )
            & ~ ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_299_in__set__conv__decomp__first,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs2 ) )
      = ( ? [Ys2: list_b,Zs2: list_b] :
            ( ( Xs2
              = ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
            & ~ ( member_b @ X @ ( set_b2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_300_in__set__conv__decomp__first,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
      = ( ? [Ys2: list_a,Zs2: list_a] :
            ( ( Xs2
              = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_301_in__set__conv__decomp__first,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( ? [Ys2: list_P1467974523212632366od_a_b,Zs2: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X @ Zs2 ) ) )
            & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Ys2 ) ) ) ) ) ).

% in_set_conv_decomp_first
thf(fact_302_mem__Collect__eq,axiom,
    ! [A2: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ( member6895035109809442519od_a_b @ A2 @ ( collec2845612350631750681od_a_b @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_303_mem__Collect__eq,axiom,
    ! [A2: list_a,P2: list_a > $o] :
      ( ( member_list_a @ A2 @ ( collect_list_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_304_mem__Collect__eq,axiom,
    ! [A2: set_Product_prod_a_b,P2: set_Product_prod_a_b > $o] :
      ( ( member1887650552762594865od_a_b @ A2 @ ( collec1744382004165596275od_a_b @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_305_mem__Collect__eq,axiom,
    ! [A2: set_a,P2: set_a > $o] :
      ( ( member_set_a @ A2 @ ( collect_set_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_306_mem__Collect__eq,axiom,
    ! [A2: b,P2: b > $o] :
      ( ( member_b @ A2 @ ( collect_b @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_307_mem__Collect__eq,axiom,
    ! [A2: a,P2: a > $o] :
      ( ( member_a @ A2 @ ( collect_a @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_308_mem__Collect__eq,axiom,
    ! [A2: product_prod_a_b,P2: product_prod_a_b > $o] :
      ( ( member1426531481828664017od_a_b @ A2 @ ( collec3336397801687681299od_a_b @ P2 ) )
      = ( P2 @ A2 ) ) ).

% mem_Collect_eq
thf(fact_309_Collect__mem__eq,axiom,
    ! [A: set_li6306622547289818894od_a_b] :
      ( ( collec2845612350631750681od_a_b
        @ ^ [X4: list_P1467974523212632366od_a_b] : ( member6895035109809442519od_a_b @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_310_Collect__mem__eq,axiom,
    ! [A: set_list_a] :
      ( ( collect_list_a
        @ ^ [X4: list_a] : ( member_list_a @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_311_Collect__mem__eq,axiom,
    ! [A: set_se3214616458824132456od_a_b] :
      ( ( collec1744382004165596275od_a_b
        @ ^ [X4: set_Product_prod_a_b] : ( member1887650552762594865od_a_b @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_312_Collect__mem__eq,axiom,
    ! [A: set_set_a] :
      ( ( collect_set_a
        @ ^ [X4: set_a] : ( member_set_a @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_313_Collect__mem__eq,axiom,
    ! [A: set_b] :
      ( ( collect_b
        @ ^ [X4: b] : ( member_b @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_314_Collect__mem__eq,axiom,
    ! [A: set_a] :
      ( ( collect_a
        @ ^ [X4: a] : ( member_a @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_315_Collect__mem__eq,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ( collec3336397801687681299od_a_b
        @ ^ [X4: product_prod_a_b] : ( member1426531481828664017od_a_b @ X4 @ A ) )
      = A ) ).

% Collect_mem_eq
thf(fact_316_Collect__cong,axiom,
    ! [P2: product_prod_a_b > $o,Q: product_prod_a_b > $o] :
      ( ! [X3: product_prod_a_b] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collec3336397801687681299od_a_b @ P2 )
        = ( collec3336397801687681299od_a_b @ Q ) ) ) ).

% Collect_cong
thf(fact_317_Collect__cong,axiom,
    ! [P2: a > $o,Q: a > $o] :
      ( ! [X3: a] :
          ( ( P2 @ X3 )
          = ( Q @ X3 ) )
     => ( ( collect_a @ P2 )
        = ( collect_a @ Q ) ) ) ).

% Collect_cong
thf(fact_318_in__set__conv__decomp__last,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
      = ( ? [Ys2: list_list_a,Zs2: list_list_a] :
            ( ( Xs2
              = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) )
            & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_319_in__set__conv__decomp__last,axiom,
    ! [X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_s6539019640784088206od_a_b,Zs2: list_s6539019640784088206od_a_b] :
            ( ( Xs2
              = ( append8009123847654438819od_a_b @ Ys2 @ ( cons_s5806323297083630014od_a_b @ X @ Zs2 ) ) )
            & ~ ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_320_in__set__conv__decomp__last,axiom,
    ! [X: set_a,Xs2: list_set_a] :
      ( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
      = ( ? [Ys2: list_set_a,Zs2: list_set_a] :
            ( ( Xs2
              = ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) )
            & ~ ( member_set_a @ X @ ( set_set_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_321_in__set__conv__decomp__last,axiom,
    ! [X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_P8504624544267313367od_a_b,Zs2: list_P8504624544267313367od_a_b] :
            ( ( Xs2
              = ( append4796217420789006444od_a_b @ Ys2 @ ( cons_P135045198529856391od_a_b @ X @ Zs2 ) ) )
            & ~ ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_322_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
      = ( ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
            ( ( Xs2
              = ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) )
            & ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_323_in__set__conv__decomp__last,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_l8778361445553822004od_a_b,Zs2: list_l8778361445553822004od_a_b] :
            ( ( Xs2
              = ( append7557446516274476873od_a_b @ Ys2 @ ( cons_l907386227682280420od_a_b @ X @ Zs2 ) ) )
            & ~ ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_324_in__set__conv__decomp__last,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_P3463435030753280173od_a_b,Zs2: list_P3463435030753280173od_a_b] :
            ( ( Xs2
              = ( append3447563168971123522od_a_b @ Ys2 @ ( cons_P8222935586017696605od_a_b @ X @ Zs2 ) ) )
            & ~ ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_325_in__set__conv__decomp__last,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs2 ) )
      = ( ? [Ys2: list_b,Zs2: list_b] :
            ( ( Xs2
              = ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) )
            & ~ ( member_b @ X @ ( set_b2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_326_in__set__conv__decomp__last,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
      = ( ? [Ys2: list_a,Zs2: list_a] :
            ( ( Xs2
              = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) )
            & ~ ( member_a @ X @ ( set_a2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_327_in__set__conv__decomp__last,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( ? [Ys2: list_P1467974523212632366od_a_b,Zs2: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X @ Zs2 ) ) )
            & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp_last
thf(fact_328_split__list__first__propE,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ? [X5: produc4115131626046251985od_a_b] :
          ( ( member3142156093895948538od_a_b @ X5 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P8504624544267313367od_a_b,X3: produc4115131626046251985od_a_b] :
            ( ? [Zs3: list_P8504624544267313367od_a_b] :
                ( Xs2
                = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: produc4115131626046251985od_a_b] :
                    ( ( member3142156093895948538od_a_b @ Xa @ ( set_Pr7079953345493216102od_a_b @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_329_split__list__first__propE,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ? [X5: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P1396940483166286381od_a_a,X3: product_prod_a_a] :
            ( ? [Zs3: list_P1396940483166286381od_a_a] :
                ( Xs2
                = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: product_prod_a_a] :
                    ( ( member1426531477525435216od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_330_split__list__first__propE,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_a,X3: a] :
            ( ? [Zs3: list_a] :
                ( Xs2
                = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: a] :
                    ( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_331_split__list__first__propE,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ? [X5: list_P1467974523212632366od_a_b] :
          ( ( member6895035109809442519od_a_b @ X5 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_l8778361445553822004od_a_b,X3: list_P1467974523212632366od_a_b] :
            ( ? [Zs3: list_l8778361445553822004od_a_b] :
                ( Xs2
                = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: list_P1467974523212632366od_a_b] :
                    ( ( member6895035109809442519od_a_b @ Xa @ ( set_li5045283583078295363od_a_b @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_332_split__list__first__propE,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ? [X5: produc2233330766403592615od_a_b] :
          ( ( member5065438833300252112od_a_b @ X5 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P3463435030753280173od_a_b,X3: produc2233330766403592615od_a_b] :
            ( ? [Zs3: list_P3463435030753280173od_a_b] :
                ( Xs2
                = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: produc2233330766403592615od_a_b] :
                    ( ( member5065438833300252112od_a_b @ Xa @ ( set_Pr5228040199190236220od_a_b @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_333_split__list__first__propE,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_b,X3: b] :
            ( ? [Zs3: list_b] :
                ( Xs2
                = ( append_b @ Ys3 @ ( cons_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: b] :
                    ( ( member_b @ Xa @ ( set_b2 @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_334_split__list__first__propE,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
            ( ? [Zs3: list_P1467974523212632366od_a_b] :
                ( Xs2
                = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: product_prod_a_b] :
                    ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Ys3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_first_propE
thf(fact_335_split__list__last__propE,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ? [X5: produc4115131626046251985od_a_b] :
          ( ( member3142156093895948538od_a_b @ X5 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P8504624544267313367od_a_b,X3: produc4115131626046251985od_a_b,Zs3: list_P8504624544267313367od_a_b] :
            ( ( Xs2
              = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: produc4115131626046251985od_a_b] :
                    ( ( member3142156093895948538od_a_b @ Xa @ ( set_Pr7079953345493216102od_a_b @ Zs3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_336_split__list__last__propE,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ? [X5: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P1396940483166286381od_a_a,X3: product_prod_a_a,Zs3: list_P1396940483166286381od_a_a] :
            ( ( Xs2
              = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: product_prod_a_a] :
                    ( ( member1426531477525435216od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Zs3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_337_split__list__last__propE,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_a,X3: a,Zs3: list_a] :
            ( ( Xs2
              = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: a] :
                    ( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_338_split__list__last__propE,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ? [X5: list_P1467974523212632366od_a_b] :
          ( ( member6895035109809442519od_a_b @ X5 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_l8778361445553822004od_a_b,X3: list_P1467974523212632366od_a_b,Zs3: list_l8778361445553822004od_a_b] :
            ( ( Xs2
              = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: list_P1467974523212632366od_a_b] :
                    ( ( member6895035109809442519od_a_b @ Xa @ ( set_li5045283583078295363od_a_b @ Zs3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_339_split__list__last__propE,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ? [X5: produc2233330766403592615od_a_b] :
          ( ( member5065438833300252112od_a_b @ X5 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P3463435030753280173od_a_b,X3: produc2233330766403592615od_a_b,Zs3: list_P3463435030753280173od_a_b] :
            ( ( Xs2
              = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: produc2233330766403592615od_a_b] :
                    ( ( member5065438833300252112od_a_b @ Xa @ ( set_Pr5228040199190236220od_a_b @ Zs3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_340_split__list__last__propE,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_b,X3: b,Zs3: list_b] :
            ( ( Xs2
              = ( append_b @ Ys3 @ ( cons_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: b] :
                    ( ( member_b @ Xa @ ( set_b2 @ Zs3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_341_split__list__last__propE,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P1467974523212632366od_a_b,X3: product_prod_a_b,Zs3: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs3 ) ) )
           => ( ( P2 @ X3 )
             => ~ ! [Xa: product_prod_a_b] :
                    ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Zs3 ) )
                   => ~ ( P2 @ Xa ) ) ) ) ) ).

% split_list_last_propE
thf(fact_342_split__list__first__prop,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ? [X5: produc4115131626046251985od_a_b] :
          ( ( member3142156093895948538od_a_b @ X5 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P8504624544267313367od_a_b,X3: produc4115131626046251985od_a_b] :
          ( ? [Zs3: list_P8504624544267313367od_a_b] :
              ( Xs2
              = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: produc4115131626046251985od_a_b] :
              ( ( member3142156093895948538od_a_b @ Xa @ ( set_Pr7079953345493216102od_a_b @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_343_split__list__first__prop,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ? [X5: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P1396940483166286381od_a_a,X3: product_prod_a_a] :
          ( ? [Zs3: list_P1396940483166286381od_a_a] :
              ( Xs2
              = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_344_split__list__first__prop,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_a,X3: a] :
          ( ? [Zs3: list_a] :
              ( Xs2
              = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: a] :
              ( ( member_a @ Xa @ ( set_a2 @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_345_split__list__first__prop,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ? [X5: list_P1467974523212632366od_a_b] :
          ( ( member6895035109809442519od_a_b @ X5 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_l8778361445553822004od_a_b,X3: list_P1467974523212632366od_a_b] :
          ( ? [Zs3: list_l8778361445553822004od_a_b] :
              ( Xs2
              = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: list_P1467974523212632366od_a_b] :
              ( ( member6895035109809442519od_a_b @ Xa @ ( set_li5045283583078295363od_a_b @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_346_split__list__first__prop,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ? [X5: produc2233330766403592615od_a_b] :
          ( ( member5065438833300252112od_a_b @ X5 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P3463435030753280173od_a_b,X3: produc2233330766403592615od_a_b] :
          ( ? [Zs3: list_P3463435030753280173od_a_b] :
              ( Xs2
              = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: produc2233330766403592615od_a_b] :
              ( ( member5065438833300252112od_a_b @ Xa @ ( set_Pr5228040199190236220od_a_b @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_347_split__list__first__prop,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_b,X3: b] :
          ( ? [Zs3: list_b] :
              ( Xs2
              = ( append_b @ Ys3 @ ( cons_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: b] :
              ( ( member_b @ Xa @ ( set_b2 @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_348_split__list__first__prop,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
          ( ? [Zs3: list_P1467974523212632366od_a_b] :
              ( Xs2
              = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Ys3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_first_prop
thf(fact_349_split__list__last__prop,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ? [X5: produc4115131626046251985od_a_b] :
          ( ( member3142156093895948538od_a_b @ X5 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P8504624544267313367od_a_b,X3: produc4115131626046251985od_a_b,Zs3: list_P8504624544267313367od_a_b] :
          ( ( Xs2
            = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: produc4115131626046251985od_a_b] :
              ( ( member3142156093895948538od_a_b @ Xa @ ( set_Pr7079953345493216102od_a_b @ Zs3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_350_split__list__last__prop,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ? [X5: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P1396940483166286381od_a_a,X3: product_prod_a_a,Zs3: list_P1396940483166286381od_a_a] :
          ( ( Xs2
            = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: product_prod_a_a] :
              ( ( member1426531477525435216od_a_a @ Xa @ ( set_Product_prod_a_a2 @ Zs3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_351_split__list__last__prop,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_a,X3: a,Zs3: list_a] :
          ( ( Xs2
            = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: a] :
              ( ( member_a @ Xa @ ( set_a2 @ Zs3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_352_split__list__last__prop,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ? [X5: list_P1467974523212632366od_a_b] :
          ( ( member6895035109809442519od_a_b @ X5 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_l8778361445553822004od_a_b,X3: list_P1467974523212632366od_a_b,Zs3: list_l8778361445553822004od_a_b] :
          ( ( Xs2
            = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: list_P1467974523212632366od_a_b] :
              ( ( member6895035109809442519od_a_b @ Xa @ ( set_li5045283583078295363od_a_b @ Zs3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_353_split__list__last__prop,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ? [X5: produc2233330766403592615od_a_b] :
          ( ( member5065438833300252112od_a_b @ X5 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P3463435030753280173od_a_b,X3: produc2233330766403592615od_a_b,Zs3: list_P3463435030753280173od_a_b] :
          ( ( Xs2
            = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: produc2233330766403592615od_a_b] :
              ( ( member5065438833300252112od_a_b @ Xa @ ( set_Pr5228040199190236220od_a_b @ Zs3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_354_split__list__last__prop,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_b,X3: b,Zs3: list_b] :
          ( ( Xs2
            = ( append_b @ Ys3 @ ( cons_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: b] :
              ( ( member_b @ Xa @ ( set_b2 @ Zs3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_355_split__list__last__prop,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P1467974523212632366od_a_b,X3: product_prod_a_b,Zs3: list_P1467974523212632366od_a_b] :
          ( ( Xs2
            = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 )
          & ! [Xa: product_prod_a_b] :
              ( ( member1426531481828664017od_a_b @ Xa @ ( set_Product_prod_a_b2 @ Zs3 ) )
             => ~ ( P2 @ Xa ) ) ) ) ).

% split_list_last_prop
thf(fact_356_in__set__conv__decomp,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
      = ( ? [Ys2: list_list_a,Zs2: list_list_a] :
            ( Xs2
            = ( append_list_a @ Ys2 @ ( cons_list_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_357_in__set__conv__decomp,axiom,
    ! [X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_s6539019640784088206od_a_b,Zs2: list_s6539019640784088206od_a_b] :
            ( Xs2
            = ( append8009123847654438819od_a_b @ Ys2 @ ( cons_s5806323297083630014od_a_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_358_in__set__conv__decomp,axiom,
    ! [X: set_a,Xs2: list_set_a] :
      ( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
      = ( ? [Ys2: list_set_a,Zs2: list_set_a] :
            ( Xs2
            = ( append_set_a @ Ys2 @ ( cons_set_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_359_in__set__conv__decomp,axiom,
    ! [X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_P8504624544267313367od_a_b,Zs2: list_P8504624544267313367od_a_b] :
            ( Xs2
            = ( append4796217420789006444od_a_b @ Ys2 @ ( cons_P135045198529856391od_a_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_360_in__set__conv__decomp,axiom,
    ! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
      = ( ? [Ys2: list_P1396940483166286381od_a_a,Zs2: list_P1396940483166286381od_a_a] :
            ( Xs2
            = ( append5335208819046833346od_a_a @ Ys2 @ ( cons_P7316939126706565853od_a_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_361_in__set__conv__decomp,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_l8778361445553822004od_a_b,Zs2: list_l8778361445553822004od_a_b] :
            ( Xs2
            = ( append7557446516274476873od_a_b @ Ys2 @ ( cons_l907386227682280420od_a_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_362_in__set__conv__decomp,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
      = ( ? [Ys2: list_P3463435030753280173od_a_b,Zs2: list_P3463435030753280173od_a_b] :
            ( Xs2
            = ( append3447563168971123522od_a_b @ Ys2 @ ( cons_P8222935586017696605od_a_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_363_in__set__conv__decomp,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs2 ) )
      = ( ? [Ys2: list_b,Zs2: list_b] :
            ( Xs2
            = ( append_b @ Ys2 @ ( cons_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_364_in__set__conv__decomp,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
      = ( ? [Ys2: list_a,Zs2: list_a] :
            ( Xs2
            = ( append_a @ Ys2 @ ( cons_a @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_365_in__set__conv__decomp,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( ? [Ys2: list_P1467974523212632366od_a_b,Zs2: list_P1467974523212632366od_a_b] :
            ( Xs2
            = ( append5335208823350062147od_a_b @ Ys2 @ ( cons_P7316939131009794654od_a_b @ X @ Zs2 ) ) ) ) ) ).

% in_set_conv_decomp
thf(fact_366_append__Cons__eq__iff,axiom,
    ! [X: list_a,Xs2: list_list_a,Ys: list_list_a,Xs3: list_list_a,Ys4: list_list_a] :
      ( ~ ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ( ~ ( member_list_a @ X @ ( set_list_a2 @ Ys ) )
       => ( ( ( append_list_a @ Xs2 @ ( cons_list_a @ X @ Ys ) )
            = ( append_list_a @ Xs3 @ ( cons_list_a @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_367_append__Cons__eq__iff,axiom,
    ! [X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b,Ys: list_s6539019640784088206od_a_b,Xs3: list_s6539019640784088206od_a_b,Ys4: list_s6539019640784088206od_a_b] :
      ( ~ ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Xs2 ) )
     => ( ~ ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Ys ) )
       => ( ( ( append8009123847654438819od_a_b @ Xs2 @ ( cons_s5806323297083630014od_a_b @ X @ Ys ) )
            = ( append8009123847654438819od_a_b @ Xs3 @ ( cons_s5806323297083630014od_a_b @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_368_append__Cons__eq__iff,axiom,
    ! [X: set_a,Xs2: list_set_a,Ys: list_set_a,Xs3: list_set_a,Ys4: list_set_a] :
      ( ~ ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
     => ( ~ ( member_set_a @ X @ ( set_set_a2 @ Ys ) )
       => ( ( ( append_set_a @ Xs2 @ ( cons_set_a @ X @ Ys ) )
            = ( append_set_a @ Xs3 @ ( cons_set_a @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_369_append__Cons__eq__iff,axiom,
    ! [X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b,Ys: list_P8504624544267313367od_a_b,Xs3: list_P8504624544267313367od_a_b,Ys4: list_P8504624544267313367od_a_b] :
      ( ~ ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
     => ( ~ ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Ys ) )
       => ( ( ( append4796217420789006444od_a_b @ Xs2 @ ( cons_P135045198529856391od_a_b @ X @ Ys ) )
            = ( append4796217420789006444od_a_b @ Xs3 @ ( cons_P135045198529856391od_a_b @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_370_append__Cons__eq__iff,axiom,
    ! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a,Ys: list_P1396940483166286381od_a_a,Xs3: list_P1396940483166286381od_a_a,Ys4: list_P1396940483166286381od_a_a] :
      ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
     => ( ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys ) )
       => ( ( ( append5335208819046833346od_a_a @ Xs2 @ ( cons_P7316939126706565853od_a_a @ X @ Ys ) )
            = ( append5335208819046833346od_a_a @ Xs3 @ ( cons_P7316939126706565853od_a_a @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_371_append__Cons__eq__iff,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b,Xs3: list_l8778361445553822004od_a_b,Ys4: list_l8778361445553822004od_a_b] :
      ( ~ ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
     => ( ~ ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Ys ) )
       => ( ( ( append7557446516274476873od_a_b @ Xs2 @ ( cons_l907386227682280420od_a_b @ X @ Ys ) )
            = ( append7557446516274476873od_a_b @ Xs3 @ ( cons_l907386227682280420od_a_b @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_372_append__Cons__eq__iff,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b,Ys: list_P3463435030753280173od_a_b,Xs3: list_P3463435030753280173od_a_b,Ys4: list_P3463435030753280173od_a_b] :
      ( ~ ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
     => ( ~ ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Ys ) )
       => ( ( ( append3447563168971123522od_a_b @ Xs2 @ ( cons_P8222935586017696605od_a_b @ X @ Ys ) )
            = ( append3447563168971123522od_a_b @ Xs3 @ ( cons_P8222935586017696605od_a_b @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_373_append__Cons__eq__iff,axiom,
    ! [X: b,Xs2: list_b,Ys: list_b,Xs3: list_b,Ys4: list_b] :
      ( ~ ( member_b @ X @ ( set_b2 @ Xs2 ) )
     => ( ~ ( member_b @ X @ ( set_b2 @ Ys ) )
       => ( ( ( append_b @ Xs2 @ ( cons_b @ X @ Ys ) )
            = ( append_b @ Xs3 @ ( cons_b @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_374_append__Cons__eq__iff,axiom,
    ! [X: a,Xs2: list_a,Ys: list_a,Xs3: list_a,Ys4: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ( ~ ( member_a @ X @ ( set_a2 @ Ys ) )
       => ( ( ( append_a @ Xs2 @ ( cons_a @ X @ Ys ) )
            = ( append_a @ Xs3 @ ( cons_a @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_375_append__Cons__eq__iff,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Xs3: list_P1467974523212632366od_a_b,Ys4: list_P1467974523212632366od_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Ys ) )
       => ( ( ( append5335208823350062147od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ X @ Ys ) )
            = ( append5335208823350062147od_a_b @ Xs3 @ ( cons_P7316939131009794654od_a_b @ X @ Ys4 ) ) )
          = ( ( Xs2 = Xs3 )
            & ( Ys = Ys4 ) ) ) ) ) ).

% append_Cons_eq_iff
thf(fact_376_split__list__propE,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ? [X5: produc4115131626046251985od_a_b] :
          ( ( member3142156093895948538od_a_b @ X5 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P8504624544267313367od_a_b,X3: produc4115131626046251985od_a_b] :
            ( ? [Zs3: list_P8504624544267313367od_a_b] :
                ( Xs2
                = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X3 @ Zs3 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_377_split__list__propE,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ? [X5: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P1396940483166286381od_a_a,X3: product_prod_a_a] :
            ( ? [Zs3: list_P1396940483166286381od_a_a] :
                ( Xs2
                = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_378_split__list__propE,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_a,X3: a] :
            ( ? [Zs3: list_a] :
                ( Xs2
                = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_379_split__list__propE,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ? [X5: list_P1467974523212632366od_a_b] :
          ( ( member6895035109809442519od_a_b @ X5 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_l8778361445553822004od_a_b,X3: list_P1467974523212632366od_a_b] :
            ( ? [Zs3: list_l8778361445553822004od_a_b] :
                ( Xs2
                = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X3 @ Zs3 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_380_split__list__propE,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ? [X5: produc2233330766403592615od_a_b] :
          ( ( member5065438833300252112od_a_b @ X5 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P3463435030753280173od_a_b,X3: produc2233330766403592615od_a_b] :
            ( ? [Zs3: list_P3463435030753280173od_a_b] :
                ( Xs2
                = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X3 @ Zs3 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_381_split__list__propE,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_b,X3: b] :
            ( ? [Zs3: list_b] :
                ( Xs2
                = ( append_b @ Ys3 @ ( cons_b @ X3 @ Zs3 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_382_split__list__propE,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ~ ! [Ys3: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
            ( ? [Zs3: list_P1467974523212632366od_a_b] :
                ( Xs2
                = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs3 ) ) )
           => ~ ( P2 @ X3 ) ) ) ).

% split_list_propE
thf(fact_383_split__list__first,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ? [Ys3: list_list_a,Zs3: list_list_a] :
          ( ( Xs2
            = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_384_split__list__first,axiom,
    ! [X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Xs2 ) )
     => ? [Ys3: list_s6539019640784088206od_a_b,Zs3: list_s6539019640784088206od_a_b] :
          ( ( Xs2
            = ( append8009123847654438819od_a_b @ Ys3 @ ( cons_s5806323297083630014od_a_b @ X @ Zs3 ) ) )
          & ~ ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_385_split__list__first,axiom,
    ! [X: set_a,Xs2: list_set_a] :
      ( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
     => ? [Ys3: list_set_a,Zs3: list_set_a] :
          ( ( Xs2
            = ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) )
          & ~ ( member_set_a @ X @ ( set_set_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_386_split__list__first,axiom,
    ! [X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
     => ? [Ys3: list_P8504624544267313367od_a_b,Zs3: list_P8504624544267313367od_a_b] :
          ( ( Xs2
            = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X @ Zs3 ) ) )
          & ~ ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_387_split__list__first,axiom,
    ! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
     => ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
          ( ( Xs2
            = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
          & ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_388_split__list__first,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
     => ? [Ys3: list_l8778361445553822004od_a_b,Zs3: list_l8778361445553822004od_a_b] :
          ( ( Xs2
            = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X @ Zs3 ) ) )
          & ~ ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_389_split__list__first,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
     => ? [Ys3: list_P3463435030753280173od_a_b,Zs3: list_P3463435030753280173od_a_b] :
          ( ( Xs2
            = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X @ Zs3 ) ) )
          & ~ ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_390_split__list__first,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs2 ) )
     => ? [Ys3: list_b,Zs3: list_b] :
          ( ( Xs2
            = ( append_b @ Ys3 @ ( cons_b @ X @ Zs3 ) ) )
          & ~ ( member_b @ X @ ( set_b2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_391_split__list__first,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ? [Ys3: list_a,Zs3: list_a] :
          ( ( Xs2
            = ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_392_split__list__first,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ? [Ys3: list_P1467974523212632366od_a_b,Zs3: list_P1467974523212632366od_a_b] :
          ( ( Xs2
            = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X @ Zs3 ) ) )
          & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Ys3 ) ) ) ) ).

% split_list_first
thf(fact_393_split__list__prop,axiom,
    ! [Xs2: list_P8504624544267313367od_a_b,P2: produc4115131626046251985od_a_b > $o] :
      ( ? [X5: produc4115131626046251985od_a_b] :
          ( ( member3142156093895948538od_a_b @ X5 @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P8504624544267313367od_a_b,X3: produc4115131626046251985od_a_b] :
          ( ? [Zs3: list_P8504624544267313367od_a_b] :
              ( Xs2
              = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_394_split__list__prop,axiom,
    ! [Xs2: list_P1396940483166286381od_a_a,P2: product_prod_a_a > $o] :
      ( ? [X5: product_prod_a_a] :
          ( ( member1426531477525435216od_a_a @ X5 @ ( set_Product_prod_a_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P1396940483166286381od_a_a,X3: product_prod_a_a] :
          ( ? [Zs3: list_P1396940483166286381od_a_a] :
              ( Xs2
              = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_395_split__list__prop,axiom,
    ! [Xs2: list_a,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_a,X3: a] :
          ( ? [Zs3: list_a] :
              ( Xs2
              = ( append_a @ Ys3 @ ( cons_a @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_396_split__list__prop,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ? [X5: list_P1467974523212632366od_a_b] :
          ( ( member6895035109809442519od_a_b @ X5 @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_l8778361445553822004od_a_b,X3: list_P1467974523212632366od_a_b] :
          ( ? [Zs3: list_l8778361445553822004od_a_b] :
              ( Xs2
              = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_397_split__list__prop,axiom,
    ! [Xs2: list_P3463435030753280173od_a_b,P2: produc2233330766403592615od_a_b > $o] :
      ( ? [X5: produc2233330766403592615od_a_b] :
          ( ( member5065438833300252112od_a_b @ X5 @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P3463435030753280173od_a_b,X3: produc2233330766403592615od_a_b] :
          ( ? [Zs3: list_P3463435030753280173od_a_b] :
              ( Xs2
              = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_398_split__list__prop,axiom,
    ! [Xs2: list_b,P2: b > $o] :
      ( ? [X5: b] :
          ( ( member_b @ X5 @ ( set_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_b,X3: b] :
          ( ? [Zs3: list_b] :
              ( Xs2
              = ( append_b @ Ys3 @ ( cons_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_399_split__list__prop,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs2 ) )
          & ( P2 @ X5 ) )
     => ? [Ys3: list_P1467974523212632366od_a_b,X3: product_prod_a_b] :
          ( ? [Zs3: list_P1467974523212632366od_a_b] :
              ( Xs2
              = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Zs3 ) ) )
          & ( P2 @ X3 ) ) ) ).

% split_list_prop
thf(fact_400_split__list__last,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ? [Ys3: list_list_a,Zs3: list_list_a] :
          ( ( Xs2
            = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) )
          & ~ ( member_list_a @ X @ ( set_list_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_401_split__list__last,axiom,
    ! [X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Xs2 ) )
     => ? [Ys3: list_s6539019640784088206od_a_b,Zs3: list_s6539019640784088206od_a_b] :
          ( ( Xs2
            = ( append8009123847654438819od_a_b @ Ys3 @ ( cons_s5806323297083630014od_a_b @ X @ Zs3 ) ) )
          & ~ ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_402_split__list__last,axiom,
    ! [X: set_a,Xs2: list_set_a] :
      ( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
     => ? [Ys3: list_set_a,Zs3: list_set_a] :
          ( ( Xs2
            = ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) )
          & ~ ( member_set_a @ X @ ( set_set_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_403_split__list__last,axiom,
    ! [X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
     => ? [Ys3: list_P8504624544267313367od_a_b,Zs3: list_P8504624544267313367od_a_b] :
          ( ( Xs2
            = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X @ Zs3 ) ) )
          & ~ ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_404_split__list__last,axiom,
    ! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
     => ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
          ( ( Xs2
            = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) )
          & ~ ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_405_split__list__last,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
     => ? [Ys3: list_l8778361445553822004od_a_b,Zs3: list_l8778361445553822004od_a_b] :
          ( ( Xs2
            = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X @ Zs3 ) ) )
          & ~ ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_406_split__list__last,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
     => ? [Ys3: list_P3463435030753280173od_a_b,Zs3: list_P3463435030753280173od_a_b] :
          ( ( Xs2
            = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X @ Zs3 ) ) )
          & ~ ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_407_split__list__last,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs2 ) )
     => ? [Ys3: list_b,Zs3: list_b] :
          ( ( Xs2
            = ( append_b @ Ys3 @ ( cons_b @ X @ Zs3 ) ) )
          & ~ ( member_b @ X @ ( set_b2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_408_split__list__last,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ? [Ys3: list_a,Zs3: list_a] :
          ( ( Xs2
            = ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) )
          & ~ ( member_a @ X @ ( set_a2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_409_split__list__last,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ? [Ys3: list_P1467974523212632366od_a_b,Zs3: list_P1467974523212632366od_a_b] :
          ( ( Xs2
            = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X @ Zs3 ) ) )
          & ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Zs3 ) ) ) ) ).

% split_list_last
thf(fact_410_split__list,axiom,
    ! [X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ X @ ( set_list_a2 @ Xs2 ) )
     => ? [Ys3: list_list_a,Zs3: list_list_a] :
          ( Xs2
          = ( append_list_a @ Ys3 @ ( cons_list_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_411_split__list,axiom,
    ! [X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ X @ ( set_se8479788141692617885od_a_b @ Xs2 ) )
     => ? [Ys3: list_s6539019640784088206od_a_b,Zs3: list_s6539019640784088206od_a_b] :
          ( Xs2
          = ( append8009123847654438819od_a_b @ Ys3 @ ( cons_s5806323297083630014od_a_b @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_412_split__list,axiom,
    ! [X: set_a,Xs2: list_set_a] :
      ( ( member_set_a @ X @ ( set_set_a2 @ Xs2 ) )
     => ? [Ys3: list_set_a,Zs3: list_set_a] :
          ( Xs2
          = ( append_set_a @ Ys3 @ ( cons_set_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_413_split__list,axiom,
    ! [X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ X @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) )
     => ? [Ys3: list_P8504624544267313367od_a_b,Zs3: list_P8504624544267313367od_a_b] :
          ( Xs2
          = ( append4796217420789006444od_a_b @ Ys3 @ ( cons_P135045198529856391od_a_b @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_414_split__list,axiom,
    ! [X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ X @ ( set_Product_prod_a_a2 @ Xs2 ) )
     => ? [Ys3: list_P1396940483166286381od_a_a,Zs3: list_P1396940483166286381od_a_a] :
          ( Xs2
          = ( append5335208819046833346od_a_a @ Ys3 @ ( cons_P7316939126706565853od_a_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_415_split__list,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ X @ ( set_li5045283583078295363od_a_b @ Xs2 ) )
     => ? [Ys3: list_l8778361445553822004od_a_b,Zs3: list_l8778361445553822004od_a_b] :
          ( Xs2
          = ( append7557446516274476873od_a_b @ Ys3 @ ( cons_l907386227682280420od_a_b @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_416_split__list,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) )
     => ? [Ys3: list_P3463435030753280173od_a_b,Zs3: list_P3463435030753280173od_a_b] :
          ( Xs2
          = ( append3447563168971123522od_a_b @ Ys3 @ ( cons_P8222935586017696605od_a_b @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_417_split__list,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( member_b @ X @ ( set_b2 @ Xs2 ) )
     => ? [Ys3: list_b,Zs3: list_b] :
          ( Xs2
          = ( append_b @ Ys3 @ ( cons_b @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_418_split__list,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ? [Ys3: list_a,Zs3: list_a] :
          ( Xs2
          = ( append_a @ Ys3 @ ( cons_a @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_419_split__list,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ? [Ys3: list_P1467974523212632366od_a_b,Zs3: list_P1467974523212632366od_a_b] :
          ( Xs2
          = ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ X @ Zs3 ) ) ) ) ).

% split_list
thf(fact_420_dtree__to__list__split__subtree,axiom,
    ! [As: list_P3463435030753280173od_a_b,R2: product_prod_a_b,E2: product_prod_a_b,Bs: list_P3463435030753280173od_a_b,T: dtree_2253494508914376569od_a_b] :
      ( ( ( append3447563168971123522od_a_b @ As @ ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ R2 @ E2 ) @ Bs ) )
        = ( dtree_983605638134528549od_a_b @ T ) )
     => ? [Xs: fset_P6924693328471822228od_a_b] :
          ( ( strict3077507913750366111od_a_b @ ( node_P1936840572307073598od_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_983605638134528549od_a_b @ ( node_P1936840572307073598od_a_b @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_421_dtree__to__list__split__subtree,axiom,
    ! [As: list_P7799544280440231939_a_b_a,R2: product_prod_a_b,E2: a,Bs: list_P7799544280440231939_a_b_a,T: dtree_1072189271477137615_a_b_a] :
      ( ( ( append3848328844090658200_a_b_a @ As @ ( cons_P8410528658686283955_a_b_a @ ( produc5056887847982525421_a_b_a @ R2 @ E2 ) @ Bs ) )
        = ( dtree_4970704345647506811_a_b_a @ T ) )
     => ? [Xs: fset_P1695881499225648404_b_a_a] :
          ( ( strict7369715241992436981_a_b_a @ ( node_P9154982556985054868_a_b_a @ R2 @ Xs ) @ T )
          & ( ( dtree_4970704345647506811_a_b_a @ ( node_P9154982556985054868_a_b_a @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_422_dtree__to__list__split__subtree,axiom,
    ! [As: list_P321204300973800749list_a,R2: list_a,E2: list_a,Bs: list_P321204300973800749list_a,T: dtree_list_a_list_a] :
      ( ( ( append622457529216362434list_a @ As @ ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ R2 @ E2 ) @ Bs ) )
        = ( dtree_6750851072906405029list_a @ T ) )
     => ? [Xs: fset_P6656299774166858224list_a] :
          ( ( strict9149861969251335199list_a @ ( node_list_a_list_a @ R2 @ Xs ) @ T )
          & ( ( dtree_6750851072906405029list_a @ ( node_list_a_list_a @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_423_dtree__to__list__split__subtree,axiom,
    ! [As: list_P8094385941167427565od_a_b,R2: product_prod_a_b > product_prod_a_b > $o,E2: list_P1467974523212632366od_a_b,Bs: list_P8094385941167427565od_a_b,T: dtree_8339642294131873593od_a_b] :
      ( ( ( append2461588412122937986od_a_b @ As @ ( cons_P3561991119763190045od_a_b @ ( produc7704224619955794135od_a_b @ R2 @ E2 ) @ Bs ) )
        = ( dtree_2881821214123584869od_a_b @ T ) )
     => ? [Xs: fset_P9180166578942778714od_a_b] :
          ( ( strict798091761169634783od_a_b @ ( node_P5897935084763704574od_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_2881821214123584869od_a_b @ ( node_P5897935084763704574od_a_b @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_424_dtree__to__list__split__subtree,axiom,
    ! [As: list_P8504624544267313367od_a_b,R2: a,E2: product_prod_a_b,Bs: list_P8504624544267313367od_a_b,T: dtree_2020077848175485859od_a_b] :
      ( ( ( append4796217420789006444od_a_b @ As @ ( cons_P135045198529856391od_a_b @ ( produc431845345726502849od_a_b @ R2 @ E2 ) @ Bs ) )
        = ( dtree_345661843391484239od_a_b @ T ) )
     => ? [Xs: fset_P8459658061218866410od_a_b] :
          ( ( strict2744672739736414409od_a_b @ ( node_a4529940054729032296od_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_345661843391484239od_a_b @ ( node_a4529940054729032296od_a_b @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_425_dtree__to__list__split__subtree,axiom,
    ! [As: list_P5700792096213675741od_a_b,R2: a,E2: list_P1467974523212632366od_a_b,Bs: list_P5700792096213675741od_a_b,T: dtree_8545934391348768041od_a_b] :
      ( ( ( append2868893624165880178od_a_b @ As @ ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ R2 @ E2 ) @ Bs ) )
        = ( dtree_4297255050927138389od_a_b @ T ) )
     => ? [Xs: fset_P8869035697996476266od_a_b] :
          ( ( strict2080577894271367375od_a_b @ ( node_a6325075177599301358od_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_4297255050927138389od_a_b @ ( node_a6325075177599301358od_a_b @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_426_dtree__to__list__split__subtree,axiom,
    ! [As: list_P1396940483166286381od_a_a,R2: a,E2: a,Bs: list_P1396940483166286381od_a_a,T: dtree_a_a] :
      ( ( ( append5335208819046833346od_a_a @ As @ ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ R2 @ E2 ) @ Bs ) )
        = ( dtree_to_list_a_a @ T ) )
     => ? [Xs: fset_P2559403164575059562_a_a_a] :
          ( ( strict_subtree_a_a @ ( node_a_a @ R2 @ Xs ) @ T )
          & ( ( dtree_to_list_a_a @ ( node_a_a @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_427_dtree__to__list__split__subtree,axiom,
    ! [As: list_P1467974523212632366od_a_b,R2: a,E2: b,Bs: list_P1467974523212632366od_a_b,T: dtree_a_b] :
      ( ( ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E2 ) @ Bs ) )
        = ( dtree_to_list_a_b @ T ) )
     => ? [Xs: fset_P5281107635120001194_a_b_b] :
          ( ( strict_subtree_a_b @ ( node_a_b @ R2 @ Xs ) @ T )
          & ( ( dtree_to_list_a_b @ ( node_a_b @ R2 @ Xs ) )
            = Bs ) ) ) ).

% dtree_to_list_split_subtree
thf(fact_428_prod__induct3,axiom,
    ! [P2: produc4115131626046251985od_a_b > $o,X: produc4115131626046251985od_a_b] :
      ( ! [A4: a,B6: a,C3: b] : ( P2 @ ( produc431845345726502849od_a_b @ A4 @ ( product_Pair_a_b @ B6 @ C3 ) ) )
     => ( P2 @ X ) ) ).

% prod_induct3
thf(fact_429_prod__cases3,axiom,
    ! [Y: produc4115131626046251985od_a_b] :
      ~ ! [A4: a,B6: a,C3: b] :
          ( Y
         != ( produc431845345726502849od_a_b @ A4 @ ( product_Pair_a_b @ B6 @ C3 ) ) ) ).

% prod_cases3
thf(fact_430_Pair__inject,axiom,
    ! [A2: list_a,B: list_a,A3: list_a,B2: list_a] :
      ( ( ( produc6837034575241423639list_a @ A2 @ B )
        = ( produc6837034575241423639list_a @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_431_Pair__inject,axiom,
    ! [A2: product_prod_a_b > product_prod_a_b > $o,B: list_P1467974523212632366od_a_b,A3: product_prod_a_b > product_prod_a_b > $o,B2: list_P1467974523212632366od_a_b] :
      ( ( ( produc7704224619955794135od_a_b @ A2 @ B )
        = ( produc7704224619955794135od_a_b @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_432_Pair__inject,axiom,
    ! [A2: a,B: product_prod_a_b,A3: a,B2: product_prod_a_b] :
      ( ( ( produc431845345726502849od_a_b @ A2 @ B )
        = ( produc431845345726502849od_a_b @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_433_Pair__inject,axiom,
    ! [A2: a,B: list_P1467974523212632366od_a_b,A3: a,B2: list_P1467974523212632366od_a_b] :
      ( ( ( produc6643264353547226055od_a_b @ A2 @ B )
        = ( produc6643264353547226055od_a_b @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_434_Pair__inject,axiom,
    ! [A2: a,B: a,A3: a,B2: a] :
      ( ( ( product_Pair_a_a @ A2 @ B )
        = ( product_Pair_a_a @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_435_Pair__inject,axiom,
    ! [A2: a,B: b,A3: a,B2: b] :
      ( ( ( product_Pair_a_b @ A2 @ B )
        = ( product_Pair_a_b @ A3 @ B2 ) )
     => ~ ( ( A2 = A3 )
         => ( B != B2 ) ) ) ).

% Pair_inject
thf(fact_436_prod__cases,axiom,
    ! [P2: produc9164743771328383783list_a > $o,P: produc9164743771328383783list_a] :
      ( ! [A4: list_a,B6: list_a] : ( P2 @ ( produc6837034575241423639list_a @ A4 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_437_prod__cases,axiom,
    ! [P2: produc3334210663054469735od_a_b > $o,P: produc3334210663054469735od_a_b] :
      ( ! [A4: product_prod_a_b > product_prod_a_b > $o,B6: list_P1467974523212632366od_a_b] : ( P2 @ ( produc7704224619955794135od_a_b @ A4 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_438_prod__cases,axiom,
    ! [P2: produc4115131626046251985od_a_b > $o,P: produc4115131626046251985od_a_b] :
      ( ! [A4: a,B6: product_prod_a_b] : ( P2 @ ( produc431845345726502849od_a_b @ A4 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_439_prod__cases,axiom,
    ! [P2: produc2639161875728825943od_a_b > $o,P: produc2639161875728825943od_a_b] :
      ( ! [A4: a,B6: list_P1467974523212632366od_a_b] : ( P2 @ ( produc6643264353547226055od_a_b @ A4 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_440_prod__cases,axiom,
    ! [P2: product_prod_a_a > $o,P: product_prod_a_a] :
      ( ! [A4: a,B6: a] : ( P2 @ ( product_Pair_a_a @ A4 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_441_prod__cases,axiom,
    ! [P2: product_prod_a_b > $o,P: product_prod_a_b] :
      ( ! [A4: a,B6: b] : ( P2 @ ( product_Pair_a_b @ A4 @ B6 ) )
     => ( P2 @ P ) ) ).

% prod_cases
thf(fact_442_surj__pair,axiom,
    ! [P: produc9164743771328383783list_a] :
    ? [X3: list_a,Y4: list_a] :
      ( P
      = ( produc6837034575241423639list_a @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_443_surj__pair,axiom,
    ! [P: produc3334210663054469735od_a_b] :
    ? [X3: product_prod_a_b > product_prod_a_b > $o,Y4: list_P1467974523212632366od_a_b] :
      ( P
      = ( produc7704224619955794135od_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_444_surj__pair,axiom,
    ! [P: produc4115131626046251985od_a_b] :
    ? [X3: a,Y4: product_prod_a_b] :
      ( P
      = ( produc431845345726502849od_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_445_surj__pair,axiom,
    ! [P: produc2639161875728825943od_a_b] :
    ? [X3: a,Y4: list_P1467974523212632366od_a_b] :
      ( P
      = ( produc6643264353547226055od_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_446_surj__pair,axiom,
    ! [P: product_prod_a_a] :
    ? [X3: a,Y4: a] :
      ( P
      = ( product_Pair_a_a @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_447_surj__pair,axiom,
    ! [P: product_prod_a_b] :
    ? [X3: a,Y4: b] :
      ( P
      = ( product_Pair_a_b @ X3 @ Y4 ) ) ).

% surj_pair
thf(fact_448_old_Oprod_Oexhaust,axiom,
    ! [Y: produc9164743771328383783list_a] :
      ~ ! [A4: list_a,B6: list_a] :
          ( Y
         != ( produc6837034575241423639list_a @ A4 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_449_old_Oprod_Oexhaust,axiom,
    ! [Y: produc3334210663054469735od_a_b] :
      ~ ! [A4: product_prod_a_b > product_prod_a_b > $o,B6: list_P1467974523212632366od_a_b] :
          ( Y
         != ( produc7704224619955794135od_a_b @ A4 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_450_old_Oprod_Oexhaust,axiom,
    ! [Y: produc4115131626046251985od_a_b] :
      ~ ! [A4: a,B6: product_prod_a_b] :
          ( Y
         != ( produc431845345726502849od_a_b @ A4 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_451_old_Oprod_Oexhaust,axiom,
    ! [Y: produc2639161875728825943od_a_b] :
      ~ ! [A4: a,B6: list_P1467974523212632366od_a_b] :
          ( Y
         != ( produc6643264353547226055od_a_b @ A4 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_452_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_a] :
      ~ ! [A4: a,B6: a] :
          ( Y
         != ( product_Pair_a_a @ A4 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_453_old_Oprod_Oexhaust,axiom,
    ! [Y: product_prod_a_b] :
      ~ ! [A4: a,B6: b] :
          ( Y
         != ( product_Pair_a_b @ A4 @ B6 ) ) ).

% old.prod.exhaust
thf(fact_454_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B: a,F: a > a] :
      ( ( member_a @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_a_a2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_455_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B: product_prod_a_b,F: a > product_prod_a_b] :
      ( ( member_a @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member1426531481828664017od_a_b @ B @ ( image_7400625786893224495od_a_b @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_456_rev__image__eqI,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,B: a,F: product_prod_a_b > a] :
      ( ( member1426531481828664017od_a_b @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_2802296252294471259_a_b_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_457_rev__image__eqI,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,B: product_prod_a_b,F: product_prod_a_b > product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member1426531481828664017od_a_b @ B @ ( image_3300603549555413765od_a_b @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_458_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B: b,F: a > b] :
      ( ( member_a @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_b @ B @ ( image_a_b2 @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_459_rev__image__eqI,axiom,
    ! [X: b,A: set_b,B: a,F: b > a] :
      ( ( member_b @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_b_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_460_rev__image__eqI,axiom,
    ! [X: b,A: set_b,B: b,F: b > b] :
      ( ( member_b @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_b @ B @ ( image_b_b @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_461_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B: list_a,F: a > list_a] :
      ( ( member_a @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_list_a @ B @ ( image_a_list_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_462_rev__image__eqI,axiom,
    ! [X: a,A: set_a,B: set_a,F: a > set_a] :
      ( ( member_a @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_set_a @ B @ ( image_a_set_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_463_rev__image__eqI,axiom,
    ! [X: list_a,A: set_list_a,B: a,F: list_a > a] :
      ( ( member_list_a @ X @ A )
     => ( ( B
          = ( F @ X ) )
       => ( member_a @ B @ ( image_list_a_a @ F @ A ) ) ) ) ).

% rev_image_eqI
thf(fact_464_ball__imageD,axiom,
    ! [F: set_Product_prod_a_b > set_Product_prod_a_b,A: set_se3214616458824132456od_a_b,P2: set_Product_prod_a_b > $o] :
      ( ! [X3: set_Product_prod_a_b] :
          ( ( member1887650552762594865od_a_b @ X3 @ ( image_5573672758305406725od_a_b @ F @ A ) )
         => ( P2 @ X3 ) )
     => ! [X5: set_Product_prod_a_b] :
          ( ( member1887650552762594865od_a_b @ X5 @ A )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_465_ball__imageD,axiom,
    ! [F: set_Product_prod_a_b > set_a,A: set_se3214616458824132456od_a_b,P2: set_a > $o] :
      ( ! [X3: set_a] :
          ( ( member_set_a @ X3 @ ( image_7003315350810844251_set_a @ F @ A ) )
         => ( P2 @ X3 ) )
     => ! [X5: set_Product_prod_a_b] :
          ( ( member1887650552762594865od_a_b @ X5 @ A )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_466_ball__imageD,axiom,
    ! [F: set_a > set_a,A: set_set_a,P2: set_a > $o] :
      ( ! [X3: set_a] :
          ( ( member_set_a @ X3 @ ( image_set_a_set_a @ F @ A ) )
         => ( P2 @ X3 ) )
     => ! [X5: set_a] :
          ( ( member_set_a @ X5 @ A )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_467_ball__imageD,axiom,
    ! [F: a > product_prod_a_b,A: set_a,P2: product_prod_a_b > $o] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ ( image_7400625786893224495od_a_b @ F @ A ) )
         => ( P2 @ X3 ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ A )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_468_ball__imageD,axiom,
    ! [F: a > a,A: set_a,P2: a > $o] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( image_a_a2 @ F @ A ) )
         => ( P2 @ X3 ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ A )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_469_ball__imageD,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,P2: a > $o] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( image_2802296252294471259_a_b_a @ F @ A ) )
         => ( P2 @ X3 ) )
     => ! [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ A )
         => ( P2 @ ( F @ X5 ) ) ) ) ).

% ball_imageD
thf(fact_470_image__cong,axiom,
    ! [M: set_a,N: set_a,F: a > product_prod_a_b,G: a > product_prod_a_b] :
      ( ( M = N )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_7400625786893224495od_a_b @ F @ M )
          = ( image_7400625786893224495od_a_b @ G @ N ) ) ) ) ).

% image_cong
thf(fact_471_image__cong,axiom,
    ! [M: set_a,N: set_a,F: a > a,G: a > a] :
      ( ( M = N )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_a_a2 @ F @ M )
          = ( image_a_a2 @ G @ N ) ) ) ) ).

% image_cong
thf(fact_472_image__cong,axiom,
    ! [M: set_se3214616458824132456od_a_b,N: set_se3214616458824132456od_a_b,F: set_Product_prod_a_b > set_Product_prod_a_b,G: set_Product_prod_a_b > set_Product_prod_a_b] :
      ( ( M = N )
     => ( ! [X3: set_Product_prod_a_b] :
            ( ( member1887650552762594865od_a_b @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_5573672758305406725od_a_b @ F @ M )
          = ( image_5573672758305406725od_a_b @ G @ N ) ) ) ) ).

% image_cong
thf(fact_473_image__cong,axiom,
    ! [M: set_se3214616458824132456od_a_b,N: set_se3214616458824132456od_a_b,F: set_Product_prod_a_b > set_a,G: set_Product_prod_a_b > set_a] :
      ( ( M = N )
     => ( ! [X3: set_Product_prod_a_b] :
            ( ( member1887650552762594865od_a_b @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_7003315350810844251_set_a @ F @ M )
          = ( image_7003315350810844251_set_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_474_image__cong,axiom,
    ! [M: set_set_a,N: set_set_a,F: set_a > set_a,G: set_a > set_a] :
      ( ( M = N )
     => ( ! [X3: set_a] :
            ( ( member_set_a @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_set_a_set_a @ F @ M )
          = ( image_set_a_set_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_475_image__cong,axiom,
    ! [M: set_Product_prod_a_b,N: set_Product_prod_a_b,F: product_prod_a_b > a,G: product_prod_a_b > a] :
      ( ( M = N )
     => ( ! [X3: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ N )
           => ( ( F @ X3 )
              = ( G @ X3 ) ) )
       => ( ( image_2802296252294471259_a_b_a @ F @ M )
          = ( image_2802296252294471259_a_b_a @ G @ N ) ) ) ) ).

% image_cong
thf(fact_476_bex__imageD,axiom,
    ! [F: set_Product_prod_a_b > set_Product_prod_a_b,A: set_se3214616458824132456od_a_b,P2: set_Product_prod_a_b > $o] :
      ( ? [X5: set_Product_prod_a_b] :
          ( ( member1887650552762594865od_a_b @ X5 @ ( image_5573672758305406725od_a_b @ F @ A ) )
          & ( P2 @ X5 ) )
     => ? [X3: set_Product_prod_a_b] :
          ( ( member1887650552762594865od_a_b @ X3 @ A )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_477_bex__imageD,axiom,
    ! [F: set_Product_prod_a_b > set_a,A: set_se3214616458824132456od_a_b,P2: set_a > $o] :
      ( ? [X5: set_a] :
          ( ( member_set_a @ X5 @ ( image_7003315350810844251_set_a @ F @ A ) )
          & ( P2 @ X5 ) )
     => ? [X3: set_Product_prod_a_b] :
          ( ( member1887650552762594865od_a_b @ X3 @ A )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_478_bex__imageD,axiom,
    ! [F: set_a > set_a,A: set_set_a,P2: set_a > $o] :
      ( ? [X5: set_a] :
          ( ( member_set_a @ X5 @ ( image_set_a_set_a @ F @ A ) )
          & ( P2 @ X5 ) )
     => ? [X3: set_a] :
          ( ( member_set_a @ X3 @ A )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_479_bex__imageD,axiom,
    ! [F: a > product_prod_a_b,A: set_a,P2: product_prod_a_b > $o] :
      ( ? [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( image_7400625786893224495od_a_b @ F @ A ) )
          & ( P2 @ X5 ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ A )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_480_bex__imageD,axiom,
    ! [F: a > a,A: set_a,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( image_a_a2 @ F @ A ) )
          & ( P2 @ X5 ) )
     => ? [X3: a] :
          ( ( member_a @ X3 @ A )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_481_bex__imageD,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,P2: a > $o] :
      ( ? [X5: a] :
          ( ( member_a @ X5 @ ( image_2802296252294471259_a_b_a @ F @ A ) )
          & ( P2 @ X5 ) )
     => ? [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A )
          & ( P2 @ ( F @ X3 ) ) ) ) ).

% bex_imageD
thf(fact_482_image__iff,axiom,
    ! [Z: a,F: a > a,A: set_a] :
      ( ( member_a @ Z @ ( image_a_a2 @ F @ A ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_483_image__iff,axiom,
    ! [Z: product_prod_a_b,F: a > product_prod_a_b,A: set_a] :
      ( ( member1426531481828664017od_a_b @ Z @ ( image_7400625786893224495od_a_b @ F @ A ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_484_image__iff,axiom,
    ! [Z: set_Product_prod_a_b,F: set_Product_prod_a_b > set_Product_prod_a_b,A: set_se3214616458824132456od_a_b] :
      ( ( member1887650552762594865od_a_b @ Z @ ( image_5573672758305406725od_a_b @ F @ A ) )
      = ( ? [X4: set_Product_prod_a_b] :
            ( ( member1887650552762594865od_a_b @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_485_image__iff,axiom,
    ! [Z: set_a,F: set_Product_prod_a_b > set_a,A: set_se3214616458824132456od_a_b] :
      ( ( member_set_a @ Z @ ( image_7003315350810844251_set_a @ F @ A ) )
      = ( ? [X4: set_Product_prod_a_b] :
            ( ( member1887650552762594865od_a_b @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_486_image__iff,axiom,
    ! [Z: set_a,F: set_a > set_a,A: set_set_a] :
      ( ( member_set_a @ Z @ ( image_set_a_set_a @ F @ A ) )
      = ( ? [X4: set_a] :
            ( ( member_set_a @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_487_image__iff,axiom,
    ! [Z: a,F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( member_a @ Z @ ( image_2802296252294471259_a_b_a @ F @ A ) )
      = ( ? [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ A )
            & ( Z
              = ( F @ X4 ) ) ) ) ) ).

% image_iff
thf(fact_488_imageI,axiom,
    ! [X: a,A: set_a,F: a > a] :
      ( ( member_a @ X @ A )
     => ( member_a @ ( F @ X ) @ ( image_a_a2 @ F @ A ) ) ) ).

% imageI
thf(fact_489_imageI,axiom,
    ! [X: a,A: set_a,F: a > product_prod_a_b] :
      ( ( member_a @ X @ A )
     => ( member1426531481828664017od_a_b @ ( F @ X ) @ ( image_7400625786893224495od_a_b @ F @ A ) ) ) ).

% imageI
thf(fact_490_imageI,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,F: product_prod_a_b > a] :
      ( ( member1426531481828664017od_a_b @ X @ A )
     => ( member_a @ ( F @ X ) @ ( image_2802296252294471259_a_b_a @ F @ A ) ) ) ).

% imageI
thf(fact_491_imageI,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,F: product_prod_a_b > product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ A )
     => ( member1426531481828664017od_a_b @ ( F @ X ) @ ( image_3300603549555413765od_a_b @ F @ A ) ) ) ).

% imageI
thf(fact_492_imageI,axiom,
    ! [X: a,A: set_a,F: a > b] :
      ( ( member_a @ X @ A )
     => ( member_b @ ( F @ X ) @ ( image_a_b2 @ F @ A ) ) ) ).

% imageI
thf(fact_493_imageI,axiom,
    ! [X: b,A: set_b,F: b > a] :
      ( ( member_b @ X @ A )
     => ( member_a @ ( F @ X ) @ ( image_b_a @ F @ A ) ) ) ).

% imageI
thf(fact_494_imageI,axiom,
    ! [X: b,A: set_b,F: b > b] :
      ( ( member_b @ X @ A )
     => ( member_b @ ( F @ X ) @ ( image_b_b @ F @ A ) ) ) ).

% imageI
thf(fact_495_imageI,axiom,
    ! [X: a,A: set_a,F: a > list_a] :
      ( ( member_a @ X @ A )
     => ( member_list_a @ ( F @ X ) @ ( image_a_list_a @ F @ A ) ) ) ).

% imageI
thf(fact_496_imageI,axiom,
    ! [X: a,A: set_a,F: a > set_a] :
      ( ( member_a @ X @ A )
     => ( member_set_a @ ( F @ X ) @ ( image_a_set_a @ F @ A ) ) ) ).

% imageI
thf(fact_497_imageI,axiom,
    ! [X: list_a,A: set_list_a,F: list_a > a] :
      ( ( member_list_a @ X @ A )
     => ( member_a @ ( F @ X ) @ ( image_list_a_a @ F @ A ) ) ) ).

% imageI
thf(fact_498_not__Cons__self2,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( cons_a @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_499_not__Cons__self2,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( cons_l907386227682280420od_a_b @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_500_not__Cons__self2,axiom,
    ! [X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( cons_P8222935586017696605od_a_b @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_501_not__Cons__self2,axiom,
    ! [X: b,Xs2: list_b] :
      ( ( cons_b @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_502_not__Cons__self2,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( cons_P7316939131009794654od_a_b @ X @ Xs2 )
     != Xs2 ) ).

% not_Cons_self2
thf(fact_503_set__ConsD,axiom,
    ! [Y: list_a,X: list_a,Xs2: list_list_a] :
      ( ( member_list_a @ Y @ ( set_list_a2 @ ( cons_list_a @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_list_a @ Y @ ( set_list_a2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_504_set__ConsD,axiom,
    ! [Y: set_Product_prod_a_b,X: set_Product_prod_a_b,Xs2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ Y @ ( set_se8479788141692617885od_a_b @ ( cons_s5806323297083630014od_a_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member1887650552762594865od_a_b @ Y @ ( set_se8479788141692617885od_a_b @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_505_set__ConsD,axiom,
    ! [Y: set_a,X: set_a,Xs2: list_set_a] :
      ( ( member_set_a @ Y @ ( set_set_a2 @ ( cons_set_a @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_set_a @ Y @ ( set_set_a2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_506_set__ConsD,axiom,
    ! [Y: produc4115131626046251985od_a_b,X: produc4115131626046251985od_a_b,Xs2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ Y @ ( set_Pr7079953345493216102od_a_b @ ( cons_P135045198529856391od_a_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member3142156093895948538od_a_b @ Y @ ( set_Pr7079953345493216102od_a_b @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_507_set__ConsD,axiom,
    ! [Y: product_prod_a_a,X: product_prod_a_a,Xs2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member1426531477525435216od_a_a @ Y @ ( set_Product_prod_a_a2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_508_set__ConsD,axiom,
    ! [Y: list_P1467974523212632366od_a_b,X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ Y @ ( set_li5045283583078295363od_a_b @ ( cons_l907386227682280420od_a_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member6895035109809442519od_a_b @ Y @ ( set_li5045283583078295363od_a_b @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_509_set__ConsD,axiom,
    ! [Y: produc2233330766403592615od_a_b,X: produc2233330766403592615od_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ Y @ ( set_Pr5228040199190236220od_a_b @ ( cons_P8222935586017696605od_a_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member5065438833300252112od_a_b @ Y @ ( set_Pr5228040199190236220od_a_b @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_510_set__ConsD,axiom,
    ! [Y: b,X: b,Xs2: list_b] :
      ( ( member_b @ Y @ ( set_b2 @ ( cons_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_b @ Y @ ( set_b2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_511_set__ConsD,axiom,
    ! [Y: a,X: a,Xs2: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ ( cons_a @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member_a @ Y @ ( set_a2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_512_set__ConsD,axiom,
    ! [Y: product_prod_a_b,X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) ) )
     => ( ( Y = X )
        | ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ) ).

% set_ConsD
thf(fact_513_list_Oset__cases,axiom,
    ! [E2: list_a,A2: list_list_a] :
      ( ( member_list_a @ E2 @ ( set_list_a2 @ A2 ) )
     => ( ! [Z2: list_list_a] :
            ( A2
           != ( cons_list_a @ E2 @ Z2 ) )
       => ~ ! [Z1: list_a,Z2: list_list_a] :
              ( ( A2
                = ( cons_list_a @ Z1 @ Z2 ) )
             => ~ ( member_list_a @ E2 @ ( set_list_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_514_list_Oset__cases,axiom,
    ! [E2: set_Product_prod_a_b,A2: list_s6539019640784088206od_a_b] :
      ( ( member1887650552762594865od_a_b @ E2 @ ( set_se8479788141692617885od_a_b @ A2 ) )
     => ( ! [Z2: list_s6539019640784088206od_a_b] :
            ( A2
           != ( cons_s5806323297083630014od_a_b @ E2 @ Z2 ) )
       => ~ ! [Z1: set_Product_prod_a_b,Z2: list_s6539019640784088206od_a_b] :
              ( ( A2
                = ( cons_s5806323297083630014od_a_b @ Z1 @ Z2 ) )
             => ~ ( member1887650552762594865od_a_b @ E2 @ ( set_se8479788141692617885od_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_515_list_Oset__cases,axiom,
    ! [E2: set_a,A2: list_set_a] :
      ( ( member_set_a @ E2 @ ( set_set_a2 @ A2 ) )
     => ( ! [Z2: list_set_a] :
            ( A2
           != ( cons_set_a @ E2 @ Z2 ) )
       => ~ ! [Z1: set_a,Z2: list_set_a] :
              ( ( A2
                = ( cons_set_a @ Z1 @ Z2 ) )
             => ~ ( member_set_a @ E2 @ ( set_set_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_516_list_Oset__cases,axiom,
    ! [E2: produc4115131626046251985od_a_b,A2: list_P8504624544267313367od_a_b] :
      ( ( member3142156093895948538od_a_b @ E2 @ ( set_Pr7079953345493216102od_a_b @ A2 ) )
     => ( ! [Z2: list_P8504624544267313367od_a_b] :
            ( A2
           != ( cons_P135045198529856391od_a_b @ E2 @ Z2 ) )
       => ~ ! [Z1: produc4115131626046251985od_a_b,Z2: list_P8504624544267313367od_a_b] :
              ( ( A2
                = ( cons_P135045198529856391od_a_b @ Z1 @ Z2 ) )
             => ~ ( member3142156093895948538od_a_b @ E2 @ ( set_Pr7079953345493216102od_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_517_list_Oset__cases,axiom,
    ! [E2: product_prod_a_a,A2: list_P1396940483166286381od_a_a] :
      ( ( member1426531477525435216od_a_a @ E2 @ ( set_Product_prod_a_a2 @ A2 ) )
     => ( ! [Z2: list_P1396940483166286381od_a_a] :
            ( A2
           != ( cons_P7316939126706565853od_a_a @ E2 @ Z2 ) )
       => ~ ! [Z1: product_prod_a_a,Z2: list_P1396940483166286381od_a_a] :
              ( ( A2
                = ( cons_P7316939126706565853od_a_a @ Z1 @ Z2 ) )
             => ~ ( member1426531477525435216od_a_a @ E2 @ ( set_Product_prod_a_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_518_list_Oset__cases,axiom,
    ! [E2: list_P1467974523212632366od_a_b,A2: list_l8778361445553822004od_a_b] :
      ( ( member6895035109809442519od_a_b @ E2 @ ( set_li5045283583078295363od_a_b @ A2 ) )
     => ( ! [Z2: list_l8778361445553822004od_a_b] :
            ( A2
           != ( cons_l907386227682280420od_a_b @ E2 @ Z2 ) )
       => ~ ! [Z1: list_P1467974523212632366od_a_b,Z2: list_l8778361445553822004od_a_b] :
              ( ( A2
                = ( cons_l907386227682280420od_a_b @ Z1 @ Z2 ) )
             => ~ ( member6895035109809442519od_a_b @ E2 @ ( set_li5045283583078295363od_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_519_list_Oset__cases,axiom,
    ! [E2: produc2233330766403592615od_a_b,A2: list_P3463435030753280173od_a_b] :
      ( ( member5065438833300252112od_a_b @ E2 @ ( set_Pr5228040199190236220od_a_b @ A2 ) )
     => ( ! [Z2: list_P3463435030753280173od_a_b] :
            ( A2
           != ( cons_P8222935586017696605od_a_b @ E2 @ Z2 ) )
       => ~ ! [Z1: produc2233330766403592615od_a_b,Z2: list_P3463435030753280173od_a_b] :
              ( ( A2
                = ( cons_P8222935586017696605od_a_b @ Z1 @ Z2 ) )
             => ~ ( member5065438833300252112od_a_b @ E2 @ ( set_Pr5228040199190236220od_a_b @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_520_list_Oset__cases,axiom,
    ! [E2: b,A2: list_b] :
      ( ( member_b @ E2 @ ( set_b2 @ A2 ) )
     => ( ! [Z2: list_b] :
            ( A2
           != ( cons_b @ E2 @ Z2 ) )
       => ~ ! [Z1: b,Z2: list_b] :
              ( ( A2
                = ( cons_b @ Z1 @ Z2 ) )
             => ~ ( member_b @ E2 @ ( set_b2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_521_list_Oset__cases,axiom,
    ! [E2: a,A2: list_a] :
      ( ( member_a @ E2 @ ( set_a2 @ A2 ) )
     => ( ! [Z2: list_a] :
            ( A2
           != ( cons_a @ E2 @ Z2 ) )
       => ~ ! [Z1: a,Z2: list_a] :
              ( ( A2
                = ( cons_a @ Z1 @ Z2 ) )
             => ~ ( member_a @ E2 @ ( set_a2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_522_list_Oset__cases,axiom,
    ! [E2: product_prod_a_b,A2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ E2 @ ( set_Product_prod_a_b2 @ A2 ) )
     => ( ! [Z2: list_P1467974523212632366od_a_b] :
            ( A2
           != ( cons_P7316939131009794654od_a_b @ E2 @ Z2 ) )
       => ~ ! [Z1: product_prod_a_b,Z2: list_P1467974523212632366od_a_b] :
              ( ( A2
                = ( cons_P7316939131009794654od_a_b @ Z1 @ Z2 ) )
             => ~ ( member1426531481828664017od_a_b @ E2 @ ( set_Product_prod_a_b2 @ Z2 ) ) ) ) ) ).

% list.set_cases
thf(fact_523_list_Oset__intros_I1_J,axiom,
    ! [X21: list_a,X222: list_list_a] : ( member_list_a @ X21 @ ( set_list_a2 @ ( cons_list_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_524_list_Oset__intros_I1_J,axiom,
    ! [X21: set_Product_prod_a_b,X222: list_s6539019640784088206od_a_b] : ( member1887650552762594865od_a_b @ X21 @ ( set_se8479788141692617885od_a_b @ ( cons_s5806323297083630014od_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_525_list_Oset__intros_I1_J,axiom,
    ! [X21: set_a,X222: list_set_a] : ( member_set_a @ X21 @ ( set_set_a2 @ ( cons_set_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_526_list_Oset__intros_I1_J,axiom,
    ! [X21: produc4115131626046251985od_a_b,X222: list_P8504624544267313367od_a_b] : ( member3142156093895948538od_a_b @ X21 @ ( set_Pr7079953345493216102od_a_b @ ( cons_P135045198529856391od_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_527_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_a_a,X222: list_P1396940483166286381od_a_a] : ( member1426531477525435216od_a_a @ X21 @ ( set_Product_prod_a_a2 @ ( cons_P7316939126706565853od_a_a @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_528_list_Oset__intros_I1_J,axiom,
    ! [X21: list_P1467974523212632366od_a_b,X222: list_l8778361445553822004od_a_b] : ( member6895035109809442519od_a_b @ X21 @ ( set_li5045283583078295363od_a_b @ ( cons_l907386227682280420od_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_529_list_Oset__intros_I1_J,axiom,
    ! [X21: produc2233330766403592615od_a_b,X222: list_P3463435030753280173od_a_b] : ( member5065438833300252112od_a_b @ X21 @ ( set_Pr5228040199190236220od_a_b @ ( cons_P8222935586017696605od_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_530_list_Oset__intros_I1_J,axiom,
    ! [X21: b,X222: list_b] : ( member_b @ X21 @ ( set_b2 @ ( cons_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_531_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_532_list_Oset__intros_I1_J,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] : ( member1426531481828664017od_a_b @ X21 @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) ) ) ).

% list.set_intros(1)
thf(fact_533_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P321204300973800749list_a,Y: $o] :
      ( ( ( wf_lis1674799249052484319list_a @ X )
        = Y )
     => ( ( ( X = nil_Pr3188421586756112173list_a )
         => ~ Y )
       => ~ ! [V: list_a,E: list_a,Xs: list_P321204300973800749list_a] :
              ( ( X
                = ( cons_P5184657343811988189list_a @ ( produc6837034575241423639list_a @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member_list_a @ V @ ( image_5971222721807453474list_a @ produc3698117735987127555list_a @ ( set_Pr2906193453920572092list_a @ Xs ) ) )
                      & ( wf_lis1674799249052484319list_a @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_534_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P8094385941167427565od_a_b,Y: $o] :
      ( ( ( wf_lis7776600765086547103od_a_b @ X )
        = Y )
     => ( ( ( X = nil_Pr7311172237979606381od_a_b )
         => ~ Y )
       => ~ ! [V: product_prod_a_b > product_prod_a_b > $o,E: list_P1467974523212632366od_a_b,Xs: list_P8094385941167427565od_a_b] :
              ( ( X
                = ( cons_P3561991119763190045od_a_b @ ( produc7704224619955794135od_a_b @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member6766302768593690647_a_b_o @ V @ ( image_9098747296146071628_a_b_o @ produc8595103656739649731od_a_b @ ( set_Pr7552351584573503100od_a_b @ Xs ) ) )
                      & ( wf_lis7776600765086547103od_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_535_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P8504624544267313367od_a_b,Y: $o] :
      ( ( ( wf_lis4492982056392339337od_a_b @ X )
        = Y )
     => ( ( ( X = nil_Pr7362181478328756183od_a_b )
         => ~ Y )
       => ~ ! [V: a,E: product_prod_a_b,Xs: list_P8504624544267313367od_a_b] :
              ( ( X
                = ( cons_P135045198529856391od_a_b @ ( produc431845345726502849od_a_b @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member_a @ V @ ( image_4017424819796311410_a_b_a @ produc6516300543326982573od_a_b @ ( set_Pr7079953345493216102od_a_b @ Xs ) ) )
                      & ( wf_lis4492982056392339337od_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_536_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P5700792096213675741od_a_b,Y: $o] :
      ( ( ( wf_lis8345819997161463695od_a_b @ X )
        = Y )
     => ( ( ( X = nil_Pr7243433834293086045od_a_b )
         => ~ Y )
       => ~ ! [V: a,E: list_P1467974523212632366od_a_b,Xs: list_P5700792096213675741od_a_b] :
              ( ( X
                = ( cons_P8483812562729421581od_a_b @ ( produc6643264353547226055od_a_b @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member_a @ V @ ( image_2222990102783010156_a_b_a @ produc7763512175944540083od_a_b @ ( set_Pr187010687427835756od_a_b @ Xs ) ) )
                      & ( wf_lis8345819997161463695od_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_537_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P3463435030753280173od_a_b,Y: $o] :
      ( ( ( wf_lis873957137097608799od_a_b @ X )
        = Y )
     => ( ( ( X = nil_Pr906174322249854893od_a_b )
         => ~ Y )
       => ~ ! [V: product_prod_a_b,E: product_prod_a_b,Xs: list_P3463435030753280173od_a_b] :
              ( ( X
                = ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member1426531481828664017od_a_b @ V @ ( image_2926283117629707974od_a_b @ produc5013601669340818563od_a_b @ ( set_Pr5228040199190236220od_a_b @ Xs ) ) )
                      & ( wf_lis873957137097608799od_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_538_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P5360711134766023690_a_b_b,Y: $o] :
      ( ( ( wf_lis4446142942163678948_a_b_b @ X )
        = Y )
     => ( ( ( X = nil_Pr5403147374433350260_a_b_b )
         => ~ Y )
       => ~ ! [V: dtree_a_b,E: b,Xs: list_P5360711134766023690_a_b_b] :
              ( ( X
                = ( cons_P2785074014425598660_a_b_b @ ( produc331601717337510060_a_b_b @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member_dtree_a_b @ V @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( set_Pr8239206241032387877_a_b_b @ Xs ) ) )
                      & ( wf_lis4446142942163678948_a_b_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_539_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P1396940483166286381od_a_a,Y: $o] :
      ( ( ( wf_list_verts_a_a @ X )
        = Y )
     => ( ( ( X = nil_Product_prod_a_a )
         => ~ Y )
       => ~ ! [V: a,E: a,Xs: list_P1396940483166286381od_a_a] :
              ( ( X
                = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member_a @ V @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xs ) ) )
                      & ( wf_list_verts_a_a @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_540_wf__list__verts_Oelims_I1_J,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: $o] :
      ( ( ( wf_list_verts_a_b @ X )
        = Y )
     => ( ( ( X = nil_Product_prod_a_b )
         => ~ Y )
       => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( X
                = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member_a @ V @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                      & ( wf_list_verts_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.elims(1)
thf(fact_541_wf__list__verts_Oelims_I2_J,axiom,
    ! [X: list_P5360711134766023690_a_b_b] :
      ( ( wf_lis4446142942163678948_a_b_b @ X )
     => ( ( X != nil_Pr5403147374433350260_a_b_b )
       => ~ ! [V: dtree_a_b,E: b,Xs: list_P5360711134766023690_a_b_b] :
              ( ( X
                = ( cons_P2785074014425598660_a_b_b @ ( produc331601717337510060_a_b_b @ V @ E ) @ Xs ) )
             => ~ ( ~ ( member_dtree_a_b @ V @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( set_Pr8239206241032387877_a_b_b @ Xs ) ) )
                  & ( wf_lis4446142942163678948_a_b_b @ Xs ) ) ) ) ) ).

% wf_list_verts.elims(2)
thf(fact_542_wf__list__verts_Oelims_I2_J,axiom,
    ! [X: list_P1396940483166286381od_a_a] :
      ( ( wf_list_verts_a_a @ X )
     => ( ( X != nil_Product_prod_a_a )
       => ~ ! [V: a,E: a,Xs: list_P1396940483166286381od_a_a] :
              ( ( X
                = ( cons_P7316939126706565853od_a_a @ ( product_Pair_a_a @ V @ E ) @ Xs ) )
             => ~ ( ~ ( member_a @ V @ ( image_3437945252899457948_a_a_a @ product_fst_a_a @ ( set_Product_prod_a_a2 @ Xs ) ) )
                  & ( wf_list_verts_a_a @ Xs ) ) ) ) ) ).

% wf_list_verts.elims(2)
thf(fact_543_wf__list__verts_Oelims_I2_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( wf_list_verts_a_b @ X )
     => ( ( X != nil_Product_prod_a_b )
       => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( X
                = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
             => ~ ( ~ ( member_a @ V @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                  & ( wf_list_verts_a_b @ Xs ) ) ) ) ) ).

% wf_list_verts.elims(2)
thf(fact_544_dtree__from__list__uneq__sequence__xs,axiom,
    ! [R0: a,Xs0: fset_P5281107635120001194_a_b_b,V1: a,Ys: list_P1467974523212632366od_a_b] :
      ( ( strict_subtree_a_b @ ( node_a_b @ R0 @ Xs0 ) @ ( dtree_from_list_a_b @ V1 @ Ys ) )
     => ? [E: b,As2: list_P1467974523212632366od_a_b,Bs2: list_P1467974523212632366od_a_b] :
          ( ( ( append5335208823350062147od_a_b @ As2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R0 @ E ) @ Bs2 ) )
            = Ys )
          & ( ( node_a_b @ R0 @ Xs0 )
            = ( dtree_from_list_a_b @ R0 @ Bs2 ) ) ) ) ).

% dtree_from_list_uneq_sequence_xs
thf(fact_545_bind__simps_I2_J,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,F: product_prod_a_b > list_P1467974523212632366od_a_b] :
      ( ( bind_P7003926910255595655od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ F )
      = ( append5335208823350062147od_a_b @ ( F @ X ) @ ( bind_P7003926910255595655od_a_b @ Xs2 @ F ) ) ) ).

% bind_simps(2)
thf(fact_546_dtree__from__list__eq__dverts,axiom,
    ! [R2: a,Xs2: list_P1467974523212632366od_a_b] :
      ( ( dverts_a_b @ ( dtree_from_list_a_b @ R2 @ Xs2 ) )
      = ( insert_a2 @ R2 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ) ).

% dtree_from_list_eq_dverts
thf(fact_547_maps__simps_I1_J,axiom,
    ! [F: product_prod_a_b > list_P1467974523212632366od_a_b,X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( maps_P4858162816583320641od_a_b @ F @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
      = ( append5335208823350062147od_a_b @ ( F @ X ) @ ( maps_P4858162816583320641od_a_b @ F @ Xs2 ) ) ) ).

% maps_simps(1)
thf(fact_548_List_Oset__insert,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( set_a2 @ ( insert_a @ X @ Xs2 ) )
      = ( insert_a2 @ X @ ( set_a2 @ Xs2 ) ) ) ).

% List.set_insert
thf(fact_549_List_Oset__insert,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( insert7736115125267272132od_a_b @ X @ Xs2 ) )
      = ( insert4534936386344385144od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ).

% List.set_insert
thf(fact_550_not__in__set__insert,axiom,
    ! [X: a,Xs2: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ( ( insert_a @ X @ Xs2 )
        = ( cons_a @ X @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_551_not__in__set__insert,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( ( insert7736115125267272132od_a_b @ X @ Xs2 )
        = ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) ) ) ).

% not_in_set_insert
thf(fact_552_dtree__from__list__sequence__dverts,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,V1: a,Ys: list_P1467974523212632366od_a_b,E1: b] :
      ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Xs2 ) @ ( dtree_from_list_a_b @ V1 @ Ys ) )
     => ? [E: b,As2: list_P1467974523212632366od_a_b,Bs2: list_P1467974523212632366od_a_b] :
          ( ( ( append5335208823350062147od_a_b @ As2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E ) @ Bs2 ) )
            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V1 @ E1 ) @ Ys ) )
          & ( ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
            = ( insert_a2 @ R2 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Bs2 ) ) ) ) ) ) ).

% dtree_from_list_sequence_dverts
thf(fact_553_subtree__from__list__split__eq__if__wfverts,axiom,
    ! [As: list_P1467974523212632366od_a_b,R2: a,E2: b,Bs: list_P1467974523212632366od_a_b,V2: a,Xs2: fset_P5281107635120001194_a_b_b] :
      ( ( wf_list_verts_a_b @ ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E2 ) @ Bs ) ) )
     => ( ~ ( member_a @ V2 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E2 ) @ Bs ) ) ) ) )
       => ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Xs2 ) @ ( dtree_from_list_a_b @ V2 @ ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E2 ) @ Bs ) ) ) )
         => ( ( node_a_b @ R2 @ Xs2 )
            = ( dtree_from_list_a_b @ R2 @ Bs ) ) ) ) ) ).

% subtree_from_list_split_eq_if_wfverts
thf(fact_554_append__is__Nil__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ Ys )
        = nil_Product_prod_a_b )
      = ( ( Xs2 = nil_Product_prod_a_b )
        & ( Ys = nil_Product_prod_a_b ) ) ) ).

% append_is_Nil_conv
thf(fact_555_Nil__is__append__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( nil_Product_prod_a_b
        = ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
      = ( ( Xs2 = nil_Product_prod_a_b )
        & ( Ys = nil_Product_prod_a_b ) ) ) ).

% Nil_is_append_conv
thf(fact_556_self__append__conv2,axiom,
    ! [Y: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( Y
        = ( append5335208823350062147od_a_b @ Xs2 @ Y ) )
      = ( Xs2 = nil_Product_prod_a_b ) ) ).

% self_append_conv2
thf(fact_557_append__self__conv2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ Ys )
        = Ys )
      = ( Xs2 = nil_Product_prod_a_b ) ) ).

% append_self_conv2
thf(fact_558_self__append__conv,axiom,
    ! [Y: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( Y
        = ( append5335208823350062147od_a_b @ Y @ Ys ) )
      = ( Ys = nil_Product_prod_a_b ) ) ).

% self_append_conv
thf(fact_559_append__self__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ Ys )
        = Xs2 )
      = ( Ys = nil_Product_prod_a_b ) ) ).

% append_self_conv
thf(fact_560_append__Nil2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( append5335208823350062147od_a_b @ Xs2 @ nil_Product_prod_a_b )
      = Xs2 ) ).

% append_Nil2
thf(fact_561_append_Oright__neutral,axiom,
    ! [A2: list_P1467974523212632366od_a_b] :
      ( ( append5335208823350062147od_a_b @ A2 @ nil_Product_prod_a_b )
      = A2 ) ).

% append.right_neutral
thf(fact_562_in__set__insert,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ( ( insert_a @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_563_in__set__insert,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( ( insert7736115125267272132od_a_b @ X @ Xs2 )
        = Xs2 ) ) ).

% in_set_insert
thf(fact_564_append1__eq__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,Y: product_prod_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) )
        = ( append5335208823350062147od_a_b @ Ys @ ( cons_P7316939131009794654od_a_b @ Y @ nil_Product_prod_a_b ) ) )
      = ( ( Xs2 = Ys )
        & ( X = Y ) ) ) ).

% append1_eq_conv
thf(fact_565_insert__Nil,axiom,
    ! [X: product_prod_a_b] :
      ( ( insert7736115125267272132od_a_b @ X @ nil_Product_prod_a_b )
      = ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) ).

% insert_Nil
thf(fact_566_transpose_Ocases,axiom,
    ! [X: list_l8778361445553822004od_a_b] :
      ( ( X != nil_li8635777946366267444od_a_b )
     => ( ! [Xss: list_l8778361445553822004od_a_b] :
            ( X
           != ( cons_l907386227682280420od_a_b @ nil_Product_prod_a_b @ Xss ) )
       => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Xss: list_l8778361445553822004od_a_b] :
              ( X
             != ( cons_l907386227682280420od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Xss ) ) ) ) ).

% transpose.cases
thf(fact_567_list_Odistinct_I1_J,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( nil_Product_prod_a_b
     != ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) ) ).

% list.distinct(1)
thf(fact_568_list_OdiscI,axiom,
    ! [List: list_P1467974523212632366od_a_b,X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( ( List
        = ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) )
     => ( List != nil_Product_prod_a_b ) ) ).

% list.discI
thf(fact_569_list_Oexhaust,axiom,
    ! [Y: list_P1467974523212632366od_a_b] :
      ( ( Y != nil_Product_prod_a_b )
     => ~ ! [X212: product_prod_a_b,X223: list_P1467974523212632366od_a_b] :
            ( Y
           != ( cons_P7316939131009794654od_a_b @ X212 @ X223 ) ) ) ).

% list.exhaust
thf(fact_570_neq__Nil__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
      = ( ? [Y3: product_prod_a_b,Ys2: list_P1467974523212632366od_a_b] :
            ( Xs2
            = ( cons_P7316939131009794654od_a_b @ Y3 @ Ys2 ) ) ) ) ).

% neq_Nil_conv
thf(fact_571_list__induct2_H,axiom,
    ! [P2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( P2 @ nil_Product_prod_a_b @ nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] : ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ nil_Product_prod_a_b )
       => ( ! [Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] : ( P2 @ nil_Product_prod_a_b @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) )
         => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
                ( ( P2 @ Xs @ Ys3 )
               => ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) )
           => ( P2 @ Xs2 @ Ys ) ) ) ) ) ).

% list_induct2'
thf(fact_572_list__nonempty__induct,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b] : ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
       => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( ( Xs != nil_Product_prod_a_b )
             => ( ( P2 @ Xs )
               => ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) ) ) )
         => ( P2 @ Xs2 ) ) ) ) ).

% list_nonempty_induct
thf(fact_573_successively_Ocases,axiom,
    ! [X: produc3334210663054469735od_a_b] :
      ( ! [P3: product_prod_a_b > product_prod_a_b > $o] :
          ( X
         != ( produc7704224619955794135od_a_b @ P3 @ nil_Product_prod_a_b ) )
     => ( ! [P3: product_prod_a_b > product_prod_a_b > $o,X3: product_prod_a_b] :
            ( X
           != ( produc7704224619955794135od_a_b @ P3 @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) ) )
       => ~ ! [P3: product_prod_a_b > product_prod_a_b > $o,X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( X
             != ( produc7704224619955794135od_a_b @ P3 @ ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) ) ) ) ).

% successively.cases
thf(fact_574_sorted__wrt_Ocases,axiom,
    ! [X: produc3334210663054469735od_a_b] :
      ( ! [P3: product_prod_a_b > product_prod_a_b > $o] :
          ( X
         != ( produc7704224619955794135od_a_b @ P3 @ nil_Product_prod_a_b ) )
     => ~ ! [P3: product_prod_a_b > product_prod_a_b > $o,X3: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc7704224619955794135od_a_b @ P3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Ys3 ) ) ) ) ).

% sorted_wrt.cases
thf(fact_575_shuffles_Ocases,axiom,
    ! [X: produc7397007428618769063od_a_b] :
      ( ! [Ys3: list_P1467974523212632366od_a_b] :
          ( X
         != ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Ys3 ) )
     => ( ! [Xs: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc5288560138351607191od_a_b @ Xs @ nil_Product_prod_a_b ) )
       => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
              ( X
             != ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) ) ) ) ).

% shuffles.cases
thf(fact_576_eq__Nil__appendI,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( Xs2 = Ys )
     => ( Xs2
        = ( append5335208823350062147od_a_b @ nil_Product_prod_a_b @ Ys ) ) ) ).

% eq_Nil_appendI
thf(fact_577_append_Oleft__neutral,axiom,
    ! [A2: list_P1467974523212632366od_a_b] :
      ( ( append5335208823350062147od_a_b @ nil_Product_prod_a_b @ A2 )
      = A2 ) ).

% append.left_neutral
thf(fact_578_append__Nil,axiom,
    ! [Ys: list_P1467974523212632366od_a_b] :
      ( ( append5335208823350062147od_a_b @ nil_Product_prod_a_b @ Ys )
      = Ys ) ).

% append_Nil
thf(fact_579_dtree__from__list__v__eq__r,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,V2: a,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( node_a_b @ R2 @ Xs2 )
        = ( dtree_from_list_a_b @ V2 @ Ys ) )
     => ( R2 = V2 ) ) ).

% dtree_from_list_v_eq_r
thf(fact_580_strict__subtree__def,axiom,
    ( strict_subtree_a_b
    = ( ^ [T1: dtree_a_b,T2: dtree_a_b] :
          ( ( is_subtree_a_b @ T1 @ T2 )
          & ( T1 != T2 ) ) ) ) ).

% strict_subtree_def
thf(fact_581_dtree__to__from__list__id,axiom,
    ! [R2: a,Xs2: list_P1467974523212632366od_a_b] :
      ( ( dtree_to_list_a_b @ ( dtree_from_list_a_b @ R2 @ Xs2 ) )
      = Xs2 ) ).

% dtree_to_from_list_id
thf(fact_582_dtree__from__list__sequence__xs,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,V1: a,Ys: list_P1467974523212632366od_a_b,E1: b] :
      ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Xs2 ) @ ( dtree_from_list_a_b @ V1 @ Ys ) )
     => ? [E: b,As2: list_P1467974523212632366od_a_b,Bs2: list_P1467974523212632366od_a_b] :
          ( ( ( append5335208823350062147od_a_b @ As2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E ) @ Bs2 ) )
            = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V1 @ E1 ) @ Ys ) )
          & ( ( node_a_b @ R2 @ Xs2 )
            = ( dtree_from_list_a_b @ R2 @ Bs2 ) ) ) ) ).

% dtree_from_list_sequence_xs
thf(fact_583_subtree__root__if__dverts,axiom,
    ! [X: a,T: dtree_a_b] :
      ( ( member_a @ X @ ( dverts_a_b @ T ) )
     => ? [Xs: fset_P5281107635120001194_a_b_b] : ( is_subtree_a_b @ ( node_a_b @ X @ Xs ) @ T ) ) ).

% subtree_root_if_dverts
thf(fact_584_rev__nonempty__induct,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > $o] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b] : ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
       => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( ( Xs != nil_Product_prod_a_b )
             => ( ( P2 @ Xs )
               => ( P2 @ ( append5335208823350062147od_a_b @ Xs @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) ) ) ) )
         => ( P2 @ Xs2 ) ) ) ) ).

% rev_nonempty_induct
thf(fact_585_append__eq__Cons__conv,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Ys @ Zs )
        = ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
      = ( ( ( Ys = nil_Product_prod_a_b )
          & ( Zs
            = ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) ) )
        | ? [Ys5: list_P1467974523212632366od_a_b] :
            ( ( Ys
              = ( cons_P7316939131009794654od_a_b @ X @ Ys5 ) )
            & ( ( append5335208823350062147od_a_b @ Ys5 @ Zs )
              = Xs2 ) ) ) ) ).

% append_eq_Cons_conv
thf(fact_586_Cons__eq__append__conv,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( cons_P7316939131009794654od_a_b @ X @ Xs2 )
        = ( append5335208823350062147od_a_b @ Ys @ Zs ) )
      = ( ( ( Ys = nil_Product_prod_a_b )
          & ( ( cons_P7316939131009794654od_a_b @ X @ Xs2 )
            = Zs ) )
        | ? [Ys5: list_P1467974523212632366od_a_b] :
            ( ( ( cons_P7316939131009794654od_a_b @ X @ Ys5 )
              = Ys )
            & ( Xs2
              = ( append5335208823350062147od_a_b @ Ys5 @ Zs ) ) ) ) ) ).

% Cons_eq_append_conv
thf(fact_587_rev__exhaust,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ~ ! [Ys3: list_P1467974523212632366od_a_b,Y4: product_prod_a_b] :
            ( Xs2
           != ( append5335208823350062147od_a_b @ Ys3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ nil_Product_prod_a_b ) ) ) ) ).

% rev_exhaust
thf(fact_588_rev__induct,axiom,
    ! [P2: list_P1467974523212632366od_a_b > $o,Xs2: list_P1467974523212632366od_a_b] :
      ( ( P2 @ nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
            ( ( P2 @ Xs )
           => ( P2 @ ( append5335208823350062147od_a_b @ Xs @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) ) ) )
       => ( P2 @ Xs2 ) ) ) ).

% rev_induct
thf(fact_589_wf__list__verts_Ocases,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( X != nil_Product_prod_a_b )
     => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
            ( X
           != ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) ) ) ).

% wf_list_verts.cases
thf(fact_590_dtree__from__list_Ocases,axiom,
    ! [X: produc2639161875728825943od_a_b] :
      ( ! [R: a] :
          ( X
         != ( produc6643264353547226055od_a_b @ R @ nil_Product_prod_a_b ) )
     => ~ ! [R: a,V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc6643264353547226055od_a_b @ R @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) ) ) ) ).

% dtree_from_list.cases
thf(fact_591_List_Oinsert__def,axiom,
    ( insert_a
    = ( ^ [X4: a,Xs4: list_a] : ( if_list_a @ ( member_a @ X4 @ ( set_a2 @ Xs4 ) ) @ Xs4 @ ( cons_a @ X4 @ Xs4 ) ) ) ) ).

% List.insert_def
thf(fact_592_List_Oinsert__def,axiom,
    ( insert7736115125267272132od_a_b
    = ( ^ [X4: product_prod_a_b,Xs4: list_P1467974523212632366od_a_b] : ( if_lis1002476807507936500od_a_b @ ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Xs4 ) ) @ Xs4 @ ( cons_P7316939131009794654od_a_b @ X4 @ Xs4 ) ) ) ) ).

% List.insert_def
thf(fact_593_subtree__from__list__split__eq__if__wfdverts,axiom,
    ! [V2: a,As: list_P1467974523212632366od_a_b,R2: a,E2: b,Bs: list_P1467974523212632366od_a_b,Xs2: fset_P5281107635120001194_a_b_b] :
      ( ( wf_dverts_a_b @ ( dtree_from_list_a_b @ V2 @ ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E2 ) @ Bs ) ) ) )
     => ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Xs2 ) @ ( dtree_from_list_a_b @ V2 @ ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E2 ) @ Bs ) ) ) )
       => ( ( node_a_b @ R2 @ Xs2 )
          = ( dtree_from_list_a_b @ R2 @ Bs ) ) ) ) ).

% subtree_from_list_split_eq_if_wfdverts
thf(fact_594_vwalk__arcs_Ocases,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( X != nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b] :
            ( X
           != ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
       => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( X
             != ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) ) ) ).

% vwalk_arcs.cases
thf(fact_595_wf__list__verts__if__wf__dverts,axiom,
    ! [T: dtree_a_b] :
      ( ( wf_dverts_a_b @ T )
     => ( wf_list_verts_a_b @ ( dtree_to_list_a_b @ T ) ) ) ).

% wf_list_verts_if_wf_dverts
thf(fact_596_wf__dverts__iff__wf__list__verts,axiom,
    ! [R2: a,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ~ ( member_a @ R2 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) ) )
        & ( wf_list_verts_a_b @ Xs2 ) )
      = ( wf_dverts_a_b @ ( dtree_from_list_a_b @ R2 @ Xs2 ) ) ) ).

% wf_dverts_iff_wf_list_verts
thf(fact_597_vwalk__to__vpath_Ocases,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( X != nil_Product_prod_a_b )
     => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
            ( X
           != ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) ) ) ).

% vwalk_to_vpath.cases
thf(fact_598_subset__eq__mset__impl_Ocases,axiom,
    ! [X: produc7397007428618769063od_a_b] :
      ( ! [Ys3: list_P1467974523212632366od_a_b] :
          ( X
         != ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Ys3 ) )
     => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Ys3: list_P1467974523212632366od_a_b] :
            ( X
           != ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Ys3 ) ) ) ).

% subset_eq_mset_impl.cases
thf(fact_599_list__exhaust__NSC,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( ! [X3: product_prod_a_b] :
            ( Xs2
           != ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
       => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
              ( Xs2
             != ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) ) ) ) ).

% list_exhaust_NSC
thf(fact_600_list__exhaust2,axiom,
    ! [Y: list_P1467974523212632366od_a_b,Ya: list_P1467974523212632366od_a_b] :
      ( ( ( Y = nil_Product_prod_a_b )
       => ( Ya != nil_Product_prod_a_b ) )
     => ( ( ( Y = nil_Product_prod_a_b )
         => ! [X212: product_prod_a_b,X223: list_P1467974523212632366od_a_b] :
              ( Ya
             != ( cons_P7316939131009794654od_a_b @ X212 @ X223 ) ) )
       => ( ( ? [X212: product_prod_a_b,X223: list_P1467974523212632366od_a_b] :
                ( Y
                = ( cons_P7316939131009794654od_a_b @ X212 @ X223 ) )
           => ( Ya != nil_Product_prod_a_b ) )
         => ~ ( ? [X212: product_prod_a_b,X223: list_P1467974523212632366od_a_b] :
                  ( Y
                  = ( cons_P7316939131009794654od_a_b @ X212 @ X223 ) )
             => ! [X21a: product_prod_a_b,X22a: list_P1467974523212632366od_a_b] :
                  ( Ya
                 != ( cons_P7316939131009794654od_a_b @ X21a @ X22a ) ) ) ) ) ) ).

% list_exhaust2
thf(fact_601_Cons__in__subseqsD,axiom,
    ! [Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) @ ( set_li5045283583078295363od_a_b @ ( subseq3159716953272533757od_a_b @ Xs2 ) ) )
     => ( member6895035109809442519od_a_b @ Ys @ ( set_li5045283583078295363od_a_b @ ( subseq3159716953272533757od_a_b @ Xs2 ) ) ) ) ).

% Cons_in_subseqsD
thf(fact_602_snoc__listrel1__snoc__iff,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,Y: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) @ ( append5335208823350062147od_a_b @ Ys @ ( cons_P7316939131009794654od_a_b @ Y @ nil_Product_prod_a_b ) ) ) @ ( listre2955263632500257159od_a_b @ R2 ) )
      = ( ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) )
          & ( X = Y ) )
        | ( ( Xs2 = Ys )
          & ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 ) ) ) ) ).

% snoc_listrel1_snoc_iff
thf(fact_603_vwalk__arcs_Oelims,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: list_P3463435030753280173od_a_b] :
      ( ( ( vertex3879532237292798496od_a_b @ X )
        = Y )
     => ( ( ( X = nil_Product_prod_a_b )
         => ( Y != nil_Pr906174322249854893od_a_b ) )
       => ( ( ? [X3: product_prod_a_b] :
                ( X
                = ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
           => ( Y != nil_Pr906174322249854893od_a_b ) )
         => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) )
               => ( Y
                 != ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y4 ) @ ( vertex3879532237292798496od_a_b @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) ) ) ) ) ) ).

% vwalk_arcs.elims
thf(fact_604_the__elem__set,axiom,
    ! [X: product_prod_a_b] :
      ( ( the_el8589169213296894365od_a_b @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) )
      = X ) ).

% the_elem_set
thf(fact_605_Cons__listrel1__Cons,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) @ ( listre2955263632500257159od_a_b @ R2 ) )
      = ( ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
          & ( Xs2 = Ys ) )
        | ( ( X = Y )
          & ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) ) ) ).

% Cons_listrel1_Cons
thf(fact_606_in__set__vwalk__arcs__append1,axiom,
    ! [X: produc2233330766403592615od_a_b,P: list_P1467974523212632366od_a_b,Q2: list_P1467974523212632366od_a_b] :
      ( ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ P ) ) )
        | ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ Q2 ) ) ) )
     => ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ ( append5335208823350062147od_a_b @ P @ Q2 ) ) ) ) ) ).

% in_set_vwalk_arcs_append1
thf(fact_607_listrel1I2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b,X: product_prod_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) )
     => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ X @ Ys ) ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) ).

% listrel1I2
thf(fact_608_append__listrel1I,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b,Us: list_P1467974523212632366od_a_b,Vs: list_P1467974523212632366od_a_b] :
      ( ( ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) )
          & ( Us = Vs ) )
        | ( ( Xs2 = Ys )
          & ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Us @ Vs ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) )
     => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Us ) @ ( append5335208823350062147od_a_b @ Ys @ Vs ) ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) ).

% append_listrel1I
thf(fact_609_vwalk__arcs_Osimps_I3_J,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( vertex3879532237292798496od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) ) )
      = ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ ( vertex3879532237292798496od_a_b @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) ) ) ) ).

% vwalk_arcs.simps(3)
thf(fact_610_in__set__vwalk__arcsE,axiom,
    ! [U: a,V2: a,P: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ U @ V2 ) @ ( set_Product_prod_a_a2 @ ( vertex_vwalk_arcs_a @ P ) ) )
     => ~ ( ( member_a @ U @ ( set_a2 @ P ) )
         => ~ ( member_a @ V2 @ ( set_a2 @ P ) ) ) ) ).

% in_set_vwalk_arcsE
thf(fact_611_in__set__vwalk__arcsE,axiom,
    ! [U: product_prod_a_b,V2: product_prod_a_b,P: list_P1467974523212632366od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ U @ V2 ) @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ P ) ) )
     => ~ ( ( member1426531481828664017od_a_b @ U @ ( set_Product_prod_a_b2 @ P ) )
         => ~ ( member1426531481828664017od_a_b @ V2 @ ( set_Product_prod_a_b2 @ P ) ) ) ) ).

% in_set_vwalk_arcsE
thf(fact_612_vwalk__arcs_Osimps_I2_J,axiom,
    ! [X: product_prod_a_b] :
      ( ( vertex3879532237292798496od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) )
      = nil_Pr906174322249854893od_a_b ) ).

% vwalk_arcs.simps(2)
thf(fact_613_Cons__listrel1E2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) @ ( listre2955263632500257159od_a_b @ R2 ) )
     => ( ! [X3: product_prod_a_b] :
            ( ( Xs2
              = ( cons_P7316939131009794654od_a_b @ X3 @ Ys ) )
           => ~ ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y ) @ R2 ) )
       => ~ ! [Zs3: list_P1467974523212632366od_a_b] :
              ( ( Xs2
                = ( cons_P7316939131009794654od_a_b @ Y @ Zs3 ) )
             => ~ ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Zs3 @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) ) ) ).

% Cons_listrel1E2
thf(fact_614_Cons__listrel1E1,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) )
     => ( ! [Y4: product_prod_a_b] :
            ( ( Ys
              = ( cons_P7316939131009794654od_a_b @ Y4 @ Xs2 ) )
           => ~ ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y4 ) @ R2 ) )
       => ~ ! [Zs3: list_P1467974523212632366od_a_b] :
              ( ( Ys
                = ( cons_P7316939131009794654od_a_b @ X @ Zs3 ) )
             => ~ ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Zs3 ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) ) ) ).

% Cons_listrel1E1
thf(fact_615_listrel1I1,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
     => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) ).

% listrel1I1
thf(fact_616_listrel1I,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,Xs2: list_P1467974523212632366od_a_b,Us: list_P1467974523212632366od_a_b,Vs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
     => ( ( Xs2
          = ( append5335208823350062147od_a_b @ Us @ ( cons_P7316939131009794654od_a_b @ X @ Vs ) ) )
       => ( ( Ys
            = ( append5335208823350062147od_a_b @ Us @ ( cons_P7316939131009794654od_a_b @ Y @ Vs ) ) )
         => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) ) ) ) ) ).

% listrel1I
thf(fact_617_listrel1E,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( listre2955263632500257159od_a_b @ R2 ) )
     => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b] :
            ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y4 ) @ R2 )
           => ! [Us3: list_P1467974523212632366od_a_b,Vs2: list_P1467974523212632366od_a_b] :
                ( ( Xs2
                  = ( append5335208823350062147od_a_b @ Us3 @ ( cons_P7316939131009794654od_a_b @ X3 @ Vs2 ) ) )
               => ( Ys
                 != ( append5335208823350062147od_a_b @ Us3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Vs2 ) ) ) ) ) ) ).

% listrel1E
thf(fact_618_set__vwalk__arcs__snoc,axiom,
    ! [P: list_P1467974523212632366od_a_b,A2: product_prod_a_b] :
      ( ( P != nil_Product_prod_a_b )
     => ( ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ ( append5335208823350062147od_a_b @ P @ ( cons_P7316939131009794654od_a_b @ A2 @ nil_Product_prod_a_b ) ) ) )
        = ( insert7668604973114392439od_a_b @ ( produc6550459592057863831od_a_b @ ( last_P8790725272581694279od_a_b @ P ) @ A2 ) @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ P ) ) ) ) ) ).

% set_vwalk_arcs_snoc
thf(fact_619_vwalk__arcs__Cons,axiom,
    ! [P: list_P1467974523212632366od_a_b,U: product_prod_a_b] :
      ( ( P != nil_Product_prod_a_b )
     => ( ( vertex3879532237292798496od_a_b @ ( cons_P7316939131009794654od_a_b @ U @ P ) )
        = ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ U @ ( hd_Product_prod_a_b @ P ) ) @ ( vertex3879532237292798496od_a_b @ P ) ) ) ) ).

% vwalk_arcs_Cons
thf(fact_620_dtree__from__list__dverts__subset__wfdverts_H,axiom,
    ! [V2: a,As: list_P1467974523212632366od_a_b,Cs: list_P1467974523212632366od_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b] :
      ( ( wf_dverts_a_b @ ( dtree_from_list_a_b @ V2 @ As ) )
     => ( ( wf_dverts_a_b @ ( dtree_from_list_a_b @ V2 @ Cs ) )
       => ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Xs2 ) @ ( dtree_from_list_a_b @ V2 @ As ) )
         => ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Ys ) @ ( dtree_from_list_a_b @ V2 @ Cs ) )
           => ( ? [As3: list_P1467974523212632366od_a_b,E12: b,Bs3: list_P1467974523212632366od_a_b,Cs2: list_P1467974523212632366od_a_b,E22: b,Ds: list_P1467974523212632366od_a_b] :
                  ( ( ( append5335208823350062147od_a_b @ As3 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E12 ) @ Bs3 ) )
                    = As )
                  & ( ( append5335208823350062147od_a_b @ Cs2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E22 ) @ Ds ) )
                    = Cs )
                  & ( ord_le817736998455962536od_a_b @ ( set_Product_prod_a_b2 @ Bs3 ) @ ( set_Product_prod_a_b2 @ Ds ) ) )
             => ( ord_less_eq_set_a @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) @ ( dverts_a_b @ ( node_a_b @ R2 @ Ys ) ) ) ) ) ) ) ) ).

% dtree_from_list_dverts_subset_wfdverts'
thf(fact_621_dtree__from__list__dverts__subset__wfdverts,axiom,
    ! [Bs: list_P1467974523212632366od_a_b,Ds2: list_P1467974523212632366od_a_b,V2: a,As: list_P1467974523212632366od_a_b,R2: a,E1: b,Cs: list_P1467974523212632366od_a_b,E23: b,Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b] :
      ( ( ord_le817736998455962536od_a_b @ ( set_Product_prod_a_b2 @ Bs ) @ ( set_Product_prod_a_b2 @ Ds2 ) )
     => ( ( wf_dverts_a_b @ ( dtree_from_list_a_b @ V2 @ ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E1 ) @ Bs ) ) ) )
       => ( ( wf_dverts_a_b @ ( dtree_from_list_a_b @ V2 @ ( append5335208823350062147od_a_b @ Cs @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E23 ) @ Ds2 ) ) ) )
         => ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Xs2 ) @ ( dtree_from_list_a_b @ V2 @ ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E1 ) @ Bs ) ) ) )
           => ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Ys ) @ ( dtree_from_list_a_b @ V2 @ ( append5335208823350062147od_a_b @ Cs @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ R2 @ E23 ) @ Ds2 ) ) ) )
             => ( ord_less_eq_set_a @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) @ ( dverts_a_b @ ( node_a_b @ R2 @ Ys ) ) ) ) ) ) ) ) ).

% dtree_from_list_dverts_subset_wfdverts
thf(fact_622_pawalk__verts_Osimps_I1_J,axiom,
    ! [U: product_prod_a_b] :
      ( ( pair_p1830641812483871010od_a_b @ U @ nil_Pr906174322249854893od_a_b )
      = ( cons_P7316939131009794654od_a_b @ U @ nil_Product_prod_a_b ) ) ).

% pawalk_verts.simps(1)
thf(fact_623_subsetI,axiom,
    ! [A: set_a,B4: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ X3 @ B4 ) )
     => ( ord_less_eq_set_a @ A @ B4 ) ) ).

% subsetI
thf(fact_624_subsetI,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A )
         => ( member1426531481828664017od_a_b @ X3 @ B4 ) )
     => ( ord_le817736998455962536od_a_b @ A @ B4 ) ) ).

% subsetI
thf(fact_625_insert__subset,axiom,
    ! [X: a,A: set_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( insert_a2 @ X @ A ) @ B4 )
      = ( ( member_a @ X @ B4 )
        & ( ord_less_eq_set_a @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_626_insert__subset,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ord_le817736998455962536od_a_b @ ( insert4534936386344385144od_a_b @ X @ A ) @ B4 )
      = ( ( member1426531481828664017od_a_b @ X @ B4 )
        & ( ord_le817736998455962536od_a_b @ A @ B4 ) ) ) ).

% insert_subset
thf(fact_627_hd__append2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( ( hd_Product_prod_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
        = ( hd_Product_prod_a_b @ Xs2 ) ) ) ).

% hd_append2
thf(fact_628_last__appendR,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( Ys != nil_Product_prod_a_b )
     => ( ( last_P8790725272581694279od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
        = ( last_P8790725272581694279od_a_b @ Ys ) ) ) ).

% last_appendR
thf(fact_629_last__appendL,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( Ys = nil_Product_prod_a_b )
     => ( ( last_P8790725272581694279od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
        = ( last_P8790725272581694279od_a_b @ Xs2 ) ) ) ).

% last_appendL
thf(fact_630_last__snoc,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( last_P8790725272581694279od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) )
      = X ) ).

% last_snoc
thf(fact_631_in__mono,axiom,
    ! [A: set_a,B4: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ B4 )
     => ( ( member_a @ X @ A )
       => ( member_a @ X @ B4 ) ) ) ).

% in_mono
thf(fact_632_in__mono,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,X: product_prod_a_b] :
      ( ( ord_le817736998455962536od_a_b @ A @ B4 )
     => ( ( member1426531481828664017od_a_b @ X @ A )
       => ( member1426531481828664017od_a_b @ X @ B4 ) ) ) ).

% in_mono
thf(fact_633_subsetD,axiom,
    ! [A: set_a,B4: set_a,C: a] :
      ( ( ord_less_eq_set_a @ A @ B4 )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B4 ) ) ) ).

% subsetD
thf(fact_634_subsetD,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,C: product_prod_a_b] :
      ( ( ord_le817736998455962536od_a_b @ A @ B4 )
     => ( ( member1426531481828664017od_a_b @ C @ A )
       => ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% subsetD
thf(fact_635_subset__eq,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B7: set_a] :
        ! [X4: a] :
          ( ( member_a @ X4 @ A5 )
         => ( member_a @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_636_subset__eq,axiom,
    ( ord_le817736998455962536od_a_b
    = ( ^ [A5: set_Product_prod_a_b,B7: set_Product_prod_a_b] :
        ! [X4: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X4 @ A5 )
         => ( member1426531481828664017od_a_b @ X4 @ B7 ) ) ) ) ).

% subset_eq
thf(fact_637_subset__iff,axiom,
    ( ord_less_eq_set_a
    = ( ^ [A5: set_a,B7: set_a] :
        ! [T3: a] :
          ( ( member_a @ T3 @ A5 )
         => ( member_a @ T3 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_638_subset__iff,axiom,
    ( ord_le817736998455962536od_a_b
    = ( ^ [A5: set_Product_prod_a_b,B7: set_Product_prod_a_b] :
        ! [T3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ T3 @ A5 )
         => ( member1426531481828664017od_a_b @ T3 @ B7 ) ) ) ) ).

% subset_iff
thf(fact_639_list_Osel_I1_J,axiom,
    ! [X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( ( hd_Product_prod_a_b @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) )
      = X21 ) ).

% list.sel(1)
thf(fact_640_image__mono,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,F: product_prod_a_b > a] :
      ( ( ord_le817736998455962536od_a_b @ A @ B4 )
     => ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ).

% image_mono
thf(fact_641_image__subsetI,axiom,
    ! [A: set_a,F: a > a,B4: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_a_a2 @ F @ A ) @ B4 ) ) ).

% image_subsetI
thf(fact_642_image__subsetI,axiom,
    ! [A: set_a,F: a > product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ( member1426531481828664017od_a_b @ ( F @ X3 ) @ B4 ) )
     => ( ord_le817736998455962536od_a_b @ ( image_7400625786893224495od_a_b @ F @ A ) @ B4 ) ) ).

% image_subsetI
thf(fact_643_image__subsetI,axiom,
    ! [A: set_Product_prod_a_b,F: product_prod_a_b > a,B4: set_a] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A )
         => ( member_a @ ( F @ X3 ) @ B4 ) )
     => ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ B4 ) ) ).

% image_subsetI
thf(fact_644_image__subsetI,axiom,
    ! [A: set_Product_prod_a_b,F: product_prod_a_b > product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A )
         => ( member1426531481828664017od_a_b @ ( F @ X3 ) @ B4 ) )
     => ( ord_le817736998455962536od_a_b @ ( image_3300603549555413765od_a_b @ F @ A ) @ B4 ) ) ).

% image_subsetI
thf(fact_645_subset__imageE,axiom,
    ! [B4: set_a,F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( ord_less_eq_set_a @ B4 @ ( image_2802296252294471259_a_b_a @ F @ A ) )
     => ~ ! [C4: set_Product_prod_a_b] :
            ( ( ord_le817736998455962536od_a_b @ C4 @ A )
           => ( B4
             != ( image_2802296252294471259_a_b_a @ F @ C4 ) ) ) ) ).

% subset_imageE
thf(fact_646_image__subset__iff,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ B4 )
      = ( ! [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ A )
           => ( member_a @ ( F @ X4 ) @ B4 ) ) ) ) ).

% image_subset_iff
thf(fact_647_subset__image__iff,axiom,
    ! [B4: set_a,F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( ord_less_eq_set_a @ B4 @ ( image_2802296252294471259_a_b_a @ F @ A ) )
      = ( ? [AA: set_Product_prod_a_b] :
            ( ( ord_le817736998455962536od_a_b @ AA @ A )
            & ( B4
              = ( image_2802296252294471259_a_b_a @ F @ AA ) ) ) ) ) ).

% subset_image_iff
thf(fact_648_subset__insertI2,axiom,
    ! [A: set_a,B4: set_a,B: a] :
      ( ( ord_less_eq_set_a @ A @ B4 )
     => ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ B4 ) ) ) ).

% subset_insertI2
thf(fact_649_subset__insertI,axiom,
    ! [B4: set_a,A2: a] : ( ord_less_eq_set_a @ B4 @ ( insert_a2 @ A2 @ B4 ) ) ).

% subset_insertI
thf(fact_650_subset__insert,axiom,
    ! [X: a,A: set_a,B4: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X @ B4 ) )
        = ( ord_less_eq_set_a @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_651_subset__insert,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ A )
     => ( ( ord_le817736998455962536od_a_b @ A @ ( insert4534936386344385144od_a_b @ X @ B4 ) )
        = ( ord_le817736998455962536od_a_b @ A @ B4 ) ) ) ).

% subset_insert
thf(fact_652_insert__mono,axiom,
    ! [C5: set_a,D: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ C5 @ D )
     => ( ord_less_eq_set_a @ ( insert_a2 @ A2 @ C5 ) @ ( insert_a2 @ A2 @ D ) ) ) ).

% insert_mono
thf(fact_653_subset__code_I1_J,axiom,
    ! [Xs2: list_a,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( set_a2 @ Xs2 ) @ B4 )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
           => ( member_a @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_654_subset__code_I1_J,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,B4: set_Product_prod_a_b] :
      ( ( ord_le817736998455962536od_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) @ B4 )
      = ( ! [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
           => ( member1426531481828664017od_a_b @ X4 @ B4 ) ) ) ) ).

% subset_code(1)
thf(fact_655_dtree__from__list__dverts__subset__set,axiom,
    ! [Bs: list_P1467974523212632366od_a_b,Ds2: list_P1467974523212632366od_a_b,R2: a] :
      ( ( ord_le817736998455962536od_a_b @ ( set_Product_prod_a_b2 @ Bs ) @ ( set_Product_prod_a_b2 @ Ds2 ) )
     => ( ord_less_eq_set_a @ ( dverts_a_b @ ( dtree_from_list_a_b @ R2 @ Bs ) ) @ ( dverts_a_b @ ( dtree_from_list_a_b @ R2 @ Ds2 ) ) ) ) ).

% dtree_from_list_dverts_subset_set
thf(fact_656_hd__in__set,axiom,
    ! [Xs2: list_a] :
      ( ( Xs2 != nil_a )
     => ( member_a @ ( hd_a @ Xs2 ) @ ( set_a2 @ Xs2 ) ) ) ).

% hd_in_set
thf(fact_657_hd__in__set,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( member1426531481828664017od_a_b @ ( hd_Product_prod_a_b @ Xs2 ) @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ).

% hd_in_set
thf(fact_658_list_Oset__sel_I1_J,axiom,
    ! [A2: list_a] :
      ( ( A2 != nil_a )
     => ( member_a @ ( hd_a @ A2 ) @ ( set_a2 @ A2 ) ) ) ).

% list.set_sel(1)
thf(fact_659_list_Oset__sel_I1_J,axiom,
    ! [A2: list_P1467974523212632366od_a_b] :
      ( ( A2 != nil_Product_prod_a_b )
     => ( member1426531481828664017od_a_b @ ( hd_Product_prod_a_b @ A2 ) @ ( set_Product_prod_a_b2 @ A2 ) ) ) ).

% list.set_sel(1)
thf(fact_660_last__ConsR,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( ( last_P8790725272581694279od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
        = ( last_P8790725272581694279od_a_b @ Xs2 ) ) ) ).

% last_ConsR
thf(fact_661_last__ConsL,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( Xs2 = nil_Product_prod_a_b )
     => ( ( last_P8790725272581694279od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
        = X ) ) ).

% last_ConsL
thf(fact_662_last_Osimps,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( ( Xs2 = nil_Product_prod_a_b )
       => ( ( last_P8790725272581694279od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
          = X ) )
      & ( ( Xs2 != nil_Product_prod_a_b )
       => ( ( last_P8790725272581694279od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
          = ( last_P8790725272581694279od_a_b @ Xs2 ) ) ) ) ).

% last.simps
thf(fact_663_longest__common__prefix,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
    ? [Ps: list_P1467974523212632366od_a_b,Xs5: list_P1467974523212632366od_a_b,Ys6: list_P1467974523212632366od_a_b] :
      ( ( Xs2
        = ( append5335208823350062147od_a_b @ Ps @ Xs5 ) )
      & ( Ys
        = ( append5335208823350062147od_a_b @ Ps @ Ys6 ) )
      & ( ( Xs5 = nil_Product_prod_a_b )
        | ( Ys6 = nil_Product_prod_a_b )
        | ( ( hd_Product_prod_a_b @ Xs5 )
         != ( hd_Product_prod_a_b @ Ys6 ) ) ) ) ).

% longest_common_prefix
thf(fact_664_hd__append,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( Xs2 = nil_Product_prod_a_b )
       => ( ( hd_Product_prod_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
          = ( hd_Product_prod_a_b @ Ys ) ) )
      & ( ( Xs2 != nil_Product_prod_a_b )
       => ( ( hd_Product_prod_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
          = ( hd_Product_prod_a_b @ Xs2 ) ) ) ) ).

% hd_append
thf(fact_665_last__in__set,axiom,
    ! [As: list_a] :
      ( ( As != nil_a )
     => ( member_a @ ( last_a @ As ) @ ( set_a2 @ As ) ) ) ).

% last_in_set
thf(fact_666_last__in__set,axiom,
    ! [As: list_P1467974523212632366od_a_b] :
      ( ( As != nil_Product_prod_a_b )
     => ( member1426531481828664017od_a_b @ ( last_P8790725272581694279od_a_b @ As ) @ ( set_Product_prod_a_b2 @ As ) ) ) ).

% last_in_set
thf(fact_667_longest__common__suffix,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
    ? [Ss: list_P1467974523212632366od_a_b,Xs5: list_P1467974523212632366od_a_b,Ys6: list_P1467974523212632366od_a_b] :
      ( ( Xs2
        = ( append5335208823350062147od_a_b @ Xs5 @ Ss ) )
      & ( Ys
        = ( append5335208823350062147od_a_b @ Ys6 @ Ss ) )
      & ( ( Xs5 = nil_Product_prod_a_b )
        | ( Ys6 = nil_Product_prod_a_b )
        | ( ( last_P8790725272581694279od_a_b @ Xs5 )
         != ( last_P8790725272581694279od_a_b @ Ys6 ) ) ) ) ).

% longest_common_suffix
thf(fact_668_last__append,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( Ys = nil_Product_prod_a_b )
       => ( ( last_P8790725272581694279od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
          = ( last_P8790725272581694279od_a_b @ Xs2 ) ) )
      & ( ( Ys != nil_Product_prod_a_b )
       => ( ( last_P8790725272581694279od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
          = ( last_P8790725272581694279od_a_b @ Ys ) ) ) ) ).

% last_append
thf(fact_669_set__subset__Cons,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b] : ( ord_le817736998455962536od_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) ) ) ).

% set_subset_Cons
thf(fact_670_in__set__vwalk__arcs__append2,axiom,
    ! [P: list_P1467974523212632366od_a_b,Q2: list_P1467974523212632366od_a_b,X: produc2233330766403592615od_a_b] :
      ( ( P != nil_Product_prod_a_b )
     => ( ( Q2 != nil_Product_prod_a_b )
       => ( ( ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ P ) ) )
            | ( X
              = ( produc6550459592057863831od_a_b @ ( last_P8790725272581694279od_a_b @ P ) @ ( hd_Product_prod_a_b @ Q2 ) ) )
            | ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ Q2 ) ) ) )
         => ( member5065438833300252112od_a_b @ X @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ ( append5335208823350062147od_a_b @ P @ Q2 ) ) ) ) ) ) ) ).

% in_set_vwalk_arcs_append2
thf(fact_671_dverts__subtree__subset,axiom,
    ! [X: dtree_a_b,Y: dtree_a_b] :
      ( ( is_subtree_a_b @ X @ Y )
     => ( ord_less_eq_set_a @ ( dverts_a_b @ X ) @ ( dverts_a_b @ Y ) ) ) ).

% dverts_subtree_subset
thf(fact_672_vwalk__arcs__append,axiom,
    ! [P: list_P1467974523212632366od_a_b,Q2: list_P1467974523212632366od_a_b] :
      ( ( P != nil_Product_prod_a_b )
     => ( ( Q2 != nil_Product_prod_a_b )
       => ( ( vertex3879532237292798496od_a_b @ ( append5335208823350062147od_a_b @ P @ Q2 ) )
          = ( append3447563168971123522od_a_b @ ( vertex3879532237292798496od_a_b @ P ) @ ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ ( last_P8790725272581694279od_a_b @ P ) @ ( hd_Product_prod_a_b @ Q2 ) ) @ ( vertex3879532237292798496od_a_b @ Q2 ) ) ) ) ) ) ).

% vwalk_arcs_append
thf(fact_673_set__vwalk__arcs__cons,axiom,
    ! [P: list_P1467974523212632366od_a_b,U: product_prod_a_b] : ( ord_le362614178720401191od_a_b @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ P ) ) @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ ( cons_P7316939131009794654od_a_b @ U @ P ) ) ) ) ).

% set_vwalk_arcs_cons
thf(fact_674_set__vwalk__arcs__append2,axiom,
    ! [Q2: list_P1467974523212632366od_a_b,P: list_P1467974523212632366od_a_b] : ( ord_le362614178720401191od_a_b @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ Q2 ) ) @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ ( append5335208823350062147od_a_b @ P @ Q2 ) ) ) ) ).

% set_vwalk_arcs_append2
thf(fact_675_set__vwalk__arcs__append1,axiom,
    ! [P: list_P1467974523212632366od_a_b,Q2: list_P1467974523212632366od_a_b] : ( ord_le362614178720401191od_a_b @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ P ) ) @ ( set_Pr5228040199190236220od_a_b @ ( vertex3879532237292798496od_a_b @ ( append5335208823350062147od_a_b @ P @ Q2 ) ) ) ) ).

% set_vwalk_arcs_append1
thf(fact_676_subset__subseqs,axiom,
    ! [X6: set_Product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ord_le817736998455962536od_a_b @ X6 @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( member1887650552762594865od_a_b @ X6 @ ( image_3302445222470734431od_a_b @ set_Product_prod_a_b2 @ ( set_li5045283583078295363od_a_b @ ( subseq3159716953272533757od_a_b @ Xs2 ) ) ) ) ) ).

% subset_subseqs
thf(fact_677_split__list__last__sep,axiom,
    ! [Y: a,Xs2: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ Xs2 ) )
     => ( ( Y
         != ( last_a @ Xs2 ) )
       => ? [As2: list_a,Bs2: list_a] :
            ( ( append_a @ As2 @ ( cons_a @ Y @ ( append_a @ Bs2 @ ( cons_a @ ( last_a @ Xs2 ) @ nil_a ) ) ) )
            = Xs2 ) ) ) ).

% split_list_last_sep
thf(fact_678_split__list__last__sep,axiom,
    ! [Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( ( Y
         != ( last_P8790725272581694279od_a_b @ Xs2 ) )
       => ? [As2: list_P1467974523212632366od_a_b,Bs2: list_P1467974523212632366od_a_b] :
            ( ( append5335208823350062147od_a_b @ As2 @ ( cons_P7316939131009794654od_a_b @ Y @ ( append5335208823350062147od_a_b @ Bs2 @ ( cons_P7316939131009794654od_a_b @ ( last_P8790725272581694279od_a_b @ Xs2 ) @ nil_Product_prod_a_b ) ) ) )
            = Xs2 ) ) ) ).

% split_list_last_sep
thf(fact_679_split__list__not__last,axiom,
    ! [Y: a,Xs2: list_a] :
      ( ( member_a @ Y @ ( set_a2 @ Xs2 ) )
     => ( ( Y
         != ( last_a @ Xs2 ) )
       => ? [As2: list_a,Bs2: list_a] :
            ( ( ( append_a @ As2 @ ( cons_a @ Y @ Bs2 ) )
              = Xs2 )
            & ( Bs2 != nil_a ) ) ) ) ).

% split_list_not_last
thf(fact_680_split__list__not__last,axiom,
    ! [Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( ( Y
         != ( last_P8790725272581694279od_a_b @ Xs2 ) )
       => ? [As2: list_P1467974523212632366od_a_b,Bs2: list_P1467974523212632366od_a_b] :
            ( ( ( append5335208823350062147od_a_b @ As2 @ ( cons_P7316939131009794654od_a_b @ Y @ Bs2 ) )
              = Xs2 )
            & ( Bs2 != nil_Product_prod_a_b ) ) ) ) ).

% split_list_not_last
thf(fact_681_split__last__eq,axiom,
    ! [As: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Bs: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ As @ ( cons_P7316939131009794654od_a_b @ Y @ Bs ) )
        = Xs2 )
     => ( ( Bs != nil_Product_prod_a_b )
       => ( ( last_P8790725272581694279od_a_b @ Bs )
          = ( last_P8790725272581694279od_a_b @ Xs2 ) ) ) ) ).

% split_last_eq
thf(fact_682_subrelI,axiom,
    ! [R2: set_Product_prod_a_b,S: set_Product_prod_a_b] :
      ( ! [X3: a,Y4: b] :
          ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) @ R2 )
         => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) @ S ) )
     => ( ord_le817736998455962536od_a_b @ R2 @ S ) ) ).

% subrelI
thf(fact_683_dtree__to__list__sub__dverts__ins,axiom,
    ! [T: dtree_a_b] : ( ord_less_eq_set_a @ ( insert_a2 @ ( root_a_b @ T ) @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ ( dtree_to_list_a_b @ T ) ) ) ) @ ( dverts_a_b @ T ) ) ).

% dtree_to_list_sub_dverts_ins
thf(fact_684_insert__subsetI,axiom,
    ! [X: a,A: set_a,X6: set_a] :
      ( ( member_a @ X @ A )
     => ( ( ord_less_eq_set_a @ X6 @ A )
       => ( ord_less_eq_set_a @ ( insert_a2 @ X @ X6 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_685_insert__subsetI,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,X6: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ A )
     => ( ( ord_le817736998455962536od_a_b @ X6 @ A )
       => ( ord_le817736998455962536od_a_b @ ( insert4534936386344385144od_a_b @ X @ X6 ) @ A ) ) ) ).

% insert_subsetI
thf(fact_686_all__subset__image,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,P2: set_a > $o] :
      ( ( ! [B7: set_a] :
            ( ( ord_less_eq_set_a @ B7 @ ( image_2802296252294471259_a_b_a @ F @ A ) )
           => ( P2 @ B7 ) ) )
      = ( ! [B7: set_Product_prod_a_b] :
            ( ( ord_le817736998455962536od_a_b @ B7 @ A )
           => ( P2 @ ( image_2802296252294471259_a_b_a @ F @ B7 ) ) ) ) ) ).

% all_subset_image
thf(fact_687_dtree_Osel_I1_J,axiom,
    ! [X1: a,X2: fset_P5281107635120001194_a_b_b] :
      ( ( root_a_b @ ( node_a_b @ X1 @ X2 ) )
      = X1 ) ).

% dtree.sel(1)
thf(fact_688_dtree_Oset__sel_I1_J,axiom,
    ! [A2: dtree_a_b] : ( member_a @ ( root_a_b @ A2 ) @ ( dverts_a_b @ A2 ) ) ).

% dtree.set_sel(1)
thf(fact_689_ssubst__Pair__rhs,axiom,
    ! [R2: a,S: b,R3: set_Product_prod_a_b,S2: b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R2 @ S ) @ R3 )
     => ( ( S2 = S )
       => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ R2 @ S2 ) @ R3 ) ) ) ).

% ssubst_Pair_rhs
thf(fact_690_append__butlast__last__id,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != nil_Product_prod_a_b )
     => ( ( append5335208823350062147od_a_b @ ( butlas8142365734376493050od_a_b @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ ( last_P8790725272581694279od_a_b @ Xs2 ) @ nil_Product_prod_a_b ) )
        = Xs2 ) ) ).

% append_butlast_last_id
thf(fact_691_distinct__adj__append__iff,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( distin2562679511629466758od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
      = ( ( distin2562679511629466758od_a_b @ Xs2 )
        & ( distin2562679511629466758od_a_b @ Ys )
        & ( ( Xs2 = nil_Product_prod_a_b )
          | ( Ys = nil_Product_prod_a_b )
          | ( ( last_P8790725272581694279od_a_b @ Xs2 )
           != ( hd_Product_prod_a_b @ Ys ) ) ) ) ) ).

% distinct_adj_append_iff
thf(fact_692_subseqs__powset,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( image_3302445222470734431od_a_b @ set_Product_prod_a_b2 @ ( set_li5045283583078295363od_a_b @ ( subseq3159716953272533757od_a_b @ Xs2 ) ) )
      = ( pow_Product_prod_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ).

% subseqs_powset
thf(fact_693_vwalk__join__split,axiom,
    ! [U: a,P: list_a] :
      ( ( member_a @ U @ ( set_a2 @ P ) )
     => ? [Q3: list_a,R: list_a] :
          ( ( P
            = ( vertex_vwalk_join_a @ Q3 @ R ) )
          & ( ( last_a @ Q3 )
            = U )
          & ( ( hd_a @ R )
            = U )
          & ( Q3 != nil_a )
          & ( R != nil_a ) ) ) ).

% vwalk_join_split
thf(fact_694_vwalk__join__split,axiom,
    ! [U: product_prod_a_b,P: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ U @ ( set_Product_prod_a_b2 @ P ) )
     => ? [Q3: list_P1467974523212632366od_a_b,R: list_P1467974523212632366od_a_b] :
          ( ( P
            = ( vertex9110485820359517623od_a_b @ Q3 @ R ) )
          & ( ( last_P8790725272581694279od_a_b @ Q3 )
            = U )
          & ( ( hd_Product_prod_a_b @ R )
            = U )
          & ( Q3 != nil_Product_prod_a_b )
          & ( R != nil_Product_prod_a_b ) ) ) ).

% vwalk_join_split
thf(fact_695_subset__code_I3_J,axiom,
    ~ ( ord_le817736998455962536od_a_b @ ( coset_3963096541349886491od_a_b @ nil_Product_prod_a_b ) @ ( set_Product_prod_a_b2 @ nil_Product_prod_a_b ) ) ).

% subset_code(3)
thf(fact_696_distinct__adj__Cons__Cons,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( distin2562679511629466758od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) ) )
      = ( ( X != Y )
        & ( distin2562679511629466758od_a_b @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) ) ) ) ).

% distinct_adj_Cons_Cons
thf(fact_697_joinable__Cons__l__iff,axiom,
    ! [P: list_P1467974523212632366od_a_b,V2: product_prod_a_b,Q2: list_P1467974523212632366od_a_b] :
      ( ( P != nil_Product_prod_a_b )
     => ( ( vertex7469968122360900522od_a_b @ ( cons_P7316939131009794654od_a_b @ V2 @ P ) @ Q2 )
        = ( vertex7469968122360900522od_a_b @ P @ Q2 ) ) ) ).

% joinable_Cons_l_iff
thf(fact_698_vwalk__join__Cons,axiom,
    ! [P: list_P1467974523212632366od_a_b,U: product_prod_a_b,Q2: list_P1467974523212632366od_a_b] :
      ( ( P != nil_Product_prod_a_b )
     => ( ( vertex9110485820359517623od_a_b @ ( cons_P7316939131009794654od_a_b @ U @ P ) @ Q2 )
        = ( cons_P7316939131009794654od_a_b @ U @ ( vertex9110485820359517623od_a_b @ P @ Q2 ) ) ) ) ).

% vwalk_join_Cons
thf(fact_699_butlast__snoc,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( butlas8142365734376493050od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) )
      = Xs2 ) ).

% butlast_snoc
thf(fact_700_joinable__Snoc__r__iff,axiom,
    ! [Q2: list_P1467974523212632366od_a_b,P: list_P1467974523212632366od_a_b,V2: product_prod_a_b] :
      ( ( Q2 != nil_Product_prod_a_b )
     => ( ( vertex7469968122360900522od_a_b @ P @ ( append5335208823350062147od_a_b @ Q2 @ ( cons_P7316939131009794654od_a_b @ V2 @ nil_Product_prod_a_b ) ) )
        = ( vertex7469968122360900522od_a_b @ P @ Q2 ) ) ) ).

% joinable_Snoc_r_iff
thf(fact_701_vwalk__join__def2,axiom,
    ! [P: list_P1467974523212632366od_a_b,Q2: list_P1467974523212632366od_a_b] :
      ( ( vertex7469968122360900522od_a_b @ P @ Q2 )
     => ( ( vertex9110485820359517623od_a_b @ P @ Q2 )
        = ( append5335208823350062147od_a_b @ ( butlas8142365734376493050od_a_b @ P ) @ Q2 ) ) ) ).

% vwalk_join_def2
thf(fact_702_image__Pow__surj,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_a] :
      ( ( ( image_2802296252294471259_a_b_a @ F @ A )
        = B4 )
     => ( ( image_7003315350810844251_set_a @ ( image_2802296252294471259_a_b_a @ F ) @ ( pow_Product_prod_a_b @ A ) )
        = ( pow_a @ B4 ) ) ) ).

% image_Pow_surj
thf(fact_703_in__set__butlastD,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs2 ) ) )
     => ( member_a @ X @ ( set_a2 @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_704_in__set__butlastD,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( butlas8142365734376493050od_a_b @ Xs2 ) ) )
     => ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ).

% in_set_butlastD
thf(fact_705_distinct__adj__ConsD,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( distin2562679511629466758od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
     => ( distin2562679511629466758od_a_b @ Xs2 ) ) ).

% distinct_adj_ConsD
thf(fact_706_distinct__adj__appendD2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( distin2562679511629466758od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
     => ( distin2562679511629466758od_a_b @ Ys ) ) ).

% distinct_adj_appendD2
thf(fact_707_distinct__adj__appendD1,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( distin2562679511629466758od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
     => ( distin2562679511629466758od_a_b @ Xs2 ) ) ).

% distinct_adj_appendD1
thf(fact_708_image__Pow__mono,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ B4 )
     => ( ord_le3724670747650509150_set_a @ ( image_7003315350810844251_set_a @ ( image_2802296252294471259_a_b_a @ F ) @ ( pow_Product_prod_a_b @ A ) ) @ ( pow_a @ B4 ) ) ) ).

% image_Pow_mono
thf(fact_709_butlast_Osimps_I2_J,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( ( Xs2 = nil_Product_prod_a_b )
       => ( ( butlas8142365734376493050od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
          = nil_Product_prod_a_b ) )
      & ( ( Xs2 != nil_Product_prod_a_b )
       => ( ( butlas8142365734376493050od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
          = ( cons_P7316939131009794654od_a_b @ X @ ( butlas8142365734376493050od_a_b @ Xs2 ) ) ) ) ) ).

% butlast.simps(2)
thf(fact_710_butlast__append,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( Ys = nil_Product_prod_a_b )
       => ( ( butlas8142365734376493050od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
          = ( butlas8142365734376493050od_a_b @ Xs2 ) ) )
      & ( ( Ys != nil_Product_prod_a_b )
       => ( ( butlas8142365734376493050od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
          = ( append5335208823350062147od_a_b @ Xs2 @ ( butlas8142365734376493050od_a_b @ Ys ) ) ) ) ) ).

% butlast_append
thf(fact_711_in__set__butlast__appendI,axiom,
    ! [X: a,Xs2: list_a,Ys: list_a] :
      ( ( ( member_a @ X @ ( set_a2 @ ( butlast_a @ Xs2 ) ) )
        | ( member_a @ X @ ( set_a2 @ ( butlast_a @ Ys ) ) ) )
     => ( member_a @ X @ ( set_a2 @ ( butlast_a @ ( append_a @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_712_in__set__butlast__appendI,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( butlas8142365734376493050od_a_b @ Xs2 ) ) )
        | ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( butlas8142365734376493050od_a_b @ Ys ) ) ) )
     => ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( butlas8142365734376493050od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_butlast_appendI
thf(fact_713_distinct__adj__singleton,axiom,
    ! [X: product_prod_a_b] : ( distin2562679511629466758od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) ).

% distinct_adj_singleton
thf(fact_714_subset__code_I2_J,axiom,
    ! [A: set_a,Ys: list_a] :
      ( ( ord_less_eq_set_a @ A @ ( coset_a @ Ys ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Ys ) )
           => ~ ( member_a @ X4 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_715_subset__code_I2_J,axiom,
    ! [A: set_Product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ord_le817736998455962536od_a_b @ A @ ( coset_3963096541349886491od_a_b @ Ys ) )
      = ( ! [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Ys ) )
           => ~ ( member1426531481828664017od_a_b @ X4 @ A ) ) ) ) ).

% subset_code(2)
thf(fact_716_distinct__adj__Cons,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( distin2562679511629466758od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
      = ( ( Xs2 = nil_Product_prod_a_b )
        | ( ( X
           != ( hd_Product_prod_a_b @ Xs2 ) )
          & ( distin2562679511629466758od_a_b @ Xs2 ) ) ) ) ).

% distinct_adj_Cons
thf(fact_717_snoc__eq__iff__butlast,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) )
        = Ys )
      = ( ( Ys != nil_Product_prod_a_b )
        & ( ( butlas8142365734376493050od_a_b @ Ys )
          = Xs2 )
        & ( ( last_P8790725272581694279od_a_b @ Ys )
          = X ) ) ) ).

% snoc_eq_iff_butlast
thf(fact_718_dtree__to__list__sub__dverts,axiom,
    ! [T: dtree_a_b] :
      ( ( wf_dverts_a_b @ T )
     => ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ ( dtree_to_list_a_b @ T ) ) ) @ ( minus_minus_set_a @ ( dverts_a_b @ T ) @ ( insert_a2 @ ( root_a_b @ T ) @ bot_bot_set_a ) ) ) ) ).

% dtree_to_list_sub_dverts
thf(fact_719_all__not__in__conv,axiom,
    ! [A: set_a] :
      ( ( ! [X4: a] :
            ~ ( member_a @ X4 @ A ) )
      = ( A = bot_bot_set_a ) ) ).

% all_not_in_conv
thf(fact_720_all__not__in__conv,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ( ! [X4: product_prod_a_b] :
            ~ ( member1426531481828664017od_a_b @ X4 @ A ) )
      = ( A = bot_bo3428410327501040244od_a_b ) ) ).

% all_not_in_conv
thf(fact_721_empty__iff,axiom,
    ! [C: a] :
      ~ ( member_a @ C @ bot_bot_set_a ) ).

% empty_iff
thf(fact_722_empty__iff,axiom,
    ! [C: product_prod_a_b] :
      ~ ( member1426531481828664017od_a_b @ C @ bot_bo3428410327501040244od_a_b ) ).

% empty_iff
thf(fact_723_Diff__iff,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
      = ( ( member_a @ C @ A )
        & ~ ( member_a @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_724_Diff__iff,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A @ B4 ) )
      = ( ( member1426531481828664017od_a_b @ C @ A )
        & ~ ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% Diff_iff
thf(fact_725_DiffI,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ A )
     => ( ~ ( member_a @ C @ B4 )
       => ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_726_DiffI,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ A )
     => ( ~ ( member1426531481828664017od_a_b @ C @ B4 )
       => ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A @ B4 ) ) ) ) ).

% DiffI
thf(fact_727_member__remove,axiom,
    ! [X: a,Y: a,A: set_a] :
      ( ( member_a @ X @ ( remove_a @ Y @ A ) )
      = ( ( member_a @ X @ A )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_728_member__remove,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( remove8198300761713201805od_a_b @ Y @ A ) )
      = ( ( member1426531481828664017od_a_b @ X @ A )
        & ( X != Y ) ) ) ).

% member_remove
thf(fact_729_image__is__empty,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( ( image_2802296252294471259_a_b_a @ F @ A )
        = bot_bot_set_a )
      = ( A = bot_bo3428410327501040244od_a_b ) ) ).

% image_is_empty
thf(fact_730_empty__is__image,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( bot_bot_set_a
        = ( image_2802296252294471259_a_b_a @ F @ A ) )
      = ( A = bot_bo3428410327501040244od_a_b ) ) ).

% empty_is_image
thf(fact_731_image__empty,axiom,
    ! [F: product_prod_a_b > a] :
      ( ( image_2802296252294471259_a_b_a @ F @ bot_bo3428410327501040244od_a_b )
      = bot_bot_set_a ) ).

% image_empty
thf(fact_732_singletonI,axiom,
    ! [A2: a] : ( member_a @ A2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ).

% singletonI
thf(fact_733_singletonI,axiom,
    ! [A2: product_prod_a_b] : ( member1426531481828664017od_a_b @ A2 @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) ) ).

% singletonI
thf(fact_734_Diff__insert0,axiom,
    ! [X: a,A: set_a,B4: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ( minus_minus_set_a @ A @ ( insert_a2 @ X @ B4 ) )
        = ( minus_minus_set_a @ A @ B4 ) ) ) ).

% Diff_insert0
thf(fact_735_Diff__insert0,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ A )
     => ( ( minus_6888070959853530735od_a_b @ A @ ( insert4534936386344385144od_a_b @ X @ B4 ) )
        = ( minus_6888070959853530735od_a_b @ A @ B4 ) ) ) ).

% Diff_insert0
thf(fact_736_insert__Diff1,axiom,
    ! [X: a,B4: set_a,A: set_a] :
      ( ( member_a @ X @ B4 )
     => ( ( minus_minus_set_a @ ( insert_a2 @ X @ A ) @ B4 )
        = ( minus_minus_set_a @ A @ B4 ) ) ) ).

% insert_Diff1
thf(fact_737_insert__Diff1,axiom,
    ! [X: product_prod_a_b,B4: set_Product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ B4 )
     => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A ) @ B4 )
        = ( minus_6888070959853530735od_a_b @ A @ B4 ) ) ) ).

% insert_Diff1
thf(fact_738_singleton__insert__inj__eq_H,axiom,
    ! [A2: a,A: set_a,B: a] :
      ( ( ( insert_a2 @ A2 @ A )
        = ( insert_a2 @ B @ bot_bot_set_a ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_739_singleton__insert__inj__eq,axiom,
    ! [B: a,A2: a,A: set_a] :
      ( ( ( insert_a2 @ B @ bot_bot_set_a )
        = ( insert_a2 @ A2 @ A ) )
      = ( ( A2 = B )
        & ( ord_less_eq_set_a @ A @ ( insert_a2 @ B @ bot_bot_set_a ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_740_set__empty,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( set_Product_prod_a_b2 @ Xs2 )
        = bot_bo3428410327501040244od_a_b )
      = ( Xs2 = nil_Product_prod_a_b ) ) ).

% set_empty
thf(fact_741_set__empty2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( bot_bo3428410327501040244od_a_b
        = ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( Xs2 = nil_Product_prod_a_b ) ) ).

% set_empty2
thf(fact_742_insert__Diff__single,axiom,
    ! [A2: a,A: set_a] :
      ( ( insert_a2 @ A2 @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) )
      = ( insert_a2 @ A2 @ A ) ) ).

% insert_Diff_single
thf(fact_743_prod_Ocollapse,axiom,
    ! [Prod: product_prod_a_b] :
      ( ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) )
      = Prod ) ).

% prod.collapse
thf(fact_744_the__elem__eq,axiom,
    ! [X: a] :
      ( ( the_elem_a @ ( insert_a2 @ X @ bot_bot_set_a ) )
      = X ) ).

% the_elem_eq
thf(fact_745_Diff__insert,axiom,
    ! [A: set_a,A2: a,B4: set_a] :
      ( ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ B4 ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ B4 ) @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ).

% Diff_insert
thf(fact_746_insert__Diff,axiom,
    ! [A2: a,A: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( insert_a2 @ A2 @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_747_insert__Diff,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ( insert4534936386344385144od_a_b @ A2 @ ( minus_6888070959853530735od_a_b @ A @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) ) )
        = A ) ) ).

% insert_Diff
thf(fact_748_Diff__insert2,axiom,
    ! [A: set_a,A2: a,B4: set_a] :
      ( ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ B4 ) )
      = ( minus_minus_set_a @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) @ B4 ) ) ).

% Diff_insert2
thf(fact_749_Diff__insert__absorb,axiom,
    ! [X: a,A: set_a] :
      ( ~ ( member_a @ X @ A )
     => ( ( minus_minus_set_a @ ( insert_a2 @ X @ A ) @ ( insert_a2 @ X @ bot_bot_set_a ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_750_Diff__insert__absorb,axiom,
    ! [X: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ A )
     => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A ) @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) )
        = A ) ) ).

% Diff_insert_absorb
thf(fact_751_remove__def,axiom,
    ( remove_a
    = ( ^ [X4: a,A5: set_a] : ( minus_minus_set_a @ A5 @ ( insert_a2 @ X4 @ bot_bot_set_a ) ) ) ) ).

% remove_def
thf(fact_752_ex__in__conv,axiom,
    ! [A: set_a] :
      ( ( ? [X4: a] : ( member_a @ X4 @ A ) )
      = ( A != bot_bot_set_a ) ) ).

% ex_in_conv
thf(fact_753_ex__in__conv,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ( ? [X4: product_prod_a_b] : ( member1426531481828664017od_a_b @ X4 @ A ) )
      = ( A != bot_bo3428410327501040244od_a_b ) ) ).

% ex_in_conv
thf(fact_754_equals0I,axiom,
    ! [A: set_a] :
      ( ! [Y4: a] :
          ~ ( member_a @ Y4 @ A )
     => ( A = bot_bot_set_a ) ) ).

% equals0I
thf(fact_755_equals0I,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ! [Y4: product_prod_a_b] :
          ~ ( member1426531481828664017od_a_b @ Y4 @ A )
     => ( A = bot_bo3428410327501040244od_a_b ) ) ).

% equals0I
thf(fact_756_equals0D,axiom,
    ! [A: set_a,A2: a] :
      ( ( A = bot_bot_set_a )
     => ~ ( member_a @ A2 @ A ) ) ).

% equals0D
thf(fact_757_equals0D,axiom,
    ! [A: set_Product_prod_a_b,A2: product_prod_a_b] :
      ( ( A = bot_bo3428410327501040244od_a_b )
     => ~ ( member1426531481828664017od_a_b @ A2 @ A ) ) ).

% equals0D
thf(fact_758_emptyE,axiom,
    ! [A2: a] :
      ~ ( member_a @ A2 @ bot_bot_set_a ) ).

% emptyE
thf(fact_759_emptyE,axiom,
    ! [A2: product_prod_a_b] :
      ~ ( member1426531481828664017od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) ).

% emptyE
thf(fact_760_DiffD2,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
     => ~ ( member_a @ C @ B4 ) ) ).

% DiffD2
thf(fact_761_DiffD2,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A @ B4 ) )
     => ~ ( member1426531481828664017od_a_b @ C @ B4 ) ) ).

% DiffD2
thf(fact_762_DiffD1,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
     => ( member_a @ C @ A ) ) ).

% DiffD1
thf(fact_763_DiffD1,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A @ B4 ) )
     => ( member1426531481828664017od_a_b @ C @ A ) ) ).

% DiffD1
thf(fact_764_DiffE,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( minus_minus_set_a @ A @ B4 ) )
     => ~ ( ( member_a @ C @ A )
         => ( member_a @ C @ B4 ) ) ) ).

% DiffE
thf(fact_765_DiffE,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( minus_6888070959853530735od_a_b @ A @ B4 ) )
     => ~ ( ( member1426531481828664017od_a_b @ C @ A )
         => ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% DiffE
thf(fact_766_in__image__insert__iff,axiom,
    ! [B4: set_set_a,X: a,A: set_a] :
      ( ! [C4: set_a] :
          ( ( member_set_a @ C4 @ B4 )
         => ~ ( member_a @ X @ C4 ) )
     => ( ( member_set_a @ A @ ( image_set_a_set_a @ ( insert_a2 @ X ) @ B4 ) )
        = ( ( member_a @ X @ A )
          & ( member_set_a @ ( minus_minus_set_a @ A @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_767_in__image__insert__iff,axiom,
    ! [B4: set_se3214616458824132456od_a_b,X: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ! [C4: set_Product_prod_a_b] :
          ( ( member1887650552762594865od_a_b @ C4 @ B4 )
         => ~ ( member1426531481828664017od_a_b @ X @ C4 ) )
     => ( ( member1887650552762594865od_a_b @ A @ ( image_5573672758305406725od_a_b @ ( insert4534936386344385144od_a_b @ X ) @ B4 ) )
        = ( ( member1426531481828664017od_a_b @ X @ A )
          & ( member1887650552762594865od_a_b @ ( minus_6888070959853530735od_a_b @ A @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) @ B4 ) ) ) ) ).

% in_image_insert_iff
thf(fact_768_subset__insert__iff,axiom,
    ! [A: set_a,X: a,B4: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X @ B4 ) )
      = ( ( ( member_a @ X @ A )
         => ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ B4 ) )
        & ( ~ ( member_a @ X @ A )
         => ( ord_less_eq_set_a @ A @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_769_subset__insert__iff,axiom,
    ! [A: set_Product_prod_a_b,X: product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ord_le817736998455962536od_a_b @ A @ ( insert4534936386344385144od_a_b @ X @ B4 ) )
      = ( ( ( member1426531481828664017od_a_b @ X @ A )
         => ( ord_le817736998455962536od_a_b @ ( minus_6888070959853530735od_a_b @ A @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) @ B4 ) )
        & ( ~ ( member1426531481828664017od_a_b @ X @ A )
         => ( ord_le817736998455962536od_a_b @ A @ B4 ) ) ) ) ).

% subset_insert_iff
thf(fact_770_Diff__single__insert,axiom,
    ! [A: set_a,X: a,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( minus_minus_set_a @ A @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ B4 )
     => ( ord_less_eq_set_a @ A @ ( insert_a2 @ X @ B4 ) ) ) ).

% Diff_single_insert
thf(fact_771_eq__snd__iff,axiom,
    ! [B: b,P: product_prod_a_b] :
      ( ( B
        = ( product_snd_a_b @ P ) )
      = ( ? [A6: a] :
            ( P
            = ( product_Pair_a_b @ A6 @ B ) ) ) ) ).

% eq_snd_iff
thf(fact_772_snd__conv,axiom,
    ! [X1: a,X2: b] :
      ( ( product_snd_a_b @ ( product_Pair_a_b @ X1 @ X2 ) )
      = X2 ) ).

% snd_conv
thf(fact_773_snd__eqD,axiom,
    ! [X: a,Y: b,A2: b] :
      ( ( ( product_snd_a_b @ ( product_Pair_a_b @ X @ Y ) )
        = A2 )
     => ( Y = A2 ) ) ).

% snd_eqD
thf(fact_774_sndI,axiom,
    ! [X: product_prod_a_b,Y: a,Z: b] :
      ( ( X
        = ( product_Pair_a_b @ Y @ Z ) )
     => ( ( product_snd_a_b @ X )
        = Z ) ) ).

% sndI
thf(fact_775_prod__eq__iff,axiom,
    ( ( ^ [Y5: product_prod_a_b,Z3: product_prod_a_b] : ( Y5 = Z3 ) )
    = ( ^ [S3: product_prod_a_b,T3: product_prod_a_b] :
          ( ( ( product_fst_a_b @ S3 )
            = ( product_fst_a_b @ T3 ) )
          & ( ( product_snd_a_b @ S3 )
            = ( product_snd_a_b @ T3 ) ) ) ) ) ).

% prod_eq_iff
thf(fact_776_prod__eqI,axiom,
    ! [P: product_prod_a_b,Q2: product_prod_a_b] :
      ( ( ( product_fst_a_b @ P )
        = ( product_fst_a_b @ Q2 ) )
     => ( ( ( product_snd_a_b @ P )
          = ( product_snd_a_b @ Q2 ) )
       => ( P = Q2 ) ) ) ).

% prod_eqI
thf(fact_777_prod_Oexpand,axiom,
    ! [Prod: product_prod_a_b,Prod2: product_prod_a_b] :
      ( ( ( ( product_fst_a_b @ Prod )
          = ( product_fst_a_b @ Prod2 ) )
        & ( ( product_snd_a_b @ Prod )
          = ( product_snd_a_b @ Prod2 ) ) )
     => ( Prod = Prod2 ) ) ).

% prod.expand
thf(fact_778_insert__Diff__if,axiom,
    ! [X: a,B4: set_a,A: set_a] :
      ( ( ( member_a @ X @ B4 )
       => ( ( minus_minus_set_a @ ( insert_a2 @ X @ A ) @ B4 )
          = ( minus_minus_set_a @ A @ B4 ) ) )
      & ( ~ ( member_a @ X @ B4 )
       => ( ( minus_minus_set_a @ ( insert_a2 @ X @ A ) @ B4 )
          = ( insert_a2 @ X @ ( minus_minus_set_a @ A @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_779_insert__Diff__if,axiom,
    ! [X: product_prod_a_b,B4: set_Product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( ( member1426531481828664017od_a_b @ X @ B4 )
       => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A ) @ B4 )
          = ( minus_6888070959853530735od_a_b @ A @ B4 ) ) )
      & ( ~ ( member1426531481828664017od_a_b @ X @ B4 )
       => ( ( minus_6888070959853530735od_a_b @ ( insert4534936386344385144od_a_b @ X @ A ) @ B4 )
          = ( insert4534936386344385144od_a_b @ X @ ( minus_6888070959853530735od_a_b @ A @ B4 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_780_singletonD,axiom,
    ! [B: a,A2: a] :
      ( ( member_a @ B @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_781_singletonD,axiom,
    ! [B: product_prod_a_b,A2: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ B @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) )
     => ( B = A2 ) ) ).

% singletonD
thf(fact_782_singleton__iff,axiom,
    ! [B: a,A2: a] :
      ( ( member_a @ B @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_783_singleton__iff,axiom,
    ! [B: product_prod_a_b,A2: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ B @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) )
      = ( B = A2 ) ) ).

% singleton_iff
thf(fact_784_doubleton__eq__iff,axiom,
    ! [A2: a,B: a,C: a,D2: a] :
      ( ( ( insert_a2 @ A2 @ ( insert_a2 @ B @ bot_bot_set_a ) )
        = ( insert_a2 @ C @ ( insert_a2 @ D2 @ bot_bot_set_a ) ) )
      = ( ( ( A2 = C )
          & ( B = D2 ) )
        | ( ( A2 = D2 )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_785_insert__not__empty,axiom,
    ! [A2: a,A: set_a] :
      ( ( insert_a2 @ A2 @ A )
     != bot_bot_set_a ) ).

% insert_not_empty
thf(fact_786_singleton__inject,axiom,
    ! [A2: a,B: a] :
      ( ( ( insert_a2 @ A2 @ bot_bot_set_a )
        = ( insert_a2 @ B @ bot_bot_set_a ) )
     => ( A2 = B ) ) ).

% singleton_inject
thf(fact_787_Pow__set_I1_J,axiom,
    ( ( pow_Product_prod_a_b @ ( set_Product_prod_a_b2 @ nil_Product_prod_a_b ) )
    = ( insert985587154976485848od_a_b @ bot_bo3428410327501040244od_a_b @ bot_bo7480059582523780564od_a_b ) ) ).

% Pow_set(1)
thf(fact_788_surjective__pairing,axiom,
    ! [T: product_prod_a_b] :
      ( T
      = ( product_Pair_a_b @ ( product_fst_a_b @ T ) @ ( product_snd_a_b @ T ) ) ) ).

% surjective_pairing
thf(fact_789_prod_Oexhaust__sel,axiom,
    ! [Prod: product_prod_a_b] :
      ( Prod
      = ( product_Pair_a_b @ ( product_fst_a_b @ Prod ) @ ( product_snd_a_b @ Prod ) ) ) ).

% prod.exhaust_sel
thf(fact_790_image__diff__subset,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] : ( ord_less_eq_set_a @ ( minus_minus_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) @ ( image_2802296252294471259_a_b_a @ F @ ( minus_6888070959853530735od_a_b @ A @ B4 ) ) ) ).

% image_diff_subset
thf(fact_791_subset__Diff__insert,axiom,
    ! [A: set_a,B4: set_a,X: a,C5: set_a] :
      ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B4 @ ( insert_a2 @ X @ C5 ) ) )
      = ( ( ord_less_eq_set_a @ A @ ( minus_minus_set_a @ B4 @ C5 ) )
        & ~ ( member_a @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_792_subset__Diff__insert,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,X: product_prod_a_b,C5: set_Product_prod_a_b] :
      ( ( ord_le817736998455962536od_a_b @ A @ ( minus_6888070959853530735od_a_b @ B4 @ ( insert4534936386344385144od_a_b @ X @ C5 ) ) )
      = ( ( ord_le817736998455962536od_a_b @ A @ ( minus_6888070959853530735od_a_b @ B4 @ C5 ) )
        & ~ ( member1426531481828664017od_a_b @ X @ A ) ) ) ).

% subset_Diff_insert
thf(fact_793_subset__singleton__iff,axiom,
    ! [X6: set_a,A2: a] :
      ( ( ord_less_eq_set_a @ X6 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
      = ( ( X6 = bot_bot_set_a )
        | ( X6
          = ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ) ).

% subset_singleton_iff
thf(fact_794_subset__singletonD,axiom,
    ! [A: set_a,X: a] :
      ( ( ord_less_eq_set_a @ A @ ( insert_a2 @ X @ bot_bot_set_a ) )
     => ( ( A = bot_bot_set_a )
        | ( A
          = ( insert_a2 @ X @ bot_bot_set_a ) ) ) ) ).

% subset_singletonD
thf(fact_795_empty__set,axiom,
    ( bot_bo3428410327501040244od_a_b
    = ( set_Product_prod_a_b2 @ nil_Product_prod_a_b ) ) ).

% empty_set
thf(fact_796_the__elem__image__unique,axiom,
    ! [A: set_Product_prod_a_b,F: product_prod_a_b > a,X: product_prod_a_b] :
      ( ( A != bot_bo3428410327501040244od_a_b )
     => ( ! [Y4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ Y4 @ A )
           => ( ( F @ Y4 )
              = ( F @ X ) ) )
       => ( ( the_elem_a @ ( image_2802296252294471259_a_b_a @ F @ A ) )
          = ( F @ X ) ) ) ) ).

% the_elem_image_unique
thf(fact_797_pawalk__verts_Osimps_I2_J,axiom,
    ! [U: product_prod_a_b,E2: produc2233330766403592615od_a_b,Es: list_P3463435030753280173od_a_b] :
      ( ( pair_p1830641812483871010od_a_b @ U @ ( cons_P8222935586017696605od_a_b @ E2 @ Es ) )
      = ( cons_P7316939131009794654od_a_b @ ( produc5013601669340818563od_a_b @ E2 ) @ ( pair_p1830641812483871010od_a_b @ ( produc8875788424405714885od_a_b @ E2 ) @ Es ) ) ) ).

% pawalk_verts.simps(2)
thf(fact_798_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
    ! [P2: a > b > $o,X: a,Y: b,A2: product_prod_a_b] :
      ( ( P2 @ X @ Y )
     => ( ( A2
          = ( product_Pair_a_b @ X @ Y ) )
       => ( P2 @ ( product_fst_a_b @ A2 ) @ ( product_snd_a_b @ A2 ) ) ) ) ).

% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_799_conjI__realizer,axiom,
    ! [P2: a > $o,P: a,Q: b > $o,Q2: b] :
      ( ( P2 @ P )
     => ( ( Q @ Q2 )
       => ( ( P2 @ ( product_fst_a_b @ ( product_Pair_a_b @ P @ Q2 ) ) )
          & ( Q @ ( product_snd_a_b @ ( product_Pair_a_b @ P @ Q2 ) ) ) ) ) ) ).

% conjI_realizer
thf(fact_800_exI__realizer,axiom,
    ! [P2: b > a > $o,Y: b,X: a] :
      ( ( P2 @ Y @ X )
     => ( P2 @ ( product_snd_a_b @ ( product_Pair_a_b @ X @ Y ) ) @ ( product_fst_a_b @ ( product_Pair_a_b @ X @ Y ) ) ) ) ).

% exI_realizer
thf(fact_801_wf__list__arcs_Oelims_I1_J,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: $o] :
      ( ( ( wf_list_arcs_a_b @ X )
        = Y )
     => ( ( ( X = nil_Product_prod_a_b )
         => ~ Y )
       => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( X
                = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
             => ( Y
                = ( ~ ( ~ ( member_b @ E @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                      & ( wf_list_arcs_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_arcs.elims(1)
thf(fact_802_wf__list__arcs_Oelims_I2_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( wf_list_arcs_a_b @ X )
     => ( ( X != nil_Product_prod_a_b )
       => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( X
                = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
             => ~ ( ~ ( member_b @ E @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                  & ( wf_list_arcs_a_b @ Xs ) ) ) ) ) ).

% wf_list_arcs.elims(2)
thf(fact_803_exE__realizer_H,axiom,
    ! [P2: b > a > $o,P: product_prod_a_b] :
      ( ( P2 @ ( product_snd_a_b @ P ) @ ( product_fst_a_b @ P ) )
     => ~ ! [X3: a,Y4: b] :
            ~ ( P2 @ Y4 @ X3 ) ) ).

% exE_realizer'
thf(fact_804_wf__list__arcs_Oelims_I3_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ~ ( wf_list_arcs_a_b @ X )
     => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
            ( ( X
              = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
           => ( ~ ( member_b @ E @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
              & ( wf_list_arcs_a_b @ Xs ) ) ) ) ).

% wf_list_arcs.elims(3)
thf(fact_805_wf__list__arcs_Osimps_I2_J,axiom,
    ! [V2: a,E2: b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( wf_list_arcs_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V2 @ E2 ) @ Xs2 ) )
      = ( ~ ( member_b @ E2 @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) ) )
        & ( wf_list_arcs_a_b @ Xs2 ) ) ) ).

% wf_list_arcs.simps(2)
thf(fact_806_is__singleton__the__elem,axiom,
    ( is_singleton_a
    = ( ^ [A5: set_a] :
          ( A5
          = ( insert_a2 @ ( the_elem_a @ A5 ) @ bot_bot_set_a ) ) ) ) ).

% is_singleton_the_elem
thf(fact_807_image__Fpow__mono,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_a] :
      ( ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ B4 )
     => ( ord_le3724670747650509150_set_a @ ( image_7003315350810844251_set_a @ ( image_2802296252294471259_a_b_a @ F ) @ ( finite3511610380200936718od_a_b @ A ) ) @ ( finite_Fpow_a @ B4 ) ) ) ).

% image_Fpow_mono
thf(fact_808_is__singletonI,axiom,
    ! [X: a] : ( is_singleton_a @ ( insert_a2 @ X @ bot_bot_set_a ) ) ).

% is_singletonI
thf(fact_809_is__singletonI_H,axiom,
    ! [A: set_a] :
      ( ( A != bot_bot_set_a )
     => ( ! [X3: a,Y4: a] :
            ( ( member_a @ X3 @ A )
           => ( ( member_a @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_singleton_a @ A ) ) ) ).

% is_singletonI'
thf(fact_810_is__singletonI_H,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ( A != bot_bo3428410327501040244od_a_b )
     => ( ! [X3: product_prod_a_b,Y4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ A )
           => ( ( member1426531481828664017od_a_b @ Y4 @ A )
             => ( X3 = Y4 ) ) )
       => ( is_sin3171834910201899932od_a_b @ A ) ) ) ).

% is_singletonI'
thf(fact_811_is__singleton__def,axiom,
    ( is_singleton_a
    = ( ^ [A5: set_a] :
        ? [X4: a] :
          ( A5
          = ( insert_a2 @ X4 @ bot_bot_set_a ) ) ) ) ).

% is_singleton_def
thf(fact_812_is__singletonE,axiom,
    ! [A: set_a] :
      ( ( is_singleton_a @ A )
     => ~ ! [X3: a] :
            ( A
           != ( insert_a2 @ X3 @ bot_bot_set_a ) ) ) ).

% is_singletonE
thf(fact_813_wf__dtree__iff__wf__list,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,R2: a] :
      ( ( ( wf_list_arcs_a_b @ Xs2 )
        & ~ ( member_a @ R2 @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) ) )
        & ( wf_list_verts_a_b @ Xs2 ) )
      = ( wf_dtree_a_b @ ( dtree_from_list_a_b @ R2 @ Xs2 ) ) ) ).

% wf_dtree_iff_wf_list
thf(fact_814_set__removeAll,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( set_a2 @ ( removeAll_a @ X @ Xs2 ) )
      = ( minus_minus_set_a @ ( set_a2 @ Xs2 ) @ ( insert_a2 @ X @ bot_bot_set_a ) ) ) ).

% set_removeAll
thf(fact_815_set__removeAll,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( remove3678230900650767028od_a_b @ X @ Xs2 ) )
      = ( minus_6888070959853530735od_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) ) ).

% set_removeAll
thf(fact_816_removeAll__id,axiom,
    ! [X: a,Xs2: list_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ( ( removeAll_a @ X @ Xs2 )
        = Xs2 ) ) ).

% removeAll_id
thf(fact_817_removeAll__id,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( ( remove3678230900650767028od_a_b @ X @ Xs2 )
        = Xs2 ) ) ).

% removeAll_id
thf(fact_818_removeAll__append,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( remove3678230900650767028od_a_b @ X @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
      = ( append5335208823350062147od_a_b @ ( remove3678230900650767028od_a_b @ X @ Xs2 ) @ ( remove3678230900650767028od_a_b @ X @ Ys ) ) ) ).

% removeAll_append
thf(fact_819_removeAll_Osimps_I2_J,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( X = Y )
       => ( ( remove3678230900650767028od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) )
          = ( remove3678230900650767028od_a_b @ X @ Xs2 ) ) )
      & ( ( X != Y )
       => ( ( remove3678230900650767028od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) )
          = ( cons_P7316939131009794654od_a_b @ Y @ ( remove3678230900650767028od_a_b @ X @ Xs2 ) ) ) ) ) ).

% removeAll.simps(2)
thf(fact_820_insert__code_I2_J,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( insert_a2 @ X @ ( coset_a @ Xs2 ) )
      = ( coset_a @ ( removeAll_a @ X @ Xs2 ) ) ) ).

% insert_code(2)
thf(fact_821_remove__code_I1_J,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( remove8198300761713201805od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( set_Product_prod_a_b2 @ ( remove3678230900650767028od_a_b @ X @ Xs2 ) ) ) ).

% remove_code(1)
thf(fact_822_SuccI,axiom,
    ! [Kl: list_a,K: a,Kl2: set_list_a] :
      ( ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 )
     => ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_823_SuccI,axiom,
    ! [Kl: list_P1467974523212632366od_a_b,K: product_prod_a_b,Kl2: set_li6306622547289818894od_a_b] :
      ( ( member6895035109809442519od_a_b @ ( append5335208823350062147od_a_b @ Kl @ ( cons_P7316939131009794654od_a_b @ K @ nil_Product_prod_a_b ) ) @ Kl2 )
     => ( member1426531481828664017od_a_b @ K @ ( bNF_Gr1699325108002784195od_a_b @ Kl2 @ Kl ) ) ) ).

% SuccI
thf(fact_824_SuccD,axiom,
    ! [K: a,Kl2: set_list_a,Kl: list_a] :
      ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ Kl ) )
     => ( member_list_a @ ( append_a @ Kl @ ( cons_a @ K @ nil_a ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_825_SuccD,axiom,
    ! [K: product_prod_a_b,Kl2: set_li6306622547289818894od_a_b,Kl: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ K @ ( bNF_Gr1699325108002784195od_a_b @ Kl2 @ Kl ) )
     => ( member6895035109809442519od_a_b @ ( append5335208823350062147od_a_b @ Kl @ ( cons_P7316939131009794654od_a_b @ K @ nil_Product_prod_a_b ) ) @ Kl2 ) ) ).

% SuccD
thf(fact_826_empty__Shift,axiom,
    ! [Kl2: set_list_a,K: a] :
      ( ( member_list_a @ nil_a @ Kl2 )
     => ( ( member_a @ K @ ( bNF_Greatest_Succ_a @ Kl2 @ nil_a ) )
       => ( member_list_a @ nil_a @ ( bNF_Greatest_Shift_a @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_827_empty__Shift,axiom,
    ! [Kl2: set_li6306622547289818894od_a_b,K: product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ nil_Product_prod_a_b @ Kl2 )
     => ( ( member1426531481828664017od_a_b @ K @ ( bNF_Gr1699325108002784195od_a_b @ Kl2 @ nil_Product_prod_a_b ) )
       => ( member6895035109809442519od_a_b @ nil_Product_prod_a_b @ ( bNF_Gr6004602585737257543od_a_b @ Kl2 @ K ) ) ) ) ).

% empty_Shift
thf(fact_828_Succ__Shift,axiom,
    ! [Kl2: set_li6306622547289818894od_a_b,K: product_prod_a_b,Kl: list_P1467974523212632366od_a_b] :
      ( ( bNF_Gr1699325108002784195od_a_b @ ( bNF_Gr6004602585737257543od_a_b @ Kl2 @ K ) @ Kl )
      = ( bNF_Gr1699325108002784195od_a_b @ Kl2 @ ( cons_P7316939131009794654od_a_b @ K @ Kl ) ) ) ).

% Succ_Shift
thf(fact_829_Cons__in__lists__iff,axiom,
    ! [X: a,Xs2: list_a,A: set_a] :
      ( ( member_list_a @ ( cons_a @ X @ Xs2 ) @ ( lists_a @ A ) )
      = ( ( member_a @ X @ A )
        & ( member_list_a @ Xs2 @ ( lists_a @ A ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_830_Cons__in__lists__iff,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( lists_9062004085038442652od_a_b @ A ) )
      = ( ( member1426531481828664017od_a_b @ X @ A )
        & ( member6895035109809442519od_a_b @ Xs2 @ ( lists_9062004085038442652od_a_b @ A ) ) ) ) ).

% Cons_in_lists_iff
thf(fact_831_in__listsI,axiom,
    ! [Xs2: list_a,A: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
         => ( member_a @ X3 @ A ) )
     => ( member_list_a @ Xs2 @ ( lists_a @ A ) ) ) ).

% in_listsI
thf(fact_832_in__listsI,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ( member1426531481828664017od_a_b @ X3 @ A ) )
     => ( member6895035109809442519od_a_b @ Xs2 @ ( lists_9062004085038442652od_a_b @ A ) ) ) ).

% in_listsI
thf(fact_833_append__in__lists__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ ( lists_9062004085038442652od_a_b @ A ) )
      = ( ( member6895035109809442519od_a_b @ Xs2 @ ( lists_9062004085038442652od_a_b @ A ) )
        & ( member6895035109809442519od_a_b @ Ys @ ( lists_9062004085038442652od_a_b @ A ) ) ) ) ).

% append_in_lists_conv
thf(fact_834_listsE,axiom,
    ! [X: a,L: list_a,A: set_a] :
      ( ( member_list_a @ ( cons_a @ X @ L ) @ ( lists_a @ A ) )
     => ~ ( ( member_a @ X @ A )
         => ~ ( member_list_a @ L @ ( lists_a @ A ) ) ) ) ).

% listsE
thf(fact_835_listsE,axiom,
    ! [X: product_prod_a_b,L: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ L ) @ ( lists_9062004085038442652od_a_b @ A ) )
     => ~ ( ( member1426531481828664017od_a_b @ X @ A )
         => ~ ( member6895035109809442519od_a_b @ L @ ( lists_9062004085038442652od_a_b @ A ) ) ) ) ).

% listsE
thf(fact_836_lists_OCons,axiom,
    ! [A2: a,A: set_a,L: list_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member_list_a @ L @ ( lists_a @ A ) )
       => ( member_list_a @ ( cons_a @ A2 @ L ) @ ( lists_a @ A ) ) ) ) ).

% lists.Cons
thf(fact_837_lists_OCons,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,L: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ( member6895035109809442519od_a_b @ L @ ( lists_9062004085038442652od_a_b @ A ) )
       => ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ A2 @ L ) @ ( lists_9062004085038442652od_a_b @ A ) ) ) ) ).

% lists.Cons
thf(fact_838_in__lists__conv__set,axiom,
    ! [Xs2: list_a,A: set_a] :
      ( ( member_list_a @ Xs2 @ ( lists_a @ A ) )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ ( set_a2 @ Xs2 ) )
           => ( member_a @ X4 @ A ) ) ) ) ).

% in_lists_conv_set
thf(fact_839_in__lists__conv__set,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ Xs2 @ ( lists_9062004085038442652od_a_b @ A ) )
      = ( ! [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
           => ( member1426531481828664017od_a_b @ X4 @ A ) ) ) ) ).

% in_lists_conv_set
thf(fact_840_in__listsD,axiom,
    ! [Xs2: list_a,A: set_a] :
      ( ( member_list_a @ Xs2 @ ( lists_a @ A ) )
     => ! [X5: a] :
          ( ( member_a @ X5 @ ( set_a2 @ Xs2 ) )
         => ( member_a @ X5 @ A ) ) ) ).

% in_listsD
thf(fact_841_in__listsD,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ Xs2 @ ( lists_9062004085038442652od_a_b @ A ) )
     => ! [X5: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ( member1426531481828664017od_a_b @ X5 @ A ) ) ) ).

% in_listsD
thf(fact_842_lists_Osimps,axiom,
    ! [A2: list_a,A: set_a] :
      ( ( member_list_a @ A2 @ ( lists_a @ A ) )
      = ( ( A2 = nil_a )
        | ? [A6: a,L2: list_a] :
            ( ( A2
              = ( cons_a @ A6 @ L2 ) )
            & ( member_a @ A6 @ A )
            & ( member_list_a @ L2 @ ( lists_a @ A ) ) ) ) ) ).

% lists.simps
thf(fact_843_lists_Osimps,axiom,
    ! [A2: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ A2 @ ( lists_9062004085038442652od_a_b @ A ) )
      = ( ( A2 = nil_Product_prod_a_b )
        | ? [A6: product_prod_a_b,L2: list_P1467974523212632366od_a_b] :
            ( ( A2
              = ( cons_P7316939131009794654od_a_b @ A6 @ L2 ) )
            & ( member1426531481828664017od_a_b @ A6 @ A )
            & ( member6895035109809442519od_a_b @ L2 @ ( lists_9062004085038442652od_a_b @ A ) ) ) ) ) ).

% lists.simps
thf(fact_844_lists_Ocases,axiom,
    ! [A2: list_a,A: set_a] :
      ( ( member_list_a @ A2 @ ( lists_a @ A ) )
     => ( ( A2 != nil_a )
       => ~ ! [A4: a,L3: list_a] :
              ( ( A2
                = ( cons_a @ A4 @ L3 ) )
             => ( ( member_a @ A4 @ A )
               => ~ ( member_list_a @ L3 @ ( lists_a @ A ) ) ) ) ) ) ).

% lists.cases
thf(fact_845_lists_Ocases,axiom,
    ! [A2: list_P1467974523212632366od_a_b,A: set_Product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ A2 @ ( lists_9062004085038442652od_a_b @ A ) )
     => ( ( A2 != nil_Product_prod_a_b )
       => ~ ! [A4: product_prod_a_b,L3: list_P1467974523212632366od_a_b] :
              ( ( A2
                = ( cons_P7316939131009794654od_a_b @ A4 @ L3 ) )
             => ( ( member1426531481828664017od_a_b @ A4 @ A )
               => ~ ( member6895035109809442519od_a_b @ L3 @ ( lists_9062004085038442652od_a_b @ A ) ) ) ) ) ) ).

% lists.cases
thf(fact_846_ShiftD,axiom,
    ! [Kl: list_P1467974523212632366od_a_b,Kl2: set_li6306622547289818894od_a_b,K: product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ Kl @ ( bNF_Gr6004602585737257543od_a_b @ Kl2 @ K ) )
     => ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ K @ Kl ) @ Kl2 ) ) ).

% ShiftD
thf(fact_847_dtree__to__list__sub__darcs,axiom,
    ! [T: dtree_a_b] : ( ord_less_eq_set_b @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ ( dtree_to_list_a_b @ T ) ) ) @ ( darcs_a_b @ T ) ) ).

% dtree_to_list_sub_darcs
thf(fact_848_dtree__from__list__eq__darcs,axiom,
    ! [R2: a,Xs2: list_P1467974523212632366od_a_b] :
      ( ( darcs_a_b @ ( dtree_from_list_a_b @ R2 @ Xs2 ) )
      = ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ).

% dtree_from_list_eq_darcs
thf(fact_849_wf__dtree_Odhead__in__dverts__no__root,axiom,
    ! [T: dtree_a_a,E2: a,Def2: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ E2 @ ( darcs_a_a @ T ) )
       => ( member_a @ ( dhead_a_a @ T @ Def2 @ E2 ) @ ( minus_minus_set_a @ ( dverts_a_a @ T ) @ ( insert_a2 @ ( root_a_a @ T ) @ bot_bot_set_a ) ) ) ) ) ).

% wf_dtree.dhead_in_dverts_no_root
thf(fact_850_wf__dtree_Odhead__in__dverts__no__root,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,E2: a,Def2: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ T ) )
       => ( member1426531481828664017od_a_b @ ( dhead_6787361551671486788_a_b_a @ T @ Def2 @ E2 ) @ ( minus_6888070959853530735od_a_b @ ( dverts6248479061738092880_a_b_a @ T ) @ ( insert4534936386344385144od_a_b @ ( root_P6085578689962329140_a_b_a @ T ) @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ).

% wf_dtree.dhead_in_dverts_no_root
thf(fact_851_wf__dtree_Odhead__in__dverts__no__root,axiom,
    ! [T: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,Def2: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ T ) )
       => ( member_a @ ( dhead_2162319049415464216od_a_b @ T @ Def2 @ E2 ) @ ( minus_minus_set_a @ ( dverts1623436559482070308od_a_b @ T ) @ ( insert_a2 @ ( root_a1460536187706306568od_a_b @ T ) @ bot_bot_set_a ) ) ) ) ) ).

% wf_dtree.dhead_in_dverts_no_root
thf(fact_852_wf__dtree_Odhead__in__dverts__no__root,axiom,
    ! [T: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,Def2: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ T ) )
       => ( member1426531481828664017od_a_b @ ( dhead_872314364254702830od_a_b @ T @ Def2 @ E2 ) @ ( minus_6888070959853530735od_a_b @ ( dverts2678499813339623674od_a_b @ T ) @ ( insert4534936386344385144od_a_b @ ( root_P6595957619955152862od_a_b @ T ) @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ).

% wf_dtree.dhead_in_dverts_no_root
thf(fact_853_wf__dtree_Odhead__in__dverts__no__root,axiom,
    ! [T: dtree_a_b,E2: b,Def2: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
       => ( member_a @ ( dhead_a_b @ T @ Def2 @ E2 ) @ ( minus_minus_set_a @ ( dverts_a_b @ T ) @ ( insert_a2 @ ( root_a_b @ T ) @ bot_bot_set_a ) ) ) ) ) ).

% wf_dtree.dhead_in_dverts_no_root
thf(fact_854_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_a_a,E2: a,Def2: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ E2 @ ( darcs_a_a @ T ) )
       => ( member_a @ ( dhead_a_a @ T @ Def2 @ E2 ) @ ( dverts_a_a @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_855_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,E2: a,Def2: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ T ) )
       => ( member1426531481828664017od_a_b @ ( dhead_6787361551671486788_a_b_a @ T @ Def2 @ E2 ) @ ( dverts6248479061738092880_a_b_a @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_856_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,Def2: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ T ) )
       => ( member_a @ ( dhead_2162319049415464216od_a_b @ T @ Def2 @ E2 ) @ ( dverts1623436559482070308od_a_b @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_857_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,Def2: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ T ) )
       => ( member1426531481828664017od_a_b @ ( dhead_872314364254702830od_a_b @ T @ Def2 @ E2 ) @ ( dverts2678499813339623674od_a_b @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_858_wf__dtree_Odhead__in__dverts,axiom,
    ! [T: dtree_a_b,E2: b,Def2: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
       => ( member_a @ ( dhead_a_b @ T @ Def2 @ E2 ) @ ( dverts_a_b @ T ) ) ) ) ).

% wf_dtree.dhead_in_dverts
thf(fact_859_wf__dtree_Odhead__in__childverts__no__root,axiom,
    ! [T: dtree_a_a,E2: a,X: dtree_a_a,E3: a,Xs2: fset_P2559403164575059562_a_a_a,R2: a,Def2: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ E2 @ ( darcs_a_a @ X ) )
       => ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ X @ E3 ) @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
         => ( ( ( node_a_a @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dhead_a_a @ T @ Def2 @ E2 ) @ ( minus_minus_set_a @ ( dverts_a_a @ X ) @ ( insert_a2 @ ( root_a_a @ X ) @ bot_bot_set_a ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts_no_root
thf(fact_860_wf__dtree_Odhead__in__childverts__no__root,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,E2: a,X: dtree_1072189271477137615_a_b_a,E3: a,Xs2: fset_P1695881499225648404_b_a_a,R2: product_prod_a_b,Def2: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ X ) )
       => ( ( member6586581021983584763_b_a_a @ ( produc4597562573236303254_b_a_a @ X @ E3 ) @ ( fset_P8686826054925190965_b_a_a @ Xs2 ) )
         => ( ( ( node_P9154982556985054868_a_b_a @ R2 @ Xs2 )
              = T )
           => ( member1426531481828664017od_a_b @ ( dhead_6787361551671486788_a_b_a @ T @ Def2 @ E2 ) @ ( minus_6888070959853530735od_a_b @ ( dverts6248479061738092880_a_b_a @ X ) @ ( insert4534936386344385144od_a_b @ ( root_P6085578689962329140_a_b_a @ X ) @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts_no_root
thf(fact_861_wf__dtree_Odhead__in__childverts__no__root,axiom,
    ! [T: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,X: dtree_2020077848175485859od_a_b,E3: product_prod_a_b,Xs2: fset_P8459658061218866410od_a_b,R2: a,Def2: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ X ) )
       => ( ( member8860753849636704977od_a_b @ ( produc9222456632128148844od_a_b @ X @ E3 ) @ ( fset_P7771937977521757707od_a_b @ Xs2 ) )
         => ( ( ( node_a4529940054729032296od_a_b @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dhead_2162319049415464216od_a_b @ T @ Def2 @ E2 ) @ ( minus_minus_set_a @ ( dverts1623436559482070308od_a_b @ X ) @ ( insert_a2 @ ( root_a1460536187706306568od_a_b @ X ) @ bot_bot_set_a ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts_no_root
thf(fact_862_wf__dtree_Odhead__in__childverts__no__root,axiom,
    ! [T: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,X: dtree_2253494508914376569od_a_b,E3: product_prod_a_b,Xs2: fset_P6924693328471822228od_a_b,R2: product_prod_a_b,Def2: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ X ) )
       => ( ( member4408059462661755003od_a_b @ ( produc8904365110017546518od_a_b @ X @ E3 ) @ ( fset_P5531643973500113333od_a_b @ Xs2 ) )
         => ( ( ( node_P1936840572307073598od_a_b @ R2 @ Xs2 )
              = T )
           => ( member1426531481828664017od_a_b @ ( dhead_872314364254702830od_a_b @ T @ Def2 @ E2 ) @ ( minus_6888070959853530735od_a_b @ ( dverts2678499813339623674od_a_b @ X ) @ ( insert4534936386344385144od_a_b @ ( root_P6595957619955152862od_a_b @ X ) @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts_no_root
thf(fact_863_wf__dtree_Odhead__in__childverts__no__root,axiom,
    ! [T: dtree_a_b,E2: b,X: dtree_a_b,E3: b,Xs2: fset_P5281107635120001194_a_b_b,R2: a,Def2: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ X ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E3 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( ( node_a_b @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dhead_a_b @ T @ Def2 @ E2 ) @ ( minus_minus_set_a @ ( dverts_a_b @ X ) @ ( insert_a2 @ ( root_a_b @ X ) @ bot_bot_set_a ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts_no_root
thf(fact_864_wf__dtree_Odhead__notin__subtree__wo__root,axiom,
    ! [T: dtree_a_a,X: dtree_a_a,E2: a,Xs2: fset_P2559403164575059562_a_a_a,P: a,R2: a,Dh: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ X @ E2 ) @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
       => ( ~ ( member_a @ P @ ( darcs_a_a @ X ) )
         => ( ( member_a @ P @ ( darcs_a_a @ T ) )
           => ( ( T
                = ( node_a_a @ R2 @ Xs2 ) )
             => ~ ( member_a @ ( dhead_a_a @ T @ Dh @ P ) @ ( minus_minus_set_a @ ( dverts_a_a @ X ) @ ( insert_a2 @ ( root_a_a @ X ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).

% wf_dtree.dhead_notin_subtree_wo_root
thf(fact_865_wf__dtree_Odhead__notin__subtree__wo__root,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,X: dtree_1072189271477137615_a_b_a,E2: a,Xs2: fset_P1695881499225648404_b_a_a,P: a,R2: product_prod_a_b,Dh: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member6586581021983584763_b_a_a @ ( produc4597562573236303254_b_a_a @ X @ E2 ) @ ( fset_P8686826054925190965_b_a_a @ Xs2 ) )
       => ( ~ ( member_a @ P @ ( darcs_4738931720465112383_a_b_a @ X ) )
         => ( ( member_a @ P @ ( darcs_4738931720465112383_a_b_a @ T ) )
           => ( ( T
                = ( node_P9154982556985054868_a_b_a @ R2 @ Xs2 ) )
             => ~ ( member1426531481828664017od_a_b @ ( dhead_6787361551671486788_a_b_a @ T @ Dh @ P ) @ ( minus_6888070959853530735od_a_b @ ( dverts6248479061738092880_a_b_a @ X ) @ ( insert4534936386344385144od_a_b @ ( root_P6085578689962329140_a_b_a @ X ) @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ) ) ) ).

% wf_dtree.dhead_notin_subtree_wo_root
thf(fact_866_wf__dtree_Odhead__notin__subtree__wo__root,axiom,
    ! [T: dtree_2020077848175485859od_a_b,X: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,Xs2: fset_P8459658061218866410od_a_b,P: product_prod_a_b,R2: a,Dh: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member8860753849636704977od_a_b @ ( produc9222456632128148844od_a_b @ X @ E2 ) @ ( fset_P7771937977521757707od_a_b @ Xs2 ) )
       => ( ~ ( member1426531481828664017od_a_b @ P @ ( darcs_113889218209089811od_a_b @ X ) )
         => ( ( member1426531481828664017od_a_b @ P @ ( darcs_113889218209089811od_a_b @ T ) )
           => ( ( T
                = ( node_a4529940054729032296od_a_b @ R2 @ Xs2 ) )
             => ~ ( member_a @ ( dhead_2162319049415464216od_a_b @ T @ Dh @ P ) @ ( minus_minus_set_a @ ( dverts1623436559482070308od_a_b @ X ) @ ( insert_a2 @ ( root_a1460536187706306568od_a_b @ X ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).

% wf_dtree.dhead_notin_subtree_wo_root
thf(fact_867_wf__dtree_Odhead__notin__subtree__wo__root,axiom,
    ! [T: dtree_2253494508914376569od_a_b,X: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,Xs2: fset_P6924693328471822228od_a_b,P: product_prod_a_b,R2: product_prod_a_b,Dh: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member4408059462661755003od_a_b @ ( produc8904365110017546518od_a_b @ X @ E2 ) @ ( fset_P5531643973500113333od_a_b @ Xs2 ) )
       => ( ~ ( member1426531481828664017od_a_b @ P @ ( darcs_380088452687597033od_a_b @ X ) )
         => ( ( member1426531481828664017od_a_b @ P @ ( darcs_380088452687597033od_a_b @ T ) )
           => ( ( T
                = ( node_P1936840572307073598od_a_b @ R2 @ Xs2 ) )
             => ~ ( member1426531481828664017od_a_b @ ( dhead_872314364254702830od_a_b @ T @ Dh @ P ) @ ( minus_6888070959853530735od_a_b @ ( dverts2678499813339623674od_a_b @ X ) @ ( insert4534936386344385144od_a_b @ ( root_P6595957619955152862od_a_b @ X ) @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ) ) ) ).

% wf_dtree.dhead_notin_subtree_wo_root
thf(fact_868_wf__dtree_Odhead__notin__subtree__wo__root,axiom,
    ! [T: dtree_a_b,X: dtree_a_b,E2: b,Xs2: fset_P5281107635120001194_a_b_b,P: b,R2: a,Dh: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ~ ( member_b @ P @ ( darcs_a_b @ X ) )
         => ( ( member_b @ P @ ( darcs_a_b @ T ) )
           => ( ( T
                = ( node_a_b @ R2 @ Xs2 ) )
             => ~ ( member_a @ ( dhead_a_b @ T @ Dh @ P ) @ ( minus_minus_set_a @ ( dverts_a_b @ X ) @ ( insert_a2 @ ( root_a_b @ X ) @ bot_bot_set_a ) ) ) ) ) ) ) ) ).

% wf_dtree.dhead_notin_subtree_wo_root
thf(fact_869_is__empty__set,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b] :
      ( ( is_emp2937470229047908218od_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( null_P9109340217170369718od_a_b @ Xs2 ) ) ).

% is_empty_set
thf(fact_870_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_a_a,E2: a,X: dtree_a_a,E3: a,Xs2: fset_P2559403164575059562_a_a_a,R2: a,Def2: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ E2 @ ( darcs_a_a @ X ) )
       => ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ X @ E3 ) @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
         => ( ( ( node_a_a @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dhead_a_a @ T @ Def2 @ E2 ) @ ( dverts_a_a @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_871_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,E2: a,X: dtree_1072189271477137615_a_b_a,E3: a,Xs2: fset_P1695881499225648404_b_a_a,R2: product_prod_a_b,Def2: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ X ) )
       => ( ( member6586581021983584763_b_a_a @ ( produc4597562573236303254_b_a_a @ X @ E3 ) @ ( fset_P8686826054925190965_b_a_a @ Xs2 ) )
         => ( ( ( node_P9154982556985054868_a_b_a @ R2 @ Xs2 )
              = T )
           => ( member1426531481828664017od_a_b @ ( dhead_6787361551671486788_a_b_a @ T @ Def2 @ E2 ) @ ( dverts6248479061738092880_a_b_a @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_872_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,X: dtree_2020077848175485859od_a_b,E3: product_prod_a_b,Xs2: fset_P8459658061218866410od_a_b,R2: a,Def2: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ X ) )
       => ( ( member8860753849636704977od_a_b @ ( produc9222456632128148844od_a_b @ X @ E3 ) @ ( fset_P7771937977521757707od_a_b @ Xs2 ) )
         => ( ( ( node_a4529940054729032296od_a_b @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dhead_2162319049415464216od_a_b @ T @ Def2 @ E2 ) @ ( dverts1623436559482070308od_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_873_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,X: dtree_2253494508914376569od_a_b,E3: product_prod_a_b,Xs2: fset_P6924693328471822228od_a_b,R2: product_prod_a_b,Def2: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ X ) )
       => ( ( member4408059462661755003od_a_b @ ( produc8904365110017546518od_a_b @ X @ E3 ) @ ( fset_P5531643973500113333od_a_b @ Xs2 ) )
         => ( ( ( node_P1936840572307073598od_a_b @ R2 @ Xs2 )
              = T )
           => ( member1426531481828664017od_a_b @ ( dhead_872314364254702830od_a_b @ T @ Def2 @ E2 ) @ ( dverts2678499813339623674od_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_874_wf__dtree_Odhead__in__childverts,axiom,
    ! [T: dtree_a_b,E2: b,X: dtree_a_b,E3: b,Xs2: fset_P5281107635120001194_a_b_b,R2: a,Def2: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ X ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E3 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( ( node_a_b @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dhead_a_b @ T @ Def2 @ E2 ) @ ( dverts_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dhead_in_childverts
thf(fact_875_child__uneq,axiom,
    ! [T: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
      ( ( member_dtree_a_b @ T @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
     => ( ( node_a_b @ R2 @ Xs2 )
       != T ) ) ).

% child_uneq
thf(fact_876_dverts__child__if__not__root,axiom,
    ! [V2: a,R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
      ( ( member_a @ V2 @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
     => ( ( V2 != R2 )
       => ? [X3: dtree_a_b] :
            ( ( member_dtree_a_b @ X3 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
            & ( member_a @ V2 @ ( dverts_a_b @ X3 ) ) ) ) ) ).

% dverts_child_if_not_root
thf(fact_877_subtree__if__child,axiom,
    ! [X: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
      ( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
     => ( is_subtree_a_b @ X @ ( node_a_b @ R2 @ Xs2 ) ) ) ).

% subtree_if_child
thf(fact_878_wf__dverts__rec,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b] :
      ( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member_dtree_a_b @ T12 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
       => ( wf_dverts_a_b @ T12 ) ) ) ).

% wf_dverts_rec
thf(fact_879_null__rec_I1_J,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ~ ( null_P9109340217170369718od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) ) ).

% null_rec(1)
thf(fact_880_darcs__child__subseteq,axiom,
    ! [X: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
      ( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
     => ( ord_less_eq_set_b @ ( darcs_a_b @ X ) @ ( darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).

% darcs_child_subseteq
thf(fact_881_dverts__child__subseteq,axiom,
    ! [X: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
      ( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
     => ( ord_less_eq_set_a @ ( dverts_a_b @ X ) @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).

% dverts_child_subseteq
thf(fact_882_subtree__child__if__dvert__notroot,axiom,
    ! [V2: a,R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
      ( ( V2 != R2 )
     => ( ( member_a @ V2 @ ( dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) )
       => ? [R4: a,Ys3: fset_P5281107635120001194_a_b_b,Zs3: fset_P5281107635120001194_a_b_b] :
            ( ( is_subtree_a_b @ ( node_a_b @ R4 @ Ys3 ) @ ( node_a_b @ R2 @ Xs2 ) )
            & ( member_dtree_a_b @ ( node_a_b @ V2 @ Zs3 ) @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Ys3 ) ) ) ) ) ) ).

% subtree_child_if_dvert_notroot
thf(fact_883_wf__dtree_Osubtree__uneq__if__arc__uneq,axiom,
    ! [T: dtree_a_b,X1: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,X2: dtree_a_b,E23: b,R2: a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X2 @ E23 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( E1 != E23 )
           => ( ( ( node_a_b @ R2 @ Xs2 )
                = T )
             => ( X1 != X2 ) ) ) ) ) ) ).

% wf_dtree.subtree_uneq_if_arc_uneq
thf(fact_884_wf__dtree_Oarc__uneq__if__subtree__uneq,axiom,
    ! [T: dtree_a_b,X1: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,X2: dtree_a_b,E23: b,R2: a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X1 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X2 @ E23 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( X1 != X2 )
           => ( ( ( node_a_b @ R2 @ Xs2 )
                = T )
             => ( E1 != E23 ) ) ) ) ) ) ).

% wf_dtree.arc_uneq_if_subtree_uneq
thf(fact_885_wf__dtree_Owf__dtree__rec,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,X: dtree_a_b,E2: b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( ( node_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( wf_dtree_a_b @ X ) ) ) ) ).

% wf_dtree.wf_dtree_rec
thf(fact_886_wf__dtree_Oroot__not__subtree,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,X: dtree_a_b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( ( node_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
         => ~ ( member_a @ R2 @ ( dverts_a_b @ X ) ) ) ) ) ).

% wf_dtree.root_not_subtree
thf(fact_887_subtree__child__if__strict__subtree,axiom,
    ! [T12: dtree_a_b,T22: dtree_a_b] :
      ( ( strict_subtree_a_b @ T12 @ T22 )
     => ? [R: a,Xs: fset_P5281107635120001194_a_b_b] :
          ( ( is_subtree_a_b @ ( node_a_b @ R @ Xs ) @ T22 )
          & ( member_dtree_a_b @ T12 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs ) ) ) ) ) ).

% subtree_child_if_strict_subtree
thf(fact_888_child__sub__if__strict__subtree,axiom,
    ! [T12: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b] :
      ( ( strict_subtree_a_b @ T12 @ ( node_a_b @ R2 @ Xs2 ) )
     => ? [X3: dtree_a_b] :
          ( ( member_dtree_a_b @ X3 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
          & ( is_subtree_a_b @ T12 @ X3 ) ) ) ).

% child_sub_if_strict_subtree
thf(fact_889_wf__dtree_Ochild__arc__not__subtree,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,X: dtree_a_b,E1: b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( ( node_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ~ ( member_b @ E1 @ ( darcs_a_b @ X ) ) ) ) ) ).

% wf_dtree.child_arc_not_subtree
thf(fact_890_subtree__child__if__dvert__notelem,axiom,
    ! [V2: a,T: dtree_a_b] :
      ( ( V2
       != ( root_a_b @ T ) )
     => ( ( member_a @ V2 @ ( dverts_a_b @ T ) )
       => ? [R4: a,Ys3: fset_P5281107635120001194_a_b_b,Zs3: fset_P5281107635120001194_a_b_b] :
            ( ( is_subtree_a_b @ ( node_a_b @ R4 @ Ys3 ) @ T )
            & ( member_dtree_a_b @ ( node_a_b @ V2 @ Zs3 ) @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Ys3 ) ) ) ) ) ) ).

% subtree_child_if_dvert_notelem
thf(fact_891_wf__dtree_Osubtree__root__not__root,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,X: dtree_a_b,E2: b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( T
          = ( node_a_b @ R2 @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( root_a_b @ X )
           != R2 ) ) ) ) ).

% wf_dtree.subtree_root_not_root
thf(fact_892_root__not__child__if__wf__dverts,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,E1: b] :
      ( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ~ ( member_a @ R2 @ ( dverts_a_b @ T12 ) ) ) ) ).

% root_not_child_if_wf_dverts
thf(fact_893_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_a_a,Dt: a > a,P: a,X: dtree_a_a,R2: a,Xs2: fset_P2559403164575059562_a_a_a,E2: a,Dh: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ ( dtail_a_a @ T @ Dt @ P ) @ ( dverts_a_a @ X ) )
       => ( ( member_a @ P @ ( darcs_a_a @ T ) )
         => ( ( T
              = ( node_a_a @ R2 @ Xs2 ) )
           => ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ X @ E2 ) @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
             => ( member_a @ ( dhead_a_a @ T @ Dh @ P ) @ ( dverts_a_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_894_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_2020077848175485859od_a_b,Dt: product_prod_a_b > a,P: product_prod_a_b,X: dtree_2020077848175485859od_a_b,R2: a,Xs2: fset_P8459658061218866410od_a_b,E2: product_prod_a_b,Dh: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member_a @ ( dtail_5621385774229617160od_a_b @ T @ Dt @ P ) @ ( dverts1623436559482070308od_a_b @ X ) )
       => ( ( member1426531481828664017od_a_b @ P @ ( darcs_113889218209089811od_a_b @ T ) )
         => ( ( T
              = ( node_a4529940054729032296od_a_b @ R2 @ Xs2 ) )
           => ( ( member8860753849636704977od_a_b @ ( produc9222456632128148844od_a_b @ X @ E2 ) @ ( fset_P7771937977521757707od_a_b @ Xs2 ) )
             => ( member_a @ ( dhead_2162319049415464216od_a_b @ T @ Dh @ P ) @ ( dverts1623436559482070308od_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_895_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,Dt: a > product_prod_a_b,P: a,X: dtree_1072189271477137615_a_b_a,R2: product_prod_a_b,Xs2: fset_P1695881499225648404_b_a_a,E2: a,Dh: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member1426531481828664017od_a_b @ ( dtail_1023056239630863924_a_b_a @ T @ Dt @ P ) @ ( dverts6248479061738092880_a_b_a @ X ) )
       => ( ( member_a @ P @ ( darcs_4738931720465112383_a_b_a @ T ) )
         => ( ( T
              = ( node_P9154982556985054868_a_b_a @ R2 @ Xs2 ) )
           => ( ( member6586581021983584763_b_a_a @ ( produc4597562573236303254_b_a_a @ X @ E2 ) @ ( fset_P8686826054925190965_b_a_a @ Xs2 ) )
             => ( member1426531481828664017od_a_b @ ( dhead_6787361551671486788_a_b_a @ T @ Dh @ P ) @ ( dverts6248479061738092880_a_b_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_896_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_2253494508914376569od_a_b,Dt: product_prod_a_b > product_prod_a_b,P: product_prod_a_b,X: dtree_2253494508914376569od_a_b,R2: product_prod_a_b,Xs2: fset_P6924693328471822228od_a_b,E2: product_prod_a_b,Dh: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ ( dtail_5302258445299794398od_a_b @ T @ Dt @ P ) @ ( dverts2678499813339623674od_a_b @ X ) )
       => ( ( member1426531481828664017od_a_b @ P @ ( darcs_380088452687597033od_a_b @ T ) )
         => ( ( T
              = ( node_P1936840572307073598od_a_b @ R2 @ Xs2 ) )
           => ( ( member4408059462661755003od_a_b @ ( produc8904365110017546518od_a_b @ X @ E2 ) @ ( fset_P5531643973500113333od_a_b @ Xs2 ) )
             => ( member1426531481828664017od_a_b @ ( dhead_872314364254702830od_a_b @ T @ Dh @ P ) @ ( dverts2678499813339623674od_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_897_wf__dtree_Odhead__in__verts__if__dtail,axiom,
    ! [T: dtree_a_b,Dt: b > a,P: b,X: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,E2: b,Dh: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_a @ ( dtail_a_b @ T @ Dt @ P ) @ ( dverts_a_b @ X ) )
       => ( ( member_b @ P @ ( darcs_a_b @ T ) )
         => ( ( T
              = ( node_a_b @ R2 @ Xs2 ) )
           => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
             => ( member_a @ ( dhead_a_b @ T @ Dh @ P ) @ ( dverts_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.dhead_in_verts_if_dtail
thf(fact_898_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_a_a,Dt: a > a,E2: a,V2: a,X: dtree_a_a,E1: a,Xs2: fset_P2559403164575059562_a_a_a,T12: dtree_a_a,R2: a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( ( dtail_a_a @ T @ Dt @ E2 )
          = V2 )
       => ( ( member_a @ E2 @ ( darcs_a_a @ T ) )
         => ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ X @ E1 ) @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
           => ( ( is_subtree_a_a @ T12 @ X )
             => ( ( ( node_a_a @ R2 @ Xs2 )
                  = T )
               => ( ( member_a @ V2 @ ( dverts_a_a @ T12 ) )
                 => ( member_a @ E2 @ ( darcs_a_a @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_899_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,Dt: a > product_prod_a_b,E2: a,V2: product_prod_a_b,X: dtree_1072189271477137615_a_b_a,E1: a,Xs2: fset_P1695881499225648404_b_a_a,T12: dtree_1072189271477137615_a_b_a,R2: product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( ( dtail_1023056239630863924_a_b_a @ T @ Dt @ E2 )
          = V2 )
       => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ T ) )
         => ( ( member6586581021983584763_b_a_a @ ( produc4597562573236303254_b_a_a @ X @ E1 ) @ ( fset_P8686826054925190965_b_a_a @ Xs2 ) )
           => ( ( is_sub5611864065441456628_a_b_a @ T12 @ X )
             => ( ( ( node_P9154982556985054868_a_b_a @ R2 @ Xs2 )
                  = T )
               => ( ( member1426531481828664017od_a_b @ V2 @ ( dverts6248479061738092880_a_b_a @ T12 ) )
                 => ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_900_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_2020077848175485859od_a_b,Dt: product_prod_a_b > a,E2: product_prod_a_b,V2: a,X: dtree_2020077848175485859od_a_b,E1: product_prod_a_b,Xs2: fset_P8459658061218866410od_a_b,T12: dtree_2020077848175485859od_a_b,R2: a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( ( dtail_5621385774229617160od_a_b @ T @ Dt @ E2 )
          = V2 )
       => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ T ) )
         => ( ( member8860753849636704977od_a_b @ ( produc9222456632128148844od_a_b @ X @ E1 ) @ ( fset_P7771937977521757707od_a_b @ Xs2 ) )
           => ( ( is_sub986821563185434056od_a_b @ T12 @ X )
             => ( ( ( node_a4529940054729032296od_a_b @ R2 @ Xs2 )
                  = T )
               => ( ( member_a @ V2 @ ( dverts1623436559482070308od_a_b @ T12 ) )
                 => ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_901_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_2253494508914376569od_a_b,Dt: product_prod_a_b > product_prod_a_b,E2: product_prod_a_b,V2: product_prod_a_b,X: dtree_2253494508914376569od_a_b,E1: product_prod_a_b,Xs2: fset_P6924693328471822228od_a_b,T12: dtree_2253494508914376569od_a_b,R2: product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( ( dtail_5302258445299794398od_a_b @ T @ Dt @ E2 )
          = V2 )
       => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ T ) )
         => ( ( member4408059462661755003od_a_b @ ( produc8904365110017546518od_a_b @ X @ E1 ) @ ( fset_P5531643973500113333od_a_b @ Xs2 ) )
           => ( ( is_sub6933245819845350814od_a_b @ T12 @ X )
             => ( ( ( node_P1936840572307073598od_a_b @ R2 @ Xs2 )
                  = T )
               => ( ( member1426531481828664017od_a_b @ V2 @ ( dverts2678499813339623674od_a_b @ T12 ) )
                 => ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_902_wf__dtree_Odarc__in__sub__if__dtail__in__sub,axiom,
    ! [T: dtree_a_b,Dt: b > a,E2: b,V2: a,X: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,R2: a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( ( dtail_a_b @ T @ Dt @ E2 )
          = V2 )
       => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
         => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
           => ( ( is_subtree_a_b @ T12 @ X )
             => ( ( ( node_a_b @ R2 @ Xs2 )
                  = T )
               => ( ( member_a @ V2 @ ( dverts_a_b @ T12 ) )
                 => ( member_b @ E2 @ ( darcs_a_b @ X ) ) ) ) ) ) ) ) ) ).

% wf_dtree.darc_in_sub_if_dtail_in_sub
thf(fact_903_disjoint__dverts__if__wf__aux,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,E1: b,T22: dtree_a_b,E23: b] :
      ( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T22 @ E23 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( ( produc331601717337510060_a_b_b @ T12 @ E1 )
             != ( produc331601717337510060_a_b_b @ T22 @ E23 ) )
           => ( ( inf_inf_set_a @ ( dverts_a_b @ T12 ) @ ( dverts_a_b @ T22 ) )
              = bot_bot_set_a ) ) ) ) ) ).

% disjoint_dverts_if_wf_aux
thf(fact_904_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_a_a,E2: a,X: dtree_a_a,E3: a,Xs2: fset_P2559403164575059562_a_a_a,R2: a,Def2: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ E2 @ ( darcs_a_a @ X ) )
       => ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ X @ E3 ) @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
         => ( ( ( node_a_a @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dtail_a_a @ T @ Def2 @ E2 ) @ ( dverts_a_a @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_905_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,E2: a,X: dtree_1072189271477137615_a_b_a,E3: a,Xs2: fset_P1695881499225648404_b_a_a,R2: product_prod_a_b,Def2: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ X ) )
       => ( ( member6586581021983584763_b_a_a @ ( produc4597562573236303254_b_a_a @ X @ E3 ) @ ( fset_P8686826054925190965_b_a_a @ Xs2 ) )
         => ( ( ( node_P9154982556985054868_a_b_a @ R2 @ Xs2 )
              = T )
           => ( member1426531481828664017od_a_b @ ( dtail_1023056239630863924_a_b_a @ T @ Def2 @ E2 ) @ ( dverts6248479061738092880_a_b_a @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_906_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,X: dtree_2020077848175485859od_a_b,E3: product_prod_a_b,Xs2: fset_P8459658061218866410od_a_b,R2: a,Def2: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ X ) )
       => ( ( member8860753849636704977od_a_b @ ( produc9222456632128148844od_a_b @ X @ E3 ) @ ( fset_P7771937977521757707od_a_b @ Xs2 ) )
         => ( ( ( node_a4529940054729032296od_a_b @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dtail_5621385774229617160od_a_b @ T @ Def2 @ E2 ) @ ( dverts1623436559482070308od_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_907_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,X: dtree_2253494508914376569od_a_b,E3: product_prod_a_b,Xs2: fset_P6924693328471822228od_a_b,R2: product_prod_a_b,Def2: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ X ) )
       => ( ( member4408059462661755003od_a_b @ ( produc8904365110017546518od_a_b @ X @ E3 ) @ ( fset_P5531643973500113333od_a_b @ Xs2 ) )
         => ( ( ( node_P1936840572307073598od_a_b @ R2 @ Xs2 )
              = T )
           => ( member1426531481828664017od_a_b @ ( dtail_5302258445299794398od_a_b @ T @ Def2 @ E2 ) @ ( dverts2678499813339623674od_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_908_wf__dtree_Odtail__in__childverts,axiom,
    ! [T: dtree_a_b,E2: b,X: dtree_a_b,E3: b,Xs2: fset_P5281107635120001194_a_b_b,R2: a,Def2: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ X ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E3 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( ( node_a_b @ R2 @ Xs2 )
              = T )
           => ( member_a @ ( dtail_a_b @ T @ Def2 @ E2 ) @ ( dverts_a_b @ X ) ) ) ) ) ) ).

% wf_dtree.dtail_in_childverts
thf(fact_909_IntI,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ A )
     => ( ( member_a @ C @ B4 )
       => ( member_a @ C @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).

% IntI
thf(fact_910_IntI,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ A )
     => ( ( member1426531481828664017od_a_b @ C @ B4 )
       => ( member1426531481828664017od_a_b @ C @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) ) ) ).

% IntI
thf(fact_911_Int__iff,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B4 ) )
      = ( ( member_a @ C @ A )
        & ( member_a @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_912_Int__iff,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) )
      = ( ( member1426531481828664017od_a_b @ C @ A )
        & ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% Int_iff
thf(fact_913_Int__insert__left__if0,axiom,
    ! [A2: a,C5: set_a,B4: set_a] :
      ( ~ ( member_a @ A2 @ C5 )
     => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B4 ) @ C5 )
        = ( inf_inf_set_a @ B4 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_914_Int__insert__left__if0,axiom,
    ! [A2: product_prod_a_b,C5: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ A2 @ C5 )
     => ( ( inf_in8976041639890736118od_a_b @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) @ C5 )
        = ( inf_in8976041639890736118od_a_b @ B4 @ C5 ) ) ) ).

% Int_insert_left_if0
thf(fact_915_Int__insert__left__if1,axiom,
    ! [A2: a,C5: set_a,B4: set_a] :
      ( ( member_a @ A2 @ C5 )
     => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B4 ) @ C5 )
        = ( insert_a2 @ A2 @ ( inf_inf_set_a @ B4 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_916_Int__insert__left__if1,axiom,
    ! [A2: product_prod_a_b,C5: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ C5 )
     => ( ( inf_in8976041639890736118od_a_b @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) @ C5 )
        = ( insert4534936386344385144od_a_b @ A2 @ ( inf_in8976041639890736118od_a_b @ B4 @ C5 ) ) ) ) ).

% Int_insert_left_if1
thf(fact_917_insert__inter__insert,axiom,
    ! [A2: a,A: set_a,B4: set_a] :
      ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ A ) @ ( insert_a2 @ A2 @ B4 ) )
      = ( insert_a2 @ A2 @ ( inf_inf_set_a @ A @ B4 ) ) ) ).

% insert_inter_insert
thf(fact_918_Int__insert__right__if0,axiom,
    ! [A2: a,A: set_a,B4: set_a] :
      ( ~ ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B4 ) )
        = ( inf_inf_set_a @ A @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_919_Int__insert__right__if0,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ( inf_in8976041639890736118od_a_b @ A @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) )
        = ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) ) ).

% Int_insert_right_if0
thf(fact_920_Int__insert__right__if1,axiom,
    ! [A2: a,A: set_a,B4: set_a] :
      ( ( member_a @ A2 @ A )
     => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B4 ) )
        = ( insert_a2 @ A2 @ ( inf_inf_set_a @ A @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_921_Int__insert__right__if1,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ( inf_in8976041639890736118od_a_b @ A @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) )
        = ( insert4534936386344385144od_a_b @ A2 @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) ) ) ).

% Int_insert_right_if1
thf(fact_922_insert__disjoint_I1_J,axiom,
    ! [A2: a,A: set_a,B4: set_a] :
      ( ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ A ) @ B4 )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B4 )
        & ( ( inf_inf_set_a @ A @ B4 )
          = bot_bot_set_a ) ) ) ).

% insert_disjoint(1)
thf(fact_923_insert__disjoint_I1_J,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ( inf_in8976041639890736118od_a_b @ ( insert4534936386344385144od_a_b @ A2 @ A ) @ B4 )
        = bot_bo3428410327501040244od_a_b )
      = ( ~ ( member1426531481828664017od_a_b @ A2 @ B4 )
        & ( ( inf_in8976041639890736118od_a_b @ A @ B4 )
          = bot_bo3428410327501040244od_a_b ) ) ) ).

% insert_disjoint(1)
thf(fact_924_insert__disjoint_I2_J,axiom,
    ! [A2: a,A: set_a,B4: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ ( insert_a2 @ A2 @ A ) @ B4 ) )
      = ( ~ ( member_a @ A2 @ B4 )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_925_insert__disjoint_I2_J,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( bot_bo3428410327501040244od_a_b
        = ( inf_in8976041639890736118od_a_b @ ( insert4534936386344385144od_a_b @ A2 @ A ) @ B4 ) )
      = ( ~ ( member1426531481828664017od_a_b @ A2 @ B4 )
        & ( bot_bo3428410327501040244od_a_b
          = ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) ) ) ).

% insert_disjoint(2)
thf(fact_926_disjoint__insert_I1_J,axiom,
    ! [B4: set_a,A2: a,A: set_a] :
      ( ( ( inf_inf_set_a @ B4 @ ( insert_a2 @ A2 @ A ) )
        = bot_bot_set_a )
      = ( ~ ( member_a @ A2 @ B4 )
        & ( ( inf_inf_set_a @ B4 @ A )
          = bot_bot_set_a ) ) ) ).

% disjoint_insert(1)
thf(fact_927_disjoint__insert_I1_J,axiom,
    ! [B4: set_Product_prod_a_b,A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( ( inf_in8976041639890736118od_a_b @ B4 @ ( insert4534936386344385144od_a_b @ A2 @ A ) )
        = bot_bo3428410327501040244od_a_b )
      = ( ~ ( member1426531481828664017od_a_b @ A2 @ B4 )
        & ( ( inf_in8976041639890736118od_a_b @ B4 @ A )
          = bot_bo3428410327501040244od_a_b ) ) ) ).

% disjoint_insert(1)
thf(fact_928_disjoint__insert_I2_J,axiom,
    ! [A: set_a,B: a,B4: set_a] :
      ( ( bot_bot_set_a
        = ( inf_inf_set_a @ A @ ( insert_a2 @ B @ B4 ) ) )
      = ( ~ ( member_a @ B @ A )
        & ( bot_bot_set_a
          = ( inf_inf_set_a @ A @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_929_disjoint__insert_I2_J,axiom,
    ! [A: set_Product_prod_a_b,B: product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( bot_bo3428410327501040244od_a_b
        = ( inf_in8976041639890736118od_a_b @ A @ ( insert4534936386344385144od_a_b @ B @ B4 ) ) )
      = ( ~ ( member1426531481828664017od_a_b @ B @ A )
        & ( bot_bo3428410327501040244od_a_b
          = ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) ) ) ).

% disjoint_insert(2)
thf(fact_930_Int__Collect__mono,axiom,
    ! [A: set_a,B4: set_a,P2: a > $o,Q: a > $o] :
      ( ( ord_less_eq_set_a @ A @ B4 )
     => ( ! [X3: a] :
            ( ( member_a @ X3 @ A )
           => ( ( P2 @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_less_eq_set_a @ ( inf_inf_set_a @ A @ ( collect_a @ P2 ) ) @ ( inf_inf_set_a @ B4 @ ( collect_a @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_931_Int__Collect__mono,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,P2: product_prod_a_b > $o,Q: product_prod_a_b > $o] :
      ( ( ord_le817736998455962536od_a_b @ A @ B4 )
     => ( ! [X3: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ A )
           => ( ( P2 @ X3 )
             => ( Q @ X3 ) ) )
       => ( ord_le817736998455962536od_a_b @ ( inf_in8976041639890736118od_a_b @ A @ ( collec3336397801687681299od_a_b @ P2 ) ) @ ( inf_in8976041639890736118od_a_b @ B4 @ ( collec3336397801687681299od_a_b @ Q ) ) ) ) ) ).

% Int_Collect_mono
thf(fact_932_IntE,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B4 ) )
     => ~ ( ( member_a @ C @ A )
         => ~ ( member_a @ C @ B4 ) ) ) ).

% IntE
thf(fact_933_IntE,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) )
     => ~ ( ( member1426531481828664017od_a_b @ C @ A )
         => ~ ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% IntE
thf(fact_934_IntD1,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B4 ) )
     => ( member_a @ C @ A ) ) ).

% IntD1
thf(fact_935_IntD1,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) )
     => ( member1426531481828664017od_a_b @ C @ A ) ) ).

% IntD1
thf(fact_936_IntD2,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( inf_inf_set_a @ A @ B4 ) )
     => ( member_a @ C @ B4 ) ) ).

% IntD2
thf(fact_937_IntD2,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) )
     => ( member1426531481828664017od_a_b @ C @ B4 ) ) ).

% IntD2
thf(fact_938_Int__insert__left,axiom,
    ! [A2: a,C5: set_a,B4: set_a] :
      ( ( ( member_a @ A2 @ C5 )
       => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B4 ) @ C5 )
          = ( insert_a2 @ A2 @ ( inf_inf_set_a @ B4 @ C5 ) ) ) )
      & ( ~ ( member_a @ A2 @ C5 )
       => ( ( inf_inf_set_a @ ( insert_a2 @ A2 @ B4 ) @ C5 )
          = ( inf_inf_set_a @ B4 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_939_Int__insert__left,axiom,
    ! [A2: product_prod_a_b,C5: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ( member1426531481828664017od_a_b @ A2 @ C5 )
       => ( ( inf_in8976041639890736118od_a_b @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) @ C5 )
          = ( insert4534936386344385144od_a_b @ A2 @ ( inf_in8976041639890736118od_a_b @ B4 @ C5 ) ) ) )
      & ( ~ ( member1426531481828664017od_a_b @ A2 @ C5 )
       => ( ( inf_in8976041639890736118od_a_b @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) @ C5 )
          = ( inf_in8976041639890736118od_a_b @ B4 @ C5 ) ) ) ) ).

% Int_insert_left
thf(fact_940_Int__insert__right,axiom,
    ! [A2: a,A: set_a,B4: set_a] :
      ( ( ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B4 ) )
          = ( insert_a2 @ A2 @ ( inf_inf_set_a @ A @ B4 ) ) ) )
      & ( ~ ( member_a @ A2 @ A )
       => ( ( inf_inf_set_a @ A @ ( insert_a2 @ A2 @ B4 ) )
          = ( inf_inf_set_a @ A @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_941_Int__insert__right,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ( member1426531481828664017od_a_b @ A2 @ A )
       => ( ( inf_in8976041639890736118od_a_b @ A @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) )
          = ( insert4534936386344385144od_a_b @ A2 @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) ) )
      & ( ~ ( member1426531481828664017od_a_b @ A2 @ A )
       => ( ( inf_in8976041639890736118od_a_b @ A @ ( insert4534936386344385144od_a_b @ A2 @ B4 ) )
          = ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) ) ) ).

% Int_insert_right
thf(fact_942_disjoint__iff,axiom,
    ! [A: set_a,B4: set_a] :
      ( ( ( inf_inf_set_a @ A @ B4 )
        = bot_bot_set_a )
      = ( ! [X4: a] :
            ( ( member_a @ X4 @ A )
           => ~ ( member_a @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_943_disjoint__iff,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ( inf_in8976041639890736118od_a_b @ A @ B4 )
        = bot_bo3428410327501040244od_a_b )
      = ( ! [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ A )
           => ~ ( member1426531481828664017od_a_b @ X4 @ B4 ) ) ) ) ).

% disjoint_iff
thf(fact_944_Int__emptyI,axiom,
    ! [A: set_a,B4: set_a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ A )
         => ~ ( member_a @ X3 @ B4 ) )
     => ( ( inf_inf_set_a @ A @ B4 )
        = bot_bot_set_a ) ) ).

% Int_emptyI
thf(fact_945_Int__emptyI,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A )
         => ~ ( member1426531481828664017od_a_b @ X3 @ B4 ) )
     => ( ( inf_in8976041639890736118od_a_b @ A @ B4 )
        = bot_bo3428410327501040244od_a_b ) ) ).

% Int_emptyI
thf(fact_946_image__Int__subset,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] : ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) ) @ ( inf_inf_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ).

% image_Int_subset
thf(fact_947_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_a_a,E2: a,Def2: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ E2 @ ( darcs_a_a @ T ) )
       => ( member_a @ ( dtail_a_a @ T @ Def2 @ E2 ) @ ( dverts_a_a @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_948_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,E2: a,Def2: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ T ) )
       => ( member1426531481828664017od_a_b @ ( dtail_1023056239630863924_a_b_a @ T @ Def2 @ E2 ) @ ( dverts6248479061738092880_a_b_a @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_949_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,Def2: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ T ) )
       => ( member_a @ ( dtail_5621385774229617160od_a_b @ T @ Def2 @ E2 ) @ ( dverts1623436559482070308od_a_b @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_950_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,Def2: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ T ) )
       => ( member1426531481828664017od_a_b @ ( dtail_5302258445299794398od_a_b @ T @ Def2 @ E2 ) @ ( dverts2678499813339623674od_a_b @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_951_wf__dtree_Odtail__in__dverts,axiom,
    ! [T: dtree_a_b,E2: b,Def2: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
       => ( member_a @ ( dtail_a_b @ T @ Def2 @ E2 ) @ ( dverts_a_b @ T ) ) ) ) ).

% wf_dtree.dtail_in_dverts
thf(fact_952_wf__dtree_Odtail__root__in__set,axiom,
    ! [T: dtree_a_b,E2: b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,Dt: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
       => ( ( T
            = ( node_a_b @ R2 @ Xs2 ) )
         => ( ( ( dtail_a_b @ T @ Dt @ E2 )
              = R2 )
           => ( member_b @ E2 @ ( image_3908709015779211070_b_b_b @ produc5748100250121904638_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) ) ) ) ) ) ).

% wf_dtree.dtail_root_in_set
thf(fact_953_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_a_a,E2: a,X: dtree_a_a,Def2: a > a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ E2 @ ( darcs_a_a @ X ) )
       => ( ( is_subtree_a_a @ X @ T )
         => ( member_a @ ( dtail_a_a @ T @ Def2 @ E2 ) @ ( dverts_a_a @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_954_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,E2: a,X: dtree_1072189271477137615_a_b_a,Def2: a > product_prod_a_b] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member_a @ E2 @ ( darcs_4738931720465112383_a_b_a @ X ) )
       => ( ( is_sub5611864065441456628_a_b_a @ X @ T )
         => ( member1426531481828664017od_a_b @ ( dtail_1023056239630863924_a_b_a @ T @ Def2 @ E2 ) @ ( dverts6248479061738092880_a_b_a @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_955_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_2020077848175485859od_a_b,E2: product_prod_a_b,X: dtree_2020077848175485859od_a_b,Def2: product_prod_a_b > a] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_113889218209089811od_a_b @ X ) )
       => ( ( is_sub986821563185434056od_a_b @ X @ T )
         => ( member_a @ ( dtail_5621385774229617160od_a_b @ T @ Def2 @ E2 ) @ ( dverts1623436559482070308od_a_b @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_956_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_2253494508914376569od_a_b,E2: product_prod_a_b,X: dtree_2253494508914376569od_a_b,Def2: product_prod_a_b > product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ E2 @ ( darcs_380088452687597033od_a_b @ X ) )
       => ( ( is_sub6933245819845350814od_a_b @ X @ T )
         => ( member1426531481828664017od_a_b @ ( dtail_5302258445299794398od_a_b @ T @ Def2 @ E2 ) @ ( dverts2678499813339623674od_a_b @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_957_wf__dtree_Odtail__in__subdverts,axiom,
    ! [T: dtree_a_b,E2: b,X: dtree_a_b,Def2: b > a] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_b @ E2 @ ( darcs_a_b @ X ) )
       => ( ( is_subtree_a_b @ X @ T )
         => ( member_a @ ( dtail_a_b @ T @ Def2 @ E2 ) @ ( dverts_a_b @ X ) ) ) ) ) ).

% wf_dtree.dtail_in_subdverts
thf(fact_958_wf__dtree_Oout__arcs__in__subarcs__aux,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,Dt: b > a,E2: b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( is_subtree_a_b @ ( node_a_b @ R2 @ Xs2 ) @ T )
       => ( ( ( dtail_a_b @ T @ Dt @ E2 )
            = R2 )
         => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
           => ( member_b @ E2 @ ( image_3908709015779211070_b_b_b @ produc5748100250121904638_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) ) ) ) ) ) ).

% wf_dtree.out_arcs_in_subarcs_aux
thf(fact_959_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_a_a,Dt: a > a,P: a,X: dtree_a_a,R2: a,Xs2: fset_P2559403164575059562_a_a_a,E2: a] :
      ( ( wf_dtree_a_a @ T )
     => ( ( member_a @ ( dtail_a_a @ T @ Dt @ P ) @ ( dverts_a_a @ X ) )
       => ( ( member_a @ P @ ( darcs_a_a @ T ) )
         => ( ( T
              = ( node_a_a @ R2 @ Xs2 ) )
           => ( ( member5016570112408633425_a_a_a @ ( produc967250717942496748_a_a_a @ X @ E2 ) @ ( fset_P1418902625193942923_a_a_a @ Xs2 ) )
             => ( member_a @ P @ ( darcs_a_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_960_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_2020077848175485859od_a_b,Dt: product_prod_a_b > a,P: product_prod_a_b,X: dtree_2020077848175485859od_a_b,R2: a,Xs2: fset_P8459658061218866410od_a_b,E2: product_prod_a_b] :
      ( ( wf_dtr5610383794032640231od_a_b @ T )
     => ( ( member_a @ ( dtail_5621385774229617160od_a_b @ T @ Dt @ P ) @ ( dverts1623436559482070308od_a_b @ X ) )
       => ( ( member1426531481828664017od_a_b @ P @ ( darcs_113889218209089811od_a_b @ T ) )
         => ( ( T
              = ( node_a4529940054729032296od_a_b @ R2 @ Xs2 ) )
           => ( ( member8860753849636704977od_a_b @ ( produc9222456632128148844od_a_b @ X @ E2 ) @ ( fset_P7771937977521757707od_a_b @ Xs2 ) )
             => ( member1426531481828664017od_a_b @ P @ ( darcs_113889218209089811od_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_961_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_1072189271477137615_a_b_a,Dt: a > product_prod_a_b,P: a,X: dtree_1072189271477137615_a_b_a,R2: product_prod_a_b,Xs2: fset_P1695881499225648404_b_a_a,E2: a] :
      ( ( wf_dtr1012054259433886995_a_b_a @ T )
     => ( ( member1426531481828664017od_a_b @ ( dtail_1023056239630863924_a_b_a @ T @ Dt @ P ) @ ( dverts6248479061738092880_a_b_a @ X ) )
       => ( ( member_a @ P @ ( darcs_4738931720465112383_a_b_a @ T ) )
         => ( ( T
              = ( node_P9154982556985054868_a_b_a @ R2 @ Xs2 ) )
           => ( ( member6586581021983584763_b_a_a @ ( produc4597562573236303254_b_a_a @ X @ E2 ) @ ( fset_P8686826054925190965_b_a_a @ Xs2 ) )
             => ( member_a @ P @ ( darcs_4738931720465112383_a_b_a @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_962_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_2253494508914376569od_a_b,Dt: product_prod_a_b > product_prod_a_b,P: product_prod_a_b,X: dtree_2253494508914376569od_a_b,R2: product_prod_a_b,Xs2: fset_P6924693328471822228od_a_b,E2: product_prod_a_b] :
      ( ( wf_dtr7064500587373793725od_a_b @ T )
     => ( ( member1426531481828664017od_a_b @ ( dtail_5302258445299794398od_a_b @ T @ Dt @ P ) @ ( dverts2678499813339623674od_a_b @ X ) )
       => ( ( member1426531481828664017od_a_b @ P @ ( darcs_380088452687597033od_a_b @ T ) )
         => ( ( T
              = ( node_P1936840572307073598od_a_b @ R2 @ Xs2 ) )
           => ( ( member4408059462661755003od_a_b @ ( produc8904365110017546518od_a_b @ X @ E2 ) @ ( fset_P5531643973500113333od_a_b @ Xs2 ) )
             => ( member1426531481828664017od_a_b @ P @ ( darcs_380088452687597033od_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_963_wf__dtree_Oarc__in__subtree__if__tail__in__subtree,axiom,
    ! [T: dtree_a_b,Dt: b > a,P: b,X: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,E2: b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( member_a @ ( dtail_a_b @ T @ Dt @ P ) @ ( dverts_a_b @ X ) )
       => ( ( member_b @ P @ ( darcs_a_b @ T ) )
         => ( ( T
              = ( node_a_b @ R2 @ Xs2 ) )
           => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
             => ( member_b @ P @ ( darcs_a_b @ X ) ) ) ) ) ) ) ).

% wf_dtree.arc_in_subtree_if_tail_in_subtree
thf(fact_964_wf__dtree_Odverts__child__subset,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,X: dtree_a_b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( ( node_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
         => ( ord_less_set_a @ ( dverts_a_b @ X ) @ ( dverts_a_b @ T ) ) ) ) ) ).

% wf_dtree.dverts_child_subset
thf(fact_965_wf__dtree_Odarcs__child__subset,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,X: dtree_a_b] :
      ( ( wf_dtree_a_b @ T )
     => ( ( ( node_a_b @ R2 @ Xs2 )
          = T )
       => ( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
         => ( ord_less_set_b @ ( darcs_a_b @ X ) @ ( darcs_a_b @ T ) ) ) ) ) ).

% wf_dtree.darcs_child_subset
thf(fact_966_disjoint__darcs__if__wf__aux2,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,E1: b,T22: dtree_a_b,E23: b] :
      ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T22 @ E23 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( ( produc331601717337510060_a_b_b @ T12 @ E1 )
             != ( produc331601717337510060_a_b_b @ T22 @ E23 ) )
           => ( ( inf_inf_set_b @ ( darcs_a_b @ T12 ) @ ( darcs_a_b @ T22 ) )
              = bot_bot_set_b ) ) ) ) ) ).

% disjoint_darcs_if_wf_aux2
thf(fact_967_suc__sub__if__strict__subtree,axiom,
    ! [T12: dtree_a_b,T22: dtree_a_b] :
      ( ( strict_subtree_a_b @ T12 @ T22 )
     => ? [X3: dtree_a_b] :
          ( ( member_dtree_a_b @ X3 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T22 ) ) ) )
          & ( is_subtree_a_b @ T12 @ X3 ) ) ) ).

% suc_sub_if_strict_subtree
thf(fact_968_dtree_Ocollapse,axiom,
    ! [Dtree: dtree_a_b] :
      ( ( node_a_b @ ( root_a_b @ Dtree ) @ ( sucs_a_b @ Dtree ) )
      = Dtree ) ).

% dtree.collapse
thf(fact_969_dtree_Osel_I2_J,axiom,
    ! [X1: a,X2: fset_P5281107635120001194_a_b_b] :
      ( ( sucs_a_b @ ( node_a_b @ X1 @ X2 ) )
      = X2 ) ).

% dtree.sel(2)
thf(fact_970_psubset__imp__ex__mem,axiom,
    ! [A: set_a,B4: set_a] :
      ( ( ord_less_set_a @ A @ B4 )
     => ? [B6: a] : ( member_a @ B6 @ ( minus_minus_set_a @ B4 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_971_psubset__imp__ex__mem,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ord_le6891031760732254900od_a_b @ A @ B4 )
     => ? [B6: product_prod_a_b] : ( member1426531481828664017od_a_b @ B6 @ ( minus_6888070959853530735od_a_b @ B4 @ A ) ) ) ).

% psubset_imp_ex_mem
thf(fact_972_psubsetD,axiom,
    ! [A: set_a,B4: set_a,C: a] :
      ( ( ord_less_set_a @ A @ B4 )
     => ( ( member_a @ C @ A )
       => ( member_a @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_973_psubsetD,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,C: product_prod_a_b] :
      ( ( ord_le6891031760732254900od_a_b @ A @ B4 )
     => ( ( member1426531481828664017od_a_b @ C @ A )
       => ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% psubsetD
thf(fact_974_dtree_Oexhaust__sel,axiom,
    ! [Dtree: dtree_a_b] :
      ( Dtree
      = ( node_a_b @ ( root_a_b @ Dtree ) @ ( sucs_a_b @ Dtree ) ) ) ).

% dtree.exhaust_sel
thf(fact_975_wf__list__arcs__if__wf__darcs,axiom,
    ! [T: dtree_a_b] :
      ( ( wf_darcs_a_b @ T )
     => ( wf_list_arcs_a_b @ ( dtree_to_list_a_b @ T ) ) ) ).

% wf_list_arcs_if_wf_darcs
thf(fact_976_disjoint__darcs__if__wf__aux4,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,E1: b,T22: dtree_a_b,E23: b] :
      ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T22 @ E23 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( ( produc331601717337510060_a_b_b @ T12 @ E1 )
             != ( produc331601717337510060_a_b_b @ T22 @ E23 ) )
           => ( E1 != E23 ) ) ) ) ) ).

% disjoint_darcs_if_wf_aux4
thf(fact_977_disjoint__darcs__if__wf__aux1,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,E1: b] :
      ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ~ ( member_b @ E1 @ ( darcs_a_b @ T12 ) ) ) ) ).

% disjoint_darcs_if_wf_aux1
thf(fact_978_disjoint__darcs__if__wf__aux3,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,E1: b,T22: dtree_a_b,E23: b] :
      ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T22 @ E23 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ~ ( member_b @ E23 @ ( darcs_a_b @ T12 ) ) ) ) ) ).

% disjoint_darcs_if_wf_aux3
thf(fact_979_psubset__insert__iff,axiom,
    ! [A: set_a,X: a,B4: set_a] :
      ( ( ord_less_set_a @ A @ ( insert_a2 @ X @ B4 ) )
      = ( ( ( member_a @ X @ B4 )
         => ( ord_less_set_a @ A @ B4 ) )
        & ( ~ ( member_a @ X @ B4 )
         => ( ( ( member_a @ X @ A )
             => ( ord_less_set_a @ ( minus_minus_set_a @ A @ ( insert_a2 @ X @ bot_bot_set_a ) ) @ B4 ) )
            & ( ~ ( member_a @ X @ A )
             => ( ord_less_eq_set_a @ A @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_980_psubset__insert__iff,axiom,
    ! [A: set_Product_prod_a_b,X: product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( ord_le6891031760732254900od_a_b @ A @ ( insert4534936386344385144od_a_b @ X @ B4 ) )
      = ( ( ( member1426531481828664017od_a_b @ X @ B4 )
         => ( ord_le6891031760732254900od_a_b @ A @ B4 ) )
        & ( ~ ( member1426531481828664017od_a_b @ X @ B4 )
         => ( ( ( member1426531481828664017od_a_b @ X @ A )
             => ( ord_le6891031760732254900od_a_b @ ( minus_6888070959853530735od_a_b @ A @ ( insert4534936386344385144od_a_b @ X @ bot_bo3428410327501040244od_a_b ) ) @ B4 ) )
            & ( ~ ( member1426531481828664017od_a_b @ X @ A )
             => ( ord_le817736998455962536od_a_b @ A @ B4 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_981_wf__darcs__rec,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b] :
      ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member_dtree_a_b @ T12 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
       => ( wf_darcs_a_b @ T12 ) ) ) ).

% wf_darcs_rec
thf(fact_982_child__uneq_H,axiom,
    ! [T: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a,V2: a] :
      ( ( member_dtree_a_b @ T @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
     => ( ( node_a_b @ R2 @ Xs2 )
       != ( node_a_b @ V2 @ ( sucs_a_b @ T ) ) ) ) ).

% child_uneq'
thf(fact_983_suc__uneq_H,axiom,
    ! [T12: dtree_a_b,T: dtree_a_b,V2: a] :
      ( ( member_dtree_a_b @ T12 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) ) )
     => ( T
       != ( node_a_b @ V2 @ ( sucs_a_b @ T12 ) ) ) ) ).

% suc_uneq'
thf(fact_984_dhead__in__child__eq__child,axiom,
    ! [T: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,E2: b,R2: a,Def2: b > a] :
      ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
     => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
       => ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
         => ( ( dhead_a_b @ ( node_a_b @ R2 @ Xs2 ) @ Def2 @ E2 )
            = ( dhead_a_b @ T @ Def2 @ E2 ) ) ) ) ) ).

% dhead_in_child_eq_child
thf(fact_985_dtail__in__child__eq__child,axiom,
    ! [T: dtree_a_b,E1: b,Xs2: fset_P5281107635120001194_a_b_b,E2: b,R2: a,Def2: b > a] :
      ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
     => ( ( member_b @ E2 @ ( darcs_a_b @ T ) )
       => ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
         => ( ( dtail_a_b @ ( node_a_b @ R2 @ Xs2 ) @ Def2 @ E2 )
            = ( dtail_a_b @ T @ Def2 @ E2 ) ) ) ) ) ).

% dtail_in_child_eq_child
thf(fact_986_dhead__in__set__eq__root,axiom,
    ! [T: dtree_a_b,E2: b,Xs2: fset_P5281107635120001194_a_b_b,R2: a,Def2: b > a] :
      ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T @ E2 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
     => ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
       => ( ( dhead_a_b @ ( node_a_b @ R2 @ Xs2 ) @ Def2 @ E2 )
          = ( root_a_b @ T ) ) ) ) ).

% dhead_in_set_eq_root
thf(fact_987_dverts__suc__subseteq,axiom,
    ! [X: dtree_a_b,T: dtree_a_b] :
      ( ( member_dtree_a_b @ X @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) ) )
     => ( ord_less_eq_set_a @ ( dverts_a_b @ X ) @ ( dverts_a_b @ T ) ) ) ).

% dverts_suc_subseteq
thf(fact_988_dverts__suc__if__not__root,axiom,
    ! [V2: a,T: dtree_a_b] :
      ( ( member_a @ V2 @ ( dverts_a_b @ T ) )
     => ( ( V2
         != ( root_a_b @ T ) )
       => ? [X3: dtree_a_b] :
            ( ( member_dtree_a_b @ X3 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ ( sucs_a_b @ T ) ) ) )
            & ( member_a @ V2 @ ( dverts_a_b @ X3 ) ) ) ) ) ).

% dverts_suc_if_not_root
thf(fact_989_disjoint__darcs__if__wf__aux5,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,T12: dtree_a_b,E1: b,T22: dtree_a_b,E23: b] :
      ( ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T12 @ E1 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
       => ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ T22 @ E23 ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
         => ( ( ( produc331601717337510060_a_b_b @ T12 @ E1 )
             != ( produc331601717337510060_a_b_b @ T22 @ E23 ) )
           => ( ( inf_inf_set_b @ ( sup_sup_set_b @ ( darcs_a_b @ T12 ) @ ( insert_b @ E1 @ bot_bot_set_b ) ) @ ( sup_sup_set_b @ ( darcs_a_b @ T22 ) @ ( insert_b @ E23 @ bot_bot_set_b ) ) )
              = bot_bot_set_b ) ) ) ) ) ).

% disjoint_darcs_if_wf_aux5
thf(fact_990_UnCI,axiom,
    ! [C: a,B4: set_a,A: set_a] :
      ( ( ~ ( member_a @ C @ B4 )
       => ( member_a @ C @ A ) )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).

% UnCI
thf(fact_991_UnCI,axiom,
    ! [C: product_prod_a_b,B4: set_Product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( ~ ( member1426531481828664017od_a_b @ C @ B4 )
       => ( member1426531481828664017od_a_b @ C @ A ) )
     => ( member1426531481828664017od_a_b @ C @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) ) ) ).

% UnCI
thf(fact_992_Un__iff,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) )
      = ( ( member_a @ C @ A )
        | ( member_a @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_993_Un__iff,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) )
      = ( ( member1426531481828664017od_a_b @ C @ A )
        | ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% Un_iff
thf(fact_994_Un__insert__left,axiom,
    ! [A2: a,B4: set_a,C5: set_a] :
      ( ( sup_sup_set_a @ ( insert_a2 @ A2 @ B4 ) @ C5 )
      = ( insert_a2 @ A2 @ ( sup_sup_set_a @ B4 @ C5 ) ) ) ).

% Un_insert_left
thf(fact_995_Un__insert__right,axiom,
    ! [A: set_a,A2: a,B4: set_a] :
      ( ( sup_sup_set_a @ A @ ( insert_a2 @ A2 @ B4 ) )
      = ( insert_a2 @ A2 @ ( sup_sup_set_a @ A @ B4 ) ) ) ).

% Un_insert_right
thf(fact_996_set__append,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
      = ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) @ ( set_Product_prod_a_b2 @ Ys ) ) ) ).

% set_append
thf(fact_997_image__Un,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( image_2802296252294471259_a_b_a @ F @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) )
      = ( sup_sup_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ).

% image_Un
thf(fact_998_UnE,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) )
     => ( ~ ( member_a @ C @ A )
       => ( member_a @ C @ B4 ) ) ) ).

% UnE
thf(fact_999_UnE,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) )
     => ( ~ ( member1426531481828664017od_a_b @ C @ A )
       => ( member1426531481828664017od_a_b @ C @ B4 ) ) ) ).

% UnE
thf(fact_1000_UnI1,axiom,
    ! [C: a,A: set_a,B4: set_a] :
      ( ( member_a @ C @ A )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).

% UnI1
thf(fact_1001_UnI1,axiom,
    ! [C: product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ A )
     => ( member1426531481828664017od_a_b @ C @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) ) ) ).

% UnI1
thf(fact_1002_UnI2,axiom,
    ! [C: a,B4: set_a,A: set_a] :
      ( ( member_a @ C @ B4 )
     => ( member_a @ C @ ( sup_sup_set_a @ A @ B4 ) ) ) ).

% UnI2
thf(fact_1003_UnI2,axiom,
    ! [C: product_prod_a_b,B4: set_Product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ C @ B4 )
     => ( member1426531481828664017od_a_b @ C @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) ) ) ).

% UnI2
thf(fact_1004_singleton__Un__iff,axiom,
    ! [X: a,A: set_a,B4: set_a] :
      ( ( ( insert_a2 @ X @ bot_bot_set_a )
        = ( sup_sup_set_a @ A @ B4 ) )
      = ( ( ( A = bot_bot_set_a )
          & ( B4
            = ( insert_a2 @ X @ bot_bot_set_a ) ) )
        | ( ( A
            = ( insert_a2 @ X @ bot_bot_set_a ) )
          & ( B4 = bot_bot_set_a ) )
        | ( ( A
            = ( insert_a2 @ X @ bot_bot_set_a ) )
          & ( B4
            = ( insert_a2 @ X @ bot_bot_set_a ) ) ) ) ) ).

% singleton_Un_iff
thf(fact_1005_Un__singleton__iff,axiom,
    ! [A: set_a,B4: set_a,X: a] :
      ( ( ( sup_sup_set_a @ A @ B4 )
        = ( insert_a2 @ X @ bot_bot_set_a ) )
      = ( ( ( A = bot_bot_set_a )
          & ( B4
            = ( insert_a2 @ X @ bot_bot_set_a ) ) )
        | ( ( A
            = ( insert_a2 @ X @ bot_bot_set_a ) )
          & ( B4 = bot_bot_set_a ) )
        | ( ( A
            = ( insert_a2 @ X @ bot_bot_set_a ) )
          & ( B4
            = ( insert_a2 @ X @ bot_bot_set_a ) ) ) ) ) ).

% Un_singleton_iff
thf(fact_1006_insert__is__Un,axiom,
    ( insert_a2
    = ( ^ [A6: a] : ( sup_sup_set_a @ ( insert_a2 @ A6 @ bot_bot_set_a ) ) ) ) ).

% insert_is_Un
thf(fact_1007_Pow__insert,axiom,
    ! [A2: a,A: set_a] :
      ( ( pow_a @ ( insert_a2 @ A2 @ A ) )
      = ( sup_sup_set_set_a @ ( pow_a @ A ) @ ( image_set_a_set_a @ ( insert_a2 @ A2 ) @ ( pow_a @ A ) ) ) ) ).

% Pow_insert
thf(fact_1008_set__union,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( union_7798659292840801954od_a_b @ Xs2 @ Ys ) )
      = ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) @ ( set_Product_prod_a_b2 @ Ys ) ) ) ).

% set_union
thf(fact_1009_Field__insert,axiom,
    ! [A2: a,B: a,R2: set_Product_prod_a_a] :
      ( ( field_a @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 ) )
      = ( sup_sup_set_a @ ( insert_a2 @ A2 @ ( insert_a2 @ B @ bot_bot_set_a ) ) @ ( field_a @ R2 ) ) ) ).

% Field_insert
thf(fact_1010_partition__set,axiom,
    ! [P2: product_prod_a_b > $o,Xs2: list_P1467974523212632366od_a_b,Yes: list_P1467974523212632366od_a_b,No: list_P1467974523212632366od_a_b] :
      ( ( ( partit8484945486742549543od_a_b @ P2 @ Xs2 )
        = ( produc5288560138351607191od_a_b @ Yes @ No ) )
     => ( ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Yes ) @ ( set_Product_prod_a_b2 @ No ) )
        = ( set_Product_prod_a_b2 @ Xs2 ) ) ) ).

% partition_set
thf(fact_1011_FieldI1,axiom,
    ! [I: a,J: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
     => ( member_a @ I @ ( field_a @ R3 ) ) ) ).

% FieldI1
thf(fact_1012_FieldI1,axiom,
    ! [I: product_prod_a_b,J: product_prod_a_b,R3: set_Pr5493900063595384711od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ I @ J ) @ R3 )
     => ( member1426531481828664017od_a_b @ I @ ( field_1126092525013176053od_a_b @ R3 ) ) ) ).

% FieldI1
thf(fact_1013_FieldI2,axiom,
    ! [I: a,J: a,R3: set_Product_prod_a_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
     => ( member_a @ J @ ( field_a @ R3 ) ) ) ).

% FieldI2
thf(fact_1014_FieldI2,axiom,
    ! [I: product_prod_a_b,J: product_prod_a_b,R3: set_Pr5493900063595384711od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ I @ J ) @ R3 )
     => ( member1426531481828664017od_a_b @ J @ ( field_1126092525013176053od_a_b @ R3 ) ) ) ).

% FieldI2
thf(fact_1015_partition__P,axiom,
    ! [P2: product_prod_a_b > $o,Xs2: list_P1467974523212632366od_a_b,Yes: list_P1467974523212632366od_a_b,No: list_P1467974523212632366od_a_b] :
      ( ( ( partit8484945486742549543od_a_b @ P2 @ Xs2 )
        = ( produc5288560138351607191od_a_b @ Yes @ No ) )
     => ( ! [X5: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ Yes ) )
           => ( P2 @ X5 ) )
        & ! [X5: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X5 @ ( set_Product_prod_a_b2 @ No ) )
           => ~ ( P2 @ X5 ) ) ) ) ).

% partition_P
thf(fact_1016_shuffles_Oelims,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xa2: list_P1467974523212632366od_a_b,Y: set_li6306622547289818894od_a_b] :
      ( ( ( shuffl3267877313157025347od_a_b @ X @ Xa2 )
        = Y )
     => ( ( ( X = nil_Product_prod_a_b )
         => ( Y
           != ( insert6792547294658701054od_a_b @ Xa2 @ bot_bo4036995510831765370od_a_b ) ) )
       => ( ( ( Xa2 = nil_Product_prod_a_b )
           => ( Y
             != ( insert6792547294658701054od_a_b @ X @ bot_bo4036995510831765370od_a_b ) ) )
         => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
               => ! [Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
                    ( ( Xa2
                      = ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) )
                   => ( Y
                     != ( sup_su3693560624713066210od_a_b @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 ) @ ( shuffl3267877313157025347od_a_b @ Xs @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) ) @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ Y4 ) @ ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Ys3 ) ) ) ) ) ) ) ) ) ).

% shuffles.elims
thf(fact_1017_concat__eq__append__conv,axiom,
    ! [Xss2: list_l8778361445553822004od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( concat6071998534011412713od_a_b @ Xss2 )
        = ( append5335208823350062147od_a_b @ Ys @ Zs ) )
      = ( ( ( Xss2 = nil_li8635777946366267444od_a_b )
         => ( ( Ys = nil_Product_prod_a_b )
            & ( Zs = nil_Product_prod_a_b ) ) )
        & ( ( Xss2 != nil_li8635777946366267444od_a_b )
         => ? [Xss1: list_l8778361445553822004od_a_b,Xs4: list_P1467974523212632366od_a_b,Xs6: list_P1467974523212632366od_a_b,Xss22: list_l8778361445553822004od_a_b] :
              ( ( Xss2
                = ( append7557446516274476873od_a_b @ Xss1 @ ( cons_l907386227682280420od_a_b @ ( append5335208823350062147od_a_b @ Xs4 @ Xs6 ) @ Xss22 ) ) )
              & ( Ys
                = ( append5335208823350062147od_a_b @ ( concat6071998534011412713od_a_b @ Xss1 ) @ Xs4 ) )
              & ( Zs
                = ( append5335208823350062147od_a_b @ Xs6 @ ( concat6071998534011412713od_a_b @ Xss22 ) ) ) ) ) ) ) ).

% concat_eq_append_conv
thf(fact_1018_lexord__same__pref__iff,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ ( append5335208823350062147od_a_b @ Xs2 @ Zs ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
      = ( ? [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
            & ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X4 @ X4 ) @ R2 ) )
        | ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ys @ Zs ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ) ).

% lexord_same_pref_iff
thf(fact_1019_concat__append,axiom,
    ! [Xs2: list_l8778361445553822004od_a_b,Ys: list_l8778361445553822004od_a_b] :
      ( ( concat6071998534011412713od_a_b @ ( append7557446516274476873od_a_b @ Xs2 @ Ys ) )
      = ( append5335208823350062147od_a_b @ ( concat6071998534011412713od_a_b @ Xs2 ) @ ( concat6071998534011412713od_a_b @ Ys ) ) ) ).

% concat_append
thf(fact_1020_lexord__cons__cons,axiom,
    ! [A2: product_prod_a_b,X: list_P1467974523212632366od_a_b,B: product_prod_a_b,Y: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ A2 @ X ) @ ( cons_P7316939131009794654od_a_b @ B @ Y ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
      = ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ R2 )
        | ( ( A2 = B )
          & ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ X @ Y ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ) ) ).

% lexord_cons_cons
thf(fact_1021_lexord__Nil__left,axiom,
    ! [Y: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Y ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
      = ( ? [A6: product_prod_a_b,X4: list_P1467974523212632366od_a_b] :
            ( Y
            = ( cons_P7316939131009794654od_a_b @ A6 @ X4 ) ) ) ) ).

% lexord_Nil_left
thf(fact_1022_shuffles_Osimps_I3_J,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) )
      = ( sup_su3693560624713066210od_a_b @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ X ) @ ( shuffl3267877313157025347od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) ) @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ Y ) @ ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys ) ) ) ) ).

% shuffles.simps(3)
thf(fact_1023_Cons__in__shuffles__rightI,axiom,
    ! [Zs: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Z: product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ Zs @ ( shuffl3267877313157025347od_a_b @ Xs2 @ Ys ) )
     => ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ Z @ Zs ) @ ( shuffl3267877313157025347od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ Z @ Ys ) ) ) ) ).

% Cons_in_shuffles_rightI
thf(fact_1024_Cons__in__shuffles__leftI,axiom,
    ! [Zs: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Z: product_prod_a_b] :
      ( ( member6895035109809442519od_a_b @ Zs @ ( shuffl3267877313157025347od_a_b @ Xs2 @ Ys ) )
     => ( member6895035109809442519od_a_b @ ( cons_P7316939131009794654od_a_b @ Z @ Zs ) @ ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ Z @ Xs2 ) @ Ys ) ) ) ).

% Cons_in_shuffles_leftI
thf(fact_1025_shufflesE,axiom,
    ! [Zs: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member6895035109809442519od_a_b @ Zs @ ( shuffl3267877313157025347od_a_b @ Xs2 @ Ys ) )
     => ( ( ( Zs = Xs2 )
         => ( Ys != nil_Product_prod_a_b ) )
       => ( ( ( Zs = Ys )
           => ( Xs2 != nil_Product_prod_a_b ) )
         => ( ! [X3: product_prod_a_b,Xs5: list_P1467974523212632366od_a_b] :
                ( ( Xs2
                  = ( cons_P7316939131009794654od_a_b @ X3 @ Xs5 ) )
               => ! [Z4: product_prod_a_b,Zs4: list_P1467974523212632366od_a_b] :
                    ( ( Zs
                      = ( cons_P7316939131009794654od_a_b @ Z4 @ Zs4 ) )
                   => ( ( X3 = Z4 )
                     => ~ ( member6895035109809442519od_a_b @ Zs4 @ ( shuffl3267877313157025347od_a_b @ Xs5 @ Ys ) ) ) ) )
           => ~ ! [Y4: product_prod_a_b,Ys6: list_P1467974523212632366od_a_b] :
                  ( ( Ys
                    = ( cons_P7316939131009794654od_a_b @ Y4 @ Ys6 ) )
                 => ! [Z4: product_prod_a_b,Zs4: list_P1467974523212632366od_a_b] :
                      ( ( Zs
                        = ( cons_P7316939131009794654od_a_b @ Z4 @ Zs4 ) )
                     => ( ( Y4 = Z4 )
                       => ~ ( member6895035109809442519od_a_b @ Zs4 @ ( shuffl3267877313157025347od_a_b @ Xs2 @ Ys6 ) ) ) ) ) ) ) ) ) ).

% shufflesE
thf(fact_1026_set__shuffles,axiom,
    ! [Zs: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member6895035109809442519od_a_b @ Zs @ ( shuffl3267877313157025347od_a_b @ Xs2 @ Ys ) )
     => ( ( set_Product_prod_a_b2 @ Zs )
        = ( sup_su3119292821646003676od_a_b @ ( set_Product_prod_a_b2 @ Xs2 ) @ ( set_Product_prod_a_b2 @ Ys ) ) ) ) ).

% set_shuffles
thf(fact_1027_concat_Osimps_I2_J,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xs2: list_l8778361445553822004od_a_b] :
      ( ( concat6071998534011412713od_a_b @ ( cons_l907386227682280420od_a_b @ X @ Xs2 ) )
      = ( append5335208823350062147od_a_b @ X @ ( concat6071998534011412713od_a_b @ Xs2 ) ) ) ).

% concat.simps(2)
thf(fact_1028_lexord__append__leftI,axiom,
    ! [U: list_P1467974523212632366od_a_b,V2: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b,X: list_P1467974523212632366od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ U @ V2 ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
     => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ X @ U ) @ ( append5335208823350062147od_a_b @ X @ V2 ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ).

% lexord_append_leftI
thf(fact_1029_Cons__shuffles__subset1,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] : ( ord_le153002609085230766od_a_b @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ X ) @ ( shuffl3267877313157025347od_a_b @ Xs2 @ Ys ) ) @ ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys ) ) ).

% Cons_shuffles_subset1
thf(fact_1030_Cons__shuffles__subset2,axiom,
    ! [Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] : ( ord_le153002609085230766od_a_b @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ Y ) @ ( shuffl3267877313157025347od_a_b @ Xs2 @ Ys ) ) @ ( shuffl3267877313157025347od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) ) ).

% Cons_shuffles_subset2
thf(fact_1031_lexord__partial__trans,axiom,
    ! [Xs2: list_a,R2: set_Product_prod_a_a,Ys: list_a,Zs: list_a] :
      ( ! [X3: a,Y4: a,Z4: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z4 ) @ R2 )
             => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) )
     => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Ys ) @ ( lexord_a @ R2 ) )
       => ( ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Ys @ Zs ) @ ( lexord_a @ R2 ) )
         => ( member8191768239178080336list_a @ ( produc6837034575241423639list_a @ Xs2 @ Zs ) @ ( lexord_a @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1032_lexord__partial__trans,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ! [X3: product_prod_a_b,Y4: product_prod_a_b,Z4: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y4 ) @ R2 )
           => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ Y4 @ Z4 ) @ R2 )
             => ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Z4 ) @ R2 ) ) ) )
     => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
       => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ys @ Zs ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
         => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Zs ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ) ) ).

% lexord_partial_trans
thf(fact_1033_lexord__append__leftD,axiom,
    ! [X: list_P1467974523212632366od_a_b,U: list_P1467974523212632366od_a_b,V2: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ X @ U ) @ ( append5335208823350062147od_a_b @ X @ V2 ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
     => ( ! [A4: product_prod_a_b] :
            ~ ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A4 @ A4 ) @ R2 )
       => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ U @ V2 ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ) ).

% lexord_append_leftD
thf(fact_1034_lexord__append__rightI,axiom,
    ! [Y: list_P1467974523212632366od_a_b,X: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ? [B8: product_prod_a_b,Z5: list_P1467974523212632366od_a_b] :
          ( Y
          = ( cons_P7316939131009794654od_a_b @ B8 @ Z5 ) )
     => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ X @ ( append5335208823350062147od_a_b @ X @ Y ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ).

% lexord_append_rightI
thf(fact_1035_concat__eq__appendD,axiom,
    ! [Xss2: list_l8778361445553822004od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( concat6071998534011412713od_a_b @ Xss2 )
        = ( append5335208823350062147od_a_b @ Ys @ Zs ) )
     => ( ( Xss2 != nil_li8635777946366267444od_a_b )
       => ? [Xss12: list_l8778361445553822004od_a_b,Xs: list_P1467974523212632366od_a_b,Xs5: list_P1467974523212632366od_a_b,Xss23: list_l8778361445553822004od_a_b] :
            ( ( Xss2
              = ( append7557446516274476873od_a_b @ Xss12 @ ( cons_l907386227682280420od_a_b @ ( append5335208823350062147od_a_b @ Xs @ Xs5 ) @ Xss23 ) ) )
            & ( Ys
              = ( append5335208823350062147od_a_b @ ( concat6071998534011412713od_a_b @ Xss12 ) @ Xs ) )
            & ( Zs
              = ( append5335208823350062147od_a_b @ Xs5 @ ( concat6071998534011412713od_a_b @ Xss23 ) ) ) ) ) ) ).

% concat_eq_appendD
thf(fact_1036_lexord__append__left__rightI,axiom,
    ! [A2: product_prod_a_b,B: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,U: list_P1467974523212632366od_a_b,X: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ R2 )
     => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ U @ ( cons_P7316939131009794654od_a_b @ A2 @ X ) ) @ ( append5335208823350062147od_a_b @ U @ ( cons_P7316939131009794654od_a_b @ B @ Y ) ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ).

% lexord_append_left_rightI
thf(fact_1037_shuffles_Opelims,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xa2: list_P1467974523212632366od_a_b,Y: set_li6306622547289818894od_a_b] :
      ( ( ( shuffl3267877313157025347od_a_b @ X @ Xa2 )
        = Y )
     => ( ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ X @ Xa2 ) )
       => ( ( ( X = nil_Product_prod_a_b )
           => ( ( Y
                = ( insert6792547294658701054od_a_b @ Xa2 @ bot_bo4036995510831765370od_a_b ) )
             => ~ ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Xa2 ) ) ) )
         => ( ( ( Xa2 = nil_Product_prod_a_b )
             => ( ( Y
                  = ( insert6792547294658701054od_a_b @ X @ bot_bo4036995510831765370od_a_b ) )
               => ~ ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ X @ nil_Product_prod_a_b ) ) ) )
           => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
                  ( ( X
                    = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
                 => ! [Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
                      ( ( Xa2
                        = ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) )
                     => ( ( Y
                          = ( sup_su3693560624713066210od_a_b @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 ) @ ( shuffl3267877313157025347od_a_b @ Xs @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) ) @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ Y4 ) @ ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Ys3 ) ) ) )
                       => ~ ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) ) ) ) ) ) ) ) ) ).

% shuffles.pelims
thf(fact_1038_listrel_Ocases,axiom,
    ! [A1: list_P1467974523212632366od_a_b,A22: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ A1 @ A22 ) @ ( listre5374766946688995057od_a_b @ R2 ) )
     => ( ( ( A1 = nil_Product_prod_a_b )
         => ( A22 != nil_Product_prod_a_b ) )
       => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( ( A1
                = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
             => ! [Ys3: list_P1467974523212632366od_a_b] :
                  ( ( A22
                    = ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) )
                 => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y4 ) @ R2 )
                   => ~ ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs @ Ys3 ) @ ( listre5374766946688995057od_a_b @ R2 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1039_listrel_Ocases,axiom,
    ! [A1: list_a,A22: list_b,R2: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ A1 @ A22 ) @ ( listrel_a_b @ R2 ) )
     => ( ( ( A1 = nil_a )
         => ( A22 != nil_b ) )
       => ~ ! [X3: a,Y4: b,Xs: list_a] :
              ( ( A1
                = ( cons_a @ X3 @ Xs ) )
             => ! [Ys3: list_b] :
                  ( ( A22
                    = ( cons_b @ Y4 @ Ys3 ) )
                 => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y4 ) @ R2 )
                   => ~ ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs @ Ys3 ) @ ( listrel_a_b @ R2 ) ) ) ) ) ) ) ).

% listrel.cases
thf(fact_1040_listrel_Osimps,axiom,
    ! [A1: list_P1467974523212632366od_a_b,A22: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ A1 @ A22 ) @ ( listre5374766946688995057od_a_b @ R2 ) )
      = ( ( ( A1 = nil_Product_prod_a_b )
          & ( A22 = nil_Product_prod_a_b ) )
        | ? [X4: product_prod_a_b,Y3: product_prod_a_b,Xs4: list_P1467974523212632366od_a_b,Ys2: list_P1467974523212632366od_a_b] :
            ( ( A1
              = ( cons_P7316939131009794654od_a_b @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_P7316939131009794654od_a_b @ Y3 @ Ys2 ) )
            & ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X4 @ Y3 ) @ R2 )
            & ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs4 @ Ys2 ) @ ( listre5374766946688995057od_a_b @ R2 ) ) ) ) ) ).

% listrel.simps
thf(fact_1041_listrel_Osimps,axiom,
    ! [A1: list_a,A22: list_b,R2: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ A1 @ A22 ) @ ( listrel_a_b @ R2 ) )
      = ( ( ( A1 = nil_a )
          & ( A22 = nil_b ) )
        | ? [X4: a,Y3: b,Xs4: list_a,Ys2: list_b] :
            ( ( A1
              = ( cons_a @ X4 @ Xs4 ) )
            & ( A22
              = ( cons_b @ Y3 @ Ys2 ) )
            & ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ Y3 ) @ R2 )
            & ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs4 @ Ys2 ) @ ( listrel_a_b @ R2 ) ) ) ) ) ).

% listrel.simps
thf(fact_1042_shuffles_Opinduct,axiom,
    ! [A0: list_P1467974523212632366od_a_b,A1: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o] :
      ( ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ A0 @ A1 ) )
     => ( ! [Ys3: list_P1467974523212632366od_a_b] :
            ( ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Ys3 ) )
           => ( P2 @ nil_Product_prod_a_b @ Ys3 ) )
       => ( ! [Xs: list_P1467974523212632366od_a_b] :
              ( ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ Xs @ nil_Product_prod_a_b ) )
             => ( P2 @ Xs @ nil_Product_prod_a_b ) )
         => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
                ( ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) )
               => ( ( P2 @ Xs @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) )
                 => ( ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Ys3 )
                   => ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) ) ) )
           => ( P2 @ A0 @ A1 ) ) ) ) ) ).

% shuffles.pinduct
thf(fact_1043_listrel__Cons2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) @ ( listre5374766946688995057od_a_b @ R2 ) )
     => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
           => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y ) @ R2 )
             => ~ ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs @ Ys ) @ ( listre5374766946688995057od_a_b @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1044_listrel__Cons2,axiom,
    ! [Xs2: list_a,Y: b,Ys: list_b,R2: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs2 @ ( cons_b @ Y @ Ys ) ) @ ( listrel_a_b @ R2 ) )
     => ~ ! [X3: a,Xs: list_a] :
            ( ( Xs2
              = ( cons_a @ X3 @ Xs ) )
           => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y ) @ R2 )
             => ~ ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs @ Ys ) @ ( listrel_a_b @ R2 ) ) ) ) ) ).

% listrel_Cons2
thf(fact_1045_listrel__Cons1,axiom,
    ! [Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) @ Xs2 ) @ ( listre5374766946688995057od_a_b @ R2 ) )
     => ~ ! [Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) )
           => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ Y @ Y4 ) @ R2 )
             => ~ ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ys @ Ys3 ) @ ( listre5374766946688995057od_a_b @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1046_listrel__Cons1,axiom,
    ! [Y: a,Ys: list_a,Xs2: list_b,R2: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ ( cons_a @ Y @ Ys ) @ Xs2 ) @ ( listrel_a_b @ R2 ) )
     => ~ ! [Y4: b,Ys3: list_b] :
            ( ( Xs2
              = ( cons_b @ Y4 @ Ys3 ) )
           => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ Y @ Y4 ) @ R2 )
             => ~ ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Ys @ Ys3 ) @ ( listrel_a_b @ R2 ) ) ) ) ) ).

% listrel_Cons1
thf(fact_1047_listrel_OCons,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
     => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( listre5374766946688995057od_a_b @ R2 ) )
       => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) @ ( listre5374766946688995057od_a_b @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1048_listrel_OCons,axiom,
    ! [X: a,Y: b,R2: set_Product_prod_a_b,Xs2: list_a,Ys: list_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Y ) @ R2 )
     => ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs2 @ Ys ) @ ( listrel_a_b @ R2 ) )
       => ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y @ Ys ) ) @ ( listrel_a_b @ R2 ) ) ) ) ).

% listrel.Cons
thf(fact_1049_shuffles_Opsimps_I3_J,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( accp_P2269134446667985584od_a_b @ shuffl3100945827408839570od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) )
     => ( ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) )
        = ( sup_su3693560624713066210od_a_b @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ X ) @ ( shuffl3267877313157025347od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) ) @ ( image_5062648198587511813od_a_b @ ( cons_P7316939131009794654od_a_b @ Y ) @ ( shuffl3267877313157025347od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys ) ) ) ) ) ).

% shuffles.psimps(3)
thf(fact_1050_splice_Opinduct,axiom,
    ! [A0: list_P1467974523212632366od_a_b,A1: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o] :
      ( ( accp_P2269134446667985584od_a_b @ splice3093269280014654292od_a_b @ ( produc5288560138351607191od_a_b @ A0 @ A1 ) )
     => ( ! [Ys3: list_P1467974523212632366od_a_b] :
            ( ( accp_P2269134446667985584od_a_b @ splice3093269280014654292od_a_b @ ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Ys3 ) )
           => ( P2 @ nil_Product_prod_a_b @ Ys3 ) )
       => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Ys3: list_P1467974523212632366od_a_b] :
              ( ( accp_P2269134446667985584od_a_b @ splice3093269280014654292od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Ys3 ) )
             => ( ( P2 @ Ys3 @ Xs )
               => ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Ys3 ) ) )
         => ( P2 @ A0 @ A1 ) ) ) ) ).

% splice.pinduct
thf(fact_1051_wf__list__arcs_Opelims_I1_J,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: $o] :
      ( ( ( wf_list_arcs_a_b @ X )
        = Y )
     => ( ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ X )
       => ( ( ( X = nil_Product_prod_a_b )
           => ( Y
             => ~ ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ nil_Product_prod_a_b ) ) )
         => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
               => ( ( Y
                    = ( ~ ( member_b @ E @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                      & ( wf_list_arcs_a_b @ Xs ) ) )
                 => ~ ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) ) ) ) ) ) ) ).

% wf_list_arcs.pelims(1)
thf(fact_1052_ImageI,axiom,
    ! [A2: a,B: a,R2: set_Product_prod_a_a,A: set_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
     => ( ( member_a @ A2 @ A )
       => ( member_a @ B @ ( image_a_a @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1053_ImageI,axiom,
    ! [A2: a,B: product_prod_a_b,R2: set_Pr3008899384982559153od_a_b,A: set_a] :
      ( ( member3142156093895948538od_a_b @ ( produc431845345726502849od_a_b @ A2 @ B ) @ R2 )
     => ( ( member_a @ A2 @ A )
       => ( member1426531481828664017od_a_b @ B @ ( image_2799180471083934717od_a_b @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1054_ImageI,axiom,
    ! [A2: product_prod_a_b,B: a,R2: set_Pr2303819121155477725_a_b_a,A: set_Product_prod_a_b] :
      ( ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ A2 @ B ) @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ A )
       => ( member_a @ B @ ( image_7424222973339957289_a_b_a @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1055_ImageI,axiom,
    ! [A2: product_prod_a_b,B: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,A: set_Product_prod_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ A )
       => ( member1426531481828664017od_a_b @ B @ ( image_7740533784927350483od_a_b @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1056_ImageI,axiom,
    ! [A2: a,B: b,R2: set_Product_prod_a_b,A: set_a] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A2 @ B ) @ R2 )
     => ( ( member_a @ A2 @ A )
       => ( member_b @ B @ ( image_a_b @ R2 @ A ) ) ) ) ).

% ImageI
thf(fact_1057_Image__singleton__iff,axiom,
    ! [B: a,R2: set_Product_prod_a_a,A2: a] :
      ( ( member_a @ B @ ( image_a_a @ R2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) )
      = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1058_Image__singleton__iff,axiom,
    ! [B: product_prod_a_b,R2: set_Pr3008899384982559153od_a_b,A2: a] :
      ( ( member1426531481828664017od_a_b @ B @ ( image_2799180471083934717od_a_b @ R2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) )
      = ( member3142156093895948538od_a_b @ ( produc431845345726502849od_a_b @ A2 @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1059_Image__singleton__iff,axiom,
    ! [B: b,R2: set_Product_prod_a_b,A2: a] :
      ( ( member_b @ B @ ( image_a_b @ R2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) )
      = ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A2 @ B ) @ R2 ) ) ).

% Image_singleton_iff
thf(fact_1060_ImageE,axiom,
    ! [B: a,R2: set_Product_prod_a_a,A: set_a] :
      ( ( member_a @ B @ ( image_a_a @ R2 @ A ) )
     => ~ ! [X3: a] :
            ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ B ) @ R2 )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1061_ImageE,axiom,
    ! [B: a,R2: set_Pr2303819121155477725_a_b_a,A: set_Product_prod_a_b] :
      ( ( member_a @ B @ ( image_7424222973339957289_a_b_a @ R2 @ A ) )
     => ~ ! [X3: product_prod_a_b] :
            ( ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ X3 @ B ) @ R2 )
           => ~ ( member1426531481828664017od_a_b @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1062_ImageE,axiom,
    ! [B: product_prod_a_b,R2: set_Pr3008899384982559153od_a_b,A: set_a] :
      ( ( member1426531481828664017od_a_b @ B @ ( image_2799180471083934717od_a_b @ R2 @ A ) )
     => ~ ! [X3: a] :
            ( ( member3142156093895948538od_a_b @ ( produc431845345726502849od_a_b @ X3 @ B ) @ R2 )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1063_ImageE,axiom,
    ! [B: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,A: set_Product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ B @ ( image_7740533784927350483od_a_b @ R2 @ A ) )
     => ~ ! [X3: product_prod_a_b] :
            ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ B ) @ R2 )
           => ~ ( member1426531481828664017od_a_b @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1064_ImageE,axiom,
    ! [B: b,R2: set_Product_prod_a_b,A: set_a] :
      ( ( member_b @ B @ ( image_a_b @ R2 @ A ) )
     => ~ ! [X3: a] :
            ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ B ) @ R2 )
           => ~ ( member_a @ X3 @ A ) ) ) ).

% ImageE
thf(fact_1065_Image__iff,axiom,
    ! [B: b,R2: set_Product_prod_a_b,A: set_a] :
      ( ( member_b @ B @ ( image_a_b @ R2 @ A ) )
      = ( ? [X4: a] :
            ( ( member_a @ X4 @ A )
            & ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X4 @ B ) @ R2 ) ) ) ) ).

% Image_iff
thf(fact_1066_rev__ImageI,axiom,
    ! [A2: a,A: set_a,B: a,R2: set_Product_prod_a_a] :
      ( ( member_a @ A2 @ A )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
       => ( member_a @ B @ ( image_a_a @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1067_rev__ImageI,axiom,
    ! [A2: a,A: set_a,B: product_prod_a_b,R2: set_Pr3008899384982559153od_a_b] :
      ( ( member_a @ A2 @ A )
     => ( ( member3142156093895948538od_a_b @ ( produc431845345726502849od_a_b @ A2 @ B ) @ R2 )
       => ( member1426531481828664017od_a_b @ B @ ( image_2799180471083934717od_a_b @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1068_rev__ImageI,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B: a,R2: set_Pr2303819121155477725_a_b_a] :
      ( ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ A2 @ B ) @ R2 )
       => ( member_a @ B @ ( image_7424222973339957289_a_b_a @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1069_rev__ImageI,axiom,
    ! [A2: product_prod_a_b,A: set_Product_prod_a_b,B: product_prod_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member1426531481828664017od_a_b @ A2 @ A )
     => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ R2 )
       => ( member1426531481828664017od_a_b @ B @ ( image_7740533784927350483od_a_b @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1070_rev__ImageI,axiom,
    ! [A2: a,A: set_a,B: b,R2: set_Product_prod_a_b] :
      ( ( member_a @ A2 @ A )
     => ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ A2 @ B ) @ R2 )
       => ( member_b @ B @ ( image_a_b @ R2 @ A ) ) ) ) ).

% rev_ImageI
thf(fact_1071_wf__list__arcs_Opelims_I3_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ~ ( wf_list_arcs_a_b @ X )
     => ( ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ X )
       => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( X
                = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
             => ( ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
               => ( ~ ( member_b @ E @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                  & ( wf_list_arcs_a_b @ Xs ) ) ) ) ) ) ).

% wf_list_arcs.pelims(3)
thf(fact_1072_wf__list__arcs_Opelims_I2_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( wf_list_arcs_a_b @ X )
     => ( ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ X )
       => ( ( ( X = nil_Product_prod_a_b )
           => ~ ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ nil_Product_prod_a_b ) )
         => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
               => ( ( accp_l5219092091033086903od_a_b @ wf_list_arcs_rel_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
                 => ~ ( ~ ( member_b @ E @ ( image_2802296252294471260_a_b_b @ product_snd_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                      & ( wf_list_arcs_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_arcs.pelims(2)
thf(fact_1073_vwalk__arcs_Opelims,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: list_P3463435030753280173od_a_b] :
      ( ( ( vertex3879532237292798496od_a_b @ X )
        = Y )
     => ( ( accp_l5219092091033086903od_a_b @ vertex3308915910108218357od_a_b @ X )
       => ( ( ( X = nil_Product_prod_a_b )
           => ( ( Y = nil_Pr906174322249854893od_a_b )
             => ~ ( accp_l5219092091033086903od_a_b @ vertex3308915910108218357od_a_b @ nil_Product_prod_a_b ) ) )
         => ( ! [X3: product_prod_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
               => ( ( Y = nil_Pr906174322249854893od_a_b )
                 => ~ ( accp_l5219092091033086903od_a_b @ vertex3308915910108218357od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) ) ) )
           => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
                  ( ( X
                    = ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) )
                 => ( ( Y
                      = ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y4 ) @ ( vertex3879532237292798496od_a_b @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) )
                   => ~ ( accp_l5219092091033086903od_a_b @ vertex3308915910108218357od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) ) ) ) ) ) ) ).

% vwalk_arcs.pelims
thf(fact_1074_wf__list__verts_Opelims_I1_J,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: $o] :
      ( ( ( wf_list_verts_a_b @ X )
        = Y )
     => ( ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ X )
       => ( ( ( X = nil_Product_prod_a_b )
           => ( Y
             => ~ ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ nil_Product_prod_a_b ) ) )
         => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
               => ( ( Y
                    = ( ~ ( member_a @ V @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                      & ( wf_list_verts_a_b @ Xs ) ) )
                 => ~ ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.pelims(1)
thf(fact_1075_wf__list__verts_Opelims_I2_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ( wf_list_verts_a_b @ X )
     => ( ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ X )
       => ( ( ( X = nil_Product_prod_a_b )
           => ~ ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ nil_Product_prod_a_b ) )
         => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
               => ( ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
                 => ~ ( ~ ( member_a @ V @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                      & ( wf_list_verts_a_b @ Xs ) ) ) ) ) ) ) ).

% wf_list_verts.pelims(2)
thf(fact_1076_wf__list__verts_Opelims_I3_J,axiom,
    ! [X: list_P1467974523212632366od_a_b] :
      ( ~ ( wf_list_verts_a_b @ X )
     => ( ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ X )
       => ~ ! [V: a,E: b,Xs: list_P1467974523212632366od_a_b] :
              ( ( X
                = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
             => ( ( accp_l5219092091033086903od_a_b @ wf_lis4731598932305523419el_a_b @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ V @ E ) @ Xs ) )
               => ( ~ ( member_a @ V @ ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ Xs ) ) )
                  & ( wf_list_verts_a_b @ Xs ) ) ) ) ) ) ).

% wf_list_verts.pelims(3)
thf(fact_1077_splice_Opelims,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xa2: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( ( splice1185298133846516737od_a_b @ X @ Xa2 )
        = Y )
     => ( ( accp_P2269134446667985584od_a_b @ splice3093269280014654292od_a_b @ ( produc5288560138351607191od_a_b @ X @ Xa2 ) )
       => ( ( ( X = nil_Product_prod_a_b )
           => ( ( Y = Xa2 )
             => ~ ( accp_P2269134446667985584od_a_b @ splice3093269280014654292od_a_b @ ( produc5288560138351607191od_a_b @ nil_Product_prod_a_b @ Xa2 ) ) ) )
         => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
                ( ( X
                  = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
               => ( ( Y
                    = ( cons_P7316939131009794654od_a_b @ X3 @ ( splice1185298133846516737od_a_b @ Xa2 @ Xs ) ) )
                 => ~ ( accp_P2269134446667985584od_a_b @ splice3093269280014654292od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ Xa2 ) ) ) ) ) ) ) ).

% splice.pelims
thf(fact_1078_splice_Osimps_I2_J,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( splice1185298133846516737od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys )
      = ( cons_P7316939131009794654od_a_b @ X @ ( splice1185298133846516737od_a_b @ Ys @ Xs2 ) ) ) ).

% splice.simps(2)
thf(fact_1079_splice_Oelims,axiom,
    ! [X: list_P1467974523212632366od_a_b,Xa2: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( ( splice1185298133846516737od_a_b @ X @ Xa2 )
        = Y )
     => ( ( ( X = nil_Product_prod_a_b )
         => ( Y != Xa2 ) )
       => ~ ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
              ( ( X
                = ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) )
             => ( Y
               != ( cons_P7316939131009794654od_a_b @ X3 @ ( splice1185298133846516737od_a_b @ Xa2 @ Xs ) ) ) ) ) ) ).

% splice.elims
thf(fact_1080_splice_Opsimps_I2_J,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( accp_P2269134446667985584od_a_b @ splice3093269280014654292od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys ) )
     => ( ( splice1185298133846516737od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ Ys )
        = ( cons_P7316939131009794654od_a_b @ X @ ( splice1185298133846516737od_a_b @ Ys @ Xs2 ) ) ) ) ).

% splice.psimps(2)
thf(fact_1081_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_a_a,A2: a,B: a] :
      ( ( order_preorder_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( member_a @ A2 @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( ord_less_eq_set_a @ ( image_a_a @ R2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) @ ( image_a_a @ R2 @ ( insert_a2 @ B @ bot_bot_set_a ) ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ B @ A2 ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1082_subset__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b,B: product_prod_a_b] :
      ( ( order_3202267353579072959od_a_b @ ( field_1126092525013176053od_a_b @ R2 ) @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ ( field_1126092525013176053od_a_b @ R2 ) )
       => ( ( member1426531481828664017od_a_b @ B @ ( field_1126092525013176053od_a_b @ R2 ) )
         => ( ( ord_le817736998455962536od_a_b @ ( image_7740533784927350483od_a_b @ R2 @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) ) @ ( image_7740533784927350483od_a_b @ R2 @ ( insert4534936386344385144od_a_b @ B @ bot_bo3428410327501040244od_a_b ) ) )
            = ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ B @ A2 ) @ R2 ) ) ) ) ) ).

% subset_Image1_Image1_iff
thf(fact_1083_successively__append__iff,axiom,
    ! [P2: product_prod_a_b > product_prod_a_b > $o,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( succes439094388111560443od_a_b @ P2 @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
      = ( ( succes439094388111560443od_a_b @ P2 @ Xs2 )
        & ( succes439094388111560443od_a_b @ P2 @ Ys )
        & ( ( Xs2 = nil_Product_prod_a_b )
          | ( Ys = nil_Product_prod_a_b )
          | ( P2 @ ( last_P8790725272581694279od_a_b @ Xs2 ) @ ( hd_Product_prod_a_b @ Ys ) ) ) ) ) ).

% successively_append_iff
thf(fact_1084_successively__mono,axiom,
    ! [P2: a > a > $o,Xs2: list_a,Q: a > a > $o] :
      ( ( successively_a @ P2 @ Xs2 )
     => ( ! [X3: a,Y4: a] :
            ( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
           => ( ( member_a @ Y4 @ ( set_a2 @ Xs2 ) )
             => ( ( P2 @ X3 @ Y4 )
               => ( Q @ X3 @ Y4 ) ) ) )
       => ( successively_a @ Q @ Xs2 ) ) ) ).

% successively_mono
thf(fact_1085_successively__mono,axiom,
    ! [P2: product_prod_a_b > product_prod_a_b > $o,Xs2: list_P1467974523212632366od_a_b,Q: product_prod_a_b > product_prod_a_b > $o] :
      ( ( succes439094388111560443od_a_b @ P2 @ Xs2 )
     => ( ! [X3: product_prod_a_b,Y4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs2 ) )
           => ( ( member1426531481828664017od_a_b @ Y4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
             => ( ( P2 @ X3 @ Y4 )
               => ( Q @ X3 @ Y4 ) ) ) )
       => ( succes439094388111560443od_a_b @ Q @ Xs2 ) ) ) ).

% successively_mono
thf(fact_1086_successively__cong,axiom,
    ! [Xs2: list_a,P2: a > a > $o,Q: a > a > $o,Ys: list_a] :
      ( ! [X3: a,Y4: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
         => ( ( member_a @ Y4 @ ( set_a2 @ Xs2 ) )
           => ( ( P2 @ X3 @ Y4 )
              = ( Q @ X3 @ Y4 ) ) ) )
     => ( ( Xs2 = Ys )
       => ( ( successively_a @ P2 @ Xs2 )
          = ( successively_a @ Q @ Ys ) ) ) ) ).

% successively_cong
thf(fact_1087_successively__cong,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > product_prod_a_b > $o,Q: product_prod_a_b > product_prod_a_b > $o,Ys: list_P1467974523212632366od_a_b] :
      ( ! [X3: product_prod_a_b,Y4: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ( ( member1426531481828664017od_a_b @ Y4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
           => ( ( P2 @ X3 @ Y4 )
              = ( Q @ X3 @ Y4 ) ) ) )
     => ( ( Xs2 = Ys )
       => ( ( succes439094388111560443od_a_b @ P2 @ Xs2 )
          = ( succes439094388111560443od_a_b @ Q @ Ys ) ) ) ) ).

% successively_cong
thf(fact_1088_successively_Osimps_I2_J,axiom,
    ! [P2: product_prod_a_b > product_prod_a_b > $o,X: product_prod_a_b] : ( succes439094388111560443od_a_b @ P2 @ ( cons_P7316939131009794654od_a_b @ X @ nil_Product_prod_a_b ) ) ).

% successively.simps(2)
thf(fact_1089_successively_Oelims_I1_J,axiom,
    ! [X: product_prod_a_b > product_prod_a_b > $o,Xa2: list_P1467974523212632366od_a_b,Y: $o] :
      ( ( ( succes439094388111560443od_a_b @ X @ Xa2 )
        = Y )
     => ( ( ( Xa2 = nil_Product_prod_a_b )
         => ~ Y )
       => ( ( ? [X3: product_prod_a_b] :
                ( Xa2
                = ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
           => ~ Y )
         => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
                ( ( Xa2
                  = ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) )
               => ( Y
                  = ( ~ ( ( X @ X3 @ Y4 )
                        & ( succes439094388111560443od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) ) ) ) ) ) ) ).

% successively.elims(1)
thf(fact_1090_successively_Oelims_I2_J,axiom,
    ! [X: product_prod_a_b > product_prod_a_b > $o,Xa2: list_P1467974523212632366od_a_b] :
      ( ( succes439094388111560443od_a_b @ X @ Xa2 )
     => ( ( Xa2 != nil_Product_prod_a_b )
       => ( ! [X3: product_prod_a_b] :
              ( Xa2
             != ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) )
         => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
                ( ( Xa2
                  = ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) )
               => ~ ( ( X @ X3 @ Y4 )
                    & ( succes439094388111560443od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) ) ) ) ) ).

% successively.elims(2)
thf(fact_1091_successively_Oelims_I3_J,axiom,
    ! [X: product_prod_a_b > product_prod_a_b > $o,Xa2: list_P1467974523212632366od_a_b] :
      ( ~ ( succes439094388111560443od_a_b @ X @ Xa2 )
     => ~ ! [X3: product_prod_a_b,Y4: product_prod_a_b,Xs: list_P1467974523212632366od_a_b] :
            ( ( Xa2
              = ( cons_P7316939131009794654od_a_b @ X3 @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) )
           => ( ( X @ X3 @ Y4 )
              & ( succes439094388111560443od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y4 @ Xs ) ) ) ) ) ).

% successively.elims(3)
thf(fact_1092_successively_Osimps_I3_J,axiom,
    ! [P2: product_prod_a_b > product_prod_a_b > $o,X: product_prod_a_b,Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( succes439094388111560443od_a_b @ P2 @ ( cons_P7316939131009794654od_a_b @ X @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) ) )
      = ( ( P2 @ X @ Y )
        & ( succes439094388111560443od_a_b @ P2 @ ( cons_P7316939131009794654od_a_b @ Y @ Xs2 ) ) ) ) ).

% successively.simps(3)
thf(fact_1093_successively__Cons,axiom,
    ! [P2: product_prod_a_b > product_prod_a_b > $o,X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( succes439094388111560443od_a_b @ P2 @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) )
      = ( ( Xs2 = nil_Product_prod_a_b )
        | ( ( P2 @ X @ ( hd_Product_prod_a_b @ Xs2 ) )
          & ( succes439094388111560443od_a_b @ P2 @ Xs2 ) ) ) ) ).

% successively_Cons
thf(fact_1094_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_a_a,A2: a,B: a] :
      ( ( order_5272072345360262643r_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( member_a @ A2 @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( ( image_a_a @ R2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
              = ( image_a_a @ R2 @ ( insert_a2 @ B @ bot_bot_set_a ) ) )
            = ( A2 = B ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
thf(fact_1095_Partial__order__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b,B: product_prod_a_b] :
      ( ( order_7408868907637916317od_a_b @ ( field_1126092525013176053od_a_b @ R2 ) @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ ( field_1126092525013176053od_a_b @ R2 ) )
       => ( ( member1426531481828664017od_a_b @ B @ ( field_1126092525013176053od_a_b @ R2 ) )
         => ( ( ( image_7740533784927350483od_a_b @ R2 @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) )
              = ( image_7740533784927350483od_a_b @ R2 @ ( insert4534936386344385144od_a_b @ B @ bot_bo3428410327501040244od_a_b ) ) )
            = ( A2 = B ) ) ) ) ) ).

% Partial_order_eq_Image1_Image1_iff
thf(fact_1096_linear__order__on__singleton,axiom,
    ! [X: a] : ( order_8768733634509060147r_on_a @ ( insert_a2 @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).

% linear_order_on_singleton
thf(fact_1097_refl__on__singleton,axiom,
    ! [X: a] : ( refl_on_a @ ( insert_a2 @ X @ bot_bot_set_a ) @ ( insert4534936382041156343od_a_a @ ( product_Pair_a_a @ X @ X ) @ bot_bo3357376287454694259od_a_a ) ) ).

% refl_on_singleton
thf(fact_1098_refl__onD2,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
      ( ( refl_on_a @ A @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
       => ( member_a @ Y @ A ) ) ) ).

% refl_onD2
thf(fact_1099_refl__onD2,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,X: product_prod_a_b,Y: product_prod_a_b] :
      ( ( refl_o7745108934136084391od_a_b @ A @ R2 )
     => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
       => ( member1426531481828664017od_a_b @ Y @ A ) ) ) ).

% refl_onD2
thf(fact_1100_refl__onD1,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
      ( ( refl_on_a @ A @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
       => ( member_a @ X @ A ) ) ) ).

% refl_onD1
thf(fact_1101_refl__onD1,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,X: product_prod_a_b,Y: product_prod_a_b] :
      ( ( refl_o7745108934136084391od_a_b @ A @ R2 )
     => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
       => ( member1426531481828664017od_a_b @ X @ A ) ) ) ).

% refl_onD1
thf(fact_1102_refl__onD,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,A2: a] :
      ( ( refl_on_a @ A @ R2 )
     => ( ( member_a @ A2 @ A )
       => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R2 ) ) ) ).

% refl_onD
thf(fact_1103_refl__onD,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b] :
      ( ( refl_o7745108934136084391od_a_b @ A @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ A )
       => ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ A2 ) @ R2 ) ) ) ).

% refl_onD
thf(fact_1104_well__order__on__domain,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,A2: a,B: a] :
      ( ( order_6972113574731384220r_on_a @ A @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
       => ( ( member_a @ A2 @ A )
          & ( member_a @ B @ A ) ) ) ) ).

% well_order_on_domain
thf(fact_1105_well__order__on__domain,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b,B: product_prod_a_b] :
      ( ( order_1514034929662316870od_a_b @ A @ R2 )
     => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ R2 )
       => ( ( member1426531481828664017od_a_b @ A2 @ A )
          & ( member1426531481828664017od_a_b @ B @ A ) ) ) ) ).

% well_order_on_domain
thf(fact_1106_refl__on__domain,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,A2: a,B: a] :
      ( ( refl_on_a @ A @ R2 )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 )
       => ( ( member_a @ A2 @ A )
          & ( member_a @ B @ A ) ) ) ) ).

% refl_on_domain
thf(fact_1107_refl__on__domain,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b,B: product_prod_a_b] :
      ( ( refl_o7745108934136084391od_a_b @ A @ R2 )
     => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ R2 )
       => ( ( member1426531481828664017od_a_b @ A2 @ A )
          & ( member1426531481828664017od_a_b @ B @ A ) ) ) ) ).

% refl_on_domain
thf(fact_1108_underS__incl__iff,axiom,
    ! [R2: set_Product_prod_a_a,A2: a,B: a] :
      ( ( order_8768733634509060147r_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( member_a @ A2 @ ( field_a @ R2 ) )
       => ( ( member_a @ B @ ( field_a @ R2 ) )
         => ( ( ord_less_eq_set_a @ ( order_underS_a @ R2 @ A2 ) @ ( order_underS_a @ R2 @ B ) )
            = ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1109_underS__incl__iff,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b,B: product_prod_a_b] :
      ( ( order_435519267815314653od_a_b @ ( field_1126092525013176053od_a_b @ R2 ) @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ ( field_1126092525013176053od_a_b @ R2 ) )
       => ( ( member1426531481828664017od_a_b @ B @ ( field_1126092525013176053od_a_b @ R2 ) )
         => ( ( ord_le817736998455962536od_a_b @ ( order_8408155384808756334od_a_b @ R2 @ A2 ) @ ( order_8408155384808756334od_a_b @ R2 @ B ) )
            = ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ R2 ) ) ) ) ) ).

% underS_incl_iff
thf(fact_1110_UNIV__I,axiom,
    ! [X: a] : ( member_a @ X @ top_top_set_a ) ).

% UNIV_I
thf(fact_1111_UNIV__I,axiom,
    ! [X: product_prod_a_b] : ( member1426531481828664017od_a_b @ X @ top_to8134405472303993176od_a_b ) ).

% UNIV_I
thf(fact_1112_range__fst,axiom,
    ( ( image_2802296252294471259_a_b_a @ product_fst_a_b @ top_to8134405472303993176od_a_b )
    = top_top_set_a ) ).

% range_fst
thf(fact_1113_range__subsetD,axiom,
    ! [F: product_prod_a_b > a,B4: set_a,I: product_prod_a_b] :
      ( ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b ) @ B4 )
     => ( member_a @ ( F @ I ) @ B4 ) ) ).

% range_subsetD
thf(fact_1114_range__eqI,axiom,
    ! [B: a,F: product_prod_a_b > a,X: product_prod_a_b] :
      ( ( B
        = ( F @ X ) )
     => ( member_a @ B @ ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b ) ) ) ).

% range_eqI
thf(fact_1115_rangeI,axiom,
    ! [F: product_prod_a_b > a,X: product_prod_a_b] : ( member_a @ ( F @ X ) @ ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b ) ) ).

% rangeI
thf(fact_1116_insert__UNIV,axiom,
    ! [X: a] :
      ( ( insert_a2 @ X @ top_top_set_a )
      = top_top_set_a ) ).

% insert_UNIV
thf(fact_1117_UNIV__witness,axiom,
    ? [X3: a] : ( member_a @ X3 @ top_top_set_a ) ).

% UNIV_witness
thf(fact_1118_UNIV__witness,axiom,
    ? [X3: product_prod_a_b] : ( member1426531481828664017od_a_b @ X3 @ top_to8134405472303993176od_a_b ) ).

% UNIV_witness
thf(fact_1119_UNIV__eq__I,axiom,
    ! [A: set_a] :
      ( ! [X3: a] : ( member_a @ X3 @ A )
     => ( top_top_set_a = A ) ) ).

% UNIV_eq_I
thf(fact_1120_UNIV__eq__I,axiom,
    ! [A: set_Product_prod_a_b] :
      ( ! [X3: product_prod_a_b] : ( member1426531481828664017od_a_b @ X3 @ A )
     => ( top_to8134405472303993176od_a_b = A ) ) ).

% UNIV_eq_I
thf(fact_1121_underS__E,axiom,
    ! [I: a,R3: set_Product_prod_a_a,J: a] :
      ( ( member_a @ I @ ( order_underS_a @ R3 @ J ) )
     => ( ( I != J )
        & ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_1122_underS__E,axiom,
    ! [I: product_prod_a_b,R3: set_Pr5493900063595384711od_a_b,J: product_prod_a_b] :
      ( ( member1426531481828664017od_a_b @ I @ ( order_8408155384808756334od_a_b @ R3 @ J ) )
     => ( ( I != J )
        & ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ I @ J ) @ R3 ) ) ) ).

% underS_E
thf(fact_1123_underS__I,axiom,
    ! [I: a,J: a,R3: set_Product_prod_a_a] :
      ( ( I != J )
     => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ I @ J ) @ R3 )
       => ( member_a @ I @ ( order_underS_a @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_1124_underS__I,axiom,
    ! [I: product_prod_a_b,J: product_prod_a_b,R3: set_Pr5493900063595384711od_a_b] :
      ( ( I != J )
     => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ I @ J ) @ R3 )
       => ( member1426531481828664017od_a_b @ I @ ( order_8408155384808756334od_a_b @ R3 @ J ) ) ) ) ).

% underS_I
thf(fact_1125_range__eq__singletonD,axiom,
    ! [F: product_prod_a_b > a,A2: a,X: product_prod_a_b] :
      ( ( ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b )
        = ( insert_a2 @ A2 @ bot_bot_set_a ) )
     => ( ( F @ X )
        = A2 ) ) ).

% range_eq_singletonD
thf(fact_1126_Refl__under__underS,axiom,
    ! [R2: set_Product_prod_a_a,A2: a] :
      ( ( refl_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( member_a @ A2 @ ( field_a @ R2 ) )
       => ( ( order_under_a @ R2 @ A2 )
          = ( sup_sup_set_a @ ( order_underS_a @ R2 @ A2 ) @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ) ) ).

% Refl_under_underS
thf(fact_1127_Refl__under__underS,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b] :
      ( ( refl_o7745108934136084391od_a_b @ ( field_1126092525013176053od_a_b @ R2 ) @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ ( field_1126092525013176053od_a_b @ R2 ) )
       => ( ( order_865818356987388173od_a_b @ R2 @ A2 )
          = ( sup_su3119292821646003676od_a_b @ ( order_8408155384808756334od_a_b @ R2 @ A2 ) @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ).

% Refl_under_underS
thf(fact_1128_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Product_prod_a_a,A2: a,B: a] :
      ( ( refl_on_a @ ( field_a @ R2 ) @ R2 )
     => ( ( antisym_on_a @ top_top_set_a @ R2 )
       => ( ( member_a @ A2 @ ( field_a @ R2 ) )
         => ( ( member_a @ B @ ( field_a @ R2 ) )
           => ( ( ( image_a_a @ R2 @ ( insert_a2 @ A2 @ bot_bot_set_a ) )
                = ( image_a_a @ R2 @ ( insert_a2 @ B @ bot_bot_set_a ) ) )
              = ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1129_Refl__antisym__eq__Image1__Image1__iff,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b,B: product_prod_a_b] :
      ( ( refl_o7745108934136084391od_a_b @ ( field_1126092525013176053od_a_b @ R2 ) @ R2 )
     => ( ( antisy9008168500843361931od_a_b @ top_to8134405472303993176od_a_b @ R2 )
       => ( ( member1426531481828664017od_a_b @ A2 @ ( field_1126092525013176053od_a_b @ R2 ) )
         => ( ( member1426531481828664017od_a_b @ B @ ( field_1126092525013176053od_a_b @ R2 ) )
           => ( ( ( image_7740533784927350483od_a_b @ R2 @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) )
                = ( image_7740533784927350483od_a_b @ R2 @ ( insert4534936386344385144od_a_b @ B @ bot_bo3428410327501040244od_a_b ) ) )
              = ( A2 = B ) ) ) ) ) ) ).

% Refl_antisym_eq_Image1_Image1_iff
thf(fact_1130_antisym__onD,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a] :
      ( ( antisym_on_a @ A @ R2 )
     => ( ( member_a @ X @ A )
       => ( ( member_a @ Y @ A )
         => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1131_antisym__onD,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,X: product_prod_a_b,Y: product_prod_a_b] :
      ( ( antisy9008168500843361931od_a_b @ A @ R2 )
     => ( ( member1426531481828664017od_a_b @ X @ A )
       => ( ( member1426531481828664017od_a_b @ Y @ A )
         => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
           => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ Y @ X ) @ R2 )
             => ( X = Y ) ) ) ) ) ) ).

% antisym_onD
thf(fact_1132_antisym__onI,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a] :
      ( ! [X3: a,Y4: a] :
          ( ( member_a @ X3 @ A )
         => ( ( member_a @ Y4 @ A )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ X3 ) @ R2 )
               => ( X3 = Y4 ) ) ) ) )
     => ( antisym_on_a @ A @ R2 ) ) ).

% antisym_onI
thf(fact_1133_antisym__onI,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ! [X3: product_prod_a_b,Y4: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A )
         => ( ( member1426531481828664017od_a_b @ Y4 @ A )
           => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y4 ) @ R2 )
             => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ Y4 @ X3 ) @ R2 )
               => ( X3 = Y4 ) ) ) ) )
     => ( antisy9008168500843361931od_a_b @ A @ R2 ) ) ).

% antisym_onI
thf(fact_1134_surj__def,axiom,
    ! [F: product_prod_a_b > a] :
      ( ( ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b )
        = top_top_set_a )
      = ( ! [Y3: a] :
          ? [X4: product_prod_a_b] :
            ( Y3
            = ( F @ X4 ) ) ) ) ).

% surj_def
thf(fact_1135_surjI,axiom,
    ! [G: product_prod_a_b > a,F: a > product_prod_a_b] :
      ( ! [X3: a] :
          ( ( G @ ( F @ X3 ) )
          = X3 )
     => ( ( image_2802296252294471259_a_b_a @ G @ top_to8134405472303993176od_a_b )
        = top_top_set_a ) ) ).

% surjI
thf(fact_1136_surjD,axiom,
    ! [F: product_prod_a_b > a,Y: a] :
      ( ( ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b )
        = top_top_set_a )
     => ? [X3: product_prod_a_b] :
          ( Y
          = ( F @ X3 ) ) ) ).

% surjD
thf(fact_1137_surjE,axiom,
    ! [F: product_prod_a_b > a,Y: a] :
      ( ( ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b )
        = top_top_set_a )
     => ~ ! [X3: product_prod_a_b] :
            ( Y
           != ( F @ X3 ) ) ) ).

% surjE
thf(fact_1138_lexord__same__pref__if__irrefl,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( irrefl3954896101477488798od_a_b @ top_to8134405472303993176od_a_b @ R2 )
     => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ ( append5335208823350062147od_a_b @ Xs2 @ Zs ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
        = ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ys @ Zs ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ) ).

% lexord_same_pref_if_irrefl
thf(fact_1139_irrefl__onD,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,A2: a] :
      ( ( irrefl_on_a @ A @ R2 )
     => ( ( member_a @ A2 @ A )
       => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ A2 ) @ R2 ) ) ) ).

% irrefl_onD
thf(fact_1140_irrefl__onD,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,A2: product_prod_a_b] :
      ( ( irrefl3954896101477488798od_a_b @ A @ R2 )
     => ( ( member1426531481828664017od_a_b @ A2 @ A )
       => ~ ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ A2 ) @ R2 ) ) ) ).

% irrefl_onD
thf(fact_1141_irrefl__onI,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a] :
      ( ! [A4: a] :
          ( ( member_a @ A4 @ A )
         => ~ ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A4 @ A4 ) @ R2 ) )
     => ( irrefl_on_a @ A @ R2 ) ) ).

% irrefl_onI
thf(fact_1142_irrefl__onI,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ! [A4: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ A4 @ A )
         => ~ ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A4 @ A4 ) @ R2 ) )
     => ( irrefl3954896101477488798od_a_b @ A @ R2 ) ) ).

% irrefl_onI
thf(fact_1143_trans__onI,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a] :
      ( ! [X3: a,Y4: a,Z4: a] :
          ( ( member_a @ X3 @ A )
         => ( ( member_a @ Y4 @ A )
           => ( ( member_a @ Z4 @ A )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Y4 ) @ R2 )
               => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y4 @ Z4 ) @ R2 )
                 => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X3 @ Z4 ) @ R2 ) ) ) ) ) )
     => ( trans_on_a @ A @ R2 ) ) ).

% trans_onI
thf(fact_1144_trans__onI,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ! [X3: product_prod_a_b,Y4: product_prod_a_b,Z4: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ A )
         => ( ( member1426531481828664017od_a_b @ Y4 @ A )
           => ( ( member1426531481828664017od_a_b @ Z4 @ A )
             => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Y4 ) @ R2 )
               => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ Y4 @ Z4 ) @ R2 )
                 => ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ Z4 ) @ R2 ) ) ) ) ) )
     => ( trans_8517530832372390472od_a_b @ A @ R2 ) ) ).

% trans_onI
thf(fact_1145_trans__onD,axiom,
    ! [A: set_a,R2: set_Product_prod_a_a,X: a,Y: a,Z: a] :
      ( ( trans_on_a @ A @ R2 )
     => ( ( member_a @ X @ A )
       => ( ( member_a @ Y @ A )
         => ( ( member_a @ Z @ A )
           => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ R2 )
             => ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ Y @ Z ) @ R2 )
               => ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1146_trans__onD,axiom,
    ! [A: set_Product_prod_a_b,R2: set_Pr5493900063595384711od_a_b,X: product_prod_a_b,Y: product_prod_a_b,Z: product_prod_a_b] :
      ( ( trans_8517530832372390472od_a_b @ A @ R2 )
     => ( ( member1426531481828664017od_a_b @ X @ A )
       => ( ( member1426531481828664017od_a_b @ Y @ A )
         => ( ( member1426531481828664017od_a_b @ Z @ A )
           => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
             => ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ Y @ Z ) @ R2 )
               => ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Z ) @ R2 ) ) ) ) ) ) ) ).

% trans_onD
thf(fact_1147_lenlex__append2,axiom,
    ! [R3: set_Pr5493900063595384711od_a_b,Us: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( irrefl3954896101477488798od_a_b @ top_to8134405472303993176od_a_b @ R3 )
     => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Us @ Xs2 ) @ ( append5335208823350062147od_a_b @ Us @ Ys ) ) @ ( lenlex8556352034305941171od_a_b @ R3 ) )
        = ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( lenlex8556352034305941171od_a_b @ R3 ) ) ) ) ).

% lenlex_append2
thf(fact_1148_lex__append__leftI,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ys @ Zs ) @ ( lex_Product_prod_a_b @ R2 ) )
     => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ ( append5335208823350062147od_a_b @ Xs2 @ Zs ) ) @ ( lex_Product_prod_a_b @ R2 ) ) ) ).

% lex_append_leftI
thf(fact_1149_lex__append__leftD,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ! [X3: product_prod_a_b] :
          ~ ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ X3 ) @ R2 )
     => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ ( append5335208823350062147od_a_b @ Xs2 @ Zs ) ) @ ( lex_Product_prod_a_b @ R2 ) )
       => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ys @ Zs ) @ ( lex_Product_prod_a_b @ R2 ) ) ) ) ).

% lex_append_leftD
thf(fact_1150_lex__append__left__iff,axiom,
    ! [R2: set_Pr5493900063595384711od_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ! [X3: product_prod_a_b] :
          ~ ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X3 @ X3 ) @ R2 )
     => ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ ( append5335208823350062147od_a_b @ Xs2 @ Zs ) ) @ ( lex_Product_prod_a_b @ R2 ) )
        = ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ys @ Zs ) @ ( lex_Product_prod_a_b @ R2 ) ) ) ) ).

% lex_append_left_iff
thf(fact_1151_Cons__in__lex,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) ) @ ( lex_Product_prod_a_b @ R2 ) )
      = ( ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ R2 )
          & ( ( size_s3956712670882376602od_a_b @ Xs2 )
            = ( size_s3956712670882376602od_a_b @ Ys ) ) )
        | ( ( X = Y )
          & ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( lex_Product_prod_a_b @ R2 ) ) ) ) ) ).

% Cons_in_lex
thf(fact_1152_inj__on__Un,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) )
      = ( ( inj_on4343330552946057671_a_b_a @ F @ A )
        & ( inj_on4343330552946057671_a_b_a @ F @ B4 )
        & ( ( inf_inf_set_a @ ( image_2802296252294471259_a_b_a @ F @ ( minus_6888070959853530735od_a_b @ A @ B4 ) ) @ ( image_2802296252294471259_a_b_a @ F @ ( minus_6888070959853530735od_a_b @ B4 @ A ) ) )
          = bot_bot_set_a ) ) ) ).

% inj_on_Un
thf(fact_1153_append__eq__append__conv,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Us: list_P1467974523212632366od_a_b,Vs: list_P1467974523212632366od_a_b] :
      ( ( ( ( size_s3956712670882376602od_a_b @ Xs2 )
          = ( size_s3956712670882376602od_a_b @ Ys ) )
        | ( ( size_s3956712670882376602od_a_b @ Us )
          = ( size_s3956712670882376602od_a_b @ Vs ) ) )
     => ( ( ( append5335208823350062147od_a_b @ Xs2 @ Us )
          = ( append5335208823350062147od_a_b @ Ys @ Vs ) )
        = ( ( Xs2 = Ys )
          & ( Us = Vs ) ) ) ) ).

% append_eq_append_conv
thf(fact_1154_inj__on__insert,axiom,
    ! [F: product_prod_a_b > a,A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ ( insert4534936386344385144od_a_b @ A2 @ A ) )
      = ( ( inj_on4343330552946057671_a_b_a @ F @ A )
        & ~ ( member_a @ ( F @ A2 ) @ ( image_2802296252294471259_a_b_a @ F @ ( minus_6888070959853530735od_a_b @ A @ ( insert4534936386344385144od_a_b @ A2 @ bot_bo3428410327501040244od_a_b ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_1155_inj__on__insert,axiom,
    ! [F: a > a,A2: a,A: set_a] :
      ( ( inj_on_a_a @ F @ ( insert_a2 @ A2 @ A ) )
      = ( ( inj_on_a_a @ F @ A )
        & ~ ( member_a @ ( F @ A2 ) @ ( image_a_a2 @ F @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_1156_inj__on__insert,axiom,
    ! [F: a > product_prod_a_b,A2: a,A: set_a] :
      ( ( inj_on8941660087544810907od_a_b @ F @ ( insert_a2 @ A2 @ A ) )
      = ( ( inj_on8941660087544810907od_a_b @ F @ A )
        & ~ ( member1426531481828664017od_a_b @ ( F @ A2 ) @ ( image_7400625786893224495od_a_b @ F @ ( minus_minus_set_a @ A @ ( insert_a2 @ A2 @ bot_bot_set_a ) ) ) ) ) ) ).

% inj_on_insert
thf(fact_1157_inj__on__image__Fpow,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ A )
     => ( inj_on1980945187301441479_set_a @ ( image_2802296252294471259_a_b_a @ F ) @ ( finite3511610380200936718od_a_b @ A ) ) ) ).

% inj_on_image_Fpow
thf(fact_1158_list__induct4,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,Ws: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o] :
      ( ( ( size_s3956712670882376602od_a_b @ Xs2 )
        = ( size_s3956712670882376602od_a_b @ Ys ) )
     => ( ( ( size_s3956712670882376602od_a_b @ Ys )
          = ( size_s3956712670882376602od_a_b @ Zs ) )
       => ( ( ( size_s3956712670882376602od_a_b @ Zs )
            = ( size_s3956712670882376602od_a_b @ Ws ) )
         => ( ( P2 @ nil_Product_prod_a_b @ nil_Product_prod_a_b @ nil_Product_prod_a_b @ nil_Product_prod_a_b )
           => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b,Z4: product_prod_a_b,Zs3: list_P1467974523212632366od_a_b,W: product_prod_a_b,Ws2: list_P1467974523212632366od_a_b] :
                  ( ( ( size_s3956712670882376602od_a_b @ Xs )
                    = ( size_s3956712670882376602od_a_b @ Ys3 ) )
                 => ( ( ( size_s3956712670882376602od_a_b @ Ys3 )
                      = ( size_s3956712670882376602od_a_b @ Zs3 ) )
                   => ( ( ( size_s3956712670882376602od_a_b @ Zs3 )
                        = ( size_s3956712670882376602od_a_b @ Ws2 ) )
                     => ( ( P2 @ Xs @ Ys3 @ Zs3 @ Ws2 )
                       => ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) @ ( cons_P7316939131009794654od_a_b @ Z4 @ Zs3 ) @ ( cons_P7316939131009794654od_a_b @ W @ Ws2 ) ) ) ) ) )
             => ( P2 @ Xs2 @ Ys @ Zs @ Ws ) ) ) ) ) ) ).

% list_induct4
thf(fact_1159_list__induct3,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o] :
      ( ( ( size_s3956712670882376602od_a_b @ Xs2 )
        = ( size_s3956712670882376602od_a_b @ Ys ) )
     => ( ( ( size_s3956712670882376602od_a_b @ Ys )
          = ( size_s3956712670882376602od_a_b @ Zs ) )
       => ( ( P2 @ nil_Product_prod_a_b @ nil_Product_prod_a_b @ nil_Product_prod_a_b )
         => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b,Z4: product_prod_a_b,Zs3: list_P1467974523212632366od_a_b] :
                ( ( ( size_s3956712670882376602od_a_b @ Xs )
                  = ( size_s3956712670882376602od_a_b @ Ys3 ) )
               => ( ( ( size_s3956712670882376602od_a_b @ Ys3 )
                    = ( size_s3956712670882376602od_a_b @ Zs3 ) )
                 => ( ( P2 @ Xs @ Ys3 @ Zs3 )
                   => ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) @ ( cons_P7316939131009794654od_a_b @ Z4 @ Zs3 ) ) ) ) )
           => ( P2 @ Xs2 @ Ys @ Zs ) ) ) ) ) ).

% list_induct3
thf(fact_1160_list__induct2,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,P2: list_P1467974523212632366od_a_b > list_P1467974523212632366od_a_b > $o] :
      ( ( ( size_s3956712670882376602od_a_b @ Xs2 )
        = ( size_s3956712670882376602od_a_b @ Ys ) )
     => ( ( P2 @ nil_Product_prod_a_b @ nil_Product_prod_a_b )
       => ( ! [X3: product_prod_a_b,Xs: list_P1467974523212632366od_a_b,Y4: product_prod_a_b,Ys3: list_P1467974523212632366od_a_b] :
              ( ( ( size_s3956712670882376602od_a_b @ Xs )
                = ( size_s3956712670882376602od_a_b @ Ys3 ) )
             => ( ( P2 @ Xs @ Ys3 )
               => ( P2 @ ( cons_P7316939131009794654od_a_b @ X3 @ Xs ) @ ( cons_P7316939131009794654od_a_b @ Y4 @ Ys3 ) ) ) )
         => ( P2 @ Xs2 @ Ys ) ) ) ) ).

% list_induct2
thf(fact_1161_inj__on__Cons1,axiom,
    ! [X: product_prod_a_b,A: set_li6306622547289818894od_a_b] : ( inj_on7271627600565608305od_a_b @ ( cons_P7316939131009794654od_a_b @ X ) @ A ) ).

% inj_on_Cons1
thf(fact_1162_range__ex1__eq,axiom,
    ! [F: product_prod_a_b > a,B: a] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ top_to8134405472303993176od_a_b )
     => ( ( member_a @ B @ ( image_2802296252294471259_a_b_a @ F @ top_to8134405472303993176od_a_b ) )
        = ( ? [X4: product_prod_a_b] :
              ( ( B
                = ( F @ X4 ) )
              & ! [Y3: product_prod_a_b] :
                  ( ( B
                    = ( F @ Y3 ) )
                 => ( Y3 = X4 ) ) ) ) ) ) ).

% range_ex1_eq
thf(fact_1163_inj__image__eq__iff,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ top_to8134405472303993176od_a_b )
     => ( ( ( image_2802296252294471259_a_b_a @ F @ A )
          = ( image_2802296252294471259_a_b_a @ F @ B4 ) )
        = ( A = B4 ) ) ) ).

% inj_image_eq_iff
thf(fact_1164_inj__image__mem__iff,axiom,
    ! [F: a > a,A2: a,A: set_a] :
      ( ( inj_on_a_a @ F @ top_top_set_a )
     => ( ( member_a @ ( F @ A2 ) @ ( image_a_a2 @ F @ A ) )
        = ( member_a @ A2 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1165_inj__image__mem__iff,axiom,
    ! [F: product_prod_a_b > a,A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ top_to8134405472303993176od_a_b )
     => ( ( member_a @ ( F @ A2 ) @ ( image_2802296252294471259_a_b_a @ F @ A ) )
        = ( member1426531481828664017od_a_b @ A2 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1166_inj__image__mem__iff,axiom,
    ! [F: a > product_prod_a_b,A2: a,A: set_a] :
      ( ( inj_on8941660087544810907od_a_b @ F @ top_top_set_a )
     => ( ( member1426531481828664017od_a_b @ ( F @ A2 ) @ ( image_7400625786893224495od_a_b @ F @ A ) )
        = ( member_a @ A2 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1167_inj__image__mem__iff,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( inj_on1230094055151127153od_a_b @ F @ top_to8134405472303993176od_a_b )
     => ( ( member1426531481828664017od_a_b @ ( F @ A2 ) @ ( image_3300603549555413765od_a_b @ F @ A ) )
        = ( member1426531481828664017od_a_b @ A2 @ A ) ) ) ).

% inj_image_mem_iff
thf(fact_1168_inj__on__Un__image__eq__iff,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ ( sup_su3119292821646003676od_a_b @ A @ B4 ) )
     => ( ( ( image_2802296252294471259_a_b_a @ F @ A )
          = ( image_2802296252294471259_a_b_a @ F @ B4 ) )
        = ( A = B4 ) ) ) ).

% inj_on_Un_image_eq_iff
thf(fact_1169_inj__on__image__eq__iff,axiom,
    ! [F: product_prod_a_b > a,C5: set_Product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ C5 )
     => ( ( ord_le817736998455962536od_a_b @ A @ C5 )
       => ( ( ord_le817736998455962536od_a_b @ B4 @ C5 )
         => ( ( ( image_2802296252294471259_a_b_a @ F @ A )
              = ( image_2802296252294471259_a_b_a @ F @ B4 ) )
            = ( A = B4 ) ) ) ) ) ).

% inj_on_image_eq_iff
thf(fact_1170_inj__on__image__mem__iff,axiom,
    ! [F: a > a,B4: set_a,A2: a,A: set_a] :
      ( ( inj_on_a_a @ F @ B4 )
     => ( ( member_a @ A2 @ B4 )
       => ( ( ord_less_eq_set_a @ A @ B4 )
         => ( ( member_a @ ( F @ A2 ) @ ( image_a_a2 @ F @ A ) )
            = ( member_a @ A2 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1171_inj__on__image__mem__iff,axiom,
    ! [F: a > product_prod_a_b,B4: set_a,A2: a,A: set_a] :
      ( ( inj_on8941660087544810907od_a_b @ F @ B4 )
     => ( ( member_a @ A2 @ B4 )
       => ( ( ord_less_eq_set_a @ A @ B4 )
         => ( ( member1426531481828664017od_a_b @ ( F @ A2 ) @ ( image_7400625786893224495od_a_b @ F @ A ) )
            = ( member_a @ A2 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1172_inj__on__image__mem__iff,axiom,
    ! [F: product_prod_a_b > a,B4: set_Product_prod_a_b,A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ B4 )
     => ( ( member1426531481828664017od_a_b @ A2 @ B4 )
       => ( ( ord_le817736998455962536od_a_b @ A @ B4 )
         => ( ( member_a @ ( F @ A2 ) @ ( image_2802296252294471259_a_b_a @ F @ A ) )
            = ( member1426531481828664017od_a_b @ A2 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1173_inj__on__image__mem__iff,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,B4: set_Product_prod_a_b,A2: product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( inj_on1230094055151127153od_a_b @ F @ B4 )
     => ( ( member1426531481828664017od_a_b @ A2 @ B4 )
       => ( ( ord_le817736998455962536od_a_b @ A @ B4 )
         => ( ( member1426531481828664017od_a_b @ ( F @ A2 ) @ ( image_3300603549555413765od_a_b @ F @ A ) )
            = ( member1426531481828664017od_a_b @ A2 @ A ) ) ) ) ) ).

% inj_on_image_mem_iff
thf(fact_1174_inj__on__image__Pow,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ A )
     => ( inj_on1980945187301441479_set_a @ ( image_2802296252294471259_a_b_a @ F ) @ ( pow_Product_prod_a_b @ A ) ) ) ).

% inj_on_image_Pow
thf(fact_1175_image__strict__mono,axiom,
    ! [F: product_prod_a_b > a,B4: set_Product_prod_a_b,A: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ B4 )
     => ( ( ord_le6891031760732254900od_a_b @ A @ B4 )
       => ( ord_less_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ) ).

% image_strict_mono
thf(fact_1176_inj__img__insertE,axiom,
    ! [F: a > a,A: set_a,X: a,B4: set_a] :
      ( ( inj_on_a_a @ F @ A )
     => ( ~ ( member_a @ X @ B4 )
       => ( ( ( insert_a2 @ X @ B4 )
            = ( image_a_a2 @ F @ A ) )
         => ~ ! [X7: a,A7: set_a] :
                ( ~ ( member_a @ X7 @ A7 )
               => ( ( A
                    = ( insert_a2 @ X7 @ A7 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B4
                     != ( image_a_a2 @ F @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1177_inj__img__insertE,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,X: a,B4: set_a] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ A )
     => ( ~ ( member_a @ X @ B4 )
       => ( ( ( insert_a2 @ X @ B4 )
            = ( image_2802296252294471259_a_b_a @ F @ A ) )
         => ~ ! [X7: product_prod_a_b,A7: set_Product_prod_a_b] :
                ( ~ ( member1426531481828664017od_a_b @ X7 @ A7 )
               => ( ( A
                    = ( insert4534936386344385144od_a_b @ X7 @ A7 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B4
                     != ( image_2802296252294471259_a_b_a @ F @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1178_inj__img__insertE,axiom,
    ! [F: a > product_prod_a_b,A: set_a,X: product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on8941660087544810907od_a_b @ F @ A )
     => ( ~ ( member1426531481828664017od_a_b @ X @ B4 )
       => ( ( ( insert4534936386344385144od_a_b @ X @ B4 )
            = ( image_7400625786893224495od_a_b @ F @ A ) )
         => ~ ! [X7: a,A7: set_a] :
                ( ~ ( member_a @ X7 @ A7 )
               => ( ( A
                    = ( insert_a2 @ X7 @ A7 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B4
                     != ( image_7400625786893224495od_a_b @ F @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1179_inj__img__insertE,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,A: set_Product_prod_a_b,X: product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on1230094055151127153od_a_b @ F @ A )
     => ( ~ ( member1426531481828664017od_a_b @ X @ B4 )
       => ( ( ( insert4534936386344385144od_a_b @ X @ B4 )
            = ( image_3300603549555413765od_a_b @ F @ A ) )
         => ~ ! [X7: product_prod_a_b,A7: set_Product_prod_a_b] :
                ( ~ ( member1426531481828664017od_a_b @ X7 @ A7 )
               => ( ( A
                    = ( insert4534936386344385144od_a_b @ X7 @ A7 ) )
                 => ( ( X
                      = ( F @ X7 ) )
                   => ( B4
                     != ( image_3300603549555413765od_a_b @ F @ A7 ) ) ) ) ) ) ) ) ).

% inj_img_insertE
thf(fact_1180_inj__image__subset__iff,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ top_to8134405472303993176od_a_b )
     => ( ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) )
        = ( ord_le817736998455962536od_a_b @ A @ B4 ) ) ) ).

% inj_image_subset_iff
thf(fact_1181_image__Int,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ top_to8134405472303993176od_a_b )
     => ( ( image_2802296252294471259_a_b_a @ F @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) )
        = ( inf_inf_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ) ).

% image_Int
thf(fact_1182_image__set__diff,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ top_to8134405472303993176od_a_b )
     => ( ( image_2802296252294471259_a_b_a @ F @ ( minus_6888070959853530735od_a_b @ A @ B4 ) )
        = ( minus_minus_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ) ).

% image_set_diff
thf(fact_1183_inj__on__image__Int,axiom,
    ! [F: product_prod_a_b > a,C5: set_Product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ C5 )
     => ( ( ord_le817736998455962536od_a_b @ A @ C5 )
       => ( ( ord_le817736998455962536od_a_b @ B4 @ C5 )
         => ( ( image_2802296252294471259_a_b_a @ F @ ( inf_in8976041639890736118od_a_b @ A @ B4 ) )
            = ( inf_inf_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ) ) ) ).

% inj_on_image_Int
thf(fact_1184_inj__on__image__set__diff,axiom,
    ! [F: product_prod_a_b > a,C5: set_Product_prod_a_b,A: set_Product_prod_a_b,B4: set_Product_prod_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ C5 )
     => ( ( ord_le817736998455962536od_a_b @ ( minus_6888070959853530735od_a_b @ A @ B4 ) @ C5 )
       => ( ( ord_le817736998455962536od_a_b @ B4 @ C5 )
         => ( ( image_2802296252294471259_a_b_a @ F @ ( minus_6888070959853530735od_a_b @ A @ B4 ) )
            = ( minus_minus_set_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( image_2802296252294471259_a_b_a @ F @ B4 ) ) ) ) ) ) ).

% inj_on_image_set_diff
thf(fact_1185_same__length__different,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( Xs2 != Ys )
     => ( ( ( size_s3956712670882376602od_a_b @ Xs2 )
          = ( size_s3956712670882376602od_a_b @ Ys ) )
       => ? [Pre: list_P1467974523212632366od_a_b,X3: product_prod_a_b,Xs5: list_P1467974523212632366od_a_b,Y4: product_prod_a_b,Ys6: list_P1467974523212632366od_a_b] :
            ( ( X3 != Y4 )
            & ( Xs2
              = ( append5335208823350062147od_a_b @ Pre @ ( append5335208823350062147od_a_b @ ( cons_P7316939131009794654od_a_b @ X3 @ nil_Product_prod_a_b ) @ Xs5 ) ) )
            & ( Ys
              = ( append5335208823350062147od_a_b @ Pre @ ( append5335208823350062147od_a_b @ ( cons_P7316939131009794654od_a_b @ Y4 @ nil_Product_prod_a_b ) @ Ys6 ) ) ) ) ) ) ).

% same_length_different
thf(fact_1186_lexord__sufE,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Qs: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Zs ) @ ( append5335208823350062147od_a_b @ Ys @ Qs ) ) @ ( lexord2574673287206998939od_a_b @ R2 ) )
     => ( ( Xs2 != Ys )
       => ( ( ( size_s3956712670882376602od_a_b @ Xs2 )
            = ( size_s3956712670882376602od_a_b @ Ys ) )
         => ( ( ( size_s3956712670882376602od_a_b @ Zs )
              = ( size_s3956712670882376602od_a_b @ Qs ) )
           => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( lexord2574673287206998939od_a_b @ R2 ) ) ) ) ) ) ).

% lexord_sufE
thf(fact_1187_lex__append__rightI,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b,Vs: list_P1467974523212632366od_a_b,Us: list_P1467974523212632366od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Xs2 @ Ys ) @ ( lex_Product_prod_a_b @ R2 ) )
     => ( ( ( size_s3956712670882376602od_a_b @ Vs )
          = ( size_s3956712670882376602od_a_b @ Us ) )
       => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Us ) @ ( append5335208823350062147od_a_b @ Ys @ Vs ) ) @ ( lex_Product_prod_a_b @ R2 ) ) ) ) ).

% lex_append_rightI
thf(fact_1188_lenlex__append1,axiom,
    ! [Us: list_P1467974523212632366od_a_b,Xs2: list_P1467974523212632366od_a_b,R3: set_Pr5493900063595384711od_a_b,Vs: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Us @ Xs2 ) @ ( lenlex8556352034305941171od_a_b @ R3 ) )
     => ( ( ( size_s3956712670882376602od_a_b @ Vs )
          = ( size_s3956712670882376602od_a_b @ Ys ) )
       => ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( append5335208823350062147od_a_b @ Us @ Vs ) @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) ) @ ( lenlex8556352034305941171od_a_b @ R3 ) ) ) ) ).

% lenlex_append1
thf(fact_1189_inj__on__iff__surj,axiom,
    ! [A: set_a,A8: set_Product_prod_a_b] :
      ( ( A != bot_bot_set_a )
     => ( ( ? [F2: a > product_prod_a_b] :
              ( ( inj_on8941660087544810907od_a_b @ F2 @ A )
              & ( ord_le817736998455962536od_a_b @ ( image_7400625786893224495od_a_b @ F2 @ A ) @ A8 ) ) )
        = ( ? [G2: product_prod_a_b > a] :
              ( ( image_2802296252294471259_a_b_a @ G2 @ A8 )
              = A ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1190_inj__on__iff__surj,axiom,
    ! [A: set_Product_prod_a_b,A8: set_a] :
      ( ( A != bot_bo3428410327501040244od_a_b )
     => ( ( ? [F2: product_prod_a_b > a] :
              ( ( inj_on4343330552946057671_a_b_a @ F2 @ A )
              & ( ord_less_eq_set_a @ ( image_2802296252294471259_a_b_a @ F2 @ A ) @ A8 ) ) )
        = ( ? [G2: a > product_prod_a_b] :
              ( ( image_7400625786893224495od_a_b @ G2 @ A8 )
              = A ) ) ) ) ).

% inj_on_iff_surj
thf(fact_1191_all__subset__image__inj,axiom,
    ! [F: product_prod_a_b > a,S4: set_Product_prod_a_b,P2: set_a > $o] :
      ( ( ! [T4: set_a] :
            ( ( ord_less_eq_set_a @ T4 @ ( image_2802296252294471259_a_b_a @ F @ S4 ) )
           => ( P2 @ T4 ) ) )
      = ( ! [T4: set_Product_prod_a_b] :
            ( ( ( ord_le817736998455962536od_a_b @ T4 @ S4 )
              & ( inj_on4343330552946057671_a_b_a @ F @ T4 ) )
           => ( P2 @ ( image_2802296252294471259_a_b_a @ F @ T4 ) ) ) ) ) ).

% all_subset_image_inj
thf(fact_1192_subset__image__inj,axiom,
    ! [S4: set_a,F: product_prod_a_b > a,T5: set_Product_prod_a_b] :
      ( ( ord_less_eq_set_a @ S4 @ ( image_2802296252294471259_a_b_a @ F @ T5 ) )
      = ( ? [U2: set_Product_prod_a_b] :
            ( ( ord_le817736998455962536od_a_b @ U2 @ T5 )
            & ( inj_on4343330552946057671_a_b_a @ F @ U2 )
            & ( S4
              = ( image_2802296252294471259_a_b_a @ F @ U2 ) ) ) ) ) ).

% subset_image_inj
thf(fact_1193_ex__subset__image__inj,axiom,
    ! [F: product_prod_a_b > a,S4: set_Product_prod_a_b,P2: set_a > $o] :
      ( ( ? [T4: set_a] :
            ( ( ord_less_eq_set_a @ T4 @ ( image_2802296252294471259_a_b_a @ F @ S4 ) )
            & ( P2 @ T4 ) ) )
      = ( ? [T4: set_Product_prod_a_b] :
            ( ( ord_le817736998455962536od_a_b @ T4 @ S4 )
            & ( inj_on4343330552946057671_a_b_a @ F @ T4 )
            & ( P2 @ ( image_2802296252294471259_a_b_a @ F @ T4 ) ) ) ) ) ).

% ex_subset_image_inj
thf(fact_1194_Cons__lenlex__iff,axiom,
    ! [M2: product_prod_a_b,Ms: list_P1467974523212632366od_a_b,N2: product_prod_a_b,Ns: list_P1467974523212632366od_a_b,R2: set_Pr5493900063595384711od_a_b] :
      ( ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ ( cons_P7316939131009794654od_a_b @ M2 @ Ms ) @ ( cons_P7316939131009794654od_a_b @ N2 @ Ns ) ) @ ( lenlex8556352034305941171od_a_b @ R2 ) )
      = ( ( ord_less_nat @ ( size_s3956712670882376602od_a_b @ Ms ) @ ( size_s3956712670882376602od_a_b @ Ns ) )
        | ( ( ( size_s3956712670882376602od_a_b @ Ms )
            = ( size_s3956712670882376602od_a_b @ Ns ) )
          & ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ M2 @ N2 ) @ R2 ) )
        | ( ( M2 = N2 )
          & ( member3051990972100052432od_a_b @ ( produc5288560138351607191od_a_b @ Ms @ Ns ) @ ( lenlex8556352034305941171od_a_b @ R2 ) ) ) ) ) ).

% Cons_lenlex_iff
thf(fact_1195_apsnd__conv,axiom,
    ! [F: b > b,X: a,Y: b] :
      ( ( product_apsnd_b_b_a @ F @ ( product_Pair_a_b @ X @ Y ) )
      = ( product_Pair_a_b @ X @ ( F @ Y ) ) ) ).

% apsnd_conv
thf(fact_1196_fst__apsnd,axiom,
    ! [F: b > b,X: product_prod_a_b] :
      ( ( product_fst_a_b @ ( product_apsnd_b_b_a @ F @ X ) )
      = ( product_fst_a_b @ X ) ) ).

% fst_apsnd
thf(fact_1197_length__removeAll__less,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_size_list_a @ ( removeAll_a @ X @ Xs2 ) ) @ ( size_size_list_a @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_1198_length__removeAll__less,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
     => ( ord_less_nat @ ( size_s3956712670882376602od_a_b @ ( remove3678230900650767028od_a_b @ X @ Xs2 ) ) @ ( size_s3956712670882376602od_a_b @ Xs2 ) ) ) ).

% length_removeAll_less
thf(fact_1199_apfst__apsnd,axiom,
    ! [F: a > a,G: b > b,X: product_prod_a_b] :
      ( ( product_apfst_a_a_b @ F @ ( product_apsnd_b_b_a @ G @ X ) )
      = ( product_Pair_a_b @ ( F @ ( product_fst_a_b @ X ) ) @ ( G @ ( product_snd_a_b @ X ) ) ) ) ).

% apfst_apsnd
thf(fact_1200_apfst__conv,axiom,
    ! [F: a > a,X: a,Y: b] :
      ( ( product_apfst_a_a_b @ F @ ( product_Pair_a_b @ X @ Y ) )
      = ( product_Pair_a_b @ ( F @ X ) @ Y ) ) ).

% apfst_conv
thf(fact_1201_fst__apfst,axiom,
    ! [F: a > a,X: product_prod_a_b] :
      ( ( product_fst_a_b @ ( product_apfst_a_a_b @ F @ X ) )
      = ( F @ ( product_fst_a_b @ X ) ) ) ).

% fst_apfst
thf(fact_1202_apsnd__apfst,axiom,
    ! [F: b > b,G: a > a,X: product_prod_a_b] :
      ( ( product_apsnd_b_b_a @ F @ ( product_apfst_a_a_b @ G @ X ) )
      = ( product_Pair_a_b @ ( G @ ( product_fst_a_b @ X ) ) @ ( F @ ( product_snd_a_b @ X ) ) ) ) ).

% apsnd_apfst
thf(fact_1203_apfst__convE,axiom,
    ! [Q2: product_prod_a_b,F: a > a,P: product_prod_a_b] :
      ( ( Q2
        = ( product_apfst_a_a_b @ F @ P ) )
     => ~ ! [X3: a,Y4: b] :
            ( ( P
              = ( product_Pair_a_b @ X3 @ Y4 ) )
           => ( Q2
             != ( product_Pair_a_b @ ( F @ X3 ) @ Y4 ) ) ) ) ).

% apfst_convE
thf(fact_1204_dtree__size__decr__aux_H,axiom,
    ! [T12: dtree_a_b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
      ( ( member_dtree_a_b @ T12 @ ( image_7065894828672115579ee_a_b @ produc697780174058963904_a_b_b @ ( fset_P783253628892185035_a_b_b @ Xs2 ) ) )
     => ( ord_less_nat @ ( size_size_dtree_a_b @ T12 ) @ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).

% dtree_size_decr_aux'
thf(fact_1205_dtree__size__eq__root,axiom,
    ! [R2: a,Xs2: fset_P5281107635120001194_a_b_b,R5: a] :
      ( ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) )
      = ( size_size_dtree_a_b @ ( node_a_b @ R5 @ Xs2 ) ) ) ).

% dtree_size_eq_root
thf(fact_1206_subtree__size__decr_H,axiom,
    ! [T12: dtree_a_b,T22: dtree_a_b] :
      ( ( strict_subtree_a_b @ T12 @ T22 )
     => ( ord_less_nat @ ( size_size_dtree_a_b @ T12 ) @ ( size_size_dtree_a_b @ T22 ) ) ) ).

% subtree_size_decr'
thf(fact_1207_dtree__size__decr__aux,axiom,
    ! [X: dtree_a_b,Y: b,Xs2: fset_P5281107635120001194_a_b_b,R2: a] :
      ( ( member4380921116106875537_a_b_b @ ( produc331601717337510060_a_b_b @ X @ Y ) @ ( fset_P783253628892185035_a_b_b @ Xs2 ) )
     => ( ord_less_nat @ ( size_size_dtree_a_b @ X ) @ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).

% dtree_size_decr_aux
thf(fact_1208_partition__on__inj__image,axiom,
    ! [A: set_Product_prod_a_b,P2: set_se3214616458824132456od_a_b,F: product_prod_a_b > a] :
      ( ( disjoi7717455583182441783od_a_b @ A @ P2 )
     => ( ( inj_on4343330552946057671_a_b_a @ F @ A )
       => ( disjoi6812529866258094733n_on_a @ ( image_2802296252294471259_a_b_a @ F @ A ) @ ( minus_5736297505244876581_set_a @ ( image_7003315350810844251_set_a @ ( image_2802296252294471259_a_b_a @ F ) @ P2 ) @ ( insert_set_a @ bot_bot_set_a @ bot_bot_set_set_a ) ) ) ) ) ).

% partition_on_inj_image
thf(fact_1209_last__zip,axiom,
    ! [Xs2: list_a,Ys: list_b] :
      ( ( Xs2 != nil_a )
     => ( ( Ys != nil_b )
       => ( ( ( size_size_list_a @ Xs2 )
            = ( size_size_list_b @ Ys ) )
         => ( ( last_P8790725272581694279od_a_b @ ( zip_a_b @ Xs2 @ Ys ) )
            = ( product_Pair_a_b @ ( last_a @ Xs2 ) @ ( last_b @ Ys ) ) ) ) ) ) ).

% last_zip
thf(fact_1210_map__append,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( map_Pr6568192469809321284od_a_b @ F @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) )
      = ( append5335208823350062147od_a_b @ ( map_Pr6568192469809321284od_a_b @ F @ Xs2 ) @ ( map_Pr6568192469809321284od_a_b @ F @ Ys ) ) ) ).

% map_append
thf(fact_1211_list_Oset__map,axiom,
    ! [F: product_prod_a_b > a,V2: list_P1467974523212632366od_a_b] :
      ( ( set_a2 @ ( map_Pr3261722632605054874_a_b_a @ F @ V2 ) )
      = ( image_2802296252294471259_a_b_a @ F @ ( set_Product_prod_a_b2 @ V2 ) ) ) ).

% list.set_map
thf(fact_1212_list_Oset__map,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,V2: list_P1467974523212632366od_a_b] :
      ( ( set_Product_prod_a_b2 @ ( map_Pr6568192469809321284od_a_b @ F @ V2 ) )
      = ( image_3300603549555413765od_a_b @ F @ ( set_Product_prod_a_b2 @ V2 ) ) ) ).

% list.set_map
thf(fact_1213_map__fst__zip,axiom,
    ! [Xs2: list_a,Ys: list_b] :
      ( ( ( size_size_list_a @ Xs2 )
        = ( size_size_list_b @ Ys ) )
     => ( ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b @ ( zip_a_b @ Xs2 @ Ys ) )
        = Xs2 ) ) ).

% map_fst_zip
thf(fact_1214_zip__Cons__Cons,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( zip_Pr9216478010423752119od_a_b @ ( cons_P7316939131009794654od_a_b @ X @ Xs2 ) @ ( cons_P7316939131009794654od_a_b @ Y @ Ys ) )
      = ( cons_P8222935586017696605od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ ( zip_Pr9216478010423752119od_a_b @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1215_zip__Cons__Cons,axiom,
    ! [X: a,Xs2: list_a,Y: b,Ys: list_b] :
      ( ( zip_a_b @ ( cons_a @ X @ Xs2 ) @ ( cons_b @ Y @ Ys ) )
      = ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ X @ Y ) @ ( zip_a_b @ Xs2 @ Ys ) ) ) ).

% zip_Cons_Cons
thf(fact_1216_zip__append,axiom,
    ! [Xs2: list_a,Us: list_b,Ys: list_a,Vs: list_b] :
      ( ( ( size_size_list_a @ Xs2 )
        = ( size_size_list_b @ Us ) )
     => ( ( zip_a_b @ ( append_a @ Xs2 @ Ys ) @ ( append_b @ Us @ Vs ) )
        = ( append5335208823350062147od_a_b @ ( zip_a_b @ Xs2 @ Us ) @ ( zip_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1217_zip__append,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Us: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Vs: list_P1467974523212632366od_a_b] :
      ( ( ( size_s3956712670882376602od_a_b @ Xs2 )
        = ( size_s3956712670882376602od_a_b @ Us ) )
     => ( ( zip_Pr9216478010423752119od_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ ( append5335208823350062147od_a_b @ Us @ Vs ) )
        = ( append3447563168971123522od_a_b @ ( zip_Pr9216478010423752119od_a_b @ Xs2 @ Us ) @ ( zip_Pr9216478010423752119od_a_b @ Ys @ Vs ) ) ) ) ).

% zip_append
thf(fact_1218_map__eq__Cons__conv,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( map_Pr6568192469809321284od_a_b @ F @ Xs2 )
        = ( cons_P7316939131009794654od_a_b @ Y @ Ys ) )
      = ( ? [Z6: product_prod_a_b,Zs2: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( cons_P7316939131009794654od_a_b @ Z6 @ Zs2 ) )
            & ( ( F @ Z6 )
              = Y )
            & ( ( map_Pr6568192469809321284od_a_b @ F @ Zs2 )
              = Ys ) ) ) ) ).

% map_eq_Cons_conv
thf(fact_1219_Cons__eq__map__conv,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,F: product_prod_a_b > product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( cons_P7316939131009794654od_a_b @ X @ Xs2 )
        = ( map_Pr6568192469809321284od_a_b @ F @ Ys ) )
      = ( ? [Z6: product_prod_a_b,Zs2: list_P1467974523212632366od_a_b] :
            ( ( Ys
              = ( cons_P7316939131009794654od_a_b @ Z6 @ Zs2 ) )
            & ( X
              = ( F @ Z6 ) )
            & ( Xs2
              = ( map_Pr6568192469809321284od_a_b @ F @ Zs2 ) ) ) ) ) ).

% Cons_eq_map_conv
thf(fact_1220_map__eq__Cons__D,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Y: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( map_Pr6568192469809321284od_a_b @ F @ Xs2 )
        = ( cons_P7316939131009794654od_a_b @ Y @ Ys ) )
     => ? [Z4: product_prod_a_b,Zs3: list_P1467974523212632366od_a_b] :
          ( ( Xs2
            = ( cons_P7316939131009794654od_a_b @ Z4 @ Zs3 ) )
          & ( ( F @ Z4 )
            = Y )
          & ( ( map_Pr6568192469809321284od_a_b @ F @ Zs3 )
            = Ys ) ) ) ).

% map_eq_Cons_D
thf(fact_1221_Cons__eq__map__D,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,F: product_prod_a_b > product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( cons_P7316939131009794654od_a_b @ X @ Xs2 )
        = ( map_Pr6568192469809321284od_a_b @ F @ Ys ) )
     => ? [Z4: product_prod_a_b,Zs3: list_P1467974523212632366od_a_b] :
          ( ( Ys
            = ( cons_P7316939131009794654od_a_b @ Z4 @ Zs3 ) )
          & ( X
            = ( F @ Z4 ) )
          & ( Xs2
            = ( map_Pr6568192469809321284od_a_b @ F @ Zs3 ) ) ) ) ).

% Cons_eq_map_D
thf(fact_1222_list_Osimps_I9_J,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,X21: product_prod_a_b,X222: list_P1467974523212632366od_a_b] :
      ( ( map_Pr6568192469809321284od_a_b @ F @ ( cons_P7316939131009794654od_a_b @ X21 @ X222 ) )
      = ( cons_P7316939131009794654od_a_b @ ( F @ X21 ) @ ( map_Pr6568192469809321284od_a_b @ F @ X222 ) ) ) ).

% list.simps(9)
thf(fact_1223_zip__eq__ConsE,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Xy: produc2233330766403592615od_a_b,Xys: list_P3463435030753280173od_a_b] :
      ( ( ( zip_Pr9216478010423752119od_a_b @ Xs2 @ Ys )
        = ( cons_P8222935586017696605od_a_b @ Xy @ Xys ) )
     => ~ ! [X3: product_prod_a_b,Xs5: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( cons_P7316939131009794654od_a_b @ X3 @ Xs5 ) )
           => ! [Y4: product_prod_a_b,Ys6: list_P1467974523212632366od_a_b] :
                ( ( Ys
                  = ( cons_P7316939131009794654od_a_b @ Y4 @ Ys6 ) )
               => ( ( Xy
                    = ( produc6550459592057863831od_a_b @ X3 @ Y4 ) )
                 => ( Xys
                   != ( zip_Pr9216478010423752119od_a_b @ Xs5 @ Ys6 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1224_zip__eq__ConsE,axiom,
    ! [Xs2: list_a,Ys: list_b,Xy: product_prod_a_b,Xys: list_P1467974523212632366od_a_b] :
      ( ( ( zip_a_b @ Xs2 @ Ys )
        = ( cons_P7316939131009794654od_a_b @ Xy @ Xys ) )
     => ~ ! [X3: a,Xs5: list_a] :
            ( ( Xs2
              = ( cons_a @ X3 @ Xs5 ) )
           => ! [Y4: b,Ys6: list_b] :
                ( ( Ys
                  = ( cons_b @ Y4 @ Ys6 ) )
               => ( ( Xy
                    = ( product_Pair_a_b @ X3 @ Y4 ) )
                 => ( Xys
                   != ( zip_a_b @ Xs5 @ Ys6 ) ) ) ) ) ) ).

% zip_eq_ConsE
thf(fact_1225_set__zip__rightD,axiom,
    ! [X: a,Y: b,Xs2: list_a,Ys: list_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Y ) @ ( set_Product_prod_a_b2 @ ( zip_a_b @ Xs2 @ Ys ) ) )
     => ( member_b @ Y @ ( set_b2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1226_set__zip__leftD,axiom,
    ! [X: a,Y: b,Xs2: list_a,Ys: list_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Y ) @ ( set_Product_prod_a_b2 @ ( zip_a_b @ Xs2 @ Ys ) ) )
     => ( member_a @ X @ ( set_a2 @ Xs2 ) ) ) ).

% set_zip_leftD
thf(fact_1227_in__set__zipE,axiom,
    ! [X: a,Y: a,Xs2: list_a,Ys: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ X @ Y ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Ys ) ) )
     => ~ ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1228_in__set__zipE,axiom,
    ! [X: a,Y: product_prod_a_b,Xs2: list_a,Ys: list_P1467974523212632366od_a_b] :
      ( ( member3142156093895948538od_a_b @ ( produc431845345726502849od_a_b @ X @ Y ) @ ( set_Pr7079953345493216102od_a_b @ ( zip_a_3848334451642478305od_a_b @ Xs2 @ Ys ) ) )
     => ~ ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
         => ~ ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1229_in__set__zipE,axiom,
    ! [X: product_prod_a_b,Y: a,Xs2: list_P1467974523212632366od_a_b,Ys: list_a] :
      ( ( member2194267517197600294_a_b_a @ ( produc5056887847982525421_a_b_a @ X @ Y ) @ ( set_Pr6132064768794867858_a_b_a @ ( zip_Pr8473376953898500877_a_b_a @ Xs2 @ Ys ) ) )
     => ~ ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ~ ( member_a @ Y @ ( set_a2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1230_in__set__zipE,axiom,
    ! [X: product_prod_a_b,Y: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ X @ Y ) @ ( set_Pr5228040199190236220od_a_b @ ( zip_Pr9216478010423752119od_a_b @ Xs2 @ Ys ) ) )
     => ~ ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ~ ( member1426531481828664017od_a_b @ Y @ ( set_Product_prod_a_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1231_in__set__zipE,axiom,
    ! [X: a,Y: b,Xs2: list_a,Ys: list_b] :
      ( ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Y ) @ ( set_Product_prod_a_b2 @ ( zip_a_b @ Xs2 @ Ys ) ) )
     => ~ ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
         => ~ ( member_b @ Y @ ( set_b2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1232_zip__same,axiom,
    ! [A2: a,B: a,Xs2: list_a] :
      ( ( member1426531477525435216od_a_a @ ( product_Pair_a_a @ A2 @ B ) @ ( set_Product_prod_a_a2 @ ( zip_a_a @ Xs2 @ Xs2 ) ) )
      = ( ( member_a @ A2 @ ( set_a2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_1233_zip__same,axiom,
    ! [A2: product_prod_a_b,B: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member5065438833300252112od_a_b @ ( produc6550459592057863831od_a_b @ A2 @ B ) @ ( set_Pr5228040199190236220od_a_b @ ( zip_Pr9216478010423752119od_a_b @ Xs2 @ Xs2 ) ) )
      = ( ( member1426531481828664017od_a_b @ A2 @ ( set_Product_prod_a_b2 @ Xs2 ) )
        & ( A2 = B ) ) ) ).

% zip_same
thf(fact_1234_pair__list__eqI,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b @ Xs2 )
        = ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b @ Ys ) )
     => ( ( ( map_Pr3261722632605054875_a_b_b @ product_snd_a_b @ Xs2 )
          = ( map_Pr3261722632605054875_a_b_b @ product_snd_a_b @ Ys ) )
       => ( Xs2 = Ys ) ) ) ).

% pair_list_eqI
thf(fact_1235_zip__map__fst__snd,axiom,
    ! [Zs: list_P1467974523212632366od_a_b] :
      ( ( zip_a_b @ ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b @ Zs ) @ ( map_Pr3261722632605054875_a_b_b @ product_snd_a_b @ Zs ) )
      = Zs ) ).

% zip_map_fst_snd
thf(fact_1236_map__idI,axiom,
    ! [Xs2: list_a,F: a > a] :
      ( ! [X3: a] :
          ( ( member_a @ X3 @ ( set_a2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_a_a @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_1237_map__idI,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,F: product_prod_a_b > product_prod_a_b] :
      ( ! [X3: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs2 ) )
         => ( ( F @ X3 )
            = X3 ) )
     => ( ( map_Pr6568192469809321284od_a_b @ F @ Xs2 )
        = Xs2 ) ) ).

% map_idI
thf(fact_1238_list_Omap__ident__strong,axiom,
    ! [T: list_a,F: a > a] :
      ( ! [Z4: a] :
          ( ( member_a @ Z4 @ ( set_a2 @ T ) )
         => ( ( F @ Z4 )
            = Z4 ) )
     => ( ( map_a_a @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_1239_list_Omap__ident__strong,axiom,
    ! [T: list_P1467974523212632366od_a_b,F: product_prod_a_b > product_prod_a_b] :
      ( ! [Z4: product_prod_a_b] :
          ( ( member1426531481828664017od_a_b @ Z4 @ ( set_Product_prod_a_b2 @ T ) )
         => ( ( F @ Z4 )
            = Z4 ) )
     => ( ( map_Pr6568192469809321284od_a_b @ F @ T )
        = T ) ) ).

% list.map_ident_strong
thf(fact_1240_map__eq__append__conv,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( map_Pr6568192469809321284od_a_b @ F @ Xs2 )
        = ( append5335208823350062147od_a_b @ Ys @ Zs ) )
      = ( ? [Us2: list_P1467974523212632366od_a_b,Vs3: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( append5335208823350062147od_a_b @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_Pr6568192469809321284od_a_b @ F @ Us2 ) )
            & ( Zs
              = ( map_Pr6568192469809321284od_a_b @ F @ Vs3 ) ) ) ) ) ).

% map_eq_append_conv
thf(fact_1241_append__eq__map__conv,axiom,
    ! [Ys: list_P1467974523212632366od_a_b,Zs: list_P1467974523212632366od_a_b,F: product_prod_a_b > product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ( append5335208823350062147od_a_b @ Ys @ Zs )
        = ( map_Pr6568192469809321284od_a_b @ F @ Xs2 ) )
      = ( ? [Us2: list_P1467974523212632366od_a_b,Vs3: list_P1467974523212632366od_a_b] :
            ( ( Xs2
              = ( append5335208823350062147od_a_b @ Us2 @ Vs3 ) )
            & ( Ys
              = ( map_Pr6568192469809321284od_a_b @ F @ Us2 ) )
            & ( Zs
              = ( map_Pr6568192469809321284od_a_b @ F @ Vs3 ) ) ) ) ) ).

% append_eq_map_conv
thf(fact_1242_image__set,axiom,
    ! [F: product_prod_a_b > a,Xs2: list_P1467974523212632366od_a_b] :
      ( ( image_2802296252294471259_a_b_a @ F @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( set_a2 @ ( map_Pr3261722632605054874_a_b_a @ F @ Xs2 ) ) ) ).

% image_set
thf(fact_1243_image__set,axiom,
    ! [F: product_prod_a_b > product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( image_3300603549555413765od_a_b @ F @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( set_Product_prod_a_b2 @ ( map_Pr6568192469809321284od_a_b @ F @ Xs2 ) ) ) ).

% image_set
thf(fact_1244_zip__eq__conv,axiom,
    ! [Xs2: list_a,Ys: list_b,Zs: list_P1467974523212632366od_a_b] :
      ( ( ( size_size_list_a @ Xs2 )
        = ( size_size_list_b @ Ys ) )
     => ( ( ( zip_a_b @ Xs2 @ Ys )
          = Zs )
        = ( ( ( map_Pr3261722632605054874_a_b_a @ product_fst_a_b @ Zs )
            = Xs2 )
          & ( ( map_Pr3261722632605054875_a_b_b @ product_snd_a_b @ Zs )
            = Ys ) ) ) ) ).

% zip_eq_conv
thf(fact_1245_inverse__permutation__of__list__id,axiom,
    ! [X: a,Xs2: list_P1396940483166286381od_a_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ ( map_Pr3897371633210041563_a_a_a @ product_snd_a_a @ Xs2 ) ) )
     => ( ( invers8098400408744078809list_a @ Xs2 @ X )
        = X ) ) ).

% inverse_permutation_of_list_id
thf(fact_1246_inverse__permutation__of__list__id,axiom,
    ! [X: product_prod_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( map_Pr7944591362074536709od_a_b @ produc8875788424405714885od_a_b @ Xs2 ) ) )
     => ( ( invers7597300294525432963od_a_b @ Xs2 @ X )
        = X ) ) ).

% inverse_permutation_of_list_id
thf(fact_1247_permutation__of__list__id,axiom,
    ! [X: a,Xs2: list_P1396940483166286381od_a_a] :
      ( ~ ( member_a @ X @ ( set_a2 @ ( map_Pr3897371633210041563_a_a_a @ product_fst_a_a @ Xs2 ) ) )
     => ( ( permut8467757748254915227list_a @ Xs2 @ X )
        = X ) ) ).

% permutation_of_list_id
thf(fact_1248_permutation__of__list__id,axiom,
    ! [X: product_prod_a_b,Xs2: list_P3463435030753280173od_a_b] :
      ( ~ ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ ( map_Pr7944591362074536709od_a_b @ produc5013601669340818563od_a_b @ Xs2 ) ) )
     => ( ( permut3539836191700332357od_a_b @ Xs2 @ X )
        = X ) ) ).

% permutation_of_list_id
thf(fact_1249_lists__image,axiom,
    ! [F: product_prod_a_b > a,A: set_Product_prod_a_b] :
      ( ( lists_a @ ( image_2802296252294471259_a_b_a @ F @ A ) )
      = ( image_5173132012296311131list_a @ ( map_Pr3261722632605054874_a_b_a @ F ) @ ( lists_9062004085038442652od_a_b @ A ) ) ) ).

% lists_image
thf(fact_1250_in__set__impl__in__set__zip1,axiom,
    ! [Xs2: list_a,Ys: list_b,X: a] :
      ( ( ( size_size_list_a @ Xs2 )
        = ( size_size_list_b @ Ys ) )
     => ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
       => ~ ! [Y4: b] :
              ~ ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X @ Y4 ) @ ( set_Product_prod_a_b2 @ ( zip_a_b @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_1251_in__set__impl__in__set__zip2,axiom,
    ! [Xs2: list_a,Ys: list_b,Y: b] :
      ( ( ( size_size_list_a @ Xs2 )
        = ( size_size_list_b @ Ys ) )
     => ( ( member_b @ Y @ ( set_b2 @ Ys ) )
       => ~ ! [X3: a] :
              ~ ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ X3 @ Y ) @ ( set_Product_prod_a_b2 @ ( zip_a_b @ Xs2 @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1252_hd__zip,axiom,
    ! [Xs2: list_a,Ys: list_b] :
      ( ( Xs2 != nil_a )
     => ( ( Ys != nil_b )
       => ( ( hd_Product_prod_a_b @ ( zip_a_b @ Xs2 @ Ys ) )
          = ( product_Pair_a_b @ ( hd_a @ Xs2 ) @ ( hd_b @ Ys ) ) ) ) ) ).

% hd_zip
thf(fact_1253_in__set__zip,axiom,
    ! [P: product_prod_a_b,Xs2: list_a,Ys: list_b] :
      ( ( member1426531481828664017od_a_b @ P @ ( set_Product_prod_a_b2 @ ( zip_a_b @ Xs2 @ Ys ) ) )
      = ( ? [N3: nat] :
            ( ( ( nth_a @ Xs2 @ N3 )
              = ( product_fst_a_b @ P ) )
            & ( ( nth_b @ Ys @ N3 )
              = ( product_snd_a_b @ P ) )
            & ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
            & ( ord_less_nat @ N3 @ ( size_size_list_b @ Ys ) ) ) ) ) ).

% in_set_zip
thf(fact_1254_nth__append__length,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,X: product_prod_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( nth_Product_prod_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ ( cons_P7316939131009794654od_a_b @ X @ Ys ) ) @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
      = X ) ).

% nth_append_length
thf(fact_1255_nth__zip,axiom,
    ! [I: nat,Xs2: list_a,Ys: list_b] :
      ( ( ord_less_nat @ I @ ( size_size_list_a @ Xs2 ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_b @ Ys ) )
       => ( ( nth_Product_prod_a_b @ ( zip_a_b @ Xs2 @ Ys ) @ I )
          = ( product_Pair_a_b @ ( nth_a @ Xs2 @ I ) @ ( nth_b @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1256_all__set__conv__all__nth,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ( ! [X4: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X4 @ ( set_Product_prod_a_b2 @ Xs2 ) )
           => ( P2 @ X4 ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
           => ( P2 @ ( nth_Product_prod_a_b @ Xs2 @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_1257_all__nth__imp__all__set,axiom,
    ! [Xs2: list_a,P2: a > $o,X: a] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_a @ Xs2 ) )
         => ( P2 @ ( nth_a @ Xs2 @ I3 ) ) )
     => ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1258_all__nth__imp__all__set,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o,X: product_prod_a_b] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
         => ( P2 @ ( nth_Product_prod_a_b @ Xs2 @ I3 ) ) )
     => ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
       => ( P2 @ X ) ) ) ).

% all_nth_imp_all_set
thf(fact_1259_in__set__conv__nth,axiom,
    ! [X: a,Xs2: list_a] :
      ( ( member_a @ X @ ( set_a2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_a @ Xs2 ) )
            & ( ( nth_a @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1260_in__set__conv__nth,axiom,
    ! [X: product_prod_a_b,Xs2: list_P1467974523212632366od_a_b] :
      ( ( member1426531481828664017od_a_b @ X @ ( set_Product_prod_a_b2 @ Xs2 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
            & ( ( nth_Product_prod_a_b @ Xs2 @ I2 )
              = X ) ) ) ) ).

% in_set_conv_nth
thf(fact_1261_list__ball__nth,axiom,
    ! [N2: nat,Xs2: list_P1467974523212632366od_a_b,P2: product_prod_a_b > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
     => ( ! [X3: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ ( set_Product_prod_a_b2 @ Xs2 ) )
           => ( P2 @ X3 ) )
       => ( P2 @ ( nth_Product_prod_a_b @ Xs2 @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_1262_nth__mem,axiom,
    ! [N2: nat,Xs2: list_a] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_a @ Xs2 ) )
     => ( member_a @ ( nth_a @ Xs2 @ N2 ) @ ( set_a2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1263_nth__mem,axiom,
    ! [N2: nat,Xs2: list_P1467974523212632366od_a_b] :
      ( ( ord_less_nat @ N2 @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
     => ( member1426531481828664017od_a_b @ ( nth_Product_prod_a_b @ Xs2 @ N2 ) @ ( set_Product_prod_a_b2 @ Xs2 ) ) ) ).

% nth_mem
thf(fact_1264_nth__append,axiom,
    ! [N2: nat,Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b] :
      ( ( ( ord_less_nat @ N2 @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
       => ( ( nth_Product_prod_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ N2 )
          = ( nth_Product_prod_a_b @ Xs2 @ N2 ) ) )
      & ( ~ ( ord_less_nat @ N2 @ ( size_s3956712670882376602od_a_b @ Xs2 ) )
       => ( ( nth_Product_prod_a_b @ ( append5335208823350062147od_a_b @ Xs2 @ Ys ) @ N2 )
          = ( nth_Product_prod_a_b @ Ys @ ( minus_minus_nat @ N2 @ ( size_s3956712670882376602od_a_b @ Xs2 ) ) ) ) ) ) ).

% nth_append
thf(fact_1265_listrel__iff__nth,axiom,
    ! [Xs2: list_a,Ys: list_b,R2: set_Product_prod_a_b] :
      ( ( member8262802279224426321list_b @ ( produc6837034579544652440list_b @ Xs2 @ Ys ) @ ( listrel_a_b @ R2 ) )
      = ( ( ( size_size_list_a @ Xs2 )
          = ( size_size_list_b @ Ys ) )
        & ! [N3: nat] :
            ( ( ord_less_nat @ N3 @ ( size_size_list_a @ Xs2 ) )
           => ( member1426531481828664017od_a_b @ ( product_Pair_a_b @ ( nth_a @ Xs2 @ N3 ) @ ( nth_b @ Ys @ N3 ) ) @ R2 ) ) ) ) ).

% listrel_iff_nth
thf(fact_1266_Sup_OSUP__cong,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,C5: product_prod_a_b > a,D: product_prod_a_b > a,Sup: set_a > a] :
      ( ( A = B4 )
     => ( ! [X3: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ B4 )
           => ( ( C5 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Sup @ ( image_2802296252294471259_a_b_a @ C5 @ A ) )
          = ( Sup @ ( image_2802296252294471259_a_b_a @ D @ B4 ) ) ) ) ) ).

% Sup.SUP_cong
thf(fact_1267_Inf_OINF__cong,axiom,
    ! [A: set_Product_prod_a_b,B4: set_Product_prod_a_b,C5: product_prod_a_b > a,D: product_prod_a_b > a,Inf: set_a > a] :
      ( ( A = B4 )
     => ( ! [X3: product_prod_a_b] :
            ( ( member1426531481828664017od_a_b @ X3 @ B4 )
           => ( ( C5 @ X3 )
              = ( D @ X3 ) ) )
       => ( ( Inf @ ( image_2802296252294471259_a_b_a @ C5 @ A ) )
          = ( Inf @ ( image_2802296252294471259_a_b_a @ D @ B4 ) ) ) ) ) ).

% Inf.INF_cong
thf(fact_1268_inj__on__image,axiom,
    ! [F: product_prod_a_b > a,A: set_se3214616458824132456od_a_b] :
      ( ( inj_on4343330552946057671_a_b_a @ F @ ( comple8492713210738191477od_a_b @ A ) )
     => ( inj_on1980945187301441479_set_a @ ( image_2802296252294471259_a_b_a @ F ) @ A ) ) ).

% inj_on_image
thf(fact_1269_size__le__if__child__subset,axiom,
    ! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R2: a,V2: a] :
      ( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
     => ( ord_less_eq_nat @ ( size_size_dtree_a_b @ ( node_a_b @ R2 @ Xs2 ) ) @ ( size_size_dtree_a_b @ ( node_a_b @ V2 @ Ys ) ) ) ) ).

% size_le_if_child_subset
thf(fact_1270_strict__subtree__subset,axiom,
    ! [T: dtree_a_b,R2: a,Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b] :
      ( ( strict_subtree_a_b @ T @ ( node_a_b @ R2 @ Xs2 ) )
     => ( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
       => ( strict_subtree_a_b @ T @ ( node_a_b @ R2 @ Ys ) ) ) ) ).

% strict_subtree_subset
thf(fact_1271_wf__darcs__sub,axiom,
    ! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R5: a,R2: a] :
      ( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
     => ( ( wf_darcs_a_b @ ( node_a_b @ R5 @ Ys ) )
       => ( wf_darcs_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).

% wf_darcs_sub
thf(fact_1272_wf__dverts__sub,axiom,
    ! [Xs2: fset_P5281107635120001194_a_b_b,Ys: fset_P5281107635120001194_a_b_b,R2: a] :
      ( ( ord_le789900035998834954_a_b_b @ Xs2 @ Ys )
     => ( ( wf_dverts_a_b @ ( node_a_b @ R2 @ Ys ) )
       => ( wf_dverts_a_b @ ( node_a_b @ R2 @ Xs2 ) ) ) ) ).

% wf_dverts_sub
thf(fact_1273_impossible__Cons,axiom,
    ! [Xs2: list_P1467974523212632366od_a_b,Ys: list_P1467974523212632366od_a_b,X: product_prod_a_b] :
      ( ( ord_less_eq_nat @ ( size_s3956712670882376602od_a_b @ Xs2 ) @ ( size_s3956712670882376602od_a_b @ Ys ) )
     => ( Xs2
       != ( cons_P7316939131009794654od_a_b @ X @ Ys ) ) ) ).

% impossible_Cons

% Helper facts (5)
thf(help_If_2_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_Itf__a_J_T,axiom,
    ! [X: list_a,Y: list_a] :
      ( ( if_list_a @ $true @ X @ Y )
      = X ) ).

thf(help_If_3_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
    ! [P2: $o] :
      ( ( P2 = $true )
      | ( P2 = $false ) ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( if_lis1002476807507936500od_a_b @ $false @ X @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Product____Type__Oprod_Itf__a_Mtf__b_J_J_T,axiom,
    ! [X: list_P1467974523212632366od_a_b,Y: list_P1467974523212632366od_a_b] :
      ( ( if_lis1002476807507936500od_a_b @ $true @ X @ Y )
      = X ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    ? [Xs7: fset_P5281107635120001194_a_b_b] :
      ( ( strict_subtree_a_b @ ( node_a_b @ r @ Xs7 ) @ t )
      & ( ( dverts_a_b @ ( node_a_b @ r @ Xs7 ) )
        = ( image_2802296252294471259_a_b_a @ product_fst_a_b @ ( set_Product_prod_a_b2 @ ( cons_P7316939131009794654od_a_b @ ( product_Pair_a_b @ r @ e ) @ bs ) ) ) ) ) ).

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